numerik pemodelan dan hidrolik

7/24/2019 Numerik Pemodelan Dan Hidrolik

http://slidepdf.com/reader/full/numerik-pemodelan-dan-hidrolik 1/143

Halaman 1

Menurut angka

Pemodelan dan

Hidrolika

Nils Olsen Reidar B.

Departemen Hidrolik danTeknik Lingkungan

Universitas NorwegiaIlmu pengetahuan dan teknologi

3rd edition, 9 Maret 2012.

ISBN 82-7598-074-7



Halaman 2

Numerik Pemodelan dan Hidrolik 1

Kata pengantar

Kelas "Numerical Modelling dan Hidrolik" adalah nama baru untukTentu saja lama "Hydroinformatics", yang ditawarkan untuk pertama kalinya dimusim semi 2001 di Universitas Norwegia Sains dan Teknologi. inikursus sarjana untuk 3/4 siswa tahun. Prasyaratadalah kursus dasar dalam mekanika hidrolika / hydromechanics / cairan, yangtermasuk derivasi dari persamaan dasar, misalnya kontinuitas persamaan dan persamaan momentum.

Ketika saya mulai kerja saya di Norwegia Universitas Sainsdan Teknologi, saya diminta untuk mengajar kursus dan membuat rencana untuk yangkonten. Dasar adalah kursus dihentikan "Sungai Hidrolik",yang juga termasuk topik pada limnologi. Saya diminta untuk memasukkan topik padakualitas air dan juga pada pemodelan numerik. Ketika menambahkan topik keTentu saja, hal ini juga diperlukan untuk menghapus sesuatu. Saya telah menghapus beberapadari hidrolika dasar tentang persamaan momentum, karena ini diajarkan dikursus lain siswa sebelumnya. Saya juga telah menghapus bagian daritopik khusus dari sungai hidrolika seperti bagian senyawa dan jembatan dan gorong-gorong analisis. Bagian senyawa hidrolik saya percayatidak dapat digunakan dalam teknik praktis pula, seperti geometri terlalu

disederhanakan dibandingkan dengan sungai alami. Analisis jembatan berdasarkan penyederhanaan model aliran 1D untuk situasi 3D. Di masa depan, saya percaya model sepenuhnya 3D akan digunakan sebagai pengganti, dan topik ini akan obso-lete. Beberapa topik pada rekayasa kelautan telah dihapus, sebagaiTentu saja baru "Marine Lingkungan Fisik" di Departemen StrukturalTeknik di NTNU meliputi mata pelajaran ini. Kursus ini juga con-tains beberapa es hidrolik dan rekayasa iklim dingin terkait, topikyang belum disertakan dalam teks ini.

Tentu saja yang dihasilkan termasuk hidrolik klasik, transportasi sedimen,numeric dan kualitas air. Itu sulit untuk menemukan satu buku yang mencakupsemua topik. Buku-buku yang juga sangat mahal, sehingga sulit untuk memintasiswa untuk membeli beberapa buku. Sebaliknya saya menulis catatan ini. Akuingin mengucapkan terima kasih kepada Departemen memberi saya waktu untuk ini, dan berharapBuku akan menarik bagi siswa.

Saya juga ingin berterima kasih kepada semua orang yang membantu saya dengan material, saran dankoreksi buku. Dr Knut Alfredsen telah memberikan saran danmateri pada solusi numerik dari persamaan Saint-Venant dan pada

pemodelan habitat. Prof. Torkild Carstens telah memberikan saran pada jet, bulu dan abstraksi air. Prof. Liv Fiksdal memberikan nasihat tentang biologi air dan Mr. Yngve Robertsen telah memberikan saran pada banjirformula gelombang. Saya juga ingin berterima kasih kepada siswa saya mengambil kursus dimusim semi tahun 2001, menemukan sejumlah besar kesalahan dan membuat saranuntuk perbaikan. Untuk versi sebelumnya, Prof. Hubert Chanson tersediakoreksi berguna.

Nama baru ini mencerminkan fokus model numerik dan hidrolika.Kata "Hydroinformatics" sangat luas dan mencakup sejumlah besartopik tidak termasuk dalam buku ini. Selain model numerik, juga beberapa topik Hidrolik tertutup, untuk gelombang misalnya banjir,transportasi sedimen, aliran stratified dan model tes fisik.

Halaman 3




Dalam memori dari Prof. Dagfinn K. Lysne

Halaman 4


Daftar isi

1. Perkenalan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.1 Motivasi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Klasifikasi program komputer. . . . . . . . . . . . . . . . . . . . . . . . . . . 6



2. Sungai hidrolika. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 aliran Uniform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 formula Gesekan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 kerugian Singular. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.4 aliran kritis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,102,5 Mantap aliran non-seragam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,132,6 Gelombang di sungai. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,162.7 Persamaan Saint-Venant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,192,8 Pengukuran debit air di sungai alami. . . . . . . . . . . . . . 0,22

2.9 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,24

3. pemodelan numerik aliran sungai di 1D. . . . . . . . . . . . . . . . . . . . . . . . . 0,263.1 aliran Mantap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,263.2 aliran goyah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,283.3 goyah aliran - gelombang kinematik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,283.4 goyah aliran - persamaan Saint-Venands. . . . . . . . . . . . . . . . . . . . . 0,313,5 routing yang hidrologi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,383,6 HEC-RAS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,393.7 Perangkat lunak komersial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,393.8 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,40

4. Penyebaran polutan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,424.1 Pendahuluan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,424.2 formula sederhana untuk koefisien difusi. . . . . . . . . . . . . . . . . . . . 0,42Dispersi 4.3 Satu-dimensi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,44

4.4 Jets dan bulu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,454,5 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,47

5. Dispersi pemodelan 2D dan 3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,485.1 Grids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,485.2 metode Discretization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,525.3 Skema Pertama-Order Hulu. . . . . . . . . . . . . . . . . . . . . . . . . . . 0,535.4 pemrograman Spreadsheet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,555.5 difusi Salah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,575.6 Skema Hulu Orde Kedua. . . . . . . . . . . . . . . . . . . . . . . . 0,58Perhitungan dan istilah sumber 5,7 Time-dependent. . . . . . . . . . . . . . . . 0,605.8 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,62

6. pemodelan numerik kecepatan air di 2D dan 3D. . . . . . . . . . . . . . . 0,646.1 persamaan Navier-Stokes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,646.2 Metode SEDERHANA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,65

6.3 model turbulensi lanjutan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,686.4 Kondisi batas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,706.5 Stabilitas dan konvergensi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,726,6 algoritma permukaan Gratis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,76

Halaman 5


6.7 Kesalahan dan ketidakpastian di CFD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,796,8 SSIIM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,816.9 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,81

7. limnologi fisik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,837.1 Pendahuluan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,837.2 Sirkulasi di danau non-berlapis. . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,837.3 Suhu dan stratifikasi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,847.4 Angin yang disebabkan sirkulasi di danau bertingkat. . . . . . . . . . . . . . . . . . . . . 0,877,5 Seiches. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,897.6 Sungai-diinduksi sirkulasi dan percepatan Coriolis. . . . . . . . . . . . . . 0,907.7 Kepadatan arus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,917.8 Intake di waduk bertingkat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,927.9 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,93



8. biologi Air. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,948.1 Pendahuluan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,948,2 reaksi biokimia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,948,3 senyawa beracun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,958.4 klasifikasi Limnological. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,978.5 Siklus hara. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,978,6 QUAL2E. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1008,7 Fitoplankton. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1018.8 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,104

9. transportasi sedimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1069.1 Pendahuluan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1069.2 Erosi. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1089.3 sedimen tersuspensi dan beban tidur. . . . . . . . . . . . . . . . . . . . . . . . . . 0,1109,4 1D formula transportasi sedimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1129,5 bentuk B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1149.6 CFD pemodelan transportasi sedimen. . . . . . . . . . . . . . . . . . . . . . . . . . 0,1169,7 Waduk dan sedimen. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1179,8 geomorfologi Fluvial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1199,9 studi Model Fisik. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,1239.10 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,126

Pemodelan habitat 10. River. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,12810.1 Pendahuluan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,12810.2 analisis habitat ikan. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,128

10.3 Zero dan model hidrolik satu-dimensi. . . . . . . . . . . . . . . . . . 0,13010.4 model hidrolik Multidimensional. . . . . . . . . . . . . . . . . . . . . . . . . . 0,13110,5 model Bioenergetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,13110.6 Masalah. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,131

Literatur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,134

Lampiran I. Source code untuk solusi eksplisit persamaan Saint-Venants. 0,139Lampiran II. Daftar simbol dan unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . 0,142Lampiran III. Solusi untuk masalah yang dipilih. . . . . . . . . . . . . . . . . . . . . . . 0,144Lampiran IV: Sebuah pengantar pemrograman di C. . . . . . . . . . . . . . . . . . 0,153

Halaman 6


1. Perkenalan

1.1 Motivasi

Di masyarakat saat ini, isu-isu lingkungan yang menjadi perhatian penting dalam proyek perencanaan yang terkait dengan sumber daya air. Pembuangan polutan

ke sungai dan danau tidak diperbolehkan, kecuali izin khusus diberikanoleh otoritas yang tepat. Dalam sebuah aplikasi untuk dibuang ke receiver sebuahing badan air, penilaian kerusakan potensial harus disertakan.Sebuah model numerik berguna dalam perhitungan efek polusi yang yangtion.

Selama tahun terakhir, banjir sungai dan keamanan bendungan telah utamamasalah di Norwegia. Peraturan baru untuk perencanaan, konstruksi dan pengoperasian bendungan telah meningkat tuntutan untuk keselamatan bendungan. Semua Norwe- bendungan gian harus dievaluasi berkaitan dengan kegagalan, danefek hilir harus dinilai. Sehubungan, zona banjir ini pemetaan yang paling sungai besar harus dilakukan, dan ini akan membuat pekerjaan yang cukup untuk insinyur hidrolik di tahun-tahun mendatang.

Dua puluh tahun terakhir juga telah melihat evolusi komputer menjadialat yang sangat berlaku untuk memecahkan masalah rekayasa hidrolik. Kebanyakan



algoritma numerik kini diciptakan pada awal tahun 1970-an.Pada saat itu, komputer masih terlalu lambat untuk digunakan untuk sebagian praktis-masalah arus cal. Namun dalam beberapa tahun terakhir munculnya cepat dankomputer pribadi yang murah telah berubah ini. Semua numerikmetode yang diajarkan dalam kursus ini diterapkan dalam program yang berjalan pada PC.

Model numerik yang paling modern sering memiliki pengguna yang canggih antarmenghadapi, menampilkan grafis warna yang mengesankan. Orang dapat dengan mudah menyebabkan pemahaman bahwa komputer memecahkan semua masalah dengan mini pengetahuan ibu pengguna. Meskipun program komputer hari inidapat menghitung hampir semua masalah, keakuratan hasilnya masih diandaikantain. Pengguna berpengalaman dapat menghasilkan meyakinkan dan mengesankan

angka warna, tapi keakuratan hasilnya mungkin masih tidak baikcukup untuk memiliki nilai dalam teknik praktis. Oleh karena itu penting bahwa pengguna program komputer memiliki pengetahuan yang cukup dari keduametode numerik dan keterbatasan mereka dan juga proc- fisikesses yang dimodelkan. Oleh karena itu buku ini memberikan beberapa chap-ters pada proses sebagai hidrolika dasar, limnologi, transportasi sedimen,kualitas air dll pengetahuan harus digunakan untuk memberikan wajarmasukan untuk model numerik, dan menilai hasil mereka. Banyak empir-formula ical diberikan, memberikan kemungkinan lebih lanjut untuk memeriksaHasil dari metode numerik untuk kasus sederhana.

Metode numerik juga memiliki keterbatasan berkaitan dengan lainnyamasalah, misalnya pemodelan curam gradien, diskontinuitas, proc-esses pada skala yang berbeda dll model numerik sendiri mungkin rentanuntuk masalah khusus, misalnya ketidakstabilan. Seringkali, sebuah program komputermungkin tidak mencakup semua proses yang terjadi di badan air. Pengguna perlu menyadari rincian metode numerik, capabili- nyaikatan dan keterbatasan untuk menilai keakuratan hasil.

Halaman 7


1.2 Klasifikasi program komputer

Terdapat sejumlah besar program komputer untuk pemodelan fluvialhidrolik dan limnologi masalah. Program memiliki berbagai tingkatkecanggihan dan kehandalan. Ilmu pemodelan numerikmaju pesat, membuat beberapa program usang sementara pro barugram yang muncul.

Program komputer dapat diklasifikasikan menurut:

- Apa dihitung- Berapa banyak dimensi digunakan- Keterangan dari metode numerik

Banyak program komputer yang dibuat khusus untuk satu aplikasi tertentu.Contohnya adalah:

- Profil permukaan air (HEC2)- Gelombang Banjir (DAMBRK)- Kualitas air di sungai (QUAL2E)- Transportasi sedimen dan tidur perubahan (HEC6)- Pemodelan Habitat (PHABSIM)

Hal ini terutama terjadi untuk model satu dimensi, dikembangkan sev-tahun eral lalu, ketika daya komputasi jauh kurang darihari ini. Ada juga program satu dimensi yang lebih modern, menggunakanuser interface yang lebih canggih dan juga termasuk modul untuk computing beberapa masalah yang berbeda. Contohnya adalah:

- HEC-RAS- MIKE11- ISIS



Dalam beberapa tahun terakhir, sejumlah program komputer multi-dimensi memilikidikembangkan. Ini juga sering termasuk modul untuk komputasi sev- proses yang berbeda eral, untuk kualitas misalnya air, transportasi sedimendan profil permukaan air.

Program multi-dimensi mungkin:

- Dua dimensi kedalaman rata-rata-- Tiga dimensi dengan asumsi tekanan hidrostatik- Sepenuhnya tiga dimensi

Ada juga ada model dua dimensi lebar-rata, tapi ini

sebagian besar digunakan untuk tujuan penelitian.

Model tiga dimensi memecahkan persamaan Navier-Stokes dalam duaatau tiga dimensi. Kadang-kadang persamaan hanya diselesaikan dalamarah horisontal, dan persamaan kontinuitas digunakan untuk mendapatkankecepatan vertikal. Ini disebut solusi dengan tekanan hidrostatikasumsi. Model sepenuhnya 3D memecahkan persamaan Navier-Stokes jugadalam arah vertikal. Hal ini memberikan akurasi yang lebih baik ketika vertikal percepatan yang signifikan.

Berbagai algoritma yang digunakan oleh jenis program dijelaskandalam bab-bab berikut, bersama-sama dengan fisika yang terlibat.

Halaman 8


2. Sungai hidrolik

Hidrolika sungai klasik yang dijelaskan dalam bab ini membentuk dasaruntuk pemodelan numerik gelombang banjir dan penyebaran polutan sungai.Dalam bab ini, tekanan hidrostatik diasumsikan dalam arah yang vertikaltion, dan juga bahwa aliran air adalah salah satu dimensi.

2.1 aliran Seragam

Definisi aliran seragam dapat divisualisasikan dengan melihat aliran airdalam flume sangat panjang, di mana kedalaman air dan kecepatan konstan pada setiap titik selama panjang flume. Di sebuah sungai alami ini tidak pernahterjadi, tapi konsep ini berguna untuk mengembangkan teknik hidrolikformula. Gambar. 2.1.1 menunjukkan bagian dari saluran lebar, dengan kekuatan padaair. Saya adalah kemiringan permukaan air dan h adalah kedalaman air.

Pasukan pada volume air dalam arah yang sejajar dengan sungai /Permukaan akan:

Geser:

Gravity:

Arah aliran yang disebut x, h adalah kedalaman air, saya adalah kemiringan permukaan air, g x adalah komponen dari gravitasi dalam arah xdan τ adalah tegangan geser di tempat tidur. Pengaturan dua kekuatan sama dengan masing-masinglain memberikan rumus untuk tegangan geser:

(2.1.1)

Densitas air dilambangkan ρ, dan g adalah percepatan gravitasi. SEBUAHmetode untuk menghitung I disajikan dalam bab berikutnya.

Profil kecepatan vertikal di sungai dengan aliran seragam dapat digambarkanteori lapisan batas. Percobaan awal dilakukan oleh Nikuradse (1933) menggunakan bola seragam, dan kemudian Schlichting (1936)menggunakan partikel dari berbagai bentuk. Percobaan yang dihasilkan tindak yang

U

g

τ

Gambar. 2.1.1 Angkatan padaVolume air di unibentuk aliran

g x

F b

τΔ x=

F g

ρ g x

V ρ Gi Δ xh= =

τ ρ GHI =



ing rumus untuk profil kecepatan vertikal untuk aliran seragam (Schlichting,1979):

(2.1.2)

U adalah kecepatan, dan itu merupakan fungsi dari jarak, y, dari tempat tidur. The parameter κ adalah konstan empiris, sama dengan 0,4. Rumus hanya berlaku untuk permukaan kasar, dan k s adalah koefisien kekasaran. Hal ini ekivalendipinjamkan kepada diameter partikel bola terpaku ke dinding untuk modelelemen kekasaran. Variabel u * adalah kecepatan geser, yang diberikan oleh:

(2.1.3)

Eq. 2.1.2 juga disebut profil logaritmik untuk kecepatan air.Formula Schlicting ini didasarkan pada data dari percobaan yang dilakukan di udara,tapi karena parameter non-dimensi yang digunakan, hasil bekerjasangat baik juga untuk aliran air. Schlichting menemukan hukum dinding berlaku untuk

g

gz

g x = G sinα = g tanα =GI untuk sudut kecil

α

U

u*-----

1κ---

30 y

k s

---------ln=

u*τρ---=

Halaman 9


semua lapisan batas, juga untuk aliran non-seragam, asalkan hanya Veloci yanghubungan sangat dekat dengan dinding dianggap.

Untuk menggunakan rumus, pertanyaan berikutnya adalah yang kekasaran untuk memilih.Terdapat sejumlah hubungan yang berbeda antara kasar-efektifness dan distribusi ukuran butir di dasar sungai. Van Rijn (1982)menemukan rumus berikut, berdasarkan 120 set data flume:

(2.1.4)

Variabel d 90menunjukkan ukuran butir saringan di mana 90% dari materilebih halus. Van Rijn melaporkan bahwa ada ketidakpastian besar di untuk- inimula, dan bahwa jumlah 3 adalah nilai rata-rata di mana mengatur datamenyarankan berbagai variasi antara 1 dan 10. Peneliti lain memilikidigunakan formula yang berbeda. Hey (1979) menyarankan rumus berikut berdasarkan data dari sungai alami dengan bahan kasar, dan laboratorium percobaan dengan kubus / elemen bola:

(2.1.6)

Kamphuis (1974) membuat formula baru berdasarkan percobaan flume nya.Yang diperoleh:

(2.1.7)

Nilai 2 bervariasi antara 1,5 dan 2,5 dalam percobaan. Kamphuismenggunakan tingkat referensi nol dari 0,7 d 90, Yang akan mempengaruhi hasil.

Schlichting (1979) dilakukan percobaan laboratorium dengan bidang dankerucut. Menggunakan 45 derajat kerucut ditempatkan tepat di samping satu sama lain, k s Nilai itu sama dengan tinggi kerucut. Dengan kata lain, sulit untuk mendapatkan perkiraan yang akurat dari k s nilai.

2.2 formula Gesekan

Sejumlah peneliti telah mengembangkan rumus untuk veloc- rataity di saluran dengan aliran seragam, mengingat kedalaman air, kemiringan air danfaktor gesekan. Rumus yang empiris, dan gesekan faktor seringtergantung pada kedalaman air. Formula yang paling umum adalah:

Rumus Manning:

(2.2.1)

k s

3 d 90

=

k s

3,5 d 84=

k s

2 d 90=

U 1n--- R

h

23--- Aku

12---

=Rumus Manning:



Jari-jari hidrolik, r h, diberikan oleh:

(2.2.2)

di mana A adalah luas penampang sungai dan P adalah dibasahi perimeter.

Seringkali nilai M Strickler ini digunakan sebagai pengganti koefisien Manning, n.Relasi adalah:

r h

SEBUAH

P ---=

Halaman 10


(2.2.3)

memberikan:

(2.2.4)

Mengingat ukuran bulir di tempat tidur, yang Manning faktor gesekandapat diperkirakan dengan rumus empiris berikut (Meyer-Peter danMüller, 1948):

(2.2.5)

Mengingat kecepatan air dan faktor gesekan, rumus dapat digunakanuntuk memprediksi kedalaman air. Bersama-sama dengan persamaan kontinuitas (2.2.6)rumus juga dapat digunakan untuk memperkirakan kemiringan permukaan air ataukerugian gesekan untuk aliran non-seragam. Sehingga ketinggian air dapat

ditemukan. Penjelasan lebih lanjut diberikan dalam Bab 3.

(2.2.6)

Pr debit air. lebar unit sungai sering dilambangkan q.

2.3 kerugian Singular

Menggunakan Energi Persamaan / Bernoulli Persamaan untuk aliran air dalamsungai, adalah mungkin untuk menghitung kehilangan energi dan air permukaanlokasi. Hilangnya gesekan diberikan oleh kekasaran dasar sungai.Ada juga kehilangan energi lainnya, yang disebut kerugian tunggal. Ini adalahdiidentifikasi dengan konstruksi tertentu di sungai, misalnya jembatandermaga, atau tikungan sungai. Kerugian kepala berhubungan dengan pusaran gener-diciptakan sekitar titik kerugian, biasanya berkaitan dengan perluasan aliran. SEBUAH bentuk zona resirkulasi, menghamburkan energi. Hilangnya kepala, h f , dapatdihitung sebagai:

(2.3.1)

di mana k adalah koefisien kerugian head yang berkaitan dengan geometri sungai /obstruksi.

Di sungai alami, seringkali sulit untuk mengidentifikasi kerugian tunggal dan menetapkannilai untuk setiap kerugian. Sebaliknya, faktor gesekan yang Manning yang berbeda seringdigunakan, di mana faktor gesekan yang efektif digunakan, sebagai kombinasi darikerugian tunggal dan kerugian gesekan. Faktor gesekan kemudian ditemukan olehkalibrasi.

Hal ini dimungkinkan untuk menggunakan Persamaan. 2.3.1 untuk kontraksi sungai, misalnya dalam con-

M 1

n---=

Koefisien beberapa Manningkoefisien:

Kaca model:

0,009-0,01Model semen:0,011-0,013

Beton saluran berjajar:0,012-0,017

Bumi berbaris saluran:0,018-0,04

Batu saluran berjajar:0,025-0,045

Bumi berbaris sungai:0,02-0,05

Sungai gunung dengan batu:0,04-0,07

Sungai dengan gulma:0,05-0,15

U Bapak h

23--- Aku

12---

=

M 26

d 90

( )16---

---------------=

q Uh=Persamaan kontinuitas:

h f

k U 2

2 g ------=



connection dengan jembatan dan dermaga jembatan. Namun, koefisien kerugian headsulit untuk menemukan tanpa menggunakan pengukuran di bidang / lab.

Halaman 11


2.4 aliran Kritis

Melihat satu dimensi aliran air di saluran, ada dua jenisenergi: energi kinematik karena kecepatan air, dan tekanan karenadengan kedalaman air, y, dan berat dari air. Jumlah ener- inigies disebut tinggi energi spesifik dari bagian, E [meter]:

(2.4.1)

Energi spesifik dapat berubah sepanjang sungai, tergantung pada debit air, kekasaran, tidur lereng dll Jika air diberikanenergi spesifik, Persamaan. 2.4.1 dapat digunakan untuk menemukan kedalaman air:

(2.4.2)

Memperkenalkan persamaan kontinuitas (2.2.6), persamaan dapat ditulis:

(2.4.3)

atau

(2.4.4)

Persamaan orde ketiga ini memiliki tiga solusi yang mungkin. Hanya dua yangsecara fisik mungkin. Jika kita memecahkan untuk energi spesifik, kita mendapatkan:

(2.4.5)

Energi spesifik minimum untuk mengangkut volume tertentu airdiperoleh derivasi dari persamaan. 2.4.5, dan pengaturan hasil nol:

(2.4.6)

(2.4.7)

Menggunakan persamaan kontinuitas, persamaan di atas dapat juga ditulis:

(2.4.8)

Istilah di sisi kiri persamaan juga disebut Froudenomor. Untuk jumlah minimum energi spesifik, jumlah Froudeadalah kesatuan, seperti yang diberikan dalam Persamaan. 2.4.8. Jika nomor Froude bawah persatuan,aliran subkritis. Alirannya superkritis jika nilai lebih tinggi dari kesatuan

Aliran superkritis sangat jarang ditemui di sungai alami. Itu ada diair terjun atau jeram. Jika aliran superkritis terjadi di sungai dengan batu di

E E P

E U

+ yU 2

2 g ------+= =

y E U 2

2 g -------=

y E q2

2 gy2------------=

y3 Ey 2-q

2

2 g -----+ 0=

E yq2

2 gy2-----------+=

Ed

yd ------ 1

q2

gy 3-------- 0= =

yc

q2

g ----3=

U c

gyc

------------ 1=

The Froude Nomor:

Fr U

gy

----------=



Halaman 12


tempat tidur, biasanya lonjakan hidrolik terbentuk. Biasanya, aliran di sungaiadalah subkritis.

Bilangan Froude penting untuk solusi numerik persamaanuntuk kedalaman air di sungai. Jika aliran kritis terjadi, ketidakstabilan sering munculdalam algoritma numerik.

Tidak teratur lintas-bagian

Derivasi di atas berlaku untuk saluran dengan silang persegi panjang bagian. Untuk saluran dengan teratur penampang, kecepatan airdigantikan oleh Q / A. Energi spesifik untuk bagian menjadi:

(2.4.9)

Energi spesifik minimum untuk bagian yang diperoleh dengan derivasisehubungan dengan y dan pengaturan ini ke nol, mirip dengan apa yang dilakukan untuk persegi panjang saluran:

(2.4.10)

Untuk perubahan kecil dalam tingkat air, incremental cross-sectionaldaerah, dA diberikan oleh

dA = Bdy (2.4.11)

Lebar atas penampang dilambangkan B. Dimasukkan ke dalam Persamaan. 2.4.10,ini memberikan:

(2.4.12)

Akar kuadrat dari sisi kiri persamaan adalah bilangan Froude untuk penampang dengan geometri yang kompleks umum:

(2.4.13)

Lonjakan hidrolik

Melompat hidrolik adalah transformasi aliran air dari supercriti-cal aliran subkritis. Visual, sepertinya gelombang berdiri di SDTV yangnel. Tingkat air tinggi hilir dari hulu.

Hal ini dimungkinkan untuk mendapatkan hubungan antara tingkat air hulu,h1, Dan hilir, h 2, Dari melompat.

Persamaan momentum memberikan gaya pada kendala seperti:

(2.4.14)

Parameter y adalah tinggi efektif dari tekanan hidrostatik.

Untuk saluran persegi panjang di mana F = 0 dan y = 1/2 y:

E yQ2

2 gA2------------+=

dE

dy

------- 1Q 2

gA 3

--------dA

dy

-------- 0= =

Q 2 B

gA 3---------- 1=

Fr Q2 B

gA 3----------

U

g SEBUAH

B---

-----------= =

F ρ Q U 2

U 1

-( ) Ρ g A2 y

2SEBUAH

1 y

1-( )+=

Halaman 13




(2.4.15)

Bagi dengan lebar dan kepadatan:

(2.4.16)

Memecahkan sehubungan dengan q2:

(2.4.17)

Gunakan definisi bilangan Froude:

(2.4.18)

Memecahkan persamaan sehubungan dengan q2, Dan menghilangkan q2 dengan

persamaan di atas:

(2.4.19)

(2.4.20)

Memecahkan sehubungan dengan y1 / y2:

(2.4.21)

Memecahkan persamaan urutan kedua:

(2.4.22)

Mengingat tingkat air dan bilangan Froude hilir melompat,tingkat air hulu melompat dapat dihitung. Hal ini dapat ditunjukkan

0 ρ Q U 2 U 1-( ) 1

2--- ρ g A

2 y2 SEBUAH 1 y1-( )+=

0 q U 2

U 1

-( ) 1

2--- Gy

22 y

12-( )+ q2 1

y2

-----1

y1

------1

2--- Gy

22 y

12-( )+= =

q2

1

2--- Gy

22 y

12-( )

1

y2-----

1

y1------

----------------------------

1

2--- Gy

22 y

12-( )

1

y1-----

1

y2------

----------------------------= =

Fr 22 U

22

gy2

--------q2

gy23

--------= =

Fr 22 gy

23

1

2--- Gy

22 y1

2-( )

1

y1

----1

y2

------

----------------------------

1

2--- Gy

2 y1-( ) Y 2 y1+( ) Y

1 y2

y2

y1

-( )-------------------------------------------------- ----------

1

2--- Gy

2 y

1+( ) Y

1 y

2= = =

Fr 2

2 1

2--- Y

2 y

1+( )

y1 y2

y23

----------1

2--- Y

2 y

1+( )

y1

y22

-----1

2--- y1 y

2

----- y

1 y

2

----- 1+= = =

y1

y2

-----2 y1

y2---- 2 Fr 2

2-+ 0=

y1

y2

----1

2--- 1 8 Fr

22+ 1-( )=

Halaman 14


bahwa formula mana indeks 1 dan 2 berubah juga berlaku.

2,5 Mantap aliran non-seragam

Hal ini dimungkinkan untuk menghitung permukaan air dari aliran non-seragam stabiloleh formula analitis, selama bentuk cross-sectional adalah rectangu-



lar. Rumus yang berasal berikut ini. Namun, jika cross-detik-tion tidak memiliki bentuk persegi panjang, lokasi permukaan air harusdihitung secara numerik. Hal ini dijelaskan dalam Bab 3.

Derivasi dari rumus untuk kedalaman air didasarkan pada Gambar. 2.5.1.Tinggi total dari garis energi dilambangkan H, sehingga:

(2.5.1)

Energi lereng, saya f , Dapat dihitung dari untuk untuk-contoh Manningmula. Untuk kasus kami, lereng dapat ditulis sebagai:

(2.5.2)

Istilah dalam kurung yang paling tepat dalam persamaan adalah energi spesifik, E, aliran. Istilah dapat ditulis:

(2.5.3)

Persamaan kontinuitas memberikan:

(2.5.4)

Derivasi terhadap y memberikan:

Fakta

Tempat tidur

Air permukaan

Garis energi

y1

y2

z 1 z 2

U 22 / 2g

U 12 / 2g

Aku0

Aku f

Gambar 2.5.1 Sebuah profil memanjang darichannel ditampilkan, antara dualintas-bagian 1 dan 2. Jarakantara bagian adalah dx. Airkedalaman ditandai y dan tingkat tidurelevasi dilambangkan z. Tempat tidur lerengdilambangkan Saya0, Dan kemiringangaris energi dilambangkan Saya f . Perhatikangaris energi ini terletak jarak di atastingkat air, sama dengan kecepatantinggi, U 2 / 2g.

dx

H zyU 2

2 g ------+ +=

Aku f

H 2 H 1-

dx-------------------

d

dx----- Zy

U 2

2 g ------+ + Aku0

d

dx------ Y

U 2

2 g ------++= = =

d

dx------ Y

U 2

2 g ------+

dE

dx-------

dE

dy-------

dy

dx-----

dy

dx------dy

dy-----

1

2 g -----

dU 2

dy---------+= = =

U 2 Q2

SEBUAH 2------=

Halaman 15


(2.5.5)

Rumus akhir yang dibutuhkan adalah untuk bentuk umum cross-sectional:

(2.5.6)

Ini dimasukkan ke Persamaan. 2.5.5, dan hasil ini dalam Persamaan. 2.5.3, memberikan:

(2.5.7)

dU 2

dy---------

d

dy-----

Q2

SEBUAH 2------ Q2 d

dA------- A2-( )dA

dy------- 2 Q2SEBUAH 3- dA

dy--------= = =

dA

dy------ B=

d ------ Y U 2

------+dy----- 1

1----- 2 Q 2- SEBUAH 3- B( )+

dy------ 1 Fr 2-( )= =



Dimasukkan ke dalam Persamaan. 2.5.2, hasilnya adalah:

(2.5.8)

yang dapat disusun kembali untuk:

(2.5.9)

Dalam lebar, saluran persegi panjang, jumlah Froude dapat ditulis:

(2.5.10)

Demikian pula, lereng gesekan, saya f , Dapat dinyatakan sebagai fungsi con-stants dan y, menggunakan Mannings Persamaan:

(2.5.11)

Kedalaman air di sini dilambangkan y. Hal ini digunakan sebagai pengganti hidrolikradius, berarti rumus ini hanya berlaku untuk lebar, saluran persegi panjang.

Eq. 2.5.10 dan Persamaan. 2.5.11 dapat dimasukkan ke dalam Persamaan. 2.5.9, menghasilkanrumus untuk perubahan kedalaman air hanya menjadi fungsi dari con-stants dan y. Ini kemudian dapat diintegrasikan analitis untuk menghitung fungsitions untuk perubahan di kedalaman air di saluran lebar dengan lebar konstan.

Untuk kasus yang lebih kompleks adalah mungkin untuk menggunakan spreadsheet untuk menghitung permukaan air. Kasus tersebut dapat saluran dengan lebar yang bervariasi ataunon-persegi panjang penampang. Contoh spreadsheet tersebut adalahdiberikan pada gambar di bawah. Catatan, untuk aliran sub-kritis kita mulai denganhilir penampang, yang tidak. 1 di meja.

dx 2 g dx 2 g dx

Aku f

Aku0dy

dx------ 1 Fr 2-( )+=

dy

dx------

Aku f

Aku0-

1 Fr

2-

-----------------=

Fr U

gy---------

Q

Dengan gy----------------= =

Aku f

U 2

M 2 y

43---

------------Q2

M 2 B 2 y

103-----

---------------------= =

Halaman 16


Juga mencatat bahwa kedalaman air dihitung dalam lintas-bagian. The parameter lain dalam tabel dihitung sebagai nilai rata-rataantara lintas-bagian. Ini berarti kecepatan air dalam tabel adalah

Silang bagian tidak.

Mendalam, y UFroude

nomor, Fr

Energikemiringan, saya f

Δy

1 (diberikan) Kontinuitas Definisi Eq. 2.5.11 Eq. 2.5.9

2 y1

+ Δy1

Eq. 2.5.10

3

4

1 1.0 1.0 0,322 0.000429 -,04213

2 0,957



fungsi dari dua kedalaman air. Iterasi karena itu diperlukan.Angka-angka dalam spreadsheet adalah contoh dengan debit air1 m3 / s di saluran lebar 1 m, nilai Manning-Strickler dari 50 kemiringan0,001 dan dx 50.

Klasifikasi profil permukaan

Bakhmeteff (1932) mengusulkan sistem klasifikasi untuk permukaan air profil, yang termasuk dalam hampir semua buku teks pada pro permukaan airfile. Sistem ini berguna untuk memahami profil permukaan air, tetapiklasifikasi itu sendiri jarang digunakan dalam praktek rekayasa. Ahli-

file diklasifikasikan menurut tidur lereng, lereng kritis dankedalaman air, seperti yang diberikan dalam Tabel 2.5.1. Kedalaman air dilambangkan y, yangkemiringan dilambangkan saya, dan E adalah energi spesifik dari air. Subscript 0menunjukkan tempat tidur dan c menunjukkan kritis lereng / kedalaman. Angka-angka dikanan meja menunjukkan profil longitudinal profil permukaan. The baris untuk kedalaman kritis dan mendalam yang normal juga diberikan. Normalkedalaman ditemukan dengan menggunakan persamaan misalnya Manning, mengingatkekasaran, tidur kemiringan dan debit air. Kedalaman kritis ditemukandari Persamaan. 2.4.7.

Sistem mengklasifikasikan setiap profil permukaan dengan suratdan nomor. Surat itu adalahhanya fungsi dari lerengsungai di dis diberikan biaya. Surat-surat yang diberikan dalamtabel di sebelah kanan adalahdigunakan.

Jumlah tersebut merupakan indeks untuk bilangan Froude yang sebenarnya disaluran. Aliran subkritisdilambangkan 1 sementara supercriti-cal aliran dilambangkan 3.Indeks 2 dapat berarti

Metode memohonlebih iterasi adalah depend-ent pada khususnyaProgram spreadsheet. UntukLotus 123, menggunakan F9 padaKeyboard berulang kali. UntukMS Excel, gunakan menuTools, Options, kalkulasitions, dan mencoret Itera-tions, dan memberikan nomordi edit-lapangan, untuk-contoh ple 50.

:

Surat Berdiri untuk Bed lereng

M Ringan Subkritis

S Curam Superkritis

C Kritis Kritis

H Horisontal HorisontalSEBUAHMerugikan Merugikan

Halaman 17


superkritis atau subkritis, tergantung pada surat itu. Ini diberikan lebihrinci dalam Tabel 2.5.1.

Tabel 2.5.1. Klasifikasi profil permukaan air

Saluranlereng

Profil jenis

Kisaran kedalamanFrdy /dx

dE /dx

Ringan

Aku0<I c

y0> yc

M1 y> y0> yc <1 > 0 > 0

M2 y0> y> yc <1 <0 <0

M3 y0> yc> y > 1 > 0 <0

Curam

Aku0> Sayac

y0<yc

S1 y> yc> y0 <1 > 0 > 0

S2 yc> y> y0 > 1 <0 > 0

S3 yc> y0> y > 1 > 0 <0

Kritis Aku0= Sayac y0= yc

C1 y> yc <1 > 0 > 0

C3 y <yc > 1 > 0 <0

Horisontal

Aku0= 0

H2 y> yc <1 <0 <0

H3 y <yc > 1 > 0 <0

Merugikan

Aku<0

A2 y> yc <1 <0 <0 yc

A2

A3

yc

H2

H3

yc

C1

C3

y0

yc S1

S2

S3

yc

y0 M1

M2

M3



Contoh profil permukaan air: Ketika sungai ringan-miring mengalirke dalam reservoir, kami memiliki kurva M1. Profil air hanya hulu sebuahspillway dapat M2, H2 atau A2. Profil sebelum melompat hidrolik bisamenjadi M3, H3 atau A3. Profil S1 dapat ditemukan setelah melompat hidrolik dalamsaluran curam.

2,6 Gelombang di sungai

Hal ini dimungkinkan untuk mendapatkan formula untuk gelombang bepergian naik atau turun SDTV sebuahnel dengan lebar konstan dan kemiringan. Estimasi kasar dari gelombang tersebut dapatdengan demikian dibuat, dan formula juga dapat digunakan untuk mengevaluasiHasil dari model numerik. Dua jenis gelombang dapat digambarkan:

- Gelombang kinematik- Gelombang dinamis

Kedua jenis berasal dan dibahas berikut ini.

Gelombang dinamis

Persamaan untuk gelombang dinamis berasal dengan melihat longitudi- yang profil nal dari gelombang yang diberikan pada Gambar. 2.6.1:

0 A3 y <yc > 1 > 0 <0

Halaman 18


Situasi stabil diperoleh jika sistem rujukan bergerak sepanjangchannel dengan kecepatan c. The kecepatan air hulu dan hilirgelombang menjadi U 1 - C dan U 2 - C, masing-masing. Momentum persamaan memberikan:

(2.6.1)

Bersama-sama dengan persamaan kontinuitas:

(2.6.2)

Memecahkan persamaan kontinuitas sehubungan dengan U 2:

(2.6.3)

Memasukkan ini ke dalam persamaan momentum, menghilangkan U 2:

(2.6.4)

Menghilangkan 's c di sisi kanan, dan bergerak istilah ke kirisisi:

Gambar sketsa 2.6.1 Definisigelombang bepergian turunsungai. Kedalaman huludan kecepatan dilambangkan h1dan U 1, Masing-masing dan

kedalaman hilir dankecepatan dilambangkan h2 danU 2masing-masing. Kecepatangelombang dilambangkan c.

y1

y2

U1

c

U2

U 1 c-( )2 y11

2--- Gy

12+ U 2 c-( )2

y21

2--- Gy

22+=

U 1

c-( ) Y 1

U 2

c-( ) Y 2

=

U 2

U 1

c-( ) y1 y2----- C +=

U 1 c-( )2 y11

2--- Gy

12 y2

2-( )+ U 1 c-( ) y

1 y2----- Cc-+

2 y2=

2 y 2 1 2 2



(2.6.5)

Menyederhanakan istilah pertama dan mengubah jabatan kedua:

(2.6.6)

Lebih menyederhanakan istilah pertama dan mengubah tanda keduaIstilah:

(2.6.7)

Membaginya dengan bagian kedua dari istilah pertama:

U 1 c-( ) y1 1 y2

---- y2- 2--- Gy 1 y2-( )( )+ 0=

U 1 c-( )2 y1

y12

y2------

1

2--- Gy

1 y2-( ) Y 1 y2+( )( )+ 0=

U 1

c-( )2 y1 y2---- Y

2 y

1-( ) 1

2--- Gy

2 y

1-( ) Y

1 y

2+( )( )- 0=

Halaman 19


(2.6.8)

Mengambil akar kuadrat dari setiap sisi, mengalikan dengan -1 dan bergerak U 1 untuksisi lain:

(2.6.9)

Jika gelombang kecil, y 2 dan y 1 kira-kira sama. Persamaankemudian menjadi:

(2.6.10)

Gelombang kinematik

Rumus untuk gelombang kinematik dengan kecepatan c, berasal dari conti- yang persamaan nuity, melihat situasi yang sama seperti pada Gambar. 2.6.1. Hal ini memberikan:

(2.6.11)

Diferensial Q dapat diturunkan dari Mannings Persamaan (Persamaan. 2.2.1)untuk lebar, saluran persegi panjang, dibedakan sehubungan dengan airmendalam:

(2.6.12)

atau ditulis ulang:

(2.6.13)

Rumus untuk daerah penampang kemudian dibedakan denganterhadap y:

(2.6.14)

U 1

c-( )2 1

2--- y2 y1---- Gy

1 y

2+( )=

c U 1 g

2---

y2

y1----- Y

2 y1+( )±=

c U gy±=Jika Persamaan. 2.6.10 dikombinasikandengan definisiBilangan Froude, fol- yangRumus melenguh adalahdiperoleh:

Jika Fr <1, c dapat menjadi positif dan negatif. Dialiran subkritis, gelombangmaka dapat melakukan perjalanan baikhulu dan bawah-streaming saluran.

Jika Fr> 1 maka c akan selalumenjadi positif. Untuk supercriti-cal aliran, gelombang bisahanya perjalanan hilir.

c U U

Fr --------±=

cdQ

dA-------=

dQ

dy-------

d

dy------ AU ( ) d

dy----- Dengan

1n--- Saya

12--- y

23--- d

dy------ B

1n--- Saya

12--- y

53--- BI

12---

n--------

5

3--- Y

23---

= = = =

dQ5

3--- BI

n------

12---

y

23---dy=

dA Bdy=



Memasukkan dA dan dQ dari dua persamaan di atas ke dalam persamaan. 2.6.11 memberikan:

(2.6.15)

yang merupakan formula untuk gelombang kinematik.

Bentuk gelombang

Eq. 2.6.10 menunjukkan bahwa kecepatan gelombang akan meningkat dengan lebih besar kedalaman.

c5

3---

1n--- Saya

12--- y

23--- 5

3--- U = =

Halaman 20


Ini juga diberikan dari Mannings Persamaan (2.2.1), sebagai meningkatkecepatan air akan menyebabkan peningkatan kedalaman. Bentuk gelombang akansehingga mengubah ketika bergerak hilir saluran. Dengan asumsidepan dan akhir gelombang memiliki kedalaman yang lebih kecil, dan maksimumkedalaman adalah pusat dari gelombang, dua fenomena akan berlangsung:

1. depan gelombang akan menjadi lebih curam2. Ekor gelombang akan menjadi kurang curam

Hal ini dapat dilihat pada Gambar. 2.6.2:

2.7 Persamaan Saint-Venant

Gelombang banjir umum untuk situasi satu dimensi digambarkan olehPersamaan Saint-Venant. Maka kecepatan air dan air kedalaman bisa bervariasi baik dalam waktu dan sepanjang satu dimensi spasial. The Saint-Venant persamaan juga disebut persamaan dinamis penuh untuk perhitungan 1DGelombang banjir.

Persamaan ini berasal dengan melihat bagian dari saluran seperti yang diberikan dalamGambar. 2.7.1. Saluran ini memiliki lereng saya, kedalaman air y, lebar B, dis air biaya Q dan kecepatan U.

Sebuah persamaan kontinuitas untuk situasi ini pertama berasal. Melihatdebit air akan keluar dan dalam volume dalam periode waktu Δ t, kitamendapatkan:

(2.7.1)

h

x

h

x

Gambar 2.6.2. Perubahangelombang bentuk dari waktu ke waktu.Gelombang awal ditampilkandi kiri. Airmengalir dari kiri ke benar. Sosok yang tepatmenunjukkan bentukgelombang setelah beberapa waktu. Thedepan curam dan ekorkurang curam.

y1

y2

Gambar 2.7.1 Sebuah profil memanjang darichannel, antara dua bagian 1 dan

2. Jarak antara bagianaku sΔx. Kedalaman air dilambangkan y.Tempat tidur kemiringan dilambangkan Saya0.

Δ x

x

P 1 P 2

Δ V Q QdQ

dx------- Δ x+- Δ t dQ

dx-------- Δ x Δ t = =



Jumlah ini air harus sama dengan perubahan volume yang disebabkan oleh membubung

Halaman 21


ing / jatuh dari ketinggian air: :

(2.7.2)

Mencatat bahwa B juga dapat berubah sebagai tingkat air naik / turun, satu kaleng bukannya menulis persamaan sebagai:

(2.7.3)

Di mana A adalah sama dengan luas penampang aliran. Catatan, di sebuah mendasichannel tangular, B adalah konstan, dan Persamaan. 2.7.2 diperoleh.

Menggabungkan Persamaan. 2.7.1 dan 2.7.3, persamaan berikut diperoleh:

(2.7.4)

Ini adalah persamaan kontinuitas sering digunakan sehubungan dengan memecahkanPersamaan Saint-Venant. Persamaan Saint-Venants sendiri berasal dariHukum kedua Newton:

(2.7.5)

Untuk bagian dalam Gambar. 2.7.1, Percepatan jangka di sisi kananmenjadi:

(2.7.6)

Kemudian kekuatan eksternal pada volume yang diberikan. Ada empat gaya:

1. Gravity komponen:

(2.7.7)

Ini adalah komponen yang sama seperti yang digunakan untuk derivasi dari rumus untukyang tegangan geser untuk aliran seragam.

2. B geser stres:

(2.7.8)

Istilah negatif, sebagai kekuatan yang berada di x-arah negatif. Seringkali,gesekan kemiringan diperkenalkan:

(2.7.9)

Gesekan kemiringan dihitung dari rumus gesekan empiris, untukcontoh persamaan Manning. Istilah kemudian menjadi:

dy

dt ------

Δ V dy

dt ----- B Δ x Δ t =

Δ V dA

dt ------- Δ x Δ t =

Persamaan kontinuitas: dA

dt ------

dQ

dx-------+ 0=

F ma=

ma ρ yB Δ xdU

dt -------=

F g

ρ gyB Δ XI 0

=

F b

τ- B Δ x=

Aku f

τ

ρ gy---------=



Halaman 22


(2.7.10)

3. Tekanan gradien:

Tekanan gradien adalah karena tingkat air yang berbeda di setiap sisielemen. Kekuatan tekanan hidrostatik pada air penampang disaluran persegi panjang diberikan sebagai:

(2.7.11)

Untuk volume control pada Gambar. 2.7.1, ada dua kekuatan, satu di setiapsisi volume. Gaya total dari tekanan gradien keharusanOleh karena itu adalah jumlah dari dua kekuatan hidrostatik.

(2.7.12)

Kedalaman dapat ditulis sebagai fungsi dari kemiringan permukaan:

(2.7.13)

(2.7.14)

Istilah terakhir berisi sejumlah kecil kuadrat, jadi ini jauh lebih kecildaripada musim lalu kedua. Istilah terakhir karena itu diabaikan. Inimemberikan:

(2.7.15)

Istilah negatif, karena positif mendalam-gradien akan menyebabkangaya tekanan di x-arah negatif.

Istilah 4. Momentum:

Persamaan momentum untuk volume control adalah:

(2.7.16)

(2.7.17)

Tanda negatif adalah karena kecepatan gradien positif akan menyebabkan lebih banyakmomentum untuk meninggalkan volume dari apa yang masuk. Hal ini menyebabkan kekuatan dix-arah negatif.

Mengatur jumlah semua empat kekuatan sama dengan is tilah percepatan, salah satumemperoleh:

(2.7.18)

F b

ρ Gyi f

- B Δ x=

F 1

2--- ρ GBY 2=

F p

1

2--- ρ GBY

12 1

2--- ρ GBY

22-=

F p

ρ gB

2---------- Y 2 y

dy

dx----- Δ x+

2-=

F p

ρ gB

2---------- Y 2

y2 2

dy

dx------ Y Δ x-

dy

dx----- Δ x-

2-=

F p

ρ- g dy

dx----- Δ x Oleh()=

F m

ρ Q U 1

U 2

-( ) ρ UBy U U dU

dx------- Δ x+-= =

F m

ρ- UBydU

dx------- Δ x=

F g F b F p F m+ + + ma=

Halaman 23




atau:

(2.7.19)

Ini dapat disederhanakan ke:

(2.7.20)

Persamaan Saint-Venant harus diselesaikan dengan metode numerik.Hal ini dijelaskan dalam Bab 3.

2,8 Pengukuran debit air di sungai alami

Ada beberapa metode untuk mengukur debit air di sungai.Metode yang paling umum adalah dengan menggunakan meteran saat ini dan mengukurlangsung pada berbagai titik dalam penampang sungai. Hal ini dapat dilakukan di beberapa pembuangan air, dan kurva Peringkat dapat diperoleh, di manadebit air sebagai fungsi dari ketinggian air diberikan. Menggunakan harian pengamatan dari permukaan air, kurva dapat memberikan serangkaian saatdebit air. Ini digunakan untuk memprediksi banjir dan air rata dis biaya di sungai untuk digunakan dalam pembangkit listrik tenaga air atau pasokan air.

Pengukuran langsung dari pembuangan di sungai saat ini biasanya dilakukan olehADCP instrumen. The ADCP adalah singkatan untuk Acoustic DopplerProfiler saat ini. The ADCP bekerja dengan mengirimkan sinyal akustikdari instrumen ke dalam air. Partikel kecil di dalam air mencerminkansinyal kembali ke penerima pada instrumen. Sinyal akan fungsidari kecepatan partikel relatif terhadap instrumen. Dengan mengukur pada

sejumlah besar poin dalam penampang, debit dapat computed.

ADCP yang biasanya dipasang di atas perahu yang diseret di air permukaan dalam penampang. Sinar poin instrumen vertikal bawah, dan langkah-langkah beberapa titik dalam profil vertikal. Sebuah echo-terdengar perangkat biasanya juga termasuk dalam instrumen, mengukurkedalaman air. ADCPs modern juga memiliki alat pelacak bawah,mengukur jarak instrumen telah dilalui. Sebuah GPS juga bisadilengkapi dengan instrumen, memungkinkan korelasi antara pengukuranKASIH dan misalnya model medan digital.

ρ gyB Δ XI 0

ρ Gyi f

- B Δ x ρ- g dy

dx------ Δ x By() Ρ UB Δ xy dU

dx-------- ρ yB Δ xdU

dt -------=

g Saya

0 Aku f

-( ) G dy

dx------ U

dU

dx------ dU

dt -------+=

Persamaan Saint-Venant:

Gambar 2.8.1 Cross-detik-tion dengan empat diukurprofil kecepatan. TheRata-rata kecepatan di setiap profil dihitung dandikalikan dengan kedalamandan lebar, w, masing-masingsektor. Ini kemudianmenyimpulkan atas semua profil untuk mendapatkan airdebit.

1 2 3 4

w3

Halaman 24




Gambar 2.8.2 Gambar perahu dengan ADCP untuk penggunaan laboratorium.

Gambar 2.8.3 Contoh hasil dari pengukuran ADCP

Halaman 25


Cara lain adalah dengan menggunakan pelacak, misalnya bahan radioaktif ataukimia yang mudah diukur untuk konsentrasi kecil di dalam air. SEBUAHkuantitas dikenal, m, pelacak yang dibuang di sungai. Lebih jauh ke bawah-streaming, di mana pelacak benar-benar dicampur dalam air, konsentrasition, c, diukur dari waktu ke waktu. Jumlah pelacak dapat dihitungsebagai:

(2.8.1)

Dengan asumsi aliran, debit air, Q, adalah konstan. Kuantitas pelacak dan integral dari pengukuran konsentrasi diketahui, sehinggadebit dapat dihitung sebagai:

m Cq td =



(2.8.2)

Metode ini mahal, sebagai pelacak hilang untuk setiap pengukuran. Juga, pelacak mungkin tidak ramah lingkungan. Oleh karena itu, metode inihanya digunakan sangat jarang, dan dalam situasi di mana sulit untuk melakukan point- pengukuran kecepatan. Kaleng ini misalnya menjadi selama banjir.

2,9 Masalah

Masalah aliran 1. Seragam

Sebuah sungai alami dengan kedalaman 2 meter, memiliki kecepatan air rata-rata 3 Nona. Apa gradien energi maksimum dan minimum? Apakah aliransuperkritis atau subkritis? Apakah ini mungkin untuk melihat langsung di lapangan?

Masalah saluran 2. Compound

Sebuah saluran dengan geometri penampang berikut dianggap:

Channel dapat dibagi dalam tiga bagian, A, B dan C. Bagian B adalahsaluran utama, dan bagian A dan C adalah overbanks. Kemiringanchannel adalah 1: 500.

Menghitung debit air di saluran, diberi Manning-Stricklerkoefisien 60 untuk seluruh channel.

Kemudian, menghitung debit air di saluran diberi Manning-Strickler koefisien 60 untuk saluran utama dan 40 untuk overbanks.

Q mc td

----------=

15 m10 m

8 m

5 m3 m

SEBUAH

B

C

Halaman 26


Soal 3. Mantap aliran non-seragam

Menghitung profil permukaan air, menggunakan spreadsheet, di belakang rendahrun-of-bendungan sungai. Lebar sungai adalah 30 meter, air dis biaya adalah 50 m3/ s, ketinggian bendungan adalah 10 meter dan kemiringan sungai1: 400.

Apa yang terjadi jika lereng adalah 1: 100? Menghitung profil muka air

untuk situasi ini juga.Soal 4: Lokasi lompatan hidrolik.

SEBUAH B C



Angka ini menunjukkan sketsa dari profil memanjang dari tempat tidur dan air profil permukaan. Air dibiarkan keluar dari outlet bawah hilir bendungan, dititik A. Pada titik B, lompatan hidrolik terjadi. Tingkat air pada titik Adan C diberikan, bersama dengan debit air. Pertanyaannya adalah untuk menemukan jarak antara A dan B.

Data:

Tingkat air (gerbang pembuka) di A: ySEBUAH = 0,4 meterTingkat air di C: y C = 3 meter

Discharge: Q = 6 m 2/ s.Manning-Strickler koefisien: 50.

Pengukuran Discharge masalah 5. menggunakan pelacak

2 kg tracer dibuang di sungai. Beberapa km di hilir, fol- yangKonsentrasi melenguh diukur:

Menghitung debit air di sungai.

c, (ppm)

waktu (menit)

55 56

0

1000

Halaman 27


3. pemodelan numerik aliran sungai di 1D

Model numerik 1D adalah alat yang paling umum digunakan di HidrolikRekayasa untuk mengevaluasi efek dari gelombang banjir di sungai. Dispersi polutan di sungai juga banyak dilakukan dengan menggunakan model 1D.

3.1 aliran Mantap

Perhitungan aliran menggunakan persamaan kontinuitas dan luas-tion untuk kerugian gesekan untuk menghitung kecepatan dan lokasi air permukaan. Rumus Manning (Persamaan. 2.2.1) ini paling sering digunakan.

Perhitungan dimulai dengan mengukur geometri dari sejumlahlintas-bagian di sungai. Jarak dan elevasi dari sejumlah poin di lintas-bagian dicatat. Jarak antar penampang juga diukur.

Untuk setiap bagian, kurva dibuat dengan daerah dibasahi sebagai fungsi daritingkat air. Kurva ini digunakan dalam perhitungan berikut.

Gambar 3.1.1. Sketsasungai dengan silangbagian.

kedalamanGambar 3.1.2. Generasikurva untuk luas penampang sebagai



Perhitungan elevasi air biasanya dimulai dengan nilai yang diberikanhilir, karena ini adalah nilai pengendali untuk aliran subkritis. Kemudianelevasi air dari hulu penampang dapat ditemukan. -Prosedur yangdure diuraikan dalam Bab 2.5 dapat digunakan, menerapkan Persamaan. 2.5.9. Alternatif-masing, variasi dari metode ini diberikan berikut ini.

Kehilangan energi antara penampang dapat ditemukan dengan memecahkanPersamaan Manning sehubungan dengan kemiringan gesekan:

(3.1.1)

Radius hidrolik, R, ditemukan dari kurva yang sama seperti yang diberikan pada Gambar.3.1.2. Melihat Gambar. 3.1.3, elevasi perbedaan air permukaan, Δ z,antara penampang kemudian dapat ditemukan dengan menggunakan Energi yang luas-tion, memberikan:

daerah

fungsi airmendalam. Kurva digunakansebagai masukan geometri untukmodel numerik.

Aku f

U 2

M 2 R

43---

--------------=

Satu masalah adalah bagaimanamenentukan gesekankoefisien, M. Biasanya, prosedur kalibrasi memilikiharus dilakukan, di manahasilnya dibandingkan dengantingkat air yang diukur,dan M nilai disesuaikanagar sesuai dengan data. Beberapa data program meliputi automatic kalibrasi-prosedur prosedur-menggunakan konsep ini.

Halaman 28


(3.1.2)

Jarak, Δ x, antara penampang diberikan oleh pengguna. Thekecepatan air di dua penampang 1 (upstream) dan 2 (bawah-stream) dihitung dari persamaan kontinuitas air:

(3.1.3)

dimana debit air, Q, adalah konstan

Prosedur ini kemudian menebak nilai Δ z, memberikan tingkat air untuk baik lintas-bagian. Kemudian A dan R diambil dari kurva untuk dualintas-bagian, dan nilai rata-rata yang digunakan dalam persamaan. 3.1.1. The kecepatanPersamaan di. 3.1.2 dihitung dari Persamaan. 3.1.3. Eq. 3.1.2 kemudian memberikan baru

z 1 z 2- Δ z Aku f

Δ xU 2

2

2 g --------

U 12

2 g ----------+= =

Fakta

Tempat tidur

Air permukaan

Garis energi

z 1

z 2

U 22 / 2g

U 12 / 2g

Aku f

Gambar 3.1.3 Sebuah profil memanjang darichannel ditampilkan, antara dua lintas bagian 1 dan 2. Jarakantara bagian adalah dx. Elevasi yangtion air dilambangkan z. Thekemiringan garis energi dilambangkan Saya f .

Δ x

U Q

SEBUAH ---=



perkiraan Δ z. Setelah beberapa iterasi, nilai-nilai Δ z harus menjadisama seperti iterasi sebelumnya, dan prosedur telah berkumpul. TheMetode solusi biasanya tidak sensitif terhadap nilai awalnya menduga, sehinggamisalnya Δ z = 0,0 dapat digunakan.

Bagian kontrol

Eq. 3.1.6 memberikan perubahan permukaan air antara dua lintas-detik-tions. Pertanyaannya kemudian yang penampang harus digunakan untuk memulai perhitungan.

Untuk aliran superkritis, permukaan air terutama ditentukan olehaliran hulu. Ingat, teori gelombang kinematik menunjukkan bahwagelombang tidak bisa menyebarkan hulu aliran superkritis. Untuk subkritismengalir, aliran biasanya ditentukan oleh tingkat air di hilir. Ini berarti bahwa perhitungan harus mulai hulu dan bergerak downaliran aliran superkritis. Untuk aliran subkritis, perhitungan harusmulai hilir dan hulu bergerak.

Sebelum memulai perhitungan, bagian pengendali harus bertekad. Sebuah contoh khas adalah bagian aliran kritis, di mana aliran

Halaman 29


pergi dari subkritis ke superkritis. Perhitungan kemudian dapat mulai di bagian ini dan bergerak hulu dalam aliran subkritis. Hal ini juga dapat bergerakhilir aliran superkritis.

Kontrol lain untuk tingkat air waduk dan danau. Kemudian permukaan air diberikan. Biasanya, aliran hulu subkritis daridanau. Kemudian perhitungan dimulai di danau dan bergerak hulu.

Salah satu masalah utama adalah ketika aliran berubah dari superkritis kesubkritis. Sebuah lompatan hidrolik kemudian akan membentuk, dan algoritma khusus berdasarkan pada persamaan momentum harus digunakan. Seringkali, algoritma initidak sangat stabil, dan itu mungkin bermasalah untuk mendapatkan solusi.

3.2 aliran goyah

Aliran air satu dimensi diatur oleh luas- Saint-Venanttions: kontinuitas air:

(3.2.1)

dan konservasi momentum:

(3.2.2)

Model 1D dapat diklasifikasikan sesuai dengan berapa banyak istilah yang digunakanPersamaan di. 3.2.2. Memecahkan penuh persamaan Saint-Venant dijelaskan dalamBab 3.4. Salah satu penyederhanaan adalah untuk mengabaikan dua istilah pertama. ini adalahdisebut persamaan untuk gelombang difusi:

(3.2.3)

Jika pertama tiga istilah dalam Pers. 3.2.2 diabaikan, gelombang kinematik yang luas-tion muncul:

(3.2.4)

Selain itu, persamaan kontinuitas (3.2.1) diselesaikan.

Solusi dari persamaan gelombang kinematik dijelaskan pada Bab 3.3.Metode Muskingum, atau Hidrologi routing, hanya menggunakan kontinyu yang persamaan ity. Metode ini dijelaskan pada Bab 3.5.

A ∂t ∂

------Q∂

x ∂-------+ 0=

U ∂

t ∂------ U

U ∂

x ∂------ g

y ∂ x ∂

----- g I b

Aku f

-( )-+ + 0=

g y ∂ x ∂

----- g I b

Aku f

-( )- 0=

g I b

Aku f

-( ) 0=



3.3 goyah aliran - gelombang kinematik

Metode paling sederhana untuk menghitung aliran goyah di sungai adalah dengan kine- yang persamaan gelombang matic. Ada beberapa metode solusi untuk luas- inition. Dua metode yang dijelaskan di sini:

- Solusi oleh perbedaan- Solusi Analitik

Solusi dengan diferensial adalah bentuk standar memecahkan persamaan gelombang. Namun, gelombang kinematik sangat sederhana yang juga memungkinkan untuk menggunakan

Halaman 30

Numerical Modelling and Hydraulics 29

an analytical solution.

Analytical solution

The solution method is based on Eq. 2.6.9, the formula for the wavevelocity, c :

(3.3.1)

In Chapter 2 this equation was derived for a wide, rectangular channel,giving K =5/3. For a natural channel, K, may vary between 1.3 and 1.6.

The algorithm is based on tracking points in the hydrograph of the wave.For each point, the water depth and water velocity is computed, basedon Manning's formula and the continuity equation. Then Eq. 3.3.1 isused to compute the wave speed. The time to travel a given distance toa downstream cross-section is then computed for each point in thehydrograph. An example is given in Table 3.3.1, taken from a spread-sheet. The spreadsheet is computed from left to right.

Table 3.3.1 Analytical computation of kinematic wave

The result is in the two columns to the right in the spreadsheet, 5000 and10 000 meters downstream. The time, T , in these columns are computed by the following equation:

(3.3.2)

The use of the Mannings formula in the table is derived using the conti-nuity equation to eliminate the water depth:

atau (3.3.3)

Time for X=0given as input

data (min)

PelepasanGiven asinput data

(m3/s)

VelocitydariManning’s

formula (m/s)

Depth, fromcontinuity persamaan

(m)

Wavespeed, c,From Eq.

3.3.1 (m/s)

Time forX=5000meters(min)

Time forX=10000meters(min)

0 100 2.30 0.58 3.83 21,8 43,5

10 200 3.03 0.883 5.05 26,5 43.0

20 300 3.56 1.126 5.94 34,0 48,1

30 400 4.00 1.330 6.66 42,5 55,0

40 300 3.56 1.12 5.94 54.0 68.1

50 200 3.03 0.88 5.05 66,5 83.0

60 100 2.30 0.58 3.83 81.8 103.5

c KU =

T T 0

x

c---+=

U MI

12--- Q

UB-------

23---

= U MI

12---Q

23---

2--------------------

35---

=



B 3

Halaman 31


Solution by differentials

There exist more involved methods to compute the kinematic wave. Thecontinuity equation and a formula for the normal depth in a reach is thendigunakan. Note that Eq. 3.2.4 for the kinematic wave gives that the energyslope is equal to the bed slope. This means the flow is uniform, and afriction formula can be used, for example Manning's formula. If the veloc-ity in this formula is replaced by Q / A , and the definition of the hydraulicradius is used, the following derivation can be made:

(3.3.4)

(3.3.5)

The equation can be differentiated with respect to time:

(3.3.6)

The continuity equation can be used:

(3.3.7)

Combining Eq. 3.3.6 and Eq. 3.3.7 gives:

(3.3.8)

Assuming the term in the bracket is constant, the equation can be solvedusing first-order differences for time and second-order differences forruang. A notation of two subscripts is used, where the first subscript, i,denotes the space direction and the second, j, the time:

(3.3.9)

(3.3.10)

Using these two equations we can transform Eq. 3.3.8 to:

Aku0 Aku f

U

Bapak

23---

----------

2QP

A A

23--- M

---------------

2

= = =

SEBUAH P

25---

Aku0

310-----

M

35---

----------------Q

35---

=

dA

dt -------

3

5---

P

25---

Aku0

310-----

M

35---

-----------------Q

25---- dQ

dt -------=

Q∂

x ∂------

A ∂t ∂

------+ 0=

Q∂

x ∂-------

3

5---

P

25---

Aku0

310-----

M

35---

-----------------Q

25---- Q∂

t ∂------+ 0=

Q∂

x ∂------

Qi 1+ j ,

Qi 1- j ,-

2Δ x--------------------------------------≈

Q∂

t ∂-------

Qij 1+. Q

ij ,-Δ t

-------------------------------≈



Halaman 32


(3.3.11)

Index i is used for the space dimension and j for the time. Given an initialsituation, Eq. 3.3.11 can be solved with respect to Q i,j+1to give a formulafor the discharge at a node as a function of the discharges at the nodesin the previous time step:

(3.3.12)

The equation can be solved numerically using a spreadsheet, if thereexist simple formulas for P as a function of the discharge. One axis in thespreadsheet is the distance x , and the other axis is the time.

Example: Solution by differentials.

Compute the water discharge at cross-section 5 at time step 11 minutes,when water discharge at time step 10 minutes are given as:

Cross-section 4: Q=203 m 3 / s

Cross-section 5: Q=195 m 3 / s

Cross-section 6: Q=188 m 3 /s.

Assume a time step of one minute, and that the cross-sections are 200meters apart. The bed slope is 1:400 and the Manning-Stricklers coeffi-cient is 66. The wetted perimeter is 54 meters at cross-section 5 for thisdischarge.

Solution: We give the numbers to equation 3.3.12:

The disharge is 208 m3/s.

Diskusi

One simplification for the kinematic wave is assuming uniform flow on areach. The water level is then a unique function of the water discharge. SEBUAHrating curve showing the water levels at a gauging station during the passing of a flood, will get different values for the rising and the fallinglimb of the hydrograph. This is shown in Fig. 3.3.1. However, because ofthe simplifications, the kinematic wave method is not able to model this

efek.

Q

i 1+ j ,

Q

i 1- j ,

-

2Δ x--------------------------------------3

5--- P

25---

Aku0

310------

M

35-------------------Q ij ,

25---- Q

ij 1+. Q

ij ,

-

Δ t -------------------------------+ 0=

Qij 1+. Q

ij ,5

3--- Aku0

310-----

M

35---

P

25---

----------------- Qij ,

25--- Q

i 1+ j , Qi 1- j ,-( )

2Δ x-------------------------------------------Δ t -=

Q 5 11. 1955

3---

1

400---------

310-----

x 6635---

54

25---

------------------------------ 195

25--- 188 203-( )

2 x 200---------------------------- 60()-=

Halaman 33




Another point to note is that the differential solution method introducessome errors, causing the maximum discharge for a wave to be damp-ened . This is not observed in the quasi-analytical solution method. Bagaimanapunever, it is noticed in field data. The damping of a real flood wave must not be confused with the damping introduced by inaccuracies in the numeri-cal algorithm.

3.4 Unsteady flow - Saint-Venant equations

Solving the full Saint-Venant equations is done for dam-break modellingand other flood problems where there is a rapid change in the waterdepth over time, and the water discharge is significantly higher than theavailable calibration data.

The approach to solving the Saint-Venant equations is more involvedthan solving the kinematic wave equation. There exist a number of differ-ent solution methods, which can be divided in two groups:

- Explicit methods- Implicit methods

When the differences in space are to be computed, the question is if thevalues in time step j or time step j +1 should be used. If the values in timestep j is used, an explicit solution is given. If the values at time step j +1are used, an implicit solution is given. An implicit solution is more stablethan an explicit solution, and longer time step can be used. An explicitsolution is simpler to program.

Q

y

observdiions

Figure 3.3.1. Rating curveduring passing of a floodwave. The kinematic wavemodel gives the same val-ues for the rising and fallingof the hydrograph. Theobservations give differentstage/discharge observa-tions in the rising and fallingof the hydrograph.

observdiions

kdiemdisaya mengertiwSebuahve

Implicit/Explicit method

Halaman 34


The explicit computational molecule is seen in Fig. 3.4.1.

Waktu

j

Figure 3.4.1 Nodes forcomputation mole-cule. The upper figureshows an explicit mole-cule and the lower figureshows an implicit mole-



Often, the gradients are computed as a combination of values at timestep j and time step j -1. A weighting factor, θ, is then used, where thefinal solution is θ times the gradients at time step j , plus (1-θ) times thegradients at time step j -1. This means if θ is 1, an implicit solution isgiven, and if θ is 0 the solution is explicit. If θ is between 0 and 1, the val-ues at both time steps will be used. The method is then still said to beimplicit. The DAMBRK program uses a default value of 0.6 for θ, equiva-lent of using 60 % of the value at time step j and 40 % of the value attime step j -1.

Both methods are described in the following.

Explicit method

The explicit solution is easier to solve than the implicit method. Hal ini dapat be seen from Fig. 3.4.1. The initial water discharge in the river is known,so the first computation starts with the next time step. For each cross-section, i , the discharge can be computed from the discharges at the previous time step. This is repeated for all cross-sections, and the dis-charges at the time step j is computed by one sweep. Then the computa-tion proceeds to the next time step.

What is needed is a formula for the water discharge at time step j as afunction of the discharges at time step j -1. This is obtained by discretiz-ing Eq. 3.2.1 and Eq. 3.2.2. For a rectangular channel with depth y andwidth B , the continuity equation, Eq. 3.2.1, becomes

(3.4.1)

The B ’s are eliminated and y and U is taken to be the value at the previ-ous time step. The following differentials are used:

Ruang

j-1

i-1 i i + 1

cule. For the explicitmolecule, the value atnode ( i,j ) is computedfrom nodes at the previ-ous time step, j -1. The inodes in the spacedirection is the differentcross-sections.

Waktu

Ruang

j

j-1

i-1 i i + 1

B y ∂

t ∂----- B

yU ( )∂

x ∂--------------+ B

y ∂

t ∂----- BU

y ∂

x ∂----- y

U ∂

x ∂------++ 0= =

Halaman 35


(3.4.2)

(3.4.3)

(3.4.4)

U ∂

x ∂------

U i 1 j , 1-+ U i 1 j ,- 1--

2Δ x-------------------------------------------------- -=

y ∂

x ∂-----

yi 1 j , 1-+ y

i 1 j ,- 1--

2Δ x------------------------------------------------=

y ∂t ∂

----- y

ij , y

ij 1-.-

Δ t ----------------------------=

yij , y

ij 1-.-Δ t

----------------------------U ij 1-.

yi 1 j , 1-+ y

i 1 j ,- 1--

2Δ x----------------------------------------------- y

ij 1-.U

i 1 j , 1-+ U i 1 j ,- 1--

2Δ x-------------------------------------------------- -++ 0=



(3.4.5)

The equation is solved with respect to the water depth at time step j:

(3.4.6)

In a similar way, the Saint-Venant equation itself (Eq. 3.2.2) can be dis-cretized as:

(3.4.7)

dimana

(3.4.8)

is taken from Mannings equation.Eq. 3.4.6 can be solved with respect to U

aku j

:

(3.4.9)

The explicit procedure then becomes:

1. Guess starting values of U and y along the channel.2. Determine inflow values of U and y3. Repeat for each time step

4. Repeat for each cross-section of a time

yij , y

ij 1-.Δ t

2Δ x---------- U

ij 1-. yi 1 j 1 -.+ y

i 1 j 1 -.--( ) yij 1-. U

i 1 j 1 -.+ U i 1 j 1-.--( )+[ ]-=

U ij , U

ij 1-.-Δ t

------------------------------- U ij 1-.

U i 1 j 1 -.+ U

i 1 – j 1-.-( )

2Δ x------------------------------------------------------

g y

i 1 j 1 -.+ y

i 1 j 1 -.--

2Δ x------------------------------------------------

+ +

g I 0 Aku f

-( )=

Aku f

U ij 1-. U

ij 1-.

M 2r i

43---

----------------------------------=

The source code in C for the explicit solution of Saint-Venant equations is

given in Appendix I.

U ij , U

ij 1-. U ij 1-.

U i 1 j 1 -.+

U i 1 – j 1-.-( )Δ t

2Δ x-------------------------------------------------- ------------=

g Δ t yi 1 j 1 -.+ y

i 1 j 1 -.--( )

2Δ x-------------------------------------------------- ------------- g Δ t I

0 Aku f

-( )+

Halaman 36


5. Compute the water level, y , from Eq. 3.4.66. Compute I f from Eq. 3.4.87. Compute the water velocity, U , from Eq. 3.4.9

End of repetition 4End of repetition 3

The explicit procedure will become unstable if the time step is chosentoo large. The Courant criteria says the time step should be smaller than

what would be required to make a water particle pass from one cross-section to another:

(3.4.10)Δ t Δ x

U c+( )---------------------<

Figure 3.4.2 Example of a flood wave routeddownstream a channelusing the full Saint-Venant equations andan explicit solver. Thechannel has a slope of1:200, and a Manning's M



Control volume approach

The explicit procedure is still fairly unstable in the form given above. Sebuahimprovement in stability can be obtained by considering a control volumeapproach when discretizing the continuity equation:

Gambar. 3.4.3 shows a longitudinal part of the river, with three cross-sec-tions: i -1, i and i +1. It also shows two water surfaces. One surface is attime step j -1, and the other is at time step j . The purpose of the algorithmis to compute the water level at section i , for time step j . This is done onthe basis of the fluxes in and out of the volume upstream of i :

value of 30 was used. Thevertical axis is the waterdischarge, and the hori-zontal axis is the time indetik. Berbedacurves show thehydrograph at 0, 1250,2500 and 5000 metersdownstream. Awalhydrograph has the trian-gular shape. A time stepof 3 seconds was used.The source code written inC to generate this figure isgiven in Appendix I.

i i + 1i-1

j-1

j

U i-1 U i

Gambar. 3.4.3. Figure forcontrol volumeapproach to discreti-zation of continuitypersamaan

Δx Δx

Halaman 37


Inflow: I = U i -1* yi -1 atau

(3.4.11)

Outflow: O = U i * yi atau

(3.4.12)

The overbar denotes average values over the time step.

The difference between the inflow and outflow is equal to the volume ofthe water between the surfaces at time j and j -1, during one time step.Hal ini memberikan:

(3.4.13)

atau

(3.4.14)

AkuU

i 1 j 1 -.-U

i 1 j ,-+( )

2-------------------------------------------------

yi 1 j 1 -.-

yi 1 j ,-

+( )

2----------------------------------------------=

OU

ij , U ij 1-.+( )

2-----------------------------------

yij , y

ij 1-.+( )

2---------------------------------U

i

yij , y

ij 1-.+( )

2---------------------------------= =

IU i

yij , y

ij 1-.+( )

2--------------------------------- Δ t 0.5 y

ij , y

ij 1-.-( )2Δ x=

yij ,

I yij 1-.

U i

2-----

Δ x

Δ t --------

Δ x

Δ t ------

U i

2-----+

-----------------------------------------------=



This equation can be used to compute the water level at section i , start-ing from the upstream end going downstream. Then the values of I andU i are known. This can be done after computing the velocities by solvingthe Saint-Venant equation. This procedure is implemented in the sourcecode given in Appendix I

Implicit method

The procedure starts with discretization of each differential terms of Eq.3.2.1 and Eq. 3.2.2 in space and time according to the figure below:

The choice of using the water discharge instead of the velocity as a vari-able has been reported to give better stability.

The equations are transformed so that all the terms only have the two

i -1 i

time step j

time step j-1

reach i

penampang lintang penampang lintangFigure 3.4.4. Discretiza-tion of terms betweencross-section i and i -1,computing for time step j . Time step j-1 is the pre-vious time step. ini adalahcalled a four-point differ-ence scheme.

Qi-1,j-1 Qi,j-1

Qi-1,j Qaku j

Halaman 38


independent variables, Q and y . The rating curve where the cross-sec-tional area is given as a function of the water depth (Fig. 3.2.2) is alsodigunakan.

After multiplication with A , the first term in Eq. 3.2.2 becomes:

(3.4.15)

Using the chain rule and the continuity equation, the second termmenjadi:

(3.4.16)

Using finite differences, the term is transformed to:

(3.4.17)

Note the weighting factor, θ, described below Fig. 3.4.1. The third termmenjadi:

(3.4.18)

The friction loss term is discretized by solving the Manning's equation:

(3.4.19)

SEBUAH U ∂t ∂

------- Qij , Qi 1- j , Q ij 1-.- Q i 1- j 1-.-+2Δ t

----------------------------------------------------------------------------------=

AU U ∂

x ∂------

SEBUAH

2---

U () 2∂

x ∂--------------

1

2---

Q2

SEBUAH ------∂

x ∂---------------= =

AU U ∂

x ∂------- θ

Q2

SEBUAH ------

ij ,

Q 2

SEBUAH ------

i 1- j ,-

2Δ x-----------------------------------------------1 θ-( )

Q2

SEBUAH ------

ij 1-.

Q 2

SEBUAH ------

i 1- j 1-.-

2Δ x------------------------------------------------------------+=

y θ y j

1 θ-( ) y j 1-+=

Ag y ∂ x ∂

----- Ag θ y

ij , yi 1- j ,-

Δ x--------------------------- Ag 1 θ-( )

yij 1-. y

i 1- j 1-.-Δ x

----------------------------------------+=

gAI f

θ gAQ j

Q j

M 2SEBUAH 2 R

43---

-------------------------1 θ-( ) gAQ

j 1-Q

j 1-

M 2SEBUAH 2 R

43---

----------------------------------------------------+=



The variables are here averaged between section i and i -1.

The same procedure is used for the continuity equation (3.2.1). The tran-sient term becomes:

(3.4.20)

The second term:

(3.4.21)

Additionally, there may be lateral inflow/outflow.

Evaluation of the equations

After the terms in the continuity and momentum equations are replaced

A ∂

t ∂------

SEBUAH ij ,

SEBUAH i 1- j ,

SEBUAH ij 1-.- SEBUAH

i 1- j 1-.-+

2Δ t -------------------------------------------------- ------------------------------=

Q∂ x ∂

------- θ Q ij , Q i 1- j ,-Δ x

------------------------------1 θ-( )Q ij 1-. Q i 1- j 1-.-Δ x

--------------------------------------------+=

Halaman 39


by Eq. 3.4.18 and 3.4.19, the two equations will be in the form:

(3.4.22)

(3.4.23)

where A, B, C, D and E are functions of constants and variables at the previous time step and the indexes c and m denote the continuity andmomentum equation, respectively. Note, all variables at time step j-1 aredikenal. The variables at time step j are unknown. The momentum equa-tion and the continuity equation are applied to all reaches in the river between the cross-sections. Given boundary conditions, there are thesame number of unknown as equations.

There are a number of methods for solving the equations. The two maingroups are direct methods and iterative methods. In direct methods, theequations are set up in a matrix, which is inverted to get the solution. Diiterative methods, guesses are made for the variables, and Equations3.4.7 and 3.4.8 are modified to get formulas for improvement of theguessed values. One of the most used method is the Newton-Raphsonformula, an iterative method where the following formula is used to get a better estimate of Q

(3.4.23)

The same equation would apply for y instead of Q when computing thewater depth.

Stability problems

Experience with solving the equations shows that convergence prob-lems occur where there is supercritical flow. Many computer programstherefore implement special algorithms to deal with this situation. Beberapaare more successful than others. If the algorithm in a given programfails, it is possible to avoid the problem by modifying the friction factor,

M , so that only subcritical flow is present in the problematic area. Theflood wave will then get a slower translation speed as the velocities aredikurangi. This must be taken into consideration when evaluating thehasil.

Boundary conditions

f c

Q y ,( ) SEBUAH cQ

i B

cQ

i 1-C

c y

i D

c y

i 1- E

c+ + + + 0= =

f m

Q y ,( ) SEBUAH m

Qi

Bm

Qi 1-

C m y

i D

m y

i 1- E

m+ + + + 0= =

Qn 1+ Q

n

f Qn

()

f ′ Qn

()----------------=



The upstream and downstream cross-section will need boundary condi-tions. There are several options:

1. User-specified values

This could typically be results from a dam break computation, whichgives the water discharge as a function of time at the upstream bound-ary. Or if the downstream value is located in a lake or in the ocean, thenthe water elevation there is known.

2. A rating curve

Halaman 40


This can be used if for example a weir is present at the downstream boundary, giving a unique relationship between discharge and watertingkat.

3. Uniform flow approximation

The Mannings formula can be used to find the discharge if uniform flowis assumed.

4. Zero gradient boundary condition.

This is typically used at the downstream boundary. The depth and/or dis-charge are assumed to be the same at the downstream boundary cross-section as the second most downstream cross-section.

The choise of boundary conditions must be made based on the availabledata and the hydraulic conditions in the river.

3.5 Hydrologic routing

A simplified river routing method is the Muskingum algorithm. ini

derived from the water continuity equation and some assumptions aboutthe volume of the water in the river:

The volume, V 0, of water in the reach is assumed to be proportional to

the steady water discharge, Q o, flowing out of the reach:

(3.5.1)

where K is a proportionality coefficient, with units [sec].

For unsteady flow, the water discharge into the reach is not alwaysequal to the discharge out of the reach. If the inflow is denoted Q i, Yangwater continuity defect is assumed to take up a volume equal to:

(3.5.2)

X is another geometrical proportionality coefficient (dimensionless).

The total volume of water in the reach is therefore: V=V 0+V D

(3.5.3)

The change in water volume in a reach between one time step j , and thenext time step j +1 becomes:

(3.5.4)

From the continuity equation, the volume change can also be written:

V 0

KQ o=

V D

KX Q i Qo-( )=

V KQ o KX Q i Q o-( )+ K XQ i 1 X ) Q o-( )+( )= =

V j 1+

V j

- K XQ j 1+i 1 X -( ) Q

j 1+o+[ ] XQ

ji 1 X -( ) Q

jo+[ ]-{ }=



(3.5.5)V j 1+

V j

-1

2--- Q

ji Q

j 1+i+( )Δ t 1

2--- Q

jo Q

j 1+o+( )Δ t -=

Halaman 41


Combining the two equations and solving with respect to Q o j+1, the fol-

lowing equation is obtained:

(3.5.6)

Expressions for the C ’s as functions of X, K and Δ t are:

(3.5.7)

(3.5.8)

(3.5.9)

By combining Eq. 3.5.4 and 3.5.5, the following expression for K isobtained:

(3.5.10)

The Muskingum method determines the C values from observedhydrographs. The numerator and denominator in Eq. 3.5.10 is computedand plotted for each time interval for different values of X . The value giv-

ing the straightest line is chosen, and K is then the slope of this line.From these values of K and X , Eq. 3.5.7-9 are used to compute C 1-3.When the C ’s are given, the water discharge can be determined fromEq. 3.5.6.

3.6 HEC-RAS

HEC is an abbreviation for H ydrologic E ngineering C enter. It is a part ofUS Army Corps of Engineers. Over the years, the organization hasmade several computer programs for water flow problems, namedHEC1, HEC2 etc. HEC2 computed the water surface profile for a steadywater flow in a natural river in one dimension. The solution procedure fol-lowed the theory in Chapter 3.2.

The original HEC2 program did not have a user interface. It read anASCII input file with all the necessary information about water discharge,friction factors, geometry etc. The result was an output file with the computed water levels. Later, a graphic user interface for the program wasmade, with interactive input of data and visualization of results. This ver-sion is called HEC-RAS. The most recent version of HEC-RAS includesalgorithms for computing unsteady flow, including mixed flow regime between supercritical and subcritical flow. It also has a dam break analy-sis module, and connection to GIS programs. Version 4 came out in2008 and has sediment transport with movable bed and water qualitycomputations.

Q j 1+o C 1Q

j 1+i C 2Q

ji C 3Q

jo+ +=

C 1

Δ t 2 KX -

2 K 1 X -( ) Δ t +-------------------------------------=

C 2

Δ t 2 KX +

2 K 1 X -( ) Δ t +-------------------------------------=

C 3

2 K 1 X -( ) Δ t -

2 K 1 X -( ) Δ t +-------------------------------------=

K 0.5Δ t Q

ji

Q j 1+i+( ) Q

jo

Q j 1+o+( )-[ ]

XQ j 1+i Q

ji-( ) 1 X -( ) Q

j 1+o Q

jo-( )+

---------------------------------------------------------------------------------------=

The parameters in theMuskingum method are based on observed dis-charges. If floods higherthan the observed valuesare to be computed, thecoefficients may not be cor-rect. The Muskingummethod can then not be

digunakan.

HEC-RAS is freeware, andcan be downloaded from theInternet. It is much used by Norwegian consulting companies and water authori-ikatan.



Halaman 42


3.7 Commercial software

Because the initial HEC2 program was limited in functionality and graph-ics, there was a period of time in the 1990's where there were a marketfor commercial software for hydraulic engineering. The purpose wasoften analysis of flooding and dam-breaks. Several commercial pro-grams were developed, and some are briefly described below.

DAMBRK

DAMBRK is made by the US National Weather Service. It consist of two bagian:

- A program for computing the outflow hydrograph from a reservoirwhere the dam is breaking

- Routing of the hydrograph downstream, by solving the Saint-Venant persamaan.

The program is in principle freeware, and the source code is publiclytersedia. The original version did not have a user interface, but com-mercial companies have made user interfaces and these are sold

together with the program. An example is BOSS DAMBRK.

There also exist two programs for separate computations of the dam break hydrograph and the flood routing, made by the same organization.The program computing the hydrograph is called BREACH. The pro-gram computing only the routing of the flood wave is called DWOPPER.

MIKE11

. MIKE 11 is made by the Danish Hydraulic Institute. It is a one-dimen-sional program with both steady state water surface profile computationand solution of the full Saint-Venant equations. The program has agraphical user interface, and program includes connections with GISsistem. MIKE 11 has a number of different add-on modules, computingfor example:

- rainfall/runoff- water quality- sediment transport

- groundwater

The modules makes MIKE 11 very well suited for a solving a number ofdifferent hydraulic river problems.

ISIS

ISIS is made by Hydraulic Research Wallingford, in the UK. It is similar infunctionality to MIKE11, with a graphical user interface and computa-tional modules for one-dimensional steady and unsteady flow. It also hasa number of add-on modules.

3.8 Problems

Problem 1. Coefficients

Derive formulas for the coefficients C 1 , C 2 , Dan C 3 di pers. 3.5.7-9. Ini

is done by combining Eq. 3.5.4 and 3.5.5, eliminating the volumes V ,and solving with respect to Q o

t+1, The resulting equation is compared

MIKE 11 is used by Norwe-gian consulting companiesand The Norwegian WaterResources and EnergyDirectorate for flood zonemapping.

ISIS is the most used tool byconsulting companies in theUK for river flow computati-tons.

Halaman 43




with Eq. 3.5.6, giving the coefficients.

Problem 2. Flood wave

Water is released from a dam, with the outflow hydrograph given in thefigure below. The river downstream is 50 m wide, and has stones withd90 of 0.2 meter. The river slope is 1:300. A town is located 2 km down-stream of the dam. When will the water start to rise in the town? Danwhen will the peak of the flood wave reach the town?

Use a computer program solving the Saint-Venant equation to computethe maximum water discharge at the town.

Time (min)

Q (m3/s)

10 70 180

60

20

Halaman 44


4. Dispersion of pollutants

4.1 Introduction

Dispersion is a combination



of the two processes: con-vection and diffusion. Convection is pollutant transport by the time-averaged watervelocity. This is relativelystraightforward to compute.For our purposes, diffusionis caused by turbulence andvelocity gradients, which aremore complicated.

The most common method

of modelling diffusion is byuse of a turbulent diffusioncoefficient, Γ, defined as:

(4.1.1)

F is the flux of a pollutantwith concentration c , pass-ing through an area A , and xis the direction of the fluxtransportasi.

4.2 Simple formulas for the diffusion coefficient

The diffusion coefficient, Γ, for diffusion of a toxic substance, can be setequal to the eddy-viscosity, ν T , of the water. Hubungan antarathe variables are given by the following formula:

(4.2.1)

Sc is the Schmidt number. This has been found to be in the range of 0.5-1.0, but more extreme values have been used. In the following, weassume a value of 1.0, meaning the turbulent diffusion coefficient isequal to the eddy-viscosity.

Classical hydraulics give a number of empirical and semi-analytical for-mulas for the eddy-viscosity in rivers or lakes. For a river, the eddy-vis-cosity, ν T , is used in the definition of the shear stress in a fluid:

(4.2.2)

The equation can be solved with respect to ν T:

The picture shows the confluence of two small rivers in Costa Rica with different kualitas air. The branch on the left part of the picture comes from an area with vol-canic activity. (Photo: N. Olsen)

Numerical algorithms forsolving the water velocityequations in 3D can be diffi-cult to learn. It is sometimeseasier to learn the algo-rithms when looking at dispersion of a pollutant thancomputing the water veloc-ity. This is the reason whythe dispersion processesare described before thecomputation of the watermengalir.

Γ

F

SEBUAH ---

dc

dx------

-----------=

Γ ν

T Sc-----=

τ νT

ρ dU dy-------=

Halaman 45


(4.2.3)

For a wide rectangular channel, Schlichting's wall laws (Eq. 2.1.2) givesthe variation of the velocity with the depth. The vertical velocity gradientcan be obtained by derivation of Eq. 2.1.2, with respect to the distanceabove the bed, y :

(4.2.4)

For a wide, rectangular channel, the shear stress increases linearly from

νT

τ

ρ dU

dy-------

-----------=

dU

dy-------

u*κ y-----=



the surface to the bed:

(4.2.5)

The water depth is denoted h . Using the definition of the shear velocity,the equation can be rewritten:

(4.2.6)

Inserting Eq. 4.2.4 and Eq. 4.2.6 into Eq. 4.2.3:

(4.2.7)

The average value over the depth is obtained by integrating Eq. 4.2.7over the depth:

(4.2.8)

Equation 4.2.8 is derived for an idealized case, with a wide s traightchannel with rectangular cross-section. Naas (1977) measured theeddy-viscosity in a number of natural rivers, and suggested the followingformula instead:

(4.2.9)

Note that this formula is based on the vertical velocity gradients, so thisgives the eddy-viscosity in the vertical direction.

The shear stress on a lake will also introduce turbulence in the water. SEBUAHsimilar approach can be used as for a river, but now the shear stress isacting on the water surface instead of the bed. The result is a formulasimilar to Eq. 4.2.8 and 4.2.9, but a different empirical coefficient may bedigunakan.

τ τ0 1

y

h----=

τ ρ u*2 1

y

h----=

νT

κ u* y 1

y

h----=

νT

1

h---

κ u*

h--------- hy y 2-( ) yd

0

hκ

6--- u

*h 0.067 u

*h= = =

ν T 0.11 u *h=

Halaman 46


4.3 One-dimensional dispersion

A typical example of a one-dimensional dispersion problem is pollutionfrom a point-source into a river where the vertical and lateral mixing is besar. Typical concentration profiles at several points in the river down-stream of the spill is given in Fig. 4.3.1:

c

t

spill x=1km x=2km x=3km x=5km

Figure 4.3.1 Time-series of concentrations at several locations downstream of a spill



The figure shows the two main processes:

- Convective movement of the point of maximum concentration- Diffusion of the spill, with reduction of the maximum concentration

The transport can be described by a convection-diffusion equation forthe pollutant concentration, c :

(4.3.1)

The problem is to find the correct value of the longitudinal diffusion coef-ficient, Γ. The coefficient is not a function of small-scale turbulent proc-esses. Instead, mixing in the longitudinal direction is often caused byconvective movements due to lateral velocity gradients. The diffusioncoefficient for a one-dimensional model of a river will be much largerthan the small-scale turbulent diffusion used in multi-dimensional mod-els.

Some researchers have developed empirical formulas for the longitudi-nal dispersion coefficient:

McQuivey and Keefer (1974):

(4.3.2)

Fisher et. al. (1979)

(4.3.3)

dc

dt ------ U

dc

dx-----+

d

dx------ Γ

dc

dx------=

Γ 0,058Q

IB-----=

Γ 0,011UB( )2

Hu*

---------------=

Halaman 47


Q is the water discharge in the river, with slope I , width B , and depth H .U is the water velocity and u * is the shear velocity.

The convection-diffusion equation can be solved analytically, assumingconstant values of velocity and diffusion coefficient, giving (Chapra,1997, p.182):

(4.3.4)

The initial concentration is denoted c 0, and the length of the spill in theriver is denoted L .

In a natural river, the simplifications used to derive Eq. 4.3.4 are not

valid. The equation may give a rough estimate of the concentration, butto get better accuracy, it is necessary to solve the convection-diffusionequation numerically.

Looking at measurements of pollution concentration in a river, the pro-files in Fig. 4.3.1 have an additional feature: a prolonged tail. It is caused by storage of pollution in recirculation zones and dead waters along theriver. This effect is difficult to take into account using one-dimensionalmodel. Instead, it is possible to model the river using a three-dimen-sional model, where the water flow field is modelled, including recircula-tion zones. This approach also have the advantage that the uncertaintywith the longitudinal diffusion coefficient is eliminated. It is computed asa part of the solution of the equations. Research is ongoing in this area.

4.4 Jets and plumes

cxt ,( )c0 L

2 πΓ t ---------------- e

x Ut -( )24Γ t

-----------------------=



Jets and plumes are water entering a reservoir or a lake, for examplefrom a river or a wastewater outlet. For an idealized case, there exist for-mulas for the dispersion of jets and plumes. The formulas can be derivedanalytically based on the momentum equation, but empirical coefficientsare required when computing dispersion because of turbulence.

Close to the outlet, the momentum of the water will be a dominatingforce determining the flow field. This is then called a jet. Gambar. 4.4.1 showsa jet close to the outlet:

Assuming a uniform velocity profile at the outlet, the jet will have a corewhere the velocity distribution changes. The length of the core is approx-imately 6 times the diameter of the outlet. In the core, the maximumvelocity is constant. The water from the jet is mixed with the surroundingwater, reducing the velocity in this area. The reduction of the velocity will

SEBUAH B C D

Figure 4.4.1 Velocity pro-

files at the core of a jetoutlet. The entrance profileis denoted A, the coreregion ends at profile C.Profile B has uniform distribution at the core, butcurved distribution at theedge. Profile D has reducedmaximum velocity, as it islocated after the corewilayah.

Halaman 48


finally take place also at the center of the jet. This point forms the end ofthe core.

After the core, the jet may moves in various directions, depending on:

- The geometry around the plume

- The density difference between the inflowing and the surroundingair- The density stratification in the surrounding water- The velocity field in the surrounding water- The turbulence in the surrounding water.

If the velocity of the surrounding water is very strong, this may also affectthe jet in the core region.

For an idealized case, it is possible to derive formulas for the velocityand effective discharge after the outlet. Assuming the receiving waterhas no velocity, turbulence or density stratification, and its density is thesame as the water in the jet, the momentum of the water stays the samein a cross-section of the jet. It is also assumed the jet will not interactwith any geometry. The momentum equation, together with experimentsthen give the following equations for a jet from a circular pipe (Carstens,1997):

(4.4.1)

(4.4.2)

(4.4.3)

The velocity is denoted u , r is the distance from the centerline of the plume, x is the distance from the pipe outlet, d 0 is the diameter of the pipe and Q 0 is the water discharge out of the pipe. Because the sur-

A flow situation dominated by the momentum of theinflowing water is oftencalled a jet . If the flow situa-tion is dominated by thedensity difference betweenthe inflowing and receivingwater, this is often called aplume .

uu

max----------- 1 r

2

0.016 x 2-------------------+

2-

=

umaxu0

----------- 6.4 x

d 0

-----1-

=

Q

Q0

------ 0.42 x

d 0-----=



rounding water will be mixed into the plume, the total water discharge, Q ,in the plume will increase with the distance from the outlet.

The formulas are empirical. Fischer et al (1979) came up with slightly dif-ferent coefficients: 6.2 instead of 6.4 in Eq. 4.4.2, and 0.28 instead of0.42 in Eq. 4.4.3.

If the receiving water does not have the same density as the water fromthe pipe, the plume will rise or s ink. A densimetric Froude number, Fr' , isoften used to derive formulas for the plume.

(4.4.4)

It is assumed that plume water density, ρ 0, has a lower value than therecipient water density, ρ soal . The formulas below are given by Rouse et.al. (1952) from experiments:

Fr ′u

0

ρ soal ρ 0-ρ soal

-------------------- gd 0

---------------------------------------=

Halaman 49


(4.4.5)

(4.4.6)

(4.4.7)

It is assumed the pipe releases the water in the vertical direction.

Detailed derivations of the formulas, together with equations for more jet/ plume cases are given by Fisher et. al. (1979).

4.5 Problems

Problem 1. Dispersion in a river

Two thousand litres of a toxic chemical is spilled f rom a factory into ariver during ten minutes. The river has a water discharge of 20 m 3/s, anaverage depth of 2 meters, and average width of 30 meters and a slopeof 1:63. Ten kilometer downstream of the factory is a city. Compute theconcentration of the chemical in the river at the city as a function of time.

Problem 2. Dispersion of a plume

A plume rises from a hydropower plant outlet into the sea. The sea waterhas a salinity of 3 %, and a density of 1023 kg/m 3. The water dischargeis 50 m 3/s from a tunnel with diameter 3 meters. The water from the tun-nel has a density of 1000 kg/m 3. The lake is 30 meters deep at the outlettitik. What is the velocity of the water 20 meters right above the outlet?Assume no initial water currents or vertical stratification.

What will happen if there are currents in the lake?

u

u0----- 4.3 Fr ′

23---- z

d 0

-----

13----

e96r 2

z 2----

=

ρ ρ soal

-

ρ0

------------------9 Fr ′

23---- z

d 0

----

53----

e71r 2

z 2----

=

Q

Q 0------ 0.18 Fr ′

23---- z

d 0

-----

53---

=



Halaman 50


5. Dispersion modelling in 2D and 3D

Dispersion modelling in 2D or 3D is conducted using a science called

computational fluid dynamics, or CFD. This chapter gives an introductionto CFD, applied to modelling dispersion of pollutants.

5.1 Grids

A basic concept of CFD is to divide the fluid geometry into elements orcells, and then solve an equation for each cell. In the following text, theword cell will be used instead of element, to avoid confusion with thefinite element method. The algorithms described in the following chap-ters are based on the finite volume method

Grids can be classified according to several characteristics:

bentukorthogonalitystruktur blocksgrid movementsnestingoutblocking

The shape of the cells is usually triangular or quadrilateral:

The orthogonality of the grid is determined by the angle between cross-ing grid lines. If the angle is 90 degrees, the grid is orthogonal. If it is dif-ferent from 90 degrees, the grid is non-orthogonal.

For non-orthogonal grids, a non-orthogonal coordinate system is oftenused to derive terms in the equations. The coordinates then follow thegrid lines of a structured grid. The three non-orthogonal coordinate linesare often called ξ,ψ,ζ, corresponding to x,y and z in the orthogonal coor-dinate system. Gambar. 5.1.3 shows the two systems in 2D. Selain itu,

“CFD is about computing abit and then guessing a bit,

for so to compute a bit moreand see if one has guessed benar. What differenti-ates the various algorithmsis where one is computing what and with which num-bers ” - student answering anexam in Hydroinformatics.

Grid classifications

Triangular and quadrilateral shapes

Gambar. 5.1.2. Grid orthogo-nality

Gambar. 5.1.1. Grid shapes

Orthogonal grid Non-orthogonal grid

Pronounciation of theGreek letters:

ξ : ksiψ: psi



third direction will be z in the cartesian system and ζ in the computationaldomain.

ζ: zeta

Halaman 51


The directions along the computational domain are often called (ξ,ψ) in2D as shown in the figure above, or (ξ,ψ,ζ) in 3D, where the last index isthe vertical direction. In the computational domain, the distance betweenthe grid lines are often set to unity, so it is easy to calculate gradients ofvariabel. It means all δξ will be unity.

An important definition is the notation of the variables at a cell. Instead ofusing x , y and z directions, the non-orthogonal cell now uses the direc-tions north, south, east, west, bottom and top. Another definition is touse indexes, as in tensor notation. Then direction 1 is east-west, direc-tion 2 is north-south and direction 3 is vertical. Using tensor notation,(ξ,ψ,ζ) can also be written (ξ 1,ξ2,ξ3).

Grids can be structured or unstructured. Often a structured grid is usedin finite volume methods and an unstructured grid is used in finite ele-ment methods. However, this is not always the case. The figure belowshows a structured and an unstructured grid. In a structured grid it is possible to make a two-dimensional array indexing the grid cells. Jika ini adalahnot possible, the grid is unstructured.

Almost all grids using triangular cells are unstructured.

It is possible to connect several structured grids. Each grid is then calleda block, and the result is called a multi-block grid.

The grid may also move during the computations. A grid that movesaccording to the solution of the equations is called an adaptive grid.Typical examples are vertical movements due to changes in water lev-

ξ

ψ

y

x

(1,1) (2,1) (3,1)

(3,3)

(2,4)

(4,4)

(4,1)

Figure 5.1.3 Body-fittedcoordinate system. Thetwo coordinate systemsare shown in two dimen-sions, where some valuesof ξ and ψ are given in brackets

Grid structure

Level2

Structured grid Unstructured grid

Figure 5.1.4 Grid struc-ture

Adaptive grid



Halaman 52


els, or changes in bed levels due to erosion or sedimentation. The gridmay also move laterally, for example modelling a meandering river.

Examples are given in Chapter 9. All these movements are due tochanges in the geometry of the computational domain. Grid movementscan also be internal. Often, a higher grid density is wanted in large gradi-ents, and algorithms have successfully been used to change the internalgrid structure to change the cell sizes according to for example velocitygradients.

High gradients may also necessitate the use of a nested grid. Inimeans that a grid with small cells are located inside a coarser grid. SEBUAHtypical example is computation of pollutants from a point source in alake. The lake itself is modelled with a coarse grid, while the concentra-tion around the point source is modelled with a finer grid, nested insidethe coarse grid.

Outblocking is a procedure where cells in a structured grid are madeinactive. This makes it easier to generate structured grids in a complexgeometri.

Grid Qualities

The accuracy and convergence of a finite volume calculation dependson the quality of the grid. Three grid characteristics are important:

- non-orthogonality- aspect ratio- expansion ratio

The non-orthogonality of the grid line intersections is the deviation from90 degrees. If the grid line intersection is below 45 degrees or over 135degrees, the grid is said to be very non-orthogonal. This is a situationone should avoid. Low non-orthogonality of the grid leads to more rapidconvergence, and in some cases better accuracy.

The aspect ratio and expansion ratio is described in the figure below:

The figure shows two grid cells, A and B . The length of the cells are Δ x SEBUAH and Δ x B.

The expansion ratio of the grid at these cells is Δx SEBUAH/ΔxB.

The aspect ratio of the grid at cell A is Δx A/Δy

SEBUAH.

The expansion ratio and the aspect ratio of a grid should not be toogreat, in order to avoid convergence problems and inaccuracies. Aspectratios of 2-3 should not be a problem if the flow direction is parallel to thelongest side of the cell. Experience shows that aspect ratios of 10-50 willgive extremely slow convergence for water f low calculations. Ekspansiratios under 1.2 will not pose problems for the solution. Experience alsoshows that expansion ratios of around 10 can give very unphysical

results for the water flow calculation.

Grid qualities

SEBUAH B

ΔxSEBUAH

ΔxΒ

ΔySEBUAHFigure 5.1.5 Expansion/aspect ratio definition:

Halaman 53




Grid generation

Some kind of geographical information is required to make a grid of anatural river or lake. Often a map can be used. The first step is usually todetermine the points at the edges of the grid. Then the internal grid inter-sections are made.

Two of the most commonly used methods to generate internal points in astructured grid are called Transfinite interpolation and Elliptic grid gener-ation . These are described in the following.

Transfinite interpolation

In a transfinite interpolation, the grid lines on two opposing edges areconnected with straight lines. An example is given in Fig. 5.1.6. Themethod is well suited for modelling rivers, as the straight lines can becross-sections. Note that the straight lines are only generated in onearah. For the right figure in Fig. 5.1.6, the lines in the longitudinaldirection are not straight.

Elliptic grid generation

Some times a smoother grid is required, than the result of the Transfiniteinterpolation. The elliptic grid generation solves a differential equation forthe location of the grid intersections:

The grid given in Fig. 5.1.7 is made by this method.

Gambar. 5.1.6. Grid generated withtransfinite interpolation.

∇ 2ξi

0=

Gambar. 5.1.7. Grid made by an ellip-tic grid generation.

Halaman 54


5.2 Discretization methods

The discretization described here is by the control volume method.

Steady state dispersion is governed by the convection-diffusion equationfor the concentration, c , of the pollutant:



(5.2.1)

The left side of the equation is the convective term, and the right side ofthe equation is the diffusive term.

The main point of the discretization is:

To transform the partial differential equation into a new equation wherethe variable in one cell is a function of the variable in the neighbour cells

The new function can be thought of as a weighted average of the con-centration in the neighbouring cells. For a two-dimensional situation, thefollowing notation is used, according to directions north, south, east andwest::

Sebuahe : weighting factor for cell e

Sebuahw : weighting factor for cell w

Sebuahn : weighting factor for cell n

Sebuah s : weighting factor for cell sSebuah p = a e+aw+an+a s

The formula becomes:

(5.2.2)

The weighting factors for the neighbouring cells a e , Sebuahw , Sebuahn and a s adalahoften denoted a nb

What we want to obtain are formulas for a nb.

In a three-dimensional computation, the same principles are involved.But two more neighbouring cells are added: t (top) and b (bottom),resulting in six neighbour cells. The simple extension from 2D to 3D isone of the main advantages of the finite volume method.

There are a number of different discretization methods available for thecontrol-volume approach. The difference is in how the concentration ona cell surface is calculated . Some methods are described in the follow-ing.

Note that the methods are based on the physics of the dispersion andflow processes. They are not mathematically based, or derived from the

convection-diffusion equation (5.2.1).

U i

∂ c

∂ xi

-------∂

∂ xi

------- Γ∂ c

∂ xi

------=

Discretization is:

cn

ce

c s

c pcw

Figure 5.2.1 Discretizationmolecule. Computation of con-centration, c , in the center cell,

p , as a function of the concentra-tion in the neighbouring cells n ,

s, e and w .

c p

Sebuahw

cw

Sebuahe

ce

Sebuahn

cn

Sebuah s

c s

+ + +

Sebuah p

---------------------------------------------------------------------=

Halaman 55


5.3 The First-Order Upstream Scheme

For a non-staggered grid, the values of the variables are given in thecenter of the cells. Using the finite volume method, it is necessary to esti-mate variable values on the cell surfaces. The main idea of the upstreammethods is to estimate the surface value from the upstream cell. Pertamaorder method uses information in only one cell upstream of the cell sur-wajah. In other words: the concentration at a cell surface for the first-orderupstream method is the same as the concentration in the cell on theupstream side of the cell side.

The control volume method is based on continuity of sediments. The basis of the calculation is the fluxes on a cell surface. The surface areais denoted A ; the velocity at the surface, normal to it, is denoted U ; c isthe concentration at the surface, and Γ is the turbulent diffusion at the

Development of CFDalgorithms was initiallydone in aeronautics.The fluid was air, andthe methods were thencalled upwind instead ofupstream. Both expres-sions are used, mean-ing the same.



permukaan.

The convective flux is calculated as: U * A * c (5.3.1)The diffusive flux is calculated as: Γ * A * dc / dx (5.3.2)

The term dc/dx is calculated as the concentration difference between thecells on each side of the surface, divided by the distance between thecentres of the cells. Looking at the west s ide of cell p , Fig. 5.3.1 explainsthe variable locations and the fluxes in the center cell.

The flux, F w, through the west side of cell P then becomes:

(5.3.3)

where A w is the area of the cell wall on the west side, equal to Δ y timesthe height of the wall. For the other s ides, the following fluxes areobtained:

cw c p ce

c s

cn

U eU w

U s

U n

Γw

Γe

Γn

Γ s

dx

dy

Figure 5.3.1 Fluxes

through the walls of the center cell in acomputational mole-cule. The cells have awidth of dx and a heightof dy . Note, the diffusioncoefficient and thevelocities are given onthe boundary betweenthe cells. The concen-trations are computed inthe centre of each cell.Also note that this is atwo-dimensional situa-tion. The area of a cellsurface is thereforeequal to dx or dy , multiplied with a unit depth inthe third dimension.

F w

U w

SEBUAH w

cw

Γw

SEBUAH w

cw

c p

-( )

dx----------------------------+=

Halaman 56


(5.3.4)

(5.3.5)

(5.3.6)

Sediment continuity means the sum of the fluxes is zero, in other words:

(5.3.7)

This gives the following equation:

(5.3.8)

F e

U eSEBUAH

ec p

Γe

SEBUAH e

c p

ce

-( )

dx---------------------------+=

F s

U s

SEBUAH s

c p

Γ s

SEBUAH s

c p

c s

-( )

dy--------------------------+=

F n

U nSEBUAH

nc

nΓ

n

SEBUAH n

cn

c p

-( )

dy

---------------------------+=

F w

F e

- F n

F s

-+ 0=

Γw

SEBUAH w

dx------ U

eSEBUAH

eΓ

e

SEBUAH e

dx------ U

sSEBUAH s

Γ s

SEBUAH s

dy------ Γ

n

SEBUAH n

dy------+ + + + + c

p

U w

SEBUAH w

Γw

SEBUAH w

dx------+ c

wΓ

e

SEBUAH e

dx------c

eU

nSEBUAH

nΓ

n

SEBUAH n

dy------+ c

nΓ s

SEBUAH s

dy-----c

s+ + +=



When we compare Equation 5.2.2 with Equation 5.3.8, we see they aresama. The concentration in Cell P is a function of the concentrationin the neighbouring cells. The resulting weighting factors are:

(5.3.9)

(5.3.10)

(5.3.11)

(5.3.12)

(5.3.13)

The water continuity equation for the grid cell is:

(5.3.14)

or:

(5.3.15)

If the above equation is inserted into the expression for a

p

, the equation

(5.3.16)

is verified to be correct.

Note that the equations above are only valid if the velocity flows in the

Sebuah p

Γw

SEBUAH w

dx------ U

eSEBUAH

eΓ

e

SEBUAH e

dx----- U

sSEBUAH s

Γ s

SEBUAH s

dy----- Γ

n

SEBUAH n

dy-----+ + + + +=

Sebuahw

U w

SEBUAH w

Γw

SEBUAH w

dx------+=

Sebuahe Γe

SEBUAH e

dx-----=

Sebuah s

Γ s

SEBUAH s

dy------=

Sebuahn

U nSEBUAH

nΓ

n

SEBUAH n

dy------+=

U w

SEBUAH w

U eSEBUAH

e- U

nSEBUAH

nU

sSEBUAH s

-+ 0=

U w

SEBUAH w

U nSEBUAH

n+ U

eSEBUAH

eU

sSEBUAH s

+=

Sebuah p

Sebuahe

Sebuahw

Sebuah s

Sebuahn

+ + +=

Halaman 57


same direction as given on the arrows in Fig. 5.3.1.

Contoh:

Particles are deposited in a river with constant width and depth. A 2D sit-uation is assumed, averaged over the width. The north-south direction isused as the vertical direction, so the indexes s and n are replaced by band t . Further simplifications are:

- uniform the water flow- negligible horizontal diffusion

Then the vertical velocity is equal to the sediment particle fall velocity, w.Also, the grid can be made orthogonal and two-dimensional, so A e = Aw= dy , and A b = At = dx . The weighting factors then become:

Sebuahe = 0.0Sebuahw = U dy

Sebuahb = Γ dx/dy (5.3.17)Sebuaht = w dx + Γ dx/dy

Sebuah p = U dy + w dx + 2 Γ dx/dy

If for example, U is 2 m/s, w is 0.01 m/s, Γ is 0.01 m 2/s, and the riverdepth is 4 meters, we may assume 10 cells in the vertical direction, giv-ing dy = 0.4 m. Modelling a reach of 1 km with 100 cells, gives dx =10 m.The coefficients becomes:

Sebuahe = 0.0Sebuahw = 0.8Sebuahb = 0.25 (5.3.18)



Sebuaht = 0.35Sebuah p = 1.4

The numbers can be inserted in a spreadsheet and the problem solved.This is described in the next chapter.

The Power-Law Scheme

The Power-Law Scheme is a first-order upstream scheme where the dif-fusive term is multiplied with the following reduction factor:

(5.3.19)

where Pe is the Peclet number, given by:

(5.3.20)

The Peclet number is the ratio of convective to diffusive fluxes. The fac-tor f is always between 1 and 0. The diffusive term will be reduced forflows where the convection is large compared with the diffusion.

5.4 Spreadsheet programming

Given formulas for the water flow field and the turbulence, it is possibleto make a spreadsheet for calculation of the pollutant concentration. Satu

f 1 0.1 Pe-( )5=

PeU Δ x

Γ-----------=

Halaman 58


application is to calculate the trap efficiency of a sand trap. Then a two-dimensional width-averaged approach is used. A structured orthogonalgrid is used, where each cell in the grid is simulated by a cell in thespreadsheet. If a uniform water velocity and turbulence field can beassumed in the vertical direction, then the same a nb coefficients can beused for all the cells. A more advanced approach is to use a logarithmic

velocity distribution, and a given distribution of the eddy-viscosity.

If the simpler approach is used, the coefficients a nb can be calculated before the programming starts. This is based on the formulas given pre-viously, and a chosen number of grid cells. The grid is structured,orthogonal and all cells have the same size.

A figure of this spreadsheet is given below, with 8 cells in the verticaldirection, and 9 cells in the horizontal direction. The size of the grid canof course be changed according to the dimensions of each problem.

SEBUAHB C D E F G H Aku J K

1 0 0 0 0 0 0 0 0 0 0

2 x Y

3 x Y

4 x Y

5 x Y

6 x Y

7 x Y

8 x Y

9 x Y



The cells marked X are inflow boundary conditions. Ini adalah A2..A10 cells. A concentration value is given in these cells. A constantvalue can be given, or it is possible to use a formula for the vertical distribution of the concentration.

The cells marked 0 is the boundary condition at the water surface. Iniis zero.

The cells marked Y is the outflow boundary condition. If the horizontaldiffusion is assumed to be zero, the values in these cells will not affect

the computation. If the horizontal diffusion is non-zero, a zero gradient boundary condition can be used. Then the formula in these cells shouldmenjadi:

Cell K2: =J2Cell K3: =J3...

Cell K10: =J10

10 x Z Z Z Z Z Z Z Z Z Y

Halaman 59


The cells marked Z is the bed boundary condition. A formula for the equi-librium sediment concentration can be used, for example van Rijn's for-mula, given in Chapter 9.6. However, often the shear stress is belowcritical at the bed of the sand trap, giving zero concentration. This will not be correct, as the sediment concentration at the bed will always behigher than the cell above. Therefore, the concentration can be set equalto the concentration in the cell above.

(Note that a more detailed calculation will give a very low diffusion coeffi-cient close to the bed of the sand trap, meaning the concentration in thecell above the bed is independent of the concentration in the bed bound-ary. For a simplified calculation, the diffusion coefficient is significant.Then the same result is obtained if zero gradient boundary condition isused.)

Cell B10 : = B9Cell C10 : = C9...Cell J10 : = J9

The discretized formula now has to be given in all the remaining interiorsel. As an example, the following data is assumed:

Sebuahw = 0.1Sebuahn = 0.2Sebuah s = 0.006Sebuahe = 0.002Sebuah p = 0.308

Starting in cell B2 , we give the following formula:

+(0.1* A2 +0.2* B1 +0.006* B3 +0.002*C 2 )/0.308

This formula is copied to all the interior cells, from cell B2 to J9 . After-wards, the calculation has to be repeated some times to get conver-gence.

The trap efficiency is calculated by first summing the inflow and the out-flow:

Inflow: sum of cells A2..A10Outflow: sum of cells K2..K10

Trap efficiency = (Inflow-Outflow)/Inflow

Running this case with varying number of grid cells will give differentresult for the trap efficiency. The next chapter explains why.

The method of invokingmore iterations is depend-ent on the particularspreadsheet program. UntukLotus 123, use F9 on thekeyboard repeatedly. UntukMS Excel, use the menuTools, Options, Calcula-tions , and cross off Itera-tions , and give a numberin the edit-field, for example 50.



5.5 False diffusion

False diffusion is due to the approximations in the convective terms inthe discretization schemes. More specifically, how the concentration onthe cell side is calculated. The effect is best shown with a coarse gridand steep gradients, for example the following situation:

A sand trap is calculated with 5 x 5 cells in the vertical and horizontaldirection, respectively. The cells are 1 meter high and 5 meters long.The water velocity is 0.3 m/s, and the sediment fall velocity is 6 cm/s.The sediments are added from a point source at the water surface, over

Halaman 60


a length of 5 meters. There is no turbulence! This gives the followingtheoretical concentration profile in the flow:

The concentration is unity along a band in the flow. The concentration iszero elsewhere.

The first-order upstream scheme is used to calculate the concentration.The following coefficients are obtained:

Sebuahn = 0.06 * 5 = 0.3

Sebuahw = 0.3 * 1 = 0.3Sebuah s = 0.0Sebuahe = 0.0Sebuah p = 0.6

This gives a n /Sebuah p = 0.5 and a w /Sebuah p = 0.5. In other words, the concentrationin a cell is the average of the concentration in the cell above and in thecell upstream. The boundary condition is a concentration of unity in onecell at the surface, and zero concentration in the other surface cells andthe inflow cells. The following result is obtained:

The maximum concentration has decreased from unity to 0.137 at the bed. It has also been smeared out over several cells at the bed.

False diffusion can be avoided by:

- aligning the grid with the flow direction- increasing the number of grid cells- using higher order schemes

U

Figure 5.5.1. Profile of the real concentrationin the geometry

0,5

0.25

0,1250.0625

0.0313

0.25

0.1230.0781 0,117 0.137 0.137

1.0

0.0

0,125 0.0625 0,0156

0.25 0.188 0,125

0.188 0.188 0,1560,125 0,156 0,156 0.137

0,117

0,0469

0.1090.0820

0.0313

0.0781

Figure 5.5.2. Kisiwith computed con-centration values



The last option is described in the following chapter.

Halaman 61


5.6 The Second Order Upstream Scheme

The Second-Order Upstream (SOU) method is based on a second-orderaccurate method to calculate the concentration on the cell surfaces. Themethod only involves the convective fluxes, and the diffusive terms arecalculated as before. The following figure shows the calculation of theconcentration on the west side of cell p : side W :

The cell on the west side of cell w is called cell ww . The concentration inthis cell is denoted c ww. The concentration in cell w is denoted c w dan

the concentration on side W of cell p is denoted c W . The SOU scheme

uses the concentration in cell ww and cell w to extrapolate linearly toside W . Given the width of the cell in the x-direction is dx , and the heightin the y-direction is dy , it is possible to derive a formula for the concen-tration on side W by triangulation:

(5.6.1)

atau

(5.6.2)

Equation 5.6.1 is only valid if the cells are of equal size. If the expansionratio is different from unity, a separate formula needs to be applied,where the coefficients 3/2 and ½ are given as a function of the expan-sion ratio.

The calculation molecule now gets nine cells, as shown in the figure tothe left

The flux through the west side of Cell P then becomes:

(5.6.3)

For the other sides, the following fluxes are obtained:

Cell pCell wCell ww

U

x

Konsentrasi

dx dx

West wall W

cww

cw

dy

cWFigure 5.6.1 Defini-

tion sketch forconcentration esti-mation at the wall,W , for the SOUskema.

cw cww

-

dx 0.5 dx+-------------------------

cw

cww

-

dx--------------------=

cw 3

2--- c

w

1

2--- c

ww-=

WW W P E EE

S

SS

N

NN

Figure 5.6.2 SOU nine-point calculation molecule F

wU

wSEBUAH

w3

2--- c

w

1

2--- c

ww- Γ

w

SEBUAH w

cw

c p

-( )

dx-----------------------------+=



Halaman 62


(5.6.4)

(5.6.5)

(5.6.6)

Again, the equations are only valid if the velocity vectors are in the samedirection as in Fig. 5.3.1. The weighting factors becomes:

(5.6.7)

(5.6.8)

(5.6.9)

(5.6.10)

(5.6.11)

(5.6.12)

(5.6.13)

(5.6.14)

For the SOU scheme, Equation 5.3.1 now becomes:

(5.6.3)

The formula is used for a two-dimensional situation. In 3D, the terms fortop and bottom is also added, giving four extra coefficients: a t , Sebuahtt , Sebuahb.

Sebuahbb.

5.7 Time-dependent computations and source terms

The derivations given previously has been made under the assumptionof a steady state condition. Often, it is necessary to compute the concen-tration changes over time. In Chapter 5.3, the fluxes into cell P wasequal to the fluxes out of the cell. In a time-dependent situation, this maynot be the case. To derive an equation for this problem, we look at whathappens during a time step Δ t , between time t and t -1. The masschange, m , in cell P would then be:

(5.7.1)

F e U eSEBUAH e3

2--- c p

1

2--- cw- Γ e

SEBUAH

e

c

p

c

e

-( )

dx---------------------------+=

F s

U s

SEBUAH s

3

2--- c

p

1

2--- c

n- Γ

s

SEBUAH s

c p

c s

-( )

dy--------------------------+=

F n

U nSEBUAH

n

3

2--- c

n

1

2--- c

nn- Γ

n

SEBUAH n

cn

c p

-( )

dy---------------------------+=

Sebuahw

3

2--- U

wSEBUAH

wΓ

w

SEBUAH w

dx------

1

2--- U

eSEBUAH

e+ +=

Sebuahww

1

2--- U -

wSEBUAH

w=

Sebuahe Γe

SEBUAH

edx-----=

Sebuahee

0=

Sebuahn

3

2--- U

nSEBUAH

nΓ

n

SEBUAH n

dy-----

1

2--- U

sSEBUAH s

+ +=

Sebuahnn

1

2--- U -

nSEBUAH

n=

Sebuah s

Γ s

SEBUAH s

dy------=

Sebuah ss

0=

c

p

Sebuahw

cw

Sebuahe

ce

Sebuahn

cn

Sebuah s

c s

+ + + Sebuahww

cww

Sebuahnn

cnn

+ +

Sebuah p

----------------------------------------------------------------------------------------------------------------------------=

m c pt , c pt 1-.-( ) V pΔ t

-----------------------------------------=

Halaman 63




The volume of cell P is then denoted V p

The fluxes in and out of cell P can then be computed as previously, but itis necessary to multiply with the time step, to get the mass instead of theflux. Equation 5.3.7 becomes:

(5.7.2)

Combining Eqs. 5.7.1 and 5.7.2, we obtain:

(5.7.3)

The left side of the equation is s imilar to the steady state situation. For atime-dependent computation, the term on the right hand side emerges.The equation can be rewritten:

(5.7.4)

The a nband a p coefficients are now the same as for the steady equation.The additional term on the right side is called a source term . Dalam sebuah computer program these has to be calculated. The most commonly way ofdoing this for Eq. 5.7.4 is by dividing the source term in two, accordinguntuk:

(5.7.5)

The source terms are:

(5.7.6)

This term only depends on known variables, as the concentration at the previous time step is known.

(5.7.7)

The final equation can then be written in the following way, where the a nband a p coefficients are the same as for the steady case:

(5.7.8)

The equation can be solved similarly as for a s teady case. Namun,

Δ tF w

Δ tF e

- Δ tF n

Δ tF s

-+ m=

Sebuahnb

cnb

Sebuah p

c p

-

nb

F w

F e

- F n

F s

-+c pt , c

pt 1-.-( ) V p

Δ t ----------------------------------------= =

Sebuah p

c p

Sebuahw

cw

Sebuahe

ce

Sebuahn

cn

Sebuah s

c s

c pt ,

c pt 1-.-( ) V

pΔ t

-----------------------------------------+ + +=

S C

S P

c pt ,

+c pt , c

pt 1-.-( ) V p

Δ t ----------------------------------------=

S C

c pt 1-. V

pΔ t

---------------------=

S P

V pΔ t -----=

c p

Sebuahw

cw

Sebuahe

ce

Sebuahn

cn

Sebuah s

c s

S C

+ + + +

Sebuah p

S P

+( )---------------------------------------------------------------------------------=

Halaman 64


because of the new dimension, time, it is difficult to do this in a spread-sheet. A computer program is often necessary.

The transient convection-diffusion equation can be written:

(5.7.9)∂ c----- U +

∂ c-------

∂------- Γ

∂ c------=



5.8 Problems

The following three line stylesare used in the sketches inthe problems 1-6:

Problem 1. Grid for reservoir

Make a sketch of a 2D structured non-orthogonal grid with 300 cells forthe geometry given below.

Problem 2. Structured grid for groyne

Make a sketch of a 2D structured non-orthogonal grid with 300 cells forthe geometry given below, without using outblocking.

Problem 3. Grid with outblocking

Make a sketch of a structured non-orthogonal grid with 300 cells for thegeometry given below, but use outblocking for the groyne.

∂ t i∂ xi ∂ xi ∂ xi

Inflow Outflow Wall

Halaman 65


Problem 4. Grid for bay

Make a sketch of a structured non-orthogonal grid with 600 cells for thegeometry given below.



Problem 5. Cylinder grid

Make a paper sketch of a structured non-orthogonal grid with 2400 cellsfor a circular cylinder with diameter 1.0 meters placed vertically in thecentre of a 4 meter wide and 7 meter long f lume.

Problem 6. Dispersion of particles

Particles are dumped in a river with depth 2 meters, velocity 1 m/s and aslope of 1/300. The particles have a fall velocity of 0.01 m/s. A waterintake is located 1 km downstream of the dumping place. What is the percentage of particles passing the intake? Assume no resuspension ofthe particles, and that the particles are added close to the water surface.

Halaman 66


6. Numerical modelling of water velocity in

2D and 3D

This chapter describes the solution procedures for the Navier-Stokes persamaan. These equations describe the water velocity and turbulence

in a river or a hydraulic system.

6.1 The Navier-Stokes equations

The Navier-Stokes equations describe the water velocity. The equationsare derived on the basis of equilibrium of forces on a small volume ofwater in laminar flow:

(6.1.1)

For turbulent flow, it is common to use the Reynolds' averaged versionsof the equations. The Reynolds' averaging is described first.

Claude Louis MarieHenri Navier was profes-sor at École Polytecniquein Paris from 1819 to1831. He derived the Navier-Stokes equationsin 1822, 23 years beforeStokes. Prof. Navier alsoworked on road and bridge constructions, andderived theories for suspension bridges.

∂ U i

∂ t --------- U

j

∂ U i

∂ x j

---------+1ρ---

∂

∂ x j

------- P δaku j

- ρν∂ U

i∂ x

j

--------∂ U

j∂ x

i

---------++=



We are looking at a time series of the velocity at a given location in turbulent flow:

The velocity, U t is divided into an average value U , and a fluctuatingvalue u . The two variables are inserted into the Navier-Stokes equationfor laminar flow, and after some manipulations and simplification the Navier-Stokes equation for turbulent flow emerges:

(6.1.2))

P is the pressure and δ aku jis the Kronecker delta, which is 1 if i = j and 0 oth-erwise. The last term is the Reynolds stress term, often modelled with

the Boussinesq' approximation:

(6.1.3)

The variable k is the turbulent kinetic energy. This is further described in

Ut

Waktu

u

U

Figure 6.1.1 Time seriesof water velocity, fordefinition of velocityfluctuations, u.

“Turbuilence is a cascadeof eddies, from largeeddies to small eddies.The largest eddies taketheir energy from the mainmengalir. The large eddiesgive energy to smallereddies etc., until the eddysize is at the Kolmogorovmicro scale. There, theenergy is killed by viscos-ity”. Prof. H. Wengle

∂ U i

∂ t --------- U

j

∂ U i

∂ x j

---------+1ρ---

∂

∂ x j

------- P δaku j

- ρ uiu j

-( )=

ρ uiu j

- ρνT

∂ U i

∂ x j

--------∂ U

j∂ x

i

---------+2

3---ρ k δ

aku j-=

Halaman 67


Chapter 6.3. Inserting Equation 6.1.3 into Equation 6.1.2 and regroupingthe variables:

(6.1.4)

There are basically five terms: a transient term and a convective term onthe left side of the equation. On the right side of the equation there is a pressure/kinetic energy term, a diffusive term and a stress term.

The convective and diffusive term are solved with the same methods asthe solution of the convection-diffusion equation for dispersion modelling in Chapter 5. The difference is that the pollution concentration is

replaced by the velocity.

The stress term is sometimes neglected, as it has very little influence onthe solution for many cases. The pressure/kinetic energy term is solvedas a pressure term. The kinetic energy is usually very small, and negligible compared with the pressure.

A difference between Eq. 6.1.4 and the convection-diffusion equation forsediments is the diffusion coefficient. Eq. 6.1.4 includes an eddy-viscos-ity instead of the diffusion coefficient. The relationship between thesetwo variables is:

νT = Sc Γ (6.1.5)

where Sc is the Schmidt number. This is usually set to unity, meaningthe eddy-viscosity is the same as the turbulent diffusivity .

∂ U i

∂ t --------- U +

j

∂ U i

∂ x j

---------1ρ---

∂

∂ x j

------- P 2

3--- k + δ

aku j- ρν

T

∂ U i

∂ x j

--------- ρνT

∂ U j

∂ xi

--------+ +=

Important note:



This leaves the problem of solving the pressure term. Several methodsexist, but with the control volume approach, the most commonly usedmethod is the SIMPLE method. This is described in the next chapter.

6.2 The SIMPLE method

SIMPLE is an abbreviation for Semi-Implicit Method for Pressure-LinkedEquations. The purpose of the method is to find the unknown pressurelapangan. The main idea is to guess a value for the pressure and use the con-tinuity defect to obtain an equation for a pressure-correction. Ketika pressure-correction is added to the pressure, water continuity is satis-fied.

To derive the equations for the pressure-correction, a special notation isdigunakan. The initially calculated variables do not satisfy continuity and aredenoted with an index *. The correction of the variables is denoted withan index '. The variables after correction do not have a superscript. The process can then be written:

P = P* + P ' (6.2.1)U k = U k * + U k ' (6.2.2)

P is the pressure and U is the velocity. The index k on the velocitydenotes direction, and runs from 1 to 3 for a 3D calculation.

Given guessed values for the pressure, the discretized version of the Navier-Stokes equations is:

Modern CFD using thefinite volume method, theSIMPLE method and thek-ε turbulence model was pioneered by a group ofresearchers in the early1970's at Department ofMechanical Engineering,Imperial College, London.The group includedresearchers as DBSpalding, BE Launder,SV Patankar and W.Rodi. The algorithms areused in most commercialCFD programs today.

Halaman 68


(6.2.3)

The convective and diffusive terms have been discretized as describedin Chapter 3. The variable B contains the rest of the terms besides theconvective term, the diffusive term and the pressure term. In the pres-sure term, A k is the surface area on the cell wall in direction k , and ξ isan index for the grid, described in Chapter 5. Looking at a pressure dif-ference between two neighbour cells, ξ will be unity.

The discretized version of the Navier-Stokes equations based on thecorrected variables can be written as:

(6.2.4)

If this equation is subtracted from Equation 6.2.3, and the two Equations6.2.2 and 6.2.3 are used, the following equation can be made for thevelocity correction in cell P :

(6.2.5)

A simplification has then been made to neglect the first term on the rightside of Eq. 6.2.4. The SIMPLEC method instead uses the following for-mula:

(6.2.6)

Sebuah p

U kp ,

* Sebuahnb

U k nb.

∗ Buk

SEBUAH k

∂ P ∗

∂ξ----------+

nb

=

Sebuah p

U kp , Sebuah

nbU

k nb. Buk

SEBUAH k

∂ P

∂ξ-------+

nb

=

U k '

SEBUAH k

Sebuah p

-----∂ P ′

∂ξ---------= Exam suggestions from students on what SIMPLE stands for:

Semi-Implicit Multiple Pres- sure Loss Equation

Surface IMplied Pressure Level Elements

Semi-Imperfect Pressure Link Estimation

U k '

SEBUAH k

Sebuah p

Sebuahnb

nb

-

-------------------------------∂ P ′

∂ξ---------=



Equations 6.2.5 and 6.2.6 give the velocity-corrections once the pres-sure-corrections are known. To obtain the pressure-corrections, the con-tinuity equation is used for cell P , where the water fluxes through eachcell side are summed up:

( k =1,2,3) (6.2.7)

Istilah dari is equal to the water continuity deficit in each cell,

from the previous iteration. We denote this expression to be Δ V .

The expression for the velocity correction from Equation 6.2.5 is insertedinto Equation 6.2.7, eliminating it as unknown. Summing up over eachside of the cell, the pressure correction gradient can be discretized forone side as follows:

SEBUAH k U

k

nb

SEBUAH k U ′

k

nb

SEBUAH k U *k

nb

+ 0= =

SEBUAH k U

*k

nb

Halaman 69


Side east: (6.2.8)

Using this formula for four sides in a 2D situation, Eq. 6.2.7 can be writ-ten:

(6.2.9)

The result is an equation where only the pressure-correction is unknown:

(6.2.10)

The index 0 is used to indicate the new set of a 0nb coefficients. The

source term, b , in Eq. 6.2.10 will be the water continuity deficit Δ V fromthe guessed velocity field. When water continuity is satisfied, this term iszero, and there are no more corrections to the pressure.

The following formula is derived for a 0e:

(6.2.11)

A similar equation holds for the other a 0nb coefficients. The index e is

then replaced by w, n, s, t or b. The a p,e factor is the average a p value incell p and cell e .

Equation 6.2.10 is solved in the same way as the other equations.

The procedure is therefore:

Guess a pressure field, P*Calculate the velocity U* by solving Equation 6.2.3Solve equation 6.2.10 and obtain the pressure-correction, P' Correct the pressure by adding P ' to P*Correct the velocities U* with U' using equation 6.2.5.Iterate from point 2 to convergence

SEBUAH k U ′

k SEBUAH -

k

SEBUAH k

Sebuah p

-----∂ P ′

∂ξ--------SEBUAH

k 2

Sebuah p

------ P ′ p

P ′e

-( )= =

SEBUAH w2

Sebuah p

------ P ′w

P ′ p

-( )SEBUAH s2

Sebuah p

------ P ′ s

P ′ p

-( )

SEBUAH e2

Sebuah p

------ P ′e

P ′ p

-( )SEBUAH

n2

Sebuah p

------ P ′n

P ′ p

-( ) Δ V

+ +

+ + 0=

a ° p P ′

pa °

nb P ′

nbb+

nb

=

a °e

SEBUAH e2

Sebuah pe ,

---------=



An equation for the pressure is not solved directly, only an equation forthe pressure- correction. The pressure is obtained by accumulative addi-tion of the pressure-correction values.

The SIMPLE method can give instabilities when calculating the pressurelapangan. Therefore, the pressure-correction is often multiplied with a number below unity before being added to the pressure. The number is a relaxa-tion coefficient. The value 0.2 is often used. The optimum factor dependon the flow situation and can be changed to give better convergencetarif. Relaxation coefficients are further described in Chapter 6.5.

Regarding the difference between the SIMPLE and the SIMPLEC

Halaman 70


method, the SIMPLEC should be more consistent in theory, as a morecorrect formula is used. Looking at Equations 6.2.5 and 6.2.6, the SIM-PLE method will give a smaller correction than the SIMPLEC method, asthe denominator will be larger. The SIMPLE method will therefore move

slower towards convergence than the SIMPLEC method. jika ada problems with instabilities, this can be an advantage.

A more detailed description of the SIMPLE method is given by Patankar(1980).

Most pressure-correction methods for incompressible flow follows algo-rithms similar to SIMPLE. There are algorithms involving more correctionsteps, for example SIMPLER and PISO. Note that the different methodwill only affect the convergence speed and the stability of the solution.The accuracy of the results will not be directly affected, as long as themethods are based on water continuity.

6.3 Advanced turbulence models

The following chapter is a brief overview of advanced turbulence mod-els. The reader is referred to White (1974) and Rodi (1980) for more

detailed description of turbulence and turbulence models.In chapter 6.1, the Boussinesq approximation was introduced for findingan expression for the Reynolds' stress term:

(6.3.1)

νT is the turbulent eddy viscosity.

Some simpler turbulence models were described in Chapter 3. Thesemodels require calibration before being used on new cases. Merekaalso based on algebraic relations, and no differential equations aresolved when computing the eddy-viscosity. The models are then oftencalled zero-equation models.

A more advanced approach is to use more complex methods to computed the eddy-viscosity. One option is to use a partial differential equa-

tion for νT . One of the more popular approaches is the Spalart-AllmarasModel:

The Spallart-Allmaras model

The model (Spallart and Allmaras, 1994) is essentially a convection-dif-fusion equation for the eddy-viscosity, where different terms are includedto take special physical phenomena into account.

(6.3.2)

The book Numerical HeatTransfer and Fluid Flow , by SV Patankar, is one

of the most readable textson CFD, and provides anexcellent introduction tothis science.

Patankar taught CFD atthe Norwegian Universityof Science and Technol-ogy in 1977.

ρ uiu j

- ρνT

∂ U i

∂ x j

--------∂ U

j∂ x

i

---------+2

3---ρ k δ

aku j-=

The kinematic viscosity isa fluid property, while theturbulent eddy-viscositydepends of the velocitylapangan.

∂νT

∂ t --------- U

j

∂νT

∂ x j

--------+ cb 1

S νT

1σ---

∂

∂ x----- ν

T

∂νT

∂ x j

--------- cb 2

νT

∂νT

∂ x j

---------2

+ cw 1

f w

νT

d -----

2-+=



The first term on the right side is the production of turbulence. Adaseveral ways this can be modelled. One is:

Halaman 71


(6.3.3)

The next term on the right side of Eq. 6.3.2 is the diffusion of turbulence.The last term is related to damping of turbulence close to the wall. Thedistance to the wall is given as d . Spalart and Allmaras suggested thefollowing function for f w:

(6.3.4)

The remaining parameters are constants:

cb1 = 0.1355, c b2 = 0.622, σ = 2/3, c w1= 3.28, c w2 = 2, c w3 = 0.3, κ=0.4

The k- ε model

Instead of solving only one equation for the eddy-viscosity, it is possibleto use two partial differential equations. The most popular two-equationmodel is the k -ε model (Jones and Launder, 1973). The k -ε model com- putes the eddy-viscosity as:

(6.3.5)

k is turbulent kinetic energy, defined by:

(6.3.6)

k is modelled as:

(6.3.7)

where P k is the production of turbulence, given by:

(6.3.8)

The dissipation of k is denoted ε, and modelled as:

(6.3.9)

The constants in the k -ε model have the following standard values:

S

∂ U j

∂ xi

---------∂ U

j∂ x

i

--------∂ U

i∂ x

j

---------+=

f

w

g 1 c

w 36+

g 6 cw 36+

--------------------

16---

= g rc

w 2

r 6 r -( )+= r ν

T

S κ2d 2

--------------=

νT

cμk 2

ε-----=

k 1

2--- u

iu

i≡

∂ k

∂ t ----- U

j

∂ k

∂ x j

-------+∂

∂ x j

------- ν

T σ

k

-----∂ k

∂ x j

------- P k

ε-+=

P k

νT

∂ U j

∂ xi

---------∂ U

j∂ x

i

--------∂ U

i∂ x

j

---------+=

∂ε

∂ t ----- U

j

∂ε

∂ x j

------+∂

∂ x j

------- ν

T σ

ε-----

∂ε

∂ x j

------- C ε1ε

k - P

k C ε2

ε2

k -----+ +=



Halaman 72


cμ = 0.09C ε1 = 1.44C ε2 = 1.92 (6.3.10)σκ = 1.0σε = 1.3

As seen from Equation 6.3.1, the eddy-viscosity is isotropic, and mod-elled as an average for all three directions. Schall (1983) investigatedthe eddy-viscosity in a laboratory flume in three directions. Dia bekerjashows that the eddy-viscosity in the streamwise direction is almost onemagnitude greater than in the cross-streamwise direction. A better turbu-lence model could therefore give more accurate results for many cases.

More advanced turbulence models

To be able to model non-isotropic turbulence, a more accurate represen-tation of the Reynolds stress is needed. Instead of using the Boussinesqapproximation (Equation 6.3.1), the Reynolds' stress can be modelledwith all terms:

(6.3.11)

The following notation is used: u is the fluctuating velocity in direction 1,v is the fluctuating velocity in direction 2 and w is the fluctuating velocityin direction 3.

The nine terms shown on the right hand side of Equation 6.3.8 can becondensed into six different terms, as the matrix is symmetrical. A Rey-nolds' stress model will solve an equation for each of the six unknownistilah. Usually, differential equations for each term are solved. Inimeans that six differential equations are solved compared with two forthe k -ε model. It means added complexity and computational time.

An alternative is to use an Algebraic Stress Model (ASM), where alge-

braic expressions for the various terms are used. It is also possible tocombine the k -ε model with an ASM to obtain non-isotropic eddy viscos-ity (Rodi, 1980).

An even more advanced method is to resolve the larger eddies with avery fine grid, and use a turbulence model only for the smaller scales.This is called Large-Eddy Simulation (LES). If the grid is so fine that sub-grid eddies do not exist because they are dissipated by the kinematicviscosity, the method is called a Direct Solution (DS) of the Navier-Stokes equations.

Note that both LES and especially DS modelling require extreme compu-tational resources, presently not available for engineering purposes.

6.4 Boundary conditions

Boundary conditions for the Navier-Stokes equations are in many ways

similar to the solution of the convection-diffusion equation. In the follow-ing text, a division in four parts is made: Inflow, outflow, water surfaceand bed/wall.

The main advantage ofthe k -ε model is the almostuniversal constants. Themodel can thereby beused on a number of vari-ous flow situations with-out calibration. For riverengineering this may notalways be the case, because when frictionalong the bed is influenc-ing the flow field, theroughness of the bed alsoneeds to be given. Jikaroughness can not beobtained from directmeasurements, it has to

be calibrated with meas-urements of the velocity.ρ u

iu j

- ρuu vu wu

uv vv wv

uw vw ww

-=

Page 73




Inflow

Dirichlet boundary conditions have to be given at the inflow boundary.This is relatively straightforward for the velocities. Usually it is more diffi-cult to specify the turbulence. It is then possible to use a simple turbu-lence model, like Equation 3.4.1 to specify the eddy-viscosity. Given thevelocity, it is also possible to estimate the shear stress at the entrance bed. Then the turbulent kinetic energy k at the inflow bed is determined by the following equation:

(6.4.1)

This equation is based on equilibrium between production and dissipa-tion of turbulence at the bed cell.

Given the eddy-viscosity and k at the bed, Equation 6.3.2 gives the valueof ε at the bed. If k is assumed to vary linearly from the bed to the sur-face, with for example half the bed value at the surface, Equation 6.3.2can be used together with the profile of the eddy-viscosity to calculatethe vertical distribution of ε.

Outflow

Zero gradient boundary conditions can be used at outflow boundaries forall variables. A boundary condition where the gradient is specified isoften called a von Neumann condition.

Water surface

Zero gradient boundary conditions are often used for ε. The turbulentkinetic energy, k , can set to zero. Rodi (1980) gives an alternativeexpression for computing k at the water surface. Symmetrical boundaryconditions are used for the water velocity, meaning zero gradient bound-ary conditions are used for the velocities in the horizontal directions. Thevelocity in the vertical direction is calculated from the criteria of zerowater flux across the water surface.

Bed/wall

The flux through the bed/wall is zero, so no boundary conditions arediberikan. However, the flow gradient towards the wall is very steep, and itwould require a significant number of grid cells to dissolve the gradientsufficiently. Instead, a wall law is used, transformed by integrating it overthe cell closest to the bed. Using a wall law for rough boundaries (Schli-chting, 1980)

(6.4.2)

also takes the effect of the roughness, k s, on the wall into account. Thevelocity is denoted U , u *is the shear velocity, κ is a coefficient equal to0.4 and y is the distance from the wall to the centre of the cell.

The wall law is used both for the velocities and the turbulence parame-ters. The use on turbulence parameters is described in more detail byRodi (1980). For the velocities, the wall shear s tress is a force on a cell,and it is computed as a sink term in the Navier-Stokes equation. The

k τ

ρ cμ

-------------=

U

u*

-----1κ---

30 y

k s

---------ln=

Halaman 74


force is computed by rearranging Eq. 6.4.2:

(6.4.3) F τ A ρ u*2SEBUAH ρ A U

1κ--- 30

y---ln

--------------------------

2

= = =



The area of the cell at the bed is denoted A , while y is the distance fromthe center of the cell to the bed. The velocity in the bed cell is denoted U .

6.5 Stability and convergence

The solution method described previously are iterative. The principle isto guess a starting value for the variables and then iterate to get a bettersolusi. The procedure is illustrated in Fig. 6.5.1.:

Convergence criteria

In an iterative procedure, some criteria has to be met to decide if thesolution is converged. Several different criteria exist, based on computa-tion of a residual. The residual is a measure of how large the deviation is between the correct value and the values in the current iteration. A lowresidual indicates that convergence is reached.

One formula for the residual, r , is given in Eq. 6.5.1:

(6.5.1)

F c is a characteristic flux, and U c is a characteristic velocity. Nilai-nilaiat the inflow boundary are often used. The total number of cells in thegrid is denoted n .

Example: We are looking at one 2D cell with the following parameters:

Sebuah

w

= 2.0, U

w

= 1.0

Sebuahe = 0.5, U e = 0.3

Sebuahn = 1.0, U n = 0,4

Sebuah s = 0.3, U s = 0.7

k s

VelocityCorrect value

Iterations1 2 3 4 5

guessednilai

Figure 6.5.1 Conver-gence graph for thevelocity in one cell.

r

Sebuah p

U p

Sebuahnb

U nb

Sumber p

--

1

n

nF cU

c

-------------------------------------------------- -----------------------------=

Halaman 75


For simplicity, we assume that the source term is zero.

The first time cell p is computed, the following values are obtained:

Sebuah p = 3.8. U p = 0.73

The velocity in cell P is computed from Eq. 5.2.2, where the concentra-tion is replaced by the velocity.

After the cell is computed, the east and north cells are recomputed and get different values. They now have values:

U e = 0.35

U n = 0.45



The contribution of this cell to the residual of Eq. 6.5.1 becomes:

= 0.075

Note that the values from the previous iteration is not used.

Another convergence criteria is based on the difference in the values between two iterations:

(6.5.2)

The disadvantage with using Eq. 6.5.2 is that the residual can go to zeroeven if the solution is not converged. This is illustrated in Fig. 6.5.2:

The reason for this can be that the velocity-correction equation from theSIMPLE method may change the velocities back to what they were before the application of the solver. Also, possible bugs in the programcan give the same problem. Many CFD programs therefore prefer to useEq. 6.5.1 instead of Eq. 6.5.2.

Instabilities

It is not always that the iterative solution method is successful in obtain-ing convergence. The system of equations may be unstable. Con yang

Sebuah p

U p

Sebuahnb

U nb

- 3.8 x 0.73 2 x 1- 0.5 x 0.35 1 x 0.45 0.3 x 0.7+ + +=

r

U i

U i 1--

1

n

nU c

-------------------------------=


Iterations1 2 3 4 5

guessednilai

Figure 6.5.2 Conver-gence graph for thevelocity in one cell.The correct value is notobtained, but still thenew values in the itera-tions are the same as inthe previous iteration.

Halaman 76


vergence graph for one cell may then look like what is given in Fig. 6.5.3.The values may oscillate, and more and more extreme values are pro-yang diinduksi. Often, one defines a solution crash as the residuals becomingabove a high value, for example 10 10.

There are several methods to prevent instabilities and accelerate con-vergence. Some are further described in the following.


Iterations1 2 3 4 5

guessednilai

Figure 6.5.3 Conver-gence graph for thevelocity in one cell,where there are insta-bilities and non-con-vergence..



Relaksasi

The main principle in the solution of the equations are to obtain animprovement of a guessed velocity field. Starting with the guessed val-ues, several iterations are done to improve the result. For each iteration,a new guess is made. Let us say that we have finished iteration i -1 and i ,and we are looking at what variables, v , we should use when startingiteration i +1. An obvious choice is of course the variables at iteration i .

However, introducing the relaxation coefficient, r , instead we use:

v = r * vi + (1-r) * vi-1 (6.5.3)

The relaxation coefficients should normally be between 0 and 1.

Relaxation will give a slower convergence speed towards the final solu-tion, but there will be less instabilities. If the solution diverges or does notconverge because of instabilities, a normal measure is to lower therelaxation coefficients.

Multigrid and block-correction

The purpose of the multigrid methods is to speed up the convergence ofsolusi. The main principle is a division of the grid several coarsersub-grids. This is shown in the figure below:

Halaman 77


The solution is first iterated once on the coarse grid. It is then extrapo-lated to the finer grid, where it is iterated further. Then it is extrapolatedto the finest grid, where more iterations are done. The solution is interpo-lated to the medium fine grid, where more iterations are done. Then thisis interpolated to the coarse grid, and computed again with more itera-tions. The sequence is repeated until convergence. There are transfor-mation functions moving the variables from one grid to the other byextrapolation/interpolation. When the finest grid has a high amount ofcells, it is possible to have several grids with varying number of gridsel.

A version of the multigrid method is called block-correction . For a two-dimensional situation, the grids then look like this:

Coarse grid Fine grid

Figure 6.5.4 Grid struc-ture for multi-gridmetode

Figure 6.5.5 Grid structure for multi-block method. Original grid to the left.



The iterations are started on the original grid. Then all variables aresummed in a slice of the grid, so that a one-dimensional grid emerges.This is solved, and the result is used to correct the original values. Iniis repeated in all directions, shown here with two coarse grids, for a two-dimensional situation.

The Rhie and Chow interpolation

Using a non-staggered variable location, all variables are calculated inthe centre of the cells. This causes oscillations in the solution and insta-tanggung. The staggered grid was invented to avoid these oscillations.

Then the pressure is calculated between the centres of the grid cells.There are several problems with this variable arrangement, especiallyfor non-orthogonal grids. The Rhie and Chow interpolation was inventedto avoid the instabilities and still use a non-staggered grid. The interpola-tion gives the velocity on the cell surface. This velocity is used to calcu-late the flux on the cell surface.

A derivation of the Rhie and Chow interpolation procedure is fairlyinvolved, and the reader is referred to Rhie and Chow (1983). Utamaidea is to use information about pressure gradients in staggered andnon-staggered positions. The resulting interpolation formula is a functionof the linearly interpolated velocity plus a term dependent on the pres-sure gradients, cell areas and the a p koefisien.

The Rhie and Chow interpolation can be interpreted as an addition of 4 thorder artificial diffusion. However, no adjustment coefficients are used.

Halaman 78


The Rhie and Chow interpolation is used in most CFD programs usingthe finite volume method, the SIMPLE algorithm and non-staggeredgrids.

Note that the Rhie and Chow interpolation has given problems for somecases where there are large source terms in the Navier-Stokes equa-

tions (Olsen and Kjellesvig, 1998b). In these cases there are significantforces on the water in addition to the pressure, for example from gravity.

Upstream methods and artificial diffusion

The discretization schemes given in Chapter 5 are all fairly stable for thecalculation of sediment concentration. However, other schemes devel-oped earlier were not so stable. The classical example is the central-dif-ference scheme. In this method, the flux on a cell wall is calculated byinterpolation between the cells on the two sides. The figure below showsthe estimation.

The fluxes through the sides are then:

(6.5.2)

Cell w Cell p

Konsentrasi

cW

cW

cPFigure 6.5.6 Estimation of concentration forthe central scheme

F w

0.5 U w

SEBUAH w

cw

c p

+( ) Γw

SEBUAH w

cw

c p

-( )

dx-----------------------------+=

( ) ΓSEBUAH

ec p

ce

-( )



(6.5.3)

(6.5.4)

(6.5.5)

Applying the continuity equation (Eq. 5.3.7), and the same method as inChapter 5.3, the coefficients become:

(6.5.6)

(6.5.7)

(6.5.8)

F e 0.5 UeSEBUAH e cp ce+ e dx--------------------------+=

F s

0.5 U s

SEBUAH s

c p

c s

+( ) Γ s

SEBUAH s

c p

c s

-( )

dy--------------------------+=

F n

0.5 U nSEBUAH

nc

nc p

+( ) Γn

SEBUAH n

cn

c p

-( )

dy--------------------------+=

Sebuahw 0.5 U wSEBUAH w Γw

SEBUAH w

dx------+=

Sebuahe

0.5 U eSEBUAH

eΓ+

e-

SEBUAH e

dx-----=

Sebuahn

0.5 U nSEBUAH

nΓ

n

SEBUAH n

dy-----+=

Halaman 79


(6.5.9)

(6.5.10)

Applying continuity, the following simplification can be done for a p:

(6.5.11)

Looking at for example a e, if the diffusivity is low compared with thevelocity, there is a chance that a e can become negative. Also, the effec-tive weighting factor is actually a e /Sebuah p. When the diffusion becomes small,the effective factor becomes very large, as a p is only a function of the dif-fusion. A large negative number for the weighing factor is not physicallyrealistic, and this causes instabilities.

The minimum value of the diffusivity before instability occur can be cal-culated by setting the weighting factor to zero.

(6.5.12)

This gives the following theoretical minimum viscosity to avoid instabili-ties:

(6.5.13)

Schemes based on the central-difference scheme or similar ill-formu-lated numerical schemes may require adding extra diffusivity to the solu-tion in order to become stable. This is called artificial diffusion , andcomes in addition to the physical diffusivity. The disadvantage with add-ing artificial diffusivity is that the increased diffusivity it may give a differ-ent final result than what the natural diffusion would give.

6.6 Free surface algorithms

The ability to compute flow with a free surface is important in hydraulicrekayasa. The free surface algorithms can be classified according tohow many dimensions are used. For two-dimensional depth-averaged

Sebuah s

0.5 U s

- SEBUAH s

Γ s

SEBUAH s

dy------+=

Sebuah p

0.5 U w

SEBUAH w

Γ+-w

SEBUAH w

dx------ 0.5 U

eSEBUAH

eΓ

e

SEBUAH e

dx------

0.5 U s

SEBUAH s

Γ s

SEBUAH s

dy------ 0.5 U

nSEBUAH

nΓ+

n-

SEBUAH n

dy------

+ + +

+

=

Sebuah p

Γw

SEBUAH w

dx------ Γ

e

SEBUAH e

dx------ Γ

s

SEBUAH s

dy----- Γ

n

SEBUAH n

dy-----+ + +=

Sebuahe

0.5 U eSEBUAH

eΓ+

e-

SEBUAH e

dx----- 0= =

Γ

e min.0.5 U

edx=Artificial/false diffusion:

Many people confuse thedifference between artificialand false diffusion. Buatandiffusion can be seen as akind of fudging factor, to geta stable solution. False dif-fusion is due to inaccurateapproximation in the discret-ization method.



computations, similar algorithms as described in Chapter 3 can be used.However, there exist a large number of different algoritms for computingthe free surface in 2D.For computing the free surface in 3D, the different algorithms can beclassified according to if an adaptive grid is used or not.

Fixed grid algorithms

The fixed grid algorithms will in general compute a two-phase flow. Thetwo phases will be water and air. The algorithms must determine wherethe location of the free surface is within the grid. Some cells will be completely filled with water, and others completely filled with air. The remain-ing cells will be partially filled with water and air.

Page 80


One of the most commonly used algorithms is called a Volume Of Fluid(VOF) method. The method introduces a variable called the volume offluid, defined as:

Vw is the volume of water in a cell and Va is the volume of air in the cell.The parameter r will therefore be 1 when the cell is completely filled withwater and 0 if a cell is completely filled with air.

The VOF ratio is computed by solving a convection-diffusion equation:

(6.6.2)

Based on the F values in all the cells, the location of the free surfacemust be determined. The reconstruction of the surface is not trivial, andthere are several different methods that can be used.

The VOF method is used in Flow-3D.

A more recent method that has attracted attention in research communi-ties is the Level Set method. Instead of solving an equation for the vol-ume of fluid, an equation for the distance, L [m], to the water surface isdigunakan. A convection equation for this distance is given as:

(6.6.2)

The equation is solved with similar methods as given in Chapter 5, asthis is a convection-diffusion equation where the diffusive term is omit-ted.

The advantage of using the level set method instead of the volume offluid method is that it is easier to compute the location of the free watersurface once the equations are solved.

Adaptive grid algorithms

Adaptive grid algorithms will change the grid so that the free surfacealways is aligned with its top. All the cells will thereby always be filleddengan air. No cells are thereby wasted by filling it with air. Metodetherefore needs less cells than the fixed grid algorithms. Another advan-tage is that inaccuracies can occur when the cells are partially filled withair. Also, the grid close to the surface will be aligned with the flow,reducing false diffusion.

A disadvantage with the adaptive grid methods is that they may be moreunstable than the fixed grid methods.

In an adaptive grid algorithm, the free surface is given an initial value.The algorithms will compute changes in the free surface, and adjust thegrid accordingly. The adjustments are done in small steps, to prevent

F Vw

Va Vw+---------------------=

∂ F

∂ t ------ U +

i

∂ F

∂ xi

------∂

∂ xi

------- Γ∂ F

∂ xi

------=

∂ L

∂ t ------ U +

i

∂ L

∂ xi

------ 0=



instabilities.

The adaptive grid methods can be further classified from which equa-tions are solved to compute the changes in the water levels. Satu

Page 81


method uses the water continuity equation in the cells close to the sur-face (Olsen and Kjellesvig, 1998). Normally, the water continuity defectis used in the SIMPLE algorithm to compute the pressure. Sebaliknya, pressure is computed by linear interpolation between the cell below andthe surface. This approach also introduces the gravity in the Navier-Stokes equations. The gravity is a large source term, causing instabili-ties in the solution. Therefore, a very short time step has to be used. Sebuahexample is given in Fig. 6.6.1, where the coefficient of discharge for aspillway is computed.

The main problems with the method is the stability and the need forextremely short time steps. When modelling a river over several weeks,it is necessary to use longer time steps. Then the gravity can not beused in the Navier-Stokes equation. The alternative is to use the Energyequation instead of the continuity equation to compute the changes inthe free surface. Then the computed pressure field is used to estimatethe location of the surface, according to the following equation:

: (6.6.3)

It is then assumed that one location in the grid is kept at a known eleva-tion. The elevation difference, dh , between this location and another cellin the grid can be computed from Eq. 6.6.3, given the pressure differ-ence dp between the cells. The method is very stable and can be usedwith large time steps. However, the water close to the surface must havea hydrostatic pressure for Eq. 6.6.3 to be valid. Therefore, very steepsurface slopes, like in Fig. 6.6.1 can not be computed. The location innatural rivers and channels can be computed.

6.7 Errors and uncertainty in CFD

As shown in previous chapters, there are a number of uncertainties inCFD modelling, and approximations in the algorithms leads to some

Figure 6.6.1 Longitudinal profile of the water level and velocities for computation of co-

efficient of discharge for a spillway. The numbers show the computed time.

0.5 sec.

5 ms

93 sec.

50 ms

dhdpρ g ------=



Page 82


errors in the results. The European Research Community on Flow, Tur- bulence and Combustion (ERCOFTAC) published Best Practice Guide-

lines for CFD, where the errors are classified according to the followingdaftar:

1. Modelling errors2. Errors in the numerical approximations3. Errors due to not complete convergence4. Round-off errors5. Errors in boundary conditions and input data6. Human errors due to inexperience of the user7. Bugs in the software

Modelling errors are errors introduced when modelling the real worldwith a number of mathematical equations. Typical modelling errors areusing a one-dimensional formulation if there are three-dimensionaleffects affecting the problem. Another example is the assumption of anisotropic turbulent eddy-viscosity made for example in the standard keModel. Non-isotropic effects may affect the results in some cases.

Errors due to numerical approximations are often introduced when dis-cretization the equations. False diffusion is a typical error in the numeri-cal approximations.

Many times an iterative solver is used for the equations. Sometimes theresults are used even if the solution is not fully converged. This could bethe case if proper convergence criteria are not used. Also, for time-dependent computations, convergence may not be reached for eachtime step.

Round-off errors are due to limitations in the accuracy of the microproc-essors of the computers. Most numerical programs nowadays use 64 bits floating point numbers with 12 digits accuracy, and then this is usu-ally not a serious problem. However, earlier 32 bits programs often usednumbers with only 6 digits accuracy, and then round-off errors could besignifikan.

Errors in the boundary conditions is one of the most common problemsin CFD modelling. Computing flow in complex geometries the grid has tofollow the water level and river bed completely. This is sometimes diffi-cult. Also there may be problems with deciding boundary conditions forexample for the roughness. Inflow boundary conditions are also uncer-tain. This applies for the distribution of the velocity at the inlet cross-sec-tion and the turbulence there. For sediment computations, the amount ofsediment inflow may be uncertain. Also, the empirical formula for sedi-ment concentration close to the bed is often not very accurate.

Human errors due to inexperience of the user is often a likely problem.Experience on CFD modelling is scarce, and it is easy to make errorswhen choosing among different parameters and algorithms in the CFDModel.

There will always be bugs in every software. An estimate often used isone bug pr. 1000-10 000 lines for a commercial program. A typical CFD program may have 100 000 - 1 million lines of code. It is therefore likelythat most CFD program has a fair number of bugs. A CFD program isusually improved relatively frequently. Every time a new algorithm ismade in the program, it is possible that bugs are introduced. This may be problematic to detect, as it is often difficult to predict how the new algo-rithm with interact with the older algorithms.

Page 83




6.8 SSIIM

SSIIM is an abbreviation for S ediment S imulation I n I ntakes with M ulti- block option. The program solves the Navier-Stokes equations in athree-dimensional non-orthogonal grid, using the k-ε turbulence modeland the SIMPLE method to compute the pressure. The program alsosolves convection-diffusion equations for various water quality constitu-ents, like sediments, temperature, algae, nutrients, pollutants etc. Time-dependent changes in bed and surface levels are computed.

The program was originally designed to compute sediment transport forhydropower intakes. Later it has been expanded to areas of river mor- phology, hydraulic structures like spillways, head loss in contractionsetc.; general water quality, density stratification, wind-induced currents,special algae algorithms etc.

The program has a graphical user interface with an interactive grid edi-tor, containing several algorithms simplifying the constructions of thegrid. The main program contains graphical presentation of results in mul-tiple dimensions, which can be run simultaneously with the solution ofthe differential equations. A separate program is included to viewcoloured surfaces in three dimensions. This is based on the OpenGLgraphics library.

The program runs on Windows, and can be down loaded from the Inter-net: folk.ntnu.no/nilsol/ssiimwin. The User's Manual gives more details,and can be downloaded from the same web page.

6.9 Problems

Problem 1. Navier-Stokes solver

Apply the SSIIM model to the example with the sand trap: Tutorial 1 inthe User's Manual. Note how many iterations are required for conver-gence of the Navier-Stokes equations, and how long time it takes onyour PC.

Problem 2. Multigrid

Repeat the calculation in Problem 1, but this time use block-correctionfor all equations. How many iterations are now needed, and how longtime does this take?

Problem 3. Relaxation coefficients

Repeat the calculation as in Problem 1, but this time change the relaxa-

tion factors first to 1.0 for all the equations. How many iterations areneeded?

Again, change the relaxation coefficients to 0.5 for all equations. Bagaimanamany iterations are needed?

Change the relaxation coefficients to 0.3 for velocity, 0.1 for pressureand 0.2 for k and ε. How many iterations are needed?

Problem 4. The Rhie and Chow interpolation

Repeat the calculation in Problem 1, but this time reduce the influence ofthe Rhie and Chow interpolation by the using the F 21 data set. Set thisto 0.5 for one run and 0.0 for the next run. How does the Rhie and Chow

There exist a large numberof CFD programs. Beberapatailor-made for hydraulicengineering, for exampleTELEMAC , SSIIM andSMS/TABS (made by USArmy Corps of Engineers).Others are general-purpose programs, that can be usedfor gas, oil, multiphase flowetc. Some of the most usedgeneral-purpose programsare PHOENICS , FLUENT ,CFX , FLOW3D , and STAR-CD . Web addresses to moreinformation about the pro-grams can be found at:folk.ntnu.no/nilsol/cfd

Page 84


interpolation affect the resulting trap efficiency? And how many iterationsare needed?

Problem 5. Upstream boundary conditions

Repeat the calculation in Problem 1, but this time use uniform upstream



and downstream velocity profile. This is done by using the G 7 data sets.How does this affect the convergence rate and the resulting trap effi-ciency? What are the reasons for the change?

Problem 6. Bed roughness

Repeat the calculation in Problem 1 twice, using a roughness of 2 cmand 0.1 mm. This is done by giving the roughness in the F 16 data set.How does the roughness affect the trap efficiency and calculation time?What is the reason for the change?

Problem 7. Initial values

Calculate the water flow for the fish farm tank example with SSIIM. Bagaimanamany iterations are needed? Remove the G 8 data set with initial veloci-ties in the control file. What happens in the calculations and why? How isit possible to make the calculations converge without lowering the initialwater velocities?

Problem 8. Stability

Implement the central scheme for Problem 1. Vary the amount of diffu-sion, and see how the result changes. What is the minimum amount ofdiffusion to get a stable solution? How does this compare with the realdiffusion?

Page 85


7. Physical limnology

7.1 Introduction

Limnology is the science of processes in lakes, including water quality,temperature, ice, water currents etc. A large number of words explainingthe processes has been made. Also various classification systems have been developed. This chapter focuses on the hydraulic and temperature proses. Biological processes and classification systems for lakes aregiven in Chapter 8.



7.2 Circulation in non-stratified lakes

The term circulation is often used for water currents in a lake. Velocitiesin a lake are often wind-induced. The water will then follow a circulation pattern, moving with the wind at the water surface and a return current isformed close to the bed. However, the term circulation is also used if thewater currents are due to inflow/outflow, and the water moves in almoststraight lines. Gambar. 7.2.1 shows a typical profile of wind-induced velocityin a shallow lake:

The magnitude of the wind-induced currents will depend on the windkecepatan. The most common approach to calculate the currents is to compute a shear stress from the wind on the surface of the lake. And thenuse this shear stress to calculate the currents. The wind-induced sheardiberikan oleh:

(7.2.1)

The wind speed is denoted U Sebuah, ρ Sebuahis the air density (around 1.2 kg/m 3)

and c 10 is an empirical coefficient. The coefficient will be differentdepending on which elevation the wind is measured. The c 10 koefisienis given for wind speeds taken 10 meters above the water surface. Di sanaare a number of empirical formulas for c 10, for example as given byBengtsson (1973)

(7.2.2)

If the wind persist for a long time, the water surface will not be horizontalany more. The slope, I , can be computed from equilibrium forces on awater element, similar to what was done for a river in Chapter 2. Theslope becomes:

Wind

Velocity profile

Return current

Figure 7.2.1. Longitudinal profile of a lake with wind-

induced currents. The veloc-ity profile is shown, wherethe water close to the sur-face moves with the wind,and a return current isformed close to the bed.

τ c10ρ

SebuahU

Sebuah2=

c10 1.1*10 3-=

Page 86


(7.2.3)

where h is the water depth.

The wind will also induce velocity gradients and turbulence in the lake.Classical hydraulics will give the following formula for the turbulent mix-

ing coefficient, Γ (derivation given in Chapter 4.2):

(7.2.4)

The shear velocity is denoted u *, h is the water depth and α is an empiri-cal coefficient. For rivers, α has been found to be 0.11. This value wasused successfully for lakes by Olsen et. al. (2000) modelling a small res-ervoir in Wales, UK. However, when modelling Loch Leven in Scotland(Olsen et. al. 1998), the formula gave too high diffusion.

7.3 Temperature and stratification

The water density in freshwater lakes and reservoirs is mainly a function

Akuτ

ρ gh----------=

Γ α u*h=

Table of water density as afunction of temperature:

Temp (0C) Density (kg/m 3)



of the temperature, as long as the sediment concentration is reasonablyrendah. Maximum density is at 4oC , with lower densities at higher and lowersuhu. Stratification of the lake/reservoir will therefore occur forsome vertical temperature distributions.

The specific heat for water is 4182 W/(kg 0Cs) The temperature changesin the water close to the surface can be computed from the energy bal-ance across the water surface. The sources/sinks of energy are:

- solar short-wave radiation- atmospheric longwave radiation- longwave black radiation from the water

- conduction- evaporation

The formula for the surface flux, I , in Watt/m 2, can be written accordingto Chapra (1997) and Henderson-Sellers (1984):

(7.3.1)

In the formula, Ir is the irradiance, B is a reduction factor and T is thetemperatur. The irradiance terms follow Stefan-Bolzmanns law, whereσ is the Stefan-Bolzman's constant 5.67x10 -8 W/(m2K4) and ε is theemissivity of water (0.97). The emissivity is a correction factor for thewater not being a perfect emitter of radiation. A is an empirical coefficient between 0.5 and 0.7 and e udarais the air vapour pressure. R is a reflection

0 999.872 999.974 1000.06 999.978 999.8810 999.7312 999.5214 999.2716 998.9718 998.6220 998.2322 997.8024 997.33

26 996.8128 996.2630 995.68

temperature flux

short-wave irradiance

long-wave irr. from atm.

long-wave irr. from water

conduction

penguapan

Aku

IrB

σ T udara

273+( )4 A 0.003 eudara

+( ) 1 R-( )

εσ T w

273+( )4-

0.136 c1

1 0.437 U 2

+( ) T w

T udara

-( )-

0.136 1 0.437 U 2

+( ) ew

eudara

-( )-

+

=

Page 87


coefficient, which usually is very small (around 0.03). The parameter c 1is Bowen's coefficient (62 Pa/ oC), U 2 is the wind speed 2 meters abovethe water surface and e w is the saturation vapour pressure at the water permukaan. The subscript w denotes water and air the air.

The short-wave irradiance is from the sun, and depends on severalfactors:

- Variation over the year- Variation over the day- Latitude of lake- Shading by clouds- Reflection from the water surface

All these parameters can be estimated and put into the factor B . Pertama

three factors can be taken from tables. The cloud shading is determined by the weather conditions. And the reflection from the water surfacedepends mostly of the solar altitude. If this is above 20 o, the reflection isless than 10 %.

The long-wave irradiance is black-body heat emission from the atmosphere and the water. The term contains attenuation from the atmosphereand reflection.

The conduction and convection term describes physical processes atthe water surface. The processes are similar to convection and diffusion,as described in Chapter 4. Eq. 7.3.1 gives an empirical formula for these proses.

Evaporation/condensation at the water surface affect the temperature,as the specific energy of a certain mass of water is different depending

Short-wave irradiance inten-sity is usually measured inμmol-photons/m2/s, orW/m2. The conversion between the units is:1 μmol photons/s = 0.3 W



on if it is in liquid or gas form. The term in Eq. 7.3.1 also gives an empiri-cal estimate of the amount of evaporation, as it is a function of the windkecepatan.

Limnological classifications

The temperature profile of a lake will vary over the year, depending onthe heat flux at the water surface. In different climates, there will be dif-ferent types of stratification. The science of limnology has given severaldefinitions to classification of lakes and stratification layers.

In a temperate climate with warm summers and cold winters, the vertical

temperature profiles are given in Fig. 7.3.1.

The upper layer close to the water surface is called epilimnion . Thelayer close to the bed is called hypolimnion . In deep lakes, the hypolim-

TT

Summer Musim dingin

Figure 7.3.1 . Vertical temper-ature profiles for a dimitic lakein the summer and the winter

Page 88


nion will hold a temperature of 4 oC throughout the year. The layer between the epilimnion and hypolimnion is called the metalimnion . Dur-ing the summer, the vertical temperature gradient is usually large in thislapisan. The thermocline is located in this layer, marking the difference between the warm upper water and the cold water close to the lake bot-

tom.In temperate climates, the winter is cold and the summer is warm. Thestratification will follow Fig. 7.3.1. In the summer, the warm water closeto the surface is lighter than the cold bottom water. In the winter, thewater close to the surface is below 4 oC , and is lighter than the bottomair. These two situations give a stable water body, and the stratifica-tion prevents mixing from taking place. However, during spring and fall,the water temperature will at some point in time be 4 oC over the wholedepth of the lake. Then vertical circulation may occur. The water fromthe bottom may rise to the surface, if wind-induced currents are present.This process is called a spring/fall overturn .

The science of limnology has also provided classifications of the lakesaccording to the overturns. If there is no overturn due to the surfacewater being too cold the whole year, the lake is called amictic . If onlyone overturn occur during the year, the lake is called monomictic . Inimay be due to the lake being so warm that no winter stratification isterbentuk. The lake is then called warm monomictic . If the lake is so cold

that no stable summer stratification occur, and only one overturn takes place in the summer, the lake is called cold monomictic . The cycledescribed in Fig. 7.3.1 with two overturns is present in dimictic lakes.There are also lakes with multiple overturns, called polymictic lakes.This is due to small changes in seasonal temperature and strong winds.

Turbulence damping

When horizontal water currents occur in a lake, there will be velocity gra-dients producing turbulence. In a stratified lake, the turbulent eddies will be dampened by the stratification. A formula for the damping of the turbulence is often given by the following formula for the turbulent diffusioncoefficient, Γ (Rodi, 1980):

(7.3.2)Γ Γ 1 β Ri+( )α=



Γ0 is the original turbulent diffusion, when not taking the s tratification into

account. Ri is the Richardson number, given by:

(7.3.3)

The formula is often used in numerical models (Olsen and Tesaker,

1995; Olsen et. al. 1999; Olsen and Lysne, 2000)

Various values of the constants a and b are used by different reserach-ers. Rodi (1980) recommends the values given by Munk and Anderson(1948):

α = -0.5 and β = 10.0, computing the diffusion for the velocity.α = -1.5 and β = 3.33, computing the diffusion for other variables.

0

Ri g ρ---

ρ∂

z ∂------

U ∂

z ∂------

2----------------=

Page 89 Numerical Modelling and Hydraulics 88

Olsen and Lysne (2000), however, found better correspondence withmeasurements when using the values:

α = -1.3 and β = 10.0, for all variables.

More advanced methods of taking the stratification into account whenmodelling turbulence is given by Rodi (1980).

7.4 Wind-induced circulation in stratified lakes

Water currents in a stratified lake will be influenced by the density varia-tion in two ways:

1. Vertical velocities will be dampened2. Turbulence will be dampened

Looking at a deep stratified lake in the summer, where wind-induced cur-rents occur, the circulation pattern given in Fig. 7.4.1 will emerge:

The slope of the water surface can be computed by looking at the forceson the water body, as given in Fig. 7.4.2:

The force from the wind is given as:

(7.4.1)

Wind

Thermocline

Figure 7.4.1. Wind-induced circulation in astratified lake. The arrowsshow the direction of thevelocity. Note this is a sta-tionary situation.

Figure 7.4.2. Forces on awater body. The hydrostatic pressure is denoted p , L isthe length of the reservoir, τis the shear stress and H isthe water depth. pl pr

τ

L H l

H r

F τ L=



The force from the hydrostatic pressure is:

(7.4.2)

We define the average water depth as:

(7.4.3)

F 1

2---ρ gH

r 2 1

2---ρ gH

r 2- ρ g

H r

H l

+( )

2----------------------- H

r H

l -( )= =

H H

r H

l +( )

2-----------------------=

Page 90


The surface slope is given as:

(7.4.4)

If Eq. 7.4.3 and Eq. 7.4.4 is inserted into Eq. 7.4.2, we obtain:

(7.4.5)

Setting this equal to the wind shear stress in Eq. 7.4.1, gives the follow-ing equation for the water surface slope:

(7.4.6)

The slope of the thermocline, I ', can be computed from looking at equilib-rium of forces in the horizontal direction on the hypolimnion (Fig. 7.4.3)

Aku H

r H

l -

L-----------------=

F ρ gHLI =

Akuτ

ρ gH ----------=

Figure 7.4.3. Forces on

the hypolimnion. The leftside is denoted L , and theright side denoted R . Thedensity difference betweenρ

1 and ρ 2 is denoted ρ'. The pressure at the bed isdenoted p . On the upper fig-ure, p is equal to ρ 1 gH,where H is the water depth.

Thermocline

ρ1

ρ2

p' L p1,L p1,R p' R

Thermocline

ρ1

ρ2

p' L (p1,R-p1,L ) p R'

hypolimnion

H

H '

Thermocline

ρ2

p' L (p1,R-p1,L ) p R'



The water below the thermocline is heavier than the water above. Thedensity difference is denoted ρ', and given by:

(7.4.7)ρ′ ρ2

ρ1-=

Page 91


It is assumed that the shear s tress between the epilimnion and thehypolimnion is negligible. The water flow direction above and below thethermocline is the same.

Looking at the forces on the epilimnion only, the situation will be similarto the derivation of the water surface slope. The difference is that thewater density is replaced by the density difference, ρ', and the wind forceis replaced by the pressure difference p R-p L. The force from the pres-sure difference becomes:

: (7.4.8)

The height from the bed to the thermocline is denoted H'.

The force from the density difference is the same as Eq. 7.4.5, only thedensity is replaced by the density difference, and the water depth isreplaced with H ':

(7.4.9)

Equilibrium of forces means the forces in Eq. 7.4.8 and 7.4.9 are thesama. Assuming H/H' can be approximated to be unity, the followingequation is obtained:

(7.4.10)

The negative sign comes from the fact that the thermocline slopes in theopposite direction of the lake water surface.

The density difference is much smaller than the water density itself, sothe slope of the thermocline is orders of magnitude larger than the watersurface slope. During strong winds it may happen that the s lope becomes so large that the cold water below the thermocline reaches thewater surface.

7.5 Seiches

Assuming we have a situation like given in Fig. 7.4.1, and the windspeed suddenly drops to zero, the thermocline and the water surface willstart to oscillate. Gambar. 7.5.1 shows the movement:

F p R

p L

-( ) H ρ1 gILH = =

F ρ′ gI ′ LH ′=

I ′ I –ρ

1ρ′-----=

Figure 7.5.1. Surface andinternal seiches in a strati-fied lake . The thermoclineis drawn with a line. The

slope of the water surface isexaggerated compared tothe slope of the thermocline.



Page 92


The movements of the water surface is called surface seiches . Themovement of the thermocline is called internal seiches .

Pada Gambar. 7.5.1, the thermocline is drawn with a straight line. This will not bethe case, as numerical and physical models shows that the water closeto the thermocline moves more like a wave, as shown in Fig. 7.5.2:

The movement of the internal seiche is associated with considerablehorizontal velocities at the thermocline.

7.6 River-induced circulation and Coriolis acceleration

A river feeding water into a lake or a reservoir will create water currentseven if there is no wind present. A particular case is an ice-covered lake,where this will be the dominant forcing mechanism for the circulation. Jikathe lake is stratified, the current may form “plume” inside the lake. Sebuahexample is given in Fig. 7.6.1, showing the velocity pattern close to thewater surface in Lake Sperillen in Norway during winter. The plume fol-

lows the right (top) side of the lake, due to the Coriolis acceleration.

Coriolis

The effect of the earth's rotation is most pronounced for large lakes with

stratifikasi. However, the effect may also be present in large non-strat-ified lakes.

Figure 7.5.2. Movement of the ther-mocline during an internal seiche .The upper figure shows the initial situa-tion. The middle figure shows the ther-mocline some time after the wind has berhenti. The lower figure shows the sit-uation later.

Speculations : Numerical experimentshas shown that under spe-cial conditions, an internalseiche may diffuseupwards to the water sur-

wajah. If this would be thecase, a sudden upwellingof water to the water sur-face would occur, withreasonably high horizontalvelocity. Local observa-tions of such phenomenain a lake with otherwisequiet water surface could be interpreted in imagina-tive ways...

Gambar. 7.6.1. Velocity vec-tors close to the surface inLake Sperillen, during thewinter. The lake is then cov-ered with ice. The river isflowing in from the right andout to the left.

Page 93




The Coriolis acceleration affect the water movement by the following for-mula:

T is the time it takes for one earth rotation, ie 24 hours or 86 400 sec., φis the latitude of the lake and U is the water velocity. In the northern hem-isphere, the acceleration will always be to the right. An example for a

lake in southern Norway, the latitude is 60 degrees, and f will be1.26x10 -4.

Slope of water surface

For a straight channel, the water surface slope would tilt s lightly due tothe Coriolis acceleration. The cross-directional slope, I c, would be equalto the ratio of the Coriolis acceleration to the gravity:

The velocity of the current is denoted U and f is the Coriolis factor. The bottom of the current will also get a cross-directional slope, I b. Looking atthe cross-directional balance of forces, similar to what was done for thethermocline, it is possible to derive a formula for I b:

As previously, ρ is the water density, and ρ' is the density difference between the water in the current and the water below.

7.7 Density currents

If the water flowing into the lake/reservoir has a different density than thelake water, a density current is formed. The density difference can bedue to:

1. Temperature variations2. Sediment concentration3. Content of salt in sea water

Combinations of the effects are also observed. If the temperature

causes the density of the inflowing water to be lower than in the lake, thecurrent will move close to the water surface. If the density is higher thanthe lake water, the current will move along the lake bed. In a stratifiedlake, it is possible that the current may move down into the water body toa temperature similar to the inflowing water.

A density current caused by high sediment concentrations is called aturbidity current . The turbidity current can transport sediment a longway into the reservoir, and even cause deposits in front of an intake of alarge reservoir. The process also redistributes sediments from the rivermouth to the deeper part of the lake/reservoir.

Sebuahf – U 4π

T ----- φsin U -= =

Akuc

fU

g -----=

Akub

ρ

ρ′----- I c ρ fU ρ′ g ----------= =

Page 94


DamWaduk

River Plunge pointFigure 7.7.1. Turbidity cur-rent entering a reservoir .The plunge point is oftenvisible at the water surface,as the river water contains



The sediment-laden river water often has a darker colour than thecleaner reservoir water. It is therefore sometimes possible to observethe plunge point at the water surface.

7.8 Intakes in stratified reservoirs

When water is abstracted from a linearly stratified reservoir, the situationgiven in Fig. 7.8.1 may occur.

Only the water in a layer at the same level as the intake is abstracted.The thickness, d , of the layer for a two-dimensional situation with a lineabstraction can be computed from:

(7.8.1)

where q is the water discharge/meter width, and k 1 is an empirical con-stant, between 3 and 5 (Steen and Stigebrandt, 1980). N is the Brunt-Väisälä frequency, given by:

(7.8.2)

If the water discharge is above a critical value, q c, then water will beabstracted from the whole depth. The following formula is used to find q

c(Carstens, 1997):

(7.8.3)

Intake

more sediments and has adifferent colour than thewater in the reservoir.

ρ' H d QU

Figure 7.8.1. Airabstraction from a strati-fied reservoir . The depth is

H , the discharge is Q and d is the height of the layer withabstracted water.

d k 1q

N ---=

N 2 g ρ---–

d ρdz -----=

qc

0.32 NH 2=

Page 95


Eq. 7.8.1 is derived theoretically from an idealized 2D situation. In prac-tice and intake will be limited in width, and 3D effects will be important.Steen and Stigebrandt (1980) developed formulas for a 3D situationwhere the abstraction gate had a significant size, of width B and heighth :

(7.8.4)

It was assumed that B was much smaller than the dam length and h wasmuch smaller than the dam height. The constant, k 3, was found byexperiments to be 0.74 (Carstens, 1997) for a gate close to the bed orthe water surface, and 1.2 for a gate midway between the water surface

d k 3Q 2

B2hN 2---------------

13---

=



and the bed (Carstens, 1997; Steen and Stigebrandt, 1980). Steen andStigebrandt (1980) also developed more complex formulas for the thick-ness of the abstraction layer for cases when the outlet size was not rela-tively small compared to the other dimensions.

7.9 Problems

Problem 1. Thermocline

A lake with depth 100 meters has a thermocline 7 meters below the

water surface. The temperature is 150C above the thermocline and 5

0C below it. A wind from the north lasts for five days. The wind speed is 15

m/s. The lake is 5 km. long in the north-south direction. Compute thedepth of the thermocline at the northern and southern side of the lake.

Problem 2. Water intake

Water is taken from a reservoir to a treatment plant to be used for munic-ipal water supply. The intake is located 10 meters below the water sur-face, with a width of 2 meters and a height of 1 meter. The discharge is 2m3/s. The temperature gradient of the reservoir is 0.5 0C/m, and the temperature at 10 meter is 11 degrees. A pollutant is spilled on the water permukaan. There is no wind at the time of spill, so on average for the wholelake, the pollutant is only mixed in the upper 1 meter layer. At the intake,water is abstracted from a layer with vertical magnitude Δh. Will this beso large that polluted water enters the intake?

Page 96


8. Water biology

8.1 Introduction

Water quality in lakes and reservoirs is often of interest for engineering

tujuan. An example is determination of location of water intakes, to prevent excessive pollutants in the inflowing water. Sometimes algaeaccumulate on one side of a lake. What is the reason for the variation inthe spatial distribution of the algal concentration? Another example is theassessment of the capacity of a lake/reservoir to receive waste water.

The dispersion of the various components was discussed in Chapter 4and 5, predicting the variation of the concentration as a function of turbu-lence, water currents etc. The variation of a component can be computed solving the convection-diffusion equation of its concentration.However, the concentration of a component is also a function of bio-reaksi kimia. However, biochemical reactions can also affect thevariation of a variable. A toxic substance may change as a function oftime due to for example sunlight. Or oxygen may be consumed by bacte-ria or generated by algae. These biochemical reactions are discussed inthe following chapters.



8.2 Biochemical reactions

Stochiometry

Stochiometry is used to quantify the different components in a biologicalor chemical reaction. An example is the photosynthesis, where carbondioxide and water are used to produce organic material and oxygen:

(8.2.1)

The balanced equation can be written:

(8.2.2)

The molecular weight of carbon is 12 g/mol and oxygen is 16 g/mol. Theweight of a carbon dioxide molecule is then 44 g/mol. The weight of theoxygen molecule is 32 g/mol. So for example 44 grams of carbon dioxidewill produce 32 grams of oxygen.

Reaction kinetics

Reaction kinetics is used to describe how fast a biochemical processtakes place. There are a number of different rate laws describing the processes, classified by the order of the kinetics:

Zero-order kinetics:

atau (8.2.3)

First-order kinetics:

Some atomic weights inunits grams/mol:

Hydrogen: 1Carbon: 12 Nitrogen: 14Oxygen: 16 CO

2 H

2O+ C

6 H

12O

6O

2+→

6 CO2

6 H 2O+ C

6 H

12O

66 O

2+→

dcdt ------ k –= c c

0kt -=

The reaction coefficients, k ,are often given in units

1/day. Remember to convertthis to 1/seconds, whenusing computer programswhere this time step is used.

Page 97


atau (8.2.4)

Second-order kinetics:

atau (8.2.5)

The equations describe a situation with only one variable. For somecases there are multiple water quality constituents. The change in onevariable may be a function of the concentration of other variables. Bagaimanapunever, often there is a limiting variable determining which process is tak-ing place. The reaction is then dependent on this variable.

Temperature dependency

Most biochemical processes depend on temperature. For example, algalgrowth will increase significantly when a lake is heated by the sun duringmusim panas. The following formula is often use to estimate the growthincrease/decrease as a function of temperature, T :

(8.2.6)

The reaction rate at 20 oC, k 20, is often used as a basis. The parameterθ is specific for the reaction. Typical values are slightly above unity.Some examples are given in Table 8.7.1. The processes in the table arefurther described in the following chapters.

Discretization

dc

dt ------ kc-= c c0e

kt -=

dc

dt ------ kc 2-= c c

01

1 kc 0t +-------------------=

k k 20

θ 20 T -( )=



Biochemical reactions are included in the convection-diffusion equationwhen computing the dispersion of a water quality parameter. ini adalahdone by including the left side of Eq. 8.2.3-5 in the source term. If first-order kinetics is used, the equation can be written:

(8.2.7)

When the equation is discretized, it becomes:

(8.2.8)

V p is the volume of the cell. The last term on the right side is due to the biochemical reactions. Note that this term is negative, so that it is in prin-ciple possible to get a negative concentration. This can lead to instabili-ties in the solution and unphysical results. To avoid this, the same procedure as in Chapter 5.7 can be used. The equation is then written:

(8.2.9)

In this way, negative concentrations will be avoided.

The time term can also be added, similarly as described in Chapter 5.7.

U i

∂ c

∂ xi

-------∂

∂ xi

------- Γ∂ c

∂ xi

------ kc-=

Sebuah p

c p

Sebuahw

cw

Sebuahe

ce

Sebuahn

cn

Sebuah s

c s

kc p

V p-+ + +=

Sebuah p

kV p

+( ) c p

Sebuahw

cw

Sebuahe

ce

Sebuahn

cn

Sebuah s

c s

+ + +=

Page 98


8.3 Toxic compounds

Toxic compounds are chemical substances not naturally occurring in theriver/lake water, causing a hazard to humans and animals drinking theair.

Individual types of toxics will not be described here, and the dangerousconcentrations need to be determined for each component. But a few ofthe processes affecting the toxic concentration are discussed.

Sorption

Toxic substances often attach to organic and inorganic sediments, suspended in the water. When the sediments settle, some of the toxic sub-stance will be removed. The total concentration of toxics, c , in a water body is therefore the sum of the dissolved toxic concentration, c d , Danthe concentration attached to particles, c p. The fractions of the two components as a function of the total concentration is given by:

(8.3.1)

(8.3.2)

The index d denotes the dissolved fraction and p denotes the fractionattached to the suspended particles. The particle concentration isdenoted c s. The partition coefficient, K d , may range between 0.0001 and1000. The value depends on the type of toxic and the composition of thesuspended particles. There exist empirical formulas for K d for sometypes of toxics/sediments. Otherwise, the coefficient must be determined by a laboratory analysis.

Photolysis

Sunlight may cause toxic chemicals to undergo a transformation to othersenyawa. The decrease in concentration is given by first-order kinet-ics (Eq. 8.2.4). The reaction rate, k , is a function of the specific chemical

F d

cd

c-----

1

1 K d c s

+--------------------= =

F p

c pc-----

K d c s

1 K d c s

+--------------------= =

In the spring of 1995 therewas a large flood in the riv-ers flowing into Lake Mjøsain Norway. The flood caused

extensive damage to infra-struktur. It was also fearedit could lead to decreasedquality of the water in thelake, which was used forwater supply to the main cit-ies in the area. This did nothappen, as much of the pol-luting compounds sorbed tosediments and deposited atthe bottom of the lake.



pand the sunlight. There exist tables for k p at the water surface. But as thelight penetrates the water body, it will be dampened. This process must be taken into account when evaluating the effect of the photolysis. Thecomputation of the irradiance damping must also take into account thateach photolysis reaction requires light at a specific wavelength. Thedamping computation must be specific for the particular light frequency.

Hydrolysis

Hydrolysis is a transformation of the toxic to other components, usually by acid or bases in the water. The reaction can be computed by first-

order kinetics, and the reaction rate, k p, is usually in the order of 10 -7 untuk0.1/day. It will be a function of the pH of the water and the chemical composition of the toxic.

Biodegradation

In biodegradation processes, toxic substances are reduced to othercompounds by organic material. Usually, different types of bacteria areinvolved in the process.

Translation to Norwegian:

Biodegradation: Biologisk nedbrytning Partition coefficient:

Partisjonskoeffisient Reaeration: Lufting Sorption: Sorpsjon

Page 99 Numerical Modelling and Hydraulics 98

8.4 Limnological classifications

Depending on the nutrition inflow to a lake, there may be more or lessorganic material and processes. A lake with little nutrients and organicmaterial is called oligotrophic . If the nutrition inflow is high and largeamount of organic material is present in the lake, it is called eutrophic .The state in between oligotrophic and eutrophic is called mesotrophic .A quantitative distinction between the types are often related to the phosphorous concentration. The mesotropic lake has a concentration between 10 and 30 mg/l. Lower phosphorous concentrations are foundin oligotrophic lakes, and higher concentrations in euthropic lakes.

The water body of a lake can be divided in several zones, depending on

the biology of the lake. The littoral zone is the shallow part of the lakeclose to land, where plants may grow. Classification also depends on theamount of light penetrating the water body. Since the algae requires lightfor photosynthesis, the production of organic components only take partin the upper zone. This is called the euphotic zone . The aphotic zoneis below. The light penetration is here so small that photosynthesis is not possible, and only decomposition processes take place.

Water quality is defined from the concentration of various components inair. For example, the water should have above a certain amountof oxygen, and the concentration of toxic components should be below athreshold value. The component is then a water quality indicator . Cer-tain species of algae has also been used for this purpose. The waterengineer is faced with the question of determining the concentration ofthe various water quality constituents.

More details on the typical processes and water quality parameters in alake is given in the next chapter.

8.5 The nutrient cycle

Most computer programs for water quality models the nutrient cycle in ariver/lake using oxygen, phosphorous, nitrogen and algae/bacteria. Gambar.8.5.1 shows a schematic view of the components involved:

Algae Oxygen

CBOD

Settling

Settling

Udara



Org. P Diss. P Org. N NH 3 TIDAK2 TIDAK3

Figure 8.5.1. A model of the components in a river/lake

Page 100


P is phosphorous, N is nitrogen and the oxygen is dissolved in the water.

Oxygen processes

Oxygen dissolved in the water is seen as a main indicator of water qual-ity. The oxygen concentration is often measured as a fraction of the sat-uration concentration, or a saturation concentration deficit. Clean waterwill usually have high oxygen concentrations, or close to zero concentra-tion deficit.

One of the most important processes for life is the photosynthesis . The process transforms carbon dioxide and water to organic substances andoxygen. In water, the process takes place in chlorophyll of the algae.Algal processes will be further discussed in Chapter 8.7.

Oxygen is used by plants and bacteria in respiration and consumption ofnutrients. Usually, there are many types of plants and nutrients, with dif-ferent types processes taking place. It is therefore difficult to derive ana-lytical formulas for the processes. Instead, an empirical approach isdigunakan. A sample is taken from the water, and the oxygen consumption ismeasured in the laboratory as a Biochemical Oxygen Demand (BOD) .The BOD will be dependent on many factors, and empirical coefficients

are required when modelling the process. Usually, the oxygen demandis from organic material. Since this is based on carbon, the term CBODis also used. BOD for Nitrogen processes is termed NBOD , and this will be discussed later.

The consumption of nutrients by bacteria/algae can be described byfirst-order kinetics:

(8.5.1)

L is the concentration of nutrients, and k d is an empirical reaction coeffi-sien.

If dissolved oxygen is consumed completely from the water, anaerobicconditions occur. Then new processes with different types of bacteriawill take place. Such processes may create toxic substances and fishdeaths. It is therefore important that an oxygen concentration above zerois maintained in rivers and reservoirs/lakes.

The bacteria/algae consuming the oxygen can be floating in the water, orit can be attached to sediments at the river/lake bed. In the last case, theterm Sediment Oxygen Demand (SOD) is often used instead of BOD.Chapra (1997) developed relatively complex algorithms to computeSOD.

The oxygen in the air above the water surface will be able to replenishthe river/lake water if the oxygen concentration, o, is below the satura-tion concentration os. A large number of formulas have been made toquantitatively determine the reaeration . The general formula is given as:

(8.5.2)

Note that this equation isindependent of the con-centration of bacteria/algae.

dL

dt ------ k

d L-=

melakukan

dt ------ k

Sebuaho s

o-( )=



The oxygen saturation concentration is denoted o s. This is the maximumoxygen concentration in the water. If the concentration goes above thisvalue, gas bubbles are formed, and the oxygen disappears to the atmosphere. The oxygen saturation concentration depends on:

Page 101


- water temperature- salinity- altitude

The value of the oxygen saturation concentration will be in the order of10 mg/liter (Chapra, 1997). Often, the oxygen deficit, D , is modelledinstead, where D = o s - o .

The reaeration coefficient, k Sebuahdepends on the oxygen flux through thewater surface film. It is also influenced by the oxygen mixing by turbu-lence below the surface. The present empirical formulas contain indirect parameters of the turbulent mixing, for example the water velocity. A for-mula for the reaeration coefficient for a river is given by O'Connor andDobbins (1958):

(8.5.3)

The formula is not dimensionless, and U is given in m/s and h is thewater depth in meters. The coefficient k Sebuahis given in units day -1.

For lakes, an empirical formula for k Sebuah, has been developed by Broeckeret al. (1978).

(8.5.4)

U w is the wind speed 10 meters above the water surface in m/s.

There exist a large number of other empirical formulas for the reaeration proses.

Nitrogen

The source of nitrogen in the water is usually organic material. Thematerial decomposes in an ammonification process, leading to forma-tion of ammonium ( NH

4+). The ammonium can react with dissolved oxy-

gen in a nitrification process to form nitrite ( NO 2-). The nitritification

process can further transform the nitrite to nitrate ( NO 3-). Since the nitri-

ficaion process uses oxygen, the process has a nitrogen oxygendemand ( NBOD ). If not, or only small amounts of dissolved oxygen is present, a denitrification process can take place. The nitrate is thentransformed to nitrite and to nitrogen in gaseous form. The nitrogen gasmay be lost to the atmosphere.

There are several implications of the processes on the water quality:

1. If nitrogen is the limiting nutrient for organic growth, abundant nitrogenmay cause increased eutrophication.

2. High concentrations of nitrate (above 10mgN/l) in drinking waters cancause disease for very young children.

3. Dissolved ammonia may form ammonia gas (NH 3), which may be beracun untuk ikan. This only happens at high temperatures and high pH in the

The development of CFDmodels enables the estima-tion of turbulence close tothe water surface. Masa depanreaeration formulas will probably be based directlyon these turbulence param-eters: the turbulent kineticenergy or the eddy-viscos-

ity.

k Sebuah

3.93U 0,5

h1,5----------=

k Sebuah

0.864 U w

h--------------------=

NH4+= ammonium

NH3= ammonia



Page 102


air. (above 20 0 C and pH above 9).

The nitrogen processes can be modelled by the equations given below,using the following subscripts: o =organic, a =ammonium, i =nitrite andn =nitrate.

(8.5.5)

(8.5.6)

(8.5.7)

(8.5.8)

Note that the stochiometry coefficients are not included. The equationscan be solved analytically for a simple system of one well-mixed lake/

reservoir. Often, an equation for dissolved oxygen is solved simultane-ously.

Fosfor

Phosphorous is usually the limiting nutrient for plant growth in freshair. The growth process is then only a function of the phosphorouskonsentrasi. The phosphorous may be dissolved in the water or itmay be present in organic material . The organic material consume thedissolved phosphorous and then settle to the bottom of the lake/river.These processes will decrease the concentration of dissolved phospho-rous in the water. However, when the organic material decomposes, the phosphorous may again be released and dissolved in the water.

Formulas for depletion of dissolved phosphorous is derived from growthformula for biological material. Stochiometry is used to determine howmuch the phosphorous concentration is reduced as a function of theorganic growth.

(8.5.9)

(8.5.10)

P organik is the organic phosphorous in the algae, PO 4 is the dissolved phosphorous. The decomposition rate of organic phosphorous isdenoted k op, and k is the algae growth factor. The algae concentration isdenoted c SEBUAH , and the fraction of phosphorous in the algae is α.

In many lakes there is relatively little phosphorous occurring naturally,leading to small concentrations of organic material. Over the last dec-ades there has been an increase in phosphorous inflow into many lakesin Europe. Often, fertilization from agriculture is the cause. Theincreased phosphorous loading leads to accelerated growth of organicmateri. This eutrophication has negative effects on the water quality.

td

d N

o() k

o Sebuah→ N o

-=

td

d N

Sebuah() k

o Sebuah→ N o

k Sebuahi→ N

Sebuah-=

td

d N

i() k

Sebuahi→ N Sebuah

k i n→ N

i-=

td

d N

n() k

i n→ N i

=

td d P

organik ( ) k op

P organik

-=

td

d PO 4( ) k

op P

organik α kc

SEBUAH -=

Page 103




8.6 QUAL2E

The QUAL2E program is made by the US Environmental ProtectionAgency. It is designed to model water quality in rivers in one dimension.The hydraulic computation is based on a steady calculation. Con yangvection-diffusion equation is solved for a number of water quality param-eter. Biological reactions are included in the equations, together withinteraction between the parameters.

The program models a number of water quality variables: Temperatureis modelled with surface fluxes as given in Chapter 7.3. Algae is mod-elled, including growth, predation, settling and scour. The cycles of nitro-gen, phosphorous, carbonaceous BOD and oxygen are modelled,according to the models described in Chapter 8.5.

The program is freeware, and can be downloaded from the Internet,including user's manuals. A Windows user interface is included. Meja8.6.1 gives default values of important input parameters:

Table 8.6.1 Parameters used in QUAL2E

8.7 Phytoplankton

The most important types of plankton for water quality is free-flowingalgae. Algae are plants with one or more cells, living in water. Duagroups of algae exist:

- periphyton : algae attached to the river/lake bed- phytoplankton : free flowing algae

The most common species of phytoplankton in freshwater can be classi-fied in three main groups:

- Cyanobacteria- Flagellates

ProsesPersamaan jumlah

Reaksirate name

Kegagalanreaction

rate (day-

1)

Max-min.reaction

rate (day -1)

Suhucoefficient θ

Algal growth 8.7.7 k 2,5 1.0-3.0 1.047Algal respiration rate 0,005 0.005-0.5 1.047

BOD decay 8.5.1 k d 0.0 0.0-10.0 1.047

Organic nitrogen decay 8.5.5+6 k NO>NH4 0.0 0.0-10.0 1.047

Organic nitrogen settling 8.5.5 σ4 0.0 0.0-10.0 1.024

Ammonia oxidation 8.5.6+7 k NH4>NO2 0.0 0.0-10.0 1.083

Nitrite oxidation 8.5.7+8 k NO2>NO3 2.0 0.0-10.0 1.047

Organic phosphorous decay 8.5.9+10 k op 0.0 0.0-10.0 1.047

Page 104


- Diatoms

Cyanobacteria often have gas vesicles, variable in size. The buoyancyof the algae can thereby be changed and vertical movement take place.The main process in the algae is photosynthesis, and an appropriateamount of light is necessary. If too little light is present, the algae will



move toward the water surface. And if the light is too strong the algaewill want to move downwards, where the turbidity of the water causedecreased light intensity. The change in phytoplankton concentrationmay be due to the rise/fall velocity together with wind-induced currentsand turbulence. Another process is algal growth and predation by zoo plankton.

Rise/sink velocity

Kromkamp and Walsby (1990) developed formulas for cyanobacteria buoyancy based on laboratory experiments:

(8.7.1)

ρSebuahis the algae density, Δ t is the time step and k 1 , k 2 , k 3 and K are con-

stants. I 24 is the average irradiance over the last 24 hours. I is the irradi-ance, with maximum value at the water surface, and decreasing valuesdownward in the water body, as shading occur. Bindloss (1976) investi-gated the damping of the irradiance for lakes in the UK, and found thefollowing relationship to compute the specific light transmission coeffi-cient, k l :

(8.7.2)

where c is the algal concentration. The irradiance is dampened by a fac-tor f , given by the following formula:

(8.7.3)

The summation is over all layers with magnitude Δ z from the surfacedown to the level y .

The fall/rise velocity, w , of the algae is calculated from Stoke's equation:

(8.7.4)

where ρ Sebuahand ρ w is the algal and water density, and ν is the kinematicwater viscosity, evaluated as:

(8.7.5)

where T is the temperature in degrees Centergrade.

The flagellates also seek optimum light intensity. Instead of changingtheir buoyancy, the flagellates have flagelles, enabling movement. Iniis similar to fins on a fish, but much lower velocities are produced. Thefollowing formula is often used for flagellates:

ρa 1,

ρa 0,

Δ t k 1

Aku IK +------------ k

2 Aku

24- k

3-+=

The formula given by Krom-

kamp and Walsby wasdeveloped from data usingan algae from the lakeGjersjøen in Norway

k l

0.0086 c 0.69+=

f e

k

l

Δ z

y=

w d 2 g

ρw

ρSebuah

-

18ρw ν-----------------=

ν 10 6- e0.55234 0.026668 T -=

Page 105


(8.7.6)

I is the actual irradiance, I memilihis the optimum irradiance, I ref is a referenceirradiance and w max is the maximum rise/sink velocity for the flagellate.

The maximum rise/sink velocity of phytoplankton are in the order of onemeter/hour.

The Diatoms have a specific weight slightly higher than water, giving aconstant fall velocity. The density can not be changed, and the diatomsdo not have flagelles. The only way to move diatoms upwards is by tur-

w wmax

Akumemilih

I –

Akuref

----------------=



bulence.

The size of the phytoplankton are often in the order of micrometer. SEBUAHmicroscope is necessary to identify the different species. Certain typesof cyanobacteria - Microcystis - form groups or colonies. The rise/sinkvelocity is a function of the algae buoyancy and the group diameter. SEBUAHlarger group will thereby get a much higher fall/rise velocity, givingincreased efficiency in search for optimum light. Groups of up to 2 mm insize have been observed.

Pertumbuhan

Time series of phytoplankton concentrations in a lake show variationsover the year. Also, the type of algae changes. Diatoms may be domi-nant in the spring, followed by cyanobacteria in the summer. This alldepends on temperature, nutrients, shading, grazing by zoo planktondan lain-lain

The algal growth can be computed by use of first-order kinetics:

(8.7.7)

The growth rate coefficient, k , may be around 1.0/day for optimum condi-tions. The biomass will then increase by 200 % in one day, and by a fac-tor 1000 in one week. Usually, optimum conditions do not exist, as thealgae need several nutrients to achieve maximum growth. The limitingnutrient for Cyanobacteria and Flagellates in freshwater is most often phosphorous. Silica is often the limiting nutrient for Diatoms.

If sufficient amount of nutrients are present, the growth can be limited by

light. Reynolds (1984) found the following formula to estimate the growthcoefficient in lakes in the UK:

(8.7.8)

The damping factor, f , is given by Eq. 8.4.3, and a and b are constants(0.4 and 9.0) as found by Reynolds (1984).

Modelling of phytoplankton

Phytoplankton can be computed in models having from zero to threespatial dimensions. A zero-dimensional model assumes complete mixingin the lake/reservoir, and predicts algae concentration over time. One-dimensional models are used for rivers and sometimes for lakes. Kemudian

c c0ekt =

k a 1 e

f b----

-=

Page 106


horizontal layers are often used, and complete mixing within each layeris assumed. Three-dimensional CFD models have been used to predictspatial variation of algae in lakes (Hedger et. al. 2000) and reservoirs(Olsen et. al. 2000). The CFD model predicts velocities in all spatialdirections on a three-dimensional grid. Wind-induced circulation can bemodelled, together with effects of inflowing/outflowing water. Con yangvection-diffusion equation for algae concentration can be solved includ-

ing sink/source terms for algal growth and settling/rise velocities. Nutrients, light and temperature can be modelled simultaneously in time-dependent calculations, enabling modelling of most of the important processes affecting the algae.

8.8 Problems

Problem 1. Completely mixed lake:

A lake with volume 1 million m 3 has received phosphorous from a sew-age plant for several years, with an amount of 2000 kg/day. Assume aloss rate of 0.1/day, and compute the average phosphorous concentra-tion in the lake.



One day an improvement of the sewage treatment plant was made,causing the loading to decrease to 300 kg/day. How long time will it take before the phosphorus concentration is below 20 % of the original value?

Problem 2. The Streeter-Phelps equations

In 1925, a study of the water quality in the Ohio river in the USA was published by HW Streeter and Earle B. Phelps. Their paper was alandmark for water quality modelling. Using first-order kinetics, theymodelled BOD and the oxygen deficit (DO) in the water as a function ofsewage released to the river. The following formulas were used:

where D is the oxygen deficit: o s-o. The concentration of BOD is denoted L , k d is a reaction coefficient for consumption of oxygen and k Sebuahis a reai-ration coefficient. The parameter r oc is the stochiometry coefficients forhow much oxygen is consumed for each unit of BOD. Assume a valueof 0.1 for the current problem.

Solve the equations using a spreadsheet for a 100 km long river with awater discharge of 100 m 3/s, where the upstream sewage outflow is 30kg/s. Present longitudinal profiles of BOD and oxygen concentration fora steady situation. Assume k d = 0.8 day -1. The river is 2 meters deepand 200 meters wide.

Compare the results with the analytical solutions of the equations:

dLdt ------ k d L-=

dD

dt ------- k –

Sebuah D r

ock

d L+=

L L0e

k d U ---- x-

=

Page 107

Numerical Modelling and Hydraulics106

Problem 3. The nitrogen cycle

Use a spreadsheet to compute the variation over time of various forms ofnitrogen in a well-mixed lake, according to Eq. 8.5.5-8.5.8. Use the reac-tion rates and initial concentrations given in Table 8.6.1.

Table 8.8.1 Parameters for problem 3

Problem 4. Stratified lake

Proses Reaksirate name

Reaksi

rate (day-1)

Variabel Awalconc.

Organic nitrogen decay k NO>NH4 0,15 Organic N 0,008

Ammonium oxidation k NH4>NO2 0,15 Nitrite 0.0

Nitrite oxidation k NO2>NO3 0,5 Nitriate 0,008

Ammonium 0.0

D D 0e

k SebuahU --- x- c

0k

d k

Sebuahk

d -

--------------- e

k d U --- x-

e

k SebuahU --- x-

-+=



A 22 meter deep lake is thermally stratified, with a 1 meter thick ther-mocline 7-8 meters below the water surface. It is assumed that the wateron both sides of the thermocline is well-mixed, and the mixing throughthe thermocline is according to a diffusion coefficient, Γ, equal to 1.3x10 -2 m2/s.

Initially, the lake is fully saturated with oxygen. Assume an oxygen con-centration of 9 mg/l. It is assumed that the wind mixes oxygen into thewater, so the water above the thermocline stays saturated with oxygen.A the lake bottom, there is a sediment oxygen demand of 0.3 (m 2day) -1.What is the concentration of oxygen below the thermocline after a long

time?

Page 108


9. Sediment transport

9.1 Introduction

Sediments are small particles, like sand, gravel, clay and silt. The waterin a river has a natural capacity of transporting sediments, given thevelocity, depth, sediment characteristics etc. Man-made structures in ariver may change the sediment transport capacity over a longer part ofthe river, or locally. Erosion may take place in connection with structures,such as bridges, flood protection works etc. The hydraulic engineer hasto be able to assess potential scour problems. During a flood, the risk forerosion damages is at its highest.

Sediments cause many problems when constructing hydropower plantsand irrigation projects in tropical countries. Deposition and filling of res-ervoirs is one problem, and the water intake has to be designed for han-dling the sediments. The sediments reaching the water turbine maycause wear on the components, as shown in the picture below.

The picture shows erosion onhydraulic machinery: the

blades leading the water towards the turbine. Photo:

N. Olsen.



In recent years, the topic of polluted sediments has received increased bunga. Organic and toxic substances may attach to inorganic sediment particles and affect the water quality. Erosion of old polluted sedimentsmay create a hazard. Sediments also affects the natural biochemistry ofshallow lakes.

The origin of sediments vary according to different climates. In tropicalcountries, rock decompose naturally. Water in form of rain and tempera-ture fluctuations, together with chemical reactions cause cracks in therock. The weathering cause a layer of particles to be formed above thesolid rock. The particles close to the surface have the longest exposure

to weathering, and have the smallest grain sizes. Larger stones areformed closer to the bedrock. Over time, the finer particles are removed by rain, causing erosion of the rock. The process is relatively slow, asvegetation cover prevents erosion from taking place. Where manremoves the vegetation, there may be accelerated erosion.

The sediments are transported by the rivers and streams through thecatchment. The annual sediment transport in a river, divided by its catch-ment area is called sediment yield. Typical numbers for tropical coun-

Page 109


The picture on theright is taken from avolcano in Costa

Rica. The fine mate-rial of the volcano lieson steep slopes. inirelatively unprotected by vegetation. Therain fall then causesrelatively extremeerosion rates. Seperti ituareas often have gul-lies, as shown on the

gambar. Photo: N.Olsen.



tries are around 100-2000 tons/km 2/year.

In some countries, like Norway, the geology is very much influenced bythe ice-age. The glaciers removed the upper layer with soil, leaving solid bedrock for a large part of many catchments. The sediment yield istherefore much lower, almost zero many places. The main sedimentsource in Norway is from the glaciers.

Sediment yield

Page 110


9.2 Erosion

The initial step to the science of sediment transport in a river is looking atforces on a sediment particle resting on the bed. The purpose is to find amethod for determining when the particle will be eroded. There are fourforces influencing the stability of the particle (Fig. 9.2.1) resting on a bedwhere the water has a velocity U .

- Gravity: G- Drag: D- Lift: L- Friction: F

Assuming the particle has diameter d , the forces can be written:

(9.2.1)

(9.2.2)

(9.2.3)

The friction force, F , is a function of the force pushing the particle down-wards, multiplied with a friction coefficient. This friction coefficient is thesame as tangens to the angle of repose of the material, α.

(9.2.4)

Some constants are used: k 1 for the shape factor of the particle, k 2 adalahdrag coefficient and C L is a lift coefficient. The critical shear s tress on the bed for movement of a particle is denoted τ c.

Equilibrium of forces along the direction of the bed gives:

(9.2.5)

Using Eq. 9.2.4 and 9.2.2, this gives

(9.2.6)

The equation is solved with respect to the particle diameter:

(9.2.7)

U

L

D

G

F

Gambar. 9.2.1 Forces on a

particle in a stream

The density of a sediment particle is often set to 2.65times the water density

G k 1

ρ s

ρw

-( ) gd 3=

D k 2u2d 2 k 2 IM 2r

43---

d 2 k 2

τ

ρ gr -------- M 2r

43---

d 2 k 3τ d 2≈= = =

L12--- C

L

ρw

d 24----π u2 1

2--- C

L

ρw

d 24----- IM 2r

43---π k 4

τ d 2≈= =

F GL-( ) α() berjemur α() k 1 g ρ

sρ

w-( ) d 3 k

4τ d 2-[ ] berjemur= =

F D=

α() k 1 g ρ

sρ

w-( ) d 3 k

4τ d 2-[ ] berjemur k

3τ d 2=

d τc

g ρ s

ρw

-( )k

1α() berjemur

k 3 k 4α() berjemur+

---------------------------------

-------------------------------------------------- ---------------τ

c g ρ

sρ

w-( )τ*-------------------------------= =



The parameter, τ*, was found experimentally by Shields (1937), and can be taken from Fig. 9.2.2:

Page 111


The value on the horizontal axis is the particle Reynolds number, givenoleh:

(9.2.8)

The viscosity of water is denoted ν, d is the particle diameter, u * adalahshear velocity and τ is the shear stress on the bed.

Shields graph can be used in two ways: If the bed shear stress, τ, in ariver is known, Eq. 9.2.7 can be used to determine the stone size that willnot be eroded. Or if the particle size on the bed is known, Eq. 9.2.7 can be used to compute the critical shear stress for movement of this parti-cle. In both cases, Eq. 9.2.7 is used, where the parameter τ* is found byGambar. 9.2.2.

If the particle is very small, under 0.1 mm, there are also electrochemicalforces occurring. The sediments are then said to be cohesive. The criti-cal shear stress then depend on the chemical composition of the sedi-ments and the water.

Example: A channel with water depth 2 meters and a slope of 1/1000 iscovered with stones of size 0.03 m. Will the stones be eroded or not?

First, the bed shear is computed:

Then the particle Reynolds number is computed.

The Shields diagram gives the Shields coefficient as 0.06. The critical shear stress for the particle is then:

Gambar. 9.2.2 Shields graph,giving the critical shearstress for movement of asediment particle

τ*=τ

g ρ s

ρw

-( ) d ---------------------------- τ *

R *u

*d

ν--------

d τ

ρ---

ν-----------= =

Cohesive sedimentsunder 0.1 mm

τ ρ ghI 1000 x 9.81 x 2 x1

1000------------20 Pa= = =

R *u

*d

ν--------

d τ

ρ---

ν-----------

0.0320

1000------------

10 6----------------------------4243= = = =

τc=τ * g ρ

sρ

w-( ) d 0.06 x 9.81 x 2650 1000-( ) x 0.03 29 Pa= =



Page 112


We see that the critical shear stress for the particle is above the actual shear stress on the bed. The particle will therefore not be eroded.

Sloping bed

The decrease, K , in critical shear s tress for the sediment particles as afunction of the sloping bed was given by Brooks (1963):

(9.2.9)

The angle between the flow direction and the channel direction isdenoted α. The slope angle is denoted φ and θ is a slope parameter. Thefactor K is calculated and multiplied with the critical shear stress for ahorizontal surface to give the effective critical shear stress for a sediment particle.

Looking at the bank of a straight channel, where the water velocity isaligned with the channel direction, α is zero. Eq. 9.2.9 is then simplifieduntuk:

(9.2.10)

The slope parameter, θ is slightly higher than the angle of repose for thematerial (Lysne, 1969). A value of 50 degrees was used by Olsen andKjellesvig (1999) computing bed movements in a sand trap.

More recently, Dey (2003) developed another formula for K:

(9.2.11)

The angles φ and α are here not defined in the same way as Brooks. Theangle α is the bed slope normal to the direction of the velocity vector.While the angle φ is the bed slope in the direction of the velocity vector.Bihs and Olsen (2011) obtained fairly good results using this formula tocompute local scour around an abutement in a channel.

K φ αsinsin

θ berjemur------------------------

φ αsinsinθ berjemur

----------------------2

cos 2φ 1φ berjemurθ berjemur

-----------2

-++=Critical shear for slopingbank

α

U

Plan viewCross-section

φ

Gambar. 9.2.3. Plan view (left) and cross-section (right) for explana-tion of the angles α and φ in the formula for the decrease of thecritical shear stress on the bed.

K φ 1 φ berjemurθ berjemur

----------2-cos=

K 0.954 1φ

θ----0.745

1α

θ----0.372

=

A very good handbook fordesign of scour protection

works in Norwegian is writ-ten by Fergus et. al. (2010).

Page 113




9.3 Suspended sediments and bed load

When the bed shear stress exceeds the critical value for the bed parti-cles, there will be a sediment transport in the river. The particles will rollalong the bed or jump up into the flow. The latter process is called salta-tion. The length of the jump will depend on the fall velocity of the parti-cles. Gambar. 9.3.1 gives the fall velocity for quartz spheres at 20 oC.

The sediments will move close to the bed or in suspension, dependingon the particle size and the turbulence in the water. The Hunter Rouse parameter (Eq. 9.3.1) is often used to determine the vertical distributionof the sediment concentration profile:

(9.3.1)

The fall velocity of the particles is denoted w , κ is a constant equal to 0.4and u * is the shear velocity. High values of z indicates the fall velocity ishigh compared with the turbulence. The sediments will then move closeto the bed. Low values of z indicates high amount of turbulence compared with the fall velocity, and the distribution becomes more uniform.Hunter Rouse also developed formulas for the vertical distribution of theconcentration, c(y) :

(9.3.2)

The water depth is denoted h , y is the distance from the bed and a is thedistance from the bed where the reference concentration, c Sebuah, is taken.Often a is set to 0.05 h . The vertical distribution of sediment concentra-tion for some values of z is given in Fig. 9.3.2, by using Eq. 9.3.2.

Sediment transport capacity

The river will have a certain sediment transport capacity, given itshydraulic characteristics and the sediment particle size. Supplying moresediments than the transport capacity leads to sedimentation, even if thecritical shear stress for the particles is exceeded. Less available sedi-ment than the transport capacity leads to erosion.

Gambar. 9.3.1 Fall velocity of quartz spheres in water.The horizontal axis is thediameter of the spheres,and the vertical axis is thefall velocity

cm / s

mm

z w

κ u*

---------=

cy ( )c

Sebuah

---------hy-

y-----------

Sebuah

ha-------------

z =

Figure 9.3.2. The verticaldistribution of sediment con-centration for some chosenvalues of z

Page 114


Initial studies of sediment transport was done by Bagnold (1973), lookingas sand transported by wind in the desert. Bagnold divided the transportinto two modes: Bed load and suspended load. The bed load rolledalong the ground, and the suspended load was transported in the air.For some reason, the same approach was used in water. The problem isthat there is no clear definition of the difference between the two trans-

Ralph A. Bagnold didmost of his sedimentresearch on sand transported by the wind. His ini-tial field work was done inthe desert of North-Africa.



port modes, agreed by most researchers. Some definitions are:

1. According to Einstein

Hans Albert Einstein, was a prominent researcher on sediment trans- pelabuhan. According to him, the bed load is transported in a distance two par-ticle diameters from the bed, as the transport mode was by sliding orrolling.

2. According to van Rijn

Van Rijn started from Bagnold's approach, and derived formulas for how

far the bed load particles would jump up into the flow. This distance is fargreater than predicted by Einstein's approach

3. According to measurements

Sediment transport in a river is often determined by measurements. SEBUAHwater bottle is lowered into the river, and water with sediments isdiekstrak. The sediment concentration is determined in a laboratory. Thewater bottle is not able to reach all the way down to the bed, so there will be an unmeasured zone 2-10 cm from the bed. Often the measured sed-iment will be denoted suspended load, and the unmeasured loaddenoted bed load.

4. According to Hunter Rouse's example

Because Hunter Rouse showed an example where the reference levelfor the concentration was 5% of the water depth, some people assumethe suspended load is above this level and the bed load is below.

As the definition of bed load and suspended load is not clear, it is neces-sary for the engineer to require further specification of the definitionwhen using sediment transport data where the terms are used

9.4 1D sediment transport formulas

There exist a large number of sediment transport formulas. Beberapaformulas are developed for bed load, and some for total load. All formu-las contain empirical constants, so the quality of the formula depend onthe data set used to calibrate the constants. In other words, some formu-las work well for steep rivers, and some for rivers with smaller slopes,finer sediments etc. The formulas give very different result for the samecase, and there is often an order of magnitude between lowest and high-est value. It is therefore difficult to know which formula to use. Berbedaresearchers also have varying opinions and preferences as to what for-mula to use.

The formulas can be divided in two groups: Bedload formulas and totalload formulas. The bedload formulas are developed for data sets whereonly bedload occur. When used in situations where the sediment transport is mainly suspended load, the formulas may give very inaccuratehasil. The total load formulas should work for both modes of transport.

When the second worldwar broke out, his knowl-edge of the desert wasused by a special regi-ment in the allied forces inEgypt, that he com-manded (Bagnold, 1990).

Hans Albert Einsteinwas son of the famousAlbert Einstein, itufounder of the theory of

relativity. One day whileHans Albert was at univer-sity, Einstein senior askedhis son what he intendedto choose as the topic forhis research. He thenanswered the science ofsediment transport. Thefather replied that he alsothought of this when hewas young, but he consid-ered it to be too difficult.

Page 115


Commonly used formulas for total load are:

Engelund/Hansen's (1967) formula:

(9.4.1)

The sediment transport, q s, is given in kg/s pr. m width. U is the velocity,ρ s is the density of the sediments, ρ w is the density of the water, τ is the

shear stress on the bed, g is the acceleration of gravity and d 50 adalah

Sediment transportformulas :

Engelund/HansenDanish researchers lookingat rivers with relatively finesediments and mild energygradient.

Ackers&WhiteBritish researchers, usingdata from 925 individualsediment transport experi-ments to find the constantsin their equation.

q s

0.05ρ s

U 2d 50

g ρ s

ρw

------ 1-

-----------------------τ

g ρ s

ρw

-( ) d 50

---------------------------------

32---

=



average sediment diameter. This version of the formula works in the SIsystem of units.

The Ackers&White (1973) formula requires five steps, given in the fol-lowing. Note that the logarithm in the function has base 10, log 10.

1. Compute a dimensionless particle size:

(9.4.2)

For uniform grain sizes, the mean particle diameter, d , is used. Untukgraded sediments, the d 35 value is used.

2. Compute four parameters, m , n , A and C , to be used later:

if D gr > 60, the particle sizes are said to be coarse:

n = 0.0 A = 0.17m = 1.5 (9.4.3)C = 0.025

if D gr is less than 60, but larger than 1, the sediments are medium sized:

n = 1.0 - 0.56 log D gr

(9.4.4)

if D gr is less than 1, the sediments are under 0.04 mm. It is assumed thatcohesive forces may occur, making it difficult to predict the transportkapasitas. However, the transport capacity is then usually much largerthan what is available for the river, so the sediment load is limited by the pasokan.

3. The mobility number is then computed (note simplification if n =0):

Mayer-Peter&MullerSwiss researchers, workingmostly on rivers with steepslopes, where most of thematerial moved close to the bed.

D gr

d

g

ρ s

ρw

-ρ

w

-----------------

ν 2--------------------------

13---

=

SEBUAH 0.23 D

gr

------------- 0.14+=

m9.66 D

gr

---------- 1.34+=

C mencatat 2,86 D gr

D gr

mencatat( )2- 3.53-mencatat=

Page 116


(9.4.5)

The water depth is denoted h .

4. The sediment concentration, c, is then given in weight-ppm:

(9.4.6)

5. The concentration is multiplied with the water discharge (in m 3/ s) dandivided by 10 3 to get the sediment load in kg/s.

Bed load formula

F gr

u*n

gd

ρ s

ρw

-ρ

w

-----------------

----------------------------------U

3210 h

d --------mencatat

---------------------------------

1 n-

=

c

d

ρ s

ρw

-

ρw

-----------------

h-------------------------- C

F gr SEBUAH

-------- 1-m U

u*----

n=



A formula for bed load was given by Mayer-Peter and Müller (1948):

(9.4.7)

The hydraulic radius is denoted r .

These are some of the most well-known formulas, together with Ein-stein's formula and Tofaletti's formula. The latter two are fairly involved,and are only used by some computer programs. There also exist a largenumber of other formulas giving more or less accurate answers.

The question remains on which formula to use. Three approaches exist:

1. Some formulas work better in particular situations, for example steeprivers etc. The problem with this approach is the difficult and inaccurateclassification of the formulas.

2. Do a measurement in the river, and use the formula that best fits theresult (Julien, 1989). The problem is the difficulty of obtaining a good pengukuran.

3. Use several formulas, and choose an estimate close to the averagenilai.

The final approach depend on the information available and the experi-ence and knowledge of the engineer.

Mayer-Peter&Muller's for-mula for bed load

q s

1

g ---

ρw grI 0.047 g ρ

sρ

w-( ) d

50-

0.25ρw

13---ρ

sρ

w-ρ s

-----------------

23---

--------------------------------------------------------------------

32---

=

Which formula to use?

Because of confusion onImperial and metric units,

and also because of mis- printing, many textbooks donot give the correct sedi-ment transport formulas

Page 117


9.5 Bed forms

Sediment particles moving on an initially flat bed may generate bedforms. Sediments forms small bumps on the bed with regular shape andinterval. The following classification system is used for different types of bed forms:

1. Ripples2. Dunes3. Bars4. Antidunes

The first three types of bed forms occur in subcritical flow. Sedimentsdeposit on the down side of the bed form and erode from the upstreamsisi. The ripples are fairly small, with a height under 3 cm, and occur

only on sediment finer than 0.6 mm. The bars are much larger, withheights similar to the water flow depth. The dunes have a size betweenthe bars and the ripples.

The antidunes are different in that they occur only in supercritical flow. SEBUAHhydraulic jump is formed between the bedforms. Deposition takes placeat the front of the dune, and the downstream side erodes. The bedformitself therefore may move upstream, even though the individual grainsizes move downstream. Antidunes may also move downstream or bestasioner.

There exist a large number of methods for prediction of bed forms, pre-dicting both the type and size. Unfortunately, as for the sediment transport formulas, the various methods give highly different answers. Sebuahexample of a bed form predictor is given by van Rijn (1987), estimatingthe bed form height, Δ of dunes:

Four different bed forms:

The Norwegian words forripples are “riller”. Dunes arecalled “dyner” or “sandbanker”. Bars are called“sandbanker” or “grusører”.

Antidunes are called “mot- banker” or “motdyner”.



(9.5.1)

where h is the water depth.

The bed form height can be used together with the bed sediment grainsize distribution to compute an effective hydraulic roughness (van Rijn,1987):

(9.5.2)

where λ is the bedform length, calculated as 7.3 times the water depth.The formula can be used to compute the velocities and the water surfaceelevation. However, the resulting shear stress is only partially used tomove sediments. When computing the effective shear stress for the sed-iment transport, the shear due to the grain roughness only should bedigunakan. This can be computed by using partition coefficients. The partitionof shear stress used to move the particles divided by the total shearstress is denoted:

(9.5.3)

Δ

h- 0.11

D50h

--------0,3

1 e

τ τc-

2τc--------------

- 25τ τ

c-

τc

--------------=

k s

3 D90 1.1Δ 1 e

25Δλ-----------

-+=

pτ

τ sτ---

τ s

τd

τ s

+----------------= =

Page 118


Here, τ s denotes the shear stress due to the roughness of the sediment particles, and τ d denotes the shear stress due to the roughness of thedunes.

Using several empirical formulas, the following equation can be derivedto compute p τ:

(9.5.4)

where p r is the partition of the roughness, given as:

(9.5.5)

In a laboratory experiment with movable sediments, the bed is often flat-tened before the experiment starts. The bedforms will grow over time,until they get the equilibrium size. The roughness will therefore also varyover time in this period.

9.6 CFD modelling of sediment transport

Sediment transport is often divided in two: suspended load and bed load.The bed load can be computed with specific bed load formulas, forexample (van Rijn, 1987):

(9.6.1)

pτ pr

() 0.25=

pr

k s particles.k s total .

------------------------3 D90

3 D90 1.1Δ 1 e

25Δλ----------

-+

----------------------------------------------------------= =

Many sediment dischargeformulas are derived fromlaboratory experiments. inivery difficult, almost impos-

sible, to scale the size of the bed forms in the laboratory.This may be one of the rea-sons why the many sedi-ment discharge formulasgive different results.

qb

D 501,5

ρ s

ρw

-( ) g

ρw

--------------------------

-----------------------------------------0.053

τ τc

-τc

-------------2.1

D500,3

ρ s

ρw

-( ) g

ρw ν 2

-------------------------0,1

----------------------------------------------------=



The sediment particle diameter is denoted D 50, τ is the bed shear stress,τc is the critical bed shear stress for movement of sediment particles, ρ w

and ρ s are the density of water and sediments, ν is the viscosity of thewater and g is the acceleration of gravity.

Suspended load is computed using the algorithms given in Chapter 5.The convection-diffusion equation for suspended load is solved:

(9.6.2)

The fall velocity for the particles is denoted w . This is a negative numberif the z direction is positive upwards. S is a source term which can beused to prescribe a pick-up flux from the bed. An alternative method tomodel resuspension of sediments, is to give aa boundary condition nearthe bed. The most commonly used method is to use van Rijn's (1987)formula:

∂ c

∂ t ----- U

i

∂ c

∂ x

i

------- w∂ c

∂ z -----+ +

∂

∂ x

i

------- ΓT

∂ c

∂ x

i

------ S +=

Page 119


(9.6.3)

The sediment particle diameter is denoted d , and a is a reference levelset equal to the roughness height

It is also possible to adjust the roughness in the computation of the watervelocities according to the computed grain size distribution at the bedand the bed forms (Eq. 9.5.2) (Olsen, 2000).

When the sediments are prescribed in the bed cell according to Eq.9.6.3, sediment continuity is not satisfied for this cell. There may there-fore be sediment deposition or erosion. The continuity defect can beused to change the bed. A time-dependent computation can computehow the geometry changes as a function of erosion and deposition ofsediments.

CFD modelling of sediment transport has currently been done by anumber of researchers on many different cases: sand traps, reservoirs,local scour, intakes, bends, meandering channels etc. However, the sci-ence is still at a research stage, and it is not yet much used in consultingkerja.

9.7 Reservoirs and sediments

A river entering a water reservoir will loose its capacity to transport sedi-KASIH. The water velocity decreases, together with the shear stress onthe bed. The sediments will therefore deposit in the reservoir anddecrease its volume.

In the design of a dam, it is important to assess the magnitude of sedi-ment deposition in the reservoir. The problem can be divided in two bagian:

1. How much sediments enter the reservoir2. What is the trap efficiency of the reservoir

In a detailed study, the sediment grain size distribution also have to bedetermined for question 1. Question 2 may also involve determining thelocation of the deposits and the concentration and grain size distribution

ctempat tidur

0,015 D50

Sebuah---------

τ τc

-τc

-------------1,5

D50

ρ s

ρw

-( ) g

ρw ν 2

-------------------------

13---

0,3-------------------------------------------------- ------=



of the sediments entering the water intakes.

In general, there are two approaches to the sedimentation problem:

1. The reservoir is constructed so large that it will take a very long timeto fill. The economical value of the project will thereby be main-tained.

2. The reservoir is designed relatively small and the dam gates areconstructed relatively large, so that it is possible to remove the sed-iments regularly by flushing. The gates are opened, lowering thewater level in the reservoir, which increases the water velocity. The

sediment transport capacity is increased, causing erosion of the

Page 120


deposits.

A medium sized reservoir will be the least beneficial. Then it will take rel-

atively short time to fill the reservoir, and the size is so large that only asmall part of the sediments are removed by flushing.

The flushing has to be done while the water discharge into the reservoiris relatively high. The water will erode the deposits to a cross-streammagnitude similar to the normal width of the river. A long and narrow res-ervoir will therefore be more effectively flushed than a short and widegeometri. For the latter, the sediment deposits may remain on the sides.

The flushing of a reservoir may be investigated by physical model stud-ies.

Another question is the location of the sediment deposits. Gambar. 9.7.1shows a longitudinal profile of a reservoir. There is a dead storage belowthe lowest level the water can be withdrawn. This storage may be filledwith sediments without affecting the operation of the reservoir.

Sediment load prediction

Rough estimates of sedimentload may be taken fromregional data. Often the sedi-

ment yield in the area isknown from neighbouringcatchments. It is then possibleto assess the seriousness ofthe erosion in the presentcatchment and estimate roughfigures of sediment yield. Theland use, slope and size of thecatchment are important fac-tor.

For a more detailed assess-ment, measurements of thesediment concentration in theriver have to be used. Sedi-ment concentrations are

HRW

LRW

Dead storage

Inflow

Figure 9.7.1 Longitudinal

profile of a reservoir. HRWis the highest regulatedwater level. LRW is the low-est regulated water level.The reservoir volume belowLRW is called the dead stor-age, as this can not bedigunakan.



measured using standardsampling techniques, andwater discharges arerecorded simultaneously. Themeasurements are taken at

Sediment sampler

Page 121


varying water discharges. The values of water discharge and sedimentconsternations are plotted on a graph, and a rating curve is made. Iniis often on the form:

(9.7.1)

Q s is the sediment load, Q w is the water discharge and a and b are con-stants, obtained by curve-fitting.

The annual average sediment transport is obtained by using a time-series of the water discharge over the year together with Eq. 9.7.1.

9.8 Fluvial geomorphology

The river plays an essential role in shaping the landscape, in the processof transporting eroded sediments to the sea. The transport mechanismsare described differently depending on the river characteristics. Dalamupper part of the catchment, the creeks will have relatively large slopes.The transport capacity will often exceed the amount of available mate-rial, leaving bedrock or larger stones on the river bed. Closer to the sea,the river gradient will be lower, and the river will not be able to transportlarge-sized material.

An alluvial river is formed where the bed material have sufficient magni-tude to enable free vertical bed fluctuations. Also, the bed material has alower size than what is given by Shields curve, so there is continuoussediment transport. The bed fluctuates depending on the sediment con-centraion. If the supply of sediment is larger than the capacity, the bedrises. If the supply is lower then the capacity, the bed is lowered. Jika

bed is in equilibrium, the river is said to be in regime . There exist theoryfor the relationship should be between different parameters for the chan-nel. This is called regime theory (Blench, 1970). An example is a rela-tionship between the water velocity, U , and the water depth, d :

(9.8.1)

The formula is given in British Imperial units, where the velocity is in feet pr. seconds and the water depth is in feet. Also formulas exist where theriver width and slope can be computed.

Q s

aQwb=

Gambar. 9.7.2 Example of sedi-ment rating curve. Thesediment load is on the ver-tical axis, and the water dis-charge on the horizontalaxis. The points are meas-ured values, and the line isEq. 9.7.1

Note : Often, most of thesediments are transportedduring the larger floods

Alluvial river

The regime theory was

developed by British waterengineers in India at the endof the 19th century.

U 0.84 d 0.64=



Page 122


Many steeper rivers are not in regime. Very little sediments may be

transported during normal and low flow. Only during large floods, the bedmaterial moves. These rivers often have large stones at the bed. Merekaare said to be paved . The paving protects the underlaying sediments.Only during very large floods the paving will be removed, and then largechanges in the bed geometry may occur. In general, the bed changesand shape of a river usually change mostly during large floods.

Meandering rivers

The river may also move sideways (laterally), as the classical meander-ing pattern evolves. As the water velocity is higher closer to the watersurface, than at the bed, a vertical pressure gradient will be formed whenthe water meets an obstacle or a river bank in a bend. The result is adownwards velocity component, causing a secondary current. (Gambar.9.8.1)

The flow pattern causes the sediment transported on the bed to move tothe inside of the curve. The shape of the resulting cross-section is given pada Gambar. 9.8.1, with the deeper part on the outside of the curve. Over time,sediments deposit on the inside of the curve and erosion will take placeon the outside. Looking at a plan of the river, the meanders will moveoutwards and downstream. There exist different classification systemsfor the plan form (Schumm et. al., 1987).

A river can be classified according to its sinuosity (Fig. 9.8.2). If the s inu-osity is below 1.05, the river is straight. Some researchers classify ameandering river by a sinuosity above 1.25.

Paving is called “dekklag”in Norwegian. Meander iscalled “elveslyng”.

Inner bankDeposition

Outer bankErosi

Figure 9.8.1 Cross-sec-tional view of a river bend ,

where the arrows indicate thesecondary current

L

CenterlineFigure 9.8.1. Definition of sinuosity of a river. Thefigure shows a plan view ofthe river. The length of thecenterline is denoted C , andthe length along the valley isdenoted L . The sinuosity isthen the ratio between thesenumbers: C / L .

Page 123




Useful terminology on meandering rivers is given in Fig. 9.8.3, giving thelocation of the apex and the cross-over. The secondary currents willhave maximum strength at the apex, giving the largest scouring andmaximum depth at this location, on the outside of the bend. Ini jugaditunjukkan pada Gambar. 9.8.4, giving a longitudinal profile of a meandering riverand explaining the word thalweg .

There are many factors affecting the meander formation, magnitude ofthe sinuosity etc. for a river: Valley slope, size of sediments, sedimentdischarge and water discharge. Cohesion of the bank material alsoseems to be important, as it affects the bank stability. Vegetation alongthe river is then also important.

Experiments on meandering channels

A natural meandering channel often do not show regular patterns. ini adalahdue to inhomogenities in bed material, local geology, vegetation, man-made scour protection etc. To study the meander formation, flume stud-ies in the laboratory have been carried out. An example is large flumestudies at Colorado State University in the 1970's. (Schumm et. al.,1987). The flume was 28 meters long, and was initially filled with sand. SEBUAHstraight channel was moulded in the sand, and a water was then runthrough it. After some time, a meandering pattern emerged. This casewas later computed using a CFD model (Olsen, 2003). The results are

Apex

Cross-over

Maximum depth

Figure 9.8.3. Plan view of ameandering river. Thelocation of the apex andcross-over is shown. Themaximum depth is at theouter side of the bend at theapex.

Cross-over is called “brekk”in Norwegian. The area atthe apex with the scour iscalled “kulp”.

Apex Cross-over

Tempat tidurtingkat

Air permukaan

Figure 9.8.4.

The upper figure shows a plan view of the meanderingriver. The broken line shows

the location of the maximumdepth of the river. ini adalahalso called the thalweg .

The lower figure shows alongitudinal profile of theriver, with the water surfaceand the bed level at the thal-weg. The thalweg levelshows a pattern with highestvalue at the cross-over andlowest level at the apex.

Thalweg

Page 124


ditunjukkan pada Gambar. 9.8.5, 9.8.6 and 9.8.7.

Figure 9.8.5 Plan view of velocity vectors in ameandering river, mod-



Note the velocity vectors close to the bed points more towards the insideof the curve than the vectors at the surface. This corresponds with Fig.9.8.1.

The water depth in Fig. 9.8.6 is largest at the outside of the bend, corre-sponding with Fig. 9.8.3

If a meander short-cuts like shown in Fig. 9.8.8, this is often called achute . Chutes can also be seen in Fig. 9.8.5 and Fig. 9.8.6 by close pemeriksaan.

elled with CFD. The blackarrows shows the vectorsat the water surface, andgrey arrows close to the bed.

5

9

1

7

5

7

3

51

3

1

31

1

1Figure 9.8.6 Plan view of water depths in a mean-dering river, modelledwith CFD. The values aregiven in cm.

Figure 9.8.7 Plan view of the meandering pattern in a channel computed with a CFD model. The case is a physical model study by Zimpfer (1975).

chute

Figure 9.8.8. Plan view of the meandering river,where a chute has formed.

Page 125


Braided rivers

Beside being straight or meandering, the river may have a third planform: braided . This usually takes place at a steeper valley slope thanthe meandering plan form. Also, braided river systems often occur inreaches where the sediment transport capacity is lower than the sedi-ment inflow, so that a net deposition occurs.

A river can evolve from a meandering planform to braided. Then chutesform first and grow larger. The braided river does not follow a regular pattern, but consists of several parallel channels, separating and joiningsatu sama lain.

The flume study described in the figures above also evolved into a braid-



ing pattern. After the meandering channel had evolved, the meander bends became very large and channel cutoffs emerged. When severalcutoffs had formed, the resulting plan form was braided (Zimpfer, 1975).CFD modelling of a braided system is shown in Fig. 9.8.9. Aslimeandering pattern can also be seen.

Smaller islands in the river are often called bars .

9.9 Physical model studies

A physical model is an important tool to estimate effects of sedimenttransport for engineering purposes. There exist a large number of scal-ing laws that has to be used according to the purpose of the study. SEBUAHdetailed description of the different methods are given by Kobus (1980).

Water flow

The water discharge in the physical model is usually determined by thethe Froude law, based on the similarity between momentum and gravita-tional forces. The Froude number should be the same in the model as inthe river:

(9.9.1)

Given the geometrical scale, the Froude number determines the waterdischarge in the model. The next problem is to determine the correctroughness in the physical model The roughness will affect the shearforces on the bed and thereby the energy slope in the model. The shearforce is affected by viscosity and the Reynolds number. The Darcy-

Figure 9.8.9 Planview of depth-aver-age water velocitiesin a meanderingriver, where cutoffshave started to

make a braided planuntuk m.

Scaling the water dis-charge in the physicalmodel

Fr U

gh---------=

Page 126


Weissbach's diagram for the friction coefficients can be used to computethe physical model friction factors. This friction factor should be the samein the physical model as in the prototype. Given the two Reynolds numbers in the physical model and the prototype, the diagram can be used tocompute the relative roughness ( k s/ r h roughness to hydraulic radius) ofthe physical model, given the similar parameter for the prototype.

The scaling of the sediments is more difficult.

Erosi

If the main topic of the investigation is the computation of maximumscour or erosion, Shield's graph may be used.

. (9.9.2)

A simplified approach is to say that τ* should be the same in the proto-type as in the physical model. Eq. 9.9.2 is then solved with respect to the particle diameter, giving the following equation:

τ*τ

g ρ s

ρw

-( ) d ----------------------------=Scaling sediments for ero-

sion studies

d m

τm

ρ sp ,

ρw

-( )



(9.9.3)

The density of the sediments in the prototype is denoted ρ s,p, and ρ s,m aku sthe density of the sediments in the model. The computation involves theshear stress and particle diameter in the prototype and in the model.Subscripts m and p are used for model and prototype, respectively. iniassumed that the particle Reynolds numbers are so great that the τ*value is the same in prototype and model. The equation is also only validfor particle sizes and bed shear stresses close to erosion. For finer parti-cles, a more involved approach must be used.

Suspended sediments

If the main topic of the investigation is suspended sediments, the HunterRouse number, z , is usually used:

(9.9.4)

If the Hunter-Rouse number is the same in the prototype and the model,then this gives the fall velocity of the particles in the physical model. Thesediment diameter and density then have to be chosen accordingly.

Scaling time

To model the time for the movement of the sediments, the ratio of sedi-ment transport to volume of sediments is used:

(9.9.5)

T is the time, Q s is the sediment load and V is the volume of the material being transported. The scaling factor for time, s t , is given as

d p

------ τ p

ρ sm. ρ

w-( )--------------------------------=

Scaling sediments forsuspended load

z w

κ u*

---------=

TQ s

V ---------

Page 127


A sediment transport formula can then be used to give the sediment dis-charge pr. unit width, q s. If s is the geometric scale (for example 1:20),then Eq. 9.5.5 and 9.5.6 can be combined to:

(9.5.6)

Example: Find the scaling time for a physical model with sediments of size 4 mm in the prototype and 2.5 mm in the physical model. The water velocity is 0.2 m/s in the model, and the water depth is 0.2 m. The proto-

type sediments have a density of 2.65 kg/dm3, and the model sedimentshave a density of 1.05 kg/dm3. Manning-Stricklers friction factor is 40,and the model scale is s=0.015 (1:66.67).

Solution: First, the sediment discharge is computed in both the prototypeand the model, using Engelund-Hansens formula. Hal ini memberikan:

q s = 0.030 kg/s/m (model)

q s = 0.191 kg/s/m (prototype)

The time scale becomes:

st

T model

T prototipe

----------------------=

st

q s prototype.q s model .

-------------------------- s g

2=

st

0,191

0.030------------0,015( )2 0.001435

1

697--------= = =



A simulation time of 1 hour in the lab will be similar to 697 hours in the prototype.

The accuracy of the scaling of the time will not be better than the accu-racy of the sediment transport formula used.

Multiple sediment processes

If the topic of the investigation involves both suspended sediments, ero-sion and sediment transport, then the different methods of scaling thesediments may give different model sediment characteristics. Mungkin

therefore not be possible to model all transport modes and processes.This has been one of the motivations for developing CFD models withsediment transport processes.

Page 128


Other problems

Scaling down finer sediments can result in particles with cohesive

Pasukan. This would be the case for particles under 0.1 mm. To avoid this,it is possible to use a distorted physical model. Then the vertical scale islarger than the horizontal scale. However, this will also distort all second-ary currents, which will not be correctly modelled.

Another problem is to scale bedforms. Dunes and ripples occur at differ-ent hydraulic conditions, and it is almost impossible to get for examplethe ratio bedform height to water depth to be the same in the physicalmodel and the prototype. Also, bedforms may occur only in the physicalmodel and not in the prototype. The bedforms will then cause differenteffects for energy loss and sediment transport capacity in the physicalmodel and the prototype.

9.10 Problems

Gambar. 9.9.1 Photo of the physical modelmade at SINTEF, Norway, for the KaliGandaki hydropower plant in Nepal. The

model was used to investigate sedimentflushing from the hydropower reservoir.The dams are shown in the lower leftcorner. The water is fed into the model atthe upper left part of the picture. Themodel is approximately ten meters longfrom the dam to the end shown on the benar. The model was built of concreteaccording to the prototype bed, and filledwith sand to the level of the spillwaycrest. The model was then carefully filledwith water, and then the flushing started by opening the dam gates. As the run-of-the-river reservoir is fairly narrow, muchof the sediments was removed by theflushing



Problem 1. Channel design

A discharge channel from a power station is 30 m wide and has a rectan-gular cross-section. The bed slope is 1:200 and the maximum water dis-charge from the power station is 100 m 3/s. To prevent erosion of thechannel, stones are used at the bed of the channel. How large must thestones be? Assume uniform grain size distribution for the stones.

Problem 2. Sediment transport

The construction of a dam in a river caused erosion downstream. Apa

would be the reason?The river authorities solved the problem by adding gravel to the river,downstream of the dam. What would be the required amount of gravel,when the water discharge was assumed to be 1500 m 3/s, the waterdepth 3.5 meters, the width 100 meters and the s lope 1:1600 The gravelsize was identical to the sediment size at the bed: a diameter of 20 mm.

Can you think of an alternative solution to the problem?

Problem 2 is taken fromthe Iffezheim dam in theRiver Rhine in Ger-many (Kuhl, 1992),where this solution waschosen.

Page 129


Problem 3. Sediment load estimation

Estimate the annual sediment load in a river, given the flow durationcurve below, and the rating curve in Fig. 9.7.1. How much of the sedi-ments are transported by the highest floods, occurring under 5 % of thetime?

Problem 4. Physical model study

A physical model study of a reservoir is conducted. The scale is 1:30.The water discharge in the prototype is 300 m 3/s. What is the dischargein the model?

The average dimensions of the physical model is 10 m long, 5 m wide,and 10 cm deep. Suggest a sediment type and size for the physicalmodel, when investigating sedimentation of prototype silt particles of 0.3

mm.Suggest a sediment type for modelling flushing of the same silt.

If a water discharge over time were to be modelled, including both sedi-mentation and erosion, what kind of sediment should be chosen?

Problem 5. HEC-6

Compute the trap efficiency over time in a water reservoir built in a riverwith slope 1:200. The water discharge is 100 m 3/s, the river width is 30meters, the dam height is 30 meters. Assume a constant reservoir widthof 200 meters. The inflow sediment is s ilt with particle diameter 0.3 mm,and the concentration is 2000 ppm.

Problem 6. Sediment load formulas

3000

2000

1000

0

m3/ s

100% 50 % 0 % (of time)

Flow duration

melengkung



Compute the sediment transport capacity pr. m. width in an alluvial chan-nel, with the following data:

U=2.5 m/sDepth, y = 1.5 mManning-Strickler coefficient: 50Sediment size: 1 mm

Use both Engelund-Hansen's and van Rijn's formulas.

Page 130


10. River habitat modelling

10.1 Introduction

The science of River Habitat Modelling has evolved over the last tentahun. The main purpose is to assess the effect of habitat for fish, mostlysalmon, in relation to river regulations. Hydropower production has often been the cause of changes in the water discharge. River Habitat Model-ling aims to quantify the effects of changes in the river flow conditionsand geometry, to assess the impact of regulations on the fish production.In Norway, this has been used to determine minimum flow regulations inrivers. Also, it is used for assessing environmental effects from hydrope-aking.

10.2 Fish habitat analysis

The basis of the currently used method is that fish will seek an optimumcondition with respect to:

- Feeding- Energy to stay in the r iver- Spawning- Protection from predation

The factors vary depending on the age of the fish. Often, the critical agefor salmon will be the juvenile stage. The studies often look at the rearingand growing areas of the fish.

Looking at river regulations, the main changes in the river will be the physical habitat. The main factors often used are:

- Water depth- Water velocity- Substrate/cover

Substrate is usually determined by a characteristic size of the stones onthe river bed. The stones provide cover/hiding places for the fish. Bagaimanapun

ever, plants with leaves extending out over the river will also providemenutupi.

The effect of feeding is neglected in simpler models. Then, it is assumedthe fish prefers optimum values of the parameters given above.

The currently most used methodology for fish habitat analysis is calledthe Instream Flow Incremental Methodology (IFIM). It can be divided inthe following stages:

1. Selection of representative reach of the river2. Counting the fish at different locations, recording habitat parameters3. Generating habitat preference functions4. Measuring habitat parameters at different discharges5. Generating spatially distributed habitat as a function of discharge6. Computing the habitat as a function of a time series of discharge

Earlier methods of evaluat-ing impacts of river regula-tions on fish was dubbedBOGSAT : B unch O f G uysS itting A round a T able, dis-cussing the effects of theregulasi. Tujuan dariRiver Habitat Modelling is toimprove the scientific back-ground for the evaluation.

The science has also lead toincreased cooperation andunderstanding between biol-ogists and engineers work-ing in rivers.

Main factors affectingthe fish:

Instream Flow Incre-mental Methodology(IFIM):



The selection of representative reach of the river has to be done withrespect to typical values of depth, velocity and substrate. Often, fish biol-ogists provide input for the selection.

Page 131


The preferred method for counting the fish is by diving. The diver coversthe whole area of the selected reach, and put a marker on the r iver bedfor each fish observation. Afterwards, the depth, velocity and substrate isrecorded for each marker. It is also possible to identify fish locationsfrom the surface, but then it is often more difficult to spot the fish. Kemudianthe velocity, depth and substrate is divided into groups, and it is countedhow many fish there is in each group. This is shown in Fig. 10.2.1:

The HSI index indicates the preferred velocity for the fish. Another ques-tion is how much area of a preferred velocity there is in the river. ini adalahcalled availability of velocity. The availability is computed the same wayas the HSI index, but on the vertical axis is the area of a given velocityinterval. This can also be scaled to unity as the maximum area.

A preference index, D , is made from the HSI index and the availabilityfunction, by using the following formula:

(10.2.1)

where p is the available habitat of a defined velocity range, and r is theHSI index. Both p and r are scaled so their values range between zeroand 1.

Three regions are made:

- Preferred habitat, where most of the fish observations are- Avoided habitat, where there are no observations- Indifferent, the region between

Preferred habitat is if D is above 0.2. If D is below -0.2, this is avoidedhabitat. The area between -0.2 and 0.2 is indifferent. Contohnya adalahgiven in Fig. 10.2.2.

It is also possible to make an index by combining the curves for velocity,

depth and substrate.Given the indexing system, a map of preferred, indifferent and avoidedhabitat is made. This is called a habitat map, and an example is given inGambar. 10.2.3. The habitat will vary according to the water discharge. Sum-ming up the preferred area in a time series of water discharge, a meas-ure of fish habitat for a given river regulation is made. Effects of changesin the regulations can thereby be computed, but using different regulateddischarges over the year.

tidak. ikan

Velocity (m/s)

12

6

5 10

HSI

Velocity (m/s)

1

5 10

0,5

Figure 10.2.1. Gener-ation of HSI indexfor velocity. The

number of fish in eachvelocity group iscounted (left figure),and scaled to maxvalue of unity to become the HSI index(right)

D rp-rp+( ) 2 rp-

------------------------------=



Page 132


The method will, however, only work if it is possible to estimate thevelocity and depth as a function of the water discharge. This has to bemade in preferably three dimensions, as the spatial variation of velocitynear the bed is required. The fish is often found near the bed. The follow-ing two chapters shows the two main methods:

- Measurements and zero/one-dimensional models- Multidimensional numerical models

This is described in the following chapters.

10.3 Zero and one-dimensional hydraulic models

Initial work on habitat hydraulics started with measuring the water veloc-ities in the characteristic reach of the river. Measurements were made in

multiple cross-sections, or transects. A two-dimensional map of thevelocities and depths were then obtained for a given discharge. Inirepeated for several water discharges. An interpolation function wasused for discharges between the measured ones.

Preference function

Velocity

1

-1

Preferred Avoided

Figure 10.2.2 Generation of preference curveas a function of velocity

Indifferent

The preference curve is abiological model for thefish habitat. There also existother biological models, based on different pendekatan.

Figure 10.2.2 Velocity vector map (right) and computed habitat (left) near the bed for juvenile salmonin a fish farm tank, seen from above. The water is entering on the left side, creating the current shown. Theoutlet is in the center of the tank, at the bottom. The fish farm tank is 1.5 meters deep and 2.7 meters wide(Olsen and Alfredsen, 1994). Most of the area has preferred velocity, except for the entrance and exit regionwhere the velocity is too high, and the corners where the velocity is lower than preferred.

Page 133




The interpolation function could be fairly involved, and often a one-dimensional backwater program was used. Various weighing functionswere calibrated for the velocity distribution in the lateral direction. -Contoh ples of computer programs using this procedure is RIMOS and PHAB-SIM.

There were two problems with this procedure:

1. A large number of field measurements had to be carried out2. The calibration of the interpolation functions were only valid for thegeometry where the measurements were taken. In other words, itwas not possible to estimate the effects of changes in the geometry.

To solve this problem, multi-dimensional models have to be used.

10.4 Multidimensional hydraulic models

The multidimensional numerical models are two-dimensional depth-averaged or three dimensional. The models solve the Navier-Stokesequations using for example the methods given in Chapter 6. The mod-els can compute the effect of changes in the bed geometry, as thegeometry is given as input for the grid. This is very useful for assessmentof fish habitat studies for river regulations, as a mean of improving thehabitat is often to create small dams or obstacles in the flow. The waterdepth is thereby increased, and also the variation in water velocity.

10.5 Bioenergetic models

Bioenergetic models are the latest in a succession of habitat assess-ment methodologies. The theory is to compute how much energy the fishuses in different locations of the flow, as a function of water velocity and possibly turbulence. As opposed to earlier studies, the food intake is alsotaken into account. The fish feed on small organisms, which have differ-ent concentrations in various locations of the river. This means the fishwill receive more food/energy in some locations than in others. The mod-els also assess how much energy is consumed by the fish. Ini adalah sebuahfunction of the velocity in the river. Using three-dimensional numericalmodels, it is possible to compute the food concentration, water velocityand turbulence over the whole three-dimensional river body. The opti-mum location for the fish can then be estimated, with the assumptionthat the fish will seek a maximum possible difference between theenergy intake and consumption.

10.6 Problems

Problem 1. Preference curves

The figures below give depth and velocity in a representative reach ofSokna River in Norway, at a discharge of 10 m 3/s. The figure with thedots provide locations of fish observations. Make preference curves forthe fish, for both depth and velocity.

At the time this book wasmade, the bioenergeticmodels were still on theresearch stage.

Page 134


4

77

7Velocity (cm/s) nearthe bed for a dis-charge of 10 m 3/s.



Problem 2. Habitat map

Make habitat maps for Sokna River, for 10 m 3/s, for both water depthand velocity.

4

4

7

7

7

7

4

7

10

1010

10

10

1010

1212

12

15

15

1,5

1,5

1,9

1,9

1,9

2.2

2.2

2.2

2.2

2.2

2.22.6

2.62.6

2.6

2.6

2.2

2,9

3.6

2.6Depth in meters, for 10 m 3/ s

Fish observations

* * ** * **** * * *** ** ** ** *'

*** * *

*

* **

* ** * *

* *

* *

* * **

Page 135


Problem 3. Habitat map for changed velocity

Compute the habitat maps when the discharge is lowered to 3 m 3/s. Thevelocity and depth is given below. Is the habitat improved or has it dete-riorated?

2.12.4

2.4

Water depth inmeters for 3 m 3/ s



Problem 4. Habitat improvement

Suggest measures to improve the habitat in the river, and methods todocument the improvements before they are carried out.

1.4

1.4

1.7

1.72.1

2.1

2.1

2.1

2.12.4 2.4

2.42.4

2.4

2.8

3,5

2

2

2

2

3

3

3

33

3

3

34

4 4

44

4 4

4

5

5

5

5

6

Velocity in cm/s nearthe bed at 3 m 3/s.

Page 136


literatur

Ackers, P. and White, RW (1973) "Sediment Transport: New Approachand Analysis", ASCE Journal of Hydraulic Engineering, Vol. 99, No.HY11.

Bagnold, RA (1973) “The nature of saltation and of 'bed-load' transportin water”, Proceedings of the Royal Society of London, A332. pp473-504.

Bagnold, RA (1988) “The physics of sediment transport by wind andwater”, A collection of hallmark papers, ASCE Publications.

Bagnold, RA (1990) “Sand, wind, and war: memoirs of a desertexplorer“, Tucson : University of Arizona Press.

Bakhmeteff (1932) “Hydraulics of open channels”, McGraw-Hill BookCompany, New York.

Bengtsson, L. (1973) "Wind Stress on Small Lakes", Tekniska Høgsko-lan, Lund, Sweden.



Bihs, H. and Olsen, NRB (2011) “Numerical Modeling of AbutmentScour with the Focus on the Incipient Motion on Sloping Beds“, Journalof Hydraulic Engineering.

Bindloss, M. (1976) "The light-climate of Loch Leven, a shallow Scottishlake, in relation to primary production of phytoplankton", FreshwaterBiology, No. 6.

Blench, T. (1970) “Regime theory design of canals with sand beds”,ASCE Journal of Irrigation and Drainage Engineering, Vol. 96, No. IR2,Proc. Paper 7381, June, pp. 205-213.

Bowles, C., Daffern, CD and Ashforth-Frost, S. (1998) "The Independ-ent Validation of SSIIM - a 3D Numerical Model", HYDROINFORMAT-ICS '98, Copenhagen, Denmark.

Brooks, HN (1963), discussion of "Boundary Shear Stresses in CurvedTrapezoidal Channels", by AT Ippen and PA Drinker, ASCE Journalof Hydraulic Engineering, Vol. 89, No. HY3.

Brethour, JM (2002) “Transient 3-D model for lifting, transporting anddepositing solid material”, Proceedings from the Fifth International Con-ference on Hydroinformatics, Cardiff, UK.

Carstens, TJ (1997) “Class notes in fluvial hydraulics - hydraulics ofreceiving waters”, Department of Hydraulic and Environmental Engineer-ing, The Norwegian University of Science and Technology. (In Norwe-gian)

Chapra, SC (1997) "Surface water-quality modelling", McGraw-Hill,ISBN 0-07-115242-3.

Demuren, AO and Rodi, W. (1984) “Calculation of turbulence-drivensecondary motion in non-circular ducts”, Journal of Fluid Mechanics, vol.140, pp. 189-222.

Dey, S. (2003) “Threshold of sediment motion on combinded transverseand longitudinal sloping beds”, Journal of Hydraulic Research, Vol. 41

Page 137


(4), pp. 405-415.

van Dorn, W. (1953) "Wind stress on an artificial pond", Journal ofMarine Research, No. 12, pp. 249-276.

Einstein, HA, Anderson, AG and Johnson, JW (1940) “A distinction between bed-load and suspended load in natural streams”, Transactionsof the American Geophysical Union's annual meeting, pp. 628-633.

Einstein, HA and Ning Chien (1955) "Effects of heavy sediment con-centration near the bed on velocity and sediment distribution", UCLA -Berkeley, Institute of Engineering Research.

Engelund, F. (1953) "On the Laminar and Turbulent Flows of GroundWater through Homogeneous Sand", Transactions of the Danish Acad-

emy of Technical Sciences, No. 3.Engelund, F. and Hansen, E. (1967) "A monograph on sediment transport in alluvial streams", Teknisk Forlag, Copenhagen, Denmark.

Fergus, T., Hoseth, KA and Sæterbø, E. (2010) “Handbook of rivers”(Vassdragshåndboka), Tapir forlag, Norway, ISBN 978-82-519-2425-2(in Norwegian).

Fisher, HB, List, EJ, Koh, RCY, Imberger, J. and Brooks, NH(1979) “Mixing in Inland and Coastal Waters”, Academic Press.

Fischer-Antze, T., Stösser, T., Bates, P. and Olsen, NRB (2001) "3Dnumerical modelling of open-channel flow with submerged vegetation",IAHR Journal of Hydraulic Research, No. 3.



Fisher, HB, List, EJ, Koh, RCY, Imberger, J. and Brooks, NH(1979) “Mixing in inland and costal waters”, Academic press, New York.

French, RH (1994) “Open-channel hydraulics”, McGraw-Hill.

Graf, WH and Altinakar, MS (1996) “Fluvial hydraulics, Vol. 2, Flowand transport processes in channels of simple geometry”, Wiley Publish-ers.

Hedger, RD, Olsen, NRB, Malthus, TJ and Atkinson, PM (2002)"Coupling remote sensing with computational fluid dynamics modelling

to estimate lake chlorophyll-a concentration", Remote Sensing of Envi-ronment, Vol. 79, No. 1, pp. 116-122.

Henderson-Sellers, B. (1984) “Engineering Limnology”, Pitman Publish-ing Limited, ISBN 0-273-08539-5.

Hervouet, JM. and Petitjean, A. (1999) “Malpasset dam-break revisitedwith two-dimensional computations”, IAHR Journal of HydraulicPenelitian, Vol. 37, No. 6.

Hey, RD (1979) “Flow resistance in gravel-bed rivers”, ASCE Journalof Hydraulic Engineering, Vol. 105, No. 4, April.

Jones, WP and Launder, BE (1972) “The prediction of Laminariza-tion with a Two-Equation Model of Turbulence”, International Journal ofHeat and Mass Transfer, Vol. 15, pp.

Kamphuis, JW (1974) ““, IAHR Journal of Hydraulic Research, Vol. 12, No. 2.

Page 138


Kawai, S. and Julien, PY (1996) “Point bar deposits in narrow sharp bends”, IAHR Journal of Hydraulic Research, No. 2.

Kobus, H. (1980) “Hydraulic modelling”, German Association for Water

Resources and Land Improvement, Hamburg, Germany.

Kreyszig, E. (1983) “Advanced Engineering Mathematics”, John Wiley &Sons Publishers.

Kuhl, D. (1992) “14 years artificial grain feeding in the Rhein, down-stream the barrage Iffezheim”, 5th Int. Symp. on River Sedimentation,Karlsruhe, Germany.

Lysne, DK (1969) “Movement of sand in tunnels”, ASCE Journal ofHydraulic Engineering, Vol. 95, No. 6, November.

Mayer-Peter, E. and Mueller, R. (1948) "Formulas for bed load transport", Proceedings from the Second Meeting of the International Associ-ation for Hydraulic Structures Research, Stockholm, Sweden.

McQuivey, RS and Keefer, TN (1974) “Simple method for predictingdispersion in streams”, ASCE Journal of Environmental Engineering, pp.997-1011.

Melaaen, MC (1992) "Calculation of fluid flows with staggered andnonstaggered curvilinear nonorthogonal grids - the theory", NumericalHeat Transfer, Part B, vol. 21, pp 1- 19.

Munk, WH and Anderson, ER (1948) “Notes on the theory of thethermocline”, Journal of Marine Research, Vol. 1.

Naot, D. and Rodi, W. (1982) “Calculation of secondary currents in chan-nel flow”, ASCE Journal of Hydraulic Engineering, No. 8, pp. 948-968.

Nikuradse, J. (1933) “Flow in rough pipes”, ?? Melaporkan tidak ada. 361.

O'Connor, DJ and Dobbins, WE (1958) “Mechanisms of reaeration innatural streams”, Trans. ASCE, no. 123, pp. 641-684.



Olsen, NRB and Melaaen, MC (1993) "Three-dimensional numeri-cal modelling of scour around cylinders", ASCE Journal of HydraulicEngineering, Vol. 119, No. 9, September.

Olsen, NRB, Jimenez, O., Lovoll, A. and Abrahamsen, L. (1994) "Cal-culation of water and sediment flow in hydropower reservoirs", 1st. Inter-national Conference on Modelling, Testing and Monitoring ofHydropower Plants, Hungary.

Olsen, NRB and Alfredsen, K. (1994) "A three-dimensional model forcalculation of hydraulic parameters for fish habitat", IAHR Conference onHabitat Hydraulics, Trondheim, Norway.

Olsen, NRB and Skoglund, M. (1994) "Three-dimensional numericalmodelling of water and sediment flow in a sand trap", IAHR Journal ofHydraulic Research, No. 6.

Olsen, NRB and Stokseth, S. (1995) "Three-dimensional numericalmodelling of water flow in a river with large bed roughness", IAHR Jour-nal of Hydraulic Research, Vol. 33, No. 4.

Olsen, NRB (1999) "Computational Fluid Dynamics in Hydraulic and

Page 139


Sedimentation Engineering", Class notes, Department of Hydraulic andEnvironmental Engineering, The Norwegian University of Science andTeknologi.

Olsen, NRB and Kjellesvig, HM (1998) "Three-dimensional numeri-cal flow modelling for estimation of maximum local scour depth", IAHRJournal of Hydraulic Research, No. 4.

Olsen, NRB and Kjellesvig, HM (1998) "Three-dimensional numeri-cal flow modelling for estimation of spillway capacity", IAHR Journal ofHydraulic Research, No. 5.

Olsen, NRB (1999) "Two-dimensional numerical modelling of flushing

processes in water reservoirs", IAHR Journal of Hydraulic Research,Vol. 1.

Olsen, NRB and Kjellesvig, HM (1999) "Three-dimensional numeri-cal modelling of bed changes in a sand trap", IAHR Journal of HydraulicPenelitian, Vol. 37, No. 2. abstract

Olsen, NRB and Lysne, DK (2000) "Numerical modelling of circula-tion in Lake Sperillen, Norway", Nordic Hydrology, Vol. 31, No. 1.

Olsen, NRB, Hedger, RD and George, DG (2000) "3D NumericalModelling of Microcystis Distribution in a Water Reservoir", ASCE Jour-nal of Environmental Engineering, Vol. 126, No. 10, October.

Olsen, NRB (2003) "3D CFD Modeling of a Self-Forming MeanderingChannel", ASCE Journal of Hydraulic Engineering, No. 5, May.

Patankar, SV (1980) "Numerical Heat Transfer and Fluid Flow",McGraw-Hill Book Company, New York.

Reynolds, CS (1984) "The ecology of freshwater phytoplankton", Cam- bridge University Press, Cambridge, UK.

Rhie, C.-M, and Chow, WL (1983) "Numerical study of the turbulentflow past an airfoil with trailing edge separation", AIAA Journal, Vol. 21, No. 11.

van Rijn, LC (1982) “Equivalent roughness of alluvial bed”, ASCE Jour-nal of Hydraulic Engineering, Vol. 108, No. 10.

van Rijn, LC (1987) "Mathematical modelling of morphological proc-esses in the case of suspended sediment transport", Ph.D Thesis, DelftUniversity of Technology.

Rodi, W. (1980) "Turbulence models and their application in hydraulics",



IAHR State-of- the-art paper.

Rouse, H. (1937) "Modern Conceptions of the Mechanics of Fluid Turbu-lence", Transactions, ASCE, Vol. 102, Paper No. 1965.

Rouse, H., Yih, CS and Humphreys, HW (1952) “Gravitational con-vection from a boundary source”, Tellus, pp. 201-210.

Schall, JD (1983) “Two-dimensional investigation of shear flow turbu-lence in open-channel flow”, PhD dissertation, Colorado State Univer-sity, USA.

Schlichting, H. (1936) “Experimental Investigations of Roughness”, Proc.Soc. Mech. Eng., USA.

Page 140


Schlichting, H. (1979) "Boundary layer theory", McGraw-Hill.

Schumm, SA, Mosley, MP and Weaver, WE (1987) “Experimentalfluvial geomorphology”, John Wiley & Sons Publishers.

Shields, A. (1936) “Use of dimensional analysis and turbulence researchfor sediment transport”, Preussen Research Laboratory for Water andMarine Constructions, publication no. 26, Berlin (in German).

Seed, D. (1997) "River training and channel protection - Validation of a3D numerical model", Report SR 480, HR Wallingford, UK.

Spallart and Allmaras (1994) “A one-equation turbulence model for aero-dynamic flows”, La Recherche Aerospatiale, no 1. pp 5-21.

Speziale, CG (1987) “On nonlinear Kl and ke models of turbulence”,Journal of Fluid Mechanics, vol. 178, pp. 459-475.

Stigebrandt, A., (1978) “Three-dimensional selective withdrawal”, Report No. STF60 A78036, The Norwegian River and Harbour Laboratory,Trondheim, Norway.

Steen, JE. and Stigebrandt, A. (1980) “Topological control of three-dimensional selective withdrawal”, Second Int. Symp. on StratifiedFlows, Trondheim, Norway, pp.447-455.

Streeter, HW and Phelps, E- B. (1925) "A study of the pollution andnatural purification of the Ohio River", US Public Health Service, Wash-ington DC, Bulletin 146.

Vanoni, V., et al (1975) "Sedimentation Engineering", ASCE Manualsand reports on engineering practice - No54.

Wu, J. (1969) "Wind Shear Stress and Surface Roughness at Air-SeaInterface", Journal of Geophysical Research, No. 74, pp. 444-455.

Yang, TC (1973) "Incipient Motion and Sediment Transport", ASCEJournal of Hydraulic Engineering, Vol. 99, No HY10.

Zimpfer, GL (1975) “Development of laboratory river channels”, PhD

Thesis, Department of Earth Resources, Colorado State University.



Page 141


Appendix I. Source code for explicit solution

of Saint-Venant equations

#include "stdio.h"#include "math.h"

main()

{

FILE *in;FILE *out;

int i,j,k,n;double timein[100];double qinn[100];double depth[200][2];double velocity[200][2];double time, y, u, alpha, beta, q, dummy;double timestep = 3.0;double deltax = 50.0;double slope = 0.005;double manning = 30.0;int sections = 99;

in = fopen("inflow","r");out = fopen("outflow","w");fclose(out);

/* reading time series */

n=0;for(j=0;j<100;j++) {

if(fscanf(in,"%lf %lf",&timein[j], &qinn[j]) != 2) break;n++;}

fclose(in);

/* initialization */

y = pow (qinn[0]/(manning*sqrt(slope)),0.6);u = qinn[0] / y;for(i=0;i<sections+1;i++) {

depth[i][0] = y;depth[i][1] = y;velocity[i][0] = u;velocity[i][1] = u;}

/* main loop */

time = timein[0];for(j=0;j<=1000;j++) {



Page 142


/* boundary conditions */

for(k=0;k<n;k++) if(timein[k]>time) break; beta = (timein[k] - time) / (timein[k] - timein[k-1]);

q = qinn[k-1] * beta + qinn[k] * (1.0-beta);y = pow (q/(manning*sqrt(slope)),0.6);u = q / y;

velocity[0][0] = u;velocity[0][1] = u;depth[0][0] = y;depth[0][1] = y;

velocity[sections][0] = velocity[sections-1][1];depth[sections][0] = depth[sections-1][1];

/* first computation of the water depth, according to Eq. 3.4.5 */

for(i=1;i<sections;i++) {depth[i][1] = depth[i][0] - timestep / (2.0 * deltax) * (velocity[i][0] *

(depth[i+1][0] - depth[i-1][0]) + depth[i][0] *(velocity[i+1][0] - velocity[i-1][0]));

}

/* compute the water velocity */

for(i=1;i<sections;i++) {dummy = - velocity[i][0] * timestep * 0.5 / deltax

* (velocity[i+1][0] - velocity[i-1][0]);dummy += - 9.81 * (timestep * 0.5 / deltax) * (depth[i+1][0] -

depth[i-1][0]);dummy += 9.81 * slope * timestep;dummy += - velocity[i][0] * velocity[i][0] * timestep * 9.81

/ (pow(depth[i][0],1.3333) * manning * manning);velocity[i][1] = velocity[i][0] + dummy;}

/* depth according to continuity - control volume approach*/

for(i=1;i<sections;i++) {u = 0.5 * (velocity[i][0] + velocity[i][1]);depth[i][1] =

(0.25 * (velocity[i-1][1] + velocity[i-1][0]) *(depth[i-1][1]+depth[i-1][0])+ depth[i][0] * (deltax / timestep - 0.5 * u ))/ (deltax / timestep + 0.5 * u );

}

/* updating variables */

for(i=1;i<sections;i++) {

velocity[i][0] = velocity[i][1];depth[i][0] = depth[i][1];}

Page 143




/* printing */

time = time+timestep;out = fopen("outflow","a");fprintf(out,"%lf ", time);fprintf(out,"%lf ",velocity[0][1]*depth[0][1]);fprintf(out,"%lf ",velocity[1][1]*depth[1][1]);fprintf(out,"%lf ",velocity[sections/4][1]*depth[sections/4][1]);fprintf(out,"%lf ",velocity[sections/2][1]*depth[sections/2][1]);fprintf(out,"%lf ",velocity[sections-1][1]*depth[sections-1][1]);fprintf(out,"\n");fclose(out);

}

}

INFLOW FILE:

0.0 10.0100.0 15.0200.0 20.0

300.0 15.0400.0 10.010000.0 10.0

Page 144


Appendix II List of symbols and units

The units are in brackets []



Latin

SEBUAH area [m 2] B width of a river/channel [m]c,C constants in k-ε turbulence model [dimensionless]c concentration [ppm, kg/m 3, dimensionless]d sediment particle diameter [m]

D habitat preference index E specific energy height [m] F flux [kg/s]g acceleration of gravity [m/s 2]h water depth [m]

Aku slope [dimensionless] Akub, Aku0 bed slope [dimensionless] Aku f, Akue, friction or energy slope [dimensionless] Aku heat fluxk friction loss coefficient [dimensionless]k turbulent kinetic energy [m 2/ s2]k coefficient for biological reactions [1/day]

K coefficient in hydrologic routing method [s] M Manning-Stricklers friction coefficient [m 1/3/s]n Mannings friction coefficient [s/m 1/3]

N Brunt-Väisälä frequency [1/s] p habitat availability index P pressure [N/m 2] P wetted perimeter [m]q water discharge/width [m 2/s]q s sediment discharge/width [kg/s/m]

Q water discharge [m 3/s]r hydraulic radius [m]r distance from centerline of plume [m]r HSI index (habitat hydraulics)t time [sec.,days]T temperature [ 0C, 0 K ]U average velocity [m/s]u fluctuating velocity [m/s]u* shear velocity [m/s]V average velocity in direction 2 [m/s]V volume [m 3]v fluctuating velocity [m/s]W average velocity in vertical direction [m/s]w fluctuating velocity in vertical direction [m/s]

x spatial variable [m] x coefficient in hydrologic routing method [dimensionless] y spatial variable [m] y water depth [m] z spatial variable - vertical distance [m]

Yunani

ε dissipation of turbulent kinetic energy [m 2/ s2]γ specific density for water [N/m 3]

Page 145


κ constant in velocity wall law (0.4) [dimensionless]μ dynamic viscosity of water [Ns/m 2] (1.3x10 -3 pada 200C) ν kinematic viscosity of water [m 2/s] (1.3x10 -6 pada 200C) ν

T turbulent eddy-viscosity [m 2/s]ρ density of water [kg/m 3]ρ

w density of water [kg/m 3]ρ

s density of sediments [kg/m 3] (often set to 2.65 kg/dm 3)τ shear stress [N/m 2]τ∗ dimensionless shear stress [dimensionless]ξ coordinate system direction 1 [dimensionless]ψ coordinate system direction 2 [dimensionless]ζ coordinate system direction 3 [dimensionless]



Γ diffusion coefficient [m 2/s]

Page 146


Appendix III Solutions to selected problems

Chapter 2, Problem 2

First, compute the cross-sectional area:

A = 8x2+10x+15x2 = 96 m 2

The wetted perimeter is:

P = 2+15+3+10+3+8+2 = 43 m

The hydraulic radius is:

Mannings formula gives the water velocity:

r SEBUAH

P ---

96

43----- 2.23 m= = =



Nona

The water discharge is:

Q = UA = 4.5x96 = 440 m 3/ s

When the Manning-Strickler friction coefficient varies, the same proce-dure and formulas have to be used for each part of the cross-section.The following table gives the results:

The total discharge is the sum of the discharges in the three parts.

Q = 78+286+39=403 m 3/s.

Section A Section B Section C

Daerah 30 50 16

Wetter perimeter 17 16 10

Hydraulic radius 1.76 3.12 1.6

Velocity 2.6 5.73 2,44

Pelepasan 78 286 39

U MI 12---r

23--- 601

500---------2.23

23--- 4,5= = =

Page 147



First, find the water level at B , from the equation for the hydraulic jump:

The velocity at C :

Nona

The Froude number at C is:

The depth is then given by:

dan

h B

hC

------1

2--- 1 8 Fr

C 2+ 1-( )=

U C

Q

yC

-----6

3--- 2= = =

Fr C

2 U C 2

gyC

--------2 2

9.81 x 3----------------0.136= = =

h B

hC

-----1

2--- 1 8 Fr

C 2+ 1-( ) 1

2--- 1 8 x 0.136 2+ 1-( ) 0,222= = =

h B

0.222 hC

0.222 x 3 0.667 m= = =



The main method to compute the distance from A to B is to start at A andcompute the water surface profile. When it reaches 0.667m, this will bethe location of the jump.

The following formula is used:

It is converted to:

This has to be solved in a table, where I f and Fr are computed. I f is computed by Manning's formula:

dy

dx------

Aku f

Aku0-

1 Fr 2------------------=

x Δ Δ y 1 Fr 2-( )

Aku f

----------------------=

Aku f

U 2

M 2 y

43---

-------------=

Page 148


The distance from A to B is 53 meters.

Chapter 2, problem 5

The integral of the concentration over time becomes:

The discharge is:

Table 1:

Δ y y U Aku f Fr 2 Δ x x

- 0,4 - - - - 0

0.05 0,425 14.11 0.25 47,8 9.4 9.4

0.05 0.475 12.63 0.172 34.24 9.7 19,1

0.05 0,525 11,43 0.123 25.36 9.9 29,0

0.05 0.575 10.43 0.0910 19.28 10,0 39.0

0.037 0.6185 9.70 0.0714 15.51 7.5 46,5

0.03 0.652 9.2 0.0599 13.24 6.13 52,6

Aku c td c Δ t 500 x 1 ppmminutes= = =

Qm

c td

----------m

Aku---

2 kg

500 min

10 6------------------

------------------4000kg

min--------- 4

m3

min--------- 0.067

m 3

s------ 67

l

s-= = = = = = =




The continuity equation gives the water velocity as:

The shear stress is:

The shear velocity is:

Computing the diffusion coefficient. Using Fischer et al. (1979): (Eq.4.3.3)

U Q

Oleh------

20

2 x 30------------ 0.33

m

s----= = =

τ ρ gyI

1000 x 9.81 x 2 x

1

63----- 313 Pa= = =

u*

τ

ρ---313

1000----------- 0.56

m

s---= = =

Page 149


The initial concentration is equal to the mass divided by the volume ofwater in the river in the time period:

These numbers we use in the general equation for the concentration

downstream of a spill: (Eq. 4.3.4)

This equation is plotted below:


Γ 0,011UB( )2

Hu*

---------------0,0110.33 x 30( )2

2 x 0.26--------------------------0.96= = =

c0m

ρ Qt ----------

2000

1000 x 20 x 10 x 60( )---------------------------------------------0.000167 167 ppm= = = =

ct ( )c0 L

2 πΓ t ---------------- e

x Ut -( )24Γ t

---------------------- 167 x 200

2 0.96π t ----------------------- e

10000 0.33 t -( )23.84 t

----------------------------------------= =



The densimetric Froude number is:

The velocity is computed from:

Fr ′u

0ρ soal

ρ0-

ρ soal

-------------------- gd 0

---------------------------------------

50

3.14 x 1.5 x 1.5( )------------------------------------

1023 1000-

1023------------------------------9.81 x 3

-------------------------------------------------- ------8,7= = =

Page 150


The velocity is equal to the initial velocity, or 7 m/s. The fact that thevelocities are the same in this case is coincidental.


The shear stress on the lake is computed from Eq. 7.2.1 as

The slope of the water surface is given by Eq. 7.2.3:

The density at 15 oC is approximately 999.12 kg/m 3 menuruttable in Chapter 7.4, and the density for 5 oC is 999.99 kg/m 3.

The slope of the thermocline is given by Eq. 7.4.10:

u

u0----- 4.3 8.7( )

23---- 20

3------

13----

e

96 02

202--------

1.01= =


τ c10ρ

SebuahU

Sebuah2 1.1 x 10 3-

x 1.2 x 152 0.3 Pa= = =

Akuτ

ρ gh----------

0,3

1000 x 9.81 x 100--------------------------------------3 x 10 7-= = =



The thermocline will tilt around an axis in the middle of the lake. The dis-

I ′ I –ρ 1ρ′----- 3 x 10 7- x 1000

999.99 999.12----------------------------------------- 3.5 x 10 4-= = =

Page 151


tance from the middle of the lake to the far ends is half the length of thelake. The rise/fall in the thermocline at these locations will be:


We start by computing the density gradient from the temperature gradi-ent. The temperature gradient at 10 meter is 0.5 degrees/meter. Di sana

will therefore be 4 meters between 10 and 12 degrees. The table inChapter 7.3 gives the densities at these temperatures as 999.73 and999.52. The density gradient becomes:

The Brunt-Väisälä frequency is given as:

(7.8.2)

This is inserted into the formula for the thickness of the abstraction layer:

(7.8.4)

The thickness of the abstraction layer is both above and below theintake. Therefore, it will only reach half the thickness above the intake, or7.5 m. This is below 9 meters, which means the intake will not take waterfrom the upper 1 meter layer.


A stable condition means that the loss is equal to the inflow: 2000 kg/hari. The loss rate is 0.1, meaning 2000 kg/day is 10 % of the total. Thetotal phosphorous is therefore 2000/0.1 = 20 000 kg. The concentrationaku s:

The steady state concentration for an inflow of 300 kg/day is similarly 3 ppm. The reduction will therefore be 17 ppm. 20 % of the original valueis 4 ppm. After time t , there will be 4-3=1 ppm left of the reduction. Thetime is described by the following equation:

Δ h L

2--- I '

5000

2------------ x 3.5 x 104- 0.87 m= = =

d ρ

dz ------

999.52kg

m 3------ 999.73

kg

m 3-------

4 m-----------------------------------------------------0.05- 25

kg

m3------= =

N 2 g ρ---–

d ρ

dz -----

9,81

1000------------ 0.0525 –( )- 0.000515 Hz = = =

d k 3

Q 2

B2hN 2---------------

13---

1.2

2m 3 s------2

2 m( )2 x 1 m 0.000515 Hz ( )-------------------------------------------------- -----------

13---

15 m= = =

c20000 kg

10 6 x 103kg

--------------------------2 x 10 5- 20 ppm= = =



Page 152



Assume uniform flow and that the bed shear stress is equal to the criticalshear stress given by Shields graph. Also, assume the channel is wide,so that the hydraulic radius is equal to the water depth. This gives sixunknown and six equations. The unknown are:

- water depth h- water velocity U - particle diameter d - bed roughness k s- Manning-Stricklers friction factor M - bed shear stress τ

The six equations are:- Water continuity: Q=BhU

(2.1.1)

(2.1.4)

(2.2.4)

(2.2.5)

From Shields graph

The equations are best solved in the following manner:

1. Guess a Manning's coefficient, M .2. Computer the water velocity using Eq. 2.2.43. Compute the water depth using the continuity equation4. Compute the shear stress using Eq. 2.2.1.5. Compute R* in Shields diagram6. Find τ∗ from Shields diagram7. Compute d from the equation in Shields graph8. Compute k s using 2.1.49. Compute Mannings M value from Eq. 2.2.5.10 Check if M is equal to what was assumed in point 1. If it is very differ-

ent, do another iteration starting from point 1, with the M value from point 9.

Using this method we get:

1. M =30 (guess)2. U =2.6 m/s3. y = 1.3 m4. τ = 60 N/m25. R * = 18 0006. τ∗ = 0.06

cc0e kt -= 1 17 e 0.1 t -= t 10 17

1------ln 28 days= =

τ ρ ghI =

k s

3 d 90=

U M r h

23--- Aku

12---

=

M 26

k s

16---

-----=

τ*=τ

c g ρ

sρ

w-( ) d

----------------------------

Page 153

152



Numerical Modelling and Hydraulics

7. d = 0.06 m8. k s = 0.18 m9. M = 3410. Iteration with M=34 gives d =0.06 m.


First, the particle fall velocity is obtained from Fig. 9.3.1 to be 0.15 m/s.

Then some basic parameters are computed:

Friction slope:

Bed shear:

= 21.42745673 N/m 2

Shear velocity:

= 0.146381203 m/s

Particle Reynolds number:

Shields curve gives the Shields parameter: C = 0.05

Critical shear stress:

= 0.809 N/m 2

The suspension parameter:

Engelund-Hansens formula:

Aku f

U 2

M 2 y

43---

-------------2,5 2

50 2 x 1.5

43---

----------------------0.001456= = =

τ ρ gyI f

1000 x 9.81 x 1.5 x 0.001456= =

u*

τ

ρ---

21,42

1000------------= =

R *

u*d

ν--------0.14638 x 0.001

10 6-----------------------------------146= = =

τc

Cg ρ s

ρw

-( ) d 0.05 x 9.81 x 2650 1000-( ) x 0.001= =

Z wκ u*

--------- 0,150.4 x 0.1464---------------------------2.5618= = =

q s

0.05ρ s

U 2d 50

g ρ s

ρw

------ 1-

------------------------τ

g ρ s

ρw

-( ) d 50

----------------------------------

32---

=

Page 154


q s

0.05 x 2650 x 2.52 0,001

9,812650

1000------------ 1-

-------------------------------------21,43

9.81 2650 1000-( )0.001-------------------------------------------------- ---------

32---

=



= 9.9 kg/m/s

Then we use van Rijn's formula for bed material load:

qb = 0.002295 m 2/ s

Van Rjin gives q b in m 2/s. This is transformed to kg/s by multiplying it

with the sediment density: 2650 kg/m 3, giving:

qb = 6.082 kg/s/m

Then we use Van Rijn's formula for suspended load. We assume the ref-erence level at the bed is equal to 5 % of the water depth, or 1.5x0.05 =0.075 m. The reference concentration is:

ctempat tidur = 0.009756758 (volume fraction)

The sediment transport caused by this reference concentration onlytakes place above the reference level. To compute the suspended sedi-ment transport, we can divide the water column in layers. For conven-ience, let us assume the first layer is twice as high as the reference level.This is then 10 % of the water depth, or 0.15 m. And that there are threemore layers of equal size:

qb

D 501,5

ρ s

ρw

-( ) g

ρw

--------------------------

-----------------------------------------0.053

τ τc

-

τc

-------------2.1

D 500,3

ρ s

ρw

-( ) g

ρw ν 2

-------------------------0,1

----------------------------------------------------=

qb

0.053

21.43 0.809-

0.809--------------------------------

2.1

0,001 0,3 2650 1000-( )9.81

1000 x 10 6-( )2---------------------------------------------

0,1---------------------------------------------------------------------------- 0.0011,5 2650 1000-( )9.81

1000---------------------------------------------=

ctempat tidur

0,015 D 50

Sebuah---------

τ τc

-

τc

-------------1,5

D 50

ρ s

ρw

-( ) g

ρw ν 2

-------------------------

13---

0,3-------------------------------------------------- ------0,015

0,001

0,075-------------

21.43 0.809-

0.809--------------------------------

1,5

0,0012650 1000-( )9.81

1000 x 10 6-( )2---------------------------------------------

13---

0,3-------------------------------------------------- -----------------------------= =

Page 155


(1.5 - 0.15)/10 = 0.45 m

We then use the Hunter Rouse formula to compute the concentration inthe center of each cell, and the logarithmic velocity profile to computethe velocities. This can then be multiplied with the height of each cell togive an estimate of the suspended load.

Table 2:

Seltidak.

Jarakfrom bed

Velocity Hunter Rousekonsentrasi

Seltinggi

Cell flux



The sum of the cell fluxes is: 0.14. This can be multiplied with the bedconcentration to give the sediment transport in m 2/s. Then multiplied

with the density, we get the sediment transport in kg/s/m:q s = 0.14* 0.009756758*2650 = 3.6 kg/s/m.

Note that the height of the cell closest to the be has only been set to0.075 m, although it is 0.15 meters. This is because the sediment transport below the reference level is considered bed load, and computed bythe bed load formula.

Total load according to van Rijn is then:

qt = 3.6 + 6.08 = 9.7 kg/s/m.

Also note that there are only four layers in the vertical direction. If 11 lay-ers had been used, we would have gotten 2 % higher result for the suspended load.

In our case, the results by van Rijn's method and Engelund-Hansens for-mula are very similar. This is normally not the case. Often, results from

different sediment discharge formulas may deviate with a factor 2-3 orlebih.

4 1.275 2,77 6.22E-06 0.45 7.7E-06

3 0.825 2,61 0.0003168 0.45 0,00037

2 0.375 2.32 0.00883 0.45 0.00923

1 0,075 1.73 1 0,075 0.130

Page 156


Appendix IV: An introduction to program-

ming in C

This chapter is written as a brief introduction to C programming for stu-dents who has not had previous courses on programming.

A computer program is written as text file using an editor. In the text file,different commands tells the computer what to do. This text file is oftencalled a source code. The text file needs to be transformed into a computer program. The file containing the computer program is often calledthe executable file. The transformation of the text file to the executablefile is done using a compiler. Depending on the language of the text file,different compilers are used. For the C language, we need a C compiler.In the present course we will use the LCC compiler, which is freeware.

The C program consists of the following structure:



main () {

Variable declarations

Commands on what to do

}

Note the brackets, which are necessary. Omitting one leads to the compiler giving an error message, and the executable is not produced.

Variable declarations

To make the program compute something, we need to declare variables.In a spreadsheet, a variable is named after its address, for example b12 .In C, we can give any name to a variable. Sebagai contoh:

int counter ;double flux ;

The variable named counter is declared as an integer, and the variableflux is declared as double , a floating point number with 12 digits preci-sion. Note the ; after each declaration.

It is also possible to declare arrays of integers or floats:

double velocity [100], depth [100];

If we have 100 cross-sections in a river, the velocity in section 14 isgiven as velocity [ 14 ].

We can also use multi-dimensional arrays:

double discharge [2][100];

In C, the first number in the array is zero, and the last array in the arraysabove is 99, if decreased with 100 elements.

Page 157


We can also declare files with different names, used to read input-datafrom and write output-data to:

FILE * input : FILE * result ;

Note the * that needs to be included for the files.

A file needs to be opened before we can read from it. The syntax for thisaku s:

input = fopen ("inflow","r");

Similar for a file we want to write to:

result = fopen ("outflow","w");

The file names are here given as inflow and outflow. These are text files.

After writing to a file, it needs to be closed for other programs to readfrom it:

fclose ( result );

To read information from a file to a variable, the following syntax is used:

fscanf ( input ,"%lf",& depth [1]);

Note the syntax. If this is not correct, the compiler will produce an errormessage, or the program will not work.



Similarly, the following syntax is used to write to files:

fprintf ( result ,"%lf ", velocity [0]);

Note the & is used when reading data and not when writing data to files.

Commands

The program is made up of a series of commands. They will be carriedout in the same order as given in the file. There are several types of perintah.

Variables can be initialized using the = operator. Sebagai contoh:

velocity [0] = 2.0;

Variables can be incremented. For example, in the following, the variable counter will increase its value by one:

counter = counter + 1;

If count was 3 before this line, it will be 4 afterwards. Incrementing aninteger with one can also be written

counter ++;

This does the same as the line before.

Page 158


A typical command is to write a formula:

velocity [1] = discharge / depth [1];

A formula can be long and complex, and one can use * for multiplication,and + and - for addition and subtraction. Also, one can use brackets incomplex formulas, to several levels:

velocity [1] = discharge / ( depth [1] + 1.0e-20);

The above formula is a trick to avoid errors if depth [1] is zero.

Repetition of formulas over many cross-sections is done in a loop. Di sanaare different types of loops, but the for loop is often used. Mengikutiloop repeats the above formula for all the 100 cross-sections in the river,starting from cross-section no. 0 to cross-section no. 99:

for ( counter = 0; counter <100; counter ++) {

velocity [ counter ] = discharge / ( depth [ counter ] + 1.0e-20);

}

Several formulas can be used inside a loop, and it is also possible tonest several loops inside each other.

An if sentence can be used to give a special logic. Sebagai contoh:

if ( froude > 1.0 ) {

depth = 2.0;

}

An if sentence can also be used to break out of a loop:

for( counter = 0; counter <100; counter ++) {



if ( depth [ counter ] < 1.0e-20) break;velocity [ counter ] = discharge / depth [ counter ];

}

This loop will be stopped if depth is below 10 -20, even if counter has notreached 99. The program will then jump out of the loop and do the nextthing in the file.

Remember that the syntax is very important. One wrong type of brack-ets, for example, will give a compiler error.

The introduction given above is only a very small fraction of all the com-mands, declarations etc. available in C. For more details, a textbook onC programming should be consulted.

numerik pemodelan dan hidrolik

Documents