H3A>3H=0<82B >5 A44 (DA5024 ;>FB Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0 J.-M. Hervouet H3A>3H=0<82B >5 A44 (DA5024 ;>FB *I>?FFCHA QCNB NB? VHCN? ?F?G?HN G?NBI> 40="8274; 4AE>D4C (&(0 &" &" c 2007 Copyright John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777 Email (for orders and customer service enquiries): [email protected] Visit our Home Page on www.wiley.com All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher. 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Pacifico British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 978-0-470-03558-0 (HB) Typeset in 10/12pt Times by Laserwords Private Limited, Chennai, India Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production. >=C4=CB !8BC >5 J6DA4B G8 !8BC >5 C01;4B G888 !8BC >5 ?;0C4B GE 2:=>F;4364<4=CB =CA>3D2C8>= 1Q?HNS S?;LM I@ >?P?FIJG?HN ;N "!# 0IG? MGIINB?L J?<<F?M 0;CHN 3?H;HN ?KO;NCIHM +;PC?L\0NIE?M ?KO;NCIHM #CHCN? ?F?G?HN N?=BHCKO?M ;H> IJNCGCT;NCIH @D0C8>=B >5 5A44 BDA5024 7H3A>3H=0<82B +IN;NCIHM ;H> =IH=?JNM CH A?IG?NLS #L?? MOL@;=? +;PC?L\0NIE?M ?KO;NCIHM +IH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM IOH>;LS =IH>CNCIHM %S>LIMN;NC= JL?MMOL? ;H> NB? IOMMCH?MK ;JJLIRCG;NCIH 0IOL=? 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JLIJ;A;NCIH /CNN?L ;H> 0NIE?L MIFONCIHM '4B>;DC8>= >5 C74 #0E84AO(C>:4B 4@D0C8>=B &HNLI>O=NCIH "KO;NCIHM QCNB BS>LIMN;NC= BSJINB?MCM #L;=NCIH;F MN?JM G?NBI> 0J;NC;F >CM=L?NCT;NCIH OCF>CHA NB? NBL?? >CG?HMCIH;F G?MB 0CAG; NL;HM@ILG ;H> A?H?L;FCT?> MCAG; NL;HM@ILG >P?=NCIH MN?J !C_OMCIH MN?J -L?MMOL?\=IHNCHOCNS MN?J +IH BS>LIMN;NC= +;PC?L\0NIE?M ?KO;NCIHM %S>LIMN;NC= MN?J IHNCHOCNS MN?J ;F=OF;NCIH I@ @L?? MOL@;=? -M?O>I Q;P? ?KO;NCIH ; P;LC;HN I@ NB? BS>LIMN;NC= MN?J 4CNB ?RJFC=CN >C_OMCIH 4CNB CGJFC=CN >C_OMCIH 1L;=?L ?KO;NCIH CH ! 1B? =IHM?LP;NCIH I@ G;MM CH ! IHM?LP;NCIH I@ G;MM I@ Q;N?L IHM?LP;NCIH I@ NL;=?L 0IOL=?M ;H> MCHEM CH NB? +;PC?L\0NIE?M ?KO;NCIHM 0IOL=?M I@ WOC> 0IOL=?M I@ NL;=?L QCNB ; >CMNLC<ONCP? M=B?G? !CLC=BF?N <IOH>;LS =IH>CNCIHM QCNB ; >CMNLC<ONCP? M=B?G? 0IOL=?M I@ NL;=?L QCNB NB? 02-$ ;>P?=NCIH M=B?G? !CLC=BF?N <IOH>;LS =IH>CNCIHM QCNB ; 02-$ ;>P?=NCIH M=B?G? 1L?;NG?HN I@ OH=IP?L?> <?>M CH ! ,JNCIH =ILL?=NCIH I@ @L?? MOL@;=? AL;>C?HNM ,JNCIH G;MECHA I@ NB? OH=IP?L?> ?F?G?HNM (>;E8=6 CA0=B?>AC 4@D0C8>=B *?NBI> I@ =B;L;=N?LCMNC=M 1L;D?=NILS =;F=OF;NCIH &HN?LJIF;NCIH *?NBI> I@ =B;L;=N?LCMNC=M CH Q?;E @ILGOF;NCIH 0NL?;GFCH? 2JQCH> -?NLIP\$;F?LECH G?NBI> IHM?LP;NCP? M=B?G? @IL NB? =IHNCHOCNS ?KO;NCIH !CMNLC<ONCP? M=B?G?M + -0& M=B?G?M CH NQI >CG?HMCIHM *2/! M=B?G? CH NBL?? >CG?HMCIHM PCCC ,+1"+10 ">34A= C427=8@D4B 8= J=8C4 4;4<4=CB !?M=LCJNCIH I@ VHCN? ?F?G?HNM 1LC;HAF? QCNB FCH?;L CHN?LJIF;NCIH .O;MC <O<<F? NLC;HAF? 1?NL;B?>LIH -LCMG *?MB >?M=LCJNCIH 1QI >CG?HMCIH;F G?MB 1BL?? >CG?HMCIH;F G?MB *;NLCR MNIL;A? !C;AIH;F MNIL;A? IGJL?MM?> 0J;LM? /IQ "" MNIL;A? ">A? <;M?> MNIL;A? ,J?L;NCIHM IH G;NLC=?M OCF>CHA NB? G;NLC=?M MM?G<FCHA ; G;NLCR @IL ?>A? <;M?> MNIL;A? MM?G<FS I@ ;H ?F?G?HN;LS P?=NIL -LI>O=N I@ ; HIH MSGG?NLC= "" G;NLCR ;H> ; P?=NIL -LI>O=N I@ ; MSGG?NLC= "" G;NLCR ;H> ; P?=NIL -LI>O=N I@ ; NL;HMJIM?> "" G;NLCR ;H> ; P?=NIL -LI=?MMCHA I@ NB? !CLC=BF?N <IOH>;LS =IH>CNCIHM -LI>O=NM <?NQ??H ; >C;AIH;F G;NLCR ;H> ; G;NLCR *;NLCR\P?=NIL JLI>O=N QCNB ?>A? <;M?> MNIL;A? *;NLCR\P?=NIL @LIHN;F JLI>O=N )CH?;L MSMN?G L?MIFONCIH *?NBI>M I@ >?M=?HN ;H> JL?=IH>CNCIHCHA IHDOA;N? AL;>C?HN G?NBI> IHDOA;N? L?MC>O;F G?NBI> IHDOA;N? AL;>C?HN IH NB? HILG;F ?KO;NCIH *CHCGOG ?LLIL G?NBI> $*/"0 G?NBI> -L?=IH>CNCIHCHA !C;AIH;F JL?=IH>CNCIHCHA 9\K JL?=IH>CNCIHCHA @IL NB? 0;CHN 3?H;HN ?KO;NCIHM FI=E >C;AIH;F JL?=IH>CNCIHCHA -L?=IH>CNCIHCHA QCNB (1 @;=NILCT;NCIH LION JL?=IH>CNCIHCHA $;OMM\0?C>?F "" JL?=IH>CNCIHCHA LION ;H> $;OMM\0?C>?F JL?=IH>CNCIHCHA @IL ?>A? <;M?> MNIL;A? %0A0;;4;8B< &HNLI>O=NCIH -;LNCNCIH I@ NB? >IG;CH /?KO?MN?> >;N; IGGOHC=;NCIH <?NQ??H JLI=?MMILM -ICHN NI JICHN =IGGOHC=;NCIH ,+1"+10 CR IFF?=NCP? =IGGOHC=;NCIH -;L;FF?FCMG FC<L;LS >;JN;NCIH I@ NB? ;FAILCNBGM !;N; MNLO=NOL? MJ?=CV= NI J;L;FF?FCMG 0=;F;L JLI>O=N *;NLCR\P?=NIL JLI>O=N 0J?=CV= =;M?M /?MOFNM FOMN?L I@ QILEMN;NCIHM %- /6 1" ,LCACH <CN &/&5 G %0A0<4C4A 4BC8<0C8>= &HNLI>O=NCIH -LCH=CJF? I@ NB? ;>DICHN G?NBI> 1L?;NG?HN I@ !CLC=BF?N =IH>CNCIHM JJFC=;NCIH NI 1?F?G;= ! ;FAILCNBG #LC=NCIH =I?^=C?HN ?MNCG;NCIH CH 1?F?G;= ! ;F=OF;NCIH I@ LCABN B;H> MC>?M I@ ;>DICHN MSMN?GM !?M=?HN G?NBI>M @IL GCHCGCTCHA NB? =IMN @OH=NCIH IHDOA;N? AL;>C?HN 3;FC>;NCIH ;H> L?MOFNM ??;820C8>=B *;FJ;MM?N >;G <L?;E Q;P? &HNLI>O=NCIH 1B? *;FJ;MM?N >;G ;==C>?HN P;CF;<F? >;N; 1QI >CG?HMCIH;F HOG?LC=;F GI>?FFCHA /?MOFNM -?L@ILG;H=?M IH P;LCIOM G;=BCH?M *IL? L?=?HN L?MOFNM IH=FOMCIH 1C>?M CH NB? "HAFCMB B;HH?F ;H> NB? NF;HNC= ,=?;H "MNCG;NCIH QCNB ;>DICHN G?NBI> "MNCG;NCIH QCNB MOLLIA;N? GI>?FM H?OL;F H?NQILEM ;H> A?H?NC= ;FAILCNBGM 0IFCN;LS Q;P? #FIQ I@ NQI WOC>M I@ >C_?L?HN >?HMCNC?M Appendix A )83464=4A0C8=6 5>A24 ,=?;HC= MN;NC= NC>? -IN?HNC;F >?J?H>CHA IH NB? A?I=?HNLC= T?HCNB >CMN;H=? -IN?HNC;F >?J?H>CHA IH NB? F;NCNO>? >?=FCH;NCIH ;H> BIOL ;HAF? !?=IGJIMCNCIH I@ BIOL ;HAF? MC>?L?;F NCG? *IP?G?HN I@ NB? GIIH *;MM I@ NB? GIIH !CMN;H=? NI NB? ";LNB FIHACNO>? I@ NB? GIIH CH CNM IL<CN R ,+1"+10 !?=FCH;NCIH ;H> LCABN ;M=?HMCIH *IP?G?HN I@ NB? MOH *;MM I@ NB? MOH !CMN;H=? NI NB? ";LNB FIHACNO>? I@ NB? MOH !?=FCH;NCIH ;H> LCABN ;M=?HMCIH I@ NB? MOH Appendix B 8QDB8>= <0CA8G F8C7 C4CA0743A0 Appendix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fFCH? 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NBCM <IIE Z;M BOA? ;M %;LLS -INN?L &3[ ;LIFCH? ;A?> Plate 1: porosity in a rectangular channel: effect on free surface and velocity flow direction discharge: 100 m3 /s depth: 5 m 20 10 0 -10 -20 -150 -100 -50 0 50 100 150 area with porosity (see the porosity profile below) 5.01 5 4.99 4.98 4.97 4.96 4.95 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 free surface (m) 0 50 100 150 200 250 300 100 150 200 250 300 porosity velocity (m/s) 0 50 Plate 2: flow in a canal with bottom friction: 2- and 3-dimensional simulation with a Manning coefficient of 0.02 100 90 80 70 60 50 40 30 20 10 0 Velocity field and coloured surfaces of u velocity (m/s) 1 0.98 0.96 0.94 0.92 0.9 0 100 200 100 90 80 70 60 50 40 30 20 10 0 velocity (m/s) 1.000000 0.8 0.75 300 400 500 0.88 0.86 0.84 0.82 0.8 0.78 0.76 0 100 200 300 400 500 mesh 0.7 0.74 0.72 0.7 0.68 0.66 0.65 free surface (m) with 2D simulation 0.6 free surface (m) with 3D simulation 0.55 0.5 0.45 0 100 200 300 Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0 400 J.-M. Hervouet 500 Plate 3: bridge pier in a canal: simulation with porosity and drag force free surface (m) free surface with pier 20 4.012 10 4.011 0 4.010 -10 4.009 4.008 -20 -100 -80 -60 -40 -20 0 20 40 60 80 100 4.007 4.006 free surface with porosity and drag force 20 4.005 10 4.004 0 4.003 -10 4.002 4.001 -20 -100 -80 -60 -40 -20 0 20 40 60 80 100 4.000 4.0125 4.01 head loss due to the pier 4.0075 profile of free surface in the longitudinal axis of the channel 4.005 with pier 4.0025 friction alone, without pier with porosity and drag force 0 50 100 150 200 Plate 4: effect of 32 marine turbines (in white rectangle) on tidal currents: drag forces modelled in shallow water equations velocity norm (m/s) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 14000 13000 12000 11000 10000 9000 velocity vectors 1.000000 8000 7000 -7000 -6000 -5000 -4000 -3000 -2000 -1000 0 1000 2000 3000 Plate 5: Lynch test case (tide on a quadratic slope): linearized shallow water equations without advection 150000 125000 100000 75000 50000 150000.0 125000.0 75000.0 50000.0 0.0 25000.0 0 100000.0 25000 free surface (m) 2.40 2.30 150000 150000 2.20 2.10 125000 125000 2.00 1.90 100000 100000 1.80 75000 1.70 1.60 50000 1.50 1.40 25000 0 1.30 1.20 0 25000 50000 75000 100000 125000 150000 quasi-bubble element 1.10 1.00 75000 50000 25000 0 0 25000 50000 75000 100000 125000 150000 linear element Plate 6: Saint-Venant equations: von Karman eddies behind bridge piers 10 5 0 -5 -10 -10 velocity 1.000000 -5 0 5 10 -2.5 0 -1 -2 -3 -4 -5 -6 -7 -8 -10 -7.5 velocity 1.000000 -5 Plate 7: Saint-Venant equations: propagation of a wave, effect of boundary conditions after 2 s, one iso-line every 1/6 m 20 20 15 15 n 10 10 5 5 0 0 0 5 10 with reflection on the domain boundaries 15 20 0 5 10 with Thompson radiation conditions 15 20 2 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 0 5 5 flow free surface with primitive equations "exact" free surface free surface with pseudo-wave equation bottom imposed discharge mesh 10 10 20 exact free surface with "non-conservative" equations 15 20 constant free surface vertical cross-section in the axis y=1 15 Plate 8: transcritical flow over a bump in a channel with a hydraulic jump Plate 9: ebb tide on a beach: free surface at different times with imposed sinusoidal elevation on the left 0.1 Saint-Venant equations t=0 s 0 t=100 s -0.1 -0.2 t=170 s -0.3 -0.4 -0.5 -0.6 t=300 s 0 0.1 10 20 -0.1 40 hydrostatic Navier-Stokes t=0s 0 30 t = 100 s -0.2 t = 170 s -0.3 -0.4 -0.5 -0.6 t = 300 s 0 10 20 30 40 0.1 non-hydrostatic Navier-Stokes t=0s 0 -0.1 t = 100 s -0.2 -0.3 t = 170 s -0.4 -0.5 -0.6 t = 300 s 0 10 20 30 40 0.1 water at rest on a slope (all equations above) 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 0 10 20 30 40 -10 -5 0 5 10 -10 -5 0 5 10 -5 0 5 10 -10 -5 0 5 10 CPU time: 11 s wave equation, linear discretisation -10 CPU time: 21.7 s primitive equations, linear discretisation -0.42 -0.39 -0.36 -0.33 -0.30 -0.27 -0.24 -0.21 -0.18 -0.15 -0.12 -0.09 -0.06 -0.03 0.00 0.03 0.06 0.09 0.12 0.15 0.18 free surface (m) -10 -5 0 5 10 -10 -5 0 5 10 -5 0 5 10 -10 -5 0 5 10 CPU time: 20.8 s wave equation, quasi-bubble velocity -10 CPU time: 54.7 s primitive equations, quasi-bubble velocity Plate 10: Saint-Venant equations and von Karman eddies behind bridge piers: effect of discretization on the free surface Plate 11: transcritical flow over a bump in a canal with a hydraulic jump: non-hydrostatic Navier-Stokes equations with five planes vertically in the mesh 2 1.5 1 0.5 0 5 10 mesh seen from above 15 20 vertical cross-section in the longitudinal axis y = 1 0.7 0.6 0.5 0.4 0.3 exact free surface with Saint-Venant equations 0.2 bottom (m) free surface with Navier-Stokes equations 0.1 0 -0.1 -0.2 0 5 10 velocity (m/s) 10 0 15 20 velocity field in a vertical cross-section 5 10 15 20 dynamic pressure in a vertical cross-section pressure in Pa divided by 100000 0.008 0.004 0.000 -0.004 -0.008 -0.012 0 5 10 15 20 Plate 12: propagation of a solitary wave in a channel depth 10 m, wave amplitude 2 m Saint-Venant equations 2.5 t=0 s 2 t=10 s t=20 s t=30 s 1.5 t=40 s 1 0.5 0 -0.5 0 100 200 2.5 300 400 500 600 400 500 600 500 600 Boussinesq equations 2 1.5 1 0.5 0 -0.5 0 100 2.5 200 300 non-hydrostatic Navier-Stokes equations 2 1.5 1 0.5 0 -0.5 0 100 200 300 400 Plate 13: Saint-Venant equations: propagation of a circular wave initial free surface 20 5 one iso-line every 25 cm 4 15 3 10 2 free surface profile (m) 5 1 0 0 5 10 15 20 0 cross-section at y = 10.05 m 0 5 10 15 20 free surface after 1 s 20 5 4 15 3 10 2 free surface profile (m) 5 1 0 0 5 10 15 20 0 cross-section at y = 10.05 m 0 5 10 15 20 Plate 14: dam break in a river, Stoker’s solution 6 5 4 3 computed depth at t = 1200 s exact depth initial depth 2 1 free surface at t = 0 and t = 1200 s 0 0 5000 10000 15000 20000 velocity computed at t = 1200 s 3 exact velocity 2 1 0 velocity at t = 1200 s 0 5000 10000 15000 20000 Plate 15: Malpasset site: contour of computational domain, position of measurement probes (6 to 14) and electric transformers (A,B,C) 6000 5000 A 6 4000 7 8 9 3000 position of dam 10 B 12 11 C 13 14 2000 1000 0 -1000 -2000 2500 5000 7500 10000 12500 15000 17500 Plate 16: Malpasset site: bathymetry and details of the mesh bottom (m) 100 90 80 70 60 dam 50 40 30 20 10 0 -10 -20 Plate 17: Malpasset dam break: velocity field after 100 s bottom (m) 100 90 80 70 mesh of 26000 elements 60 50 40 velocity (m/s) 10 30 20 10 0 -10 -20 mesh of 104000 elements (1 vector every 4 is shown) Plate 18: Malpasset dam break: depth and velocity after 20 minutes depth (m) 36 34 32 30 28 26 24 22 20 18 16 14 12 10 velocity (m/s) 10 8 6 4 2 0 Plate 19: Malpasset dam break: the flood wave reaches the sea after 36 minutes depth (m) 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 mesh of 104000 elements 5 4 3 2 1 0 10000 12500 15000 17500 12 14 16 -2000 -1000 0 1000 2000 3000 4000 2500 5000 7500 10000 Navier-Stokes 6 planes 12500 15000 17500 0 2 4 6 8 7500 10 5000 18 5000 2500 Shallow Water Equations 20 6000 -2000 -1000 0 1000 22 3000 3000 -2000 -1000 0 1000 2000 3000 4000 5000 6000 -2000 -1000 0 1000 2000 4000 4000 depth (m) 24 5000 5000 2000 6000 6000 5000 7500 10000 2500 5000 7500 10000 Navier-Stokes 10 planes 2500 Navier-Stokes 2 planes Plate 20: Malpasset dam break: flood after 2200 s 12500 12500 15000 15000 17500 17500 Plate 21: semi-diurnal tide in the Channel: tidal range and phase tidal range (m) 2.0 800000 1.9 tidal range 1.8 1.7 1.6 600000 1.5 1.4 1.3 400000 1.2 1.1 1.0 200000 0.9 0.8 0.7 0 0.6 0.5 0.4 -200000 0.3 -600000 -400000 -200000 0 200000 400000 0.2 0.1 0.0 phase 800000 phase (degrees) 360.0 330.0 300.0 600000 270.0 240.0 210.0 400000 180.0 150.0 120.0 200000 90.0 60.0 30.0 0 0.0 -200000 -600000 -400000 -200000 0 200000 400000 Plate 22: semi-diurnal tide in the Channel: bathymetry, mesh and friction coefficients bathymetry (m) 0 800000 bathymetry and mesh -250 -500 -750 600000 -1000 -1250 -1500 400000 -1750 -2000 200000 -2250 -2500 0 -2750 -3000 -3250 -200000 -3500 -3750 -400000 -4000 -4250 -600000 -4500 -4750 -1.25e+06 -1e+06 -750000 -500000 -250000 0 800000 250000 500000 -5000 5 600000 Chézy coefficient 100.0 400000 1 200000 0 95.3 4 76.5 62.2 54.3 3 47.2 2 -200000 zonation and associated friction coefficients -400000 -600000 -1.25e+06 -1e+06 -750000 -500000 -250000 0 250000 500000 Plate 23: mixing of two fluids with different density: vertical cross-section in the longitudinal axis of the canal heavy fluid 0 light fluid -1 salinity -2 1.00 -3 0.95 -4 0.90 0 5 10 15 20 25 30 initial conditions 0.85 0.80 0.75 0.70 0 0.65 -1 0.60 -2 0.55 0.50 -3 -4 0.45 0 5 10 15 20 25 30 Navier-Stokes equations with hydrostatic assumption 0.40 0.35 0.30 0.25 0.20 0 0.15 -1 0.10 -2 0.05 -3 0.00 -4 0 5 10 15 20 25 non-hydrostatic Navier-Stokes equations Plate 24: automatic estimation of the friction coefficient in a canal (data simulated with a Strickler coefficient of 35, initial guess 5) 0.9 0.8 0.7 0.6 0.5 bottom (m) 0.4 free surface (m) 0.3 0.2 0.1 0 -0.1 100 90 80 70 60 50 40 30 20 10 0 0 0 mesh 100 200 300 100 200 400 300 500 400 500 35 friction (Strickler coefficient) 30 25 20 15 10 5 1.25 iterations 0 1 2 3 4 5 6 7 error on depth 1 0.75 0.5 0.25 0 iterations 0 1 2 3 4 5 6 7 30 70?C4A =CA>3D2C8>= )F4=CH H40AB >5 34E4;>?<4=C 0C 0CH=? 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NL;=?L ? 12 HYDRODYNAMICS OF FREE SURFACE FLOWS The pressure gradient term in the first two momentum equations is then developed to first order: 1 1 ∂p ∆ρ ∂p ≈− (2.30) − 1− ρ ∂x ρo ρo ∂x 1 ∂p ∆ρ ∂p 1 − 1− ≈− (2.31) ρ ∂y ρo ρo ∂y which gives by using Equation 2.27: 1 1 ∂p ≈− − ρ ∂x ρo Zs ∆ρ ∂Zs ∂ ∆ρ ∆ρ ∂patm −g 1− −g dz (2.32) 1− ρo ∂x ρo ∂x ∂x ρo z 1 1 ∂p ≈− − ρ ∂y ρo Zs ∆ρ ∂Zs ∂ ∆ρ ∆ρ ∂patm −g 1− −g dz (2.33) 1− ρo ∂y ρo ∂y ∂y ρo z These two gradients are broken down into a slope effect of the free surface − g ∇(Zs ) and into two source terms due to buoyancy, the first one called barotropic and the second baroclinic, which can be written along x: Zs ∆ρ ∂ ∆ρ ∂Zs −g dz (2.34) g ρo ∂x ∂x ρo z and along y: ∂ ∆ρ ∂Zs −g g ρo ∂y ∂y z Zs ∆ρ dz ρo (2.35) and to which is added the effect of atmospheric pressure. They are subsequently integrated in the source terms Fx and Fy . 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IGCNN?> B?L?;@N?L EQUATIONS OF FREE SURFACE HYDRODYNAMICS 31 Recapitulation: the sum of all these terms, a number of which disappear because of the impermeability of the bottom and open surface, gives the following equation: −−→ ∂ ∂ ∂Zs ∂(hu) + (huu) + (huv) = −gh + hFx + div(hν e grad(u)) ∂t ∂x ∂y ∂x (2.128) In the same manner, the equation for V gives: −−→ ∂(hv) ∂ ∂ ∂Zs + (huv) + (hvv) = −gh + hFy + div(hν e grad(v)) ∂t ∂x ∂y ∂y (2.129) ν e is an effective diffusion which will model turbulent viscosity and dispersion. The W equation has not been retained (it has already been used to obtain the hydrostatic pressure hypothesis). With the mass conservation equation, these equations constitute the two-dimensional set of Barré de Saint-Venant equations, in what is called the “conservative” form. 2.3.4 Different forms of equations Conservative form Equations 2.119, 2.128 and 2.129 are rewritten by stating: Qx = hu and Qy = hv is the vector of the components Qx and Qy . The “discharge” Q is in fact per Q unit length, in m2 /s: Continuity: ∂h =0 + div(Q) ∂t (2.130) −−→ ∂Qx ∂Zs + div(Qx u) = −gh + hFx + div(hν e grad(u)) ∂t ∂x (2.131) −−→ ∂Qy ∂Zs + div(Qy u) = −gh + hFy + div(hν e grad(v)) ∂t ∂y (2.132) Momentum: Source terms in the continuity equation: In the presence of water intake or release at the free surface, inside the domain or at the bottom, or if one takes into account rain or infiltrations in the ground, the right-hand side of the continuity equation is not zero but equal to a term that we shall denote as “Sce” for “Source” (it can also be a sink). Sce is expressed in m/s and is positive for an injection of water into the domain. It is for example the rainfall, expressed in metres per second. We should multiply Sce by a surface to obtain a real discharge. The mass conservation equation is thus written as: ∂h = Sce + div(Q) ∂t (2.133) %6!/,!6+*& 0 ,# #/"" 02/# " #),40 0CGCF;LFS CH NB? GIG?HNOG ?KO;NCIHM Q? MBIOF> N;E? CHNI ;==IOHN NB? 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MJ;=? >CG?HMCIH ?KO;F NI IL CM NB? <IOH>;LS L?AOF;L ; HILG;F P?=NIL =;H <? >?VH?> IH CN I@ N NB? =IHNCHOOG F?P?F JBSMC=;F KO;HNCNC?M JL?MMOL? N?GJ?L;NOL? =IGJIH?HNM I@ NB? P?FI=CNS P?=NIL ?N= ;L? @OH=NCIHM I@ CH *IL?IP?L Q? L?KOCL? NB;N NB?M? @OH=NCIHM ;L? MKO;L? CHN?AL;<F? ;H> ( >?HIN?M NB? M?N I@ MKO;L? CHN?AL;<F? @OH=NCIHM >IN JLI>O=N =;H <? >?VH?> IH NBCM M?N ;M . K L KL : MO<M?N I@ ( CM =IGJIM?> I@ @OH=NCIHM QBIM? >?LCP;NCP?M ;FIHA NB? * >CL?= NCIHM I@ MJ;=? ;L? ;FMI MKO;L? CHN?AL;<F? 1BCM AO;L;HN??M NB;N ; @OH=NCIH AL;>C?HN Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0 J.-M. Hervouet %6!/,!6+*& 0 ,# #/"" 02/# " #),40 QIOF> ;FMI <? MKO;L? CHN?AL;<F? 1BCM MO<M?N CM >?HIN?> $ CM >?VH?> CH $ ;M . K L KL =H7: K =H7: L : H?Q >IN JLI>O=N 1B? MJ;=?M ( ;H> $ QCNB NB?CL >IN JLI>O=N ;L? =;FF?> %CF<?LN MJ;=?M %CF<?LN MJ;=?M ;L? P?LS MCGCF;L NI NB? "O=FC>?;H P?=NIL MJ;=?M ,H <INB I@ NB?G ;H> QCNB NB? 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I<M?LP?> WOR CM E8I;HL;: < BKN J 07< ;H> ;M 0 < 07 07< 0 < " (1 !40 ? 174 HYDRODYNAMICS OF FREE SURFACE FLOWS a priori added mass = ∆Tin Ωn Ψi dΩn = −∆t FLUEXT(i) Tiimposed − Tin − ∆T n we can check that we recognize the required property: a priori added mass + a posteriori added mass + observed f lux = −Tiimposed ∆t F LU EXT (i) which is the wanted flux. 5.8 5.8.1 Treatment of uncovered beds in 3D Option 1: correction of free surface gradients This option used in 2D has been considered impossible for a long time in 3D, on account of severely distorted elements which appear when, in the absence of water, the free surface and the bottom become identical. It appeared later that the work with these severely distorted elements was manageable on condition that some divisions by 0 were avoided. For example, to calculate the vertical average of the horizontal velocity, calculation with the trapezoidal rule should be replaced on the exposed beds by an arithmetic average. Apart from a few numerical problems, for which a remedy exists, the correction of the free surface gradients is formally the most elegant because, when a free surface gradient is treated with a resolution of the Saint-Venant equations, nothing else is left in the actual 3D part of the resolution for both the hydrostatic hypothesis and the non-hydrostatic equations. Plate 9, shown earlier, shows that exposure on a beach can be treated in 3D, in this case with 10 planes along the vertical, with zero water depth on the right side of the calculation domain. In the non-hydrostatic case, we notice a trough in the free surface. This trough becomes more prominent when the flow at the exit of the pool which is emptied, becomes supercritical. The water trapped in the puddle seems to take longer to return to rest. This is because the dynamic pressure gradient is not subject to any specific treatment on the exposed banks. 5.8.2 Option 2: masking of the uncovered elements As in the case of 2D, the columns of dry or partially dry elements have to be extracted from the calculation domain. This is done with the help of an array equal to 1 or 0, called a mask. The detailed description of this technique is mainly a catalogue of dissuasive difficulties. Untimely masking–unmasking In situations at the limit between covering and exposing, with fluxes of water changing signs from one time step to another, the elements balance endlessly between a normal situation and a situation where they are removed from the mesh. In the presence /"0,)21&,+ ,# 1%" +3&"/\01,("0 ".21&,+0 I@ ; <INNIG MFIJ? NB? WIQ CM AOC>?> <S NB? MFIJ? ;H> NBIM? ?F?G?HNM QBIM? <INNIG ?F?P;NCIH CM NB? BCAB?MN QCFF >LS VLMN ,H NB? INB?L B;H> IH ; W;N <INNIG NB? WIQ CM GO=B GIL? ?LL;NC= ;H> NB? JLI<F?G CM GIL? ;=ON? &N CM NB?L?@IL? 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NB? @IFFIQCHA FIIJ !, &0"$ +0"$ 5$),0"$&0"$ 5$),0"$&0"$ 5&0"$ 6$),0"$&0"$ 5$),0"$&0"$ 5$),0"$&0"$ 5&0"$ 6$),0"$&0"$ "+!!, MODERN TECHNIQUES IN FINITE ELEMENTS 223 This loop is a little more complex for rectangular matrices because some values of X are not initialized by the contribution of the diagonal, and need to be set to 0 first. However, as the edge-based storage provides assembled matrices, it allows a more efficient matrix–vector product, the frontal product, detailed in the following subsection. 7.4.10 Matrix–vector frontal product The idea is to write the matrix–vector product directly in its classical form: X(i) = npoin Aij Y (j) j=1 As the matrix A is sparse, we do not want really to write loops from 1 to NPOIN, but we want to use an optimal storage, like the edge-based storage. For each point, a data structure will be defined containing the number of neighbours and their addresses, and the entry of the matrix related to it. In the classical versions of this kind of matrix–vector product, the internal loop is a loop on the number of neighbours for each point, which inhibits any attempt at vectorization. With the frontal product, the points are numbered according to the number of their neighbours, in ascending order. This results in long sequences of nodes, with the same number of neighbours, for which a specific loop is written. The maximum number of neighbours for one point, denoted MXPTVS, is thus restricted by what has been specified in the program. A 2-dimension table, denoted LIMVOI, determines the arrangement of the nodes: LIMVOI(K,1) is the global number of the first point having K neighbours. LIMVOI(K,2) is the global number of the last point having K neighbours. The address of the neighbours of points is denoted IY, with size (NPOIN,MXPTVS). The address of the elements of the matrix, related to the neighbours, is denoted IW, which has the same dimension as IY. The table X is, as usual, initialized by the contribution of the diagonal DA Y . The loop for those nodes having three neighbours is given below. In this case, the extra-diagonal terms are stored in the table W: DO I=LIMVOI(3,1),LIMVOI(3,2) X(I)=X(I)+W(IW(I,1))*Y(IY(I,1))+W(IW(I,2))*Y(IY(I,2)) +W(IW(I,3))*Y(IY(I,3)) ENDDO Unfortunately, W changes when the matrix is symmetric, and IW must be written in two versions. Transposed products, like the computation of X = t A Y , requires two other versions. The whole data structure is thus stored in an array named IKLEM1 with four dimensions: NPOIN, MXPTVS, 4 and 2. The third dimension is coded as follows: 1: IW for a normal product 2: IY for a normal product 3: IW for a transposed product 4: IY for a transposed product %6!/,!6+*& 0 ,# #/"" 02/# " #),40 1B? @IOLNB >CG?HMCIH C ? CM =I>?> ;M @IFFIQM HIH MSGG?NLC= G;NLCR MSGG?NLC= G;NLCR !8=40A BHBC4< A4B>;DC8>= 4? ACP? B?L? ;H IP?LPC?Q I@ CN?L;NCP? 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NB;H ; L;NCI I@ NQI ?P?H C@ Q? IP?LFIIE NB? =IGGOHC=;NCIH NCG? 1BCM CM >O? NI NB? @;=N NB;N NB? L?MOFNM I@ ?;=B JLI=?MMIL B;P? NI <? ;MM?G<F?> 1B;N F?;>M NI ; Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0 J.-M. Hervouet 248 HYDRODYNAMICS OF FREE SURFACE FLOWS new cost which limits drastically the overall time to be gained. This is known as a “granularity” problem, the size of the tasks to be parallelized being too small. It is unlikely that progress in the algorithm would help in solving this theoretical bottleneck. From that, the idea emerges to break up the problem in another way: not by isolating “small” tasks, but in a kind of geographical way by decomposing the domain of computation. This idea of domain decomposition aims at assigning to each processor one part of the domain over which it would solve the fluid mechanics problem. The results of the other processors would help in defining artificial boundary conditions arising from the partition. The assignments given to each processor are the same, but the data differ. According to the Flynn classification (see [51]), this corresponds to SMPD (Single Program Multiple Data) parallelism. The implementation is quite easy in the case of explicit algorithms: each equation is only a function of the variables related to the nodes in the immediate vicinity, computed at the earlier time step. Each processor is then in charge of the equations related to a group of nodes, and of the data concerning the neighbours of these nodes, resulting from the previous time step. This is executed merely by a partition of the domain, with an overlap of one element, and data transmission at the end of each time step. In the case of our implicit algorithms, and especially with linear systems solved on the whole domain, the implementation is more complex, though possible. In fact, we can guarantee that the results of a parallel computation will be the same as a scalar computation, except for truncation errors because computations will not be performed in the same order. Parallelism (initially implemented on our algorithms by Reinhard Hinkelmann at the University of Hanover [101]) can be limited to the following problems: • Partition of the domain. • Communication between processors. • Implementation of some basic algorithms: vector assembly, dot product, computation of normal vector and so on. We shall examine these three points hereafter. 8.2 Partition of the domain In the case of finite element meshes and implicit algorithms, it is better to partition without overlapping, edge to edge. Though the techniques of partition are beyond the scope of this book, we shall mention some of them and give the original references. Then we shall specify the kind of additional data structures necessary to describe the domain in order to implement parallelization. The specifications for an efficient domain decomposer would be the following: • Realizing a partition of meshes of triangles (the sub-domains of three-dimensional meshes of prisms will be made from sub-domains of triangles). • Partitioning the domain into blocks in order to ensure a balance between the processors. Because the main algorithms are made of loops over the elements -/))")&0* ;H> HI>?M NB? J;LNCNCIH QCFF AO;L;HN?? ; <;F;H=? <?NQ??H NB? HOG<?LM I@ ?F? G?HNM IL MCGCF;LFS <?NQ??H NB? HOG<?L I@ HI>?M QCNBCH ?;=B MO< >IG;CH *CHCGCTCHA NB? HOG<?L I@ HI>?M MB;L?> <S >C_?L?HN MO< >IG;CHM ;H> @IL QBC=B =IGGOHC=;NCIHM <?NQ??H JLI=?MMILM QCFF <? H?=?MM;LS ,NB?L =LCN?LC; @IOH> CH NB? FCN?L;NOL? ;CG?> ;N MCGJFC@SCHA NB? L?MIFONCIH I@ ; FCH?;L MSMN?G IP?L ?;=B MO< >IG;CH ;L? HIN L?F?P;HN NI IOL ;JJLI;=B M?? 89 1B? MCGJF?MN N?=BHCKO? CM EHIQH ;M L?=OLMCP? =IIL>CH;N? <CM?=NCIH 1B? >IG;CH CM J;LNCNCIH?> CHNI NQI ;L?;M <S ; P?LNC=;F IL BILCTIHN;F FCH? QBIM? 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ACP?H NI !CLC=BF?N ;>DICHN P;LC;<F?M >I?M HIN ;_?=N NB? INB?LM ;H> NB;N CN =;H <? =BIM?H CH ;H ;L<CNL;LS G;HH?L #IL NBCM Q? ;L? AICHA NI MBIQ NB;N NB? LCABN B;H> MC>? I@ NB? ?KO;NCIH I@ ; HILG;F ;>DICHN P;LC;<F? >I?M HIN >?J?H> IH NB?M? ;L<CNL;LS P;FO?M 1B? N?LG 7B 7B 7B 7B 4 , 7B 7B 47> 47> NB;N Q? B;P? ;>>?> NI NB? LCABN B;H> MC>? NI N;E? CHNI ;==IOHN !CLC=BF?N =IH>CNCIHM CM T?LI @IL ; >?AL?? I@ @L??>IG ?F 1B? N?LG 7B 7B 7B 7B 4 , 7B 7B 47> 47> CM <LIE?H >IQH CHNI 1B? N?LG 4 7B , 4 7B , , 7B NB? N?LG % , 7B ;H> , 7B , <S =B;HACHA NI <?=IG?M 7B 7B 47> 4 7B , 7B >I?M HIN >?J?H> IH 4 1BCM N?LG CM ?KO;F NI 7B 7B 47> 4 7B , 7B 0I CN CM M??H NB;N !CLC=BF?N ;L<CNL;LS P;FO?M I@ , 7B QCFF <? =;H=?FF?> <S NB? JLI>O=N QCNB 7B &@ Q? FIIE ;N NB? N?LG , CN HIQ <?=IG?M , 7B 7B % 4 7B 7B 47> % 4 7B , 7B %6!/,!6+*& 0 ,# #/"" 02/# " #),40 IL ?P?H 7B 7B , 7B 47> 7B 7B % 7B 47> 4 7B , 7B <?=;OM? =NO;FFS , , % 4 7B 7B , 7B 47> 4 7B % 7B 47> , 7B CM ; T?LI P?=NIL ?R=?JN C@ NB? JICHN HOG<?L ?F CM I@ !CLC=BF?N NSJ? &H NBCM =;M? NB? >IN JLI>O=N CM ?KO;F NI NB? =IGJIH?HN ?F I@ % , 7B QBC=B G?;HM 7B ,7> NB? ;>DICHN P;LC;<F? I@ ; !CLC=BF?N P;FO? B?H=? T?LI 4? NB?H QLCN? 7B 7B % 7B 47> 4 7B , 7B 7B 7B % 7B 47> 4 7B , 7B ;H> CN CM ;A;CH @IOH> NB;N !CLC=BF?N ;L<CNL;LS P;FO?M I@ , 7B QCFF <? =;H=?FF?> <S NB? JLI>O=N QCNB &N L?G;CHM NI =B?=E NB;N CH NB? ?RJL?MMCIH M?F?=N?> ;L<CNL;LS P;FO?M >I HIN =IG? CH ?CNB?L &@ CH NB? ?RJL?MMCIH Q? =B;HA? NI ;H> NI 7B 7B 7B ; ;4 /7A /7A 7> <?=IG?M 7B 7B % 4 7B ; /7A % 4 7B 7B ; /7A % 4 7B 7> FF NB? P?=NILM % 4 7B B;P? HIH T?LI N?LGM IHFS @IL !CLC=BF?N P;FO?M QBC=B >I HIN >?J?H> IH NB? J;L;G?N?L / MI NB?M? N?LGM ;L? ?FCGCH;N?> 4B;N L?G;CHM CM IHFS 7B 7B % 4 7B 7B 4 7B ; ; /7A /7A 7> CH QBC=B NB?L? CM ;FQ;SM ; JLI>O=N <S IL>?L NI I<N;CH <7B , & ; ; /7A 7B IH NB? F?@N NB;N =;H <? <LIOABN CHNI , 7B CH 7B 7B % , ; /7A 7B 4 7B 7B 7B 4 ; /7A CH QBC=B ;A;CH !CLC=BF?N P;FO?M I@ ;>DICHN P;LC;<F?M >I HIN =IG? CH ;HS FIHA?L @IL , 7B =IH=?LHM IHFS HILG;F P;FO?M &H =IH=FOMCIH I@ NBCM M?=NCIH CN CM NB?L?@IL? M??H NB;N ;>DICHN MSMN?GM QCFF B;P? NB? M;G? NL?;NG?HN I@ !CLC=BF?N =IH>CNCIHM ;M >CL?=N MSMN?GM ;H> NB;N !CLC=BF?N ;>DICHN P;LC;<F?M =;H <? M?F?=N?> ;L<CNL;LCFS -/*"1"/ "01&*1&,+ ??;820C8>= C> )4;4<02 0;6>A8C7< A82C8>= 2>4P284=C 4BC8<0C8>= 8= )4;4<02 &H NB? =;M? I@ 1?F?G;= ! @LC=NCIH =IG?M CHNI NB? GIG?HNOG ?KO;NCIH ;H> CM NL?;N?> CH ;H CGJFC=CN @ILGOF;NCIH 1B? NQI =IGJIH?HNM I@ @LC=NCIH @IL=? ;L? L?MJ?=NCP?FS 5 K "F 4 K L > 5 L "G 4 K L > CH QBC=B > CM NB? Q;N?L >?JNB 4 ; >CG?HMCIHF?MM @LC=NCIH =I?^=C?HN ;H> K ;H> L NB? NQI =IGJIH?HNM I@ BILCTIHN;F P?FI=CNS 1B?M? @IL=?M ;L? >CM=L?NCT?> L?MJ?=NCP?FS CH NB? @ILG I@ "F "G 5 K< 4 K > 5 L< 4 K > L L L <?CHA N;E?H ;N NB? NCG? J< 1B?M? @IL=?M NB? G;AHCNO>? I@ NB? P?FI=CNS K QLCNN?H CH ;H CGJFC=CN @ILG =IG? NI GI>C@S NB? G;NLCR I@ NB? VH;F FCH?;L MSMN?G L?G;CHCHA OH=B;HA?> 4? @OLNB?L =BIIM? NI >I ; ZG;MM FOGJCHA[ I@ NB?M? @LC=NCIH @IL=?M QBC=B G?;HM NB;N @IL NB? J;LN =IH=?LHCHA P?FI=CNC?M CM =B;HA?> NI " QB?L? " A% % CM NB? C>?HNCNS G;NLCR ;H> A NB? P?=NIL ?KO;F NI . 5 A L 7 : 4 K > QB?L? 7 CM NB? <;MCM L?F;NCP? NI NB? JICHN QCNB CH>?R ? 0I Q? B;P? <> <7B , , & ; 7B ,7> ; /7A 7B 7> 7B 7B 7B ; ;4 /7A /7A 7> <> <7B , , 7B 7B ,7> 7> 7B 7B ;4 /7A 7> @IL >I?M HIN >?J?H> IH / IL ?P?H <7B , & ; ; /7A 7B , 7B 7B 7B ;4 /7A <7B , 7B , 7B A 7B ;4 /7A &N CM NB?L?@IL? H?=?MM;LS NI EHIQ NB? >?J?H>?H=? I@ A CH N?LGM I@ / NBOM MJ?=C@SCHA 9 NB? F;Q OM?> #IL ; BfTS IL 0NLC=EF?L @ILGOF; A M=;F?M IH / ;H> ' NBOM B;M ; P;FO? I@ 9 ' @IL CH>?R ? ?I ;H> @IL NB? INB?LM 1BOM Q? A?N <7B 7B , & ; 7B A47A , 7A ; ; /7A /7A 7B 268 HYDRODYNAMICS OF FREE SURFACE FLOWS Note: we have assumed so far that a friction coefficient was estimated per point. If we look for a limited number of coefficients by dividing the mesh into different zones, or arranging points class-wise, then the gradient becomes: nit ∂J m = m ∂Sis it=1 i ∈ classe(is) Piit 2kXiit m Sis (9.40) m is in which index i covers all the points to which the same friction coefficient Sis assigned. 9.4.2 Calculation of right-hand sides of adjoint systems Principle Let us recall that the right-hand side of the adjoint system has for component ip: ∂Ait+1 it+1 ∂B it+1 it+1 ∂Ait it ∂B it it it it 2Wip (Xip − Mip ) − X − ,P X − ,P − it it it it ∂Xip ∂Xip ∂Xip ∂Xip (9.41) it ), let us recall that the term 2Wip With a cost function of the formulation f (Xip ∂f it it (Xip − Mip ) is replaced by ∂X it . A and B originate from a variational formulation, ip which gives to each point a weight proportional to the integral of its basis Ω Ψi dΩ. For the homogeneity, it would be necessary to take a cost function J which is of nitsake np it it 2 − Mip ) but rather: not it=1 ip=1 Wip (Xip nit it=1 Ω it it 2 Ψi Wip (Xip − Mip ) dΩ which complicates the calculation of its gradient. Without changing the calculation of the gradient, it can also be considered that Wip is equal not to 1 or 0 but to Ω Ψi dΩ or 0, which restores the homogeneity of the right-hand side. In any case, it is easy to it it − Mip ). calculate 2Wip (Xip In the case of Telemac-2D, the matrix A is composed of blocks and vector X contains the water depth and the two components of the velocity. The three blocks of the right-hand sides are marked CV 1, CV 2 and CV 3: AM 1 BM 1 BM 2 H CV 1 0 , X = U , B = CV 2 A = CM 1 AM 2 CM 2 0 AM 3 V CV 3 Note: strictly speaking, the unknown H is in fact in Telemac-2D: δH = H n+1 − H ; we consider here for more clarity and symmetry that the unknown is H n+1 . 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QBC=B ACP?M 0 7> "77C 7B K/28 >7B 7> 7> K7B 7> C "7> 7> K7B /28 >7B 7> 7> K7B 7> C@ NB?L? CM HI >?J?H>?H=? I@ 4 CH L?F;NCIH NI > QBC=B CM HIN NB? =;M? @IL NB? 0NLC=EF?L @ILGOF; @IL QBC=B Q? QIOF> NB?H B;P? 0 7> 7B C "7> 7> K/28 >7B 7> 7> K7B 7> 276 HYDRODYNAMICS OF FREE SURFACE FLOWS So we have: T 15 = − 1 u it+1 it+1 4 it+1 F Uadj U (Chézy) or T 15 = − F u Uadj U it+1 (Strickler) H it 3H it (9.73) and: T 17 = − 1 u it+1 it+1 4 it+1 F Vadj V (Chézy) or T 17 = − F u Vadj V it+1 (Strickler) it H 3H it (9.74) Terms T 21 and T 31: T 21 = it+1 Hadj , ∂CV 1it+1 it ∂Uip The dependences of CV 1 in relation to the velocity are in the transport term (θh − 1)(Txh + Tyh )it H it , in the term (θu − 1)Bxuh U it and the boundary term T B1. By x as the matrix of the general term: denoting Tadj ∂hit u h x (i, j) = Tadj Ψi ϕj Ω ∂x and: y (i, j) = Tadj Ω ∂hit u h Ψ ϕ ∂y i j we find: it+1 it+1 x Hadj + (θu − 1)t Bxuh Hadj + T B1uadj T 21 = (θh − 1)Tadj (9.75) y it+1 it+1 Hadj + (θu − 1)t Byuh Hadj + T B1vadj T 31 = (θh − 1)Tadj (9.76) by denoting: T B1uadj ip =− Γ n u hit+1 adj h Ψip nx dΓ and Terms T 22 and T 32: T 22 = it+1 Uadj , T B1vadj ∂CV 2it+1 it ∂Uip ip =− Γ n u hit+1 adj h Ψip ny dΓ The dependence in relation to the velocity u in CV 2 originates from: M u it U and (θu − 1)(Txu + Tyu )it U it ∆t M u it M u it+1 ∆t U gives ∆t Uadj ∂((Txu +Tyu )it U it ) ) equal to: it ∂Uip where it+1 and (θu − 1)(Txu + Tyu )it U it gives a term (θu − 1)(Uadj , -/*"1"/ "01&*1&,+ C 7B 1/28 + 7B 7B C =H7: K : K 7 7B 17> ,L ?P?H C . 7B C 7B K 7> 1/28 : N C . L 7B C7> 7B : 1 O /28 ;H> NB?L?@IL? ) C 7B 0 1 J /28 C . 7B C 7B K 7> 1/28 : N C . L 7B . K7B C7> 7B : 1 O /28 4? VH> CH NB? M;G? 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CM ; MCHAF? =LCN?LCIH JLI<F?G NB? =IMN @OH=NCIH < N CM NB? MOG I@ MKO;L?> ;GJFCNO>? >?PC;NCIHM ;N NB? ?CABN G?;MOLCHA JICHNM 1B? M?=IH> IH? CM ; >O;F =LCN?LCIH JLI<F?G ;>>CNCIH;FFS ;CGCHA ;N GCHCGCTCHA NB? MOG < N I@ MKO;L?> JB;M? >?PC;NCIHM 1B? VLMN JLI<F?G CM MIFP?> QCNB >C_?L?HN G?NBI>M NB? @ILG?L ZL;Q @IL=?[ CG JF?G?HNM ; L;H>IG M?;L=B NB;N CM @I=OMM?> IH NB? GIMN JLIGCMCHA ;L?;M CH NB? M?;L=B >IG;CH ;M NB? CHP?MNCA;NCIH JLI=??>M NB? F;NN?L OM?M A?H?NC= ;FAILCNBGM 0CH=? NB? A?H?NC= ;FAILCNBGM CH>O=? =B;HA?M CH ; JIJOF;NCIH I@ CH>CPC>O;FM L?JL?M?HNCHA NB? JI N?HNC;F MIFONCIHM NI NB? JLI<F?G HIN NB? GCHCGOG I@ @OH=NCIH < <ON NB? M?N I@ JICHNM N G??NCHA < N < CM M?;L=B?> @IL G;S <? CHN?LJL?N?> ;M ;H IL>?L I@ G;AHCNO>? I@ NB? ?LLIL =;OM?> <S NB? MOLLIA;N? GI>?F 1B?L? QCFF HIN <? ; MCHAF? MIFONCIH <ON ; M?N I@ JICHNM CH NB? PC=CHCNS I@ NB? MIFONCIH #LIG NB? L?MOFNM I@ NB;N IJNCGCT;NCIH CN QCFF ;JJ?;L NB;N NB? NIJIFIAS I@ NB;N PC=CHCNS QCFF JLIPC>? P?LS OM?@OF CH@ILG;NCIH ;<ION NB? <?B;PCIOL I@ NB? MIFONCIHM 1B? M?=IH> JLI<F?G CM MIFP?> <S G?;HM I@ A?H?NC= ;FAILCNBGM "R;GCHCHA NB? -;L?NI @LIHN QCFF <? I@ NB? ONGIMN CGJILN;H=? @IL ;H;FSMCHA NB? JLI<F?G 1B? -;L?NI @LIHN CM =IGJIM?> I@ MIFONCIHM QBC=B ;JJ?;L NI <? NB? <?MN ;M L?A;L>M IH? I@ NB? =LCN?LC; ;GIHA M?NM I@ MIFONCIHM QBC=B ACP? NB? M;G? J?L@ILG;H=? IH INB?L =LCN?LC; #IL NB? MCHAF? =LCN?LCIH =;M? NB? @IFFIQCHA JLI<F?G CM MIFP?> #CH> NB? JICHNM N G??NCHA < N < QCNB < ;H> %6!/,!6+*& 0 ,# #/"" 02/# " #),40 cost friction of zone 4 #CAOL? =FOMN?L I@ MIFONCIHM @IL =I?^=C?HN I@ ;L?; 1B;N JLI<F?G CM =F;MMC=;FFS @ILGOF;N?> ;M ;H IJNCGCT;NCIH JLI<F?G OH>?L =IH MNL;CHN 1B? MIFONCIHM NI NB;N JLI<F?G ;L? >CMJF;S?> CH NB? @ILG I@ M=;NN?L JFINM &H #CAOL? NB? @LC=NCIH =I?^=C?HN I@ TIH? CM NB? ;<M=CMM; ;H> NB? =IMN CM NB? IL>CH;N? %?L? NB? @LC=NCIH =I?^=C?HN I@ H?;LFS ;FF MIFONCIHM CM =F?;LFS CH NB? L;HA? 8 9 ,NB?L =;M?M MG;FF TIH?M @IL ?R;GJF? 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GCHCG; I@ ?;=B @OH=NCIH ;L? I<N;CH?> @IL KOCN? >C_?L?HN M?NM I@ @LC=NCIH =I?^=C?HNM ,H INB?L ?R;GJF?M CN ;JJ?;LM NB;N NB? @OH=NCIHM ?RBC<CN >CM=IHNCHOCNC?M ;H> ;L? HIN GIHINIHC= '4BD;CB 0=3 0=0;HB8B 1B? M?=IH> ;H> NBCL> =IFOGHM I@ 1;<F? ACP? NB? 0NLC=EF?L @LC=NCIH =I?^=C?HNM I<N;CH?> L?MJ?=NCP?FS QCNB ; L;H>IG M?;L=B ;H> QCNB A?H?NC= ;FAILCNBGM #IL NB? F;NN?L ; JIJOF;NCIH I@ CH>CPC>O;FM ;H> A?H?L;NCIHM B;P? <??H OM?> QBC=B --)& 1&,+0 friction of zone 5 friction of zone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error on phase error on tidal range #CAOL? -;L?NI @LIHN CH ; >O;F =LCN?LCIH JLI<F?G MG;FF ?_?=N IH NB? ;==OL;=S <?CHA ;=BC?P?> NBIOAB NB?S F?;> NI L;NB?L >C_?L?HN M?NM I@ =I?^=C?HNM ;H> NBCM IH=? ;A;CH L;CM?M NB? KO?MNCIH I@ FI=;F GCHCG; 0BC@NCHA OJ NI ?CABN ;L?;M IL NI >O;F =LCN?LCIH IJNCGCT;NCIH ;=NO;FFS <LCHAM FCNNF? CGJLIP?G?HN ;H> L?KOCL?M GIL? ?RJ?LNCM? >=2;DB8>= 0=3 ?A>B?42CB 1BIM? G?NBI>M <;M?> IH ; GCHCGCT;NCIH @LIG ; L?MJIHM? MOL@;=? L?P?;F B?L? NB;N ;N NB? M;G? =IGJON;NCIH;F =IMN NB?S =;H JLIPC>? <?NN?L L?MOFNM #CH>CHA M?NM I@ VP? =I?^=C?HNM NB;N ;L? <?NN?L NB;H ;H IJNCGCT;NCIH CHPIFPCHA ;H ?RJ?LN @IL MCR GIHNBM ;H> ; NB?MCM CM ; P?LS JLIGCMCHA ;=BC?P?G?HN 1B? JLCG;LS ;>P;HN;A? I@ NB? H?Q G?NBI>M <?CHA N?MN?> B?L? CM NB;N NB?S =IHMC>?L NB? ?P;FO;NCIH I@ NB? =LCN?LC; ;H> NB? =IGJON;NCIH;F ;FAILCNBGM ;M <F;=E <IR?M QB?L?;M NB? ;>DICHN G?NBI>M MBIOF> @IFFIQ NB?M? ;FAILCNBGM ;M =FIM?FS ;M JIMMC<F? NBOM B?;PCFS <OL>?HCHA <INB >?P?FIJG?HN ;H> G;CHN?H;H=? 1B?M? H?Q G?NBI>M BIQ?P?L ;FMI CGJFS ; BOG;H MECFF ?MJ?=C;FFS QCNB M?P?L;F =LCN?LC; ? A <S >CMJF;SCHA ;H> ;H;FSMCHA NB? -;L?NI @LIHNM 1B? ION=IG? I@ NB;N ;JJLI;=B QCFF NB?H >?J?H> IH ?CNB?L NB? OM?L @LC?H>FCH?MM IL NB? L?KOCL?> EHIQF?>A? F?P?F @IL CGJF?G?HNCHA NB?G CH L?;F FC@? =IH>CNCIHM (>;8C0AH F0E4 0IFCN;LS Q;P?M JLIJ;A;NCHA QCNBION >CMNILNCIH ;L? HINC=?> ;N NCG?M CH NB? M?; IL CH ; =;H;F ;@N?L NB? MO>>?H =FIMCHA I@ ; FI=E &H );CNIH? 89 JLIJIM?> ;H ;JJLIR CG;NCIH NI NB? VLMN IL>?L =IGJ;L?> QCNB M?? <?FIQ I@ MO=B ; Q;P? QBC=B CM ; J;LNC=OF;L FCGCN =;M? I@ ; =HIC>;F Q;P? &H ; =;H;F QCNB ; =IHMN;HN >?JNB ;FIHA ;RCM APPLICATIONS 295 Table 10.13: magnitude of wave M2 obtained with surrogate models. Measurement point Zeebrugge Calais Boulogne Dieppe Cherbourg Saint-Malo Buoy C1 Ouessant Measurement 1.59 m 2.46 m 2.93 m 3.08 m 1.87 m 3.74 m 1.29 m 2.06 m Random 1.63 2.39 2.93 3.03 1.94 3.65 1.29 1.98 search m m m m m m m m Genetic algorithm 1.66 m 2.42 m 2.95 m 3.04 m 1.93 m 3.65 m 1.29 m 1.98 m Table 10.14: phase of wave M2 Measurement point Measurement (degrees) Random search Genetic algorithm Zeebrugge 14 14.79 14.63 Calais 345 343.78 343.94 Boulogne 331 332.31 332.84 Dieppe 311 313.49 314.300 Cherbourg 228 226.56 226.95 Saint-Malo 178 175.70 178.18 Buoy C1 103 105.41 105.40 Ouessant 111 114.68 114.77 Ox, this solution is as follows: U (x, t) = A gH0 H0 cosh( W (x, z, t) = 3gH0 A H0 3/2 1 3A (x 4H03 − ct))2 3A tanh( (x − ct)) 4H 3 z 0 H0 cosh( 3A3 (x − ct))2 4H (10.1) (10.2) 0 h(x, t) = H0 + cosh( A 3A (x 4H03 − ct))2 (10.3) where H0 is the reference depth. The magnitude of the wave is A, and its celerity is: c = g(H0 + A) Second-order [20] and third-order approximations [180] have also been found since Laitone’s first publication. With a depth of 10 m and a magnitude A of 2 m, we are in the application domain of the Boussinesq equations. On the other hand, the Saint-Venant equations reveal non-physical behaviour and the solitary wave is deformed and tends to break. A series of tests has been carried out in a 600 m long canal. The initial condition represents a %6!/,!6+*& 0 ,# #/"" 02/# " #),40 MIFCN;LS Q;P? QBIM? G;RCGOG CM ;FL?;>S G ;Q;S @LIG NB? F?@N ?HNLS I@ NB? =;H;F C ? ;N ; >CMN;H=? MO=B NB;N NB? Q;N?L >?JNB ;N NB? ?HNLS CM F?MM NB;H NB I@ CNM G;AHCNO>? 1B? =;F=OF;NCIH =IHMCMNM I@ CN?L;NCIHM I@ M ;@N?L QBC=B NB? Q;P? A?NM =FIM? NI NB? IONF?N IH NB? LCABN I@ NB? =;H;F 1B? MCT? I@ ?F?G?HNM CM ;<ION G QBC=B ACP?M ; IOL;HN HOG<?L =;F=OF;N?> @LIG NB? Q;P? =?F?LCNS I@ NB? 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F; !CL?=NCIH >?M "NO>?M ?N /?=B?L=B?M 0fLC? +O=Ff;CL? %S>L;OFCKO? 1B?LGCKO? \ 89 3&,))"1 - ) ,H NB? HOG?LC=;F GI>?FFCHA I@ MNL;NCV?> WIQM -BSMC=;F JLI =?MM?M CH ?MNO;LC?M ' !LIE?LM 4 3;H )?OMM?H ?>M 89 3&,))"1 - ) %/! ' - "0-,0&1, - )2/"+ " ! *f=;HCKO? >?M WOC>?M ;JJFCKOf? -L?MM?M >? FY"=IF? +;NCIH;F? >?M -IHNM ?N B;OMMf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ndex ADI (Alternating-Direction Implicit), 85 advection, 20, 91, 141, 177 ALE (Arbitrary Lagrangian–Eulerian), 134 assembly, 215, 216 baroclinic, 12, 18, 57, 133 barotropic, 12, 18, 57, 133 bisection, 249 Boltzman equation, 88 bottom shear stress, 13 boundary conditions k–ε model, 72 Navier–Stokes, 9 Saint-Venant, 58, 99 tracer, 24, 61 Boussinesq, 24, 44 approximation, 11, 57, 156 equations, 44, 108 model, 62 buoyancy, 12, 17, 57, 143 celerity, 33, 39, 58 centrifugal force, 14 Chézy coefficient, 50, 54, 70 formula, 13, 51, 53, 146, 267, 276, 289 characteristic curves, 35, 91, 177 cnoidal wave, 45 conditioning, 104, 109, 111, 225 conjugate gradient, 226, 228, 278 conjugate residual, 227 connectivity table, 199, 250 conservation of energy, 118 conservation of mass Navier–Stokes, 9, 27, 153, 165 Saint-Venant, 27, 42, 43, 96, 98, 114, 153 tracer, 115, 166 conservation of momentum Navier–Stokes, 9 Saint-Venant, 31, 96, 98, 116 consistency, 84 continuity, see conservation of mass Coriolis coefficient, 16 force, 14, 55 cost function, 258 Crank–Nicholson scheme, 86 Crout preconditioning, 241, 245 CSR (Compressed Sparse Row), 203 dam break, 130, 281 damping functions, 65 density of water, 17, 56, 297 diagonal storage, 202 diffusion, 92 dilatation coefficients, 11 Dirac function, 102, 135 Dirichlet boundary conditions (imposed values), 4, 95, 97, 100, 102, 105, 112, 158, 166, 171, 173, 220, 264 dispersion, 29 dissipation, 70 distributive scheme, 169, 183 divergence, 21, 41 drag force, 53 EBE storage (element-by-element), 203 edge-based storage, 205 eigenvalue, 87, 225 eigenvector, 87 Elder model, 63 entropy, 117 Hydrodynamics of Free Surface Flows: Modelling with the Finite Element Method ß 2007 John Wiley & Sons, Ltd. ISBN: 978-0-470-03558-0 J.-M. Hervouet 340 INDEX Euler, 33 equations, 33 fetch, 14 finite element, 77, 195 finite volume, 87 fractional steps, 90, 136 friction, 13, 50, 103, 146, 267 friction velocity, 73 Froude number, 36, 58 Galerkin method, 89 Gauss–Seidel preconditioning, 245 genetic algorithms, 290 GMRES, 230 Goutal, 95, 97 gradient, 41 Haaland, 70 head loss, 37, 53 heat, 16 heat capacity ratio, 34 Helmholtz equation, 78, 111 Hilbert space, 78 hydraulic jump, 36, 116, 128 impermeability, 26, 148 incident wave, 59 inf–sup condition, 81 isoparametric transformation, 207 Jacobian, 19, 22, 87, 106, 143, 207–209, 211, 318 Janin, 3 Jankowski, 4 jet, 66 Julian calendar, 309 k–ε model, 44, 55, 70, 111 kernel, 88, 135 kinematic viscosity, 9 kinetic schemes, 88 Lagrangian, 10, 83, 133, 135, 177, 257, 259 Laplacian, 123 Leibnitz, 26 logarithmic layer, 69 M-matrix, 210 MAC (Marker And Cell), 134 Malpasset, 281 Manning coefficient, 44 formula, 14, 52 mass lumping, 104, 124, 127 matrix–vector product, 220, 222, 253 Mercator projection, 39 minimum error, 229 mixing length model, 64, 71 momentum, see conservation of momentum monotonicity, 85 MURD scheme, 190 N scheme, 184, 188 nabla, 8 Navier, 5 Navier–Stokes equations, 7, 133, 155 Neumann, 74 Newton, 5 second law, 7 Newtonian fluid, 9 Nezu and Nakagawa model, 64 Nikuradse formula, 52, 70 no-flux boundary condition, 24, 61 parallelism, 247 parameter estimation, 257, 288 partition, 248 Poisson equation, 78 porosity, 43, 94 Prandtl, 64, 71 preconditioning, 34, 224, 234 pressure, 12 atmospheric, 11, 17, 56 hydrostatic, 11, 25 prism, 196 production of turbulent energy, 71 propagation, 33, 92 pseudo wave equation, 83, 94, 123, 158 PSI scheme, 184, 189 quasi-bubble, 123, 126, 195, 212 Quetin model, 65 radiation conditions, 105 Ramette, 70 INDEX 341 Reichard law, 70 relaxation coefficient for diffusion, 94 coefficient for the tracer, 113 coefficient for velocity, 93 coefficient for water depth, 93 Reynolds equations, 61 number, 54, 61, 96 stress, 30, 61, 62, 75 tensor, 30, 63, 65 Riemann invariants, 35, 59, 105 problem, 87 Ritter solution, 130 Roe scheme, 87 Rossby, 16 rough flow, 69, 101, 146 Saint-Venant, 24 equations, 24, 83 salinity, 11, 17, 63 Schmidt number, 63 sediment, 11, 23, 63 Serre equations, 45 shallow water equations, see Saint-Venant sigma transform, 18, 140 sinks, 168 Smagorinski model, 68 smooth flow, 69, 70, 101 Sobolev space, 78 solitary wave, 294 source terms, 31, 49, 56, 92 sources, 168 space discretization, 95 SPH (Smoothed Particles Hydrodynamics), 135 stability, 84 Stoker solution, 130 Stokes, 5 strain rate tensor, 9, 64 stress tensor, 9, 62 Strickler coefficient, 51, 54, 279 formula, 13, 51, 53, 146, 267, 276 subcritical, 36, 58 supercritical, 36, 38, 58, 89, 129, 174 SUPG (Streamline Upwind Petrov Galerkin), 141, 172, 181 surrogate model, 290 Telemac, 1, 195 2D, 89, 256, 267, 281 3D, 255 temperature, 11, 17, 23, 63 test function, 80 tetrahedron, 196, 317 Thompson method, 105 time discretization, 90, 92, 95 tracer, 23, 60, 113, 115, 165, 166 triangle, 195 Tsanis model, 66 tsunami, 33 turbulence, 61 turbulent fluctuation, 71 turbulent kinetic energy, 70 uncovered beds 2D, 121 3D, 174 uniqueness, 84, 105 variational formulation, 80, 95, 112, 157 viscous layer, 69 VOF (Volume Of Fluids), 134 von Karman constant, 64 eddies, 84, 100 weir, 128, 129 wind, 14, 55