A. HAUGUEL (l), G. LABADIE (2) and B. LATTEUX (3)

Transcription

1 A FINITE ELEMENT METHOD FOR THE SHALLOW WATER EQUATIONS by A. HAUGUEL (l), G. LABADIE (2) and B. LATTEUX (3) ABSTRACT The paper reprts the current prgress in develping a finite element methd fr the shallw water equatins. The main feature f the methd is the special care given t the advective and diffusive parts f the equatins, s that it can be f interest t use it when dealing with flws strngly influenced by cnvective and bundary layer effects. The slutin prcedure has been chsen s as t allw a calculatin with a big number f ndes. Sectin 3 f the paper utlines the methd. In sectin 4, is detailed the prcedure fr the advective terms, invlving the determinatin f the characteristic curves. Sectin 5 is devted t the diffusin and prpagatin terms. Finally numerical results are presented in sectin INTRODUCTION Envirnmental hydraulics ften requires the calculatin f flws in cmplex dmains. Mrever, ne ften wishes a finer descriptin fr sme znes f particular interest, while mst f the dmain can be apprximated by a rather carse grid. The finite element methd prvides such a flexibility and a grwing interest is taken in its ptential. Hwever the size f the resulting discrete system restricts the pssibilities f using it. An interesting apprach, presented here, leads t cnsider separate prblems n each f the scalar variables (water level and flw discharge cmpnents) and hence allws t reduce the strage requirement. (1) Head f the Research Hydraulic Divisin, Labratire Natinal d'hydraulique, E.D.F., Chatu, France. (2) Research Engineer, Labratire Natinal d'hydraulique, E.D.F., Chatu, France. (3) Research Engineer, Labratire Natinal d'hydraulique, E.D.F., Chatu, France. 617

2 618 COASTAL ENGINEERING NOTATION F = Crilis and frictin frces (cmpnents F^) g - acceleratin due t gravity h = depth f water n m = average f h in time K = mmentum diffusin cefficient Q = uh flw rate per unit length (cmpnents Q-[) u - velcity field (cmpnents u-^) z - free surface elevatin. 3. EQUATIONS The shallw water equatins cnsist f the mass cnservatin equatin and tw mmentum equatins I dz + V.Q = 0 3t ^ (1) 9Qi + V.(Qi u ) + gh ~ KAQi * Fi 9t ^ 9xi A fractinal step algrithm is used t decmpse the equatins int a pure cnvectin system and the remainder. Q n, Z n being given at time t n, the unknwns at time t n+ l, Q n+ 1, Z n+ 1 are cmputed by the tw fllwing successive steps ; - Determinatin f auxiliary unknwn Q n+ "2 slutin f the cnvectin equatins ; (2) ^_Qi + V.(Qi u n ) - 0 3t ^ - Reslutin f the diffusin and prpagatin prblem (3) 2-?. + V.Q = 0 JL5j " K AOi + ghjlz = Fi 3t 3xi *ith initial cnditins Z n, Q n+ y

3 SHALLOW WATER EQUATIONS ADVECTION STEP It invlves the reslutin f equatins (2), which can be transfrmed int equatins n velcity, by use f the cntinuity equatin ; (4) Jui + u 11 '. Vu{ = 0 9t ^ This frm has been prefered t the riginal frm (2) as explained in sectin 6. A characteristics methd has been used t slve thse equatins (4). It requires tw steps : - cmputatin f the characteristic curves passing thrugh the ndal pints ; - interplatin at the ft f thse curves. The methd is uncnditinally stable and the time step has just t be small enugh t catch the time variatin f the velcity field u. a, Mrever the amunt f cmputatin needed is very reasnable. In particular, there is n nn-symmetric system t assemble and slve at each time step, as wuld appear in a standard implicit methd. As fr accuracy, it is quite satisfactry, althugh smewhat depending n the time-step : the nly significant errr being made when perfrming the interplatin at the ft P; f the characteristic, the accuracy decreases when Pj is far frm a mesh-pint, particularly when using a linear interplatin. Finally, cnservativity is nt ensured by the scheme ; this is prbably the main drawback f the methd. This technique has been applied fr a lng time in the slutin f the Navier Stkes equatins and is described in mre detail in [1]. Sme mre research is dne n the subject [2], 5. DIFFUSION AND PROPAGATION STEP An implicit discretizatin has been chsen fr system (3) except fr the nn linear term ghvz which is partly explicited. There is n need fr a secnd rder scheme in time fr such lng waves as the tide. If we drp, fr sake f simplicity, the superscripts n+1, the system t slve in this step is f the frm ; (5) az + V.Q = R

4 620 COASTAL ENGINEERING 1982 (6) aq - KAQ + gh m Vz = T a, ^ % with ;f,= ij R=iz n, and T = J Q + F^ - g (fcn -h m )V z n DT DT a, DT ^ ^ (h m being the average in time f h, that is a functin f space crdinates nly). The discretizatin f equatins (5), (6) with their bundary cnditins (Q given) yields t a large size linear system. A natural idea t reduce this size, is t eliminate Q between equatins (5) and (6). This is achieved by applying the divergence peratr t (6) and plugging (5) int the result. The equatin btained is ; (7) 2 a z - V. (Ka + g h m )V z = W with ; W = a R " KA. R "V.T. a, Equatin (7) is an elliptic equatin, giving by discretizatin a symmetric matrix. It can easily be slved if a bundary cnditin is prvided fr z. If s, Q can be straight frwardly cmputed frm equatin (6) where the tevm gh m Vz is nw a knwn functin and can be rejected int the right hand side. With Dirichlet bundary cnditins, equatin (6) decmpses int tw decupled equatins, ne fr each cmpnent Qi f Q leading t the same symmetric matrix fr the discrete system. ^ The prblem then amunts t the determinatin f the trace f z n the bundary. The prcedure used fr this purpse is an extensin f the technique prpsed by Glwinski-Pirnneau [3] fr the Stkes prblem. It is described in detail in [5]. This technique is advantageus n the pint f view f cmputatinal efficiency. Tw "dmain" matrices have t be stred, ne fr the variable z, anther fr each cmpnent f Q (it is the same matrix fr the tw cmpnents). They are symmetric and sparse and small enugh t be prcessed by an in-cre slver such as an incmplete Chleski precnditined cnjugate gradient. In additin, the matrix f the bundary peratr has t be factrized and stred. NUMERICAL RESULTS Several applicatins have been perfrmed, frm a very schematic case, in rder t have a first view n the pssible prblems, t a realistic cmplex ne, t give a mre cmprehensive test f the methd.

5 SHALLOW WATER EQUATIONS Time (hurs) k Theretical value *~\s Cmputed value Fig.1.Finite elements discretizatin Fig.2. Average surface elevatin Fig.3 Velcity fields and free surface elevatin lines Lw water High water SCHEMATIZED MASSACHUSSETS BAY

6 622 COASTAL ENGINEERING Schematized Massachussets Bay In this first experiment, emphasis has been laid n the general behaviur f the methd rather than the cmparisn with actual data. Thus neither the frictin and Crilis frce, nr the varying bttm tpgraphy were simulated. The dmain was carsely apprximated by a rectangle (fig.l) divided int 192 triangular elements with respectively 119 and 429 ndes fr piecewise linear and quadratic apprximatins. Bth cmpnents f the flw discharge were prescribed n the bundaries ; this discharge was taken t be zer n the land limits, and was schematized n the cean limit by a sinusidal unifrm flw. Results as regards current fields and free surface elevatin (fig. 2 and 3) were fund t be quite satisfactry ; in particular glbal mass cnversatin has been insured at less than 1% f the ttal mass flux ver a whle tide. Teachings f this first applicatin are as fllws ;. linear and quadratic apprximatins fr the water height, tgether with quadratic apprximatin fr the flw discharge, have given clsely the same results ;. there exists sme stability cnditin fr the diffusin step ; the nndimensinal number (K.DT/DX^) has t be mre than abut 10"^" t avid scillatins f the cmputed slutin (prbably because f the cnflict arising between a nearly hyperblic system t slve and bundary cnditins related t the diffusin, i.e. prescribed velcity n the limits) ;. sme kind f "versht" can appear in the advectin step near the bundary when the flw gets ut, due t a bad prperty f the quadratic interplatin n the flw discharge Dunkirk uter harbur The secnd applicatin, which is less schematic, cncerns the cmputatin f the tidal flw pattern in the vicinity f the new uter harbur f Dunkirk, lcated n the french cast f the Nrth Sea. This example has been chsen fr tw main reasns.". the shape f the dmain and the cmplexity f the laying ut f the limits give the typical case where a finite clement methd turns ut t be specially suitable (gd bundary representatin and lcal refinement facilities) and prvide a rather severe test f the mdel ;

7 SHALLOW WATER EQUATIONS 623 a. Mesh f the whle area km I 1.5 km b. Detail f the harbur Fig.A_FINITE ELEMENTS MODEL OF DUNKIRK OUTER HARBOUR

8 624 COASTAL ENGINEERING 1982 this case has been previusly studied by means f a scale mdel and f a finite difference mdel ; s there exist references t which cmpare the results f the finite element mdel. The finite element grid (fig.4) has been generated autmatically by a prgram develped in ur labratry. As it can be nticed frm this illustratin, a special mesh refinement has been intrduced near the limits, in rder t btain a better cnsideratin f the bundary cnditins (which is particularly useful fr the land bundaries). In this experiment a piecewise quadratic apprximatin, has been used fr each variable ; there are abut nine hundred triangular elements and tw thusand calculatin ndes. The characteristics f the cmputatin are as fllws ;. tide range is abut five meters ;. maximum velcity is abut 1,5 m/s ;. tide and current are in phase (that is t say that the tide wave is here quite purely prgressive) ;. eddy viscsity cefficient is equal t 5 m^/s. time step is sixty secnds and Curant number fr prpagatin varies lcally frm 1 t 5. Finally bth cmpnents f the flw discharge are prescribed n the bundaries. In fact tw cmputatins have been carried ut successively in this case f Dunkirk ; the first ne under schematic cnditins and the secnd ne in a mre realistic way Sc_hema_t i_c mpjjt a_t_in In the first trial the flw discbarge is assumed t be sinusidal and parallel t the upper limit ; the bttm is flat and bed frictin is nt cnsidered, just as in the example f Massachussets Bay. Sme instabilities appeared in the current field during the cmputatin ; the prblem turned ut t cme frm the advectin step, slved at first n the flw discharge Q ; 3Qi + u 11. V Qi = " Qn V -u n The discntinuity f the term in the right hand side was fund t be respnsible fr the instabilities. When using the frmulatin n velcity nly, thse instabilities actually disappeared. Fig. 5 shws the evlutin f the flw pattern in the vicinity f the harbur at the beginning f the fld current (slack ccurs abut three hurs befre high tide) : the flw deflectin in frnt f the harbur and the frmatin f tw eddies can be nticed ;

9 '7/'''/''»: SHALLOW WATER EQUATIONS 625

10 626 COASTAL ENGINEERING after HW 5:00 after KW 0 I 2 3km 6:00 after HW Fig. 6 i i i i m/s SCHEMATIC MODELISATION OF DUNKIRK OUTER HARBOUR COMPUTED CURRENTS IN THE WHOLE DOMAIN

11 SHALLOW WATER EQUATIONS 627. The first ne in the harbur entrance ; it becmes weaker and weaker as the harbur fills in.. The secnd eddy dwnstream the east jetty expands gradually eastwards and in fact it will fill at the end f the fld the whle eastern part f the dmain because f the absence f bed frictin and because the bundary cnditins dn't allw it t get ut. Fig. 6 pints ut in the whle dmain the same phenmenn during the ebb ; an eddy appears at the beginning f the ebb n the left f the west jetty and expands prgressively till the end f the ebb. Finally mass cnservatin was fund t be as satisfying as fr the first example ; here als the glbal errr regarding the free surface elevatin ver ne tide is less than 1 % f the tide range Rea.l_is_tic cmputatin A secnd cmputatin has been carried ut after the insertin in the mdel f the bed frictin and f the variability f the bed tpgraphy. Fig. 7 exhibits the bathymetry in the vicinity f the harbur : the depths are quite variable, especially because f the navigatin channel fr il tankers ; the steepness f the edges f the channel (abut 1;10) is in fact a severe cnditin fr the shallw water equatins. Bundary cnditins are nw deduced frm the results f a previus cmputatin made with a finite difference methd n a larger dmain. The cmputatin has been wrked ut during mre than ne tide, and the re suits have been cmpared t the data btained by phtgraph s f surface flats n the scale mdel, built abut twelve years ag fr the study f the new uter harbur. Fig. 8 t 10 display this cmparisn during the mst interesting perid, frm the beginning f the fld till high water. On fig. 8, at the beginning f the fld, an eddy appears in a similar way n bth mdels, just behind the end f the west jetty, while the harbur fills in ; half an hur later, bth eddies have shifted and spread ut likewise. Afterwards (fig. 9) the eddy expands gradually till it cvers the whle entrance in bth mdels, with perhaps a slight difference in the shape. The flw deflectin due t the harbur can be nticed farther than ne kilmeter away frm the entrance in bth cases.

12 628 COASTAL ENGINEERING 1982 > X Q. a: a. Q UJ m _i n n z> m c X a: LU «O qc z Q I r-

13 SHALLOW WATER EQUATIONS 629 s x. " _l - O z u a /) Q Z UJ u. '.".'.''Ufa

14 630 COASTAL ENGINEERING 1982 E CO _1 LU Q O I UJ _l a 9k Q z a: u. s x

15 SHALLOW WATER EQUATIONS x cn _i LU Q O Z UJ cn Q Z 5 OH LU b Z> cn LU cc LU X cn ac. &, \ \\v\\\ \\ il

16 632 COASTAL ENGINEERING 1982 Frm ne hur befre high water till high water (fig. 10) the eddy remains stable ; dwnstream the harbur, a slight flw separatin can be nticed, but the cmputatin des nt develp a big eddy as in the case f the first cmputatin (fig. 5). The mass cnservatin is less accurately btained than in the previus experiments ; the glbal level errr ver ne tide is abut 3 % f the tide range ; this seems t be due t the steepness f the sea bed near the breakwaters ; the prblem can prbably be vercme by taking a mre severe threshld f cnvergence in the numerical reslutin f the linear systems. 7. CONCLUSION Cmpared t the classical finite difference methd, which is nw currently used t slve the shallw water equatins, this finite element methd presents many advantages, such as a better descriptin f the bundaries and a greater flexibility f the grid ; n the ther hand, the implementatin f the mdel is rather lnger and mre cmplex, and invlves a bigger cnsumptin f prcessing time ; nevertheless, due t the special care given t reduce the size f the systems t be slved, the in-cre memry requirements are quite reasnable. As an example cmputatin ver ne tide (12 h 24 mn) using nearly ndes f discretizatin and a time step f sixty secnds has required abut 2 hurs f CRAY 1 prcessing time and K bytes f in-cre memry. Numerical experiments have clearly shwn that even under rather severe cnditins the mdel has a nice behaviur and exhibits a fair cmparisn with experimental data. At present, imprvements are in prgress, with the implementatin f an incident wave cnditin and the cnsideratin f the influence f wind stress and atmspheric pressure field. The mdel will sn be used n a dmain cvering the whle English Channel up t the edge f the cntinental shelf (fig. 11), with tw purpses : - firstly t examine the impact f a pssible tidal pwer plant n the tidal pattern in this area ; - secndly t study the generatin and prpagatin f strm surges in the English Channel.

17 SHALLOW WATER EQUATIONS 633 Ul X X (Q _J z UJ Ul X t- LL O _J UJ 2: (f) z UJ Ul Ul I z u. O) u.

18 634 COASTAL ENGINEERING 1982 REFERENCES 1. J.P. Benque, G. Labadie, B. Ibler A finite element methd fr Navier-Stkes equatins - First Internatinal Cnference n numerical methds fr nn linear equatins - Swansea, U.K., sept, 2nd - 5th J.P. Benque, J. Rnat Quelques difficultes des mdeles numeriques en hydraulique - Fifth Internatinal Sympsium n cmputing methds in applied sciences and engineering - INRIA, December 1981, Versailles, France. 3. M.O. Bristeau, R. Glwinski, J. Periaux, P. Perrier, 0. Pirnneau and G. Pirier Applicatin f ptimal cntrl and finite element methds t the calculatin f transnic flws and incmpressible viscus flws - Rapprt de recherche , IRIA LABORIA, Le Chesnay, France. 5. G. Labadie, J.P. Benque, B. Latteux A finite element methd fr the shallw water equatins - 2nd Internatinal Cnference n numerical methds fr laminar and turbulent flw, Venice, Italy, July G. Labadie, B. Latteux Reslutin des equatins de Saint-Venant par une methde d'elements finis - Rapprts E.D.F. HE/41/82.15 et HE/42/82.34.