IMPLICIT FORMULATION FOR 1D AND 2D ST VENANT EQUATIONS PRESENTATION OF THE METHOD, VALIDATION AND APPLICATIONS

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1 IMPLICIT FORMULATION FOR 1D AND 2D ST VENANT EQUATIONS PRESENTATION OF THE METHOD, VALIDATION AND APPLICATIONS Tierry Lepelletier 1, Quentin Araud 2 1 Hydratec Immeuble Central Seine - 42 quai de la Râpée Paris - lepelletier@ ydra.setec.fr 2 Hydratec 1 rue de la Course Strasbourg araud@ydra.setec.fr KEY WORDS St Venant equations, Implicit formulation, solving procedure, sock capture, bencmarking. ABSTRACT St Venant equations form te teoretical basis to ydraulic modeling in te field of river flow and urban ydraulics. Numerical scemes may prove difficult to implement due to te yperbolic nature of tese equations and to te variety of configurations met in practice. Existing ydro-informatic software are usually specialized in a given domain. Optimized solving scemes are selected to best fit te practical performance wic is required in tis domain of application. In order to meet te needs to a large variety of domains an original formulation as been developed to solve te full St Venant equations in one and two dimensions, wit optional simplifications to optimize some computations, wile keeping te same code arcitecture. Te underlying formulation is based upon an original metod for solving full St Venant equations troug finite volume space discretization and implicit time marcing solving procedure. Tis metod is unconditionally stable, time step is variable and is adjusted automatically witin calculation in order to preserve numerical accuracy in case of discontinuities, suc as sock formation. Tis numerical sceme is implemented in Hydra software, wic is developed by HYDRATEC. Various 1D and 2D tests, dam break problems and recirculation in 2D domains are presented and are compared wit oter referenced software in teir respective domain of application, suc as Telemac (developed by EDF). 1 DISCRETIZATION OF TWO DIMENSIONAL ST VENANT EQUATIONS AND SOLVING PROCEDURE St Venant equations are first expressed in classical integral form witin a cell : UU Ωii ddωii + (FF Γ ii nn xx + GGnn yy )ddγ ii = SS ff Ωii ddωii + ggrad(z b ) dωi wit : (1) Ωi UU = qq xx uu xx qq yy uu yy qq xx qq xx ² FF = + gg ² 2 GG = qq xx qq yy qq yy ² qq yy qq xx qq yy + gg ² 2 n Notations are defined in appended glossary. Pressure and gravity terms are combined and are approximated as follows : gg ( Γ ²)nnddΓ ii 2 ii + ggrad(z b )dωi gg ıııı Ω jj zz ii 2 jj zz ii nnddγ Γ iiii were ıııı = ii+ jj iiii 2 (2)

2 Tis expression is valid as long as differences between water depts between two adjacent cells remain small. It turns out tat it remains valid in presence of socks as will be sown in section 2 below. zi i zj j Mes i Mes j Spatial discretization of equation (1) consists in decomposing te wole domain into quadrangular or triangular cells and to average components of te unknown U vector witin eac cell. Connectivity condition between cells must be ensured. Equation (1) is rewritten as follows in discretized form, using te approximation of equation (2) : AA ii UU ii FF = + AA iiss ffff + jj FFnn xx + GGnn yy ll iiii = 0 (3) were : qq xx ² qq xx + 0.5gg zz ıııı jj zz ii GG = qq xx qq yy qq yy ² qq yy qq xx qq yy + 0.5gg zz ıııı jj zz ii Te terms jj FFnn xx + GGnn yy ll iiii Flux ij represent te flux terms across two adjacent cells. Tis alternative formulation yields following advantages : - gravity terms are source terms but tey are included in te line integral terms F and G wic are expressed along eac cell boundary : expression (3) is best fitted to link togeter equations between different domains, suc as 2D and 1D domain. Linkage is provided by te flux terms Flux ij wic ave different expressions according to te pysical link type. It is also easy to combine simplified and complete formulations witin a model. Simplified treatment of a sub domain just requires to ignore te convective terms in vectors F and G. - It provides numerical robustness, - It is easy to introduce singularities witin adjacent cells. Witin a 2D domain te flux components are treated by : - centered formulation for te volume fluxes, - upwind formulation for te momentum fluxes. 0.5(qq ii + qq jj ) 0 FF iiii. nn xx + GG iiii. nn yy = αα ii uu xxxx qq ii + αα jj uu xxxx qq jj + 0.5gg zz ıııı jj zz ii nn xx wit : αα ii uu yyyy qq ii + αα jj uu yyyy q jj 0.5gg zz ıııı jj zz ii nn yy uu xxxx uu yyyy : average flow velocity witin one cell i qq ii q xx,ii. nn xx + q yy,ii. nn yy qq jj q xx,jj. nn xx + q yyyy. nn yy αα ii = 1 si qq ii > 0 αα ii = 0 si qq ii < 0 αα jj = 1 si qq jj > 0 αα jj = 0 si qq jj < 0

3 Time integration of equation (3) is treated in a fully implicit way : Flux terms : FF nn+1 iiii = FF nn iiii + FF iiii UU UU ii + FF iiii UU ii UU jj jj GG ii nn+1 = GG ii nn + GG iiii UU ii UU ii + GG iiii UU jj UU jj UU ii nn+11 = UU ii nn + UU ii UU jj nn+11 = UU jj nn + UU jj Te Jacobian matrices : FF iiii UU ii, FF iiii UU jj, GG iiii UU ii, GG iiii UU jj are ranked 3x3 et contain differentiated terms wit respect to natives variables defined in vector U Inertia term: UU AA ii ii = AA ii UU dddd ii Friction term : SS nn+1 ffi = SS nn ffff + SS ffff UU UU ii ii Te final set of equations to be solved at eac time step is put in matrix form : MM iiii UU ii + KK iiii jj UU ii = FF ii (4) In practice tis matrix system is formed by calculating contribution of eac boundary to flux terms, ten contributions of volume terms witin eac cell. Solution of matrix system (4) is calculated using Pardiso Matrix Solver. Tis formulation encompasses te 1D case, it can terefore be specialized to 1D problems or extended to mixed problems including 1D and 2D sub domains witin a model witout difficulty. 2. SHOCK CAPTURE Te above formulation is based on native conservation laws of yperbolic St Venant equations. It contains all te terms wic are supposed to produce discontinuities depending on pysical configurations. In order to demonstrate te ability of te above algoritm to describe socks te restricted 1D case wit 3 adjacent cells is considered : M1 M2 M3 Te discretized equation (3) along te x axis is written as follows in steady state flow regime for te cell M2. Te following equation is obtained by neglecting friction terms : QQuu 1 + QQuu 2 + gg 2 (zz 2 zz 1 ) 12 + gg 2 (zz 3 zz 2 ) 23 = 0 (5) 12 = 0.5( ) 23 = 0.5( )

4 Let us assume tat a ydraulic jump occurs between cell 1 and 2 : One as : zz 2 zz 3 z 1 z z 3 3 Q ( flow rate) Equation (5) becomes : QQuu 1 + QQuu 2 + gg ( ) 12 = 0 QQuu 2 + gg = QQuu 1 + gg (6) Equation (6) describes a jump equation wic is linearized wit respect to term 12. Tis sows tat our formulation allows to capture socks, albeit introduction of some distortion for strong socks. Te practical validity or tis simplification is discussed wit te tests case presented in capter 4. Te above analysis can be extended easily to transient sock waves, suc as in te dam break problem, by taking in account te inertia terms of equation (3) : it can be cecked tat te rigt expression for te sock velocity of te moving sock wave is obtained. Tis sock analysis can extended to te 2D case. A furter simplification is introduced by assuming tat te local direction of a 2D sock wave is perpendicular to te boundary between two adjacent cells. Starting from equation (3) one obtains te equations for te oblique ydraulic jump, assuming tat te boundary line between two cells is a sock line. Tis additional assumption yields wrong results locally but it as been found tat te distortion levels out wen one considers te resulting wave pattern on a scale involving a few cells togeter. 3. EXTENDED FORMULATION WITH HYDRAULIC SINGULARITIES So far te formulation above applies to an omogenous 2D domain : two adjacent cells are connected by relations involving flux terms. It is fairly straigtforward to generalize tis concept by adapting te flux terms to ydraulic excange laws involving singular ead losses: Excange between te two cells is controlled by te flow rate Q et te momentum flux vector F. In case of a singularity linking te cells, Q and F are expressed as : Q(Hi, Hj) and F(Hi, Hj) were H stands for te averaged total ead witin one cell. Te expressions for Q and F depend on te nature of te singularity.

5 Assuming a flow from cell i to cell j te flux leaving cell i is written as : Q qq FFFFFFFF iiii = ii uu ii qq ii vv ii And te flux entering flux j is written as : FFFFFFFF jjjj = Q FF xx FF yy As an example te gate let us consider te gate type singularity wic is governed by a variety of laws according to te gate geometry and ydraulic upstream and downstream conditions. Figure 1 : definition sketc for te gate/weir singularity Next diagrams illustrates all te laws wic govern tis singularity and te range of validity of eac. Te flow equations are defined suc tat flow continuity is ensured at every transition between two flow regimes. Equations are given below for 4 basic regimes: Weir flow regime: HH 1 pp 1 < 3 2 vv : QQQQ DD = 2 3 CC ccbb 2gg(HH 1 pp 1 ) 3 2 QQQQ NN = bb 2gg[(HH 1 pp 1 ) (HH 2 pp 2 )] 1 2(HH 2 pp 2 ) Gate flow regime: HH 1 pp 1 > 3 2 vv QQQQ DD = bbcc cc vv 2gg(HH 1 pp 1 ) QQQQ NN = KKKK vv 2gg((HH 1 pp 1 ) (HH 2 pp 2 )) 1/2 Figure 2: delimitation of flow regimes for te gate/weir singularity Special consideration must be given to te momentum flux just downstream of te gate. Iterative calculations are necessary to determine te proper flow regime and to apply te corresponding flow equations. In case of an incident flow oblique to te gate te upstream momentum flux vector is decomposed into two components, one normal to te gate, te oter one parallel to te gate. Te flux parallel to te gate is supposed uncanged, wile te oter flux component is treated by te 1D gate equation system.

6 3. BENCHMARKING Te above formulation as been implemented into te Hydra computational software and as been evaluated against analytical solutions, field data and existing referenced codes suc as Telemac ( EDF) and Rubard (IRSTEA). Some selected examples are presented below to demonstrate te capabilities of te code: 3.1 Hydraulic jump in rectangular cannel Cannel widt is discretized into 3 meses 5m eac. Cannel lengt is discretized into 60 meses : - 20 cells 10x5m in te upstream part of te canal wic as a 2% slope, - 40 cells 5x5m in te downstream part wic is flat. Flow rate is 60 m 3 /s. Figure below sows te computed water profile along te centerline. It is compared wit te analytical solution: Analytical solution Canal invert 2D simulation Figure 3: water profile elevation along a rectangular cannel : computed and analytical solutions Te two curves are in good agreement, given te spatial discretization used for te computations. Difference on te upstream end is due to te fact tat no upstream limit condition is imposed : te code selects critical flow regime. Te water profile ten connects gradually towards uniform flow. 3.2 flow past a gate Cannel discretization and flow conditions are te same as above but a gate is introduced at abscissa 0.4km. Gate opening dept is 50cm. te grap below sows te computed water profiles using complete formulation and simplified formulation ( witout convective terms) :

7 Ca na l in ve rt Canal invert Complete numerical solution Simplifiednumerical solution Figure 4: computed water profile elevation along a rectangular cannel wit a gate singularity Water profile computed wit te full equations exibits a ydraulic jump in te upper section of te cannel and anoter one about 25m downstream of te gate. Te flow past te gate is not influenced by te downstream conditions. In te simplified calculations convective terms are ignored : super critical flow is not modelled, wic results in a significant over estimates of te water profiles upstream of te gate D flow in a river past a navigation weir. A section of te Aisne river as been modelled to analyze velocity field past a navigation weir and to investigate resulting geo-morpological consequences. Te model consists of about meses arranged as follows:

8 Navigation Weir Flow input ndition z(q) Figure 5 : 2D mesing for te river modelling pas a navigation weir Te weir is closed except along a 25m section next to te rigt bank. It is oblique to te upstream flow. River flow rate is 100 m 3 /s. Te weir elevation is adjusted so tat te flow past it is critical. A close up view of te flow velocity distribution is sown below : ydra code Figure 6 : flow distribution computed by ydra numerical code One can see clearly te two recirculation zones induced by te jet flux past te weir. Tis pattern was actually observed on te site. Te same calculations was performed using Telemac code. Te computed flow distribution exibit a very similar pattern as sown below. In te Telemac model te

9 mes size around te weir is muc more refined tan in te Hydra model because te weir is modelled as a geometric singularity : it is totally immerged witin te 2D mesing. Telemac code Figure 7: flow distribution computed by telemac numerical code 3.4 Transient dam break problem in a rectangular cannel. Tis problem is classic and as an analytical solution. Te case investigated is defined by te following definition sketc : Mesing Dam position expansion wave Sock wave Figure 8 : definition sketc for te dam break problem

10 Initially water dept is 5m upstream of te dam and 1m downstream. Te dam breaks at time=0. Figures below sow te longitudinal water and velocity profiles along te canal at time t = 288seconds. Numerical solution Analytical solution Figure 9 : water elevation and velocity profiles at time 288seconds after sudden dam break. Examination of tese graps sow near perfect agreement between computation and analytical solution. Tis good agreement suggests tat te approximation introduced in equation (3) is valid to solve practical problems involving socks.

11 4 DISCUSSION AND CONCLUSION Te implicit formulation presented in tis paper to solve full equations of free surface flow is believed to mark a progress wic departs from existing metods. It combines several advantages: unconditional numerical stability, computational robustness, accuracy, flexibility for solving problems in a wide variety of fields: estuary and river ydraulics, flood propagation and networks ydraulics. Finite volume formulation is based on te native conservative form of te St Venant equations witout any assumptions on te solution structure across a ydraulic sock. It includes any kind of ydraulics singularities in transient as well as steady state flow regimes and to combine simplified and full resolution of equations in te same model. Formulation is also quite suited to model 1D-2D interactions witin a single model. Calculations are fast: inclusion of singularities and constraints lines into a model enables to coose fairly loose mesing as compared wit full 2D codes for te same accuracy. Tis results in considerable time savings in computations. Tis upgraded formulation is implemented into Hydra software and as sown particular efficiency in all modelling problems involving floods and inundations in urban areas. Hydra is now coupled to a user s interface immerged in te Open Source Qgis GIS ( ref[4]). Coupling between tis interface and te computational code, sould make Hydra attractive among te ydro informatic software available on te market. BIBLIOGRAPHY 1. J.M. Hervouet - Hydrodynamics of free surface flow. Modelling wit Finite Element metod - Wiley ed A Paquier - Rupture de barrage : validation des modèles numériques du CEMAGREF dans le cadre de CADAM Ingenierie IRSTEA edition E. Toro Sock capturing metods for free surface flows Wiley ed Development of a ydraulic modelling platform witin an Open Source environment C Duran, T. Lepelletier, V. Mora Symydro symposium. GLOSSARY ( ux uy ) : velocity components in te x and y direction. g : acceleration of gravity : water dept i : mean water dept witin cell i lij : segment lengt between two adjacent cells Ai : cell area zb : bottom elevation U : vector composed of unknown variables ( qx qy ) ( qx qy ) : = ( ux uy ) H : pressure ead = + uu xx 2 +uu2 yy 2gg SS ff : bottom friction vector wit components : SS ff,xx = gg (nn² xx uu xx uu 2 xx + uu2 yy 4/3 ) et SS ff,yy = gg(nn² yy uu 2 xx + uu2 yy 4/3 ) ( nx ny ) : Manning coefficients in Ox et Oy direction.