FullSWOF a software for overland flow simulation

Transcription

1 FullSWOF a software for overland flow simulation O. Delestre, S. Cordier, F. Darboux, M. Du, F. James, C. Laguerre, C. Lucas & O. Planchon LJA Dieudonn e & Polytech Nice Sophia SIMHYDRO 2012 Sophia-Antipolis 13 September 2012

2 Problem context

3 Shallow Water (Saint-Venant) system z, z + h h v u u O z y x Data : topography z, rain R, infiltration I Unknowns : velocities u, v, water height h th + x (hu) + y (hv) = R I t (hu) + x ( hu 2 + gh 2 /2 ) + y (huv) = gh( xz S f x ) t (hv) + x (huv) + y ( hv 2 + gh 2 /2 ) = gh( y z S f y )

4 Strategy Properties of the 1D Shallow Water system Choice of the method depending on the properties Validation : analytical solutions and laboratory experiment Application : field data

5 1D Shallow Water system z, z + h R(t, x) q(t, a) q(t, b) I(t, x) O a b x Data : topography z, rain R, infiltration I Unknowns : velocities u, water height h A system of conservation laws { th + x(hu) = R I t(hu) + x(hu 2 + gh 2 /2) = gh ( xz S f ) (1)

6 System properties (I) : Hyperbolicity Setting q = hu ( ) ( h U =, F (U) = q compact form Hyperbolicity if h > 0 : q q 2 /h + gh 2 /2 ) (, B = tu + xf (U) = tu + F (U) xu = B, R I gh ( xz S f ) ), λ (U) = u gh, λ +(U) = u + gh Saint-Venant gaz dynamic Froude number Fr = u Mach number u c c c = gh free surface waves celerity c = p (ρ) sound speed 1 subcritical Fr < 1 subsonic supercritical Fr > 1 supersonic 1. p(ρ) = ρrt perfect gaz

7 Numerical method (I) Finite volume method t t n+1 t n O we get x t n x i 1 x i 1/2 x i x i+1/2 x i+1 x U n+1 i = Ui n t x with the interface flux approximation We integrate tu + xf (U) = 0 on the volume [t n, t n+1 [ ]x i 1/2, x i+1/2 [, U n i = 1 x and we set xi+1/2 x i 1/2 [ F n i+1/2 F n i 1/2], U(t n, x) dx F n i+1/2 = F(U n i, U n i+1) 1 t t n+1 t n F (U(t, x i+1/2 )) dt.

8 Numerical method (I) For each choice of F(U G, U D ) we have a different finite volume scheme : second Order HLL, kinetic, Rusanov, VFRoe-ncv, (suliciu,...) in space : MUSCL, ENO, modified ENO, (UNO, WENO, DG,...) in time : Heun/RK2 TVD (RK3, RK4,...)

9 Numerical method (I) For each choice of F(U G, U D ) we have a different finite volume scheme : second Order HLL, kinetic, Rusanov, VFRoe-ncv, (suliciu,...) in space : MUSCL, ENO, modified ENO, (UNO, WENO, DG,...) in time : Heun/RK2 TVD (RK3, RK4,...) Coupling with the source term (topography xz) Necessity : compatibility with steady states

10 System properties (II) : Steady states

11 System properties (II) : Steady state { th + x(hu) = R I t(hu) + x(hu 2 + gh 2 /2) = gh ( xz S f ) th = tu = tq = 0 { xhu = R I x(hu 2 + gh 2 /2) = gh ( xz S f ). (2)

12 System properties (II) : Steady states Lac at rest equilibrium { u = 0 g(h + z) = Cst. z, z + h H sur = z + h = Cte O x

13 Hydrostatic reconstruction (II) [Audusse et al., 2004] We define and z = max(z G, z D ) U G = (h G, h G u G ), U D = (h D, h Du D ) h G = max(h G + z G z, 0) h D = max(h D + z D z, 0) Thus, we have ( ) F G (U G, U D, Z) = F(UG, UD) 0 + ( g(h 2 G (hg ) 2 )/2 ), F D (U G, U D, Z) = F(UG, UD) 0 + g(h 2 D (hd) 2 )/2 where F(U G, U D ) is the numerical flux..

14 Friction treatment [Bristeau and Coussin, 2001] Shallow Water system with friction f { th + x(hu) = 0, t(hu) + x(hu 2 + gh 2 /2) + h xz = hf, (3) f = f (h, u) friction force (on the bottom) Several friction laws possible Manning : f = n 2 u u h 4/3 Darcy-Weisbach : f = F u u 8gh Semi-implicit treatment [Bristeau and Coussin, 2001] q n+1 i + F qn i q n+1 i t = qi n 8hi nhn+1 i t ( ) Fi+1/2 F i 1/2 x

15 Validation on analytical solutions SWASHES Semi-implicit (subcritical-supercritical) 1 0 topography free surface critical level -1 z, z+h (m) x (m)

16 Summary of the chosen numerical method Numerical flux : HLL Second order scheme : MUSCL Friction : semi-implicit treatment Shallow Water system with rain R { th + x(hu) = R t(hu) + x(hu 2 + gh 2 /2) + h xz = hf time splitting/explicit treatment (4)

17 Validation on experiments INRA rain simulator

18 A simulation result (Manning) 9 8 Measures n= q (g/s) Simulation time (s)

19 Thies parcel Senegal ([Tatard et al., 2008], IRD) SVG velocity measure [Planchon et al., 2005] Number of meshes : (4 m 10 m)

20 Thies parcel Senegal ([Tatard et al., 2008], IRD)

21 Thies parcel Senegal ([Tatard et al., 2008], IRD) Hyetogramme P (mm/h) t (s) Hydrogramme q (mm/h) t (s) Full SWOF Mesure Thies

22 Thies parcel Senegal ([Tatard et al., 2008], IRD) FullSWOF 2D PSEM 2D v simulees (cm/s) v mesurees (cm/s)

23 Thies parcel Senegal ([Tatard et al., 2008], IRD) Green-Ampt infiltration parameters θ i (resp. θ s) initial (saturation) volumetric water content K s soil saturated hydraulic conductivity q [mm/h] measures θ s θ i = θ s θ i = 0.08 θ s θ i = 0.1 θ s θ i = t [s] q [mm/h] measures K s = K s = K s = K s = t [s] (a) (b)

24 Thies parcel Senegal ([Tatard et al., 2008], IRD) (a) FullSWOF 2D (b) MIKE SHE (DHI software)

25 FullSWOF & conclusion code in C++ under under CecCILL-V2 (GPL compatible) license and free sources available from https ://sourcesup.cru.fr/projects/ possibilities to modify the code, add numerical methods, physical models, customize inputs/outputs,... object and inheritance variables encapsulation vector class (2d) objects distributor fixed CFL and fixed t Doxygen documentation : html, latex,... documentations GUI Watersheds & FullSWOF paral

26 Thank you

27 HLL flux F (U G ) if 0 < c 1 F(U G, U D ) = F (U D ) if c 2 < 0 c 2F (U G ) c 1F (U D ) + c1c2(u D U G ) else c 2 c 1 c 2 c 1 with two parameters For c 1 and c 2, we take c 1 < c 2., c 1 = inf ( inf λ j(u)) and c 2 = sup ( sup λ i (U)). U=U G,U D j {1,2} U=U G,U D i {1,2} with λ 1(U) = u gh and λ 2(U) = u + gh. retour

28 Audusse, E., Bouchut, F., Bristeau, M.-O., Klein, R., and Perthame, B. (2004). A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput., 25(6) : Bristeau, M.-O. and Coussin, B. (2001). Boundary conditions for the shallow water equations solved by kinetic schemes. Technical Report 4282, INRIA. Planchon, O., Silvera, N., Gimenez, R., Favis-Mortlock, D., Wainwright, J., Le Bissonnais, Y., and Govers, G. (2005). An automated salt-tracing gauge for flow-velocity measurement. Earth Surface Processes and Landforms, 30(7) : Tatard, L., Planchon, O., Wainwright, J., Nord, G., Favis-Mortlock, D., Silvera, N., Ribolzi, O., Esteves, M., and Huang, C.-h. (2008). Measurement and modelling of high-resolution flow-velocity data under simulated rainfall on a low-slope sandy soil. Journal of Hydrology, 348(1-2) :1 12.