Numerical Modeling of Stratified Shallow Flows: Applications to Aquatic Ecosystems

Transcription

1 Numerical Modeling of Stratified Shallow Flows: Applications to Aquatic Ecosystems Marica Pelanti 1,2, Marie-Odile Bristeau 1 and Jacques Sainte-Marie 1,2 1 INRIA Paris-Rocquencourt, 2 EDF R&D Joint work with E. Audusse (Univ. Paris XIII), B. Perthame (Univ. Paris VI, INRIA) N. Goutal, M.-J. Salençon (EDF R&D) BANG Presentation Day, INRIA Paris-Rocquencourt, September 22, 2009

2 Outline 1 Motivations 2 A variable-density multilayer Saint-Venant model 3 A Reservoir model based on Navier Stokes: OPHÉLIE c EDF 4 Numerical simulations 5 Perspectives

3 Motivations Environmental flows Modeling of complex free surface environmental flows Aquatic ecosystems: lakes, estuarine waters, lagoons,... Related applications: water quality, sustainability of ecosystems. Lake of Geneva (Lac Léman) Étang de Berre Important vertical stratification (temperature, salinity); Column structure result of complex 3D forcing mechanisms: Natural: rivers inflows, wind, solar radiation,... Man-made: inflow/outflow in hydraulic reservoirs, pollutant spills,... Bio-dynamics primarily controlled by thermo-chemical status.

4 Motivations Environmental flows Typical lake stratification wind rain solar radiation Lake stratification with estimates of energy transfer time scales [s] Temperature vertical profile [ C] [J. Imberger, Hydrobiologia 125, 7 29, 1985] (

5 A variable-density multilayer Saint-Venant model Saint-Venant models A Multilayer Saint-Venant Model with Variable Density Saint-Venant models Main assumption: ε = H 0 /L 0 1 (shallow flow) Classical Saint-Venant equations - Uniform density and vertically uniform velocity; - Hydrostatic pressure (vertical velocity = 0). Multilayer Saint-Venant models - Allow non-uniform velocity along the vertical. Immiscible layers [Ovsyannikov 79, Vreugdenhil 79, Parés et al. 00,01,03...] Multilayer model with mass exchange [Audusse Bristeau Perthame Sainte-Marie, 2009] - Allows fluid circulation between layers. Goal: Introduce variable density description via the multilayer strategy.

6 A variable-density multilayer Saint-Venant model Mathematical model Variable-Density Multilayer Saint-Venant System Hydrostatic Euler equations with ρ = ρ(t ), T = active tracer ρ t + (ρu) x + (ρw) z = 0, (ρu) t + (ρu2 ) x + (ρuw) z p z = ρg, + p x = 0, (ρt ) t + (ρut ) x + (ρwt ) z = 0. + B.C. z z 4+1/2 = η(x, t) 0 z 3+1/2(x, t) z 2+1/2(x, t) Free surface u4(x, t) η(x, t) u3(x, t) H(x, t) u2(x, t) T4(x,t), h4(x,t) T3(x,t), h3(x,t) T2(x,t), h2(x,t) x h α = l α H, N α=1 l α = 1 u(x, z, t) N α=1 1 z h α (z)u α (x, t) z 1+1/2(x, t) z 1/2 = zb(x, t) zb(x, t) u1(x, t) Bottom T1(x,t), h1(x,t) T (x, z, t) N α=1 1 z h α (z)t α (x, t)

7 A variable-density multilayer Saint-Venant model Mathematical model Variable-Density Multilayer Saint-Venant System Unknowns: H, u α, T α, α = 1,..., N ( N t α=1 ρ ) αh α + N x( α=1 ρ ) αh α u α = 0, t (ρ αh α u α ) + x ( ρα h α u 2 α + h α p α ) = u α+1/2 G α+1/2 u α 1/2 G α 1/2 + z α+1/2 x t (ρ αh α T α ) + x (ρ αh α u α T α ) = T α+1/2 G α+1/2 T α 1/2 G α 1/2, α = 1,..., N. h α = l α H, z α+1/2 (x, t) = z b (x) + α j=1 h j(x, t), p α+1/2 z α 1/2 x p α 1/2, p α = g 2 ρ αh α + g N j=α+1 ρ jh j, p α+1/2 = g N j=α+1 ρ jh j, G α+1/2 = t( α j=1 ρ ) jh j + α x( j=1 ρ ) jh j u j, ρα = ρ(t α ). Hyperbolicity? Typically hyperbolic in practice.

8 A variable-density multilayer Saint-Venant model Numerical method Numerical Method System of 2N +1 equations in X = (H, {T α, q α } 1 α N ), q α = ρ α h α u α : t W (X) + x f(x) =S G(X, t,x X) + S p (X) Gα±1/2 pα±1/2. Conservative portion and S G : Finite Volume Kinetic Scheme [Perthame Simeoni, 2001] - Positivity preserving; - Avoids eigenvalues computation. Source S p : Hydrostatic Reconstruction Method [Audusse et al., 2004] - Well-balancing of equilibrium states. Particular case at rest: u α = 0, H + z b = const., ρ α = const., α = 1,..., N. Additional implemented features: 2nd order in space (MUSCL) and time (Heun); Vertical viscosity for velocity and temperature (implicit method).

9 A Reservoir model based on Navier Stokes: OPHÉLIE c EDF OPHÉLIE Hydrodynamic reservoir model based on Navier Stokes. [M.-J. Salençon, J.Y. Simonot, 1989] c EDF 2D laterally averaged hydrostatic incompressible Navier Stokes eqs. Rigid lid hypothesis: zs x = zs y = 0. (Bu) x + (Bw) z = 0, B(x, z) = lake width, ( x νx B u ) ( x + 1 B z νz B u ) z p x g u u ( x Dx B T ) ( x + 1 B z Dz B T ) z 1 Q ρ 0 C p z, u t + u u x + w u z = 1 B T t + u T x + w T z = 1 B BC 2, p=p s + gρ 0 (z s z) κgρ 0 zs z (T T 0) 2 dz, ρ(t )=ρ 0 (1 κ(t T 0 )) 2. Includes surface thermodynamics. Rigid lid vs. free-surface: much less restrictive CFL constraint. (BUT limited range of applications.) Model developed for specific application: Lake of Sainte-Croix.

10 Numerical simulations Wind stress Lake response to wind stress wind Homogeneous lake Stratified lake M.-J. Salençon and J.-M. Thébault, Modélisation d écosystème lacustre, (After J. Imberger, 1979, 1987).

11 Numerical simulations Multilayer model Numerical Tests - Multilayer Saint-Venant model Wind stress modeling Additional source term: S W = C 10 ρ a ρ w V W V W applied on surface layer N. V W = wind velocity, C 10 = wind stress coefficient for wind at 10 m. ρ a = air density, ρ w = water density. T = temperature, ρ(t ) = ρ 0(1 κ(t T 0)) 2, T 0 = 4 C. T1 > T2 T1 T2 ν 0 wind T 1 = 20 C, T 2 = 7 C, ν = 0.03 m/s 2. Test 1. V W = 10 m/s H meta = 2.5 m. (a) Passive tracer. ρ = const. (b) ρ = ρ(t ). Num. layers = 56, x-grid cells = 24. CFL = 0.8.

12 Numerical simulations Multilayer model Test 1 V W = 10 m/s, H meta = 2.5 m. Uplifting of the thermocline until steady position.

13 Numerical simulations Multilayer model Test 2: Upwelling V W = 15 m/s, H meta = 20 m. Upwelling of the middle layer (metalimnion).

14 Numerical simulations Multilayer model Test 3: Variable topography V W = 15 m/s, H meta = 20 m. Simulation with wind stopped after 18 hours.

15 Numerical simulations Multilayer model Limitations of the hydrostatic assumption Model lacks description of unstable stratification. Static equilibrium stable if ρ z 0. Test 2 at later times: Whereas physically: cold water front propagation with vertical isotherms. (cf. results OPHÉLIE) Non-hydrostatic terms need to be included in the model: t (ρw) + x (ρuw) + z (ρw2 ) + p z = ρg. - Preliminary results with modeling of z (ρw2 ) (Exp. Rayleigh). OPHÉLIE: Stabilization of temperature profile by mixing.

16 Perspectives Current and future work Current studies Variable-density multilayer Saint-Venant model Non-hydrostatic terms. Surface thermodynamics. Prospected work Comparison multilayer S.-V. / OPHÉLIE (time scale days). Coupling hydrodynamics with biological processes. OPHÉLIE application to realistic environment (Lac de Grangent).