Towards 2D overland flow simulations Olivier Delestre Laboratory - PowerPoint PPT Presentation

Towards 2D overland flow simulations Olivier Delestre Laboratory J.A. Dieudonn´ e & Polytech Nice Sophia University of Nice – Sophia Antipolis CEMRACS 2013

Problem context

Preventing overland flow and erosion From upstream... (Photos : Yves Le Bissonnais, INRA)

...to downstream.

Downstream zones modifications (watersheds) ◮ Where is the water coming from ? ◮ Where is it flowing ? Use of physical models is required to : ◮ simulate flow (volumes and location) ◮ suggest changes (grass strip). � to carry out improvements

Shallow Water (Saint-Venant) system z, z + h h v u � u y z O x Data : topography z , rain P , infiltration I Unknowns : velocities u , v , water height h  ∂ t h + ∂ x ( hu ) + ∂ y ( hv ) = P − I  hu 2 + gh 2 / 2 � � ∂ t ( hu ) + ∂ x + ∂ y ( huv ) = gh ( − ∂ x z − S f x ) hv 2 + gh 2 / 2 � � ∂ t ( hv ) + ∂ x ( huv ) + ∂ y = gh ( − ∂ y z − S f y ) 

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

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

System properties (I) : Hyperbolicity Setting q = hu � h � � � � P − I � q U = , F ( U ) = , B = , q 2 / h + gh 2 / 2 gh ( − ∂ x z − S f ) q compact form ∂ t U + ∂ x F ( U ) = ∂ t U + F ′ ( U ) ∂ x U = B , Hyperbolicity if h > 0 : � � 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 p ′ ( ρ ) sound speed 1 � c = subcritical Fr < 1 subsonic supercritical Fr > 1 supersonic 1. p ( ρ ) = ρ RT perfect gaz

System properties (II) : Conservation laws Integral of equation (1) in x ∂ t h + ∂ x q = P − I , gives � b � b d h ( t , x ) dx = q ( t , a ) − q ( t , b ) + P ( t , x ) − I ( t , x ) dx , dt a a Mass conservation of water. Second equation : momentum equation

System properties (III) : Steady states

System properties (III) : Steady state � ∂ t h + ∂ x ( hu ) = P − I (2) ∂ t ( hu ) + ∂ x ( hu 2 + gh 2 / 2) = gh ( − ∂ x z − S f ) ∂ t h = ∂ t u = ∂ t q = 0 � ∂ x hu = P − I . ∂ x ( hu 2 + gh 2 / 2) = gh ( − ∂ x z − S f )

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

Numerical method (I) Finite volume method ∆ x t We integrate ∂ t U + ∂ x F ( U ) = 0 t n +1 on the volume ∆ t n [ t n , t n +1 [ × ] x i − 1 / 2 , x i +1 / 2 [, t n and we set � x i +1 / 2 1 U n U ( t n , x ) dx i = ∆ x x i − 1 / 2 O x i − 1 x i − 1 / 2 x i x i +1 / 2 x i +1 x we get i − ∆ t U n +1 = U n F n i +1 / 2 − F n � � , i − 1 / 2 i ∆ x with the interface flux approximation � t n +1 1 F n i +1 / 2 = F ( U n i , U n i +1 ) ∼ F ( U ( t , x i +1 / 2 )) dt . ∆ t t n

Numerical method (I) ◮ For each choice of F ( U G , U D ) we have a different finite volume scheme : HLL, kinetic, Rusanov, VFRoe-ncv, suliciu, ... ◮ second Order ◮ in space : MUSCL, ENO, modified ENO ◮ in time : Heun

Numerical method (I) ◮ For each choice of F ( U G , U D ) we have a different finite volume scheme : HLL, kinetic, Rusanov, VFRoe-ncv, suliciu, ... ◮ second Order ◮ in space : MUSCL, ENO, modified ENO ◮ in time : Heun ◮ Coupling with the source term (topography ∂ x z ) Necessity : compatibility with steady states

Steady states (II) � ∂ t h + ∂ x ( hu ) = 0 (3) ∂ t ( hu ) + ∂ x ( hu 2 + gh 2 / 2) = − gh ∂ x z ∂ t h = ∂ t u = ∂ t q = 0 � hu = Cst . u 2 / 2 + g ( h + z ) = Cst We consider � u = Cst . g ( h + z ) = Cst

Hydrostatic reconstruction (II) [Audusse et al., 2004] We define z ∗ = max ( z G , z D ) and U ∗ G = ( h ∗ G , h ∗ G u G ) , U ∗ D = ( h ∗ D , h ∗  D u D )  h ∗ G = max ( h G + z G − z ∗ , 0) . h ∗ D = max ( h D + z D − z ∗ , 0)  Thus, we have  � � 0 F G ( U G , U D , ∆ Z ) = F ( U ∗ G , U ∗ D ) +  2 − ( h ∗  G ) 2 ) / 2 g ( h G  , � � 0 F D ( U G , U D , ∆ Z ) = F ( U ∗ G , U ∗ D ) +  2 − ( h ∗  D ) 2 ) / 2 g ( h D  where F ( U G , U D ) is the numerical flux.

Friction treatment Shallow Water system with friction f � ∂ t h + ∂ x ( hu ) = 0 , (4) ∂ t ( hu ) + ∂ x ( hu 2 + gh 2 / 2) + h ∂ x z = − hf , f = f ( h , u ) friction force (on the bottom) Several friction laws possible f = n 2 u | u | ◮ Manning : h 4 / 3 f = F u | u | ◮ Darcy-Weisbach : 8 gh

Friction treatment ◮ Apparent topography [Bouchut, 2004] z app = z + b n We consider : with ∂ x b n = S n f

Friction treatment ◮ Apparent topography [Bouchut, 2004] z app = z + b n We consider : with ∂ x b n = S n f ◮ Semi-implicit [Bristeau and Coussin, 2001] i | q n +1 + F | q n i + ∆ t q n +1 ∆ t = q n i � � F i +1 / 2 − F i − 1 / 2 i i h n +1 8 h n ∆ x i with q n +1 ∗ for the right part, we have i q n +1 ∗ q n +1 i = i 1 + ∆ t F | u n i | 8 h n +1 i

Validation on analytical solutions – SWASHES New test cases : ◮ Saint-Venant/shallow water : ◮ data z ◮ unknowns h et u (and so q )

Validation on analytical solutions – SWASHES New test cases : ◮ Saint-Venant/shallow water : ◮ data z ◮ unknowns h et u (and so q ) ◮ test cases ◮ data h and q (and so u ) ◮ unknown z

Validation on analytical solutions – SWASHES New test cases : ◮ Saint-Venant/shallow water : ◮ data z ◮ unknowns h et u (and so q ) ◮ test cases ◮ data h and q (and so u ) ◮ unknown z ◮ Several possibilities ◮ several friction laws ◮ diffusion source term [Delestre and Marche, 2010] ◮ rain source term

1 1 topographie topographie surface libre surface libre 0 niveau critique 0 niveau critique -1 -1 -2 z, z+h (m) z, z+h (m) -2 -3 -4 -3 -5 -4 -6 -5 -7 -8 -6 0 200 400 600 800 1000 0 200 400 600 800 1000 x (m) x (m) 1 2 topographie topographie surface libre surface libre 0 0 niveau critique niveau critique -2 -1 -4 z, z+h (m) z, z+h (m) -2 -6 -8 -3 -10 -4 -12 -5 -14 -6 -16 0 200 400 600 800 1000 0 200 400 600 800 1000 x (m) x (m)

Validation on analytical solutions – SWASHES Apparent topography (subcritical-subcritical) 1 topographie surface libre 0 niveau critique -1 -2 z, z+h (m) -3 -4 -5 -6 -7 -8 0 200 400 600 800 1000 x (m)

Validation on analytical solutions – SWASHES Semi-implicit (subcritical-subcritical) 1 topographie surface libre 0 niveau critique -1 -2 z, z+h (m) -3 -4 -5 -6 -7 -8 0 200 400 600 800 1000 x (m)

Validation on analytical solutions – SWASHES Semi-implicit (subcritical-supercritical) 1 topographie surface libre 0 niveau critique -1 z, z+h (m) -2 -3 -4 -5 -6 0 200 400 600 800 1000 x (m)

Validation on analytical solutions – SWASHES Semi-implicit (supercritical-subcritical) 1 topographie surface libre 0 niveau critique -1 z, z+h (m) -2 -3 -4 -5 -6 0 200 400 600 800 1000 x (m)

Summary of the chosen numerical method ◮ Numerical flux : HLL ◮ Second order scheme : MUSCL ◮ Friction : semi-implicit treatment ◮ Shallow Water system with rain P � ∂ t h + ∂ x ( hu ) = P (5) ∂ t ( hu ) + ∂ x ( hu 2 + gh 2 / 2) + h ∂ x z = − hf time splitting/explicit treatment

Validation on experiments – INRA rain simulator

Settings of the experiment z 5 cm P ( t ) L c = 4 m q ( t, L ) x O L 0 ≤ t ≤ 250s � 50 mm/h if ( x , t ) ∈ [0 , L ] × [5 , 125] R ( x , t ) = 0 else

Analytical solutions and simulations 7 f=0.12, numerique f=0.12, cinematique 6 f=0.12, exact f=0.34, numerique f=0.34, cinematique 5 f=0.34, exact 4 q (g/s) 3 2 1 0 0 50 100 150 200 250 Temps de simulation (s)

Water height and velocity at equilibrium 0.12 0.0006 0.1 0.0005 0.08 0.0004 u (m/s) h (m) 0.06 0.0003 0.04 0.0002 f=0.12, numerique f=0.12, numerique f=0.12, cinematique f=0.12, cinematique f=0.12, exact f=0.12, exact 0.02 0.0001 f=0.34, numerique f=0.34, numerique f=0.34, cinematique f=0.34, cinematique f=0.34, exact f=0.34, exact 0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 x (m) x (m)

What about reality ? 9 8 7 6 q(.,L) (g/s) 5 4 3 2 1 0 0 50 100 150 200 250 t (s)

”Calibration” Darcy-Weisbach Manning 9 9 Mesures Mesures f=0.1 n=0.008 8 8 f=0.11 n=0.009 f=0.12 n=0.01 7 7 f=0.13 n=0.011 f=0.14 n=0.012 6 6 f=0.15 n=0.013 f=0.16 n=0.014 5 f=0.17 5 n=0.015 q (g/s) q (g/s) f=0.18 n=0.016 4 4 3 3 2 2 1 1 0 0 0 50 100 150 200 250 0 50 100 150 200 250 Temps de simulation (s) Temps de simulation (s)

A simulation result (Manning) 9 Mesures n=0.013 8 7 6 5 q (g/s) 4 3 2 1 0 0 50 100 150 200 250 Temps de simulation (s)

Parcels in Niger ([Esteves et al., 2000], IRD)