Introduction Modeling and simulation of bedload transport with viscous effects E. Audusse, L. Boittin, M. Parisot, J. Sainte-Marie Project-team ANGE, Inria; CEREMA; LJLL, UPMC Université Paris VI; UMR CNRS 7958 January 30, 2018 L.Boittin (ANGE, Inria) 30.01.18 1 / 23
Introduction Context L.Boittin (ANGE, Inria) 30.01.18 2 / 23
Introduction Keywords A variety of processes: erosion, deposition, suspension, collision... A variety of models, made by hydraulicians/mathematicians: Reduced Complexity Models Shallow-Water-Exner equation triple-deck model bi-fluid model RANS, LES, DNS Choose something that works at the scale of a river... L.Boittin (ANGE, Inria) 30.01.18 3 / 23
Introduction Classical bed load transport model: Exner equation Clear water Saint-Venant equations Sediment layer Exner equation Based on mass conservation Solid discharge given by empirical relationships Einstein (1942), Meyer-Peter and Müller (1948), Nielsen (1992) The solid discharge depends on the water depth and velocity: no intrinsic mechanism in the sediment Used in industrial codes: TELEMAC, MIKE HYDRO River L.Boittin (ANGE, Inria) 30.01.18 4 / 23
Introduction Goals: derive and simulate a new bedload transport model Mathematicians attempts: make reasonable assumptions and derive a new model (similar to an Exner model, but different) Formal derivation of a model, from the Navier-Stokes equations Coupled model with an energy equation E.D. Fernández-Nieto et al. Formal deduction of the Saint-Venant-Exner model including arbitrarily sloping sediment beds and associated energy. In: Mathematical Modelling and Numerical Analysis (2016) Try to include the sediment rheology in the model L.Boittin (ANGE, Inria) 30.01.18 5 / 23
Model derivation Strategy for the model derivation kinematic condition η H 1, U 1, W 1 no penetration kinematic condition H 2, U 2, W 2 no penetration Navier-Stokes equations ζ B thin layer approximation high viscosity, high friction in the sediment layer h 1, ū 1 κ ζ h 2, ū 2 κ B Saint-Venant equations, sediment transport model Two separated phases, no mass exchange L.Boittin (ANGE, Inria) 30.01.18 6 / 23
Model derivation Inviscid model with high friction Assume the following scaling: H = εl W = εu, Fr = 1, Re 1 = 1 ε, Re 2 = 1, K ζ = ε, K B = 1. Assume that the space variations of the µ k are of the order ε 2. Then, for ε small enough, the system { t h 1 + x (h 1 ū 1 ) = 0, t (h 1 ū 1 ) + x (h 1 ū 1 ū 1 + g h2 1 2 Id) = gh 1 x (h 2 + B) ρ2 ρ 1 κ ζ (ū 1 ū 2 ), { t h 2 + x (h 2 ū 2 ) = 0, κ B (ū 2 ) = gh 2 x (h 2 + ρ1 ρ 2 h 1 + B) + κ ζ (ū 1 ū 2 ), is derived from the bilayer Navier-Stokes system with the following modeling errors: H 1 h 1 = O(ε), H 2 h 2 = O(ε 2 ), U 1 ū 1 = O(ε), U 2 ū 2 = O(ε 2 ). L.Boittin (ANGE, Inria) 30.01.18 7 / 23
Model derivation Proof Momentum eq. and boundary conditions: z (µ 2 z U 2 ) = O(ε 2 ), µ 2 z U 2 = O(ε 2 ), at z = ζ µ 2 z U 2 = εκ B (U 2) + O(ε 2 ), at z = B Imposes that U 2 = Ū 2 + O(ε 2 ) = εũ 2 + O(ε 2 ). µ 2 z U 2 B = O(ε 2 ) Model similar to [Fernandez-Nieto 15], obtained with different arguments Fine, but similar to an Exner model, and no rheology... L.Boittin (ANGE, Inria) 30.01.18 8 / 23
Model derivation Viscous model Assume the following scaling: H = εl W = εu, Fr = 1, Re 1 = 1 ε, Re 2 = ε, K ζ = ε, K B = 1. Assume that the space variations of the µ k are of the order ε 2. Then, for ε small enough, the system { t h 1 + x (h 1 ū 1 ) = 0, t (h 1 ū 1 ) + x (h 1 ū 1 ū 1 + g h2 1 2 Id) = gh 1 x (h 2 + B) ρ2 ρ 1 κ ζ (ū 1 ū 2 ), { t h 2 + x (h 2 ū 2 ) = 0, κ B (ū 2 ) = gh 2 x (h 2 + ρ1 ρ 2 h 1 + B) + κ ζ (ū 1 ū 2 ), with κ B (ū 2 ) = κ B (ū 2 ) x (ν 2 h 2 D x ū 2 ) is derived from the bilayer Navier-Stokes system with the following modeling errors: H 1 h 1 = O(ε), H 2 h 2 = O(ε 2 ), U 1 ū 1 = O(ε), U 2 ū 2 = O(ε 2 ). L.Boittin (ANGE, Inria) 30.01.18 9 / 23
Model derivation Proof Momentum eq. and boundary conditions: z (µ 2 z U 2 ) = O(ε 2 ), µ 2 z U 2 = O(ε 2 ), at z = ζ µ 2 z U 2 = O(ε 2 ), at z = B Vertically integrated horizontal momentum equation: U 2 = Ū 2 (x, t) + O(ε 2 ) t (H 2 Ū 2 ) + x (H 2 Ū 2 Ū 2 ) + H 2 Fr 2 x( ρ 1 ρ 2 H 1 + H 2 + B) Main order terms: = κ B (Ū 2 ) ε + 1 ε x (ν 2 H 2 D x Ū 2 ) κ ζ (Ū 2 U 1 ) + O(ε), κ B (Ū 2 ) x (ν 2 H 2 D x Ū 2 ) = O(ε). Imposes that Ū 2 = O(ε) = εũ 2 + O(ε 2 ). L.Boittin (ANGE, Inria) 30.01.18 10 / 23
Model derivation Threshold for incipient motion Classical laws for sediment transport used in hydraulic engineering have a threshold for incipient motion (eg. Meyer-Peter and Müller). Critical shear stress: τ c Effective shear stress: τ eff = κ ζ ū 1 gh 2 x ( ρ 1 ρ 2 h 1 + h 2 + B) Velocity equation: κ B (ū 2 ) = τ eff Take κ B ( ) such that κ B (ū 2 ) = ( κ ū 2 α + τ c ) ū2 ū 2 x (ν 2 h 2 x ū 2 ), with κ = { κ if τ eff τ c, κ R + if τ eff < τ c L.Boittin (ANGE, Inria) 30.01.18 11 / 23
Model analysis Model analysis Dissipative energy balance for the bilayer model For smooth enough solutions: t (K 1 + E) + x (K 1 u 1 + h 1 φ 1 ū 1 + h 2 φ 2 ū 2 ) = ρ 2 ū 2 κ B (ū 2 ) ρ 2 κ ζ ū 1 ū 2 2, with K 1 = 1 2 ρ 1h 1 ū 1 2 : kinematic energy of the water E = g ( ( ρ 1 h h1 1 2 + h 2 + B ) ( + ρ 2 h h2 2 2 + B )) : potential energy φ 1 = ρ 1 g(h 2 + h 1 + B), φ 2 = ρ 2 (h 2 + ρ1 ρ 2 h 1 + B): potentials Sediment layer only Positivity Maximum principle for smooth solutions (without forcing) L.Boittin (ANGE, Inria) 30.01.18 12 / 23
Numerical tests Objectives Simulate the sediment layer alone { t h 2 + x (h 2 ū 2 ) = 0, κ B (ū 2 ) = gh 2 x (h 2 + ρ 1 ρ 2 h 1 + B) + κ ζ (ū 1 ū 2 ) without forcing with forcing Simulate the coupled system L.Boittin (ANGE, Inria) 30.01.18 13 / 23
Numerical tests A first idea for the numerical scheme Simplified model: Sediment layer alone No viscosity No topography nonlinear heat equation t h 2 g κ B x (h 2 2 x h 2 ) = 0. Explicit scheme: parabolic CFL condition t C( x) 2 Try an implicit scheme L.Boittin (ANGE, Inria) 30.01.18 14 / 23
Numerical tests 3 different schemes Implicit (linearized) finite volume schemes, staggered grid h n+1 2,i = h2,i n t x (hn 2,i+1/2 un+1 2,i+1/2 hn 2,i 1/2 un+1 u n+1/2 2,i+1/2 ν 2 ( x) (h n 2 2,i+1 (un+1 2,i+3/2 un+1 2,i+1/2 ) hn 2,i (un+1 = gh2,i+1/2 n h n+1 2,i+1 hn+1 2,i x, This scheme dissipates the discrete energy. Reconstruction of h i+1/2 : upwind with respect to u. { h n h2,i+1/2 n = 2,i if u n+1 2,i+1/2 0 h2,i+1 n if un+1 2,i+1/2 < 0 2,i 1/2 ) 2,i+1/2 un+1 2,i 1/2 )) A fixed point is needed. But the positivity of h n+1 2,i+1/2 is ensured. L.Boittin (ANGE, Inria) 30.01.18 15 / 23
Numerical tests Influence of the viscosity 1 0.9 0.8 2 =1 2 =10-2 2 =10-4 2 =10-6 h 2 0.7 0.6 0.5 0 0.2 0.4 0.6 0.8 1 x L.Boittin (ANGE, Inria) 30.01.18 16 / 23
Numerical tests Simulations with a threshold in the friction coefficient L.Boittin (ANGE, Inria) 30.01.18 17 / 23
Numerical tests Simulations with a threshold in the friction coefficient Non-flat stationary states L.Boittin (ANGE, Inria) 30.01.18 17 / 23
Numerical tests Strategy for the coupled scheme Requirements on the scheme in the water layer: staggered-grid, entropy dissipative, implicit friction term A good candidate: scheme by Herbin, Latché, Gunawan Implementation: 1 Resolution in the water layer, without friction 2 Resolution in the sediment layer 3 Correction of the water velocity due to the friction L.Boittin (ANGE, Inria) 30.01.18 18 / 23
Numerical tests How does the model behave? Without viscosity L.Boittin (ANGE, Inria) 30.01.18 19 / 23
Numerical tests How does the model behave? With viscosity L.Boittin (ANGE, Inria) 30.01.18 19 / 23
Numerical tests Convergence 1.2 1.1 1 0.9 100 200 500 1000 2000 5000 10000 20000 50000 0.8 0.7 0.6 0 2 4 6 8 10 Initial condition BCs: (hu)(0) = 0.5, h(l) = 0.5 L.Boittin (ANGE, Inria) 30.01.18 20 / 23
Numerical tests Convergence 0.8 0.75 0.7 100 200 500 1000 2000 5000 10000 20000 50000 1.2 1.1 1 0.9 0.8 100 200 500 1000 2000 5000 10000 20000 50000 0.65 0.7 0.6 0.6 0 2 4 6 8 10 Sediment layer 0.5 0 2 4 6 8 10 Water and sediment surfaces L.Boittin (ANGE, Inria) 30.01.18 20 / 23
Numerical tests Convergence 0.76 0.74 0.72 0.7 0.68 100 200 500 1000 2000 5000 10000 20000 50000 1.2 1.1 1 0.9 100 200 500 1000 2000 5000 10000 20000 50000 0.66 0.8 0.64 0.62 0.7 0.6 0 2 4 6 8 10 Sediment layer 0.6 0 2 4 6 8 10 Water and sediment surfaces L.Boittin (ANGE, Inria) 30.01.18 20 / 23
Numerical tests Influence of the flowrate (left BC) Sediment layer Water and sediment surfaces L.Boittin (ANGE, Inria) 30.01.18 21 / 23
Conclusions and perspectives Conclusions A new model for sediment transport with viscosity Formal derivation Preliminary analysis Numerical analysis and checks Parametric study of the behaviour Future work Comparison with co-located schemes Nonlinear friction law More physical rheologies (Bingham?) Comparison with experiments L.Boittin (ANGE, Inria) 30.01.18 22 / 23
Conclusions and perspectives Why doesn t the order of approximation in the water layer ruin the approximations in the sediment layer? t (H 2 Ū 2 ) + x (hū 2 Ū 2 ) + H 2 Fr 2 x (rh 1 + H 2 + B) = κ BŪ2 ε + 1 ε x (µ 2 h 2 D x Ū 2 ) κ ζ (Ū 2 ū 1 ) + O(ε 2 ), t (H 2 Ū 2 ) + x (hū 2 Ū 2 ) + H 2 Fr 2 x (rh 1 + H 2 + B) = κ BŪ2 ε + 1 ε x (µ 2 h 2 D x Ū 2 ) κ ζ (Ū 2 ū 1 ) + O(ε), And then κ B Ū 2 x (µ 2 h 2 D x Ū 2 ) = O(ε). Imposes Ū 2 = O(ε) = εũ 2 + O(ε 2 ). Then: t H 2 + ε x (H 2 Ũ 2 ) = 0, t H 2 + ε x (H 2 ũ 2 ) = O(ε 2 ) Then h 2 = H 2 + O(ε 2 ). L.Boittin (ANGE, Inria) 30.01.18 23 / 23