The Navier-Stokes system with temperature and salinity for free surface flows Part II: Numerical scheme and validation

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The Navier-Stokes system with temperature and salinity for free surface flows. Numerical scheme and validation

Léa Boittin, François Bouchut, Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, Fabien Souillé

To cite this version:

Léa Boittin, François Bouchut, Marie-Odile Bristeau, Anne Mangeney, Jacques Sainte-Marie, et al..

The Navier-Stokes system with temperature and salinity for free surface flows. Numerical scheme and validation. 2021. �hal-02510722v2�

The Navier-Stokes system with temperature and salinity for free surface flows

- Numerical scheme and validation

L. Boittin

, F. Bouchut

, M.-O. Bristeau

, A. Mangeney

, J.

Sainte-Marie

, and F. Souillé

1

Inria Paris, 2 rue Simone Iff, CS 42112, 75589 Paris Cedex 12 and Sorbonne Université, Univ. Paris Diderot, SPC, CNRS, Laboratoire

Jacques-Louis Lions, LJLL, F-75005 Paris

2

Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), CNRS, Univ. Gustave Eiffel, UPEC, F-77454, Marne-la-Vallée, France

3

Univ. Paris Diderot, Sorbonne Paris Cité, Institut de Physique du Globe de Paris, Seismology Group, 1 rue Jussieu, Paris F-75005, France

4

Risk Management Solutions, Peninsular House, 30 Monument Street, London, EC3R 8NB, UK

5

EDF R&D LNHE - Laboratoire National d’Hydraulique et Environnement, 6 quai Watier, F-78400 Chatou

September 23, 2021

Abstract

In this paper, we propose a numerical scheme for the layer-averaged Euler with variable density and the Navier-Stokes-Fourier systems presented in part I (Boittin et al., 2020). These systems model hydrostatic free surface flows with density variations. We show that the finite volume scheme presented is well balanced with regards to the steady state of the lake at rest and preserves the positivity of the water height. A maximum principle on the density is also proved as well as a discrete entropy inequality in the case of the Euler system with variable density.

Some numerical validations are finally shown with comparisons to 3D analytical solutions and experiments.

Keywords: Navier-Stokes equations, free surface flows, variable density flows, layer- averaged formulation, finite volume scheme

Contents

1 Introduction 2

2 The layer-averaged models 4

2.1 The layer-averaged Navier-Stokes-Fourier model . . . 6

2.2 The layer-averaged Euler system . . . 10

3 Numerical scheme for the layer-averaged Euler system 10 3.1 Strategy for the time discretization . . . 11

3.2 Semi-discrete (in time) scheme . . . 11

3.3 Eigenvalues for the advection and pressure part . . . 13

3.4 Finite volume formalism for the Euler part . . . 15

3.4.1 The horizontal fluxes and the pressure terms . . . 17

3.4.2 The hydrostatic reconstruction technique . . . 18

3.4.3 The vertical exchange terms . . . 19

3.4.4 The variable update . . . 20

3.4.5 Properties of the numerical scheme . . . 21

3.5 Kinetic fluxes . . . 24

3.6 Discrete entropy inequality . . . 25

4 Numerical scheme for the layer-averaged Navier-Stokes-Fourier sys- tem 36 4.1 Semi-discrete (in time) scheme . . . 36

4.2 Spatial discretization of the diffusion terms . . . 36

5 Numerical validation 38 5.1 Parabolic bowl . . . 38

5.2 Lock exchange . . . 42

5.3 Comparison with the Boussinesq assumption . . . 43

5.3.1 Temperature diffusion . . . 44

5.3.2 Thermal equilibrium . . . 47

6 Conclusion 48

1 Introduction

In this paper we present a numerical scheme for the 3D incompressible Navier-Stokes- Fourier system with variable density and free surface, as well as numerical test cases.

This model describes variable density flows with free surface, the density variations coming from differences in temperature and/or salinity. The model is presented in the companion paper (Boittin et al., 2020), in which a layer-averaged formulation is also given. The layer-averaged formulation suppresses the need for moving meshes (Decoene

and Gerbeau, 2009), (Donea et al., 2004). It allows to perform 3D simulations with a 2D fixed mesh.

Variable density flows are frequently studied by oceanographers. Different systems of coordinates exist, among which terrain-following coordinates and isopycnal coordi- nates. For a discussion of the advantages and disadvantages of the various coordinates frequently used in ocean models, the reader is referred to (Griffies et al., 2000) and (Song and Hou, 2006). The layer-averaged model presented here is not a terrain-following co- ordinate model. Though the layer thicknesses are defined as fractions of the total water height, it is not aσ-coordinate system. The model also differs from isopycnal coordinate models because in the layer-averaged formulation, the layers exchange mass between themselves, which means that the internal layer boundaries are actually not physical.

For the Euler part of the Navier-Stokes-Fourier system with variable density, a finite- volume formalism is adopted. The hydrostatic reconstruction technique is used (Audusse et al., 2004). Therefore, the topography is accurately represented and the scheme is well- balanced. Yet the discretization of the nonconservative pressure terms demands special care. For the viscosity terms, we use finite elements as in (Allgeyer et al., 2019).

In (Audusse et al., 2011), a similar model was studied and simulated. The scheme presented in (Audusse et al., 2011) was a 2D (x−z) scheme and relied on a kinetic interpretation. In the present work we present a fully 3D scheme which is more flex- ible in a certain sense, because the kinetic flux is only one of the possible choices for the numerical flux. With any flux consistent with the semi-discrete in time Euler sys- tem, the resulting scheme is well-balanced and preserves the nonnegativity of the water depth. A maximum principle on the density is satisfied. In order to prove an in-cell entropy inequality, we adopt a kinetic flux, already used in the context of the Shallow Water equations in (Perthame and Simeoni, 2001), (Audusse et al., 2011). The entropy inequality is satisfied for a constant topography and includes third-order rest terms.

Moreover, the unknowns in (Audusse et al., 2011) were not the same, which resulted in a complicated numerical scheme - nonlinear systems were solved at each step and a Newton fixed-point method was used. The present scheme is simpler and does not involve nonlinear systems. Finally, the present scheme is more stable than the scheme in (Audusse et al., 2011), the CFL condition of which could actually degenerate and give a time step equal to zero. With the proposed scheme, the computational cost of the simulation of a non-Boussinesq flow is not greater than that of the simulation of a Boussinesq flow.

The proposed numerical scheme is validated on three test cases simulated with the Freshkiss3d code Freshkiss3d (2020). Each of these test cases allows to validate an aspect of the numerical scheme: wet/dry interface treatment, buoyancy terms, diffusion effects, second order extensions (space and time). . . A real application of hydrodynamics in a river with chlorides entries will be presented in a forthcoming paper (the research report is available (Souillé et al., 2017), in French).

The first test is a convergence test towards an analytical solution (Bristeau et al., 2020) for the Euler system with variable density. In the second test, a lock exchange simulation is performed and the results are compared with experimental data available

from the literature (Adduce et al., 2012). Finally, in two diffusion cases, the differ- ences between the Navier-Stokes-Fourier with variable density and Boussinesq models are evidenced.

The paper is organized as follows. In section 2, the layer-averaged Navier-Stokes- Fourier system and the Euler with variable density introduced in (Boittin et al., 2020) are recalled. A numerical scheme for the layer-averaged Euler model with variable density is presented in Section 3, its properties are studied. An extension of this scheme for the layer-averaged Navier-Stokes-Fourier model with variable density is presented in section 4. The numerical test cases are presented in section 5.

2 The layer-averaged models

hsec:reminder_modelsiFrom now on, we just call the Euler system, the Euler system with variable density. We briefly recall the features of the multilayer models presented in (Boittin et al., 2020) and studied here. The multilayer Navier-Stokes-Fourier model is a layer-averaged version of the incompressible, hydrostatic Navier-Stokes-Fourier system

∇ ·U=−ρ⁰(T)

ρ²c_p ∇ ·(λ∇T) +µ|∇_x,yu|²+µ

∂u

∂z

2!

, (1) eq:NSF_1_p2

∂ρ

∂t +∇ ·(ρU) = 0, (2)?eq:NSF_2_p2?

∂(ρu)

∂t +∇_x,y·(ρu⊗u) + ∂(ρuw)

∂z +∇_x,y Z η

z

ρgdz₁ =µ∆_x,yu+µ ∂

∂z ∂u

∂z

,(3) eq:NSF_3_p2

whereU(t, x, y, z) = (u, v, w)^T is the velocity,u= (u, v)^T is the horizontal velocity vector and ρ is the density. The notation ∇ denotes ∇ =

∂

∂x,(_∂y^∂,(_∂z^∂T

, ∇_x,y corresponds to the projection of ∇ on the horizontal plane i.e. ∇_x,y =

∂

∂x,_∂y^∂ T

and the quantity

|∇_x,yu|² means |∇_x,yu|² = (∇_x,yu) : (∇_x,yu)^T.

The temperature T is linked to the density ρ via the state equation T =T(ρ). The viscosity coefficient is denoted by µ, the heat conductivity by λ and the specific heat capacity at constant pressure byc_p.

The energy balance for model (1)-(3) is

∂

∂t

ρ|u|² 2 +ρe

+∇_x,y·

u

ρ|u|² 2 +

Z η z

ρgdz₁+ρe

−µ∇|u|² 2

=∇.(λ∇T), with e is the internal energy of the fluid governed by

∂(ρe)

∂t +∇_x,y·(ρeu) =Z η z

ρgdz₁

∇_x,y·u+µ|∇_x,yu|²+µ

∂u

∂z

2

+∇.(λ∇T).

We consider a free surface flow, therefore we assume

zb(x, y)≤z ≤η(t, x, y) :=h(t, x, y) +zb(x, y),

Figure 1: Flow domain with water height h(t, x, y), free surface η(t, x, y) and bottom z_b(x, y).

hfig:notations3d_rhoi

with z_b(x, y)the bottom elevation and h(t, x, y)the water depth, see Fig. 1.

Let n_b and n_s be the unit outward normals at the bottom and at the free surface respectively defined by

n_b = 1

p1 +|∇_x,yz_b|²

∇_x,yz_b

−1

, and n_s= 1

p1 +|∇_x,yη|²

−∇_x,yη 1

. On the bottom we prescribe an impermeability condition

U.n_b = 0, (4) eq:bottom_p2

and a friction condition given e.g. by a Navier law µ

q

1 +|∇_x,yz_b|² ∂u

∂n_b =−κu, (5)?eq:fric_p2?

with κ a Navier coefficient.

On the free surface, the kinematic boundary condition

∂η

∂t +u(t, x, y, η)· ∇_x,yη−w(t, x, y, η) = 0, (6) eq:free_surf_p2

is satisfied, along with the no stress condition µ∂u˜

∂ns

= 0, (7)?eq:bound3ns_p2?

with u˜ = (u,0)^T.

On solid walls, we prescribe a slip condition

U·n = 0, (8)^?eq:slip?

coupled with an homogeneous Neumann boundary condition µ∂u

∂n = 0, n being the outward normal to the considered wall.

Boundary conditions for the temperature also have to be considered, we can choose either Neumann or Dirichlet conditions namely at the bottom

λ∂T

∂n_b =φ_T|_zb, (9) BC:neumann_bottom_p2

or

Tb =T_b⁰ (10) BC:dirichlet_bottom_p2

and at the free surface

λ∂T

∂n_s =φ_T|_η (11) BC:neumann_surface_p2

or

Ts=T_s⁰ (12) BC:dirichlet_surface_p2

where φ_T|_zb, φ_T|_η are two given temperature fluxes and T_b⁰, T_s⁰ are two given tempera- tures. SinceT =T(ρ), the boundary conditions forρnaturally ensue from the boundary conditions forT.

The system is completed with some initial conditions

h(0, x, y) =h⁰(x, y), ρ(0, x, y) =ρ⁰(x, y), U(0, x, y, z) =U⁰(x, y, z).

The system (1)-(3) was derived from the compressible Navier-Stokes system in (Boittin et al., 2020). More specifically, the derivation consisted in performing the incompressible limit. This model respects the second principle of thermodynamics (non-decreasing entropy).

2.1 The layer-averaged Navier-Stokes-Fourier model

?hsec:nsf_multilayeri?We consider a discretization of the fluid domain by layers (see Fig. 2) where the layer α contains the points of coordinates (x, y, z) with z ∈ L_α(t, x, y) = (zα−1/2, z_α+1/2) and {z_α+1/2}_α=1,...,N is defined by

z_α+1/2(t, x, y) = z_b(x, y) +Pα

j=1h_j(t, x, y), α ∈[0, . . . , N],

h_α(t, x, y) = z_α+1/2(t, x, y)−zα−1/2(t, x, y) =l_αh(t, x, y), (13) ^eq:layer and PN

α=1l_α = 1.

Using the notations (13), let us consider the spaceP^N,t0,h of piecewise constant functions defined by

P^N,t0,h =

1z∈Lα(t,x,y)(z), α∈ {1, . . . , N} , (14)?eq:P0_space?

Figure 2: Notations for the layerwise discretization.

hfig:free_p2i

where 1z∈Lα(t,x,y)(z) is the characteristic function of the layer L_α(t, x, y). Using this formalism, the projection of ρ, u, v and w on P^N,t0,h is a piecewise constant function defined by

X^N(t, x, y, z,{z_α}) =

N

X

α=1

1z∈Lα(t,x,y)(z)X_α(t, x, y), (15)?eq:ulayer?

for X ∈(ρ, u, v, w). When the quantities {ρ_α(t, x, y)}_α=1,...,N are known, if the function T =T(ρ)is known, it is possible to recover the temperature using the formula

T^N(t, x, z,{z_α}) =

N

X

α=1

1z∈Lα(t,x,y)(z)T(ρ_α(t, x, y)).

The layer-averaged Navier-Stokes-Fourier system introduced in (Boittin et al., 2020) reads

∂h

∂t +

N

X

α=1

∇x,y·(hαuα) =−

N

X

α=1

ρ⁰(T_α)

ρ²_αc_p (ST ,α− Sµ,α), (16) eq:massesvml_ns_p2

∂ρ_αh_α

∂t +∇x,y·(ραhαuα) =ρ_α+1/2G_α+1/2−ρα−1/2Gα−1/2, α = 1, ..., N, (17)?eq:massesvml1_ns_p2?

∂ρ_αh_αu_α

∂t +∇_x,y·(ρ_αh_αu_α⊗u_α) +∇_x,y h_αp_α

=p_α+1/2∇_x,yz_α+1/2−p_α−1/2∇_x,yz_α−1/2 +u_α+1/2ρ_α+1/2G_α+1/2−uα−1/2ρα−1/2Gα−1/2+∇_x,y·(µh_α∇_x,yu_α)

+Γ_α+1/2(u_α+1−u_α)−Γ_α−1/2(u_α−uα−1)−κ_αu_α, α= 1, ..., N, (18) eq:mvtsvml_ns_p2

with

G_α+1/2 =−

N

X

j=1

X^α

p=1

l_p−1j≤α

∇_x,y·(h_ju_j) +

α

X

j=1

ρ⁰(Tj)

ρ²_jc_p (S_{T ,j}−S_µ,j), (19)?eq:Qalphabis_T_p2?

κ_α =

(κ if α= 1 0 if α 6= 1 ,

S_T,α =

λ∇_x,y·(h_α∇_x,yT_α) + 2λ_α+1/2T_α+1−T_α

h_α+1+h_α −2λ_α−1/2T_α−Tα−1

h_α+hα−1

, (20) ^S_T_alpha λ_α+1/2 =λ for α= 1, . . . , N −1,

T_α =T(ρ_α).

Forα= 0,2λ_α+1/2^T_h^α+1−Tα

α+1+hα =φ_T|_zb if the Neumann boundary condition (9) is chosen, or h₀ =h₁,T₀ =T_b⁰if the Dirichlet boundary condition (10) is chosen. Likewise, forα=N, 2λ_α+1/2^T_h^α+1−Tα

α+1+hα = φT|η with the boundary condition (11), or hN+1 = hN, TN+1 = T_s⁰ with the boundary condition (12). The dissipation term due to the viscous effects is

S_µ,α =−h_αµ|∇_x,yu_α|²−Γ_α+1/2|u_α+1−u_α|²

2 −Γ_α−1/2|u_α−uα−1|²

2 −κ_α|u_α|², (21) ^S_mu_alpha with

Γ_α+1/2 = 2µ_α+1/2 hα+1+hα

, (22) Gamma_alpha_plus

µ_α+1/2 =







0 if α= 0

µ if α= 1, . . . , N −1 0 if α=N.

(23)?mu_alpha_plus?

The term |∇x,yuα|² actually denotes

|∇_x,yu_α|² = (∇_x,yu_α) : (∇_x,yu_α)^T

=

∂uα

∂x 2

+ ∂uα

∂y 2

+ ∂vα

∂x 2

+ ∂vα

∂y 2

. The velocitiesu_α+1/2 and the densities ρ_α+1/2 at the interfaces are defined by

v_α+1/2 =

v_α if G_α+1/2 ≤0

vα+1 if Gα+1/2 >0 (24)?eq:upwind_uT_p2?

forv =u, ρ.

The pressure termsp_α,p_α+1/2 are given by

p_α=g ρ_αh_α 2 +

N

X

j=α+1

ρ_jh_j

!

and p_α+1/2 =g

N

X

j=α+1

ρ_jh_j. (25)?eq:palpha1_p2?

The pressure is hydrostatic. The terms G_α+1/2 represent the mass exchanges between the layers. Notice that some of the viscous and diffusion terms have been simplified, see (Boittin et al., 2020). Since the right-hand side of the total height conservation equation (16) is nonzero, we expect to observe dilatation and contraction due to the temperature diffusion and to the viscosity.

We recall here the following result, obtained in (Boittin et al., 2020).

?hcontinuous_energy_balance_nsi?Proposition 2.1 The system (16)-(18) completed with the equation

∂

∂t(ρ_αh_αe_α) +∇_x,y·(ρ_αh_αu_αe_α) =ρ_α+1/2e_α+1/2G_α+1/2−ρα−1/2eα−1/2Gα−1/2

+p_αρ⁰(T_α)

ρ²_αc_p (S_{T ,α}− S_µ,α) +S_{T ,α}− S_µ,α admits, for smooth solutions, the energy balance

∂

∂tE_α+∇_x,y·(u_α(E_α+h_αp_α−µh_α∇_x,yu_α)) +Γ_α+1/2|uα+1|²− |uα|²

2 −Γα−1/2

|uα|²− |uα−1|² 2

= ρ_α+1/2u²_α+1/2

2 +gρ_α+1/2z_α+1/2

!

G_α+1/2+p_α+1/2 G_α+1/2−∂zα+1/2

∂t

− ρα−1/2

u²_α−1/2

2 +gρα−1/2zα−1/2

!

Gα−1/2−pα−1/2 Gα−1/2− ∂z_α−1/2

∂t

−1

2 ρ_α+1/2(u_α+1/2−u_α)²+gh_α(ρ_α+1/2−ρ_α)

G_α+1/2 +1

2 ρα−1/2(uα−1/2 −u_α)²−gh_α(ρα−1/2−ρ_α)

Gα−1/2+S_{T ,α}, (26) eq:energy_mcl_ns_p2

with

E_α =ρ_αhα|uα|²

2 + ραghαzα

2 +e_α. (27)?eq:energ_al_NS_p2?

Note that in (26), we use the notation u_α∇_x,yu_α =

u_α^∂u_∂x^α +v_α^∂v_∂x^α u_α^∂u_∂y^α +v_α^∂v_∂y^α

=∇_x,y|u²_α| 2 . The sum of Eqs. (26) for α= 1, . . . , N gives

∂

∂t

N

X

α=1

E_α+

N

X

α=1

∇_x,y·u_α(E_α+h_αp_α)

= −

N

X

α=1

ρ_α+1/2|uα+1−uα|²

2 |G_α+1/2|+

N

X

α=1

S_{T ,α}

−g 2

N

X

α=1

h_α(ρ_α+1/2−ρ_α) +h_α+1(ρ_α+1/2−ρ_α+1)

G_α+1/2. The sum of S_{T ,α} over the layers gives

N

X

α=1

S_{T ,α} =λ

N

X

α=1

∇_x,y·(h_α∇_x,yT_α)− ∇T|_s·n_s+∇T|_b·n_b.

As explained in (Boittin et al., 2020), the terms on the last line of the right-hand side are third-order terms.

2.2 The layer-averaged Euler system

What we refer to hereafter as the layer-averaged Euler system is the system (16)-(18) without viscosity and without diffusion terms, i.e.

∂h

∂t +

N

X

α=1

∇x,y·(hαuα) = 0, (28) eq:massesvml_p2

∂ραhα

∂t +∇x,y·(ραhαuα) =ρ_α+1/2G_α+1/2−ρα−1/2Gα−1/2, α = 1, ..., N, (29)?eq:massesvml1_p2?

∂ρ_αh_αu_α

∂t +∇_x,y·(ραhαuα⊗uα) +∇_x,y hαpα

=pα+1/2∇_x,yzα+1/2−pα−1/2∇_x,yzα−1/2

+u_α+1/2ρ_α+1/2G_α+1/2−uα−1/2ρα−1/2Gα−1/2, α = 1, ..., N. (30) eq:mvtsvml_p2

The quantity G_α+1/2 (resp. Gα−1/2) corresponds to mass exchange accross the interface z_α+1/2 (resp. zα−1/2) and G_α+1/2 is defined here by

G_α+1/2 =

α

X

j=1

∂h_j

∂t +∇_x,y·(h_ju_j)

=−

N

X

j=1

X^α

p=1

l_p−1^j≤α

∇_x,y·(h_ju_j), (31) eq:Qalphabis_p2

forα = 1, . . . , N. Notice that the mass conservation for the layer α writes

∂hα

∂t +∇_x,y·(h_αu_α) = G_α+1/2−Gα−1/2, (32) eq:exchange_terms

and the sum (total or partial) of the previous equations with G_1/2 = G_N+1/2 = 0 (see (Boittin et al., 2020, Prop. 3.1)) gives Eqs. (28) and (31).

The energy balance verified by the system (28)-(30) is given in (Boittin et al., 2020).

It is very similar to the balance (26), obviously without the viscosity and diffusion terms. In the balance for the Euler system, the internal energy eα does not intervene.

It is actually equal to 0 because there is no volume variation.

3 Numerical scheme for the layer-averaged Euler sys- tem

hsec:euler_numi

In this section, a numerical scheme for the layer-averaged Euler system is designed and analyzed. It extends the work done by some of the authors in (Allgeyer et al., 2019;

Audusse et al., 2011). Before specifying the scheme for the Euler system, a common strategy for the time discretization of the Navier-Stokes-Fourier and Euler systems is presented. The discretization of the diffusion terms does not present any additional difficulty, however including these terms considerably lengthens the equations. This is why it seems preferable to explain the numerical scheme for the Euler system first.

The advantages of the numerical scheme are the following

• it gives a 3D approximation of the Navier-Stokes-Fourier system, while only 2D situations were considered in (Audusse et al., 2011)

• it can be implemented with any flux that is consistent with the homogeneous Saint- Venant system; the kinetic flux is used only for the discrete entropy property stated for a constant topography

• the scheme is endowed with strong stability properties (well-balanced, positivity of the water depth)

• convergence curves towards a 3D non-stationary analytical solution with wet-dry interfaces are obtained, see paragraph 5.1.

3.1 Strategy for the time discretization

The system (16)-(18) has the form

∂U

∂t +∇_x,y·F(U) = S_p(U, z_b) +S_e(U, ∂_tU, ∂_xU) +S_v,f(U), (33) ^eq:glo_p2 where the vector of unknowns is

U= (h, ρ1h1, . . . , ρNhN, qx,1, . . . , qx,N, qy,1, . . . , qy,N)^T ,

with q_x,α = ρ_αh_αu_α, q_y,α = ρ_αh_αv_α. We denote by F(U) = (F_x(U), F_y(U))^T the fluxes of the conservative part and by

S_p(U, z_b) =

0, . . . , p_3/2∂z3/2

∂x −p_1/2∂z1/2

∂x , . . . , p_3/2∂z3/2

∂y −p_1/2∂z1/2

∂y , . . . T

, the non-conservative part of the pressure terms. The source terms areS_e(U, ∂_tU, ∂_xU) and S_v,f(U), representing respectively the mass and momentum exchanges and the vis- cous and friction effects. Notice that, as a consequence of the layer-averaged discretiza- tion, the system (33) is made of only 2d(x, y) partial differential equations with source terms. Hence, the spacial approximation of the considered PDEs is performed on a 2d planar mesh.

We consider discrete times tⁿ with tⁿ⁺¹ = tⁿ+ ∆tⁿ. For the time discretisation of the layer-averaged Navier-Stokes-Fourier system (33) we adopt the following scheme

Uⁿ⁺¹ =U−∆tⁿ(∇_x,y·F(U)− S_p(U, z_b)) + ∆tⁿS_eⁿ⁺¹+ ∆tⁿS_v,f^n+l, (34) ^eq:glo_dis where the integer l = 0,1/2,1 will be precised below. In (34) and wherever there is no

ambiguity the superscript ⁿ has been omitted.

3.2 Semi-discrete (in time) scheme

From now on and until the end of this section, the system considered is the Euler system.

Similarly to (Allgeyer et al., 2019), the semi-discrete in time scheme (34) withS_v,f^n+l = 0 writes

h^n+1/2_α = h_α−∆tⁿ∇_x,y(h_αu_α) (35) eq:nscont_layer_d

(ρ_αh_α)^n+1/2 = ρ_αh_α−∆tⁿ∇_x,y·(ρ_αh_αu_α) (36) eq:nsml22_dbis

(ρ_αh_αu_α)^n+1/2 = ρ_αh_αu_α−∆tⁿ

∇_x,y·(ρ_αh_αu_α⊗u_α) +∇_x,y h_αp_α

−p_α∇_x,yh_α+ρ_αgh_α∇_x,yz_α

, (37) eq:nsml22_d

hⁿ⁺¹ = h^n+1/2 =

N

X

α=1

h^n+1/2_α =h−∆tⁿ

N

X

α=1

∇_x,y·(h_αu_α), (38) eq:nsml11_d

(ρ_αh_α)ⁿ⁺¹ = (ρ_αh_α)^n+1/2+ ∆tⁿ

ρⁿ⁺¹_α+1/2G_α+1/2−ρⁿ⁺¹_α−1/2Gα−1/2

, (39) eq:nsml33bis_d

(ρ_αh_αu_α)ⁿ⁺¹ = (ρ_αh_αu_α)^n+1/2+ ∆tⁿ

uⁿ⁺¹_α+1/2ρⁿ⁺¹_α+1/2G_α+1/2−uⁿ⁺¹_α−1/2ρⁿ⁺¹_α−1/2G_α−1/2

(40), eq:nsmlbis_d

with

G_α+1/2 = −

N

X

j=1

X^α

p=1

l_p−1j≤α

∇_x,y·(h_ju_j). (41) ^eq:nsf_6d

Since the relations

pα±1/2 =p_α∓ ρ_αgh_α

2 , z_α = z_α+1/2+z_α−1/2 2

hold, in (37) we often use the identity

p_α+1/2∇_x,yz_α+1/2−pα−1/2∇_x,yzα−1/2 =p_α∇_x,yh_α−ρ_αgh_α∇_x,yz_α. (42) eq:S_P_alpha

Notice that hⁿ⁺¹_α =l_αhⁿ⁺¹ and hⁿ⁺¹_α 6=h^n+1/2α , see Eq. (32).

The first three equations (35)-(37) consist in an explicit time scheme where the horizontal fluxes and the pressure terms are taken into account whereas in Eqs. (39)-(40) an implicit treatment of the exchange terms between layers is proposed. The implicit part of the scheme requires to solve a linear problem, see paragraph 3.4.3 and Allgeyer et al. (2019).

Following (34), Eqs. (35)-(38) also writes

U^n+1/2 =U−∆tⁿ(∇_x,y·F(U)− S_p(U, z_b)), (43) eq:glo_dis1

these computations being detailed in paragraphs 3.4.1 and 3.4.2. Equations. (38)-(40) can be reformulated under the form

Uⁿ⁺¹ =U^n+1/2 + ∆tⁿS_eⁿ⁺¹, (44)?eq:glo_dis2?

this implicit step being precised in paragraph 3.4.3.

3.3 Eigenvalues for the advection and pressure part

In order to propose a finite volume discretisation for Eqs. (35)-(38), an estimation of the eigenvalues is necessary and given below.

Without the exchange terms, the system (28)-(30) writes

∂h

∂t +

N

X

α=1

∇_x,y·(h_αu_α) = 0, (45) eq:massesvml_se

∂(ρ_αh_α)

∂t +∇_x,y·(ρ_αh_αu_α) = 0, (46)?eq:massesvml1_se?

∂(ρ_αh_αu_α)

∂t +∇_x,y·(ρ_αh_αu_α⊗u_α) +p_α∇_x,yh_α−ρ_αgh_α∇_x,yz_α = 0, (47) eq:mvtsvml_se

with

p_α∇_x,yh_α−ρ_αgh_α∇_x,yz_α=gh_α

N

X

j=α+1

∇_x,y(ρ_jh_j) + 1

2∇_x,y(ρ_αh_α)

!

+ ρ_αgh_α

2 ∇_x,yh_α+ρ_αgh_α

α−1

X

j=1

∇_x,yh_j+∇_x,yz_b

! . This system is the continuous version of system (35)-(38).

We rewrite the system (45)-(47) under the form

∂h

∂t +

N

X

j=1

(l_jh∇_x,y·u_j) +

N

X

j=1

(l_ju_j)· ∇_x,yh= 0, (48) eq:massesvml_se1

∂ρ_α

∂t +u_α.∇_x,yρ_α = 0, (49)?eq:massesvml1_se1?

∂u_α

∂t + (u_α· ∇_x,y)u_α+ 1

ρ_αh_α(p_α∇_x,yh_α−ρ_αgh_α∇_x,yz_α) = 0, (50) eq:mvtsvml_se1

and the quasilinear form of the system (48)-(50) writes

∂U˜

∂t +A( ˜U)∇_x,yU˜ =s_b( ˜U), (51) eq:quasi_lin

with U˜ = (h,u₁, . . . ,u_N, ρ₁, . . . , ρ_N)^T, and

A( ˜U) =

A₁( ˜U) A₂( ˜U) A₃( ˜U) A₄( ˜U)

,

A₁( ˜U) =



















 PN

j=1l_ju_j l₁h . . . l_Nh

˜

p₁ u₁ 0 . . . 0

˜

p₂ 0 u₂ 0 . . . 0 ... 0 ... u_j ... 0

... 0 ... 0 ... 0

˜

p_N 0 0 . . . 0 u_N



















 ,

A₂( ˜U) =

















0 . . . 0

gh²₁

2 0 ... 0

gh₂h₁ ^gh₂²² 0 0 ... 0 ^gh

2 j

2 0

gh_Nh₁ 0 0 ^gh₂^2N















 ,

A₃( ˜U) = 0_N×N+1, A₄( ˜U) =diag(u_j),

˜ p_j = g

ρ_j ρ_jl_j+ρ_j

j−1

X

i=1

l_i+

j−1

X

i=1

ρ_il_i

! .

For the sake of simplicity, the expression of the matrixA( ˜U)given below corresponds to the 1D case i.e. forv_i = 0,i= 1, . . . , N. Notice thatA₂( ˜U)and A₃( ˜U)^T are rectangular matrices with N + 1 rows and N columns.

The following proposition holds, consisting in a version of the Cauchy’s interlace theorem (Hwang, 2004) in the case of non symmetric matrix.

Proposition 3.1 The system (51) is strictly hyperbolic for h > 0 and the eigenvalues of A( ˜U) belong to the interval (λ_min, λ_max) with

λ_min = min

j {u_j, v_j} − max{ρ_j} min{ρ_j}

pgh, λ_max = max{u_j, v_j}+max{ρ_j}

min{ρ_j} pgh.

hprop:hyperi

Proof of prop. 3.1 Since A( ˜U) is a block-matrix, its eigenvalues consist in the eigen- values ofA₁( ˜U) completed with the set{u_i}^N_i=1. Writing the characteristic polynomial of A₁( ˜U) under the form (development e.g. along the first row)

P_A1 = Π^N_i=1(λ−u_i) λ−

N

X

j=1

l_ju_j−

N

X

i=1

l_ih˜p_i λ−u_i

! ,

the eigenvalues of A₁( ˜U) satisfy λ−

N

X

j=1

ljuj =QA1(λ),

Figure 3: The two functions λ 7→ Q_A1(λ) and λ 7→ λ−PN

j=1l_ju_j, each intersection of the two curves is an eigenvalue of A₁( ˜U).

hfig:pol_caraci

withQ_A1(λ) = PN i=1

lihp˜i

λ−ui. ForN = 3, the functions λ7→Q_A1(λ) andλ7→λ−PN j=1l_ju_j are depicted over Fig. 3 and it is easy to see that the four eigenvalues λi exists with the interlacing

λ₁ < u₁ ≤λ₂ ≤. . .≤u₃ < λ₄. Moreover we have

Q_A1(λ_max) =

N

X

i=1

l_ih˜p_i λmax−ui

≤ min{ρ_j} max{ρj}

N

X

i=1

l_ih˜p_i

√gh ≤ max{ρ_j} min{ρj}

pgh≤λ_max−

N

X

j=1

l_ju_j, and likewise

Q_A1(λ_min) =

N

X

i=1

l_ihp˜_i λmin−ui

≥ −min{ρ_j} max{ρj}

N

X

i=1

l_ihp˜_i

√gh ≥ −max{ρ_j} min{ρj}

pgh≥λ_min−

N

X

j=1

l_ju_j,

therefore the eigenvalues {λ_i}^N_i=1 of A₁( ˜U) satisfy

λ_min ≤λ_i ≤λ_max, i= 1, . . . N, proving the result.

3.4 Finite volume formalism for the Euler part

hsubsec:fviIn this paragraph, we propose a space discretization for the model (35)-(40) completed with (41). We first recall the general formalism of finite volumes on unstructured meshes.

LetΩdenote the computational domain with boundaryΓ, which we assume is polygonal.

LetT_h be a triangulation of Ω for which the vertices are denoted byP_i with S_i the set of interior nodes and G_i the set of boundary nodes. The dual cells C_i are obtained by joining the centers of mass of the triangles surrounding each vertex P_i. We use the following notations (see Fig. 4):

• K_i, set of subscripts of nodesP_j surrounding P_i,

• |Ci|, area of Ci,

• Γ_ij, boundary edge between the cells C_i and C_j,

• L_ij, length of Γ_ij,

• n_ij, unit normal toΓ_ij, outward to C_i (n_ji =−n_ij).

IfP_i is a node belonging to the boundary Γ, we join the centers of mass of the triangles adjacent to the boundary to the middle of the edge belonging toΓ (see Fig. 4) and we denote

• Γ_i, the two edges of C_i belonging to Γ,

• Li, length ofΓi (for sake of simplicity we assume in the following thatLi = 0 if Pi

does not belong to Γ),

• n_i, the unit outward normal defined by averaging the two adjacent normals.

(a) (b)

Figure 4: (a) Dual cell C_i and (b) Boundary cellC_i.

hfig:meshi

We define the piecewise constant functionsUⁿ(x, y)on cellsC_i corresponding to time tⁿ as

Uⁿ(x, y) =Uⁿ_i, for (x, y)∈C_i,

with Uⁿ_i = (hⁿ_i, ρⁿ_1,ihⁿ_1,i, . . . , ρⁿ_N,ihⁿ_N,i, q_x,1,iⁿ , . . . , q_x,N,iⁿ , q_y,1,iⁿ , . . . , qⁿ_y,N,i)^T i.e.

Uⁿ_i ≈ 1

|C_i| Z

Ci

U(tⁿ, x, y)dxdy.

For the topography, we choose a piecewise constant approximation under the form zi ≈ 1

|C_i| Z

Ci

zb(x, y)dxdy.

We will also use the notation

Uⁿ_α,i ≈ 1

|Ci| Z

Ci

U_α(tⁿ, x, y)dxdy, with U_α defined by

Uα = (hα, ραhα, ραhαuα, ραhαvα)^T. (52) ^eq:u_alpha 3.4.1 The horizontal fluxes and the pressure terms

hsubsubsec:horizontal_discreteiA finite volume scheme for solving the system (35)-(38) is a formula of the form U^n+1/2_i =U_i− X

j∈K_i

σ_i,jF_i,j−σ_iF_e,i+X

j∈K_i

σ_i,jS_p(U_i,U_j, z_b,i, z_b,j), (53) ^eq:upU0 where using the notations of (34)

X

j∈Ki

L_i,jF_i,j ≈ Z

Ci

∇_x,y·F(U)dxdy, (54) eq:flux_dis1

with

σ_i,j = ∆tⁿL_i,j

|C_i| , σ_i = ∆tⁿL_i

|C_i| . Here we consider first-order explicit schemes where

F_i,j =























 F_i,j^h F_i,j^ρ1h1

...

F_i,j^ρNhN F_i,j^u1

...

F_i,j^uN

























. (55) eq:flux_def

and

F_i,j^h =

N

X

α=1

F_i,j^hα, (56) ^eq:flux

and for the boundary nodes

F_i,e=























 F_i,e^h F_i,e^ρ1h1

...

F_i,e^ρNhN F_i,e^u1

...

F_i,e^uN

























. (57) ^eq:fluxbis

The fluxesF_i,j^hα,F_i,j^ραhα,F_i,j^uα appearing in expressions (55),(56), (57) are numerical fluxes such that

F_i,j^mα =F^mα(U_α,i,U_α,j,n_i,j), with m=h, ρh,u, α= 1, . . . , N.

Relation (53) tells how to compute the values U^n+1/2_i knowing U_i and discretized valuesz_b,i of the topography. Following (54), the term F_i,j in (53) denotes an interpola- tion of the normal component of the flux F(U).n_i,j along the edge C_i,j. The functions F(U_i,U_j,n_i,j)∈R^2N+1 are the numerical fluxes, see (Bouchut, 2004).

Until now, the expression for the numerical fluxes is not detailed since any numerical fluxes (Rusanov, HLL,. . . ) can be used (Bouchut, 2004). In paragraph 3.5 we define F(U_i,n_i,j)using kinetic fluxes and we prove a discrete entropy inequality for the system.

The computation of the value U_i,e, which denotes a value outside C_i (see Fig. 4-(b)), defined such that the boundary conditions are satisfied, and the definition of the bound- ary flux F(U_i,U_e,i,n_i) are described in (Allgeyer et al., 2019). Notice that we assume a flat topography on the boundaries i.e. z_b,i =z_b,i,e.

For the discretization of the pressure source term S_p(U, z_b), we adopt a strategy defined below.

3.4.2 The hydrostatic reconstruction technique

hsubsubsec:HRiThe hydrostatic reconstruction scheme (HR scheme for short) for the Saint-Venant sys- tem has been introduced in (Audusse et al., 2004) in the 1d case and described in 2d for unstructured meshes in (Audusse and Bristeau, 2005). The HR in the context of the kinetic description for the Saint-Venant system has been studied in (Audusse et al., 2016).

In order to take into account the topography variations and to preserve relevant equilibria, the HR leads to a modified version of (53) under the form

U_i^n+1/2 =Ui −X

j∈K_i

σi,jF_i,j^∗ −σiFi,e+ X

j∈K_i

σi,jS_p,i,j^∗ , (58) ^eq:upU0_HR where

F_i,j^∗ = F(U_i,j^∗ , U_j,i^∗ ,n_i,j), (59)?eq:flux_HR?

S_p,i,j^∗ = S_p(U_i, U_i,j^∗ , z_b,i, z_b,j,n_i,j)

=

















0

˜

p^∗_1,i,j(h_1,i,j−h_1,i)n_i,j −gρ_1,i˜h^∗_1,i,j(z_1,i,j−z_1,i)n_i,j ...

˜

p^∗_α,i,j(h_α,i,j−h_α,i)n_i,j −gρ_α,i˜h^∗_α,i,j(z_α,i,j−z_α,i)n_i,j ...

















(60) ^eq:Sp_HR

with

z_b,i,j^∗ = max(z_b,i, z_b,j), h^∗_i,j = max(h_i+z_b,i−z_b,i,j^∗ ,0),

U_i,j^∗ = (h^∗_i,j, ρ_1,il₁h^∗_i,j, . . . , ρ_N,il_Nh^∗_i,j, ρ_1,il₁h^∗_i,ju_1,i, . . . , ρ_N,il_Nh^∗_i,ju_N,i, . . .)^T, z_α,i,j^∗ =z_b,i,j^∗ +

lα

2 +Pα−1 j=1 l_j

h^∗_i,j, z_α,i,j =z_α,j,i= ^z

∗

α,i,j+z_α,j,i^∗

2 ,

h_α,i,j =h_α,j,i= ^h

∗

α,i,j+h^∗_α,j,i

2 ,

˜h^∗_α,i,j = ^hα,i+h

∗ α,i,j

2 ,

˜

p^∗_α,i,j = ^pα,i+p

∗ α,i,j

2 ,

(61) eq:state_HR

and

zα =zb+ l_α 2 +

α−1

X

j=1

lj

!

h, pα = ρ_αgh_α

2 +

N

X

j=α+1

ρjghj. Throughout this work, the^∗ refers to the HR technique.

Remark 3.2 Since the quantity S(U, z_b) appearing in (33) contains non conservative terms, its integration over the cell C_i is not straightforward and we have used the result proposed by Bouchut (Bouchut, 2004, Proposition 5.3) to obtain the expression (60). 3.4.3 The vertical exchange terms

hsubsubsec:exchangesiWe give the fully discrete expression of the step for the vertical exchanges, described by equations (39)-(40). The step for the vertical exchanges consists in

U_iⁿ⁺¹ =U_i^n+1/2+ ∆tⁿG_iⁿ⁺¹, (62) ^eq:upU1_HR

with

G_iⁿ⁺¹ =











































0 ρⁿ⁺¹_3/2,iG_3/2,i ρⁿ⁺¹_5/2,iG_5/2,i−ρⁿ⁺¹_3/2,iG_3/2,i

...

ρⁿ⁺¹_N−1/2,iGN−1/2,i−ρⁿ⁺¹_N−3/2,iG_N−3/2,i

−ρⁿ⁺¹_N−1/2,iGN−1/2,i

uⁿ⁺¹_3/2,iρⁿ⁺¹_3/2,iG3/2,i

uⁿ⁺¹_5/2,iρⁿ⁺¹_5/2,iG_5/2,i−uⁿ⁺¹_3/2,iρⁿ⁺¹_3/2,iG_3/2,i ...

uⁿ⁺¹_N−1/2,iρⁿ⁺¹_N−1/2,iG_N−1/2,i−uⁿ⁺¹_N−3/2,iρⁿ⁺¹_N−3/2,iG_N−3/2,i

−uⁿ⁺¹_N−1/2,iρⁿ⁺¹_N−1/2,iG_N−1/2,i









































 .

The mass conservation for the layerαis governed by Eq. (31) and its discretization gives hⁿ⁺¹_α,i =h^n+1/2_α,i + ∆t(G_α+1/2,i−G_α−1/2,i).