Modélisation, analyse mathématique et simulations numériques de ...

tel-00656013, version 1 - 3 Jan 2012
3.2 Numerical experiments 75
where ũ(t,x,a) = u(t,x,Z(t,x,a)) and ˜w(t,x,a) = w(t,x,Z(t,x,a)). Boundary conditions
are obtained in the same way. This formulation allows to perform a finite volume
scheme.
We take periodic boundary conditions, no friction and the computational domain is
[0,1]. We run the multilayer code for 15 layers (h = 0.006) and 40 layers (h = 0.002), with
the following initial conditions
⎧
⎨
⎩
η(0,x) = 0.1+0.01 1+0.5 cos(2πx/L)+0.2 sin(4πx/L) ,
ui(0,x) ≡ 0 for 1 i N ,
(3.2.5)
and corresponding initial datum for the Euler system. In Figures 3.4 and 3.5 we output
the profiles of the free surface and the mean velocity at different times. We can observe
that curves fit well. Nevertheless, since the Lagrangian formulation of the Euler equations
is only valid on short times, the hydrostatic code becomes unstable and starts to oscillate
after around 100 iterations.
0.105
0.104
0.103
0.102
0.101
0.1
0.099
0.098
0.097
hydro
15 layers
0.096
0 1 2 3 4 5 6 7 8 9 10
0.106
0.104
0.102
0.1
0.098
0.096
0.094
hydro
15 layers
0 1 2 3 4 5 6 7 8 9 10
(a) (b) (c)
Figure 3.4: Evolution of the free surface , t = 10 (a), t = 40 (b), t = 80 (c)
0.106
0.104
0.102
0.1
0.098
0.096
0.094
hydro
15 layers
0 1 2 3 4 5 6 7 8 9 10
The velocity fields are also plotted in Figure 3.6 at time t = 50 for both models. The
profiles are in good agreement.
3.2.3 Test 3: perturbation of rest in velocity, flat bottom
In this section, we again consider a flat bottom, the spatial domain is [0,1], viscosity
µ = 0.0001, no friction, periodic boundary conditions and we perturb the lake at rest in