Figure 5: 3D axisymmetrical parabolic bowl: error between the analytical water depth (a), the analytical horizontal velocity (b) and the simulated ones with the five unstructured meshes. The curves for the first order scheme (space and time) and its second order extension (space and time) are compared to the first and second order theoretical curves (dashed lines).
a lot of wet/dry interfaces where the second order reconstruction in space cannot be applied. This can explain the differences between the theoretical and observed slopes for the convergence tests of the second order schemes, see Tab. 2 and Fig. 5.
6.2.2 Draining of a tank
Considering the Navier-Stokes system (1)-(3) completed with the boundary conditions (5)-(8), the following proposition holds, see [25] for more details about the proposed analytical solution.
Proposition 6.2 For some $t_0 \in \mathbb{R}$, $t_1 \in \mathbb{R}^*_+$, $(\alpha, \beta) \in \mathbb{R}^2_+$ such that $\alpha\beta > L$, let us consider the functions $h, u, v, w, p, \phi$ defined for $t \ge t_0$ by
where $f(t) = 1/(t - t_0 + t_1)$ and with a flat bottom $z_b(x,y) = z_{b,0} = \text{cst}$ and $p^a(t,x,y) = p^{a,1}(t)$, with $p^{a,1}(t)$ a given function.
Then $h, u, v, w, p$ as defined previously satisfy the 3d hydrostatic Navier-Stokes system (1)-(3) completed with the boundary conditions (6), (4), (7), (5) and $\kappa = \frac{2\nu\alpha\beta}{h(t,x,y)[\alpha\beta - 2(x\cos^2(\theta) + y\sin^2(\theta))]}$ in (6), $W = \nu\beta/2$ with $t_s = \frac{1}{\sqrt{2}}(1,1,0)^t$ in (7). The appropriate boundary conditions for $x \in {-L/2, L/2}$ or $y \in {-L/2, L/2}$ are also determined by the expressions of $h, v, u, w$ given above.
Choosing the viscosity $\nu = 0$, the variables $h, u, v, w, p$ become analytical solutions of the 3d hydro- static Euler system (12)-(13) completed with the boundary conditions (5), (4) and $p(t, x, y, \eta(t, x, y)) =$ 0.
Proof of prop. 6.2 The proof of prop. 6.2 relies on very simple computations and is not detailed here. ■