# Benchmarks

## Friction test cases

Here we test the friction source term in the case of fluvial/torrential flows for three differents source terms. The original test case is the one of MacDonald[1] and you can find more similar test cases in Delestre & al.[2]. The principle is to set the topography in order to have an analytical solution for h(x). The source term is treated semi-implicitly at the same order than the Saint-Venant function, which is at order 2, except for the Darcy 1 order case.

### Case description

We consider a 1D topography ${z}_{b}\left(x\right)$ which is 1000 metres long. At steady state, the momentum equation of Saint-Venant can be written as :

${\partial }_{x}{z}_{b}\left(x\right)=\left(\frac{{q}^{2}}{g{h}^{3}}-1\right){\partial }_{x}h\left(x\right)-{S}_{f},$

where ${S}_{f}$ is the friction term. By chosing an analytical function $hex\left(x\right)$ for $h\left(x\right)$, we can find the resulting topography with a r-k method.

### Fluvial

For the sub-critical case, the analytical water height at steady state is :

$hex\left(x\right)={\left(\frac{4}{g}\right)}^{1/3}\left(1+0.5×exp\left(-16×\left(\frac{x}{1000}-0.5{\right)}^{2}\right)\right).$

The slope is initially dry ($h\left(x\right)=u\left(x\right)=0$) and we chose as boundaries conditions :

$q\left[left\right]=1.5{m}^{2}.{s}^{-1}andh\left[left\right]=hex\left(0\right),$

$q\left[right\right]=1.5{m}^{2}.{s}^{-1}andh\left[right\right]=hex\left(1000\right).$

We plot an example of the resulting topography for the Darcy friction :

### Torrential

For the super-critical case, the analytical water height at steady state is :

$hex\left(x\right)={\left(\frac{4}{g}\right)}^{1/3}\left(1-0.2×exp\left(-36×\left(\frac{x}{1000}-0.5{\right)}^{2}\right)\right).$

The slope is initially dry ($h\left(x\right)=u\left(x\right)=0$) and we chose as boundaries conditions :

$q\left[left\right]=2.5{m}^{2}.{s}^{-1}andh\left[left\right]=hex\left(0\right).$

We plot an example of the resulting topography for the Darcy friction :

### Order treatment

We compare the convergence of $\mid h{\mid }_{1}$ for the different source treatment (order 1 and 2), for the torrential case with the darcy friction term :

Note that the convergence depends strongly on the chosen source term.

### Subcritical to super-critical

The flow is subcritical upstream and super-critical downstream. The water height is :

${h}_{ex}\left(x\right)={\left(\frac{4}{g}\right)}^{1/3}\left(1-3×tanh\left(-3×\left(\frac{x}{1000}-0.5{\right)}^{2}\right)\right).$ for $0 and : ${h}_{ex}\left(x\right)={\left(\frac{4}{g}\right)}^{1/6}\left(1-6×tanh\left(-3×\left(\frac{x}{1000}-0.5{\right)}^{2}\right)\right).$ for $500.

• Darcy with $q0=2m.{s}^{-1}$ and $f=0.042$

## Rain test cases

We add a rain term to the MacDonald test case, as seen in Delestre et al.[2]. Write R the constant rain intensity, the flux along the channel is written :

$q\left(x\right)=q0+x*R.$

So the topography is now :

${\partial }_{x}{z}_{b}\left(x\right)=\left(\frac{q\left(x{\right)}^{2}}{g{h}^{3}}-1\right){\partial }_{x}h\left(x\right)-\frac{2q\left(x\right)R}{gh\left(x{\right)}^{2}}-{S}_{f}$

We take the same function for hex(x) seen with the friction test cases in both torrential and fluvial case and we use the Darcy friction term :

## References

[1] I. MacDonald, M. Baines, N. Nichols, and P. G. Samuels, “Analytic Benchmark Solutions for Open-Channel Flows,” no. November, pp. 1041–1045, 1997.

[2] O. Delestre, C. Lucas, P. A. Ksinant, F. Darboux, C. Laguerre, T. N. T. Vo, F. James, and S. Cordier, “SWASHES: A compilation of shallow water analytic solutions for hydraulic and environmental studies,” Int. J. Numer. Methods Fluids, vol. 72, pp. 269–300, 2013.