# sandbox/geoffroy/friction/README

# 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 ${z}_{b}(x)$ which is 1000 metres long. At steady state, the momentum equation of Saint-Venant can be written as :

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

where ${S}_{f}$ is the friction term. By chosing an analytical function $hex(x)$ for $h(x)$, 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 :

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

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

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

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

- Manning with $n=0.033$
- Poiseuille with $\nu =1e-6$
- Darcy with $f=0.093$
- Darcy with a 1 order treatment for the source term with $f=0.093$

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 :

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

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

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

- Manning with $n=0.04$
- Poiseuille with $\nu =1e-6$
- Darcy with $f=0.065$
- Darcy with a 1 order treatment for the source term with $f=0.065$

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 :

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

- 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 :

$$q(x)=q0+x*R.$$

So the topography is now :

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

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.