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 zb(x) which is 1000 metres long. At steady state, the momentum equation of Saint-Venant can be written as :

xzb(x)=(q2gh31)xh(x)Sf ,

where Sf 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)=(4g)1/3(1+0.5×exp(16×(x10000.5)2)).

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

q[left]=1.5m2.s1 and h[left]=hex(0),

q[right]=1.5m2.s1 and h[right]=hex(1000).

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

Torrential

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

hex(x)=(4g)1/3(10.2×exp(36×(x10000.5)2)).

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

q[left]=2.5m2.s1 and h[left]=hex(0).

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

Order treatment

We compare the convergence of h1 for the different source treatment (order 1 and 2), for the torrential case with the darcy friction term :

Error Convergence

Error Convergence

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 :

hex(x)=(4g)1/3(13×tanh(3×(x10000.5)2)). for 0<x<500 and : hex(x)=(4g)1/6(16×tanh(3×(x10000.5)2)). for 500<x<1000.

  • Darcy with q0=2 m.s1 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 :

xzb(x)=(q(x)2gh31)xh(x)2q(x)Rgh(x)2Sf

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.

Return to home page