/** # 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 : $$hex(x) = \left( \frac{4}{g} \right)^{1/3} \left( 1 + 0.5\times exp\left( -16\times (\frac{x}{1000}-0.5)^2 \right) \right).$$ The slope is initially dry ($h(x) = u(x) = 0$) and we chose as boundaries conditions : $$q[left] = 1.5 m^2 .s^{-1} \ and \ h[left] = hex(0) ,$$ $$q[right] = 1.5 m^2 .s^{-1} \ and \ h[right] = hex(1000).$$ * [Manning](fricfluv_man.c) with $n = 0.033$ * [Poiseuille](fricfluv_poiseuille.c) with $\nu = 1e-6$ * [Darcy](fricfluv_darcy.c) with $f = 0.093$ * [Darcy with a 1 order treatment for the source term](fricfluv_darcy_0.c) with $f = 0.093$ We plot an example of the resulting topography for the Darcy friction : ![Resulting topography](/topofluvial_darcy.png) ### Torrential For the super-critical case, the analytical water height at steady state is : $$hex(x) = \left( \frac{4}{g} \right)^{1/3} \left( 1 - 0.2\times exp\left( -36\times (\frac{x}{1000}-0.5)^2 \right) \right).$$ The slope is initially dry ($h(x) = u(x) = 0$) and we chose as boundaries conditions : $$q[left] = 2.5 m^2 .s^{-1} \ and \ h[left] = hex(0).$$ * [Manning](frictor_man.c) with $n = 0.04$ * [Poiseuille](frictor_poiseuille.c) with $\nu = 1e-6$ * [Darcy](frictor_darcy.c) with $f = 0.065$ * [Darcy with a 1 order treatment for the source term](frictor_darcy_O.c) with $f = 0.065$ We plot an example of the resulting topography for the Darcy friction : ![Resulting topography](/topotorrential_darcy.png) ### Order treatment We compare the convergence of $|h|_1$ for the different source treatment (order 1 and 2), for the torrential case with the darcy friction term : ![Error Convergence](/tor_darcy10_error.png) 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 : $$h_{ex}(x) = \left( \frac{4}{g} \right)^{1/3} \left( 1 - 3\times tanh\left( -3\times (\frac{x}{1000}-0.5)^2 \right) \right).$$ for $0 < x < 500$ and : $$h_{ex}(x) = \left( \frac{4}{g} \right)^{1/6} \left( 1 - 6\times tanh\left( -3\times (\frac{x}{1000}-0.5)^2 \right) \right).$$ for $500 < x < 1000$. * [Darcy](fricsubsup_darcy.c) with $q0 = 2\ m. s^{-1}$ 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 : $$\partial_x z_b(x) = (\frac{q(x)^2}{g h^3}-1) \partial_x h(x) -\frac{2q(x) R}{g h(x)^2}- S_f$$ 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 : * [Rain in a fluvial flow with Darcy friction](fricfluv_rain_darcy.c) * [Rain in a torrential case with Darcy friction](frictor_rain_darcy.c) ## 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](http://basilisk.fr/sandbox/geoffroy/README) */