diff --git a/examples/shallowwaterfriction/doc/_intro.md b/examples/shallowwaterfriction/doc/_intro.md index 9b616a9fb8df5bd0ab4de1f6aac50b7b481f0ec9..d5328920d47a4c9ad1989988d5d0daa1727b92bf 100644 --- a/examples/shallowwaterfriction/doc/_intro.md +++ b/examples/shallowwaterfriction/doc/_intro.md @@ -26,18 +26,18 @@ The model domain is given by a rough channel with a slope of 0.001. The domain is 500 meters long and 5 meters wide. The bottom altitude is 10 m at the inflow and hence 9.5 m at the outflow. Bottom friction is considered by applying -Manning's law ($`n`$ = 0.025). +[Manning's law](#mannings-law) ($`n`$ = 0.025). ### Boundary conditions -At the lateral sides a no-flow boundary condition is applied. Also no friction is +At the lateral sides a no-flow boundary condition is applied. Also, no friction is considered there and therefore a no slip boundary -condition is applied. These are the default boundary condition for the shallow -water model. At the left border a discharge boundary condition +condition is applied. These are the default boundary conditions for the shallow +water model. At the left border, a discharge boundary condition is applied as inflow boundary condition with $`q = -1.0 m^2 s^{-1}`$. -At the right border a fixed water depth boundary condition +At the right border, a fixed water depth boundary condition is applied for the outflow. Normal flow is assumed, therefore the water depth at the right border is calculated using the equation -of Gauckler, Manning and Strickler. +of [Gauckler, Manning and Strickler](#analytical-solution). ### Initial conditions The initial water depth is set to 1 m, which is slightly higher than the normal flow @@ -46,9 +46,9 @@ water level during the simulation until the normal flow condition is reached in the entire model domain. The initial velocity is set to zero. ## Model description -As mentioned above, this examples uses the shallow water equations (SWEs) to solve the problem. -These are a depth averaged simplification of the Navier-Stokes equations. To calculate the -bottom friction Manning's law is used. An alternative is Nikuradse's law, which is also implemented +As mentioned above, this example uses the shallow water equations (SWEs) to solve the problem. +These are a depth-averaged simplification of the Navier-Stokes equations. To calculate the +bottom friction, Manning's law is used. An alternative is Nikuradse's law, which is also implemented in DuMu<sup>x</sup>. ### Shallow water model @@ -60,16 +60,16 @@ The shallow water equations are given as: \frac{\partial \mathbf{G}}{\partial y} - \mathbf{S_b} - \mathbf{S_f} = 0 ``` -where $`\mathbf{U}`$, $`\mathbf{F}`$ and $`\mathbf{G}`$ defined as +where $`\mathbf{U}`$, $`\mathbf{F}`$ and $`\mathbf{G}`$ are defined as ```math \mathbf{U} = \begin{bmatrix} h \\ uh \\ vh \end{bmatrix}, \mathbf{F} = \begin{bmatrix} hu \\ hu^2 + \frac{1}{2} gh^2 \\ huv \end{bmatrix}, -\mathbf{G} = \begin{bmatrix} hv \\ huv \\ hv^2 + \frac{1}{2} gh^2 \end{bmatrix} +\mathbf{G} = \begin{bmatrix} hv \\ huv \\ hv^2 + \frac{1}{2} gh^2 \end{bmatrix}, ``` -$`h`$ the water depth, $`u`$ the velocity in x-direction and $`v`$ the velocity in y-direction, -$`g`$ is the constant of gravity. +$`h`$ is the water depth, $`u`$ and $`v`$ are the velocities in x- and y-direction, respectively, +and $`g`$ is the gravitational acceleration. The source terms for the bed slope $`\mathbf{S_b}`$ and friction $`\mathbf{S_f}`$ are given as @@ -77,10 +77,10 @@ $`\mathbf{S_f}`$ are given as ```math \mathbf{S_b} = \begin{bmatrix} 0 \\ -gh \frac{\partial z}{\partial x} \\ -gh \frac{\partial z}{\partial y}\end{bmatrix}, -\mathbf{S_f} = \begin{bmatrix} 0 \\ghS_{fx} \\ghS_{fy}\end{bmatrix}. +\mathbf{S_f} = \begin{bmatrix} 0 \\ghS_{fx} \\ghS_{fy}\end{bmatrix}, ``` -with the bedSurface $`z`$. $`S_{fx}`$ and $`S_{fy}`$ are the bed shear stess +with the bed surface $`z`$. $`S_{fx}`$ and $`S_{fy}`$ are the bed shear stess components in x- and y-direction, which are calculated by Manning's law. ### Mannings law @@ -100,13 +100,13 @@ Since normal flow conditions are assumed, the analytic solution is calculated us of Gauckler, Manning and Strickler: ```math -v_m = n^{-1} R_{hy}^{2/3} I_s^{1/2} +v_m = n^{-1} R_{hy}^{2/3} I_s^{1/2}, ``` -Where the mean velocity $`v_m`$ is given as +where the mean velocity $`v_m`$ is given as ```math -v_m = \frac{q}{h} +v_m = \frac{q}{h}, ``` $`I_s`$ is the bed slope and $`q`$ the unity inflow discharge. @@ -117,7 +117,7 @@ Hence, the water depth $`h`$ can be calculated by h = \left(\frac{n q}{\sqrt{I_s}} \right)^{3/5} ``` -### Discretisation +### Discretization For this example, a cell-centered finite volume method (cctpfa) is applied to solve the SWEs in combination with a fully-implicit time discretization. For cases where no sharp fronts or traveling waves occur it is possible to apply time steps larger than CFL number = 1 to reduce