diff --git a/dumux/freeflow/shallowwater/model.hh b/dumux/freeflow/shallowwater/model.hh index db984d34c95c68506bcda92d6eb3c4f2c8006b60..85d702fb9c29a553124b02ee6ba756ee1bece1d1 100644 --- a/dumux/freeflow/shallowwater/model.hh +++ b/dumux/freeflow/shallowwater/model.hh @@ -21,40 +21,48 @@ * \ingroup ShallowWaterModel * * \brief A two-dimensional shallow water equations model - * The two-dimensonal shallow water equations (SWEs) can be written as + * + * The two-dimensional shallow water equations (SWEs) can be written as: * * \f[ * \frac{\partial \mathbf{U}}{\partial t} + - * \frac{\partial \mathbf{F}}{\partial x} + \\ + * \frac{\partial \mathbf{F}}{\partial x} + * \frac{\partial \mathbf{G}}{\partial y} - \mathbf{S_b} - \mathbf{S_f} = 0 * \f] * - * with U, F, G are defined as + * The first equation is the water balance equation (volume balance) and the following two equations balance the + * momentum in x-direction and y-direction. \f$\mathbf{U} \f$, \f$\mathbf{F} \f$ and \f$\mathbf{G} \f$ are defined as: * * \f[ * \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{F} = \begin{bmatrix} hu \\ hu^2 + \frac{1}{2} gh^2 - \nu\frac{\partial uh}{\partial x} \\ huv - \nu\frac{\partial vh}{\partial x} \end{bmatrix}, + * \mathbf{G} = \begin{bmatrix} hv \\ huv - \nu\frac{\partial uh}{\partial y} \\ hv^2 + \frac{1}{2} gh^2 - \nu\frac{\partial vh}{\partial y} \end{bmatrix} * \f] * - * Z is the bedSurface, h the water depth, u the velocity in - * x-direction and v the velocity in y-direction, g is the constant of gravity. + * \f$h \f$ is the water depth (in \f$m \f$), \f$u \f$ the velocity in + * x-direction and \f$v \f$ the velocity in y-direction (in \f$ms^{-1} \f$). \f$g \f$ is the constant of gravity (in \f$ms^{-2} \f$). + * \f$\nu \f$ is the effective turbulent viscosity (in \f$m^2s^{-1} \f$). + * By default the shallow water model neglects the viscous terms, but they can be enabled by setting the parameter ShallowWater.EnableViscousFlux = true. * - * The source terms for the bed friction S_b and bed slope - * S_f are given as + * The source terms for bed slope \f$\mathbf{S_b} \f$ and bottom friction \f$\mathbf{S_f} \f$ + * are given as: * \f[ * \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 \\ -\frac{\tau_{x}}{\rho} \\ -\frac{\tau_{y}}{\rho}\end{bmatrix}. * \f] * - * A cell-centered finte volume method (cctpfa) is applied to solve the SWEs - * in combination with a fully-implicit time discretization. For large time - * step sizes (CFL > 1) this can lead to a strong semearing of sharp fronts. - * This can be seen in the movement of short traveling waves (e.g. dam break - * waves). Nevertheless the fully implicit time discretization showes often - * no drawbacks in cases where no short waves are considered. Thus the model - * can be a good choice for simulating flow in rivers and channels, where the + * \f$z \f$ is the bed surface (in \f$m \f$), \f$\tau_{x} \f$ and \f$\tau_{y} \f$ the bottom shear stress (in \f$Nm^{-2} \f$) in x- an y-direction, respectively. + * \f$\rho \f$ is the water density (in \f$kgm^{-3} \f$). The bed slope source term \f$\mathbf{S_b} \f$ is covered by the hydrostatic reconstruction + * within the flux computation and must therefore not be treated separately within the computation of the source terms. + * On the contrary, the bottom friction source term must be implemented in the problem (have a look at the shallow water example). + * + * When using a fully implicit Euler time discretization, note the following: + * Large time step sizes (CFL > 1) can lead to a strong smearing of sharp fronts. + * This can be seen in the movement of fast traveling waves (e.g. dam break + * waves). Nevertheless, the fully implicit time discretization shows + * good results in cases where slowly moving waves are considered. Thus, the model + * is a good choice for simulating flow in rivers and channels, where the * fully-implicit discretization allows large time steps and reduces the * overall computation time drastically. * diff --git a/examples/shallowwaterfriction/README.md b/examples/shallowwaterfriction/README.md index 2b98d77446b541d14c8e02fce6da44ceee21bbe2..e5327b6a8b4b457244aa43525537b530af0ec8a3 100644 --- a/examples/shallowwaterfriction/README.md +++ b/examples/shallowwaterfriction/README.md @@ -73,29 +73,27 @@ where $\mathbf{U}$, $\mathbf{F}$ and $\mathbf{G}$ defined as $h$ the water depth, $u$ the velocity in x-direction and $v$ the velocity in y-direction, $g$ is the constant of gravity. -The source terms for the bed slope $\mathbf{S_b}$ and friction +The source terms for bed slope $\mathbf{S_b}$ and bottom friction $\mathbf{S_f}$ 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 \\ -\frac{\tau_x}{\rho} \\ -\frac{\tau_y}{\rho} \end{bmatrix}.  -with the bedSurface $z$. $S_{fx}$ and $S_{fy}$ are the bed shear stess -components in x- and y-direction, which are calculated by Manning's law. +with the bed surface $z$. $\rho$ is the water density. $\tau_x$ and $\tau_y$ are the bottom shear stress components in x- an y-direction, respectively. +The bottom shear stress is calculated by Manning's law. ### Mannings law -The empirical Manning model specifies the bed shear stress by the following equations: +The empirical Manning model specifies the bottom shear stress by the following equation math -S_{fx} = \frac{n^2u}{R_{hy}^{4/3}} \sqrt(u^2 + v^2), - -S_{fy} = \frac{n^2v}{R_{hy}^{4/3}} \sqrt(u^2 + v^2) +\mathbf{\tau} = \frac{n^2 g\rho}{h^{1/3}} \sqrt{u^2 + v^2} \begin{bmatrix} u \\ v \end{bmatrix}  -$n$ is Manning's friction value and $R_{hy}$ is the hydraulic radius, -which is assumed to be equal to the water depth $h$. +$n$ is Manning's friction value. +In addition, the dumux shallow water model extends the water depth by a roughness hight to limit the friction for small water depth. ### Analytical solution Since normal flow conditions are assumed, the analytic solution is calculated using the equation @@ -105,7 +103,8 @@ of Gauckler, Manning and Strickler: v_m = n^{-1} R_{hy}^{2/3} I_s^{1/2}  -Where the mean velocity $v_m$ is given as +$R_{hy}$ is the hydraulic radius, which is assumed to be equal to the water depth $h$. +The mean velocity $v_m$ is given as math v_m = \frac{q}{h}