Shallow water flow

Two-dimensional shallow water flow (depth-averaged) More...

Description

A two-dimensional shallow water equations model.

The two-dimensional shallow water equations (SWEs) can be written as:

\[ \frac{\partial \mathbf{U}}{\partial t} + \frac{\partial \mathbf{F}}{\partial x} + \frac{\partial \mathbf{G}}{\partial y} - \mathbf{S_b} - \mathbf{S_f} = 0 \]

The first equation is the water balance equation (volume balance) and the following two equations balance the momentum in x-direction and y-direction. \( \mathbf{U} \), \( \mathbf{F} \) and \( \mathbf{G} \) are defined as:

\[ \mathbf{U} = \begin{bmatrix} h \\ uh \\ vh \end{bmatrix}, \mathbf{F} = \begin{bmatrix} hu \\ hu^2 + \frac{1}{2} gh^2 - \nu\frac{\partial uh}{\partial x} \\ huv - \nu\frac{\partial vh}{\partial x} \end{bmatrix}, \mathbf{G} = \begin{bmatrix} hv \\ huv - \nu\frac{\partial uh}{\partial y} \\ hv^2 + \frac{1}{2} gh^2 - \nu\frac{\partial vh}{\partial y} \end{bmatrix} \]

\( h \) is the water depth (in \( m \)), \( u \) the velocity in x-direction and \( v \) the velocity in y-direction (in \( ms^{-1} \)). \( g \) is the constant of gravity (in \( ms^{-2} \)). \( \nu \) is the effective turbulent viscosity (in \( m^2s^{-1} \)). 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 bed slope \( \mathbf{S_b} \) and bottom friction \( \mathbf{S_f} \) are given as:

\[ \mathbf{S_b} = \begin{bmatrix} 0 \\ -gh \frac{\partial z}{\partial x} \\ -gh \frac{\partial z}{\partial y}\end{bmatrix}, \mathbf{S_f} = \begin{bmatrix} 0 \\ -\frac{\tau_{x}}{\rho} \\ -\frac{\tau_{y}}{\rho}\end{bmatrix}. \]

\( z \) is the bed surface (in \( m \)), \( \tau_{x} \) and \( \tau_{y} \) the bottom shear stress (in \( Nm^{-2} \)) in x- an y-direction, respectively. \( \rho \) is the water density (in \( kgm^{-3} \)). The bed slope source term \( \mathbf{S_b} \) 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.

Files
file	boundaryfluxes.hh
	Compute boundary conditions for the Riemann Solver.
file	freeflow/shallowwater/fluxvariables.hh
file	freeflow/shallowwater/indices.hh
file	freeflow/shallowwater/iofields.hh
	Add I/O fields specific to shallow water.
file	freeflow/shallowwater/localresidual.hh
file	freeflow/shallowwater/model.hh
	A two-dimensional shallow water equations model.
file	freeflow/shallowwater/problem.hh
file	freeflow/shallowwater/volumevariables.hh

Classes
class	Dumux::ShallowWaterFluxVariables< TypeTag >
	The flux variables class for the shallow water model. More...
struct	Dumux::ShallowWaterIndices
	The common indices for the shallow water equations model. More...
class	Dumux::ShallowWaterIOFields
	Adds vtk output fields for the shallow water model. More...
class	Dumux::ShallowWaterResidual< TypeTag >
	Element-wise calculation of the residual for the shallow water equations. More...
struct	Dumux::ShallowWaterModelTraits
	Specifies a number properties of shallow water models. More...
struct	Dumux::ShallowWaterVolumeVariablesTraits< PV, FSY, MT >
	Traits class for the volume variables of the shallow water model. More...
class	Dumux::ShallowWaterProblem< TypeTag >
	Shallow water problem base class. More...
class	Dumux::ShallowWaterVolumeVariables< Traits >
	Volume variables for the shallow water equations model. More...