Shallow water flow with bottom friction

In this example, the shallow water flow model is applied to simulate a steady subcritical flow including bottom friction (bed shear stress).

You will learn how to

• solve a shallow water flow problem including bottom friction
• compute and output (VTK) an analytical reference solution

Result. The numerical and analytical solutions for the free surface will look like this:

Table of contents. This description is structured as follows:

• Shallow water flow with bottom friction
  • Problem set-up
    • Model domain
    • Boundary conditions
    • Initial conditons
  • Model description
    • Shallow water model
    • Mannings law
    • Analytical solution
    • Discretisation
• Implementation
  • Folder layout and files
  • Part 1: Shallow water flow simulation setup
  • Part 2: Main program flow

Problem set-up

Model domain

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).

Boundary conditions

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 is applied as inflow boundary condition with q = -1.0 m^2 s^{-1}. 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.

Initial conditons

The initial water depth is set to 1 m, which is slightly higher than the normal flow water depth (0.87 m). Therefore, we expect a decreasing water level during the simulation until the normal flow condition is reached in the entire model domain. The inital 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 in DuMu^x.

Shallow water model

The shallow water equations are given 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

where \mathbf{U}, \mathbf{F} and \mathbf{G} defined as

\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}

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 \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 \\ghS_{fx} \\ghS_{fy}\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.

Mannings law

The empirical Manning model specifies the bed shear stress by the following equations:

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)

n is Manning's friction value and R_{hy} is the hydraulic radius, which is assumed to be equal to the water depth h.

Analytical solution

Since normal flow conditions are assumed, the analytic solution is calculated using the equation 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

v_m = \frac{q}{h}

I_s is the bed slope and q the unity inflow discharge.

Hence, the water depth h can be calculated by

h = \left(\frac{n q}{\sqrt{I_s}} \right)^{3/5}

Discretisation

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 the computation time. Even if a steady state solution is considered, an implicit time stepping method is applied.

Implementation

Folder layout and files

└── shallowwaterfriction/
    ├── CMakeLists.txt          -> build system file
    ├── main.cc                 -> main program flow
    ├── params.input            -> runtime parameters
    ├── properties.hh           -> compile time configuration
    ├── problem.hh              -> boundary & initial conditions
    └── spatialparams.hh        -> spatial parameter fields

Part 1: Shallow water flow simulation setup

Part 2: Main program flow