FEATool Multiphysics  v1.14 Finite Element Analysis Toolbox
Shallow Water Equations

Saint-Venant shallow water equations is a simplified model of fluid flow with a free surface. The non-conservative form of the equations read

$\left\{\begin{array}{l} \frac{\partial h}{\partial t} + (u\frac{\partial h}{\partial x} + v\frac{\partial h}{\partial y} ) + (h+H)(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y}) = 0 \\ \frac{\partial u}{\partial t} + (u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} ) = -g\frac{\partial h}{\partial x} \\ \frac{\partial v}{\partial t} + (u\frac{\partial v}{\partial x} + v\frac{\partial v}{\partial y} ) = -g\frac{\partial h}{\partial y} \end{array}\right.$

where h is the unknown free surface height relative to the mean level H.

# Tutorial

This model is available as an automated tutorial by selecting Model Examples and Tutorials... > Classic PDE > Shallow Water Equations from the File menu. Or alternatively, follow the step-by-step instructions below.

1. To start a new model click the New Model toolbar button, or select New Model... from the File menu.

Both the Custom Equation and Convection and Diffusion physics modes are used in this example.

1. Select the Custom Equation physics mode from the Select Physics drop-down menu.
2. Enter h into the Dependent Variable Names edit field.
3. Press OK to finish the physics mode selection.
4. Select Rectangle from the Geometry menu.
5. Enter 0 into the xmin edit field.
6. Enter 5 into the xmax edit field.
7. Enter 0 into the ymin edit field.
8. Enter 1 into the ymax edit field.
9. Press OK to finish and close the dialog box.
10. Switch to Grid mode by clicking on the corresponding Mode Toolbar button.
11. Enter 0.2 into the Grid Size edit field.
12. Press the Generate button to call the grid generation algorithm.
13. Switch to Equation mode by clicking on the corresponding Mode Toolbar button.
14. Press the edit button.

Define the equation for the height with a small disturbance as initial condition.

1. Enter h' + (u*hx_t + v*hy_t) + (h+H)*(ux_t+vy_t) = 0 into the Equation for h edit field.
2. Press OK to finish and close the dialog box.
3. Enter 0.2*exp(-((x-4)^2)) into the Initial condition for h edit field.

Add two Convection and Diffusion physics modes for the two velocity components. Although the Custom Equation physics mode could also be used, the Convection and Diffusion physics modes feature pre-defined equations and convective stabilization which is necessary for these equations.

1. Switch to the + tab.
2. Select the Convection and Diffusion physics mode from the Select Physics drop-down menu.
3. Enter u into the Dependent Variable Names edit field.
4. Press the Add Physics >>> button.
5. Enter 0 into the Diffusion coefficient edit field.
6. Enter u into the Convection velocity in x-direction edit field.
7. Enter v into the Convection velocity in y-direction edit field.
8. Enter -g*hx into the Reaction rate edit field.
9. Press the Artificial Stabilization button.
10. Select the Check to enable shock capturing check box.
11. Select High order from the Shock capturing discretization order drop-down menu.
12. Press OK to finish and close the dialog box.
13. Switch to the + tab.
14. Select the Convection and Diffusion physics mode from the Select Physics drop-down menu.
15. Enter v into the Dependent Variable Names edit field.
16. Press the Add Physics >>> button.
17. Enter 0 into the Diffusion coefficient edit field.
18. Enter u into the Convection velocity in x-direction edit field.
19. Enter v into the Convection velocity in y-direction edit field.
20. Enter -g*hy into the Reaction rate edit field.
21. Press the Artificial Stabilization button.
22. Select the Check to enable shock capturing check box.
23. Select High order from the Shock capturing discretization order drop-down menu.
24. Press OK to finish and close the dialog box.
25. Press OK to finish the equation and subdomain settings specification.

Define the two constants for the mean fluid level H = 1, and the gravitational constant g = 9.8.

1. Switch to Boundary mode by clicking on the corresponding Mode Toolbar button.

The height h should not be constrained and should therefore be prescribed homogeneous Neumann conditions everywhere

1. Select 1, 2, 3, and 4 in the Boundaries list box.
2. Select the Neumann, g_h radio button.
3. Switch to the cd tab.

A wall with zero velocities is prescribed at the right end, and the top and bottom boundaries are assumed to be symmetric (zero normal flow).

1. Select 4 in the Boundaries list box.
2. Select Concentration from the Convection and Diffusion drop-down menu.
3. Switch to the cd2 tab.
4. Select 1, 3, and 4 in the Boundaries list box.
5. Select Concentration from the Convection and Diffusion 2 drop-down menu.
6. Press OK to finish the boundary condition specification.
7. Switch to Solve mode by clicking on the corresponding Mode Toolbar button.
8. Press the Settings Toolbar button.

Select the Time-Dependent solver with the following parameters.

1. Enter 1e-3 into the Non-linear stopping criteria for solution differences (changes) between iterations edit field.
2. Enter 0.01 into the Time step size edit field.
3. Enter 1.5 into the Duration of time-dependent simulation (maximum time) edit field.
4. Press the Solve button.

Once the solver is done, turn on the height plot for h and visualize the free surface for different times.

1. Press the Plot Options Toolbar button.
2. Select the Height Expression check box.
3. Press OK to plot and visualize the selected postprocessing options.

The shallow water equations fluid dynamics model has now been completed and can be saved as a binary (.fea) model file, or exported as a programmable MATLAB m-script text file, or GUI script (.fes) file.