The Shallow Water Equations, Time Explicit Interface

The interface (

), found under the branch (

) when adding a physics interface, is used to solve the Shallow Water equations in a 1D or 2D geometry. These equations model the flow of a free surface in a fluid under the assumption that the horizontal scale is much greater than the vertical length scale. They are frequently used for modeling both oceanographic and atmospheric fluid flow. Models of such systems can be used to predict areas affected by pollution, coastal erosion, and polar ice-cap melting, provided the fluid layer is shallow enough.

Comprehensive modeling of such phenomena using physical descriptions such as the Navier–Stokes equations can often be problematic, due to the scale of the modeling domains as well as the resolution of free surfaces. The shallow water equations, of which there are a number of representations, provide an easier description of such phenomena.

A typical configuration for the flow of fluid in a shallow layer is shown in Figure 10-1.This physics interface approximates free surface problems where the thickness h of the fluid layer, or water depth, is small compared to the lateral dimensions of the geometry. The lower boundary of the fluid is treated as a nonpenetrable wall and has a height hb(x, y) over a reference xy-plane placed at z = 0. The topography of the bottom is assumed constant in time. Using the shallow water equations, the water depth h and water flux q are computed in a reduced dimension of the problem. Free surface problems that would require a 3D geometry when modeled with the Navier–Stokes equations can be modeled in 2D instead. 2D problems can also be reduced to 1D.

Figure 10-1: An example illustrating a typical configuration for shallow water equations. A water layer of thickness h flows over a nonflat bottom with topography represented by hb. The total height of the free surface over the reference xy plane at z = 0 is represented by H. The gravity points downward (−z direction) and gives rise to a hydrostatic pressure ranging from 0 atm at the free surface to a value of gρh at the bottom.

When this physics interface is added, the following default nodes are also added in the — , , and . Then, from the toolbar, you can add other nodes that implement, for example, boundary conditions. You can also right-click to select physics features from the context menu.

The is the default physics interface name.

The is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers, and underscores (_) are permitted in the field. The first character must be a letter.

The default (for the first physics interface in the model) is swe.

Select a value for (SI unit m/s). The default value is gconst. It should be a global quantity.

To display this section, click the button (

) and select in the dialog box. Normally these settings do not need to be changed.

Select the . This CFL number will be used when defining the swe.wtc used in the if the is set to . Note that the method will be unstable for CFL numbers larger than 1.

The Shallow Water Equations, Time Explicit interface uses functions with .

The dependent variables (field variables) are the h (SI unit: m) and the q (SI unit: m2 /s). The names can be changed but the names of fields and dependent variables must be unique within a component.