Template:Gravity Waves Courant Number

Gravity Waves Courant Number

The Courant number is a spatially varied (dataset) dimensionless value representing the time a particle stays in a cell of a mesh/grid. This is based on the size of the element and the speed of that particle. A Courant number of 1.0 implies that a particle(parcel or drop of water) would take one time step to flow through the element. Since cells/elements are not guaranteed to align with the flow field, this number is an approximation. This dataset is computed at nodes so it is uses the average size of the cells/elements attached to the node. Since the typical Courant number calculation requires as an input a velocity as shown in the equation below:

${\displaystyle CourantNumber=TimeStep*(Gravity*Depth)^{0.5}/NodalSpacing}$

• Time Step: a user specified constant
• NodalSpacing: dependent on the computational grid/mesh
• Velocity: typcially computed by hydrodynamic simulation

As noted above, the velocity is typically output from a simulation, so this tool approximates velocity based on the speed of a gravity wave through the water column.

The speed of wave propogation is depentdent on the depth of the water column. For purposes of this tool, the speed is assumed to be:

$Velocity=TimeStep*(Gravity*Depth)^{0}.5$

Note: the variable in the equation above is "Depth", not "Elevation".

Depth is also typically output from a hydrodynamic simulation. For this tool it can be approximated as:

$Depth=Max(WaterLevel-GroundElevation,MinimumDepth)$

For coastal applications the water level can be assumed as 0.0 (Mean Sea Level) or a constant offset from MSL.

For riverine appliations the water level is more difficult to approximate. It may require computing a sloped plane. In this case, negative values should be filtered out.

In all cases, the depth should be truncated to a minimum value. Velocity only exists in a water column with positive depth.

The tool computes the Courant number at each node in the selected geometry based on the specified time step.

If the input velocity magnitude dataset is transient, the resulting CourantNumber dataset will also be transient.

For numerical solvers that are Courant limited/controlled, any violation of the Courant condition, where the Courant number exceeds the allowable threshold could result in instability. Therefore, the maximum of the Courant number dataset gives an indication of the stability of this mesh for the specified time step parameter.

This tool is intended to assist with numerical engine stability, and possibly the selection of an appropriate time step size.

The Gravity Waves Courant Number tool dialog has the following options:

Input Paramters

• Input dataset – This is a required input parameter. Specify which depth dataset will be used to generate the gravity wave Courant number dataset.
• Gravity – Enter the gravity value.
• Use time step – Enter the time step value.

Output Parameters

• Gravity waves time step dataset – Enter the name for the new gravity wave Courant number dataset.