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25.10.1 Changing the Courant Number

For FLUENT's density-based solver, the main control over the time-stepping scheme is the Courant number (CFL). The time step is proportional to the CFL, as defined in Equation  25.5-14.

Linear stability theory determines a range of permissible values for the CFL (i.e., the range of values for which a given numerical scheme will remain stable). When you specify a permissible CFL value, FLUENT will compute an appropriate time step using Equation  25.5-14. In general, taking larger time steps leads to faster convergence, so it is advantageous to set the CFL as large as possible (within the permissible range).

The stability limits of the density-based implicit and explicit formulations are significantly different. The explicit formulation has a more limited range and requires lower CFL settings than does the density-based implicit formulation. Appropriate choices of CFL for the two formulations are discussed below.

Courant Numbers for the Density-Based Explicit Formulation

Linear stability analysis shows that the maximum allowable CFL for the multi-stage scheme used in the density-based explicit formulation will depend on the number of stages used and how often the dissipation and viscous terms are updated (see Section  25.13). But in general, you can assume that the multi-stage scheme is stable for Courant numbers up to 2.5. This stability limit is often lower in practice because of nonlinearities in the governing equations.

The default CFL for the density-based explicit formulation is 1.0, but you may be able to increase it for some 2D problems. You should generally not use a value higher than 2.0.

If your solution is diverging, i.e., if residuals are rising very rapidly, and your problem is properly set up and initialized, this is usually a good sign that the Courant number needs to be lowered. Depending on the severity of the startup conditions, you may need to decrease the CFL to a value as low as 0.1 to 0.5 to get started. Once the startup transients are reduced you can start increasing the Courant number again.

Courant Numbers for the Density-Based Implicit Formulation

Linear stability theory shows that the density-based implicit formulation is unconditionally stable. However, as with the explicit formulation, nonlinearities in the governing equations will often limit stability.

The default CFL for the density-based implicit formulation is 5.0. It is often possible to increase the CFL to 10, 20, 100, or even higher, depending on the complexity of your problem. You may find that a lower CFL is required during startup (when changes in the solution are highly nonlinear), but it can be increased as the solution progresses.

The coupled AMG solver has the capability to detect divergence of the multigrid cycles within a given iteration. If this happens, it will automatically reduce the CFL and perform the iteration again, and a message will be printed to the screen. Five attempts are made to complete the iteration successfully. Upon successful completion of the current iteration the CFL is returned to its original value and the iteration procedure proceeds as required.

User Inputs

The Courant number is set in the Solution Controls panel (Figure  25.10.1).

Solve $\rightarrow$ Controls $\rightarrow$ Solution...

Figure 25.10.1: The Solution Controls Panel for the Density-Based Explicit Formulation

Enter the value for Courant Number under Solver Parameters. (Note that the panel shown in Figure  25.10.1 is for the density-based explicit formulation.)

When you select the density-based explicit formulation in the Solver panel, FLUENT will automatically set the Courant Number to 1; when you select the density-based implicit formulation, the Courant Number will be changed to 5 automatically.

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