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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 26.7: Changing the Multi-Stage Scheme). 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.

4.3.3 Modeling and Basics of Fluid Flow

This chapter describes the basic physical models that ANSYS FLUENT provides for fluid flow and the
commands for defining and using them.
The information in this chapter is presented in the following sections:
w Inviscid Flows

w Compressible Flows

w Swirling and Rotating Flows

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