(Pseudo-) Time-Marching Approach

The coupled flow solver solves the system of conservation equations using a pseudo-time-marching approach.

For steady simulations, the coupled solver employs a time marching scheme to drive the unsteady form of the governing equations to a steady state. In this case, a pseudo-transient term replaces the physical time-derivative. The solution advances in pseudo-time to drive this term to zero. The solution in each cell is advanced independently with an optimal pseudo-time step computed locally according to stability constraints. In this way, convergence to steady state is achieved in the most efficient manner.

For transient simulations, two different time integration schemes are available:

You choose between these two by activating either the Implicit Unsteady Model or the Explicit Unsteady Model, while also selecting the Coupled Flow model.

Implicit Unsteady

The Implicit Unsteady approach is appropriate if the time scales of the phenomena of interest are either of the following:

  • The same order as the convection and/or diffusion processes (for example, vortex shedding).
  • Related to some relatively low frequency external excitation (for example, time-varying boundary conditions or boundary motion).

In the Implicit Unsteady approach, each physical time-step involves some number of inner iterations to converge the solution for that given instant of time. These inner iterations can be accomplished using implicit spatial integration or explicit spatial integration schemes. You specify the physical time-step size that is used in the outer loop. The integration scheme marches inner iterations using optimal pseudo-time steps that are determined from the Courant number.

With the Implicit Unsteady approach, you are required to set the physical time-step size, the Courant number, and the number of inner iterations at each physical time-step.

The transient phenomena being modeled generally governs the physical time-step size. The time step must at least satisfy the Nyquist sampling criterion: more than two time-steps per period are required.

In general, the guidelines for setting the Courant number in steady-state integration schemes also apply to the transient integration schemes.

The number of inner iterations per physical time-step is harder to quantify. Generally, you determine this number by observing the effect that it has on results. Select a number of inner iterations, plot a monitor for one or more specific quantities against iteration, and see whether these monitors are converging within each time-step.

Smaller physical time-steps generally mean that the solution is changing less from one time step to the next; fewer inner iterations are then required. There is an optimal balance of time-step size and number of inner iterations for a given problem and desired transient accuracy.

Explicit Unsteady

If the unsteady time scales are of the order of the acoustic processes (for example, shock front tracking), the Explicit Unsteady approach is the proper choice.

The Explicit Unsteady approach is effectively the same as the explicit spatial integration scheme, but using the same physical time step for all cells in the domain (rather than local time-stepping). In this case, the solver automatically determines the size of the time-step as Eqn. (968) such that one value satisfies the Courant condition at all points; the minimum allowable time-step is used everywhere. Thus, each iteration becomes a time-accurate advancement of the solution. In this case, the Courant number must be 1 or less, resulting in a physical time-step that varies from one iteration to the next as the flow field changes. Furthermore, preconditioning of the governing equations is omitted; the Explicit Unsteady approach is therefore unsuited for incompressible flow simulations.