
For timeaccurate calculations, explicit and implicit timestepping schemes are available. (The timeimplicit approach is also referred to as "dual time stepping''.)
Explicit Time Stepping
The explicit time stepping approach, is available only for the explicit scheme described above. The time step is determined by the CFL condition. To maintain time accuracy of the solution the explicit time stepping employs the same time step in each cell of the domain (this is also known as globaltime step), and with preconditioning disabled. By default, ANSYS FLUENT uses a 4stage RungeKutta scheme for unsteady flows.
Implicit Time Stepping (DualTime formulation)
The implicittime stepping method (also known as dualtime formulation) is available in the densitybased explicit and implicit formulation.
When performing unsteady simulations with implicittime stepping (dualtime stepping), ANSYS FLUENT uses a low Mach number timederivative unsteady preconditioner to provide accurate results both for pure convective processes (e.g., simulating unsteady turbulence) and for acoustic processes (e.g., simulating wave propagation) [ 350, 263].
Here we introduce a preconditioned pseudotimederivative term into Equation 18.51 as follows:
where denotes physicaltime and is a pseudotime used in the timemarching procedure. Note that as , the second term on the left side of Equation 18.521 vanishes and Equation 18.51 is recovered.
The timedependent term in Equation 18.521 is discretized in an implicit fashion by means of either a first or secondorder accurate, backward difference in time.
The dualtime formulation is written in semidiscrete form as follows:
where { } gives firstorder time accuracy, and { } gives secondorder. is the inner iteration counter and represents any given physicaltime level.
The pseudotimederivative is driven to zero at each physical time level by a series of inner iterations using either the implicit or explicit timemarching algorithm.
Throughout the (inner) iterations in pseudotime, and are held constant and is computed from . As , the solution at the next physical time level is given by .
Note that the physical time step is limited only by the level of desired temporal accuracy. The pseudotimestep is determined by the CFL condition of the (implicit or explicit) timemarching scheme.
Table 18.5.1 summarizes all operation modes for the densitybased solver from the iterative scheme in steadystate calculations to timemarching schemes for transient calculations.
Solution Method  DensityBased Solver  Explicit Formulation  DensityBased Solver  Implicit Formulation 
SteadyState 
 3stages RungeKutta
 local time step  timederivative preconditioning  FAS 
 local time step
 timederivative preconditioning 
Unsteady  Explicit Time Stepping 
 4stages RungeKutta
 global time step  no timederivative preconditioning  No FAS 
N/A 
Unsteady  Implicit Time Stepping (dualtime formulation) First Order 
 dualtime formulation
 Physical time: first order Euler backward  preconditioned pseudotime derivative  inner iteration: explicit pseudotime marching, 3stage RungeKutta 
 dualtime formulation
 Physical time: first order Euler backward  preconditioned pseudotime derivative  inner iteration: implicit pseudotime marching 
Unsteady  Implicit Time Stepping (dualtime formulation) Second Order 
 dualtime formulation
 Physical time: second order Euler backward  preconditioned pseudotime derivative  inner iteration: explicit pseudotime marching, 3stage RungeKutta 
 dualtime formulation
 Physical time: second order Euler backward  preconditioned pseudotime derivative  inner iteration: implicit pseudotime marching 