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26.4.4 Temporal Discretization for Unsteady Flows

For time-accurate calculations, explicit and implicit time-stepping schemes are available. (The implicit approach is also referred to as "dual time stepping''.)

Explicit Time Stepping

In the explicit time stepping approach, the explicit scheme described above is employed, using the same time step in each cell of the domain, and with preconditioning disabled.

Dual Time Stepping

When performing unsteady simulations with the implicit-time formulation (dual-time stepping), FLUENT uses a low Mach number preconditioner (for all flow regimes that span from zero speed to Mach one). To provide for efficient, time-accurate solution of the preconditioned equations, FLUENT employs a dual time-stepping, multi-stage scheme to produce accurate results both for pure convective processes (e.g., simulating unsteady turbulence) and for acoustic processes (e.g., simulating wave propagation) [ 85, 288]. Here we introduce a preconditioned pseudo-time-derivative term into Equation  26.4-1 as follows:

 \frac{\partial }{\partial t} \int_V {\mbox{\boldmath$W$}}\, ... ...t d{{\mbox{\boldmath$A$}}} = \int_V {\mbox{\boldmath$H$}}\, dV (26.4-18)

where $t$ denotes physical-time and $\tau$ is a pseudo-time used in the time-marching procedure. Note that as $\tau \rightarrow \infty$, the second term on the LHS of Equation  26.4-19 vanishes and Equation  26.4-1 is recovered.

The time-dependent term in Equation  26.4-19 is discretized in an implicit fashion by means of either a first- or second-order accurate, backward difference in time. This is written in semi-discrete form as follows:

\begin{eqnarray*} \left[\frac{{\Gamma}}{\Delta \tau} + \frac{\epsilon_0}{\Delta ... ...ldmath$W$}}^{n} + \epsilon_2 {\mbox{\boldmath$W$}}^{n-1} \right) \end{eqnarray*}

where { $\epsilon_0 = \epsilon_1 = 1/2, \epsilon_2 = 0$} gives first-order time accuracy, and { $\epsilon_0 = 3/2, \epsilon_1 = 2, \epsilon_2 = 1/2$} gives second-order. $k$ is the inner iteration counter and $n$ represents any given physical-time level.

The pseudo-time-derivative is driven to zero at each physical time level by a series of inner iterations using either the implicit or explicit time-marching algorithm. Throughout the (inner) iterations in pseudo-time, ${\mbox{\boldmath$W$}}^{n}$ and ${\mbox{\boldmath$W$}}^{n-1}$ are held constant and ${\mbox{\boldmath$W$}}^{k}$ is computed from ${\mbox{\boldmath$Q$}}^{k}$. As $\tau \rightarrow \infty$, the solution at the next physical time level ${\mbox{\boldmath$W$}}^{n+1}$ is given by ${\mbox{\boldmath$W$}}({\mbox{\boldmath$Q$}}^k)$.

Note that the physical time step $\Delta t$ is limited only by the level of desired temporal accuracy. The pseudo-time-step $\Delta \tau$ is determined by the CFL condition of the (implicit or explicit) time-marching scheme.

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© Fluent Inc. 2005-01-04