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25.5.5 Unsteady Flows Solution Methods

For time-accurate calculations, explicit and implicit time-stepping schemes are available. (The time-implicit 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 global-time step), and with preconditioning disabled. By default, FLUENT uses a 4-stage Runge-Kutta scheme for unsteady flows.



Implicit Time Stepping (Dual-Time formulation)


The implicit-time stepping method (also known as dual-time formulation) is available in the density-based explicit and implicit formulation.

When performing unsteady simulations with implicit-time stepping (dual-time stepping), FLUENT uses a low Mach number time-derivative 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) [ 376, 276].

Here we introduce a preconditioned pseudo-time-derivative term into Equation  25.5-1 as follows:


 \frac{\partial }{\partial t} \int_V {\mbox{\boldmath$W$}}\, ... ...t d{{\mbox{\boldmath$A$}}} = \int_V {\mbox{\boldmath$H$}}\, dV (25.5-20)

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 left side of Equation  25.5-20 vanishes and Equation  25.5-1 is recovered.

The time-dependent term in Equation  25.5-20 is discretized in an implicit fashion by means of either a first- or second-order accurate, backward difference in time.

The dual-time formulation 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.

Table  25.5.1 summarizes all operation modes for the density-based solver from the iterative scheme in steady-state calculations to time-marching schemes for transient calculations.


Table 25.5.1: Summary of the Density-Based Solver
Solution Method Density-Based Solver - Explicit Formulation Density-Based Solver - Implicit Formulation


Steady-State
- 3-stages Runge-Kutta
- local time step
- time-derivative preconditioning
- FAS
-
- local time step
- time-derivative preconditioning

Unsteady -
Explicit Time Stepping
- 4-stages Runge-Kutta
- global time step
- no time-derivative preconditioning
- No FAS
N/A

Unsteady -
Implicit Time Stepping
(dual-time formulation)
First Order
- dual-time formulation
- Physical time: first order Euler backward
- preconditioned pseudo-time derivative
- inner iteration: explicit pseudo-time marching, 3-stage Runge-Kutta
- dual-time formulation
- Physical time: first order Euler backward
- preconditioned pseudo-time derivative
- inner iteration: implicit pseudo-time marching

Unsteady -
Implicit Time Stepping
(dual-time formulation)
Second Order
- dual-time formulation
- Physical time: second order Euler backward
- preconditioned pseudo-time derivative
- inner iteration: explicit pseudo-time marching, 3-stage Runge-Kutta
- dual-time formulation
- Physical time: second order Euler backward
- preconditioned pseudo-time derivative
- inner iteration: implicit pseudo-time marching


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© Fluent Inc. 2006-09-20