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25.5.4 Steady-State Flow Solution Methods

The coupled set of governing equations (Equation  25.5-6) in FLUENT is discretized in time for both steady and unsteady calculations. In the steady case, it is assumed that time marching proceeds until a steady-state solution is reached. Temporal discretization of the coupled equations is accomplished by either an implicit or an explicit time-marching algorithm. These two algorithms are described below.

Explicit Formulation

In the explicit scheme a multi-stage, time-stepping algorithm [ 161] is used to discretize the time derivative in Equation  25.5-6. The solution is advanced from iteration $n$ to iteration $n + 1$ with an $m$-stage Runge-Kutta scheme, given by

\begin{eqnarray*} {\mbox{\boldmath$Q$}}^{0} &=& {\mbox{\boldmath$Q$}}^{n} \\ \D... ...1} \\ {\mbox{\boldmath$Q$}}^{n+1} &=& {\mbox{\boldmath$Q$}}^{m} \end{eqnarray*}

where $\Delta {\mbox{\boldmath$Q$}}^{i} \equiv {\mbox{\boldmath$Q$}}^{i} - {\mbox{\boldmath$Q$}}^{n}$ and $i = 1,2,\ldots,m$ is the stage counter for the $m$-stage scheme. $\alpha_i$ is the multi-stage coefficient for the $i^{th}$ stage. The residual ${{\mbox{\boldmath$R$}}}^{i}$ is computed from the intermediate solution ${\mbox{\boldmath$Q$}}^{i}$ and, for Equation  25.5-6, is given by

 {\bf R}^{i} = \sum^{N_{\rm faces}} \left({\mbox{\boldmath$F$... ...) \right) \cdot {\mbox{\boldmath$A$}}- V {\mbox{\boldmath$H$}} (25.5-13)

The time step $\Delta t$ is computed from the CFL (Courant-Friedrichs-Lewy) condition

 \Delta t = \frac{2 \mbox{CFL} \cdot V}{\sum_f {\lambda_{\rm f}}^{max} A_f} (25.5-14)

where $V$ is the cell volume, $A_f$ is the face cell, and ${\lambda_{\rm f}}^{max}$ is the maximum of the local eigenvalues defined by Equation  25.5-9.

For steady-state solutions, convergence acceleration of the explicit formulation can be achieved with the use of local time stepping, residual smoothing, and full-approximation storage multigrid.

Local time stepping is a method by which the solution at each control volume is advanced in time with respect to the cell time step, defined by the local stability limit of the time-stepping scheme.

Residual smoothing, on the other hand, increases the bound of stability limits of the time-stepping scheme and hence allows for the use of a larger CFL value to achieve fast convergence (Section  25.5.4).

The convergence rate of the explicit scheme can be accelerated through use of the full-approximation storage (FAS) multigrid method described in Section  25.6.4.

By default, FLUENT uses a 3-stage Runge-Kutta scheme based on the work by Lynn [ 225] for steady-state flows that use the density-based explicit solver.

Implicit Residual Smoothing

The maximum time step can be further increased by increasing the support of the scheme through implicit averaging of the residuals with their neighbors. The residuals are filtered through a Laplacian smoothing operator:

 \bar{R}_i = R_i + \epsilon \sum (\bar{R}_j - \bar{R}_i) (25.5-15)

This equation can be solved with the following Jacobi iteration:

 \bar{R}_i^{m} = \frac{R_i + \epsilon \sum \bar{R}_j^{m-1}} {1 + \epsilon \sum 1} (25.5-16)

Two Jacobi iterations are usually sufficient to allow doubling the time step with a value of $\epsilon=0.5$.

Implicit Formulation

In the implicit scheme, an Euler implicit discretization in time of the governing equations (Equation  25.5-6) is combined with a Newton-type linearization of the fluxes to produce the following linearized system in delta form [ 394]:

 \left[{\rm D} + \sum_{j}^{N_{\rm faces}} {\rm S}_{j,k} \righ... ... \Delta {\mbox{\boldmath$Q$}}^{n+1} = -{\mbox{\boldmath$R$}}^n (25.5-17)

The center and off-diagonal coefficient matrices, ${\rm D}$ and ${\rm S}_{j,k}$ are given by

$\displaystyle {\rm D}$ $\textstyle =$ $\displaystyle \frac{V}{\Delta t} \, {\Gamma} + \sum_{j}^{N_{\rm faces}} {\rm S}_{j,i}$ (25.5-18)
$\displaystyle {\rm S}_{j,k}$ $\textstyle =$ $\displaystyle \left(\frac{\partial {\mbox{\boldmath$F$}}_j}{\partial {\mbox{\bo... ...{\partial {\mbox{\boldmath$G$}}_j}{\partial {\mbox{\boldmath$Q$}}_k}\right) A_j$ (25.5-19)

and the residual vector ${\mbox{\boldmath$R$}}^n$ and time step $\Delta t$ are defined as in Equation  25.5-13 and Equation  25.5-14, respectively.

Equation  25.5-17 is solved using either Incomplete Lower Upper factorization (ILU) by default or asymmetric point Gauss-Seidel algorithm, in conjunction with an algebraic multigrid (AMG) method (see Section  25.6.3) adapted for coupled sets of equations.

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