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The coupled set of governing equations (Equation 18.5-6) in ANSYS 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. To learn how to apply the two formulations, see this section in the separate User's Guide.
Explicit Formulation
In the explicit scheme a multi-stage, time-stepping algorithm [
146] is used to discretize the time derivative in Equation
18.5-6. The solution is advanced from iteration
to iteration
with an
-stage Runge-Kutta scheme, given by
where
and
is the stage counter for the
-stage scheme.
is the multi-stage coefficient for the
stage. The residual
is computed from the intermediate solution
and, for Equation
18.5-6, is given by
The time step
is computed from the CFL (Courant-Friedrichs-Lewy) condition
where
is the cell volume,
is the face area, and
is the maximum of the local eigenvalues defined by Equation
18.5-9.
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 18.5.4).
The convergence rate of the explicit scheme can be accelerated through use of the full-approximation storage (FAS) multigrid method described in Section 18.6.4.
By default, ANSYS FLUENT uses a 3-stage Runge-Kutta scheme based on the work by Lynn [ 211] 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:
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(18.5-15) |
This equation can be solved with the following Jacobi iteration:
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(18.5-16) |
Two Jacobi iterations are usually sufficient to allow doubling the time step with a value of
.
Implicit Formulation
In the implicit scheme, an Euler implicit discretization in time of the governing equations (Equation 18.5-6) is combined with a Newton-type linearization of the fluxes to produce the following linearized system in delta form [ 370]:
The center and off-diagonal coefficient matrices,
and
are given by
and the residual vector
and time step
are defined as in Equation
18.5-13 and Equation
18.5-14, respectively.
Equation 18.5-17 is solved using either Incomplete Lower Upper factorization (ILU) by default or symmetric point Gauss-Seidel algorithm, in conjunction with an algebraic multigrid (AMG) method (see Section 18.6.3) adapted for coupled sets of equations.
Explicit relaxation can improve the convergence to steady state of the implicit formulation. By default, explicit relaxation is enabled for the implicit solver and uses a factor of
0.75. You can specify a factor
to control the amount that the solution vector
changes between iterations after the end of the algebraic multigrid (AMG) cycle:
By specifying a value less than the default value of
1 for
, the variables in the solution vector will be under-relaxed and the convergence history can be improved. For information on how to set this value, see
this section in the separate
User's Guide.
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Note that explicit relaxation is available for the density-based implicit solver in steady state mode only.
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