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When running a well-posed CFD simulation, there are many factors that
govern your decision to declare the solution done. Residual reports, force
monitors, and overall balances are just a few of the factors that can
be used to make this judgment call. More important than the measuring
devices is what you want to achieve by running the simulation in the first
place. For example, deep convergence is required if you are an aerodynamicist
and want to get accurate predictions of lift and drag coefficients. Rough
convergence is acceptable, on the other hand, if you are simply looking
for approximate flow features.
The reason that solution convergence is an issue with all CFD software
is due to the iterative nature of the solution procedures used. In particular,
iteration is necessary to handle the non-linearity of the equations that
govern fluid flow, heat transfer, and related processes. For any given
conservation equation, an approximate solution is obtained at each iteration
that results in a small imbalance in the conservation statement. During
the course of the iterative solution algorithm, the imbalance in each
cell is a small, non-zero value that, under normal circumstances, decreases
as the solution progresses. This imbalance is called the residual. The
total residual for each variable across the entire solution domain is
the sum of the absolute values of the individual cell residuals. This
total residual is often scaled so that the residuals of different variables
can be compared or combined. Scaling factors are taken from the bulk flow
quantities or from the error at the start of the calculation.
These residuals have met the convergence criteria, but are still in decline.
A plot of drag coefficient during a simulation shows oscillation in the
early stages followed by a gradual leveling-off to a constant value as
the solution converges.
The convergence criteria are pre-set conditions on the residuals that
indicate that a certain level of convergence has been achieved. For example,
a criterion that the scaled residual for the x-momentum drop to 0.001
indicates that the overall error in this variable is about three orders
of magnitude less than the bulk x-momentum in the system. Variations on
this definition exist for certain variables or solver techniques, but
a common thread for all is that as the net residual declines, so does
the error in the solution.
Unfortunately, the reduction in residuals is not the only indicator of
convergence. A truly converged solution is one that is no longer changing
with successive iteration. If the residuals for all problem variables
fall below the convergence criteria but are still in decline, the solution
is still changing, to a greater or lesser degree. A better indicator occurs
when the residuals flatten in a traditional residual plot (of residual
value vs. iteration). This point, sometimes referred to as convergence
at the level of machine accuracy, takes time to reach, however, and may
be beyond your needs. For this reason, alternative tools such as reports
of forces, heat balances, or mass balances can be used instead.
In FLUENT, reports of integrated quantities at surfaces and boundaries
are often used to judge convergence. For example, mass flow rates at all
flow boundaries will add to zero for a converged solution. Note, however,
that if the outflow boundary condition is used, the net flux will be
zero even if the solution is not converged. This is because a guessed
mass outflow, based on the total inflow, will be temporarily set at these
boundaries to help resolve the flow field during the solution. If the
discrete phase model (DPM) is being used and there is mass transfer from
the discrete phase to the gas phase, an imbalance will be reported, even
if the solution is converged. The difference can be accounted for by changes
in the discrete phase mass flow rate through the relevant boundaries in
the particle tracking summary report. Heat balances are also computed
from integral reports, but the reports should include all boundaries,
not just flow boundaries. Volumetric sources of heat inside the solution
domain can be accounted for through the DPM summary or through heat source
reports in conducting wall zones.
Force and moment reports can be used to compute lift and drag on a body
in an external flow simulation, or to compute torques in rotational flows.
Numerical reports can be examined when the solution process has been halted,
or plots can be generated during the calculation and updated with each
iteration. During the solution, graphical reports such as these tend to
exhibit oscillating behavior. As the solution approaches convergence,
the plotted curves flatten, indicating that the flow field has stabilized
with regard to the selected flow features. Graphical reports can also
be used to monitor the progression of solution-sensitive quantities in
selected regions inside the domain.
Reducing the underrelaxation factors to extremely low values, say 0.01,
will cause the residuals to drop. Don't be fooled by this behavior,
since it does not usually mean that a correct converged solution has been
reached. The relaxation factors are included in the scaling, and changes
to these values will cause step changes in the scaled residual plots.
Use the tools described above to ensure that the variables in your solution
are appropriately balanced to suit your needs.
As you consider these issues during your next simulation, keep in mind
that the accuracy of your well-converged solution is also dependent on
the quality of your grid and the sophistication of the models you choose.
A coarsely-gridded two-dimensional simulation of a complex three-dimensional
system will not provide you with precise flow features, no matter how
deeply converged the solution is!
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