336
11. Efficiency and Accuracy Improvement
independent solution within five significant digits, although the errors in the
solutions are an order of magnitude different.
Note that the error can be computed for integral quantities (drag, lift, etc.)
as well as for field values but the order of convergence may not be the same
for all quantities. It is usually equal to the theoretical order (e.g. second)
for problems with smooth solutions, e.g. laminar flows. When complicated
models (for turbulence, combustion, two-phase flow etc.) or schemes with
switches or limiters are used, the definition of order may be difficult. However,
it is not absolutely necessary to compute quantities like the order p or what
Roache (1994) calls the grid convergence index; it is sufficient to show the
change in the computed quantity of interest for a series of grids (preferably
three). If the change is monotonic and the difference decreases with grid
refinement, one can easily estimate where the grid-independent solution lies.
Of course, one should also use the Richardson extrapolation to estimate the
grid-independent solution.
Note also that the refinement need not extend over the whole domain. If
the estimate indicates that the error is much smaller in some regions than
elsewhere, local refinement can be used. This is particularly true for flows
around bodies, where high resolution is needed only in the vicinity of the
body and in the wake. Methods using local refinement strategies are very efficient. However, care is needed; if the grid is not refined where the truncation
errors are large, large errors may occur elsewhere, since the errors are subject
to the same transport processes (convection and diffusion) as the variables
themselves.
Estimation of Modeling Errors. Modeling errors are the most difficult
ones to estimate; to do so, we need data on the real flow. In most cases, data
are not available. Therefore, modeling errors are usually estimated only for
some test cases, for which detailed and accurate experimental data are available, or for which accurate simulation data exist (e.g. large-eddy or direct
numerical simulation data). In any case, before one can compare a computation with experiment, the iteration and discretization errors should be
analyzed and shown to be small enough. In some cases the modeling and
discretization errors cancel each other, so that results on a coarse grid may
agree better with experimental data than ones obtained on a finer grid. Experimental data should therefore not be used to verify the code; one must
use systematic analysis of the results. Only when it is proven beyond reasonable doubt that the results do converge towards a grid-independent solution
and that the discretization errors are small enough, can one proceed with
comparison of numerical solution and experimental data.
It is also important to bear in mind that the experimental data are only
approximate, and that the measurement and data processing errors can be
significant. They may also contain significant systematic errors. However,
they are indispensable for the validation of models. One should compare
computational results only with experimental data of high accuracy. Analysis
11. Efficiency and Accuracy Improvement
independent solution within five significant digits, although the errors in the
solutions are an order of magnitude different.
Note that the error can be computed for integral quantities (drag, lift, etc.)
as well as for field values but the order of convergence may not be the same
for all quantities. It is usually equal to the theoretical order (e.g. second)
for problems with smooth solutions, e.g. laminar flows. When complicated
models (for turbulence, combustion, two-phase flow etc.) or schemes with
switches or limiters are used, the definition of order may be difficult. However,
it is not absolutely necessary to compute quantities like the order p or what
Roache (1994) calls the grid convergence index; it is sufficient to show the
change in the computed quantity of interest for a series of grids (preferably
three). If the change is monotonic and the difference decreases with grid
refinement, one can easily estimate where the grid-independent solution lies.
Of course, one should also use the Richardson extrapolation to estimate the
grid-independent solution.
Note also that the refinement need not extend over the whole domain. If
the estimate indicates that the error is much smaller in some regions than
elsewhere, local refinement can be used. This is particularly true for flows
around bodies, where high resolution is needed only in the vicinity of the
body and in the wake. Methods using local refinement strategies are very efficient. However, care is needed; if the grid is not refined where the truncation
errors are large, large errors may occur elsewhere, since the errors are subject
to the same transport processes (convection and diffusion) as the variables
themselves.
Estimation of Modeling Errors. Modeling errors are the most difficult
ones to estimate; to do so, we need data on the real flow. In most cases, data
are not available. Therefore, modeling errors are usually estimated only for
some test cases, for which detailed and accurate experimental data are available, or for which accurate simulation data exist (e.g. large-eddy or direct
numerical simulation data). In any case, before one can compare a computation with experiment, the iteration and discretization errors should be
analyzed and shown to be small enough. In some cases the modeling and
discretization errors cancel each other, so that results on a coarse grid may
agree better with experimental data than ones obtained on a finer grid. Experimental data should therefore not be used to verify the code; one must
use systematic analysis of the results. Only when it is proven beyond reasonable doubt that the results do converge towards a grid-independent solution
and that the discretization errors are small enough, can one proceed with
comparison of numerical solution and experimental data.
It is also important to bear in mind that the experimental data are only
approximate, and that the measurement and data processing errors can be
significant. They may also contain significant systematic errors. However,
they are indispensable for the validation of models. One should compare
computational results only with experimental data of high accuracy. Analysis