306
9. Turbulent Flows
The turbulent diffusion terms are usually modeled using a gradient diffusion type of approximation. In the simplest case, the diffusivity is assumed to
be isotropic and is simply a multiple of the eddy viscosity used in the models
discussed earlier. In recent years, anisotropic and nonlinear models have been
suggested. Again, no attempt is made to discuss them in detail here.
In three dimensions, Reynolds stress models require the solution of seven
partial differential equations in addition to the equations for the mean flow.
Still more equations are needed when scalar quantities need to be predicted.
These equations are solved in a manner similar to that for the k-E equations.
The only additional issue is that when the Reynolds-averaged Navier-Stokes
equations are solved together with a Reynolds stress model they are even
stiffer than those obtained with the k-E equations and even more care is
required in their solution and the calculations usually converge more slowly.
While there is no doubt that Reynolds stress models have greater potential
to represent turbulent flow phenomena more correctly than the two-equation
models (see HadiiC, 1999, for some illustrative examples), their success so far
has been moderate. Excellent results have been obtained for some flows in
which k-E models perform badly (eg., swirling flows, flows with stagnation
points or lines, flows with strong curvature and with separation from curved
surfaces, etc.); however, in some flows their performance is hardly better at
all. There is a lot of current research in this field, and new models are often
proposed. Which model is best for which kind of flow (none is expected to
be good for all flows) is not yet clear, partly due to the fact that in many
attempts to answer this question numerical errors were too large to allow clear
conclusions to be reached (Bradshaw et al., 1994). In many workshops on the
subject of evaluation of turbulence models, the differences between solutions
produced by different authors using supposedly the same model are often
as large if not larger than the differences between the results of the same
author using different models. This is one reason why numerical accuracy
is emphasized in this book; its importance can not be overemphasized and
constant attention to it is required.
9.6 Very Large Eddy Simulation
Researchers have attempted to build LES from the ground up starting with
simple flows and going on to increasingly more complex flows in small steps. In
most of these simulations, a large fraction of the energy of the turbulence was
in the resolved scales and good results were achieved. Success is not assured
for complex flows; this would be the case if sufficient computer resources were
available, but that is not always the case.
The objective of flow simulation is usually to obtain a few selected properties of the flow a t minimum cost. It is wise to use the simplest tool that
will provide the desired results but it is not easy to know in advance how
well each method will work. Clearly, if RANS methods are successful, there
9. Turbulent Flows
The turbulent diffusion terms are usually modeled using a gradient diffusion type of approximation. In the simplest case, the diffusivity is assumed to
be isotropic and is simply a multiple of the eddy viscosity used in the models
discussed earlier. In recent years, anisotropic and nonlinear models have been
suggested. Again, no attempt is made to discuss them in detail here.
In three dimensions, Reynolds stress models require the solution of seven
partial differential equations in addition to the equations for the mean flow.
Still more equations are needed when scalar quantities need to be predicted.
These equations are solved in a manner similar to that for the k-E equations.
The only additional issue is that when the Reynolds-averaged Navier-Stokes
equations are solved together with a Reynolds stress model they are even
stiffer than those obtained with the k-E equations and even more care is
required in their solution and the calculations usually converge more slowly.
While there is no doubt that Reynolds stress models have greater potential
to represent turbulent flow phenomena more correctly than the two-equation
models (see HadiiC, 1999, for some illustrative examples), their success so far
has been moderate. Excellent results have been obtained for some flows in
which k-E models perform badly (eg., swirling flows, flows with stagnation
points or lines, flows with strong curvature and with separation from curved
surfaces, etc.); however, in some flows their performance is hardly better at
all. There is a lot of current research in this field, and new models are often
proposed. Which model is best for which kind of flow (none is expected to
be good for all flows) is not yet clear, partly due to the fact that in many
attempts to answer this question numerical errors were too large to allow clear
conclusions to be reached (Bradshaw et al., 1994). In many workshops on the
subject of evaluation of turbulence models, the differences between solutions
produced by different authors using supposedly the same model are often
as large if not larger than the differences between the results of the same
author using different models. This is one reason why numerical accuracy
is emphasized in this book; its importance can not be overemphasized and
constant attention to it is required.
9.6 Very Large Eddy Simulation
Researchers have attempted to build LES from the ground up starting with
simple flows and going on to increasingly more complex flows in small steps. In
most of these simulations, a large fraction of the energy of the turbulence was
in the resolved scales and good results were achieved. Success is not assured
for complex flows; this would be the case if sufficient computer resources were
available, but that is not always the case.
The objective of flow simulation is usually to obtain a few selected properties of the flow a t minimum cost. It is wise to use the simplest tool that
will provide the desired results but it is not easy to know in advance how
well each method will work. Clearly, if RANS methods are successful, there