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9. Turbulent Flows
9.4 RANS Models
Engineers are normally interested in knowing just a few quantitative properties of a turbulent flow, such as the average forces on a body (and, perhaps,
its distribution), the degree of mixing between two incoming streams of fluid,
or the amount of a substance that has reacted. Using the methods described
above to compute these quantities is, to say the least: overkill. These methods
should only be used as a last resort, when nothing else succeeds or, occasionally, to check the validity of a model of the type described in this section which
produces less information. Because it is based on ideas proposed by Osborne
Reynolds over a century ago, it is called the Reynolds-averaged method.
In Reynolds-averaged approaches to turbulence, all of the unsteadiness
is averaged out i.e. all unsteadiness is regarded as part of the turbulence.
On averaging, the non-linearity of the Navier-Stokes equations gives rise to
terms that must be modeled, just as they did earlier. The complexity of
turbulence, which was discussed briefly above, makes it unlikely that any
single Reynolds-averaged model will be able to represent all turbulent flows so
turbulence models should be regarded as engineering approximations rather
than scientific laws.
9.4.1 Reynolds-Averaged Navier-Stokes (RANS) Equations
In a statistically steady flow, every variable can be written as the sum of a
time-averaged value and a fluctuation about that value:
where
Here t is the time and T is the averaging interval. This interval must be large
compared to the typical time scale of the fluctuations; thus, we are interested
in the limit of T co, see Fig. 9.10. If T is large enough, $ does not depend
on the time at which the averaging is started.
If the flow is unsteady, time averaging cannot be used and it must be
replaced by ensemble averaging. This concept was discussed earlier and is
illustrated in Fig. 9.10:
where N is the number of members of the ensemble and must be large enough
to eliminate the effects of the fluctuations. This type of averaging can be
applied to any flow. We use the term Reynolds averaging to refer to any of
9. Turbulent Flows
9.4 RANS Models
Engineers are normally interested in knowing just a few quantitative properties of a turbulent flow, such as the average forces on a body (and, perhaps,
its distribution), the degree of mixing between two incoming streams of fluid,
or the amount of a substance that has reacted. Using the methods described
above to compute these quantities is, to say the least: overkill. These methods
should only be used as a last resort, when nothing else succeeds or, occasionally, to check the validity of a model of the type described in this section which
produces less information. Because it is based on ideas proposed by Osborne
Reynolds over a century ago, it is called the Reynolds-averaged method.
In Reynolds-averaged approaches to turbulence, all of the unsteadiness
is averaged out i.e. all unsteadiness is regarded as part of the turbulence.
On averaging, the non-linearity of the Navier-Stokes equations gives rise to
terms that must be modeled, just as they did earlier. The complexity of
turbulence, which was discussed briefly above, makes it unlikely that any
single Reynolds-averaged model will be able to represent all turbulent flows so
turbulence models should be regarded as engineering approximations rather
than scientific laws.
9.4.1 Reynolds-Averaged Navier-Stokes (RANS) Equations
In a statistically steady flow, every variable can be written as the sum of a
time-averaged value and a fluctuation about that value:
where
Here t is the time and T is the averaging interval. This interval must be large
compared to the typical time scale of the fluctuations; thus, we are interested
in the limit of T co, see Fig. 9.10. If T is large enough, $ does not depend
on the time at which the averaging is started.
If the flow is unsteady, time averaging cannot be used and it must be
replaced by ensemble averaging. This concept was discussed earlier and is
illustrated in Fig. 9.10:
where N is the number of members of the ensemble and must be large enough
to eliminate the effects of the fluctuations. This type of averaging can be
applied to any flow. We use the term Reynolds averaging to refer to any of