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Old   November 18, 1998, 03:51
Default Diffusive k-e ?
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It is often said that the standard k-e model is very diffusive. I take this to mean that it often overpredicts eddy viscosity. Does any one know under what conditions this occurs, why this is so and what modifications to the standard k-e model have been implemented and tested to resolve this problem.

Many thanks.
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Old   November 18, 1998, 04:11
Default Re: Diffusive k-e ?
Jonas Larsson
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It commonly occurs in flows with large normal strain, for example in stagnation regions, in strongly accelerated regions and across shocks. This problem affects most models based on the Boussinesq assumption, but it is especially severe for k-epsilon models since the epsilon equation in addition also has a tendency to convect too large length-scales into the boundary layers in stagnation flows.

There are many ad-hoc fixes to this, which you should use depends on your applications. A few things that have been done are:

Different types of varying C_mu (can be base on realizability a'la Shih and others or on RNG theory or on EASM ideas tracing back to Rodi's thesis).

Many other limiters to avoid unphysical length or time scales are also often used. One common limiter is the one developed by Durbin. It has been used succefully for many kinds of flows, including heat transfer simulations (where this is a big problem).

The lenght-scale problem with the epsilon equation can be reduced by using for example the "Yap-correction", which is a source term in the epsilon eqation which drives the lenght scale to its eqilibrium value.

Many people have also succefully used more advanced non-linear models to solve these problems - many different variants exist, but I suspect you're not looking for something like this so I'll stop here.

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Old   November 18, 1998, 09:44
Default Re: Diffusive k-e ?
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The k-e model is diffusive because it models turbulent transport as a diffusive process like laminar viscosity (ie it will act to reduce mean gradients). In many cases this is not an appropriate model. I would doubt the size of the eddy viscosity coefficient was being criticized more the fact it was being used in the first place (but I do not know the context so I may be wrong). However, below is an illustration of a simple mechanism for generating an eddy viscosity coefficient that is too big.

Consider a plane stagnating 2D flow (no mean gradients in z) where dU/dx = -dV/dy from continuity. For isotropic turbulence (no shear and uu=vv=ww) the mean turbulent production term is given by:

P = -uu.dU/dx - vv.dV/dy

which is zero from continuity.

However, an eddy viscosity model would give you:

P = -(mt.2.dU/dx - 2/3.q)dU/dx - (mt.2.dV/dy - 2/3.q)dV/dy

= -mt.2.( (dU/dx)**2 + (dV/dy)**2 )

which, in the present of significant turbulence (ie reasonably big turbulent viscosity, mt) is nothing like zero, feeds upon itself to generate turbulence, and can produce quite spectacular nonsense. If the level of turbulence being convected into the region is low the effect does not really get going in practice. Note that this has nothing to do with failings in the dissipation rate equation (the popular source for all failings with the k-e model).

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Old   November 18, 1998, 17:43
Default Re: Diffusive k-e ?
Jonas Larsson
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Which models/limiters/tricks would you recommend in order to reduce these problems?
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Old   November 19, 1998, 07:37
Default Re: Diffusive k-e ?
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As a general recommendation - none. I would recommend gaining knowledge about how your simulation is affected by your set of assumptions and extracting the desired information in that light. To seek to improve the accuracy of simulations is certainly a worthwhile objective but it is a very different activity to extracting information for engineering purposes.

In my experience, reliable accuracy is only introduced into a simulation by introducing more physics (but I would view the peformance of the discretized convection term, for example, in the light of its physical properties). For example, a large proportion of the work in the 80s on Reynolds stress transport models did not really do this and was not particularly successful. Significant advantages of using a RST model over an eddy viscosity model are really only seen when the unmodelled terms (production/convection) are important to the physics of the problem (eg curved flows but not simple boundary layers). They all become unreliable when the flow involves, for example, substantial large scale turbulent motion (handled by modelled terms). The physics requires a more appropriate set of modelling assumptions, for example Large Eddy Simulation (which is also developing its fair share of non-physics based models...)

Having said that, one can obviously tune up models for certain situations. A good example of that would be some of the very simple turbulence models developed for aerofoils. But can one resist the urge to use them outside the very limited range in which they were designed work?

I seem to be wandering off topic because I cannot really answer the question. So I will stop before it degenerates into a rant.
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