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February 2, 1999, 02:56 
Wall functions

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February 2, 1999, 14:42 
Re: Wall functions

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The law of the wall approach is a general procedure, therefore, in principle, it can be used in many other different problems. First of all, the method assumes that a oneparameter family of solutions ( curves) exist right next to the wall and extending to some distance away from the wall. Normally, this family of curves ( solutions) are expressed in terms of the combination of the linear sublayer profile, transition zone profile and the loglaw profile. This profile can be piecewise continuous depending upon the equations ( formula ) used. So, once you know the value of the parameter, the solution ( or the curve) is uniquely determined. So the missing link is the parameter in the formula. Second, we assume that the numerical solution with a turbulence model will contain the exact same law of the wall solution behavior near the wall. This is a big assumption. Assuming this is true, then, there will be a overlapping region between the numerical solution and the oneparameter family soultion. In this commonly valid region, the solution should be continuous as long as the value of the parameter is known. In principle, this region must be shared by the law of the wall solution and the numerical solution. In the actual implementation of the method, if the numerical solution contains only the high Reynolds number turbulence model solution, then, it can produce only the loglaw portion of the solution. Therefore, in this case, the matching must be carried out in this commonly valid region ( say 20 < y+ > 200 )only. Notice that there is a gap between this matching point and the wall which is replaced by the analytical law of the wall profile. The complete solution is now replaced by the numerical solution away from the wall up to the matching point and the analytical solution from the matching point to the wall. Suppose that a low Reynolds number turbulence model is used in the numerical solution procedure, and assuming that the near wall solution shares the same identical solution as the law of the wall analytical solution, then the matching point can be anywhere in this commonly valid region. So far, published low Reynolds number turbulence models still have some difficulties in meeting this requirement.( related to the common problem of kepsilon two equation low Re model and the epsilonequation model). So, as long as you have a low Re model which can produce the same solution as the law of the wall curve, the wall function procedure can be applied to the low Re modelling. In summary, for the law of the wall procedure to work, both the numerical solution and the analytical law of the wall solution must be identical in the matching "region". You can not match a logprofile numerical solution to a linear analytical solution near the wall. Up to now, the law of the wall profile( curves) is derived experimentally for boundary layer flows. The ability for a low Re turbulence model to reproduce the identical law of the wall profile has been only demonstrated for boundary layer flow with zero pressure gradient.( this is the case where the turbulence model parameters are adjusted). Near wall turbulence modelling is still the most important problem in CFD.


February 3, 1999, 02:06 
Re: Wall functions

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The wall function method is a crude technique to satisfy the wall boundary condition for high Reynolds number flows. My experience with this type of approach is that it is highly unreliable in the transition region as Mr. Chien has mentioned that there is a wide transitional region where the matching is to be done. On the other hand for low Reynolds number kepsilon model such type of technique (wall function) is not needed because we can apply the no slip boundary condition at the wall. That is u=k=epsilon=0 at y=0. There are numerous low Reynolds number models available in the literature, but from my experience I would recommend the original model by Jones and Launder (1972). I have applied this model to oscillatory boundary layers (contrary to what Mr. Chien thinks about these models) and an excellent agreement was found not only with the experimental data but DNS data as well. For a review of some of these models please refer to February 03, 1999
1. Tanaka, H. and Sana, A.(1995), Numerical study on transition to turbulence in a wave boundary layer, in Sediment Transport Mechanisms in Coastal Environments and Rivers , Ed. M. Belorgey, R.D. Rajaona and J.F.A. Sleath , World Scientific, pp.1425. 2. Sana, A. and Tanaka, H., (1996), The testing of low Reynolds number kepsilon models by DNS data for an oscillatory boundary layer, The 6th Int. Symp. on Flow Modelling and Turbulence Measurements, Sep. 1996, Florida, U.S.A. pp.363370. I hope it would be better if you try low Reynolds number model in your study first and look for the modifications to be made for the particular case of your study. 

February 3, 1999, 13:50 
Re: Wall functions

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I'm glad to know that excellent results for the oscillating boundary layer problem have been obtained using the original Jones and Launder model ( low Re model). But we are not so lucky for flows with strong adverse pressure gradient, or flow separations. Also, in 3D flows, wall function approach is still the only practical approach.


February 3, 1999, 20:47 
Re: Wall functions

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I agree with you about the performance of low Reynolds number models in the flow under strong adverse pressure gradients. But in my opinion that should not be the excuse to use wall function method. Of course this method is practical because it is less accurate. Nowadays the term 'practical' is used in this sense. We should look for better models. Why I think low Reynolds number model is better? Because of its ability to reproduce flow properties in the whole region under consideration.


February 4, 1999, 14:02 
Re: Wall functions

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The "practical approach " I mentioned simply means that " the approach can be used on the routine basis". In this case, " the wall function approach is being used on the routine basis for solving 3D problems ". The reason behind is that the use of low Reynolds number model to cover the whole flow field in 3D problems requires large computer memory and fast CPU. Based on my experience, a medium size 3D problem can easily take 500meg to 1 giga memory using wall function approach. The CPU time required on a typical workstation is about a couple of days for incompressible flows and a couple of weeks for compressible flows calculation. For a 3D design code ( compressible NavierStokes), the CPU time can be reduced to one to three days because of the well defined geometry. When the low Re model is used in a 3D calculation, it can easily become a research project by itself ( or something like a PhD dissertation). Based on my experience ( I started using the low Re model in my code in early 70's), in many cases the wall function results are more reliable than the low Re model results. Sometimes, the results from the low Re model is unpredictable.( here we are not talking about the finetuned low Re model for a particular application.)


February 5, 1999, 02:16 
Re: Wall functions

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Hi, A low Reynolds number model essentially does away with the requierment of wall functions. This is not preferred sometimes because of the enormous computer time required to solve near wall flows( near the wall where the gradients are very steep, one needs to have very fine grids to resolve them). If you are solving an industrial problem in three dimensions, it is advisable to use wall function to save computer time.


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