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 Nick Georgiadis February 10, 2000 16:51

help with wall functions

I am trying to use the wall-function approach of Ota and Goldberg, in which none of the flow variables are reset, and in fact, a no-slip surface is used. The essence of the model is that you use a law of the wall (they propose using the White-Cristoph compressible law of the wall) to solve for the wall shear stress, and then simply use this shear stress in the solution scheme. Any turbulence model (even an algebraic model like I am using, Cebeci-Smith) may then be employed with this method.

My questions to anyone familiar with this (or a similar) approach is:

(1) When solving the law of the wall expression, the iterated quatity is friction velocity. This quantity is then squared and multiplied by the density to get the wall shear stress. How can this (especially through the squaring process) be expected to get an accurate wall shear stress?

(2) Apart from solving for the wall shear stress, the (solved) shear stress throughout the rest of the bdy. layer should be constant, particularly near the wall. Can this happen in practice when using such a wall function? especially with a higher order scheme that would use the flow variables at the wall point and one off the wall?

This wall function approach has been used with success by several authors, so I am sure my doubts are due to my ignornace on the topic. As a result, any explanations to my 2 questions will be very appreciated.

 John C. Chien February 10, 2000 17:56

Re: help with wall functions

(1). Why not send the e-mail directly to the author(s) of the method? (2). Unless you have the name of the reference paper, it is somewhat difficult to know what you are talking about. (3). In general, the law of the wall is a one parameter family of curves, see Schlichting's Theory of Boundary Layer book. The unknown parameter is the v*, the friction velocity. It is the dimensionless wall shear stress. (4). This log-law of the wall can be derived from the assumption of constant shear stress throughout the boundary layer. I think, the book also covers this part. (5). So, the constant shear stress assumption, and the universal law of the wall are well-known. (6). Exactly how this should be used in a turbulence model varies greatly depending upon the model and the numerical method used. So, the best place is the author(s) of the model, I think.

 Bob Anderson February 19, 2000 03:23

Re: help with wall functions

Your first question confuses me. u_\tau is by definition equal to sqrt(\tau_w/\rho). So if you can find u_\tau and you know \rho, you know \tau_w. Its that simple.

The answer to your second question is that if your pressure gradients are small, constant stress is a good assumption through the log layer, in fact the law-of-the-wall is based upon this assumption. If your pressure gradients are not small, then you have to do something else, like add a first order correction to \tau_w.

Bob

 Nick Georgiadis February 20, 2000 10:28

Re: help with wall functions

Bob,

Thank you for your response. Regarding my first question, I am not challenging whether the formulation is physically correct, I know u-tau = sqrt(tauw / rho).....the problem is when you rewrite this, tauw = rho * (u-tau)**2 - I am wondering as to how accurate this can be expected to be in a computational sense, since you are indirectly solving for u-tau, then squaring this value (and as a result, any innacuracies of u-tau are greatly magnified) Then you use this value for the wall-shear where it very much determines the bdy. layer characteristics.

I should mention that at the time I posted the first message, my bdy. layers (Mach 1.36 and Mach 1.91 flat plates with zero pressure gradient) were not correct - but then I took the suggestion of a co-worker to completely turn off the artificial dissipation at the points near the wall, and this substantially improved the results. The behavior of the first point off the wall seems critical in getting the wall function method to work - you have to be careful with your solution scheme (how you do the differencing) and any artificial dissipation.

 Bob Anderson February 20, 2000 18:07

Re: help with wall functions

If you converge your iteration for u_\tau to machine accuracy, then it should be no less accurate than any other squared quanitity in your calculation, of which there are likely dozens or hundreds, so I would not worry about it.

Artificial dissipation is an important issue for boundary layer calculations with or without wall functions, so yes, you do have to be careful there. Seems like you have it under control now.

Out of curiousity, how do you handle the v component of velocity?

Bob

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