Modification of Mixing Length Damping Function to include Surface Roughness
 

Surface roughness has been incorporated into the van Driest damping function in studies by van Driest himself (1956) and more recently by Krogstad (1991) and Crimaldi et al (2006).

These models appear to be directed primarily at external boundary layer flows with minimal pressure gradients. I get poor results when I apply them to pipe flow and do not recover the friction factors corresponding to the Colebrook model for rough pipes. The problem is exacerbated for high k+ values.

Are there any other studies that have considered the issue of how to tweak the mixing length to accommodate surface roughness in internal flows with strong streamwise pressure gradients?

Thanks

References:

1) E. Van Driest, “On turbulent flow near a wall,” J. Aeronaut. Sci. 23, 1007, 1956
2) P-A Krogstad, "Modification of the van Driest Damping Function to Include the Effects of Surface Roughness" AIAA Journal, vol 29, 1990.
3) J.P Crimaldi, J.R. Kossef and S.G. Monismith, "A mixing-length formulation for the turbulent Prandtl number in wall-bounded flows with bed roughness and elevated scalar sources", Physics of Fluids, vol 18 095102, 2006

Does it need to be the mixing length specifically? Otherwise you could give a look at the works on the Analytical Wall Function at Manchester and those derived from them.

For example, I haven't read it but, you could give a look at this:

https://hal.archives-ouvertes.fr/hal-01879569/document

The point is that including non equilibrium effects into the mixing length model is already a task in itself. I know of this work only:

https://www.researchgate.net/publica...ssure_gradient

On top of this, you need to add the roughness effect.

Admittedly, I am no expert (or a fan) of mixing length models, maybe it has been done already...