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.
The general form of Prandtl's Mixing length model is
$$ l_m = F(y)D(y_+) $$
where $y$ and $y_+$ are distances from the wall in physical and wall coordinates. For pipe flow, the Nikuradse quadratic model for $F(y)$ is suggested. The damping function $D(y)$ as proposed by van Driest has been modified to account for surface roughness as $$ D(y_+)=1-G(k_+)e^{-\frac{y_+}{A_+}}+H(y_+,k_+) $$
where $A_+=26$ seems to be an almost universally accepted value, and the roughness term $H(y_+,k_+)$ has been modeled by various researchers as follows:
van Driest (1956): $\hspace{2cm}G(k_+)=1\hspace{1cm}H(y_+,k_+)=e^{-\frac{60}{k_+}\frac{y_+}{A_+}}$
Krogstad (1991): $\hspace{2cm}\hspace{2cm}G(k_+)=1\hspace{1cm}H(y_+,k_+)=e^{-\frac{70}{k_+}^{1.5}\frac{y_+}{A_+}}\sqrt{1+e^{-\frac{70}{k_+}}}$
Crimaldi et al. (2006): $\hspace{2cm}H(y_+,k_+)=0\hspace{1cm}\hspace{2cm}G(k_+)=\frac{60-k_+}{55}$
Using all of these models and solving the ODE for the turbulent velocity profile in fully developed pipe flow, and then ensuring mass conservation enables the estimation of a pressure gradient from which a friction factor can be calculated and compared against the established Colebrook friction factor for a given Reynolds number and relative roughness (normalised with pipe diameter). Here, $k_+=\frac{U_\tau k}{\nu}$, where $k$ is the engineering (Colebrook-White) roughness.
As can be seen in the plots below, the comparisons are not good at all especially for high values of $k_+$ which correspond to higher Bulk Reynolds numbers and/or relative roughness. The $y-$axis is the ratio of the computed friction factor to that predicted by the Colebrook formula for the same given Reynolds Number and relative roughness. Ideally the value should be very close to unity.
This is not by any means a knock against the authors above, who almost certainly did not intend for it to be used in such a comparison. My question however is - 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
