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Theory:

We derived our turbulence wall roughness formulation from "Viscous Fluid Flow" Second Edition, by Frank M. White, ISBN: 0-07-069712-4; Pages 426-429.

Basically, the law of the wall is given by

u+ = 1/k ln( y+ ) + B(k+) + Pc(Grad_P)

Where
u+ = u / u*
y+ = y * mu / u*
k+ = k * u* / mu
u = velocity
y = distance to the wall
mu = kinematic viscosity
k = average roughness height
u* = sqrt( tau_w / dens )
tau_w = shear at the wall
dens = density
k = 0.41 (Karman constant)
B = wall function constant
Pc = pressure correction

For smooth walls, B = 5.5. For rough walls, B is a function of k+, which is a function of wall shear (or u* to be exact). The exact equations are written in White's book above.

Given the above, effectively the roughness shifts the u+ curve down.

Practice:

In practice, one must consider the following:

1. The first mesh point MUST be larger than the roughness height. That is, other than the nodal point on the wall, all other points need to have y > k. Otherwise, the theory is incorrect; since the u+(y+) curve will go below zero.

2. On the other hand, we like to mesh such that y+ of the first node should not be over 300. At times, this criteria will contradict y > k condition. In this case, y+ of 300 should be sacrificed in favor of y > k.

3. All of the roughness theory and measurement come from experiments performed in air over sand paper (usually associated with the aero-space field). Hence, there is an assumption of Gaussian distribution of roughness, leading to a self similarity solution.

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