The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

CHAPTER 4. NUMERICAL IMPLEMENTATION 56
Thus, while ω is infinite at the wall, finite values may be specified near the
wall according to Equation 4.15. However, the gradient of ω near the wall
is excessively large (approximately two orders of magnitude greater than
gradients on ˜ε), so it is not practical to solve the ω transport equation in the
near wall region. The approach recommended by Wilcox [91] is to specify
ω according to the limiting behaviour suggested in Equation 4.15 within the
first 7 to 10 cells, where y + < 2.5. Transport equations on 〈U〉 and k use
specified values of ω in this region, and the transport equation of ω is only
solved outside of this very near-wall region.
4.4.3 The Logarithmic Law of the Wall
A wall function mesh incorporates a near-wall cell that is large enough to
fully encompass the buffer layer and extend into the region where the log
law is valid. This is shown schematically in Figure 4.2. Because flows of low
Reynolds number are considered in this thesis, the buffer layer is very large.
The near-wall node location was set to y/δ = 20% or y + = 36.0. The main
grid of PASSABLE uses vertexes centred between nodes, so the near-wall
node is positioned far from the centre of its cell. The remaining grid was
made up of 18 nodes. An expansion factor of 1.00 was used for these 18 cells.
The total number of grid cells used with wall functions was thus much less
than with the low-Reynolds-number solution.
Numerically, the log law is implemented by severing the link to the wall
in the near-wall cell, and then manipulating the 〈U〉, k, ε source terms in
the near-wall cell to reflect the log law. A full discussion of the numerical
implementation of log laws in CFD can be found in the book of Versteeg &