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

CHAPTER 2. TURBULENCE MODELS 29
In applying a wall-function boundary condition to k, the goal is not simply
to prescribe a value of k at the near-wall cell node. Rather, better accuracy
is achieved by applying sources to the k equation at the near-wall cell that
represent cell-averaged production and dissipation terms (Pk and ε). Various
methods exist for estimating these quantities. The Launder & Spalding [34]
wall function is employed in this work. In this wall function,
∂ 〈U〉
Pk = τw
∂y
(k)
ε = ρCµ
2
∂
〈U〉
∂y
τw
(2.45)
(2.46)
The transport equation for ε does not make use of ε. Instead, ε is prescribed
in the near-wall cell from k according to the mixing length hypothesis,
C 3/4
3/2 k
µ = κy (2.47)
ε
The log law appears in Figure 2.1, plotted together with DNS and experi-
mental data at various Reynolds numbers. It can be seen that the log law
provides reasonable agreement with DNS and experimental results beyond
the buffer layer (y + > 30), and particularly when y + is not excessively large.
The log law produces good results when applied as a boundary condition in
a code whose near-wall cell extends to a height that fully encompasses the
buffer layer. Since y + is a function of the flow rate as well as y, the optimum
choice of near-wall cell size may be influenced by the flow rate considered.
However, it is usually sufficient to choose a large enough near-wall cell to
fully encompass the buffer layer for the lowest flow rate anticipated.