Method of wall functions

14.3 Method of wall functions

Basic relations of k " model were derived under assumption that the local p turbulent Reynolds number Re t D is very high. This is not the case in the wall flow. Due to damping effect of the wall, the velocity pulsations and the turbulent kinetic energy are small. Application of the k " model is not valid. An efficient solution of this problem was proposed by Spalding who developed the method of wall functions. The wall function is an ana- lytical representation of the solution close to the wall which is matched with the numerical solution far from the wall. Below we derive basic relations of the wall function method following to the textbook [15].

It is assumed that the turbulence is in equilibrium, i.e. P D ":

Let us consider two dimensional flow along a plate:

@N u The derivative x

can be expressed from the mixing length model of Prandtl:

@y

t =l

@y

@y

Equations of the k – ε Model

From analysis of dimension we obtain

t D (14.39)

From (14.36) and (14.37) it follows

F rom (14.36) and (14.39) we have: @N 1=2 u

for wall stress:

and kinetic energy:

kD w %c 1=2

Let y p be the ordinate of the first grid node near the wall. Using approximation of Prandtl for l we obtain the dissipation rate at y p :

(14.40) and turbulent kinetic energy

" p D c 3=4 k 3=2 p

k p D w 1=2 %c

Equations of the k – ε Model

w can be found from the assumption that the first node is in the logarithmic range:

Values " p w (14.41) are used within finite volume method when the k " equations are written for the volumes adjacent to the wall. For instance the generation term is calculated as

@u

@u

P dxdy D

ij

dxdy D

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