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88 3. Turbulence Models

3.4 Two-Equation Models

There are several two-equation models. Three of the more popular, accurate
and widely used models are the k-e model of Jones and Launder [16], the k-uj
model of Wilcox [1] and the SST model of Menter which blends the k-e model
in the outer region and k-uj model in the near wall region [17]. All three models
can be used for a range of flow problems with good accuracy as discussed in
[1-3]. Here we discuss the k-e model which is the most popular and widely used
two-equation eddy viscosity model. For a discussion of the other two models,
the reader is referred to [1-3].
In this model, e m is given by Eq. (3.1.4). The kinetic energy k and rate of
dissipation e are obtained from the turbulence kinetic energy equation written
as _ _
— = — IYi/+ — ^ — 1 +£ (— + ^ \ — - £ (3 4 1)
Dt dxk \\ &k ) dxk J m \ dxj dxi J dxj
and the dissipation equation

— =—\(v+—}—] +c -e f^i^^l\^i_ c £^ (342)
x
Dt dxk |A °e) d k\ £l k m \dxj dxi) dxj £2 k
For boundary-layer flows at high Reynolds number, Eqs. (3.4.1) and (3.4.2)
become
dk dk _ d femdk\ (du\ 2 (<IAO\
U V m
dx dy dy\a kdyj \dyJ
de de d /e mde\ e (du\ 2 e 2 . A A.
U +V = + £m ( 3 A 4 )
0- X d-y 0-y{^dy) ^k {0-y) ~ ^T
The parameters c^, c £l , c £2 , a^ and a £ are given by

Cy, = 0.09, c £l = 1.44, c £2 = 1.92, a k = 1.0, a £ = 1.3 (3.4.5)

The set of equations comprising conservation of mass and momentum, Eqs.
(2.2.2) to (2.2.4) together with Eqs. (3.4.1) and (3.4.2), or with boundary-layer
approximations, Eqs. (2.3.33), (2.3.34), (3.4.3) and (3.4.5) for two-dimensional
flows together with the boundary conditions discussed below, represent a closed
set in which the equations for mean momentum, turbulence energy, and dissipa-
tion rate have the same form and can generally be solved by the same numerical
method.
The above equations given by Eqs. (3.4.1) to (3.4.5) apply only to free shear
flows. For wall boundary-layer flows, they require modifications to account for
the presence of the wall. Without wall functions, it is necessary to replace the
true boundary conditions at y — 0 by new "boundary conditions" defined at
some distance yo outside the viscous sublayer to avoid integrating the equations
through the region of large y gradients near the surface as discussed in [2, 3].