Page 103 - Computational Fluid Dynamics for Engineers

3.4 Two-Equation Models 89

There are several approaches that can be used to model the turbulence ki-
netic energy and rate of dissipation equations near the wall region. For example,
in one approach, wall functions are introduced into Eqs. (3.4.3) and (3.4.4) so
that the model equations are applicable throughout the whole layer. Launder
and Sharma [18] modify Eq. (3.1.4)

k 2
"1 = 0^— (3.4.6)
£
and rewrite Eqs. (3.4.3) and (3.4.4) as

1 2 2
dk dk d (u tdk\ fdu\ 2 n fdk / ^
U V Ut V
dx dy dy\a kdyj \dy) \ dy I

2
de de d fu tde\ e (du\ 2 £ e 2 n (d u\ 2 , o .

where
3.4 2
U = exp 2 , R t l = — (3.4.9a)
(l + J 4 /50) J' ve
i
= l - 0.3exp(-i2?) (3.4.9b)
f 2
The boundary conditions are

y = 0, k = £ = 0 (3.4.10a)

y —> 6, k —> k e, s —> e e (3.4.10b)

To avoid numerical problems, k e and £ e should not be zero. In addition, k e and
can not be prescribed arbitrarily because their development is governed by
£ e
the transport equations (3.4.7) and (3.4.8) written at the boundary-layer edge,
dk
u e—- = -£ e (3.4.11a)

The above equations can be integrated with respect to x with initial conditions
corresponding to k eo and £ eo at XQ. The solution provides the evolutions of k{x)
and e(x) as boundary conditions for the k- and ^-equations.