Page 106 - Computational Fluid Dynamics for Engineers

92 3. Turbulence Models

w(rj) = 1 — cos(7T7/) (3.5.7)

Here 77 is a free parameter, depending on flow history in a way that is usually
unknown, but constant and equal to about 0.55 in zero pressure gradient for
R e > 5000, so that Eq. (3.5.6) defines the function F x in Eq. (3.5.5).
Equation (3.5.6) gives du/dy nonzero at y — 6. To remedy the difficulty,
a number of expressions have been proposed. A convenient one proposed by
Granville [21] uses a modification of Eq. (3.5.6) written as
u 1
— - - [In y + + c + 77[(1 - cos TTT?) + {rj 2 - rf)} (3.5.8)
U T K
From Eq. (3.5.8) and from the definitions of 6* and 9 it can be shown that

l
fu e — uu Tl u T (\\ \ . „
= / - -d V = ^ - ( - + Il), 3.5.9a
6 JO U T U e KU e \Y1 }

1 + -Si(7T)
6 J 0 U e \ U eJ KU e V 12 / \ KU e J { 7T
7
1
r2
+ 1.5IP + — - — - 0.12925/7 f (3.5.9b)
From Eq. (3.5.9b), taking Si(?r) = 1.8519, write

X
RQ U T ( 11 \ ( U
, _ + 7 7 - — (1.9123016 + 3.05603/7 + 1.577'
\
Rft KU e 12 / \KU e
Evaluating Eq. (3.5.8) at 77 = 1,
?/_ \
1
r
Ue_ _ 1_ , . //S?/„ u, . n „ 1
8u e
ln[ — — ) +C + 277 (3.5.10)
For given values of cj and RQ, Eqs. (3.5.9b) and (3.5.10) can be solved for 6
and 77 so that the streamwise profile u can be obtained from Eqs. (3.5.8) and
+
(3.5.9a) in the region y+ > 30, with fi(y ) given by Eq. (3.5.2).
The expression given by Eq. (3.5.6) can be extended in several ways. One
method uses Eq. (3.5.3) and adds the wake component
n(x)

to it. Another method uses an expression due to Thompson [3]

+
y , y + < 4 (3.5.11a)
u + =
+ 3
ci + c 2 lny + + c 3 (lny+) 2 + C4(lny ) , 4 < y+ < 50 (3.5.11b)
where c\ = 1.0828, c 2 = -0.414, c 3 = 2.2661, c 4 = -0.324.