Page 98 - Computational Fluid Dynamics for Engineers

84 3. Turbulence Models

(£m)o = aUe<5*7tr7 (3.2.3)

Here <5* denotes the displacement thickness
u
1 dy (3.2.4a)
-j
Jo
and 7 accounts for the intermittency of the outer region and is represented by
[y-Y]
7 = 1 - e r f (3.2.4b)
y/2a
where Y and a are general intermittency parameters, with Y denoting the
value of y for which 7 = 0.5, and a the standard deviation. The dimensionless
intermittency parameters Y/6* and a/6* are expressed as functions of H as
shown in Fig. 3.1a. The variation of the ratio of boundary-layer thickness 6 to
<5* with H is shown in Fig. 3.1b. The parameter a is calculated from

°-° 168 (3.2.5)
a = 1.5
1
-'(£'£
Here subscript m denotes the location where turbulent shear is maximum. The
parameter f3 is given by
6
Rt < 1.0
l + 2Rt(2-Rt)
0=< (3.2.6)
l + Rt
Rt > 1-0
Rt
Here Rt denotes the ratio of wall shear to maximum Reynolds shear stress [2,3],

Rt = = — (3.2.7)
(-Q(u'v'))m
In Eqs. (3.2.1) 7t r is an intermittency factor which represents the streamwise
region from the onset of transition to turbulent flow. It is defined by the following
expression
rx dx
7tr 1 — exp -G(x-x tr) I (3.2.*
u e
JXu
where xt Y is the location of the start of the transition and the factor G is given
empirically by
G R M 3 2 9a
= &$ £ ( - - )
with R XtT denoting Reynolds number, R Xtr = (u ex/v) tr, and C a constant with
a recommended value of 60 for high Reynolds flows. For lower Reynolds number,
C is given by [2, 3]
C 2 - 213(log Rx tr - 4.7323) (3.2.9b)