Page 100 - Computational Fluid Dynamics for Engineers

86 3. Turbulence Models

In Navier-Stokes calculations, Baldwin and Lomax replace the absolute value
of the velocity gradient du/dy in Eqs. (3.2.1) and (3.2.11) by the absolute value
of the vorticity |CJ|,
\ \ \du dv\
M = ( 3 2 1 3 a )
| ^ ~ ^ | - '
and the intermittency factory 7 in Eq. (3.2.4b) is written as

csy
7 1 + 5.5 (3.2.13b)
2/max
with C3 = 0.3. The studies conducted by Stock and Haase [14] clearly demon-
strate that the modified algebraic eddy viscosity formulation of Baldwin and
Lomax is not a true representation of the CS model since their incorporation of
the length scale in the outer eddy viscosity formula is not appropriate for flows
with strong pressure gradients.
Stock and Haase proposed a length scale based on the properties of the
mean velocity profile calculated by a Navier-Stokes method. They recommend
computing the boundary-layer thickness S from
6 = 1.936y max (3.2.14)

where 2/ max is the distance from the wall for which y\du/dy\ or F in Eq. (3.2.11)
has its maximum. With 6 known, u e in the outer eddy viscosity formula, Eq.
(3.2.3), is the u at y = <5, and 7 is computed from
-1
7 1 + 5.5 (3.2.15)

based on Klebanoff's measurements on a flat plate flow and not from Eq.
(3.2.13b). The displacement thickness 6* for attached flows is computed from
its definition, Eq. (3.2.4a), and, for separated flows from
u
6* = f l l - — )dy (3.2.16)
/
either integrating the velocity profile from y = 0, or y = y u=o to <5, or using the
Coles velocity profile, see Eq. (3.5.6). The results obtained with this modifi-
cation to the length scale in the outer CS eddy viscosity formula improve the
predictions of the CS model in Navier-Stokes methods as discussed in Stock
and Haase [14].
A proposal which led to Eq. (3.2.14) was also made by Johnson [15]. He
recommended that the boundary-layer thickness 6 is calculated from
6 = 1.2y 1/2 (3.2.17)
where
0
5
Vl/2=y a t -B = - " (3.2.18)
For a discussion regarding other length scales, the reader is referred to [1,3].