Page 101 - Computational Fluid Dynamics for Engineers

3.3 One-Equation Models 87

3.3 O n e - E q u a t i o n M o d e l s

Of the several methods that fall in this group, we only consider the method due
to Spalart and Allmaras [2, 9]. This method employs a single transport equation
for eddy viscosity, is popular for wall boundary-layer and free-shear flows and is
used in both boundary-layer and Navier-Stokes methods. Its defining equations
are as follows.
(3.3.1)

1 d
— = c bl [1 - f t2] Sv t - (c Wlf w - ~J>ft 2) {y + h)
K* + a dxk L dx k\
Cfr 2 dv t dv t
(3.3.2)
+ a dxk dxk
Here

= 0.1355, = 0.622, (3.3.3a)
c bl c b2 c Vl = 7.1,
3
CW2 — U.O, ^Ws = 2, K = 0.41 (3.3.3b)
1/6
X 1 + c! '^3
Ui = 3 U 2 = i - fw = 9 6 (3.3.3c)
X + c^ 1 + Xfu, ' 2 + c?
^ 3
vt .6 ^
X = —, 0 = ^ + c 1(;2(r - r ) , r = — (3.3.3d)

(3.3.3e)

c
= c t3e~ ^\ 2 = 1.1, = 2 (3.3.3f)
- Q 4 X
f t2 c t3 ct 4
where d is the distance to the closest wall and S is the magnitude of the vorticity,
QJ = 2 (,3^ ~ a^tj*
The wall boundary condition is vt = 0. In the freest ream and as initial con-
dition 0 is best, and values below ^ are acceptable [9].
For boundary-layer flows, Eq. (3.3.2) can be written as

di>t dv t dh\ 2
c 6i ( l - / t 2 ) ^ + +
dx dy dy c h2 dy)

(3.3.4)
I c W l jw 2 ^ 2 )
where
\du
S = (3.3.5)
dy + KW JU2
For a detailed discussion of this model and its accuracy compared to other
turbulence models, the reader is referred to [1-3].