Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

where c is a constant equal to 0.09, is the turbulent kinetic energy and l m is the
Prandtl mixing length (ˆ 0:4y, where y is the distance from the nearest wall). The
Prandtl mixing length l m is often related to the length scale of the turbulence L as
l m ˆ c03
C D
1=4
L …5:29†
where C D and c 0 are constants.
The turbulent kinetic energy is calculated from the following transport equation
@
@t ‡ @u i
@x j
ÿ @
@x i
‡ T
@
ÿ
@xi R ij
where k is a constant generally equal to unity. Further,
" ˆ C D
3=2
L
@ui ‡ " ˆ 0 …5:30†
@xj …5:31†
Two-equation models ( -" and -! models)
Here in addition to the equation given above, another transport equation of the
form
@"
@t ‡ @ui" @x j
ÿ @
@x i
‡ T
"
@"
ÿ C "1
@x i
" Rij
@ui "
‡ C "2
@xj 2
ˆ 0 …5:32†
is solved and here C "1 is a constant ranging between 1.45 and 1.55, C "2 is a constant in
the range 1.92±2.0 and " is also a constant equal to 1.3.
In the above two-equation model, T is calculated as
T ˆ c
"
2
…5:33†
These models are not valid near walls. To model wall e€ects, either wall functions or
low Reynolds number versions have to be employed. For further details on these
models the reader can refer to the relevant works. 56;62 We give the following low
Reynolds number versions for the sake of completeness.
Low Reynolds number models
For the one-equation model, the following form is suggested by Wolfstein 56
and
t ˆ c 1=4 1=2 l m f …5:34†
" ˆ C D
3=2
Lf b
f ˆ 1 ÿ e ÿ0:160Rk ; fb ˆ 1 ÿ e ÿ0:263R p y
k ; Rk ˆ
Turbulent ¯ows 163
…5:35†
…5:36†
where y is the distance from the nearest wall.
For two-equation models, the coe cients c , C "1 and C "2 appearing in the twoequation
model discussed above are multiplied by damping functions f , f "1 and f "2