Page 107 - Computational Fluid Dynamics for Engineers

References 93

Some models also require the initial shear stress profile r. To generate this,
one can use an expression given in [3] modified for the pressure gradient effect:

rv du fy du '
du e 7 n
d
d
i-r-y + u I ^~ y- 2 u ^~ v (3.5.12)
dx Jo Ox Jo Oy
Following [3], assume
u = u eg(r]), 6/6 = const. (3.5.13)

and from the momentum integral equation, Eq. (2.4.36), we write

(3.5.14)
ax g ax

Using Eqs. (3.5.13) and (3.5.14), it can be shown that Eq. (3.5.12) can be written
as
= 1 + 8 g2dri 9 gdri
-e{[ - Fo
2 8 du e dr 2
k - W 9 l+-j(j 9 dr]-gj gdrj (3.5.15)
dx
Cf u e
Once the streamwise velocity profile u and shear stress r are computed by
the expressions discussed above, the dissipation rate e can be computed from
the assumption
T
-P=e (3.5.16)
gdy
which is accurate in the log-law region and adequate in the rest of a boundary
layer.

References

[1] Wilcox, D. C: Turbulence Modeling for CFD. DCW Industries, Inc., 5354 Palm Drive,
La Canada, Calif., 1998.
[2] Cebeci, T.: Turbulence Models and Their Application. Horizons Pub., Long Beach,
Calif, and Springer, Heidelberg, 2003.
[3] Cebeci, T.: Analysis of Turbulent Flows. Elsevier, London, 2003.
[4] Durbin, P. A. and Petterson Reif, B. A.: Statistical Theory and Modeling for Turbulent
Flows. John Wiley and Sons, New York, 2001.
[5] Prandtl, L.: Uber ein neues Formelsystem fur die ausgebildete Turbulenz, Nachrichten
von der Akad. der Wissenschaft in Gottingen, 1945.
[6] Boussinesq, J.: Theorie de l'ecoulement tourbillant, Mem. pres. Acad. Sc. XXIII, 46,
1877.
[7] Kolmogorov, A.N.: Equations of Turbulent Motion of an Incompressible Fluid, Izv.
Akad. Nauk. SSR Ser. Phys. 6, 56, 1942. (English translation, Imperial College, Mech.
Eng. Dept. Report ON/6, 1968).