Page 74 - Computational Fluid Dynamics for Engineers

2.4 Reduced Forms of the Navier-Stokes Equations 59

approximation to Eqs. (2.2.45)-(2.2.47), with 77 locally normal to the surface
and all the viscous terms associated with derivatives neglected, we obtain

dQ dE OF
—- H 1 Re drj (2.4.7)
dr <9£ dr\
where

0
r] xmi + rj ym2
- 1
J (2.4.8a)
Vxm2 + r} ym3
um
rj x(umi + vrri2 + 777,4) + Vy( 2 + vrri3 + 777,5)
with
m i = 2r v
3 ^ ^ ~ 1y v)

7773 = - ( - 2 7 7 ^ + 477^^)
(2.4.8b)
M 5 / 2x
7774 =
Pr( 7
V 9 ( 2N
7775
P r ( 7 - 1 ) '^77
Equation (2.4.7) is strictly for laminar flows. As is shown in Chapter 3, its
extension to turbulent flows is usually carried out by using eddy viscosity (e m)
and turbulent Prandtl number (Pr$) concepts. This is done by replacing the
coefficient of viscosity /i by
/i + QS m (2.4.9)

(
and by replacing the coefficient of thermal conductivity k = fiC p/Pr) with
(2.4.10)

The relations can easily be incorporated into Eq. (2.4.7) by replacing \i in the
definitions of 777,1, ^7,2, 777,3 by

/7(1 +£m/v) (2.4.11)
and [/i/Pr(7 — 1)] in the definitions of 7774 and 7775 by

M f 1, P r £m
(2.4.12)
- 1) V Prt v
Pr( 7
Unlike /i and fc, however, the parameters e m and Pr^ are not properties of the
fluid but depend on the flowfield and are related to the velocity and temperature
field by empirical formulas.