Page 78 - Computational Fluid Dynamics for Engineers

2.4 Reduced Forms of the Navier-Stokes Equations 63

elliptic, the boundary-layer equations are parabolic with disturbances propagat-
ing only downstream and not upstream. This property of the boundary-layer
equations significantly reduces the complexity of the solution procedure.
For two-dimensional flows, the boundary-layer equations simplify further
and can be written as + (2433)
! £-° --

2
du du 1 dp d u d ,— 1— i. , n A nA.
u— + v— = —-f- + v—g z - — wV 2.4.34
ox oy g ax oy oy
2
dT dT q w k d T d / 7 — x , „ „ „ N
dx dy gC p gC p dy 2 dy y
As will be discussed in detail in Chapter 7, the solutions of the boundary-
layer equations can be obtained in their partial-differential equation form with
assumptions made for the Reynolds stress and heat flux terms. This approach
is called the differential approach, in contrast to the integral approach based
on the solutions of momentum and energy integral equations, which are ordi-
nary differential equations. These integral equations result from integrating the
boundary-layer equations across the shear layer [3, 7] and introducing defini-
tions of boundary-layer parameters. For two-dimensional incompressible flows,
the momentum and energy integral equations are, respectively, given by

dO 9 /TT ^du e Cf . A n .

dOr + e Ldu 1 = St
ax u e ax
Here, #, <5*, i7, Cf in the momentum integral equation denote momentum thick-
ness, displacement thickness, shape factor and local skin-friction coefficient,
respectively, and are defined by

0= r — (1 - — )dy (2.4.38a)
V U eJ
J 0 U e <2A38b)
*"=r ('-;)*

c (2A38d)
' = u^U

In the energy integral equation, the parameter 9T and Stanton number St are
defined by
9 T= r^Ll^dy (2.4.39a)
Jo u e i w — ± e