Page 80 - Computational Fluid Dynamics for Engineers

2.5 Stability Equations 65

the mean velocity and pressure satisfy the two-dimensional equations of motion,
the equations simplify further and can be written as

..
dv! + dv' ^ (2 5 1} x
-
/rt
^ % = ° --
at ox ox oy oy g ox
dv' ,dv dv' ,dv dv' 1 dp' ^ 2 2 / / n r o ,
— + u'— + u— + v'— + v— = —JL + vV v' 2.5.3
at ox ox oy oy g oy
These equations, with the overbars on u and v dropped for convenience, can be
simplified further by noting that all velocity fluctuations and their derivatives
are of the same order of magnitude and by assuming that the mean flow velocity
u is a function of y only so that Eq. (2.4.33) gives v = 0, that is,

u = u(y), v = 0 (2.5.4)

This assumption is known as the parallel flow approximation; which, with the
introduction of dimensionless quantities defined by

u' , v' __ u p' tuo
f f
^ * = > ^* = = •> u * ~ 5 P* = o? £* ~ —7~~
u 0 u 0 uo gufi L (2.5.5)
x y UQL
x* = - , 2/* = - , R=
L L v
allows Eqs. (2.5.1) to (2.5.3) to be written as

dv! dv' ^ /^ ^ ^x
^ + % = 0 (2 5 6)
- -
dv! du' ,du „r ^ . „ ^ „ „, . , _ _

^ - + ^ = - ^ + 0 ( ^ + ^ 1 (2-5.8)

For convenience, the subscript * on the dependent and independent variables
has been dropped.
Equations (2.5.6) to (2.5.8) form a set of coupled partial-differential equa-
tions with solutions that describe how disturbances originate near the surface
y = 0 and spread out through the boundary layer and beyond as they are con-
vected along the local streamlines. To study the properties of these equations,
we apply the standard procedure of stability theory, namely separation of vari-
ables. Assume that the small disturbance is a sinusoidal traveling wave and
represent a two-dimensional disturbance as
i
q'(x,y,t)=q(y)e ^-^ (2.5.9)