Page 91 - Computational Fluid Dynamics for Engineers

76 2. Conservation Equations

iau -\—— = 0 (P2.7.1)
dy
2
d v! 2 f r
a u! — iR (au — u))v! — iv ——h ap (P2.7.2)
dy 2 dy
2
d v' 2 / x / -dp'
- a v' = iR (P2.7.3)
2
dy [au — oj)v — i——
dy
(c) For wall boundary-layer flows, Eqs. (2.5.6) to (2.5.8) or (P2.7.1) to (P2.7.3)
are subject to the following boundary conditions at y = 0
u' = v 1 = 0 (P2.7.4)

Show that u\ v' and p' behave exponentially as y —> oc with a typical asymptotic
representation
(P2.7.5)
where A\ and A 2 are constants and £i and £2 are defined by
z-2 2 (P2.7.6a)
i{ = a
2
£2 = OL + iR[au e — u] (P2.7.6b)
with the restriction that their real parts are not negative.

2-8. The equations in Problem 2.7 can also be expressed in other forms. Show
that if we represent the amplitude functions of the perturbation quantities u'\
/
v 1 and p' by , <\> and 77, respectively, and introduce them into Eqs. (P2.7.1) to
(P2.7.3), we get
d(p
= -iaf (P2.8.1)
dy
2
a.'- 2 ad f
<*f i(au — co) + + a—(j> + ilia - - - 4 r z = 0 (P2.8.2)
R dy R dy
= - , ( a t i - w ) 0 - - 0 + - ^ (P2.8.3)
dy

2-9. Show that the equations in Problem 2.7 can also be expressed as a fourth-
order differential equation in (j) in the form given by Eq. (2.5.13).

2-10. The boundary conditions for the Orr-Sommerfeld equation follow from
the relations given by Eqs. (P2.7.4) and (P2.7.5). At the wall, it follows from
Eq. (P2.7.1) that
;
y = 0, 0 = 0 = O (P2.10.1)
At the edge of the boundary layer, Eq. (2.5.13) reduces to