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wise flow, two different velocity scales are used for the velocity perturb- .. ations. ..... au(y ) - w vanishes. As the wave speed c '" 0)/(a2 + 82)1/2, thts turning. _ c.
NASA-CR-173282 19840009464 \

A Reproduced Copy OF

, .

Reproduced for NASA

by the

I'IASA

Scientific and Technical Information Facility

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.LJ\!'lGLEY RESEARCH CE!'ITEH LIBRARY, NASA .t!AMPTO~L ~lf!GI~.I~

FFNo 672 Aug 65

, .,

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o

.' ~ '~"':""'-'"

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OEPAKTr~ENT OF HATHE~~i\TICAL SCIENCES

SCHOOL OF SCIENCES AND HEALTH PROFESSIONS

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OLD DOI·UNION UNIVERSITY NORFOLK, VIRGINIA

;; ~

STABILITY OF THE LAMINAR BOUNDARY LAYER IN A STREAMWISE CORNER

:r: U

~ UJ

By William D. Lakin, Principal Investigatol'

(I)

Final Report FOt' the per"iod

ending September 30. 1983

Prepared fmo the

National Aeronautics and Space Administration Lan!ll ey Res-car'ell Center Hamilton, Virginia

Under Research Grant NAGl-297 John R. Dagenhart~ Technical fllOnitor Ai rfoil Ael'odynanri cs Branch

(~ASA-CR-1"lJLtl:':)

R 0,

- u'(Yc)

the reduc:ed equation

R2

.. V ..,

(u ..

obtaint::d by formally letti.ng

2

a)(I'· - a

R + ...

a•~

2

2 ..

a )v -

-

u"

(2.27)

v. "" 0

in (2.24) with

(1 + ~)1/2 c

a

(2.2R)

a

will have a regular singular point at the

full

fourth-order

equation

Yc

whereas

for

Yc

is a regular point of

Dealing

with

this

spurious

ningularity provides one of the main challenges in deriving uniform approxlma-

In this rE!gard, (2.24) is similar to the usual

tions to solutions of (2.24). Orr-Sommerfeld equation

for

the stability of

ttJo-dimensional boundary J AND WELL-BALANCED TYPE

The uniform first. approxirnation to the well-balaneed solution is siMply

(6.1)

First approximations to the three solutions of balanced type have the farms

ORt01Nf..L rr:,G)~ t~ OF pOOR QUAUI\'

-25-

(6.2)

where !~o

G, A, B, and

C are regular at the tllrning point and

is a constant

A

be determined. Substitution of (6.2) into (4.S) shows that

same differential equations as

a. b. and

c.

A, B, and

C

satisfy the

Consequently

{A(n),B(n),C(n)} • A(O){a(n),b(n),c(n)}.

The ,. . ell--balanced part

.-

G(n)

(6.3)

of (6.2) is found to satisfy the equation

R2 G • A{4n(A + nC)' + A - 2n(nC)' + fO(A + nC) - g1 ne}

!

which, on simplification, becomes

(6.5)

A comparison of this equation with (4.14) now shows that the solution of (6.5)

which is regular at

n· 0

Is of the form

(6.6)

where

bO is an arbitrary constant,

provl~ed

t!1i$R¥%ill'iW:::;::;;U;P &.'!'!'!~ ·f~~"\,.,,

JWn~!iffJi;;;pl@4$2;;;;::;

hi. . g ,W

.;

l>

Fig.~.

,..(~

£;~

r- !'t4

The Stokes lines (left) and the anti-Stokes lines (right) in

tne ?-plane ,dth

.:,

rcal and

=i0'!

-('1a-".,

ph ~= tn.

i J' Ii ~t;.i,.: