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 \
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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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.' ~ '~"':""'-'"
",
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,.: