ON THE INVISCID SOLUTIONS OF THE ORR-SOMMERFELD EQUATION
D. A. Nield
(received 19 January, 1971;
Abstract.
revised 14 July, 1971)
Modified Heisenberg solutions which are analytic continuations
of the Tollmien solutions are constructed.
These Heisenberg solutions
are then linked with the long wave-length solutions of Drazin and Howard
for unbounded flows.
Explicit solutions for symmetric channel flows and
boundary layer flows are found.
Earlier asymptotic theory for viscous
flows by Lin and Reid is clarified and confirmed.
1. Introduction
The Orr-Sommerfeld equation arises when one considers the stability
of a basic parallel flow
(i/Q/), 0,0), y 1 5 y 5 y , to disturbances
with perturbation stream-function
(p(y)eia(-x ~
.
From the Navier-
Stokes equation for an incompressible fluid one obtains (see e.g. [7 10])
the linearised equation
(£/ - c) (Z?2 - a2) cp - £/"<p = (iai?)-1 (Z?2 - ot2)2<p,
where D = d/dy.
(1.1)
For inviscid fluids the Reynolds number R is infinite,
and the fourth-order Orr-Sommerfeld equation (1) reduces to the secondorder Rayleigh equation
(U - o)(Dz - a2) cp -
p = 0.
(1.2)
The solutions of the latter equation are of interest not only in their own
right, but are required as a first step towards the solution of the OrrSommerfeld equation.
A feature of the Rayleigh equation is that it is
singular at any value yQ for which U = c, U" £ 0.
Math. Chronicle 2(1972), 43-52.
43
Solutions of (1.2) as series in powers of a2 were introduced by
Heisenberg [4].
They have the form [7]
<
1K y ) = (U - o){qQ(y) + a 2q (y) + . . .
qQ(y) = 1 or
where
+ a2nqn (y) + . . .
>
(1.3)
(V - o)~2dy
and
In each integral the lower limit is arbitrary, but traditionally it has
been taken as y .
The path of integration must then pass below the crit
ical point for nearly real values of a.
As shown by Lin [6], for finite
values of y these Heisenberg expansions are entire functions of a2 .
How-
ever, they do not converge uniformly as y *■°°, and the implicit presence
of the singularity has hindered efforts to relate them to other pairs of
solutions of Rayleigh's equation but Georgescu
[3] did manage to obtain
some formal relationships.
An alternative approach, used by Tollmien [ll], is to seek solutr
ions of equation (1.2) in the form of power series in y - yq) where
£/Q/c) = o and
t 0.
Then y
is a regular singular point, and the
method of Frobenius leads to expressions of the form
<p1 (y ) = (y - yC.J
and
<p (?/) =
2
where
p
series with leading term unity.
and
<p2 ,
P2 (t/-z/c)
be zero, which ensures that
and
P2
denote power
one may specify that the coefficient of
The series
\y - yq \ = r,
where
P1
r
and
cp2
y - yQ
in
cp2
essentially contains no
P2
are convergent within the
is the distance from
singular point, wherever that may be.
singularity in
Pj
( 1 . 6)
In order to remove the ambiguity in
defining
circle
(1.5)
, iy - ycJ
1
U "
(y - 2/ ) + jf-rvty) lQg (y - y J *
2
n
a
U ' = U '(y s) , U " = £/" (t/fl)
multiple of
p
y^
to the next
The presence of the logarithmic
can cause difficulty.
In the neutral case
y
is real, and care must be taken in selection of the correct branch.
44
Another set of solutions of the Rayleigh equation was used by
Drazin and Howard [l] in their study of unbounded flows.
These were of
the form
«P (j/) = e±ay {XQ(y) + X (i/)a + ... + Xn (y) a 1 + ... },
(1.7)
where X («) = £/(°
°
) - o and X (°
°
) = 0, n > 1 . Under certain conditions
0
(see §3 below) these series were shown to be convergent.
In this paper a modified form of the Heisenberg expansion is intro
duced.
The lower limit of integration is taken to be y . and where neces
sary the singular part of an integral is explicitly extracted.
Two part
icular modified expansions are identified with the Tollmien solutions.
The solutions for two-dimensional channel flows, and for plane Poiseuille
flow as a special case, are treated in §2.
Boundary layer flows, with the
asymptotic suction profile as an example, are similarly considered in §3.
Here the modified Heisenberg solutions are related to the appropriate
Drazin-Howard solution.
cid
In §4 we make some comments on the use of invis-
solutions in the study of viscous flows by Lin [6,7] and Reid [8].
In particular, the earlier work on the determination of the asymptote to
the upper branch of a neutral curve is clarified and confirmed.
(This
involves determining the behaviour of expressions as a and o tend to zero).
2. Solutions for channel flows
The Tollmien solutions are
Vo"
1
<p (zd = 0/ - yQ) + 2jj- t & " V * 2 + *
(rr
iy - y J 3 +.• •
( 2 . 1)
and
U "
Q
<P (*/) = --- <P (#) log (z/ - y ) + 1 +
2
77 r 1
U
(U ,;)2
O
C
-----------+. i 9
2U ' (U ')2
a
a
iy - yc V +---
•
(2 . 2)
As a preliminary to finding modified Heisenberg expansions which are iden
tical with <p and (p near w . we note that
1
2
^
45
(2.3)
v - O = Uo ’ ty - yQ) + UQ"(y - yQ) 2 + 0(y - yQ) 3;
and
V "
a
C«c ')
(U - a )2
(y - yQ) 2
(2.4)
+ 0 ( 1).
VQ '{y - yQ)
Hence we define
(2.5a)
?,„W
- 20
\y J < V >2
1
(U - a ) 2
U "
c
,
(y - yQ) 2
u "
+ ^Z/
dy +
- »«>
y - y,
i/ "
l0 8
(» *
•y
(£/ - o )‘ 2 dy
- 2^
(2.5b)
'
(U - a )2 q > (y) dy,
Jn
(j = 1 ,2 ),
(2.5c)
and
'V 0' 5 = V o7 1
+ ^
+ “2<7'7'1 *
qjn +
K U =
1>2)'
(2 . 6 )
(The final constant in equation (2.5b) is determined by the requirement
that <p must contain no multiple of <p .)
the cp
2
1
and <p
2
One may now verify that
l
defined by equations (2.5) and (2.6) are identical with the
Tollmien solutions (2.1) and (2 .2 ).
The modified Heisenberg expansions for cp
tions of y in any domain D
and <p
are analytic func
which is bounded and does not contain y 3
°
since each is a uniformly convergent sum of analytic functions. The uni
1
form convergence follows since constants K and L can be found such that
|a2n q
| < K\a(y-y ) |2n/ ( 2 n ) !
°
in
46
and
\°
-2n 4oJ
2n' < L \a (y - y J
12n_1/C2n
- i)!
A general solution (normalized so that its value at y
is unity)
of the Rayleigh equation is
$(2/) = Ay (y) + (p (2/),
(2.7)
1
2
where A is independent of y. For symmetric channel flow the boundary
condition $'(i/ ) = 0 must be satisfied.
Thus
cp 1 Cy )
A = - — 2___ 2_
*•(*)
1
2
(2-8)
After some algebra, one finds that
A(a,c) = a(o)£2 - b(c) + 0( a2).
(2.9)
(IM)
where
a(c) =
rz/2
(£/-<?)2 dy
Vi
'{U - a y q20 (y)dy
and
He)
=
•y2
(U - o')2 dy
ty
CV )2jzi
{U ~ c)2
dy
( 2 . 10 )
| J*2 (17 - a)2 dy^
47
Further manipulation leads to the result
V
Ai a,a) = y — oT2
1 + 2
2
a - b
u 2 • (S,')
2___ ™ 2 J.
av)
(2.11)
*2 U2 dy, and
where
/-»2
o nj n 2
^
■MO).
(2.12)
For plane Poiseuille flow, for which
V{y) = l - y2, y = -l, y - 0 ,
one obtains
719
11575
3.
2
log 2
15
a2 + 0 ( a \ c 2j
.(2.13)
Solutions for boundary layer flows
For a flow occupying the space y > 0, the disturbance amplitude
function cp(^) must be bounded as y -*■
iently slowly as y -» °
°
, then cp ~ e °^.
In fact, if Uiy) varies suffic
Hence Drazin and Howard [l,2]
introduced the 'long wave-length expansion'
cp+ U/) -
(3.1)
I
n= o
where
x„(») =
£ % ) - o, Xn+I(y) - (£/ - e)
(1/ - o)~2dy
2(U - o)Xn ' dy.
(3.2)
It follows from their work that <p (z/) is an analytic function of y within
1
a domain D2 throughout which (V - a)
is bounded and |d[(£/-e)2] / dy |
-
< Me ^ , y > 0
(where M and a are positive constants), provided that a<ha.
For values y
of v within the domain D = D fl D , wherein cp and the
y
m
i
2
+
modified Heisenberg expansions (p and cp are analytic and the Rayleigh
1
2
equation is regular, there exist A and B (independent of y) such that
48
Alp (y) + cp (y) = B<p (y).
(3.3)
1
2
+
The values of A and B can be found by satisfying equation (3.3), and its
derivative, at some fixed y , One thus obtains, using the abbreviations
W(.y) = U(y) - e and ^
A =
M2
(V)
1 +
aW
|
- W(°°) , that
f ( y ) d y + | ml f i . y)
+ 0 '( & ) ] < & - S’ G^ )} 01 + 0 (<*2)
(3.4)
and
U
5 =
'
r~
r
fm
° 'l +1
aW
1
W2
dy
-
y a W2
o (a2)
(3.5)
0
where
fiy) =
W 2
_2_
W2
-
W2
—
W 2
and
giy)
(V )2
--- + _c_ log [y~y _) - _o_
y-y
u '
c
2U
a 3a
o
a(3.6)
(It is readily seen that A is indeed independent of y
.)
For the case of the asymptotic suction profile, £/(z/) = 1 - e
equation (3.4) gives
/ ! = £ ♦ § + 0(a),
(3.7)
which agrees with the result of Hughes and Reid [5].
is here independent of a is peculiar
(The fact that A
to this special profile.)
One also
finds that
1
+1
B = aTT^o) + ITT-
4.
*
•
^3.8)
Applications to viscous flows
Two independent solutions <p and cp of the Rayleigh equation are
1
2
-1
also approximations (with error of order (a/?) ) to two solutions (com-
49
monly called the 'inviscid solutions') of the Orr-Sommerfeld equation.
A basic set of solutions of the latter equation is completed by the ad
dition of two ’
viscous solutions' cp and cp . The one usually labelled
3
4
<p becomes exponentially large as y increases, and hence must be rejected.
The solutions of the Orr-Sommerfeld equation must thus be of the
form
cp = $ + Cep ,
(4.1)
3
where $ = Ap
+ cp is an inviscid solution. For symmetric channel flow
l
2
the boundary conditions are <p = tp1 = 0 at the wall {y = y ), and
* <pw = 0 in the middle of the channel (v = y ).
implies that
(w ) = 0.
2
As in §2 above, A
2
can be chosen so that $'(w ) = 0.
For an even solution, symmetry then
Also <p is exponentially small as y -*■y *and
2
3
so $ + Cep also satisfies these conditions.
3
C using the boundary conditions at y
2
Elimination of the constant
leads to the characteristic equation
QO
cp ' (y )
---- L = _2--- L_ .
*ty )
<p (y )
1
3
(4.2)
1
The left hand side depends only on the inviscid solutions;
hand side depends only on the viscous solution cp^.
the right
An identical equation,
but with a different inviscid solution $, holds for boundary layer flows.
(The previously existing difficulty of expressing $ in terms of Tollmien
series for cp and cp has, as we have seen in §3, been overcome.) Dependl
2
ing on the approximation used for cp , one then obtains a characteristic
equation relating the inviscid solution $ to a single viscous function
such as the Tietjens function F(z) or the function G(Z) used by Reid [8].
Here
1
z = (aM/J )3 Q/^-^ )
ancl z = 5 (cxP/C^j')2)
equations are
r, ,
™
where
1
•
The alternative
•,
• -<■
.
U '
X = _J_ {y -y ) - 1,
C" 1
50
«.»>
or
V
*Q/i)
G(Z) -------------a $' (z^ )
(4.3b)
To investigate the asymptotes of the neutral curve it is first nec
essary to examine the behaviour of the inviscid solutions for small val
ues of a.
Lin [6;7,p .43-45] did this by re-ordering the terms in the
(old) Heisenberg solutions.
A somewhat simpler (but still non-rigorous)
treatment by Reid [8] was based on a transformation of the Rayleigh equa
tion
into a Riccati equation.
With the aid of the modified Heisenberg
expansion, we can proceed in a more systematic way.
For example, it is
a straightforward, if somewhat tedious matter to find an expression for
G(Z ) for small a and a, and hence to calculate the asymptote to the upper
branch of the neutral curve.
The results of Reid are confirmed.
Further,
the calculation can now be carried out to higher order.
The author wishes to thank Professor W.H.Reid for many suggestions
and for arranging his stay at the University of Chicago.
Financial sup
port came from National Science Foundation Grant No. GP-8620.
51
1.
P.G. Drazin and L.N. Howard,
The instability to long waves of
unbounded parallel invisoid flow, J. Fluid Mech. 14(1962), 257283.
2.
P.G. Drazin and L.N. Howard,
Hydrodynamic stability of parallel
flow of invisoid fluid, Advances in Applied Mechanics 9(1966),
1-89.
3.
A. Georgescu,
solutions,
4.
5.
Uber Stabilitat und Turbulenz von Flussigkeits-
Ann. Physik 74(1924), 577-627.
T.H. Hughes and W.H. Reid,
suction
6.
Rev. Roumaine Math. Pures Appl. 14(1969), 991-998.
W. Heisenberg,
stromen,
On a relationship between Heisenberg and Tolhnien
On the stability of the asymptotic
boundary layer profile,
C.C. Lin,
J.Fluid Mech. 23(1965), 715-735.
On the stability of two-dimensional parallel flows.
Parts I, II, III, Quart. Appl. Math. 3(1945), 117-142, 218-234,
277-301.
7.
C.C. Lin, The theory of hydrodynamic stability,
Cambridge Univers
ity Press, London, 1955.
8.
W.H. Reid,
The stability of parallel flows, in M. Holt(ed.) Basic
developments in fluid dynamics, Academic Press, New York, 1965.
9.
S.F. Shen, Stability of laminar flows,
of laminar flows,
in F.K. Moore (ed.), Theory
Princeton University Press, Princeton, N.J.,
1964.
10. J.T. Stuart,
Hydrodynamic stability,
Laminar boundary layers,
11. W. Tollmien,
in L. Rosenhead (ed.),
Clarendon Press, Oxford, 1963.
Uber die Entstehung der Turbulenz, Nachr. Ges. Wiss.,
Gottingen, Math. phys. Klasse (1929), 21-44.
52
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