On the spectrum of natural oscillations of two-dimensional laminar flows


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On the spectrum of natural oscillations of two-dimensional laminar flows
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Physical Description:
34 p. : ill ; 27 cm.
Grohne, D
United States -- National Advisory Committee for Aeronautics
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Washington, D.C
Publication Date:


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Aerodynamics   ( lcsh )
Laminar flow -- Research   ( lcsh )
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )


If the stability of a laminar flow is investigated by the method of small oscillations, a complicated eigenvalue problem is encountered. Heretofore, attention has been directed toward the isolation of those normal modes of oscillation which are either neutrally stable or which grow in time. The present paper deals with the entire spectrum of characteristic values and the form taken by the associated oscillations.
Includes bibliographic references (p. 27-28).
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by D. Grohne.
General Note:
"Report date December 1957."
General Note:
"Translation of "Über das spektrum bei eigenschwingungen ebener laminarströmungen." Zeitschrift für angewandte Mathematik und Mechanik, vol. 34, no. 8-9, August-September 1954, pp. 344-357."

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University of Florida
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*"lijber das Spektrum bei Eigenschwi~g~ungen ebener Lamirnars tr~jmanlgen."
Zeitschrift fiir angewandte Mathemiatik und Mechanik, vol. Sk, no. 8-9,
August-Se~ptember 1954, pp. 344-557.





By D. Croh-ne


In the investigation of stability of a two-dimensional la1ina3r flow
with respect to small disturbances, we describe a disturbance of the
stream function moving downstream (in the direction of the x-axis) by, the
"partial wave formula"

9 = g(y)eia(x-ct)


and obtain then for the distribution of the disturbance amplitude Q(y)
at right angles to the main flow the so-called stability differential
equation of the fourth order

(U -c) 9" a29).- U~IyF =a -1- c

- 222 n+ a 0)


where U(y) designates the velocity profile of the basic larrinar flow
In addition, we enforce certain boundaryr conditions,, in the specific
case of the parallel channel

9(+1) = 0 9'(+1) = 0


whiich express the fact that even the disturbed flow adheres to the
bound ing walls. In these equations, the velocities U and c are
referred to a velocity of reference UO; furthermore, the lengths x,
y, and 1,13 to half the channel width b, and finally the time t to
the tihe unit b/UO. The Reynolds number R is defined by


The boundary-value problem consisting of differential equation and
boundary conditions determines, for each pair of parameters a and R,
a spectrum of an infinite number of eigenvalues ep. The associated
dIs~turbances (1.1) are damped when Emi(c,) < O, and are excited when
Imly > ; ais assumed to be positive an~d real. A basic flow is
called stable for a value of R when the entire eigenvalue spectrum
n,t for all possible values of a, contains olnly damped disturbances.
Th-us- the range of' thle Reynolds number R is divided up into a region
o-f t ab ili~ty I < R < RR and a region of instability R > R", which are
separated from one another by the stabilityr boundary R*.

Since, in the literature published up till now almost exclusively
neutral oscillstions at most, excited oscillations have been investi-
gated, we shall investigate in the present report, following a suggestion
of Pro~f. Djr. W. Tollkien, the entire spectrum of the eigenvalues c, as
a function of a and R; for simplification, we shall emphasize the
dependence on ari. A general solution of this problem is possible in
the fojllowinC two special cases: (1) in the case U = const. which is
equivalent to U .We deal here with the "oscillations of a fluid
at rest" already treated by Lord Rayrleigh. The solution is possible in
thc domar~in of elementary and transcendental functions. The second
special case co~nerns the rectilinear Couette flow U =y investigated
by L Hof fef.5).The solution can be reduced to tabulated Bessel
funci~t ions.

For more general velocity profiles U(y), the eigenvalues on can
be detersined app~r-xima3tively' analytically in the following limiting

I. In the lirmitintg case aR 0( for arbitrary order n of the

II. In the liimitin~g case n -,om for constant aR

III. In th-e limiltintg case aR -so for restricted order n.

NACA TM 1417

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A continuus transformation of the three cases into one another
for constant subscripts is possible in the above named special cases
U = O and U = y. The assignment of subscripts of the eigenva~lues cn
can be made in the cases I and II according to increasing damping, that
is, ac.cordingt to the rule

Im en+1) Im en (1.4)

Ho3wever, this rule is not always applicable to the case III whn the subt-
scripts used ar~e to remain constant for continuous variation of a and

The boundary-value problem formlated in (1.,2), (1.3) is, generally,
not self-adjoijnt; thus, th~e re~ducltion to the well3-knw statements andl
estimates of thec Sturm-Liouille theory is eliminated. The eigeat~une-
tions generally do not .formu an orthogonal system. They do form, however,
as 0. Haupt (ref. 3) has shown, under certain assumptions, a system of
functions that is complete with respect to each of th~e functions which.
satisfyl the boLundary conditions (1_.3) and are four times continuously
differenztia~ble. This system of functions can. be transformed into an
orthogonal system.



As already found by Lo~rd Raylieigh (ref. 8), the entire system of
eigenftunctions and eigeEnvalue-s in th~e case of the basic flow U = 0,
that is, for a mediumr at rest, can be given as a closed system. Since
thee eigenvalues are suitable foir approximative representations in
the -ase of' mo~re general basic flows also, we shall. derive themn here
briefly;. In the case of the basic flow U = O, the stability differential,
equation (1.2) is simplified to

;-(4) 2/f" +I a 4$ iaR C 4i" 4) = 0 (2.1)

wherr wt tshall den;ote- thel eigenv.alues by C, to dis~tinguishi them from~
thec eigenvalulea c ofr the; general stability differential. equation.
The equation is solve~d by each of thre functions

cosh as y inli; sinh a i~
O4I(;) = cajsh -my cosh; Q, 4 y sn y- ihm
cosh 2.I SiT sr a


NACA TMr 1417

if we put
m2 = a2 R C(2 .5)

The part cp(4l) = 0 of the boundary conditions is .identically satisfied.
The remaining boundary conditions #'(+l) = 0 lead to the related branches
of the eigenvalue: equation:

a tanh a = a> tanh a> in the case I

a coth a = m coth nu in the case II 24

The equations (2.4) have, for positive a, no roots Lo outside the
ima~ginary axis of the complex au-plane. With my, also iu, is an eigen-
value associated with the same eigenfunction. Thus it is sufficient
to consider only the positive imaginary eigenvalues a). If we designate
even. eigeafunctions by even subscripts and odd eigeaf'unctions by odd
subscripts, the equation (2.4) may have the solutions UjO, 02, my4, ..
in the case I and cul, a3 ,5 up in the case II. The eigenvalues
can always be made to form a monotonic sequence

CUO n1 03 2L;
0<--<-< <...

The associated eigenvalues Cn are according to equation (2-.5)

Cn iaR.(2.5)

They are, therefore, arranged in the order of increasing damping. The
nth eigenvalue may be estimated upward and downward by

a2 +<2 2< iaR ., C< a.2+% 26

From the representation (2.5), it follows that the eigenvalues C
become~ very large for aR ->0 as well as for n -sm. The same behavior
occurs, also, for more general velocity profiles U(y) because the main
parts of the stability differential equation (1.2) are then represented

NACA TM 14171

by the3 equation (2.1). We shall, now express this train of thought more
accurately by subjecting t~he difference c C of the eigenvalues c
to a more accurate estimate compared to the eigeavatlue C of equa-
tion. (;2.1), for more general profiles. With introduction. of the differ-
ential operators

NCrpl = (p(4) 2a~" + a4q~

LC~I= U((pl' a2~p>- U"'4~

nLPT1=cp"- a2~


theF stability differe-ntial equation (1.2) may be written in the form

cM 9 = LP N 9 -

Correspondingly, equation (2.1) reads


Utilizing thre .fact that the operators M and N ar~e self-azdjoint, we
obtain from these two equations thie relationship


PM 4 dy;- = + L day


If the normalization which is still open for Q, is fixed by the rule



Ir~-1 =


there follows, after introduction of the auxiliary quantities

j_, +1 ld

QM, m,. dy


i Q M # dy


(PMCe dy

NACA TM~J 1417

from equattion. (2.9) the representation

c -C = + q (2.12)

In this equation, C an may be :regarded as known by virtue of the
functions a, represented in equations (2.2). The eigenvalues Cn
have already been delimited in expression (2.6). For Q we obtain
directly the estimate

Q -1


In connection with a simultaneous estimate of the function
(9 4)" a2(9 (g), we obtain for q

for -- ->

If we substitute both inrto equation (2.12), we obtain, with consideration
of equation (2.5) and expression (2.6), the two partial statements

on Cn n ~+ 0 -

for CLR -->0 and for arbitrary order n


e, Cn 2 -1

U *dy + O for

n ->mo for fixed aiP


The latter estimate indicates thazt the eigenvalues o~n of the stability
differential equation. for sufficientlyr high order n finally tend toward
the eigenvalues C, of the "zero flow" withini the real part increased by
the mean velocity of the basic flow). (Comp~are eq. (2.1).) A mutual
coordination of the ei~cnvalues on to the eigenvalues Cn, however,
is by virtue of equation (2.15b), meaningful only due to the fact that
the difference Cen Cn+1l of the appro~ximation eigenovalues comes out
consl~cscideral larger than the estimated remiainder in equation (2.15b):
~For, b~ea~use ::' (2.5) and (2.6i)

U y + 0

@ >> 1

Q = O --


SCn Cn+1 ; ostn (2.16)

is valid. It should be mentioned that F. Noether (ref. 7, p. 239, for-
mula (28)) has already indicated an asymptotic representation for slightly
differently defined eigentvalues for unlimitedly increasing order,
although only intimating an argument which, leads one to expect con-
siderabDle difficulties.

We mention, furthermre, an estimate for the eigenvalues e indi-
cated by C. S. Morawetz (ref. 6, p. 580)

lc- e~n <:6 *(aR)1/2

where on is an approx:imative eigenvalue which (in our notation) is
determined by the equation


and corresponds more or less to our approximate eigenvlalue C, intro-
duced in. equattion (2.1). In the above estimae of Morawets, neither aR
nor n. ma become arbitrarsily large; in the first case, the eigenvalus
would shift into the exc-luded neighborhood of a = v yl vw = U; yl des-
ignates the wall); in the other case the estimate would become meaning-
lejs .;ince the be-havior of the quantity a for -nlimitedly increasing
n is not given.



In the special case of the basic flow at rest, U = 0, the behavior
of the eigenavalues on for un~limiftedly increasing aR~ is described
by the formulla (2.S) in. which the quantities an, no longer depend on
aR. InI deviation from this lawz, there results for more general velocity
pr~ufiles a behavior like

NACA TM 141'1

c (-1 --- for aR -tm (3.1)

where the complex valued qutn-tities Pn n~o longer depend on aRi. If
these eigenvalues are adjoined to the eigenvlues in equation (2.15a),
by continuous transition of a and aR, the ordering principle (1.4)
according to increasing damnping is lost even in special cases like the
rectilinear Couette flow. If we therefore desire that the subscripts
of Lthe eigenvalues on remain. unchaned for coninuous transformation
of the limiting cases aR1 ->0O and aR -,am into one another, we must
actually carry out this procedure which presupposes a general solution
of the eigenvalue problem or a solution which is a~pproxim~ate only inso-
far as the individual eigenvalues still remain distinguishable from one
another. We succeeded in, obtaining a solution in this sense only in the
special case of the rectilinear Couette flow. It will, therefore, form
the subject of the following section.

After in~sertionl of the velocity profile U = y of the rectilinrear
Couette flow into the stability differential equation (1.2), the latter
can be reduced, by means of the substitution

to the Bessel differential equation in the auxiliary form

v" iaR(y c)C +~V sq= (3.5)

In order to arrive, through the boundary conditions, at the eigenvalues,
we must invert equation (3.2) in the form

cp(y) = : t(?) sinh a(y ?) d j4

The boundary conditions 9(1 '-)=Othen are identically satis-
fied; the remaining boundary conditions cP(+1) = '9'(+1) = 0 require
that the two eluationrs

.+1 +1

j -1 J( sinh ay dy = 0 1 4(y)cosh amy dy = 0 (3.5)


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By means of' the substitution

and yk =c +

=El with E = GR,-


y yk

equation (73.) mayr be transformed into the differential

the differential

i a;k+ qq~ = o


*I(s) and q,(n)


are assumed to be two suitable fundamental solutions of this equation,
and 4-1 81 is assumd to designate the values which are, because of
equations (3.6;), associatted with the walls y = -1 and y = +1

'1-1 = 1

- c -

1 1= 1 c ~2


there follows from equations (35.) the eigenvalue equation






. -1


smn acq dq

.I a

4 I(?)COSh aE d? -

Q (?)cosh ae? d'l = 0


For further treatment of
sequence of functions Ani 1)

this equation, the introduction of a
by the Laplace integral

qz+1 -

.n-1 z

A,() = 1


10 NACA TM 1417

is advis~able in which -the path of integration Ai runs from infinity
to infinity in the manner drawn in figure 1. The functions Ant?)
~stisfy the differential formla

=( An+( q ( 5.12)

and the recursion formla

i A +3 q *I An+1 + n An = 0 (3.13)

by means of which. all1 the functions An(4) and their integrals and
derivatives may be constructed recursively from the three b!asic functions

AO( r) Al( ?) A2( q) (5.14)

The significance of LhZEe functions An(4T) for the stability prob-
lem lies in the f'act thatt the two particular solutions of the differential
equation (3.7) needed in the eigenvalue equation (5.10) can be repre-
sented in the form

2xi 2n'i

(?)n() = AlII('I) =e *A e(.5

Th~at the differential equation (3.7) is satisfied follows from the
formlas (3.12) and (3.13). The linear independence of the two functions
follows frocm the fact that the Wronski determinant, which is constant of
course, does not disappear at the point n =- 0.

The basic functions Al( 1) and A2(?) = Al'(Bl) have been numeri-
cally tabulated (in somewhat different notation) for a quadratic point
grid with the mesh width 0.1 within the circle /T 6 of the complex

r)-plane by H. H. Aiken (ref. 1). The basic functionAO)= Algd
can be determined from it by a numrical integration.

Outside of this table, the b~ehav.ior of th~e functions Ar,(?) may be
infe~rredl from the s3:-rmp~totic series representation


NACA TM4 1417 11

which is valid for Ir -em ini the ang~le space 1 + 8~ arg il9 8
with arbitrarily small 5 > 0. AcTCOrding to H., Holstein (ref. 4), the
first coefficient of the series i3

-G~d (3.16a)

a 1 e

TIhe asympto~ti c eries are obtainable directly from the Lalace integral
(3.11) bjy means of Riemann~'s saddle-point method.

For the representation of the eigenvalues, the zeros rlN of the
function AO(?) are necessary,. An ajsymptotic calculation of these seros
for I ip> s not possible directly by means of expression (7.16),
since the zeros would move ouit of' the range of validity of this repre-
sentation. We avoid this difficulty by applying the second relations
obtainable from the integral ($.11)

An ie e


. A( / e


(* = conjugate-conplex values)

4a= Pu(l)

. n
+ e

i n '
n al

. Ann

Anfl?) + e


P (g) o~f the degree -n, satisfy the same recur-

where the polynomriias
sion fo~rmula (3r.15)


1 -Pn,+) n+,1 + n-Fn = O

with the initial elements

PO = 1

P1 O

P2 = 0

Cor.binatiion of fo~rmulas (j.16~) and (3.18) then yields the asymp.totic,
represe--ntati:n valid fo~r I_, in the angle space larg zl < n 6

NACA TM 1417

cosj z5/2 x ~

i 6


Hence, there follows, for the zeros 5N of AO ~N) = 0 which, according
to equation (3.17)r lie in pairs symmetrically with respect to the
straight line arg = =ix/6, the asymnptotic representation

?N (jEn)2/ 5 6 3n -

n = 1, 2, 3..... (j.21)

The value of the lowest pair of zeros was calculated to be


according: to the table of Aiken (ref. 1).

For further treatment of the eigenvalue equation. (5.10), it is
advisable to expand the functions sinlh asy1 and cosh ac q into their
Taylor series, and then to interchange the summtion with integration
which is justified by a theorem of Bromwich. (Compare ref. 2, p. 398.)
The series obtained converge, according to theorems of the Laplace trans-
formation, for each value of arE. If these series are broken off after
the first terms, provided with residual terms, and substituted into the
eigenvalue equ~ation (3.10), the latter is, for this reason, and with
conslideat ic~n of equation (3.1rl), simplified to

A ~-?f A Y-]*1l
= O a2E )

fOr I ? constant


What happens now when aR increases beyond all limits, that is,
when E tenyds toward 0? Because of the relatio-nship following from
equation (jt.9), pl- -1 = 2/E, at least one of the two quantities
~11 -1 must te~nd toward infinity for E 40t, on a parallel to the
real axis. It is sufficient to assume~ this regarding 71, because in
the other case everything would form a mirror image with respect to the
i-aginrsly alxis of the rl-plane (ss essentiallyr occurs in. the Couette flow

?N = 4.257 ei x 0.270 ]

AO 91 )- AO -1 )

A2 T1 A2 -1

NACA TM 1417

where with C, also, -CX. is an eigenvalue). With consideration of
the asymptotic behavior (eq. (3.16)), the eigenvalue equattion (3.24)
then is simplified to

iAO ~-1) E OE2 For 6 << 1 (3.25)
A2 r-1) 2

From this formula, we recognize that q-1 for E -+0 must tend

toward the zeros qN of the function AO('1) estimated in expres-
sionls (3.21). We thus obtain for the eigenvalues c, with considera-
tion of equations (3.9), the asymrptotic representation

c + 1 -Eq+ O( E
with qN from AO N)= O

Thuts we hav:e proved the previously given eigenvalue formula (3.11)
for the special case of the Couette flow.

In order to follow the variation of the eigenvalues c over the
entire range i0 4 aR < I~, we must go back to the eigen,alue equa-
tion (5.10) or its approxtimate for in equation (3.24), with the fune-
tions AO(1), Al( 1), A2('1) to be assumed as known. We have accord-
ingly calculated the 1;2 lowest eigenvalues c as :functions of aLR for
a fixed value a = 1 an represented themo in figure 2. The variability
of the eigenvalue curves with a is only slight and becomes, for
instance, for aLR ->mn with E Small Of thne order 0 (a2 E ) .

The eigenvalue spectrum of the rectilinear Couette flow has been
discussed already by L. Ho~pf (ref. 5). 'Hopf replaced, more or less on
the level of our eigenvalue equation (3.10), the solutions $ JII
represented by him by :Hank~el functions of the order 1/3 by the first
terms of their asymrptotic series (3.16), whereby the eigenvalue equa-
tiojn was simplified to an algiebraic equation of auxiliary arguments and
circular and hyp~erbolic functions. However, since Hopf committed certain
errors in the asymptotic representations of the Hankel functions, his
results require partial corrections. Although these changes are hardly
significant for small values of arR, the vatlues of, for instance, qJ
in a folrm:ula correspondilng to (3.26) undergo a considerable change. Th~e
topological connection of the eigenva~lue curves c = e(adaR) a~lso
appears different to us from what it appeared to Hopf1. However, the


qualitative p~icture of the eigerafunctions, the physical conclusions
drawn from it, and the main result that all2 o~scillatio~ns are damped -
remain the samne.



For a basic flow with symmtrical velocity profile

U(y) = U(-y)


the stability differential equation (1.2)ua~lwayrs has a f~undamnental
system of four solutions (1, I21 (P3 so th~at

$2 y

1 (y



If a linear combination, of these solutions is to satisfy~ the boun~dary,
.-jnditions of equl~ations (1.5) in thle sequence W-)=0 '-)=O
9(+1) = O, Q'(+1) = 0, the following determinant, simplified with
considrac-tion of the s:,mme~tries (4.2t), mst disappear:


i( -1)




Since this determinant ma be written as the products ofT the two-co~lumnJ


$(y) are even functions ofy

5(y) are odd function; o~f


cp (-1) (1)
6 (-1 9( 1)


NcACA TMr 1417

one obtains, by equstinig one of the two factors to zero, one branch of
the eigen-.alue equation each time. For this reason the~ eigeznfulnctions
can be either only even or only odd, with the respective ei~genvalues c
6enerslly being different.

In order to arrive fromr these equations at asymnptotic eigenvalue
form-ula;, we shall determine the: four fu.ndame-ntal solutions (4.2)
91 Sl in such a manner that they are available for appropiriate
asymptotic expansions. We find that the fundamental solutions desccribed
by W. Tollkien (ref. 12), "Asym-nptatic Inegrati~on of the Stability Differ-
enltial. EquaLtion", the asymrptotic representations of which are provided
with residual-term estimates, are suited to the problem.

In order to establish the connection of these fundamental solutions
with ours, it is indispensable to discuss first the concept of "friction-
less approxiimatiorn." The quest for solutions of the complete stability
differential equatio;Cn (1.2) which for aR -+, my together with their
derivatives with respect to y, tend toward a limiting function

lim cP(y,aR) = X(y) (4.4)
aR -,

leads to the so-called "frictionless differential equation"'

(U c)(X" a.2X) U'"X = 0 (4.5)

which must necessarily be satisfied by such limiting functions. If we
want to use the solutions of this frictionless differential equation
fr'_r th~e approximation of the solutions of thre completed~ differential
equationr for aR my, we must not disregard the orange of validity of the
ti~L-raunday-value statements in, equation (4.4) in the comrplex y-plane.
A3ccor~ding to W. Wasow (ref. 15), the following theorem is valid with
respect toi this:

"lOfr t~he four~2 fundamental solutions (4.;2), one even and one odd
solution can be determined in, each. case so that with two appropriate
frictri unless solutions 1i(y) anid j2(;y) the approximations

9 1 1()+OX1(y) = odd function of

1 () =12() +0 2(y) = even function of y 46

NACA TM 1417

in each fixed interior of a io~uble region (I + II) or (II + III) or
(III + I) are valid and become invalid, in each case, in the comple-
mentary third region. III or I or II. The sam is true for the derivra-
tives with respect to y." (Compare fig. 3.)

The boundaries between the regions I, LII, and III satisfy the

Re 1l~ J-U e = 0

if yki denotes the "critical point" defined by

U Yk) = c Re yTk < O (6.7)

For more details regarding the regions I, II, and III see W~asow (ref. 15).

The frictionless differential equation (4.5) has at thle critical
point, U yk) = c, a singular point with regard to dete~rminateness. Two
fundamental solutions take the form

X1tY) = (v k) 1 Y- Yh)

X2 7) = Pl ;Yk) U1 7 ~ k) ( Ik)1n~! y Yk) 49

if P and P denote power series with the begiruning
-1 -2

Pl(z) = 1. + z + O z2) P (z) = 1 + O iz2) (4.9a)
1 2Uk -

(Co~r.pare W. Tcolrie~n, ref. 12, p. 35.) The commn radius of convergence
of these power ;Frles is limited either by the radius of convergencee of
a corresponindl series for U c or by the next-adjacent zero of U c
as a cingulzr point of the differential equastion.

NACA TM 1417

For the further developmnt it is a~dvisable to introduce a sequence
of factions Bn1(T) by the Laplace integral

By(D)~~ ~~ =-e .E1(In a + ^)dz (.0

which is comparable to equation (3.11). In it, 8 = 0.5772 .. denotes
the Euler constant. The path of integration B runs, in the manner
indicated in figure 4, in the complex z-plane cut open along (O,-ioo)
from infinity to infinity.

TIhe functions Bl(tl) satisfy the differential formula

---n- = Bu1(B)(4.11)

an the recursion formla

i Bn+3 + '1 Bn+1 + n3 Bn= n (4.12)

in which Pn(rl) are the polynomials defined in equation (3.19).

By means of these two formulas all the3 fuctions B (rl) and their
derivatives anrd integrals can be constructed recursivel~y from the three
basic functions

BO(A) Bl(4) B2( t) (4.13)

By means of the representation

B (l) = 2ni AOt7) (1n~-)A(1) A ~(-I*) (4.13a)

(* = conjugate-complex value), the basic functions B1 and B2 can be
reduced to the funtions An. (Comare W. Tollmien, ref'. 10, p. 27.)

The significance of the functions En(q) for our stability eigen-
value problems lies in the fact that the function Bl(?), because of
equations (4.11) and (4.12), satisfies the differential equation

NACA TM 1417

d B, d2 B1
i + p 1
du dq2


which, with the designation "differential equation for the friction
correction," has been introduced as an essential constituent into the
asymptotic integration of the stability differential equation bjy
W. Tollkien refss. 10 and 12).

After these preparations, we turn to the four funmdam~ental solutions
97,I r7 (P777, CPV of the complete stability differential -Iquation
constructed by Tollmien, regarding its ability to be expanded asymp-
totically. According to W. Tollmien (ref. 12, p. 77) these four solu-
tions ma~y be determined, with. use of the substitution

with 6 = anRd y

frOm U Yk) = c


y ;Yk = E'

in such a manner that they have in a fixed interior of the q-plaLne (com-
plex for reasons of analytic continuation) as well as in every fixed
interior of the region II of the frictionless approximation (compare
eq. (4.6a)) the following asymptotic representations:

cp (y) = X1(Y) + 0 6 )


cp,,() = P2 E )

or in every fixed

+ 6) G (4) p+ q n e +E In

id interior of II(.1b

rII(Y) = x2(Y) + o0e5)


'SIII(Y) = A-1(;1) + O(E)


is valid. Finally, there applies, according to W. Wasow (ref. 15),
quo-tient -asymp~totically in every fixed interior of II ( compare eqs. (4.6))

in ) I constant

NJACA ?TM 1417

777 (y), Wy(y) constant --5-c)


Corresponding formlas are valid for the derivatives.

For further treatment of the eigenvalue equation (4.5), we must
express the fundamental solutions $1 ''4 used in it by the above
fundamental solutions ... If for the latter, the represen~ta-
tions (4.17) are used~ immediately, and with the residual terms in each
circle @constant for a ->0 being valid, the result reads

'1,21Y -d1,2 PIII(y) = 1+E *~ InE +8, B-()+0e1

U~ X2(0) U; X (0)
with --S S = -X i
Ukl 1 x O U 2,

X1 and X2. (Compare eqs. (4.9))

fromr the frietrionlless solutions


E = A-1(?) + 0(E)

$ l(y) = cPI(y) + OEi e


is valid. Corresponding formulas are valid for the derivatives.

If we nowcl write the two eigenvalue equations obtained in equa-
tion (4.5) as a product of three factors, for instance,

*exp idR re U -e

cp (-1)

;20 NACA TM 1417

(9 = arbitrary constant), the zeros of the two first factors do not
make a contribution to the eigenvalue configuration since the; are
compensated gy corresponding poles of the third factor unless the
derivative rp (-1) should disappear simultaneously. It is therefore
sufficient to find only the zeros of the third factor. After insertion
of the approximations (4.21_) we thus obtain

f UIn t6+S1,+81,2 +BO

1+e.~ 1 n+1+S1,2) +B-d1 il

~III -

+ O(E In E


( = p-1

-0 (E n E)

e n E + 1+ 81,2 + BO 1)]

1 f ln + c1 E + 1 + 1,2) -1L ~

AO( )

with n = q1 = -1 k)/e and S1,
from the frictionless solutions. The
next-~;igher approximation in equation

82 according to equations (4.21a)

fun~ct ion JI(1)
(4.21b) reads

stermming from a


A,1 (4)

and ma be reduced, by means of the formulas (3j.12) and
three tabulated basic functions AO0, A1, A2'

(5.15), to the

How do the eigenvalueis c behave if in the eigenvalue equa-
tion (4.22) we let aR -sm, that is E ->0O? Evidentlyr r-1 then tends
towi~ardj the zeros ?[ of the function AO(?). By Tayrlor series expanded
about these zeros, there follows more exac-:ly

A1 I C +

U ~

n = ] +
-1 11

NACA TM 1417

The eigenvalues then behave asymptoticall~y like

c U(-1) = -U~ EN -1 1 6186+ 6 4.

with r from AO(6 N = As a supplement to equations (5.21) we shall
give here a few zeros rlN and values iA2 Al:

5N ) A?l N

-4.1;22 + i 1.065 -1.686 1 1.222 For >> ]1
-2.983 + i .037 +1.902 + 1 0.851
-6.8 + i .2.5 -2.2 1.5 there applies
-5.5 + i. 4.5 +2.4 + i .1.14 A
Al UN) V

The remarkble fact about the asymrptotic eigenvalue formula (4.23)
is that it is transformd into the corresponding :formula (3.;26) after
substitution of the velocity p~rofile U = y of the Couette flow, although
the two formuilas were derivred under completely different assum~ptions.

The asymtotic eigenlvalue formula (4.23) is already so greatly
reduced that it no longer permits a distinction of the eigenvalues c
which are associated with even or odd eigen functions. For this, we
crust go back to the moe exact formlat (4.22) in which the character-
istics "even"" or "odd"" of thre eigenfunctions are taken into consider-
ation by rreans of the constants S1 atnd 82' to be determined "without

We. have used the eigenvlu equation in. the form (4.22) also for
the n~umerical. calculation of the eigenvalues c in the examples treated.
We selected as examples the tw-dimenrsional Poiseuille flow and a flo
with- an inf'lect~ion-poin~t profile. We represented the variation of the
four lowest eigenva~lues as funtions of R for a fixed value of a
in figures- 5 and 6. The numericlal calculation itself is after redue-
tiocn of the nonalgebraic elements contained in. the eigenvlue equation
to the three tabuclated basic functions AO, Al," A2 and to the fric-
tionless solutionss a prob~len involving numerical methods, the details
of which cannot be discussed here. We shall mention only the following
approx>icate representation of the frictionless constant S2

NACA TM 1417

U.S2 A .a-2 + O(1) for a << 1 with A = 0----~---
U' I (U c) dy.

(Compare W. Tollmien, ref. 11, p. 100), which may be applied ad:anta-
geously for small values of a.

The subscripts for the eigenvalues c obtained frcm equation (4.22),
in. the sense of a continuous connection with the limiting5 case aR -0,O
remain an open problem here. In the range of validity of equation (4.22)
alone, a generally valid choice of subscripts according to the rule
Im en+1) Lm en), that is, according to increasing damping, cannot in
principle be carried out, either. The zeros in equation (4.25) can be
ordered according to the increasing imaginary part, but the~ Im 0 )
curves may penetrate one another if a is changed.


Determination of the Extcited Eigenvalues

Let the approximation (4.6) by means of thle frictiornless solutions
be suited either to the double region I + II (compare fig. 3) or to the
double region II + III whereby the logarithmic term is alwayse uniquely
determined in the frictionless solutions. Applyring the approximations
(4.17), we then obtain, by way of the eigenvalue equations (4.3), eigen-
value s c which, for aLR --, tend toward the so-called "frictioniess
eigen values" c(0)(a) which are defined by the boundary co~ndition
X1(1) Oor ji2(-1) = 0 of the odd or even frictionless solutions
X1' X2. Thne following general statements may be made .regarding these
frictionless eigenvalues, limited by the range of validity; of the
bioulndry-value expressions (4.6), according .to W. Tol~mien (ref. 11),
partlly on the basis of the "Rayleigh-Tlollmien theojrems:"

"For velocity profiles without turning points, no excited friction-
less eigen.alu~es are possible. The approximation (4.6), associated
with the damed frictionless eigenvaluies, must always take place in the
interior of the double region I + II."

"For inflection-poin~t profiles, there always exist excited friction-
less leienvalues associated with an even eigerafunction."

NACA TM 1417

Beyond these general statements, frictionless eigenvalues associ-
ated with an oIddl eigenfunction were not fou~ind in any of the examples;
neither did we find eigenvalues such that the associated approxima-
tion (4.6i) would have taken Iplace in the interior of the double region
II + III. As examples, we chose the two-dimensional Poiseuille flow as
representative of a profile without an inflection point, and the inflection-
point profile U =(2e 1 + 2 (E) xos 3T~y. The frictionless
eigenvalues c, fouLnd only associated with an even eigenfunc-tion, are
represented in figures 7 and 8. The range of existence of these eigen-
values is always given by an interval O < osat o h rc
tionless eigenvalue c, T~ollien (ref. 11, p. 100) has set up the
following ap~proximte formulas:

crU'( -1) JtU"
a~~~adC r 1iq (cr)l =@ -- with U yK)==cr

O (5.1)

We now seek the connection between the frictionless eigenvalues
and those discussed up till now. Th-e closed solution, in the case of
the Couette flow, canot give an answer to this problem because the
frictiolnless eigenvalues in question do not exist there at all. :How-
ever, it is po3ssle to insert the frictionless eigenvalues into the
equation (4.22) and thus to interpret them as a limiting case within
the eigenval~ues (eq. (4.23)).

Let us, therefore, perform on the eigenrvalue equa~tiojn (4.22) the
limiting process aR! -smo, that is, E ->0 for constant (-1 yk) = Eq Jl
thle ,Istificiation of this procedure is based on the equation preceding
(6.2).By, means of the asymptotic formlas (3.16) there then. follows
for aR -o

I + -1 yk) I 1-y ,

Th-ese are, however, precisely the first Taylor term of thet frictionless
eigenvalue equation X1-)=Oor X2(-1) = O which would be obtained,
according to the significance (equation (4.21a)) of .S1, S2J in thei case
of Taylor expansion in the sense of the series (4.9).

24 NACA TM 1417

On the basis of this finding, the determination of the excited
eigenvalues (Ime > 0) can be simplified. Since, according to the results
of the second section, excited eigenvalues can1 appear only within the
first eigen~values of finite number and for sufficiently large values
of R, it suffices to examine equation (4.22) -with respect to excited
eigenvalues. For this, the following alternative is valid: Excited
eigenvalues can be (a~pproxi~mately-) d~etermined either by the friction-
less boundary-value problem in combination with sufficiently large values
of R, or they lie in a neighborhood of c = U(-1) and can be determined
by means of one of the equations (4.22) or (4.23) as associated with
finite values of q-l'

As follows from this for sufficiently- large values of R, but as
was confirmed in the examples for th~e smaller values of R also, the
greatest excitation for inflection-point profiles is always combined with
the frictionless eigenvalue or its continuation toward smaller R
values. Hence, there follows the well-known fact that, in the case of
turning-point profiles, the stability behavior mayr be concluded even
from the frictionless differential equation alone. Let us compare to
this the calculation of the frictionzless eigenvalues of' G. Rosenbrook
(ref. 9) for an inflection-point profile which he had measured in a diver-
gent channel.



In order to Judge the variation, with. time of a disturbance, we shall
3ecomposeju the latter into -partial waves of the type in equation (1.1).
It is then. necessary to know the variation of the amplitude cF(y) over
the channel ~widthi. We consider here ornly the case of very large values
of a~R.

For the Couette flowr, there follows, by~ equation (5.4), in the
notation of equations (3.6), (3.9)t and (3.15) for E << 1, that is,
aR >> 1, as approximaste expression for ~the eigenfunction

iW(y) = F(7) F qe (6.1)

F(7) = (6.2)

NACA TM 14c17

The boundary conditions 9(p(1) = O are identically satisfied; the
remuaining boundary conditions are identical with the eigenivalue equa-
tion. (3.24). As follows even from the differential equation alone,
?*~( -y) is an eigenfunction associated with the eigenvalue -c0. We
calculated accordingly for a = 1 and R = 103 the eigenfunction
associated with the eigenvalue e = -0.7 iO.3 and represented it in
figure 9.

It is striking in this figure that the essential changes of the
eignfncionocurin a layr -1 y yO which could be defined
pe-rhaps by the angle spac-e arg nr = x6 of the strong increase of
A-1(?). In the variable y this "inner friction layer" is, according
to equations (3.6) and (3.26), approximately

-1 y 6-1+E-e]+ 1 <165

This representation shows that the width of the lawyer increases with
grow ing order n. of the eigerafuLnctions; the magnitude~ of damping
increasing simultaneously. The velocity of the associated disturbance
wae is approximately equal to the velocity of the basic flow in the
center of the layer. Furthermore, the thickness of the layer tends with
a towad zero. The physicaEl interpretation of this situation signifies
according to Hopf (ref. 5, p. 57) "that any arbitrary disturbance for
lartIe v.aluesJ of R is damped in such a manner that, finally, distances
seem; to manate only from the walls, without mutual interference a
behavior which reminds one of frictionless fluids."

For more general basic flows, Tollkien (ref. 12) set up an approxi-
F~ate expression for the eigenfunction; in it, one~ can recognize again,
in the case of damping, an "inner friction layer" which would have to
be defined by the angle space %/6 I arg rl 6 5Jrl6 of the great changes
in increase of B-1(q) or A().In the variable ;y, this layer is,
accolrdinlg to equations (4.16),

or~ +\ -Soi r Sci
-1 + =y =-1 + ----- E ->0 (6.9)
U'(-1) U'(-1)

whenc-e we obtain. for the higher eigenvalues, according to equation (4.23),
again the formula (6.3) for the C~ouette flow. If, however, frictionless
dam~ped e~igenva~lues e in the sense of section 5 exist, the inner fric-
tion lawyer, expressions. (6.9), retains also for the limting process
S- a finite thickness and a finite distance from the wall. We calcu-
lated, for this latter case, the even eigeafunction in the example of
th-e Poilseuiille flow, for a = 1 anld R = 7.7 x 105, and represented it

26 NACA TM 1417

in. figue 10. The eigenvalue c = 0.1178 i x 0.049 h~ardlyi deviates
from the frictionless eigenvalue associated with a = 1.

Comparing the inner friction layer with the boundary lawyer, we muay
say that the boundary layer represents that flow region in which the
behavior of the lamninar basic flow is decisively influenced b; the inner
friction, whereas the inner friction layer indicates the region where
the disturbance is decisively subject to the influence of the f'rictionr,
since outside this layer the disturbance can be determined without

Translated by Mary L. Mahler
National Advrisory Committee
for Aeronautics

NACA TM 1417


1., Aiken, H. H-.: Tables of the Modified Hankel Functions of Order One-
Third and of Their Derivatives. Harvard University Press, Candbridge,
rlass., 19345.

2. Doetsc~h, G.: Thleorie und Anwendung der La~place-Transformation .
(Berlin), 19537.

3. Haut, 0.: Uber die E~ntwicklung einer wilkiirlichen Funkrtion nach den
Eienfun~kt~ionen des Turbulenz-projblem~s Sitzber. d. M~inchener
.Akad., Mathem. phys. K1. 1912.

4. Holstein, H.: Uber die aii~ere und innere Reibungschicht bei
Stijrungen lamninarer Strijmungen. Z.1 angev. Mah. Mech. 1950,
pp. 2=1-49.

5. Hopf, L.: D~er Verlauf kleiner Schwringnen auf einer Stridmung
reiblender Fltasigk~eit. Ann. d. Phys. 1914, pp. 1-60.

6. l!orawetz,J G. S.: Thne Eigenvalues of Somet Stability Problems In~volvingt
Viscosity. J~ourn. of Ration. M~echan. and Analysis. Vol. 1, 1952,
PP. 579-603.

7. Wdoeth~er, F.: Zur .Asymptotischen Behandlung; der statiolEiren Lb'sungen
im Turbulenloproblem.m Z. angew. Math. Mech., 1926, pp. 232-243.

8. Losrd Rayleigh: Scient. Papsers III, pp. 575-584.

9. Rocsenbrrook, G.: Instabilitait der Gleitschicht imn schwach divergenrten
Kanal. Z. ang~ew. MYath. Miiech., 1937, pp. 8-24.

10. olkenW.:Uber die Entstehung de~r Turbulenz. Nachr. d. Ges. d.
Wissensch., Gottingen 1929.

11. TolLnien, WJ.: Ein allgemeines Kriteriumn der lInstatbilitait lampina~rer
Ge schwindigkeitsverteilunen. Nachr. d. Ges. d. Wissensch. Cittingen

12. Toltr-.ie~n, W.: Asymtotische Integration. der Storungsdifferential-
gleichunzg ebener laninarer Stromungen bei hohen Reynoldssehen
Zahlen. Z. angew. Miath. Mech. 1987, pp. 37-50 and 70-83.

28 NACA TM 1417

15. Wasow, W.: The Complex Asymptotic Theo~crie of a Fourth O~rder
Differential Equation of H~ydrodynamics. Ann. of f-lathi. 49,J
1968, pp. 852-871.

NACA TM 11417


Figure 1. Path of intgratIon A in the complex z-plane.

Figure 2. Riectillnear Cojuette flow. The twelve lowest eigen-
values c as functions of R for a = 1.

NACA TM 1417

Fig~iure 3. The regions I, II, and III in the complex. y-plane.

Figure 4. Path of integration B in the complex z-plane cut open
along (0, -ioo.

NACA TM 1417r

0.4 t-





O 4


105 106 R

-0. 1


Figure 5. Two-dimensional Poiseuille flow. The four lowest eigen-
vaues c as functions of RZ for as = 0.87.

32 NACA TMI 141[

2b ~Um
0.2~ Umb tUs

O 105 106 R

-0. 1-

Figure 6. Inflection-pointi profile. U=(r 1) + i2 !cos- yr .T
The four lowest eigenl-a.1ues c as fu~nctions of Rt for o = 0.5.

NACA TM 1-417

r= 0.5

0.4 Cr

-0.1 -

-0.2 `


Figure 7. Two~3-d~ilen pension P~oiseuille flowv. The frictionless eigen-
Ialuez c associated wit~h an even~ -ilna Mucti:n, as a function


0 2 0 4 0.6 Cr
-0.05 -a

-0. 1-

U = J 1) + (2 \J)cos y ~.

Figure 8. Inflection-point profile.

The frictionless eigenva-lue c associated wit an even eigen-
functio~n, as a fun~ct~ion of a.

NACA TM 1417

Figure 9. Redziinea~r Couette~ Gowl. El a-en fun~et in
at = 1 and R= 10o associated wiithi the eigen;value
c = -0.70 ix 0.30.



-01 01

P '(yr) for

2 y

Figure 10:.- Tw~o-dimensioinal Poiseuille flow. Eigenlunction Ip'(y)
for a = 1 and R = 7i.7 x 10" associated with the eigenvalue
c = 01.178 i x 0.049.


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