The excitation of unstable perturbations

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Title:
The excitation of unstable perturbations
Series Title:
NACA TM
Physical Description:
63 p. : ill. ; 27 cm.
Language:
English
Creator:
Pretsch, J
United States -- National Advisory Committee for Aeronautics
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NACA
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Washington, D.C
Publication Date:

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Subjects / Keywords:
Aerodynamics   ( lcsh )
Laminar flow   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
With the aid of the method of small oscillations which was used successfully in the investigation of the stability of laminar velocity distributions in the presence of two-dimensional perturbations, the excitation of the unstable perturbations for the Hartree velocity distributions occurring in plane boundary-layer flow for decreasing and increasing pressure is calculated as a supplement to a former report. The results of this investigation are to make a contribution toward calculation of the transition point on cylindrical bodies.
Bibliography:
Includes bibliographic references (p. 39).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by J. Pretsch.
General Note:
"Report date September 1952."
General Note:
"Translation of "Die anfachung instabiler störungen in einer laminaren reibungsschicht." Bericht der Aerodynamischen Versuchsanstalt Göttingen E.V., Institut für Forschungsflugbetrieb und Flugwesen, Jahrbuch der deutschen Luftfahrtforschung, August 1942."

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University of Florida
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sobekcm - AA00006196_00001
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Full Text
)cr~h-13Wq













NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1343


THE EXCITATION OF UNSTABLE PERTURBATIONS

IN A LAMINAR FRICTION LAYER*

By J. Pretsch


With the aid of the method of small oscillations which was used
successfully in the investigation of the stability of laminar velocity
distributions in the presence of two-dimensional perturbations, the
excitation of the unstable perturbations for the Hartree velocity
distributions occurring in plane boundary-layer flow for decreasing and
increasing pressure is calculated as a supplement to a former report.
The results of this investigation are to make a contribution toward
calculation of the transition point on cylindrical bodies.


OUTLINE


I. STATEMENT OF THE PROBLEM
II. THE GENERAL DIFFERENTIAL EQUATION DESCRIBING THE PERTURBATION
III. THE SOLUTIONS cp, c2* OF THE DIFFERENTIAL EQUATION DESCRIBING
THE FRICTIONLESS PERTURBATION FOR FINITE EXCITATION
(al Binomial Velocity Distribution (Pressure Decreasel
(b) Sinusoidal Velocity Distribution (Pressure Increase)
IV. THE FRICTION SOLUTION p3* FOR FINITE EXCITATION
V. STATEMENT AND SOLUTION OF THE EICENVALUE PROBLEM
VI. RESULTS OF THE CALCULATION
VII. DISCUSSION OF THE RESULTS
VIII. SUMMARY
IX. REFERENCES







*"Die Anfachung instabiler Stbrungen in einer laminaren Reibungs-
schicht." Bericht der Aerodynamischen Versuchsanstalt Gottingen E. V.,
Institute fur Forschungsflugbetrieb und Flugwesen, Jahrbuch der deutschen
Luftfahrtforschung, August 1942, p. 154-171.







NACA TM 1343


I. STATEMENT OF THE PROBLEM


If one wants to make a theoretical calculation of the profile drag
of bodies in a flow for a certain direction of air flow, one must know -
in addition to the pressure distribution the position of the transition
zone where the laminar boundary layer becomes turbulent. The separation
point of the laminar layer forms a rearward limit for the transition
point on a section of this body surrounded by the flow. It lies in the
region of the pressure increase at the point where the velocity distri-
bution in the boundary layer has the wall shearing stress zero. This
separation point is a fixed point of the profile in the flow, the
position of which does not shift due to a variation of the Reynolds

number Re = U0t (U, = velocity of air flow, t = chord of the body).
V
One may calculate it according to the well-known approximation method
of Pohlhausen which, for prescribed pressure distribution, provides for
every profile point a velocity distribution of the boundary layer with
a certain form parameter X (\ = -12 separation. As forward limit for
the transition point, one may take the stability limit of the laminar
layer with respect to small two-dimensional perturbations which were
calculated in references 2 and 3 according to the method developed by
W. Tollmien (ref. 1). According to this method, there exists for every
form of velocity distribution in the boundary layer a so-called critical

Reynolds number Re*cr = -) (Ua = local potential velocity,
'cr
6* = local displacement thickness) below which all perturbations are
damped; its value increases as the form of the velocity profile becomes

fuller (with increasing X). Where the Reynolds number Re* = Uab

formed with the actual potential velocity Ua and the actual displace-
ment thickness 6* exceeds this critical Reynolds number, there begins
the instability of the boundary layer. In contrast to the laminar
separation point, this stability limit is therefore not fixed on the
profile for a normal pressure distribution but travels forward toward
the stagnation point on bodies in flows with increasing Reynolds
number Re (see fig. 1).

The actual transition point which lies between these two limits -
stability limit and separation point is known to likewise shift forward
with increasing Reynolds number Re. That it lies only a certain
distance behind the stability limit (as shown by a comparison of experi-
mental transition points and theoretical stability limits) is plausible,
because the excitation of the unstable perturbations starts only at this
limit point of the stability and must obviously have attained a certain
degree before the instability further downstream leads to the breakdown
of the laminar flow configuration.







NACA TM 1343


For this reason it seemed necessary to calculate, or at least to
estimate, for the velocity distributions in the laminar boundary layer
in the entire instability range of the latter, the excitation of the
unstable perturbations as well, with the objective of making an improved
calculation of the transition point possible.

For the velocity distribution on the flat plate in longitudinal
flow (Blasius profile), H. Schlichting (ref. 4) has already determined
the excitation quantity as a function of perturbation frequency and
Reynolds number in a part of the instability range; and for the velocity
distributions in the region of pressure increase, W. Tollmien (ref. 5)
has explained the behavior of the excitation (in first asymptotic
approximation for very large Reynolds numbers) in a very general manner,
neglecting the effects of internal friction.


II. THE GENERAL DIFFERENTIAL EQUATION

DESCRIBING THE PERTURBATION


Since the bases for the method of stability investigation have been
discussed in detail in an earlier report (ref. 3), we can refer to the
results attained there.

Let U(x,y) and V(x,y) be the tangential and normal components
of a plane steady boundary-layer flow; let x denote the length of the
arc and y the normal to the profile contour. The stream function of
the two-dimensional perturbation motion which we superpose on this
basic flow is assumed to be

t(x,y,t) = cp(x,y)ei(x-ct) = c(x,y)e itei(x-rt)

c = cr + ici pi = cia Pr = cr(


here t denotes the time, a the spatial circular frequency of the
perturbation, the real part cr of c its phase velocity, and the
imaginary part ci a measure for its excitation (ci > O) or damping
(ci < 0); besides pi is the logarithmic increment of the excitation
of the perturbation amplitude, and Or the circular frequency in time
of the perturbation.







NACA TM 1343


If we substitute the motion originating by superposition of the
boundary-layer flow U, V, and the perturbation motion

u = q'eia(x-ct)
u = e -
ay
S(2)

v- =- = i + acp(x-t)



into Navier-Stokes' differential equations, we obtain as was proved
in reference 3 in detail the differential equation describing the
perturbation in the form


(U UpIV 2a2p, + (p
(3)
Re* }


Here the prime (') denotes differentiation with respect to the wall
distance y; the velocities are referred to the local velocity Ua at

the boundary-layer limit, and the wall distance as well as the wave
2ir
length A = are referred to the local displacement thickness b*.

In order to avoid misunderstandings it should again be emphasized
that in spite of the assumption that U, V, and cp be functions of
the arc length x only the form of the local velocity distribution is
decisive for the stability investigation as one recognizes from
equation (3). The immediate effect of the pressure gradient, however,
is negligibly small as is also the influence of the x-dependence of the
perturbation amplitude q.

The boundary conditions of the differential equation (3) result
from the condition that the perturbation velocities u, v vanish at
the wall and that the friction effect at the outer boundary-layer limit
(U'' = 0) has disappeared. With their aid, the calculation of the
excitation of the unstable perturbations may be reduced to an eigenvalue
problem, the solution of which is discussed in section V.

We deal first with establishing the particular solutions of the
differential equation (3), limiting ourselves to small values of the







NACA TM 1343


excitation quantity ci; thus, the general solution of the differential"
equation describing the perturbation (3) may be represented in the form


cp* = Cv*qv; = C V- ici UO' = U' (4)
v=1 v=l UO


There p, denotes the particular solutions for ci = 0 which are
obtained in the calculation of the limiting curve of the instability
range (neutral curve) in the a, Re*-plane (ref. 3), and the wy
signifies additional functions for ci > 0.

We turn first to calculating the integrals cpl* CP2* and the
additional functions ai 2*.


III. THE SOLUTIONS qpl*, C2* OF THE DIFFEREIITIAL

EQUATION DESCRIBING THE FRICTIOIILESS

PERTURBATION FOR FII!ITE EXCITATION


If aRe* is assumed to be very large, the differential equation (3)
is simplified to the so-called frictionless-perturbation equation


(U c)(q" a2p) U"c = 0 (5)


This differential equation has a pole of the first order at the location
U = c = cr + ici to which we coordinate the point yc* of the complex
y-plane. In the neighborhood of this singularity, a fundamental system
may be easily indicated by series development. In order to establish
the connection with the case of the purely real c treated before
(see ref. 3), we first give the relation between the complex yc* and
the wall distance yc of the "critical" layer U = cr.

From


U(yc) = Cr







6 NACA TM 1343


and


U(yc*) = r + ici (7)


there follows


U(yc) U(yc) = ici = (* Y)UO (8


and, with limitation to the terms linear in ci, therefore

ici
Y* = Yc + (9)
U '


We shall now indicate the construction of the solutions pl", *
of equation (5) for those special velocity distributions U(y' by which
re shall approximate the laminar boundary profiles of Hartree (ref. 6)
for the calculation of the excitation of the unstable perturbations
in the same manner as we did before in reference 3 for the calcula-
tion of their critical Reynolds numbers.


(a) Binomial Velocity Distribution
(Pressure Decrease)

In the region of the pressure decrease, we used the approximation
function


U = 1 (a y)n n = 2,3,4, (10)


where a denotes the coordinate of the point of junction to the potential
velocity.

With the new variable

Y Yc
Yl = (11)
a Yc






NACA TM 1343


and with


al = a(a Yc)


the perturbation equation (5) for indifferent perturbations then reads


1 (1 iyni( 0129
> I


+ n(n 1)(1 yl)n-2c = 0


Using the relations (9' and (11), we now introduce the complex variable


Sy Yc
Y1 = = Y
a Yc


c Y- Yc ici
a Yc Uo'(a Yc)


With the abbreviation


ci ci
fl- i =-i-
Uo'(a yc) n(cr 1)


(15)


the differential equation describing the frictionless perturbation for
nondisappearing excitation then is transformed into the form


-1 -yl n + nif 1l- (1 y)n-l 12 *)
_~ L dyl


n(n 1) 1 y1*)"-2 + (n 2)ifl1( yl)n-3 = 0


(16)


in which, according to our presupposition, only the terms linear in
fl have been taken into consideration.


(12)


(13)







NACA TM 1343


If one writes equation (16) after multiplication by
*2
1n-1
1- (i yln + nifl -(I y

and after division in the form


yl.2 dq*
dyl*2


+ oi iy*i = O
i=l


the first solution 'ji* is given by


v=0
V--O


a Y 1
a ye ~


(19)


CO
= o g1V
V=o


The series coefficients ev
to the recursion formula


v(v +


The solution 2 p


are obtained from the B *


v=1

r is f t


for all y1 is found to be


= 1 + by *=2 V +
v =2


2e* al In y7
a Yc


(17)


a -
a ye


with


ici
U0


(18)


according


(20)


eo = 1


2 1 dy
2 =T1IP *







NACA TM 13h3


In this equation, the logarithms are, however, ambiguous; one must cut
the complex yl*-plane in such a manner that

3 *
Sit < arg y <
2 1 2


If we put


bv*l*v=2
v=2


byy + if1 hyy1V
v=l


(22)


2 may be represented for
form


92* = 1 +
V=2


yl = real part of yl* > 0



byV + if! hvy +
V=1


ifl)- / Yc


00
+ if, Z
V=0


ici
= qp---2 ,2
UO


a2 Yc hVy +
a V- t1


"-elc- -l in y +


2el In yl VOgy1Y
V=o


at first in the


2el(l -



(n yl


gVYl)


with


(23)


(24)







NACA TM 1343


In the transformation of equation (21) into equation (23) the
relation


el = e(l ifl) (25)


is put to use; it is easily obtained from the perturbation equation (5)
if U(y) and 'l* for ci = 0, and for ci = 0 in the neighborhood
of the singularity U = c are developed into a Taylor series, since


2e = (a yc) (26)
Uo'


and

UO'''
2el = (a yc)----
UO' + ici O
Uo'


U'" ici/U0" Uo'__
~ (a yc)o 1 __-
UOK' O' UO' Uo"



= (a yc)O (1 ifl) (27)
U0'


If we express in equation (27) the exact value


UO"' UO''
UO' UO"/







IIACA TM 13143


again by the approximate value
that due to the small value of
term for the value of
term for the value of -


fl, this step is motivated by the fact
ci, one deals here only with a correction


For y1 < D, 'p2 is represented by the expression


r9* = 1 + bv y*
V =2


ici
UI"I


+ 2el*a l ln yl* + i arg y*



(28)


w2 = -y hyV + I- 2- Iy
a yc +a yel 1)


(29)


The term with i arg yl* in equation (28) was obtained by
H. Schlichting (ref. 4) and W. Tollmien (ref. 5) by a discussion of the
general perturbation equation (3) in the neighborhood of the singularity
U = cr + ici in a similar manner as the "transition substitution" in
the critical layer for purely real c by taking the friction effect
into consideration.

Since we are, in the calculation of the transition substitution,
concerned with a representation of el of maximum accuracy, we shall
replace el by the expression (26) of the Taylor development of the
exact Hartree profile.


with







NACA TM 1343


For the linear and the parabolic approximate distribution
n = 1, n = 2 in equations (10) and (16)] H. Schlichting (ref. 4)
has already given the solutions p1*, 2* This calculation was con-
tinued for n = 3 and n = 4. The coefficients Bv*, ev*, by*,
g hb are compiled numerically in table 1. For the convergence of
the power series, the reflections made in reference 3 are valid.

(b) Sinusoidal Velocity Distribution
(Pressure Increase)

In the region of pressure increase, the Hartree velocity distri-
butions in the boundary layer (see ref. 3) were approximated by
formulation in terms of a sine formula introduced by W. Tollmien
(ref. 5).

U = Us + (1 Us)sin( (30)


where s denotes the wall distance of the inflection point.

With the new real variable

y2 = y (31)
Y2 2 a- s

and with


a2 = |(a s)a (32)


the perturbation equation (5) for the indifferent perturbations then
read


sin y + sin y2]( _2 2 + sin y2 y2)s = 0 (33)
S \ (Y2 Ysdy22







NACA TM 1343


where y2s


was put equal to


Y2s
2 a-s


(34)


We now introduce the complex variable


y Yc* Yc Yc* ici
-2 2 a s 22(a s)
2 a-s 2 a-s U0' 2(a- s)


(35)


With the abbreviation


f2 Ci _1 ci
UO' 2(a s) (1 Us)cos y2s
t h e f r i c t i o n l e s d i f r e t al e q a i n d e c i i g h e t u b t o


(36)


the frictionless differential equation, describing the perturbation
for non-vanishing excitation, then is transformed into


[in 2* Y2s) + sin y2s f2cos Y2* 2s cos y2s


(d2 *


- c2*) + [sin 62* Y2s)-if2 cos (y2 Y2s) P* = O


(37)


If (37) is written like equation (16) in the form


2*2 d2p*
dy2*2


+ i
i=l


the solutions pl*, q1 are determined
(18), (23), and (28). One has


Bii*Y2*i = 0


(38)


by analogy with the equations


2 iv* ( (* I ici )
n(a 1 s Y2 e, *y2 2(a s) 1
2(a s) v=0 2a ) U


(39)







NACA TM 1343


with


00
all v 2 v
v- =
V =0


(40)


For y2 > 0, the solution cp2 is o


p2 = 1 + > b *y2*


bt i ned


+ 2el* 2(a- n y2
2(a s


3 < arg y <
2 2


If one puts


v= bv*y2*v
v =2


00
= bVy2
V=2


00
+ if2 h 'y2
v=l


for y2 > 0 may be represented in the form


o00
1 + bI bvy2 + if2
v=2


0o
11- hy2v
V=1


+ 2el + if2
sin y2s cos Y2s


2(a s)


ici
= P2 u 0 "2
U


if2 vY2 I1n Y2 +
V=0 V) (


S+ 21ell (1
2(a s) y2


in Y2
+ +
sin Y2s cos Y2s,


2el In Y2 v Y2
v=0


(41)


CP2 =


(42)


if2
y2 /


with


(43)


(44)






NACA TM 1343


In transforming (41i into (43'), one makes use of the relation


el* = el 1 + 2
sin Y2s cos Y2s
eif (+ if Y

which follows from the two equations

a s Uo''
el
UO'


S+ ic Us
el 1T U0 f
L + U "
lUO + ici0
UUO


(a s)
if


UO'',
U0O


(a s) Uo
= ---
n UO


+ if2 s
sin Y2s cos Y2s


(47)


for y2 < O, is represented by the expression

S= + b y2* + eln y + i
v=2 + 1 (a s) Tl (n Y2* + i


ici
2 UO I 2
U0',


3
2 n


arg Y2*)


< arg Y2


15




(45)


(46)


V
< F)(48







16 NACA TM 1343


with


Sel1 CP Y12 In Y2
2 2(a s) 1 h2 a s y sin y2s cos y2s


2e, In y2 gYy2 + 2iel 1 pl
21 Y + 2 (a s) sin y2s COs Y2s


5- VY2 >1 (49)
V=O


For the calculation of the transition substitution, we shall
replace the term el, as in section III(a), by the accurate value
(equation (46)) of the Taylor development of the exact Hartree profile.

In a comparison of the relations (24) and (44) or (27) and (47) or
(29) and (49), it is striking that, in the expressions for el* and a2
for the sinusoidal boundary-layer profiles in case of pressure increase,
the product sin y2s cos y2s appears in the denominator of several
terms. The sign of this product is negative when the inflection point
of the velocity distribution lies more closely to the wall than the
critical layer (y2 < 0); the critical layer thus lies in the part of
the velocity profile showing concave curvature. The sign of the expres-
sions divided by sin y2s cos y2s then is the same as the sign of the
corresponding terms in the solutions for the velocity distributions in
case of pressure decrease which, as is known, have concave curvature at
every wall distance. If, in contrast, the critical layer is located
between point of inflection and wall (y2s > 0), then, in the part of the
velocity profile having convex curvature the signs discussed are reversed.
In that case, when the critical layer shifts to the point of inflection
itself (Y2s = 0O, the behavior of el* and m2 is regular since then


el ici if2 (50)
2(1 Us) 2







NACA TM 1343


and


2(yc = s)



a2(yc


(a s) hY2 +In y21
2(a s =1 2(a s)


(2 > 0)


(51)


= s)


hy2 2(~ -s) In --
YV + 1Pl i 1nlY2_
hVY2 2(a s)


2 < 0) (52)


The series coefficients


PV*, ev*, bv*,


sinusoidal basic velocity
convergence of the series
report (ref. 3) are again


are given numerically in table
developments, the explanations
valid.


2. For the
of the earlier


IV. THE FRICTION SOLUTION cP3*


FOR


FINITE EXCITATION


Besides playing a role in the critical layer U = c, the friction
of the fluid is of importance also in the neighborhood of the wall where
it occasions two more solutions cp*, 'p of the general differential
equation describing the perturbation (3).

If we introduce the variable


Y Yc*
=
E*




( U -1/3
e* = R + ici 1-
Uo I


(53)





(54)


gv, hv


for the


with







NACA TM 1343


we obtain from equation (3) in the limiting process E --0
Reynolds numbers the differential equation


id43,4*
d *
CL114


for large


d2 *
+ 9 3,4 = 0
dT1*2


(55)


The solutions of this differential equation
the form of the velocity distribution U(y)
solutions (P3, P4 for the excitation zero
corresponding to reference 2:


P3,4* =


are just as independent of
as was the case for the
(ref. 1); they read


dT.1* *1/2 1/3


where H(1),(2) signifies the Hankel function of the first and second
kind, respectively.

Since the Hankel function of the first kind increases for large
wall distance beyond all limits, it cannot be contained in the general
integral (4) so that we there may put C4 = 0.


V. STATEMENT AND SOLUTION OF

THE EIGENVALUE PROBLEM


After having found the particular solutions of the differential
equation describing the perturbation (3), we now state the eigenvalue
problem, (which results from the boundary conditions of the perturbation
equation) for investigation of the excitation of the unstable pertur-
bations.

At the wall, the tangential as well as the normal component of the
perturbation velocity disappear; thus one has


Cl* l w + C2 2 w + C3* 3 w = 0


Cl 1lw' + c2 (2 w + C3*'3*w' = 0


(57)


(58)


: L 2 i*3/2) *
|j ii d







NACA TM 1343


At the point of auction y =a to the region of constant velocity,
U" is zero and therewith cp9" a29 = O, that is cp* = e-ay, so that
the third boundary condition reads


C1 *D a + C2 2 *a = 0


(59)


with


v a = a'


+ cP a'


V = 1, 2, .


(60)


A term 03a* does not appear because the particular friction solution
, has already been damped at the point of junction.
J


In order that the three homogeneous equations (57),
may have a solution different from zero, the determinant


Pl"w 2 w 3 w

l 2' 3w' = 0

la 2a 0


The solution of this determinant yields the equation


3w
3* '
3 w


2*w '1a -


"'*w
lw 2a
ilw2a


or with Schlichting's abbreviations


1 93L D((O*) = F = F
0*nO CP3*w 1 *0


(58), and (59)
must disappear


(61)


(62)


(63)


I







NACA TM 1343


_1 2*wla Pl*w2*a E*(a,cr,cij (64)
Yc* wP'rDi*a P1 w 2*a


finally


F* = E*(,cr,ci) (65)


Therein

0 Ye*
*IO (66)


The complex equation (65) is equivalent to two real equations in which
the parameters a, Re*, Cr, and ci are contained. If one limits
oneself to the case of indifferent perturbations (ci = 0), one obtains
from them after elimination of cr a relation between the Reynolds
number Re* and the wave length 2 of the perturbation. That is the
equation of the neutral curve or the curve of constant zero excitation
by which the stable and unstable perturbation states are separated and
the lowest Reynolds number of which is the so-called critical Reynolds
number Re*cr; when this critical Reynolds number is exceeded, the
excitation of the unstable perturbations begins.

These neutral curves were determined in a previous report (ref. 3)
for the velocity profiles for decreasing and for increasing pressure
calculated by Hartree.

In the present investigation, we shall assume the excitation
quantity ci to be different from zero and investigate the curves of
constant excitation enclosed by this neutral curve.

For this purpose, we shall solve the complex equation (65') in a
somewhat simpler manner than Schlichting (ref. 4), by determining not
the.differential quotients -i and at the location of the
8a 6 Re*
neutral curve, but the curves of constant excitation ci > 0 directly
by a method similar to that applied for the neutral curve ci = 0;
however, without making use of this curve itself. We consider first
the left side of equation (65).







NACA TM 1343


Since we intend to limit ourselves to small values of excitation,
we develop F* in the form


F F ( = F(0o)


o 0/ dF \
) 0 dTl 0 1


+ .


(67)


where aO as before in reference 1 is defined by the relation


0= yc(Re*Uo,) +1/3


(68)


According to equations (9), (54), (66), and (68) one now has


= Yc1 +


~ YycRe*Uo


L--
Yc UO
3 UO
UO


(69)


Thus equation (67) becomes


F* = F (O*)


F(T0)


ici U
UO Yc


U_ dF
uo '?0


+ Y
3


= G(TD0,crci)


O( *


(70)







NACA TM 1343


One sees therefore that the form of velocity distribution which does
not enter into the exact solution (?q 0 according to equations (63)
and (56) does appear explicitly in the development of F (O*) with
respect to ci.

The differential quotient

dF dFr dFi
= -4 1--
d0IO d-0i dTl0


was determined by graphic differentiation of the function F q). The
numerical values of its real and imaginary part are given numerically
in table 3. Since real and imaginary parts nowhere disappear simultane-
ously, the function F(TO*) will be free from singularities in region
around the function F(qO) and the development (671 will be thus per-
missible.

We now consider the right side of equation (65. which is defined
by the relation (64).

Remembering the splitting of the frictionless solutions
(pvV = 1,2) in the form of equation (4), we first put

0 (D ici
V a va "va
UO
with

Iva = Uva + ava
furthermore (71)

Pvw t ici


'V* ici '
v -vw -- vw
U0







NACA TM 1343


For the binomial velocity distribution
values of the additional functions mVa WVw

mv' Ua,' are given by the equations
a w


U = 1 (a y)n the
and their derivatives


Wla = gv
V =0


-pa 1LF hv +

a yc v=


2el'-1
a Yc


1 I 1
J2a (
(a yc,)2 v=1


w1w =T gVY1wv
V=O


2w 1= b
iwa yc V=1 w


2el
vhv + 2
(a yc)2


+ 2elqlw/
a Yc Ylw


2 la
a Yc


+ Cla + W-


- in |yiw| wi-) +


2e In y lwl + il UO -- w_ lw
U1 a Yc


00
lw a Yc V=l


gvYlwy-1


3 vhVYl V-0 +
w (a yc)2 VVI


(a Yc


1 \" In y 1


Ul y
UO a Yc lw


>(72)


+ 0w
Ylw


+ In IYlw l (a


-( 1 11
S Ylw2 yw/


-Yc) +







NACA TM 1343


For the sinusoidal velocity distribution

U = Us + (1 Us) sin( s


in contrast, the corresponding formulas read



la = gVY2a
v=O


V hv f $elqj1a +
2a= -ah ---
2a 2(a s) v=1 a- s \2a

2e1 In y2amla

la ~n Vl 2aV-1
(la a s) v=1

I2 iw V + r2e1
(a a 2 v h2a- 2(a s


Y2a sin y2s co y2s + 2a

0la 2(a s) + in Y2a li'
Y2a .

al = ; g2wV
v=O


In Y2a +
Bin y2s cos y2s


)2L(a s)( y

In Y2a
sin y2s cos Y2s


_- n t h.v V+ elPlwl 1 In y2w
_s \Y2w in cos
2 2(a s) 1 2w Y2w sin y2s cos y2s) +
U0. _____________


I UO lw
2e1 In Y2j Uw + in -- ...
I d Uo' 2(a s)sin Y2s cos Y2s


WIw)


UIw = Z_ VgVY 2wVg 1
S2(a s) 0

032 vhvY2wv-1 + 2(a2el- a +
2 (a s)2 v=l 2(a s)2(a ) 2w2


S, 1 In Y2w +
Y2w sin 2s cos Y2s) 2w sin Y2s cos y2

"lw 2(a s) + UO cp'
y2w+ n y2w i + 2(a s)sin- l lw
y2v f 1 '-1 UO 2(a s sin y2, Cos yVsa


(731







NACA TM 1343


If the critical point shifts to the inflection point
distribution (Yc = s, Y2s = 0), the equations for w2a,
m2w' (equation (74)) are simplified to


2a i" hVY2a V
2(a -s) L =il


la n y2aI
2(a s)


w2a Va2- S)2 vb y2a


2w (a s)2 hvy2v
-4(a S)L v=




inkl/
4(a s)2


V-l + Pla
S2(a s)Y2a


~rPlw in |Y2w
2(a s)


2+ sylw
2(a s)Y2w


+ la n y2a


i IT3Ciw


We now split P2w, 92w', )2w,
parts according to the formulas


Cp2w = A + iB1


P2w = A2 + iB2


'2w


into real and imaginary


)2w = M + iN1


W2w = M2 + iN2


then one obtains


p2* = A1


Ci
+ '7


N1 + i(BI


'2 w A2+ N2


of the
"2a ,


velocity
"2w,


S(74)


(75)


-U M


- M
c i j
0 /


+ iB2


(76)







NACA TM 1343


Then we put the expression (64) into the form


E*(C,cr,ci) = E(a,cr) 1 + ci(zl + iz2)


(77)


where E, as previously defined in reference 1, is defined by


1
E -
yc


cP2wcla ~lw 2a
P2w' la ~lw 2a


First, limiting oneself to.the terms linear in ci


Al la +


1 [l la


(78)


one has


c .
U (N L la + Bl la


+ rii~i~j


ici
~ lw 2a 1u--lv a


~ A2 la + 1(N2la


+ B21l2a)

a C2 )


i 2 la


- ,(A2
U0


+ M2la)J


P1 2 "a ~ cPlw 2a


- ci/ ,P 1
--r(alw"I'2a + ;Plw'^al
U-


*
'P1 w 2 a



2 w la


(79)


UCi (Alj
U0






NACA TM 1343


Hence, there results after a short intermediate calculation


E* ~E 1 ci +
UO 'Yc

Ci (Illa + Bli2la) + i(wlw'f2a + tPlw2a Alila Ml0la)
UO0 Al la cPlwo2a + iBl1la


(N2'Tla + B2lal + i(clw' 2a + q(lw'2a Ala M2la) (80)"
A20la 'Plw'2a + iB2 la




We put for abbreviation


Nl1la + Blnla = m

Wlw 2a + P1wI2a Al2la Mlla = nI

N2 la + B2ala = m2
(81)

alw'2a + Tlw22a A2nla M2 la = n2

Al la clw2a = K1

A2 la qlw' 2a = K2







NACA TM 1343


Then there follows from equation (80)


ci ml + in1 "2 + in
+ --
UJ' l + iB]Yl K2 + iB2 la


ci ilKl + nlB1"la
++ B12a2
Uo [Kl + Bl2la2


m2K2 + n2B2la
K22 + B22"la2j


n2K2 m2B2laj
K22 + B22la2 j


(82)


icilnlK1 mlBlla
UO' Kl2 + B12.la2


If we finally introduce the designations


S1 mlK + nlBlla
UO' 0K2 + Bi2 la2


1
Z2 -.-


m2K2 + n2B2Dla
K22 + B2 (la


+ 1 nlK mlBl la
UO' Ki2 + B12la2


n2K2 mB2 la
K22 + B22 la2]


the representation (77) is attained.

By means of the equations (70), (77), and (83) we thus have divided
the two functions F* and E* (a,cr,ci) appearing in equation (65) into
real and imaginary parts and can now graphically solve this equation.
For this purpose, we plot for constant ci for several values ye the
imaginary part of the function F*


I (F) = (I o0 + T Y1
UO Ic


against its real part


F*) = R[F( O] 0 1
UYC(


(84)


(85)


I
+ U 'd
3 Ug ydio


ici
1-
UO'Yc

ici
1 UoYc
UO IYc


E* E



E


(83.1


+3 Uo'"'
0 ddjo







NACA TM 1343


and plot in this polar diagram to the same values ci and yc, the
imaginary part of the function E*(M,,cr,ci)


I(E*) = I(E)(1 + cizl) + ciz2R(E) (86)


against its real part


R(E*) = R(E)(1 + ciZl) ciz2I(E) (87)


The points of intersection of the functions F* and E* connected one
to the other by the same pairs of values cr, ci yield first the values
TI pertaining to the corresponding a-values, and then with (68) the
curve a(Re*) of the constant excitation ci. This plotting and cal-
culation then are repeated for other values ci.


VI. RESULTS OF THE CALCULATION


As immediate result of the rather extensive calculations, the polar
diagrams with the curves E*(a,ci,cr) and F* = F(lO*) for several
Hartree velocity distributions U(y) in the laminar boundary layer are
represented in figures 2 to 21. These boundary-layer profiles belong
to the special class of pressure distributions

Ua = const. xm (88)

and are rigorous solutions of Prandtl's boundary-layer equations which
were obtained with the aid of a Bush apparatus (ref. 6). We use,
according to Hartree, for characterization of these boundary-layer pro-
files the parameter

P= 2m (89)
1 + m

The profiles with m > 0, P > 0 occur in case of decreasing pressure;
the profiles with m < O, 0 < 0 in case of increasing pressure. The
profile 0 = 1 is the exact Hiemenz profile at the stagnation point of
a cylindrical body, the profile P = 0 is the exact Blasius profile at
the flat plate in longitudinal flow, and the profile 0 = -0.198 is a
separation profile (wall shearing stress zero).










NACA TM 1343


I


0 0
o ,





- 0 r0
S El a? 4*



*o 'd
0g4)4


4 94 0





-i 4.- + Pl
oi O M
04A 0 z







c 4 ca


lol
4- a, 4) a






P "- 0 A 4-
(a 0 P P4




ori
l > r-4 o r 4
P RA -
0 A A $-4
a 0 cc 0
,c 3 T*H o


S 4) D o



H 'd P4 0
4- r-4 4-0 4
4 H H 4)o
1 0 10










--44
Pb D < 'd02




se-m! (




+' H O-4
+K A 4)


-r- m
S)-1 0 M LH 4

4-: 0 ri 0 U






W- cU 0
-H 0
4)4) *HIs



+)n 02 h4-$
CC ri 1


ON *i 0 F

H O r-\ FL 4-



r .O *-1 co


4P 4 0 r
CO CkP












*H H O 4 0
1 4.) 4 '


\0
r-4 r-4
0 CO1

II II

cl cd
a-n A-n


0



( )


a-,

U
.-
C'J
0
II
>i 4$


UJ'



II

a-A





Ca
CO
O
0
































C(I
I






II










a
II









111


COL


.-4
CU





bid


































r0
II









*~I

II
C


4-A
'Y)
-t
m


II


























(11


I
Cl





i-I
II












dl


- I I
Ca





II
NL


a-i
0\
ON
CI
0
II































81 4$

cIa














0
II
O









II


.-r
Cu
0
II





4$

b;l








.D





r-
0


0








O
C

























cl
0-4





0
II


Ir
0


I



0-





+
r


II

D







NACA TM 1343


is the dimensionless wall distance used by Hartree and the coordinate
a of the point of junction to the potential velocity Ua is connected
with the displacement thickness by the relation

a _
= -** (92)
kg

the quantity k6*(B) may be found plotted graphically in figure 2 of
reference 3.

The wall distance Cc of the critical layer indicated in figures 2
to 21 is obtained if in equation (91) y is replaced by yc.

The following basic remarks should be made concerning the graphic
solution of the eigenvalue problem in figures 2 to 21:

Since ci has been presupposed so small that only the variations
of the particular solutions (p* linear in ci need to be taken into
consideration compared to the solutions pV for ci = 0, the curves E*
and F* appear for equal ci-interval as "equidistant" curve families
in the sense of equations (77) and (70). Actually, this "equidistance"
will be lost in case of higher values of ci; however, the calculation
expenditure would increase intolerably even if only, for instance, the
terms quadratic in ci were to be taken into consideration. In this
sense, the curves

Pi5* cia&*
Ua Ua

against a6b represented in figures 22 to 26 (from now on we use dimen-
sional quantities) which result from the evaluation of the polar diagrams
also are to be interpreted as approximations (in this and the following
figures, the value Pi which is, according to equation (1), physically
more important and which characterizes the logarithmic increment of the
excited perturbation amplitude has been plotted instead of ci). Here,
as in Schlichting's report (ref. 4) mentioned before, only the derivative


d
d(a*)\Ua /







rACA TM 1343


Pi5* i5*
at the location = O, that is, the slope of the Ua curve at
Ua
the a&*-axis is rigorously correct. H. Schlichting interpolated
between the base points at which he had determined the slope directly,
although with a much higher calculation expenditure, with a curve of the
third degree

2.i6* (e*\2
U- = a0 + a1(ab*) + a2(a5*)2 + a (a*13
Ua
where the four constants are fixed by the coordinates of the base
points and the values of the curve slope in them. Thereby,

Schlichting obtained for the Blasius profile higher values than
Ua
those occurring in figure 25 whereas the values of the slope are in

agreement. Actually, the curves ---(a&*) in the center part of the
Ua
(a8*)-region enclosed in each case will run somewhat higher or lower
than indicated by figures 22 to 26. At any rate, however, they may be
interpreted as a first approximation in the usual sense.

These curves i* over ab represent sections through the
Ua
"excitation mountain range" enclosed by the neutral curve as base curve,
along the lines yc = Const. or cr = Const. By interpolation one
obtains from them the maps of excitation represented in figures 27 to 32,

in which the lines of equal excitation i = Const., can be interpreted
Ua
as "contour lines" of the "excitation mountain range." Instead of the
lines of intersection cr = Const., we plotted in figures 27 to 32 the
lines

prV cra5*
rV = c = Const.
Ua2 UaRe*

the significance of which will be discussed later in section VII. Even
at first consideration of these excitation maps, a fundamental difference
in the shape of the excitation mountain range is conspicuous according
to whether the velocity profiles of the laminar friction layer lie in
the region of decreasing pressure (P > 0) or of increasing pressure
(< <0).

In the region of decreasing pressure, the "excitation mountain
range" has the form of a mountain with pronounced peak which is steeply
ascending for a small Reynolds number Re*, slowly flattens after a
larger Re* and shrinks to zero width and height for Re*-->o. The
absolute height of the peak, that is, the maximum excitation increases
with decreasing B.







NACA TM 1343


In the region of increasing pressure, in contrast, the excitation
mountain range changes behind the peak with growing Re* into a
"mountain ridge" of constant width and constant contour profile.

The properties of this "ridge" have been thoroughly investigated
by W. Tollmien (ref. 5). Since we made use of Tollmien's theory, we
must now briefly represent its results.

Searching for a general instability criterion, W. Tollmfen estab-
lished that the frictionless perturbation equation (5) possesses for the
laminar velocity profiles in the region of pressure increase which
have a point of inflection in contrast to those of the region of
decreasing pressure for Re*--- aside from the neutral solution
existing for all profiles with the parameters


a = 0


cr = 0


9,nI = U


(93)


in addition, the neutral solution with the parameters


a = as


cr = Us


Pnll = Ps


(94)


the subscript s therein denotes the point of inflection.


W. Tollmien calculated the eigenvalue as and the
4qs for the sinusoidal velocity distribution (30) which
from the frictionless perturbation equation (5)


(aa)s = p cot p


sin p y
Ts -
sin ps
a


sin p aa (
Ps= pe a
sin --
a


eigenfunction
we also used


sp 1 )
sin p 2( s
\ a)


(0




(>a)


(95)


(96)






NACA TM 1343


For this second neutral eigensolution, the phase velocity cr therefore
equals the velocity of the basic flow at the point of inflection Us.

For the excitation Pi and the circular frequency Or in the
neighborhood of the neutral frequencies a5* = 0 and a5* = (aB*),,
W. Tollmien has derived the following formulas


a = 0 U' Ua3 )-Re* -- (a*)2 (97)
Us Uw' 1 2 Ua2 U 5


where the subscript w signifies that the values have to be taken at
the wall y = 0, and that

c = a: B [ *)3 Re*
Us U a

= aUS BE U 2 a6*)3 (as*26 (98)





B = f Jo (99)dy
U12 Ua*2 2 + .(99


where
E = lim -2dY 2
E urn U 200 Ulf 2 d (100)
E O (U -Us2 S s+6 (U -Us'2 s


We have calculated the "ridge contour profiles" of the excitation
and the circular frequency according to the formulas (97) and (98) for
the Hartree velocity distributions P = -0.10, -0.14, -0.16, -0.18,
-0.198; the values for the derivatives of the velocity were not taken
from the sinusoidal approximation distribution but from the exact
velocity distribution. These "ridge contour profiles" of the excitation
and the circular frequency have been plotted in figures 33 and 34. The
curves corresponding to the formulas just mentioned have been drawn in
solid lines; the transitions between the two curve arcs, interpolated
somewhat arbitrarily, have been drawn in dashed lines. In the range







TACA TM 1343


of very large Re*, the lines of constant excitation and constant cir-
cular frequency were transferred from figures 33 and 34 into figures 31
and 32.

Figure 3 represents the variation of the maximum excitation



\ Ua max

against the form parameter P, separately for the excitation for finite
Re* (peak of the excitation mountain range) and for Re* ) o ridgee
of the excitation mountain range). For 0 < -0.10 these two values
seem more and more to approach one another so that with decreasing P
the peak becomes less and less pronounced, and the excitation starting
from small Re* monotonically increases to the values of the "ridge
contour profile."

This figure and the preceding ones show clearly that the maximum
excitation and accordingly the excitation in general is considerably
larger in the region of pressure increase than in that of pressure
decrease for smaller Re* as well.

After thus having estimated the magnitude of excitation in the
entire instability range of all velocity distributions occurring in the
laminar friction layer, we shall discuss the physical conclusions
resulting from our calculations for the position of the transition point.


VII. DISCUSSION OF THE RESULTS


Let the pressure distribution Ua(x) against the arc length of
the cross section profile, and the oscillation of circular time frequency
r be prescribed. Then this oscillation superposed on the boundary-
layer flow travels downstream on a curve


U2 = f(x)
Ua2

As we mentioned at the beginning, there pertains to every point x of
the profile in the flow a fixed value of the Pohlhausen parameter X
which characterizes by way of approximation the velocity distribution
in the boundary layer at this point. According to figure 34 in refer-
ence 3 one may coordinate to this parameter X the Hartree parameter 0







NACA TM 1343


and therewith one of the excitation maps calculated in the present
report (figs. 27 to 32). In order to obtain the excitation of the
perturbation pr at a certain point x of the profile, one has there-
0i8*
fore to read off the excitation (number of the "contour line")
Ua
on the corresponding excitation map at the point of the map determined
by the pair of values of the Reynolds number


Re* =- g(x)
V


and the dimensionless circular frequency

prv
= f(x)
Ua2


We shall start the discussion of the results of our above excitation
Umt
calculations with the limiting case of a small Reynolds number Re .
According to the explanations in section I (compare fig. 1) the stability
limit lies, for small Reynolds numbers only, at the point where the
laminar boundary layer separates. If the perturbation waves are so
long that the curve


S= f(x)
Ua2


intersects the instability region for the separation profile (i = -0.198),
it is very violently excited when entering this zone and leads quickly
to transition to the turbulent flow pattern. The transition point then
practically coincides with the separation point which we had denoted
its rearward limit. If, on the other hand, the perturbation waves are
very short so that the curve


Orv
S= f(x)
Ua2


does not intersect the instability region, the laminar layer separates
without transition.







NACA TM 1343


If the Reynolds number Re = -- increases, the stability limit

shifts forward in the direction toward the pressure minimum (fig. 1),
the perturbation enters a region of instability further upstream for a
larger value of the parameter p or the Pohlhausen parameter X, and
is there initially excited to a degree which decreases the more the
stability limit shifts forward but which increases due to the fact that
the perturbation downstream (with decreasing P or X) reaches insta-
bility zones with rapidly increasing excitation. The transition point
shifts frontward corresponding to the excitation which started earlier
and is still strong.
Umt
If we finally increase the Reynolds number Re so that the

stability limit shifts ahead of the pressure minimum, the excitation in
turn starts accordingly sooner; however and this must be regarded as
the most important result of our calculations for the time being in
the region of decreasing pressure, the excitation is so slight that it
generally attains amounts equalling the excitations produced in the
cases treated just now only after having passed the pressure minimum.

In this manner, one may easily give the theoretical explanation
for the fact proved by many experiments, that the transition point even
in case of very high Reynolds numbers rarely ever shifts ahead of the
pressure minimum. Only in cases of very long acceleration sections
and perturbations with very long waves where the perturbation does not
too soon leave the instability region again, the limited excitation of
the region of decreasing pressure will be sufficient to induce the
transition still in the region of pressure decrease. However, these
cases are rare in technical application.

A detailed calculation of the degree of excitation which causes
the transition is meaningful only in connection with corresponding
experiments. Therefore, it will be postponed until these experiments,
now in the preparatory stage, have been carried out. Probably one will
have to regard the form of the pressure distribution in the region of
pressure increase as the most important test condition since, according
to our theoretical deliberations, the contributions to the excitation
of perturbations in the region of pressure decrease are insignificant.
The aim of this experimental investigation and of the excitation cal-
culation to be performed simultaneously on the basis of the present
report will be to find a connection between the pressure-distribution
form and the degree of excitation attained at the measured transition
point so that it will be possible to calculate, inversely, the transition
point for a prescribed pressure distribution from this relation.







NACA TM 1343


VIII. SUMMARY


As a contribution to the solution of the important problem of the
calculation of the transition point of a plane laminar flow, we had
first determined (in an earlier report, according to Tollmien's method
of small oscillations for the Hartree velocity distributions appearing
in the boundary layer in case of decreasing and of increasing pressure)
only the critical Reynolds number beyond which the perturbations super-
posed on the laminar flow are excited. In connection with those cal-
culations now, the excitation itself in the entire instability range
of the perturbations was calculated. The excitation in the narrow
instability range of decreasing pressure turns out to be very much
smaller than the excitation in the more extensive instability range of
increasing pressure; thus the known fact that the transition point
generally does not shift ahead of the pressure minimum even in case of
high Reynolds numbers may be explained on a theoretical basis, as shown
in tables 1, 2, and 3.

Systematic experimental measurements of the transition point,
together with calculations to be performed on the basis of the results
given here, are to establish the connection between the variation of
the pressure gradient and the degree of excitation which produces the
transition and thereby a basis for determination of the transition point
for prescribed pressure variation by calculation.


Translated by Mary L. Mahler
National Advisory Committee
for Aeronautics







NACA TM 1343


REFERENCES


1. Tollmien, W.: Uber die Entstenhung der Turbulenz. Pachr. d. Ges.
d. Wiss. zw Gattingen Math.-Phys. Kl., 1929, pp. 21-44. (Available
as NIACA TM 609.1

2. Schlichting, H., and Bussmann, K.: Zur Berechnung des Ums.hlages
laminar = turbulent (Preisausschreiben 1940 der Lilienthal-
Gesellschaft fir Luftfahrtforschung), dieses Jahrbuch.

3. Pretsch, J.: Die Stabilitat einer ebenen Laminarstr6mung bei
Druckgefalle und Druckanstieg (Preisausschreiben 1940 der Lilienthal-
Gesellschaft fur Luftfahrtforschung). Jahrbuch 1941 der Deutschen
Luftfahrtforschung, p. 58.

4. Schlichting, H.: Zur Entstehung der Turbulenz bei der Plattenstromung.
rlachr. Ges. Wiss. Gottingen Math. Phys. Klasse, 1933, pp. 181-208.

5. Tollmien, W.: Ein allgemeneines Kriterium der Instabilitat laminarer
Geschwindigkeitsverteilungen. Nachr. d. Ges. d. Wiss. zw G6ttingen,
Math.-Phys. Y1., fleue Folge, Vol. 1, No. 5, 1935, PP. 79-114.
(Available as NACA TM 792.)

6. Hartree, D. R.- On an Equation Occurring in Falkner and Skan's
Approximate Treatment of the Equation of the Boundary Layer. Proc.
Phil. Soc. Cambridge, Vol. 33, 1937, pp. 223-239.







NACA TM 1343


TABLE 1
SEIES COFCICIEWTS OF THE SOUIONS 9 2, F* FOR TH BASIC FIlD

U =1 (a- y)n n=1, 2, 3, 4

n=l


Yc y*
Ye Y W


s ain 5^ asin tCo- o(aY)=1 tV


= = cos (a a cos sin (gy) = !


n=2

oo0 = l (1 l) X =* (-1 ivf)


(v 3)


e1 = +1 = 2I-




eC*=-- -(- 2l) e ) -




e2







>* m 1212 192 23
=- (1 2. fl) 21) + f= (1 51) 0 1 ) L




b21 = -1 61) + 31f2 11 20if


b (1 7fl) 37;1 2fl) + 3 b5 51 ) + 1 31 fi) 1 6 6 to



b2 (1 fl) 2000, 1( -0(1 33f1) + 1 if,)

1920 1512000


9D = 1
ao=O

1 a212


g2 = 2 .2
* ll 1 + I 1

8 1
1 2 +101 a 4 __
6=36001 64800 720



ho =0




-4 IL+ 431M1i2 t U14
192 1440 1 5

S T 1 27 2 6959 1 71 6
= o1920 432002 7 7


gl= -1


23=- 12

S 1 23a14
5 3600 1800

1 2 141 .4 la16
= 6300 1587600 o 33075



h1 = 2 + a12




1 511 3 2 2291 1 a!.6
h560 36Doo C' 54000 Mi i











NACA TM 1343


TABiE 1
IE M2 LC EFICIE2T OF THE SOUITIOS I*, q* FOR THE BASIC FIDO

U 1 -(a .y )"n;a 1, 2, 1, Conti-ed

n=3

." 0 2 2( ifl)


" "_ : ( 1) (3 6v)it









ea,.' :-. L : .2



:e i L *e (1 -t1)

k.lf -(1-fi)

- l -i l e n 1 3 l) 1 )



l, l _. 1


I 6 Z rli- (1 .a1 2



_3
1 .* 1 i





2" -I 5 I 3il) 1 ( 21fl)






*- 7f rfLE. 1)+ (1--311)+ 7 61 i


r I -1'4 _____1)+ 71_

-8 .. -+' I -41-Y-, a f1)4- 1 4i1f) + 1
7640320


Aj = I






S .14



t 4 8











-L I "W7 6
W1 L69 o 37x''.
_1 &6 ,<87 i ,
C 190 1 .'.O TC


S= 2





2 22

1- 13 42 2 16







h 6 + a?




3 7 -o n 7 L + -
b5 6 6 7-, l'89nao 3 12


^ 378 2 ayc3 E 1707: + P1 T76.:'00 5040









NACA TM 1343


TABZ 1
s8BR CCsOICISm OF T28E aOnWS ', w' FOR I BASIC FOW

U ( y.); y 1, 2, 3, 4 Concluded



0 0 1* 3(1 Itl)


.* = (1 2tfl) I2 (1 ")









-(1 lifl) 1)

* (1= ) 9 1 fi)

S10* k _1 =0I)


e2 (1 21fi) -* ( 31f) _(1 if)


,t 5 l5 1 21fl) + e l (1 3_1 1) alk 2 14 -1?1,


e 2 1 41fl) + 31 f'1- l)) -


=7 I 51 fl) 31 BB5sB T1I if)-






21 x33 x 5 x 7 2 x 33 x 53 2 2 3 53 7

563 8
28 x 35 53 x 72




S 1 ) 1 21 ) 1 1) (1 31 ) 1 .




o2k 111 T
b -~ 1 61t1) -1 1 i 4-3 2 2,- -)


S 1 -T- f 711) s (1 51tl) 3t1) 1761(1 itl)


b 1 U76 1 6417) 3 51 if1 -16 f)
b _e 39 1 15731 20 35 29 32 Pi1 6



go 1 h 3



3k 9 2 I +21 9 23 k

16+. +8 .k
1 2 2 1 k 2 1 3OT10 17 0 I 6

2Ul ll9o14 +1516 '18



bo 0 hl 12 *o12
237 12 23 13 k

T01 21712o 69k7
h.o,0 120 23
I 64 48o 15 W L 1 1231 1 I + O6

6 15"9 2+ 561 4+ 71 6 307 1*67 274 i k "'.31 6
6*0 2W=00 259LW1 '250o'1 hr W&980l -3 067D 375 A 3








NACA TM 1343


TABLE 2
.EUIE COXEFFICIENTS OF THE SOLUTIONS I1*, f*9 FOR THE BASIC FLOW
U = U l Us) sin ). ABBREIATION: p = cs
\-- sin y2. COB Y25
B = 0 8" = tn yi t* = 1 + I tanS2ys(l + ip) a2
tar .r j L- lP' lar3 I'(l + 31p)

B = tanyar-ys p i .p I S .'(1 + 4ip)
e* -- t- n y y lp n tan3y2s(l + 3ip) tan5e(l1 + 51p)

8 t&O ?'ll Ip ) nrly2s(1 + 4 p + ki tan m 2 (1 61p)
87 L l11 p l tanY2(1 + 3ip) 1 tan5y,(1+ 5ip) tanTy2s(1 + Tip)
6= tanydl lp canya(l + + ip) +-- tan6y2(1 + + i tan8y2s(1 + 8ip)


e' el = tar I, p e2* = + as2
3 Il
e'= t- n yt 1 Ipl -n ran y2s(l + ip)


es 2 = 5 ,al 1 Ip tan ys(1 + ip) tan"25(1 + 31i + 24tan y2(1 ip)
e 4 2* _1 ta 1 + 2ip) ta+y2s(1 + ip "2 + tan2 + 2
e7 t l + p tan y27(1 + ip) + 2 t3 1 + ip) + 2 ta5 +

tan y+ 1 1pi 2 ta[12 (1 + 1i8 + tn ta y2 1 + ipp
e = y=li -a _(1 + ,2p) + 3P1 + 0 P) + ia 2(1 + 6ipj +
6 -- ta Is + 4 p) n 2(1 + Y 2 t s(1 + +

o -= ++ ta21p) 3 + 270p + a+2
b-e 2 [67 tan a (1 + ip) 9 tan3ytany(l+ 31p tan yt y(1 + ip)+
c2 = -+ 5 2s(1 + 21 p) + s + p + a s1 + 2ip] + (







b = 3tan y2s1 + Lp) + tan 2s(1 + 5ip + a22 tan y2a(1 + 1p) t tan^y2s(1 + 36p8 -
_2 t tan r1 yi-a-1 + pp)






b 1 t n yp t85 2eip(1 + ti y + tani +a(1 + 0ip) + tan y2(1 + 6ip51 +
2a3 + 164tan260(0 + p+2






1 = o tan y2-(1 +) tanS21 + 3ip) ay1 + 16) 3 tany2s(1 + Tit)
ta. ta7 2(1 + ip) + tan3ya(1 + 2ip) + t5y( + i +
2 t.1 y<1 + ip) 2 + tay( + 3 tan y2s( + ip)

S= tany2 a(1 + +) + i) +


Sta2Y(l1 + 81p) + p22 tan22(1 + .ip) ta2y+(1 +p) -

b 4. tan y2(1l + ip) + ta 32o + +1P ]) tan 2s(1 + ip) +



02 tan tan1 1 + +
583 tan2Y2,(l + 21p)1 + 1Fi +a ( 13589 41p) + -L-tan(2sp(l + 17173 +








NACA TM 1343


TABLE 2
SERIES COEFFICIENTS OF THE SOLUTIONS *, V2 OBR THE BASIC FLCW

0 = U *+ (1 Ue) in L-.- ABBREVIATION: p Sin 2 ConciuQed
a --a Sjsin y2, coo ;,a

2
g= 1 g1 = tan y2 g = + tan2y 3 = tan y + tan

g + tan, Y2, 1 + tan y2.



94 = tan y2s + tan y tan3ya 1 + tan y22

S tan y 4 tan3y t +24)
tan y2. tan3y, + 23 tan Y2
120 2a 50\ 72 1800






6 5 = + l tan2y5 2s + t? + t2 Io t oo + 4 11 + 101tan y + -
720 n3 2 5400E 12 72


o = 50 -1 tan ya + -2 +a22 1 2 9 tan Yy2 a +tan3 a 2 tan yt


S( 12 3 tan 2.e + 3 2 + 11 2 tan + 2)








t =- I tan y 2s + tan~yL2! a + +a 27 tant y2 + 451 j- a2 t+ 1
g6 -53a + a_ ++
4 306720 35 84o0






















h ^ ta y- t anBy2 + 61tany + ttan am + %s n yB 1 a -rB tn
a22 = 0 h+ 222 1 a26 tan yy2









S72 28 tan 72856 any2 -n +
4 81 6

4 25 121 18 6 )6

hi,6= 89 5 a 0 ^ t + tan+I yB + 1 e 2 tan Y + 1 a 1
2884 3 64 108 4 6) 30

h5_0_23 h12ta.2Y22.+aY2. t6 Y2- +an 6019 ta192.253 n t2tnsy2

4288 720 60


h3 an2Y(53 3 tan26 y7 tan2Y2a) + 1., c

6 157 tan y 5 ta 3yp2, + 59 t 2 ( an 2a + 1 a 22 [6 5y



720 c2 tan y ta + 321883 1800 L Y 6.
1 81 291


IiT 97 t_2y2. I +.tan4y2 7 ta.6y2 29 t-8y2. +
1 25920 362880 2880 1440 134o0

1 2 [_ 3253 1069 t ,+ 221 tan1y2s tan6 Y2d +
23520 90 4725 tan y26 14 3 j


1 6 -9 t2y2 111899 tan-4y2 + 1 IM 6 [ 49 t91 +!8
25200 2 914580 33075 IT 10. a 50







NACA TM 1343


TABLE 3

VALUES OF THE DIFFERENTIAL QUOTIENTS OF THE REAL AND OF THE INAGIIIARY


PART OF THE


FUlICTICOI F(T0) WITH


RESPECT TO


Fr Fi Fr 6Fi
o 0- "no -'- 0
a0o n j o ao


-0.226
-.230
-.235
-.239
-.242
-.245
-.246
-.246
-.243
-.237
-.226
-.210
-.188
-.162
-.133
-.102


-3.6
-3.7
-3.8
-3.9
-4.0
-4.1
-4.2
-4.3
-4.4
-4.5
-4.6
-4.7
-4.8
-4.9
-5.0


0.396
.399
.395
.379
.360
.330
.297
.250
.180
.080
.020
-.026
-.045
-.057
-.062


-0.066
-.030
-.010
-.055
-.100
-.155
-.205
-.270
-.338
-.341
-.282
-.226
-.182
-.150
-.118


-2
-2.1
-2.2
-2.3
-2.4
-2.5
-2.6
-2.7
-2.8
-2.9
-3.0
-3.1
-3.2
-?.3

-3.4
- .5


0. 135
.121
.117
.118
.121
.128
.135
.145
157
.172
.191
.219


.38
. 34'1-'
. i3









NACA TM 1343















0

-4








(I




0r-













0







o
i









r'-
Coa) -
a- /







NACA TM 1343


-0.4


-0.4 -0.2


Figure 3


).4 0.6 0.8
- R (F), R (E)


-0.2 0.2 0.4 0.6 0.8
R (F*), R (E*)
Figure 4


Figures 2, 3, and 4.- Polar diagrams for determination of the curves of con-
stant excitation.


Figure 2


= 0.10
C,= 0.118


u- -


0.4
6 8 o0xl0'3
S2 9 Z 1.0
Ci=o
02


0.2 0


(F (E*E






NACA TM 1343


-OA -0.2 0.2 0.4 06 0.8

Figure 5 R (F),R (E


) 1 (F ),(= o0.15

Ci = O = 0.142
04 0246 I0-3
S.8=0.6
0.2



-0.4 -0.2 0.2 0.4 0.6 0.8

Figure 6 F), (E)






Figure 6
0.6 (F'),, (Ej
I tc= 0.20
S04 4x-3 Cr, =0.187
Ci= o 0.6

02


-0.4 -Q2 0.2 0.4 0.6 0.8
w R(F),R (E*)
Figure 7

Figures 5, 6, and 7.- Polar diagrams for determination of the curves of con-
stant excitation.







NACA TM 1343


0.2 0.4 0.6 U.8
R (F,R (Ee
Figure 8


Figure 9


Figure 10


Figures 8, 9, and 10.- Polar diagrams for determination of the curves of con-
stant excitation.


96 I
I(F), (E*) t 0.05
SCr= 0.033



4 X 10-3j90


Ci=o ---- ---


-0.4 -0.2


A






NACA TM 1343


Figure II


-- R (F ),R(E*)
Figure 12


I(F),I(E) Sc- 0.40
Cr= 0.188

20 x 10-3 3 = 0



Ci= o

-0.4 -0.2 0.2 0.4 0.6 0.8
R(F)), R(E)
Figure 13

Figures 11, 12, and 13.- Polar diagrams for determination of the curves of
constant excitation.






NACA TM 1343


R (F),R(E*)
Figure 14

0.6
I (Fi*), (E*) c=0.70
8 l0o Cr-0.327
6 12 x10-3IO'

P=o
Ci=





-0.4 -0.2 0.2 0.4 0.6 0.8
SR (F ), R(E*)
Figure 15


'(F*), I(E ) c=0.80
Ci= Cr= 0.372
04 3xio- 3



0.2



-0.4 -2 0.2 0.4 0.6 0.8
-- R (F*),R(E*)
Figure 16

Figures 14, 15, and 16.- Polar diagrams for determination of the curves of
constant excitation.






NACA TM 1343


-V R (F ),R (E')


Figure 17


Figure 18


Figures 17 and 18.- Polar diagrams for determination of the
stant excitation.


curves of con-


-0R (F*), R (E*)






NACA TM 1343


0.6


0.4
Gi=40xIO-3


-0.2


I(F*), (E) tc =.12
Cr=0.410

--- =-0.1


0.2 0.4 0.6
- R (F*) ,R (E*)


Figure 19


Figure 20


Figures 19, 20, and 21.-


-) R(F) ,R(E*)
Figure 21

Polar diagrams for determination
constant excitation.


of the curves of


i







NACA TM 1343


5xI00
Cr= 0.060



0 0.05 0.10 0.15 0.20


Figure 22

IO0
i 8* =0.6 c ,r
U 0.15 0.20 0.10 0.097
0.15 0.142
50.20 0.187
5xl104




0 0.05 0.10 0.15 0.20 0.25
_- a8

Figure 23


as* 0.3 C Cr
Ua 9=0.2 0.05 0.033

3 0.4 0.10 0.068
O -- 0.20 0.133 -
0.2 0.30 0.197
0.40 0.258

5x10-4
C=0.05

0.I

0 0.05 0.10 0.15 0.20 025
a 8 *

Figure 24


Figures 22, 23, and 24.- The excitation i as a function of the reciprocal
Ua
perturbation wave length a = for constant critical velocity cr,
A







NACA TM 1343


0.15 0.20
--Fig a u
Figure 25


0.25 0.:


tc
0.80
1.00
1. 1 2
1.20
1.30



.6


0.2 0.3 0.4 0.5
F.u 6*
Figure 26
Figure 26


Cr
0.285
0.363
0.410
0.442
0.462
* Asymptotic value
for Re -- oo
- Interpoloted


Figures 25 and 26.- The excitation as a function of the reciprocal
2pV Ua
perturbation wave length a = for constant critical velocity cr.


4xl0"3
3xl0-3

2x10-3


0
0


.05


Cr
0.188
0.281
0.327
0.372


0.10


2 x 10-



10-2



0
0


U 1.12


B /

I


.1I


I `


i 8S* 0=o 0.6
U"a -- 0.7
- 0 Y4-

---/^,~-04.- --A ^ ;-


0






NACA TM 1343


103 10 4 15 106 1

Re=p
Figure 27

Q3
,3 0.6
a1














Figure 28


Figures 27 and 28.- The curves of constant excitation in the instability regions
of a few laminar boundary-layer profiles.
pie=; 9.1 X0-4







103 104 105 106 107

Re Rz eU

Figure 28

Figures 27 and 28.- The curves of constant excitation in the instability regions
of a few laminar boundary-layer profiles.








NACA TM 1343 57










o
f-


o0

cti
II- 0 00J




0 0//" ,d 0



-o
00 C






Sa




-0
0
uIP" 0 o .
I_ a-



Q1Ic







a 0
o 1






0 M..
^








58 NACA TM 1343











Co



O .
0



o




o *J-
a .
--O
4"1



2 0 C










.o
2 e


0O OI
0 / b c "'




S*~
*" ^/^ A/-- A .2s















I------\ ---__________
*0. NMo
0 0-r
o *' M








NACA TM 13:, 59












0 Cd
00


110
\D


S. ... c
t s .'50




If 7 1 "


O S-
Cd
-4-



SCL o
> -
a i

oa) ro


c



A 4





mgo o






"'4











5x10-3

,/


S=-0.198


ai*
- = 0
aU


-9 1 I I (1 f 11


a8


t


10 10I


4

5

6


7


----- 0.0,

10-21
I I j o.o
S I I0.0.


o0.6I
0. I.

I I 0.0
041 1 0 o.o f

SI I

S1 I
o.61 1 o.
o I I


I I

I I I
-/-/-/,-


0.07
0.06
-0.05
0.04


Uo*
-- Re
7,


Figure 32.- The curves of constant excitation in the instability regions of a
few laminar boundary-layer profiles.


NACA TM 1343


3


I mI I


10-3


108







NACA TM 1343
























































cL
8 -

*"
_o
c?_ _

$I 3


0
0
bD

4-l
ci)








0)
ci

4-













o
0
4
r *
i







nc-i
3o


4 mi





" -






0 ,

0D




ro
ii)






NACA TM 1343


--ow 8*


Re
U02
t~


0.2 0.4 0.6 0.8 02 04 0.6 0.8 1.0
Sa*
-^-Q0


Figure 34.- The circular time frequency Re* for boundary-layer pro-
Ua'
files in the region of increasing pressure for Re* .






NACA TM 1343


\ Uo/max

0.06


Figure 35.- The maximurr excitation \ as a function of the form
parameter of he ainr bondary-layer profiles.
parameter 0 of the laminar boundary-layer profiles.


NACA-Lngley 9-5-52 -1000









- N II e 0
* .P S S -
>- .0 3' 3 S n-
.c > 5 *-. w-! M



CJ A .2g~ g ^ L
0. .0 CM 4 \
I.w 440-


- 0 ^d a -
S14
u 0 on

C 4 6 d~d O~O
0 ~u ,. cU14
0 'L~m4 .' Z.
c.ld .o


w a o
Zb





0 o u
w< O




OLz gii<
3 C rt cl






"[wo6 s .
S .!52^iS








E- 0<< 0c E So
-. eM gSin 'Sg
^*^ %,am1=*,
<|=z | 0 |
"f A5 r^ P0u^^ ~^


y j- 0, aw

5 0 0"
.0 4" O )
o44




C r.- L.

0.* ed s
0 0 00 m
MO 0 'a





o rL e U 8



S.O w .





^^?S-?L~


L-
br, fLo

u -
0



:3
o4 4
C 0
la
*a 9
O^ <


4 4 q d


:- a = z
w 0 4 .
S O 1 S3 U| ^9U
.0 2 **4*

S-- HM B-- 44 .^ s e 'S r C B
30 P 6.^ .1
o~~~; ~ LU g 41,
C c Cd 0
c, L) I'MmI


ot 0,4 >
JO 0" r3 t
Ln.


- ie | | l I
Ci 0 a B9 & i t

U ,0 0m ~
04-00 m0
< 0r
~UU~''P ~Cd
Lk z ui V6 3
L-i -;s ~


Z4

0 U






i .0 4
0 L'- r' .



, r ,-. -






c-75< 0 o s. 4 w
0 F- ., -. W0

w-. 0- -o 0
zm =- IvU "C
| |i
S^J^Ss^ S
sgzP-igi.
|3 S <||
Isy. d
Ii 'id.iio
EMeI II^S
Z^Lsaria


0 4 c 04
M r
U5 < 4a



.0 .0



14r~.4ci
"a. 0
O e *^ c


OMil-iS ^
SMB 0U


Ii|w-1 1

.o E m- c G m
.C ;



uwE o 1
V0 0 0




c Z
u !2 Z

0. 0 I u
, o44a

;

4 a- S 32 gi
0 .0 44 14
@3 C..O


-l
c-




bD
0

14-
oc ^
1.. -4


0~c@







w. rxd:,0 f~~.
4W9 Q454

O OU d g ".o 1
ll Q >



ro o4 i i






.00
gcilVeeoa .
wh~ .a 3^:1









E-4 -S .U g i
o0 .a 6 0
usaco
el Z.M-Imu
R"|se 5^sl-




gcass !-!!
^lauga l-sl"-















































































a




U

E
a

'C


0
8*
U
-j


0

0

0.












0
*a
*E e





br.0
St



"3



D U
5g






*H.0




OL4U
S- 0
01


t


0.



0
79


Eb~


Ut-
> .-I to..

0
n B0

Z k...
*to


0
a
0
V.



s *





a
U


0
0


(0
0
SK e
*R


0
9.





0
bD 0
>
U t
e









-0.8
a 0
..- *o


0
.0
0~
ca'o
1


0i
S3

*R

es
a'.


0


b-*2
SS -S



.0

III0
5 *5
i* S


I0


I '
0





0
3 "
3
4. *




a -






-* eq i 0
0 ctC0 r0
.5 i- LoS
^I "a t 002 JZ^.0 3.j <>3^
S W 0r ,0u0) Bi00gla .
.5 *3 SZ Li
S -i



S a
C '
o zut -< 42-


; ci d


0 .





s0 b s
t g









u0 mm 0.








z z 2 z. i s


.0 mW
o- Li 2 w

:0- 2Aii.

; S2 S .0 S c

w Cl C:"-
oa o, R a, Li
o3 LDZ.0 Ofl 2
- r 0 Li
u W w f
g 5 U .2 a, -
.c ua :a U* r_ E
C. E] (U 0
aw 0W
W -005.0




W L. '0
z 'iicu


- eq i 0
lOf4~0 0
02 Cv) LocE .0 ~'*


4r(i H d


o
St 0


0
w eo I


u-W u
,3 US *
0 *?*
000. <
05.0 8
"'00 0


0' .
.-^t
Uy'3 I
010 -
'5 fc83
fr..., U-


S- 0 3


= 'r
-.- 5 > .0 =

-m a, 0 l

0 0 ~ E:
e ~ L =g toT 1, i
Li 4 = =


S w aut' r 44
i- L.W o.0^ .y tEO,

- eJ -


(fl .000
4 ._0 00..



1 ..J Lii 2 "!

3: .0 U. ,






.W 0 ,.. r.

I; g.. L'-
EMis u g




Gn.^ S-u g in
L.S 00 LL. -

3.U L S So n
025

fr--a ^ -= -s rLo
z.C J S A^ -d
z ) EfiiJ.
n^E rilgg


r- == -W


o :: ci T
*:2 o 8 0 S,
-d m C 0L w.

Cc S cc 0 Cc C 0,
o at A 0 n c' 5

S 0 a' L' .5 0
oa 0
0 0. -
M2 W
00
El~ E. -M awm

.0-w
0 0 0
^ S Li 2. g Wa.

600


M o aj v
.5 1 t

































0
0
us g


a
S Q









> m
S'a


a


*ss
s


a
3=
So

SL.

81
0.


C1
Oa L
Cu
.2 -


;g0
to c

w 14
bD C.-

i-'
014a 01

* a-

41.i
1.J L


a
.0

l0~
. 0









-S a
U'-
- 0
a
* a.
I.J l








UNIVERSITY OF FLORIDA


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