Amplitude distribution and energy balance of small disturbances in plate flow

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Title:
Amplitude distribution and energy balance of small disturbances in plate flow = Amplitudenverteilung und energiebilanz der kleinen störungen bei der plattenströmung
Portion of title:
Amplitudenverteilung und energiebilanz der kleinen störungen bei der plattenströmung
Physical Description:
44 p. : ; 28 cm.
Language:
English
Creator:
Schlichting, H
United States -- National Advisory Committee for Aeronautics
Publisher:
National Advisory Committee for Aeronautics
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aeronautics   ( lcsh )
Force and energy   ( lcsh )
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federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
The distribution of the correlation coefficient and of the amplitude of the disturbance velocities is calculated as a function of the distance from the wall for two neutral disturbances, one at the lower and one at the upper branch of the neutral stability curve. The energy balance of the disturbance motion is also investigated and it is found that as required for neutral stability the energy of the disturbance motion that is dissipated by viscosity is equal to the energy transferred to the disturbance motion from the main flow during one cycle.
Bibliography:
Includes bibliographic references (p. 27).
Statement of Responsibility:
by H. Schlichting.
General Note:
"Technical memorandum 1265."
General Note:
"Report date April 1950."

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University of Florida
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sobekcm - AA00006221_00001
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Full Text

fA TM 2







T7b .29


NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1265


AMPLITUDE DISTRIBUTION AND ENERGY BALANCE OF SMALL

DISTURBANCES IN PLATE FLOW*

By H. Schlichting


SUMMARY


In a previous report by W. Tollmlen, the stability of laminar flow
past a flat plate was investigated by the method of small vibrations,
and the wave length X = 2x/a, the phase velocity cr, and the Reynolds
number R of the neutral disturbances established. In connection with
this, the present report deals with the average disturbance veloc-

ities and \ V72 and the correlation coefficient u
u72 l2
u v
as function of the wall distance y for two special neutral disturbances
(one at the lower and one at the upper branch of the equilibrium curve
in the aR plane). The maximum value of the last two quantities lies
in the vicinity of the critical layer where the velocity of the basic
flow and the phase velocity of the disturbance motion are equal. The
energy balance of the disturbance motion is investigated. The transfer
of energy from primary to secondary motion occurs chiefly in the neigh-
borhood of the critical layer, while the dissipation is almost completely
confined to a small layer next to the wall. The energy conversion in
the two explored disturbances is as follows: In one oscillation period,
half of the total kinetic energy of the disturbance motion on the lower
branch of the equilibrium curve is destroyed by dissipation and replaced
by the energy transferred from the primary to the secondary motion. For
the disturbance on the upper branch of the equilibrium curve, about a
fourth of the kinetic energy of the disturbance motion is dissipated and
replaced in one oscillation period. The requirement that the total
energy balance for the neutral disturbances be equal to zero is fulfilled
with close approximation and affords a welcome check on the previous
solution of the characteristic value problem.


*"Amplitudenverteilung und Energiebilanz der kleinen Storungen bei
der Plattenstromung." Nachrichten von der Gesellschaft der Wissenschaften
zu Gbttingen, Neue Folge, Band 1, No. 4, 1935, PP. 47-78.






NACA TM 1265


1. Introduction

The numerous efforts, within the last decade, to solve the problem
of turbulence (reference 1) have at least produced some satisfactory
results for a certain class of boundary-layer profiles when their
stability was investigated by the method of small vibrations, with due
consideration to the friction of the fluid and the profile curvature
(reference 2). Referring to Tollmien (reference 3), who treated the
example of laminar flow past a flat plate, the writer investigated
several other cases: the Couette flow (reference 4), the amplification
of the unstable disturbances in plate flow (reference 5), and the
stabilizing effect of a stratification by centrifugal forces (reference 6),
and temperature gradients (reference 7). Every one of the investigations
was restricted to the solution of the corresponding characteristic-value
problems, without calculating the characteristic function itself. In
that manner, the wave lengths of the unstable, hence "dangerous",
disturbances were identified as function of the Reynolds number. In
most cases, only the disturbances situated right at the boundary,
between amplification and damping, were determined. For these, just
as much energy is transferred from the primary to the secondary flow,
as secondary-motion energy is dissipated by the friction so that the
total energy balance is zero.

All the stability studies made up to now were, for reasons of
mathematical simplicity, based upon an assumedly plane fundamental flow,
which depends only on the coordinate transverse to the direction of the
flow, and a plane superposed disturbance motion which propagates in form
of a wave motion in the primary-flow direction. While there is no
objection to the limitation to the plane fundamental flow, since it is
frequently realized experimentally, objection may be raised to the plane
disturbance motion because the disturbances accidentally produced in
practice are almost always three-dimensional. Accordingly, it might
appear as if the limitation to two-dimensional disturbances was all
too special. However, H. B. Squire (reference 8) recently demonstrated
on the Couette flow this theorem is equally applicable to boundary-
layer profiles that precisely the specific case of the two-dimensional
disturbance motion is particularly suitable for the stability study in
the following sense: According to Squire, a two-dimensional flow, which
is unstable against three-dimensional disturbances at a certain Reynolds
number, is unstable against two-dimensional disturbances even at a lower
Reynolds number. The two-dimensional disturbances are therefore "more
dangerous" for a flow than the three-dimensional. The critical Reynolds
number, which is defined as lowest stability limit, is thus obtained
precisely from the two-dimensional, not the three-dimensional, disturbances.

To gain a deeper insight into the mechanism of the turbulence
phenomena from small unstable disturbances, a more detailed knowledge
of the properties of these small disturbances is necessary. The present







NACA TM 1265


report, therefore, deals first, with the distribution of the amplitude
of the disturbance over the flow section, that is, the calculation of
the characteristic functions and second, with the study of the energy
distribution and energy balance of the disturbance motion. The investi-
gations are based upon the disturbances of the laminar flow past a flat
plate which are situated exactly at the boundary between amplification
and damping (neutral oscillations).


Chapter I

AMPLITUDE DISTRIBUTION

2. Discussion of the differential equation of disturbance.


Let U(y) be the velocity distribution of the fundamental flow
(fig. 1) and the flow function of the superposed disturbance motion,
which is assume as a wave motion moving in the x direction (direction
of primary flow), whose amplitude p is solely dependent on y, hence


y(x,y,t) = ((y)el(a-t) = ((y)eia(x-ct)

a is real and X = 2n/a is the wave length of the disturbance;
P = ,r + iPi and c = P/a are, in general, complex; T = 29/Pr is
the period of oscillation; Pi indicates the amplification or the lamping,
depending upon whether positive or negative; cr = Pr/C is the phase
velocity of the disturbance. For the disturbance amplitude (p, after
introduction of dimensionless variables from the Navier-Stokes differential
equations, it results in a linear-differential equation of the fourth
order, the differential equation of the disturbance


(U c)(q" a29) U"= (p"" 2a2p" + aqp) ()

(R = Umb/V = Reynolds number, Um = constant velocity outside of the
boundary layer, 8 = characteristic length of the boundary-layer
profile = boundary-layer thickness, v = kinematic viscosity.) The
general solution (p of the disturbance equation is built up from four
particular solutions (pl, *..' 4


p Cl= 1 + C2P2 + C3q(3 + C(P4






NACA TM 1265


The boundary conditions (p = qP' = 0 for y = 0 and y = o give
(as explained in the report cited in reference (5))


C4 = 0 (3a)

and for CI, C2, C3 the system of equations


CllO + C220 + C 330 = 0

ClP10' + C2cP20' + C3(30' = 0
(4)
Clala + C202a = 0

(va = +va' + cPva; v = 1, 2)

The subscript 0 indicates the values at the wall y = 0, subscript a
the values in the connecting point y = a to the region of constant
velocity. From (4) the equation of the characteristic value problem
follows as

Pl0 q20 qC30

Po' ^20' 30 = 0 (5)
o 0 30 0
la 2a 0

This equation is discussed in the earlier reports for several cases.
It contains, aside from the constants of the basic profile, the
parameters a, R, cr, and ci. The complex equation (5) is equivalent
to two real equations, and, if limited to the case of neutral disturbances
Ici = 0), these two equations give, after elimination of cr, one
equation between a and R. This is the equation of the neutral curve
in the aR plane, which separates the unstable from the stable disturbance
attitudes, and was originally computed by Tollmien for the plate flow.
They are assumed to be known for the subsequent study (fig. 2).1

Lihe writer computed the neutral curve for the plate flow cited
under reference 5 again and found some differences with respect to
Tollmien's report. The newly obtained values are used in this study.






NACA TM 1265 5


In the following, the amplitude and energy distribution is computed
for two neutral oZcillations, one of which lies on the lower, the other
on the upper branch of the equilibrium curve (fig. 2). The parameters
of these two neutral oscillations, obtained from the earlier calculation,
are indicated in table 1.

For the calculation of the integration constants Cl, C2, C3
from (4), we put

Cl = 1 (3b)

because the amplitude of disturbance remains inleterminate up to a
constant factor, the intensity factor of the disturbance motion. Thus,
for the other two constants

Sla
2 2a

(6)

SI la 1 la (j
j =P 0 (0 10 B P20 10

The particular solutions qpl(y) and qP2(y) are readily obtainable
by expansion in series from the so-called frictloaless differential
equation of disturbance of the second order, which follows from the
general equation (1) by omission of the terms on the right-hand side
afflicted with the small factor 1/aR. The point U = cr, where phase
velocity of disturbance motion and basic flow velocity agree, and which
is termed the critical point y = yk, is a singular point of the
frictionless differential equation of disturbance, which plays a
prominent part in the investigations.






NACA TM 1265


Putting


Y k *Y k
= -- = v2 ; k= a; a(a y ) = a2
S I a k 1 k 2


where subscript k denotes the values at the critical points, the
frictionless solutions for linear2 velocity distribution read


p = a.-1 sinh(ay1); q2 = cosh(Caly)


and for parabolic velocity distribution

1 2 2
= al2 + a2 2 + a2 + ...3
a1 2 2k


P2 =\ b' + bl + b 2 + "2



=2 O + b 2 + 2'2 + +
q)2 bo blY2 -+ -1 P


Uk"
+ -- ( log 2 for 2 > 0
k

U "
- 1(logly2 in) for y2 < 0
Uk'


TThe Blasius profile of the plate flow is approximated by a linear
and a quadLratic function (fig. i), namely

0 5 y/= 0.175: U/Um = 1.68 y/l


0.175 5 y/b 1.015: U/U, = 1 (1.015 -y/&)2


(7a)


y//6 1.015: U/Um = 1

For the connection between 6 and the displacement thickne3s b* used
in figure 2, b* = 0.3415 Is applicable.






NACA TM 1265


According to earlier data, the coefficients are given by the equations

1 2 aL
al = 1; a2 = ; a = --; a
2 4 0004 a2

a5 = 0.0013 a22 + 0.0083 24; a6 = 0.0024 a22 + 0.000 a2

Op-
bO = 1; b1 = O; b2 = -1 + -2- (10)

b = 0.125 + 0.056 22: b4 = 0.021 0.141 a22 + 0.042 m4

b = 0.005 + 0.005 a.2 + 0'004 a4)


b6 = 0.0015 + 0.0012 a22 0.0038 a24 + 0.0014 a26

The particular solution p1, with its derivatives, is regular
throughout the entire range of flow (-I <1 < 0, O y2 +i), and can
be numerically computed with these data. But the particular solution (2
has a singularity, in which (2' becomes logarithmically infinite in
the critical layer y = yk. The more detailed discussion has shown that
the friction at the wall, and in a restricted vicinity of the critical
layer, must be taken into consideration. The first gives the friction
solution p3, the other, the friction correction for (p2. Introducing
the new variable


(uk)1/3 y Yk
T = y yk) (aRUk) (11)

gives (only the greatest terms from (1) for p(n) being taken into
account) the differential equation


ip" + T" = U (12)
UkI

from which follows the correction for p2 near the critical layer, as
well as the third friction-affected solution .3 The calculation of
these two solutions merits a little closer study.






8 NACA TM 1265


3. The friction solution p,
J

The friction solution %cp, is obtained from the differential
equation (12) when the inhomogeneous terms encumbered with the small
factor e are omitted; hence, from the differential equation

i "" + j(P3" = 0
-A-

Unusual in this equation is that, in contrast to the complete disturbance
equation (1) ard to the frictionless.equtation of disturbance, the
depenience of the parameters a, R, and U' by (ll) enters only as
scale factor for y, and that it is not at all affected by 'U. As a
result, P93(Tl) can be computer once for all entirely independent from the
velocity profile. In this instance, Tietjens' report (reference 9)
constitutes a valuable support. A fundamental system 1" = F "
and 32" = F of equation (13) is given by



1/2 J 3/2 4-it1 and Ij/2 1 3/2 e-i/4 (14)



The expansion in series of the Bessel functions



Jp(Z) (z)P- (iz/2)2V
p 2, V=0 V!(p + V + 1)






NACA TM 1265


gives, when the constant factors (-1/3 (-1) /6ir (4


and (-)/ (-1) /r (2) are omitted

7 _13 19
2!3 -4*7 413 *4-7-10-13 6136.-47-10-13-16-19

S 4 10 16
+i -- + + .o +
1314 3,33.4*7.10 5!35 *47*10*13*16

6 12
2!3 2.25 4134.2.5.8-11


+ 1 54 + ***
1,31.2 333-2.5-8 5135-*2*5*8*111*4

Owing to the boundary condition r3 = 0 for T = + o only the solution
aggregate PIF1 + 02F2" approaching zero for great positive real n
(01, 02 = integration constants) comes into question. For great 1, this
can be represented by the Hankel function of the second kind with the

subscript and the argument q3/2 ei/4, hence by



QP" F (n) = n1/2 H/(2) 3/2 e-i/4]
4 : 1/3 1 31









Herewith, the looked-for solution of (13) the constant
put as cp'(n ) = 3 for the sake of simplicity3 is
expansion in power series near r = 0

II
S311 F.1 1
30 = F+ 2F2)
cP30


NACA TM 1265


factor being
given as




(16a)


and as asynnptotic expansion for great Tj


ScP3 = BF
30 4


(B30' = 1)


Integrating between the limits To and n gives


cp0'
9301


= P1 Fl "d 2 + 02
Jo o0


F2'"dn + 1


- '1 f
To l0


F "dTidn + 2 F 2"dndTn + n n -D
0 o0


where B and D are additional integration constants dependent
on Toj; o is the i coordinate at the wall, y = 0; hence by (1)


1/3
1o ="k/= -k (aRTUk'


(lla)


The boundary condition j1/ 0' = 1 at y = 0, that is, T = io, is
fulfilled by adding the term 1 in equation (16b).


S-C3C30' is the gliding speed of the frictionless solution at the
wall.


-30


(16b)


(16c)






NACA TM 1265


Tietjens computed the integration constants Pi, 02, B, D from
the fact that at a point q = 1, which lies in the range of validity
of both expansions, asymptotic expansion and power series expansion of
the solution of (13) up to and including the third differential quotient
must agree. The resultant equation system, set up and solved by Tietjens,
reads


B F1111 + P2F2"' = BF41'

0P1F1 + 2F2 = BFq"


P1 Fl"dTI + P 2 F2"dTi + 1 = BF4'
STo oq 0


F2"'dndy + T D = BF4


(c) (17)



(d)


0 F1 dnd + P2
ou o o 0


The integration constants 01, P2, B, and D can also be computed
by a simpler method than Tietjens', with the aid of the transition
formula from Hankel's to vessel's functions, which reads4



H /(2)(z) = e Ji/3 J /(z) J (Z)
1/3 sin R/3 [ 1/3 -1/3

This obviates the joining of the two expansions, makes (17) superfluous,
and (17b) gi-es the exact values immediately:


2 = -e-i/6 32/3 and

= -1.o + .687r
= -1.190 + 0.6871


3 3
S~.7 313 + t sin


= 0.789(1 + i)


(18)


Tollmien, who pointed this out to the writer, had this representation
as far back as 1929, but summarily took over the data by Tietjens for the
sake of simplicity.






NACA TM 1265


The factors P1 and D as function of no can then also be indicated
explicitly, namely


B =



D = -


1 ei/3 1 /3/ r() + Fld +


+


+ % o


o1


TI

Jo


- Bq3(O)


(19)


with


Bp3(0) = i0i

Table 2 contains the results of a new calculation of Tietjens' value
carried out by these formulas. The differences from Tietjens' figures are


insignificant. The values of F1", F2", Fl"d1 = FI',
0 10


F2"dn = F2',


J F1'dT = FlI, F2'dn = F2 as function of q are indicated separately

as real and imaginary part in table 3. Since, according to (15), these
quantities are either symmetrical or antisymmetric functions of n, this
table can be continued immediately according to the negative values of q.

For the two neutral oscillations, whose amplitude and energy distri-
bution is to be computed, it is

TI = -2.63 and To = -4.05

The corresponding values of E, according to (lla), are given in table I.
along with the integration constants P1, 2', D obtained by interpolation
for these no values.

This takes care of all the data necessary for computing the friction
solution cp with its first and second derivative as function of I by
the equations (16a, b, and c). Table 4 gives the thus obtained values






NACA TM 1265


(P" (P"1
of ---
30 30
is, according


--- as function of n. The connection between i and y
30
to equation (ll),


Y = Vk + E

In all cases, the friction solution p., from the wall toward the inside
of the flow is very quickly damped out; but it still extends a little
beyond the critical layer for both oscillations.


h. The friction correction of 9c2 in the intermediate layer

The second frictionless solution 92 behaves singularly at the critical
Uk
layer y = yk' namely through equation (9) as j-(y7 yk)log(y yk),
iC


so that q2' behaves as

Uk" I
as --
Uk ( -k)


U II
Uk


From the differential equation (12), in which only the greatest
friction terms are taken into account, follows a solution q2 modified
by the friction, which joins the frictionless solution at some distance
from the critical point. For this purpose cp2 is expanded in powers
of the previously introduced small quantity e = (aRUk)-1/3


(20)


P2 = P20+ E'21 + **


p20 being chosen equal to unity. From (12) follows the inhomogeneous
differential equation


(21)


21 ir21" = --i
Uk'


for (21 with reference to n.






NACA TM 1265


On account of the very small value of C, i can assume great values
even at small y yk values. An attempt is made to find such a
solution P21" of this equation, which for large -j, but small (y yk)
joins on to the frictionless solution

d C2 Uk 1
dY2 Uk' Yk

For large n there shall be:
2
d '21 Uk"
2=-T (22))
dy2 Uk Ti
The corresponding homogeneous equation appeared earlier in the
calculation of cp.. (equation (13)). It has the fundamental system FI"(i)
and F2 "() (equation (15)). A particular solution of (21) is


Uk n flT
S -Uk' F1i" F2"d F2" F d

which can be verified easily by substitution, and the general solution
of (21) is


P21() = iF2" T F1"dl iF1" L F2"d + clFl" + c2F2" (23)


The integration constants c1 and c2, which can be complex, are
evaluated from the boundary conditions. The quantity 921" is complex
and shall join the real frictionless value (equation (22)) for large






NACA TM 1265 5


values of 1. By decomposition in real and imaejinary parts, the four
equations defining the integration constants read


U '" "Lr
U,'21r = G"(r) = -F21"iF' F2r "Fli' + li"F2r' + Flr"F2i





Uk '
F F 1 F
Cr C li" 2rFr 2iF21

k (24)
U, 21i = H"( ) = F21"li + F2r"Flr' + FlF -FrF2r (2


+" c iF" c F + c = 0

+clrFli+t +cliF n+ C2rF21 + c2iF2r


for n = +A


From (21), with the boundary condition (22), it follows that P21r" is
an antisymmetrical, and 1in a symmetrical function of T1. Moreover,
since Flr', F21', F 2, Fli are symmetrical and Fli, 2r', F21i F r
antisymmetrical functions of ri, the following must be true

li = c2r = 0 (25a)
for reasons of symmetry. The other two constants clr, c21 are obtained
by solving the above equation system for i = n,. For the present
calculation, il = 4 was chosen. The series for the Bessel functions
are still fairly convergent for n = 4; but since differences of very
large numbers occur, the separate terms in (24) must be computed to
five digits (table 3). For c1r and c2i


clr = 1.2852; c21 = 0.9373 (25b)

Uk' U'
so that P21r" = G"(1) and P2 = () can be calculated. The
k k
values are given in table 5.






NACA TM 1265


d)2
The values of Q2 and -- in the intermediate layer are obtained
immediately by quadratures, namely

dp2r Uk_ _
SUk(2a)
d r= (1 + log e) + G'() (26a)

and

d.22 Uk" Uk" n Uk
--= U H )d = U- ( = ) (26b)
dy~ Uk' UkuJ4 Uk

A check on this numerical calculation is given by the fact that
1921
for at transition from large positive to large negative j the
transition substitution for q2 deduced by Tollmien (reference 3) must
result again (compare equation (9)), which he obtained by discussion of
the asymptotic representation of the Hankel functions. Tollmien's
transition substitution gives

(p21 9 2i Uk
d) /y=+ dy y=-m Uk

the present numerical calculation gives


2i) 21) Uk Is
= P + -- H"(')dr
d, ,/y=+ \ dy /y=-oo Uk1 -

and the graphical integration gives


+0 T"(l)dl ,;T 4 H"C(Tl)drl = 3.14 (27)

that is, complete agreement within the scope of mathematical accuracy.






NACA TM 1265


The intermediate layer near the critical point, which by the
present calculation reaches from about q = -4 to I = +4, is already
so wide at the first neutral oscillation that it reaches up to the wall
(vall ro = -2.63); at the second oscillation with To = -4.05, the
boundary of the intermediate layer is reached exactly at the wall. With
this, all data needed for the numerical calculation of the solution (p2'
corrected by the friction with p2' and cp ", are available.


5. The numerical values of the integration constants

All three particular solutions Clp' P2' (3 are numerically known.
To build up the required solution ( from it, the numerical values of
the Integration constants C and C, must be accertained (equation (6)).
First of all, equation (2) is rewritten In a more suitable form, namely

3
9 = C1 + C '2 + C,' 9- (2a)
P30

where equations (3a) and (3b) were reported to and. 9 was replaced by
the quantity ?(P 0 which follows immediately from the numerical
calculations. Comparison with (6) gives

3' '- la O -( C920'+ 1') (6a)
2a; L 2 20

This method of writing has the advantage that the two integration
constants C2 and C,' in (2a) are dependent only on the values of
the frictionless solutions Q1 and 92, hence are relatively simple
to compute.

The values of p la' 92a' la' q2a' ,la' 2a and the values
of C2 and C thus computed by (6) and (6a) for both neutral
oscillations are given irn ttble 1. Table 6 and figures 3 and 4 give
the values of pr' i p r I' computed with it, hence the desired
amplitude distribution as function of y/b.






NACA TM 1265


Outside of the boundary layer, at y/6 > 1.015 the simple formula


Sr = C*e-aC r = -aC*e-aY

(pi = 0

is valid for the amplitude distribution. The constant C* is so chosen
that the value of (pr' joins the already found value in y/6 = 1.015.
(Table i.)


6. The average fluctuation velocities and the correction factor

(compare reference 10)

Changing to the real method of writing

u' =- = K p' cos(ax Prt) p' sin(cx Ort) UM

(30)
= = Kca. sin(ax Prt) + PI cos(cx Prt Um

K is a freely available intensity factor. According to figures 3 and 4,
the one phase (qr or pr') predominates in both neutral oscillations.

Tne amplitude distr-ibution of u' and v' can be represented
most appropriately by forming, in analogy with the turbulent fluctuation

velocity, the dimensionless quantities and where the dash
U U
m Um
denotes the time average value formation over a period T at a fixed
point x, y, or in other words

uT
u12 = ; u'2dt (T = vibration period)
T t=o






NACA TM 1265 19

The result is


S r2 K 2 +2. v2 Ka
,2 +uM = M r2 + (pr 2 (31)

and


u'2 +'- K2f ,2 ,2 22 2(
u 2 -2 TLr + Pi' + C (2Pr + ,2) (32)


The last quantity gives the mean kinetic energy of the motion
disturbance. (See eq. 36.) These averages, which are independent of x,
are represented in figures 5, 6, and 7 and table 7 for both neutral
vibrations as functions of y/6. The intensity factor itself was so
chosen that the average value of 'u2 in the boundary layer is equal

to 0.05Um ( '2 y = 0.05 (table 1). The maximum amplitude
k 10 5
for both neutral vibrations lies near the critical layer. The correlation
factor between u' and v', which is completely independent from the
intensity of the motion disturbance, can then also be calculated. It is


1 1 T___
k(u', v') = vt =1
_u,2 ,2 u'v'd \u2 v,2



k(u', v') = (33)
\ (r'2 + c,'2) (Pr2 + 12)

The correlation factor is likewise dependent on y/6 only; its
variation is indicated in figure 8 and table 7. It is negative almost
throughout the entire range of the flow, for both neutral vibrations,
as is to be expected, since, owing to the positive dU/dy, positive u'
is usually coupled with negative v' and negative u' with positive v'.
The maximum value of k is -0.17 and -0.19, respectively. It is inter-
esting to compare the theoretically established correlation coefficient
with Townend's data in a developed turbulent flow (reference 11). The k
values of -0.16 to -0.18, obtained for the flow in a channel of square






NACA TM 1265


cross section at various distances from the axis, are of the same order
of magnitude as those obtained by the present calculation for the
incipient turbulence.


Chapter II

ENERGY DISTRIBUTION

7. The kinetic energy of the disturbance motion.


Having established the amplitude distribution for the two neutral
vibrations, the energy of the disturbance motion can be computed. The
total kinetic energy of the disturbance motion in a layer of unit height,
which, in x direction, extend over a wave length X and in y direc-
tion from the wall to infinity is


E = (u'2 + v'2)dxdy
2 x=0 J=0


P= X r2U 2 2 + Ti 12 + 2(q)p2 + + 2 d(y/6) (34)

The energy dE of the secondary motion in a strip of width dy
and length X is accordingly


E pX 2K 2 1 '2 i2 + 2 r2 + ) (35)
= 22 Pr + ai +

Besides,


5 dE u2 + .12
0.533 = (36)
Eo dy Um2


E is the basic-flow energy in a layer of unit height, length X, and
width 6 (compare equation (32)). Figure 7 shows the dimensionless
energy distribution by equations (35) and (36). The energy is strongly
concentrated near the critical layer. To obtain the total energy E,
the integral (34) must be evaluated. Dividing it in two parts with the
limits 0 = y/56 1.015 and 1.015 y/b < m, the first portion is






NACA TM 1265


obtained by graphical integration based on the computed amplitude
distribution. The second portion is obtained analytically by (29),
namely


S 2 2 (pr2 + i2) d(y/b) = abCe-2.03a
Jy/6=1.01i r +1

The results of the evaluation are given in table 1.

Now the energy of the disturbance motion is compared with the basic-
flow energy Eo in the space of unit height and surface area X X .
It is, by equation (7a)


E = U(y)cdy = 0.533 2X6 (37)
x=0 y=0

Hence, for the ratio of energy of the secondary motion to the energy
of the basic flow E/E, the values presented in table 1 are obtained.


8. The energy balance of the disturbance motion.

Consider the time variation of the secondary-motion energy of a
particle that moves with the basic flow, hence

2 ,2 o -: 'u 2 + v'2(38)
u'2 + = + U ) 2)} (8)

For stable disturbances, the total change of energy of the secondary-
motion is 'J / D(u12 + v'2) dv < 0, for unstable disturbances > 0,
m2 s j J Dt
and for neutral disturbances = 0, the integration extending over the
entire range of the particular flow. Participating on the variation
of the secondary-motion energy are: first, the transfer of kinetic
energy from the primary to the secondary flow, or vice versa; second,
the pressure variation; and, third, dissipation. For neutral vibrations,
the total energy balance is not only equal to zero for the entire space
in question, but for every point y of the cross section, the energy






NACA TM 1265


increase per vibration period T = 2ir/r is also equal to zero. This
is easily confirmed in the following manner: It is


Pf Dr(u'2 + v,2)dt
2 ftO=0



= -(u2 +v'2)dt + uf T + v' 2)dt
2 It=0 6t 2 t=O 6x


= P u2 + + pU u' PU v' ( dt = 0
2 -0 0 x 6x

The first term disappears by reason of the periodicity of u' and v'.
The same holds true when the last term for u' and v' is entered
according to equation (30). Thus, the energy increase per vibration
period T is equal to zero at every point x, y for a neutral
vibration.

It is interesting to see how the several factors enumerated above
participate on the energy conversion in a specific case. For both
specific cases of neutral disturbance the calculation of the energy
is carried out for a plane basic flow and a plane disturbance motion
according to Lorentz (reference 12)


Dt Cu2 + vt2)= -Pu'v' p

fu 22 dy

6x- -C)

where
iv' bu'
= = coefficient of viscosity
ox dy






NACA TM 1265


The first term gives the transfer of energy from the primary to the
secondary flow, the second gives the contribution resulting from the
pressure variations, and the third and fourth terms, the loss of energy
by dissipation. After integration of this term with respect to y over
the total width of the laminar flow from y = 0 to y = and with
respect to x over a wave length X of the disturbance, the second and
fourth terms disappear, since u' and v' disappear for y = 0
and y = and with respect to x have the period X. Thus, the
growth of the energy per unit time in a layer of unit height and base
area 0 < y <, O < x < is:


'v' dxdy -
dy


y=0


(39)


= x dxdy~
(/=0 \ax by


The first integral gives the total energy passing from the primary to
the secondary motion; the second, the total dissipation. The portion
of the energy due to pressure variation is removed by the integration.
The two energy portions for the two neutral disturbances are evaluated.
Through substitution of (30), followed by integration with respect
to x, we find


((rI,l rcpi,) dy IUm2
T=0 dy


2 DE = -QaL2U 2
X Dt m


DE = 2nK2(e1 + e2) X5 2
Um Dt 1 2 2) 2 tm

eI and e2 denoting the dimensionless energy integral


(40)


(41a)


= -p X
Ix=0 y=0


+ (C~p a2)2 d


(Pr'i Pr') d(/) d(y/b)= Y el'd(y/6)
di(y/6) Uo


el *- o
0


Un=0 I






NACA TM 1265


)o 00
e2 = 1 Pr" 2r)2 + (pi" a2i)2 d(y/6) = e2'd(y/6) (41b)



To find the energy change of the disturbance motion in the vibration
period T(T = X/cr, or = phase velocity), this energy change is referred
to the total kinetic energy E of the secondary motion which is given
by equation (34). From (40) follows then the specific energy change of
the disturbance motion as


T DE -m 2 (Cel' + e2')d(y/b) (42)
Eo Dt cr 0.533Z J (2)

where Z = 0.432 and Z = 0.810 for the first and second neutral
vibration, respectively, while 'Jm/cr = 2.86 for both neutral vibrations.

The local energy transfer from primary to secondary motion (1) and
the local dissipation (2) for the two neutral vibrations is then


12 = 78.0e l, or = 4i. 12' (43a, b)

when

T DE =
Dt

The values of e, (equations 4la, b) can be obtained (table 8) on
1,2
the basis of the computed amplitude distribution for both neutral
vibrations. Figure 9 represents the local energy conversion. The
dissipation in vall proximity is seen to be extremely great, while the
critical layer is of no particular importance for the dissipation. But
the energy transfer from the primary to the secondary motion is greatest
in the neighborhood of the critical layer, while at the wall and farther
outside it is very small. The curve is similar to that of the correla-
tion (fig. 8), as anticipated.

The graphical integration of e' ani e2' gives the values
DE
indicated in table 1. The energy balance D- = 0, or e + e2 = 0







I'ACA Tl1 1265


for the neutral vibrations is therefore fulfilled with satisfactory
approximation5, and constitutes a very welcome check on the rather
complicated solution of the characteristic value problem.


The total energy transferred in vibration period
primary to the secondary motion is


T from the


(LE)1/E = 78.0el or 41.6e1

and the total energy dissipated

(1E)2/E = 78.0e2 or b1.6e2

These figures are also shown in table 1. Thus, at the first neutral
*vibration, about half of the secondary-motion energy is destroyed by
dissipation during one vibration; at the second neutral vibration, the
energy conversion is only about half as great.

At the second neutral vibration, the vibration period is a little
greater than at the first, that is, as is readily obtainable .from the
data of tale 1, is


T (2)
1 = 'O)


= 10.1 x 104 -V-;
U2
m


T 2n = 14.8 x 104
2 r2 Um
m


To illustrate; For a plate flow in water with

v = 0.01 cm2sec-1; Um = 20 cm sec-1


5According to the present
vibrations is somewhat greater
primary to the secondary flow.
stability calculation only the


calculation, the dissipation for both
than the transfer of energy from the
This is due to the fact that in the
dissipation of the friction solution P3


was taken into account, while the dissipation of the frictionless vibra-
tion ((p, (2) was ignored. But, in the energy equation, the dissipation
of frictionless and frictional vibration was computed and is therefore a
little greater. Thus, the "neutral vibrations" have, exactly computed,
still a little damping, and the indifference curve (fig. 2) is, as a
result, shifted a little toward the inside.






26 NACA TM 1265


the periods of vibration are

T1 = 2.50 sec; T2 = 3.70 sec

Thus, vibrations of comparatively great periods are involved.


Translated by J. Vanier
National Advisory Committee
for Aeronautics







IlACA Th 12'65


REFERENCES


1. Prandtl, L.: Bemerkungen uber ile Entstehung der Turbulenz. Ztschr.
f. angew. Math. u. Mech. Bd. 1, p. 431, 1921.
Uber die Entstehung der Turbulenz. Zeitschr. f. angew. Math. u.
Mech. Bd. 11, p. 407, 1931.

2. Schlichting, H.: Netiere Untersuchunrgen 'iber lie TurbuLlerLzentztehung.
Die Naturwissen.rchaften Bd. 22, p. 37., 1934.

3. Tollmien, W.: Uber die Entstehung der Turbulenz. Naclr. d. Ges. d.
Wiss. zu Gottingen, Math.-Phys. Klasse 1929, p. 21 and Verhdig.
d. III. Intern. Kongr. f. techn. Mech. Stockholm 1930, p. 105.
(Available as NACA TM 609.)

4. Schl-ichting, H.: Uber die Stabilitat der CouettestrimunLg. Ann. d.
Phvysik, V. Folge, Bd. 14, p. 905, 1932.
And Verhdlg. d. Intern. Mathematikerkongresses Ziurich 1932, p. 283.

5. Schlichting, H.: Zur Entstehung der Turbulenz bel der Plattenstrimunzg.
Nachr. d. Ges. d. Wiss. zu Gottingen, Math.-Phys. Klasse 1933,
p. 181 and Ztschr. f. angew. Math. u. Mech. Bd. 13, p. 171, 1933.

6. Schlichting, H.: Uber die Entstehung der Turbulenz in eimen
rotlerenden Zylinder. Nachr. d. Ges. d. Wiss. zu G.ittingen,
Math.- Phys. Klasse 1932, p. 160.

7. Schlichting, H.: Turbulenz bei Warmeschichtung. Summarj of results
in report cited under reference 2 in Verhilg. d. IV. Intern.
Kongr. fe. techn. Mechanik Cambridge 1934, p. 245.


8. Squire, H. B.: Stability for Three-Dimensional Disturbances of
Viscous Fluid Flow between Parallel Walls. Proc. Roy. Soc.
(London), ser. A. Vol. 142. 1933, pp. 621-628.

9. Tletjens, 0.: Beitrage zum Turbulenzproblem. Diss. Gottingen 1922,
and Ztschr. d. angew. Math. u. Mech. Bd. 5, p. 200, 1925.

10. Tollmien, W.: Uber die Korrelation der Geschwiniigkeitokomponenten in
perlodlsch schwankenden Wirbelverteilunpen. Zeitschlr. f. angew.
Math. u. Mech. Bd. 13, p. 95, 1935.

11. Townend, H. C. H.: Statistical Measurements of Turbulence in the
flow of air through a pipe. Proc. Roy. Soc. A. Vol. 14.5 p. 180,
1934.

12. Lorentz, H. A.: Abhandlungen uber thoretis.he Physik. Bd. I, p. 43,
Leipzig, 1907.






NACA TM 1265


TABLE 1


THE PARAMETERS OF THE TWO NEUTRAL VIBRATIONS OF THE PLATE FLOW

First neutral Second neutral
vibration vibration


cr/Um





1/E



B2
D

(la
('la

CP2a


la
I/..











cla


q 2a
C2

10,
C20


11.r015r2 + p1
Pr2 + '
o0


+ a2((pr2 + (i2)


S0015
S1 ,:i


0.466
2.62 x
.163
.350
.209
1.625
-.494
-2.63
12.6
.0695
+.1021
f-.1526
L-. 0736i
(1.374
+.2001
.416
.040
.211
-2.425
.234
-2.327
.101
1.005
1 -.046
+1.563i
r-i1 O0
S-.157i
.706
.1454


.371

.090


0.737
6.08 x 103
.258
.350
.209
1.625
-.494
-4.05
19.4
t -.0470
+.02761
S.0368
-.06501
.395
+1.241
.435
.097
S.306
-2.240
.417
-2.014
.207
1.011
-.114
+1.563i
j -.988
L -.3251
1.075
.1166

.681

.183






NACA TM 1265


TABLE 1


THE PARAMETERS OF THE TWO NEUTRAL VIBRATIONS OF THE PLATE FLOW Concluded


First neutral vibration Second neutral vibration

E 0.432K2 0.810K2
Eo

E 0.00913 0.0110
E
o

e x 103 5.75 6.39

ex 103 -6.16 -7.10

(6E)1
0.447 0.265
E
( E)2
-0.479 -0.294
E






NACA TM 1265


TABLE 2


NEWLY CALCULATED VALUES OF


AND D


-%o 1l D

o 0.387 + 0.6721 0.672 0.3871
0.5 0.341 + 0.3661 0.770 0.3801
1.0 0.262 + 0.2131 0.892 0.3501
1.5 0.192 + 0.1421 1.023 --0.2611
2.0 0.132 + 0.1131 1.202 0.1351
2.5 0.0822 + 0.10311 1.358 + 0.1241
3.0 0.0332 + 0.09721 1.397 + 0.5191
3.5 -0.0165 + 0.07821 1.139 + 1.0231
4.0 -0.0465 + 0.03231 0.493 + 1.2141






NACA TM 1265


TABLE 3


THE PARTICULAR SOLUTIONS OF p3 AS FUNCTION OF

[Compare equation (15)]


SFr FI F2 F2 F F F F '
Ir 11 2r 21 Ir ii 2r 21
0 0 0 0 0 0 0 0 0
0.5 0.021 0.000 0.125 0.000 0.1250 0.0005 0.5000 0.0026
1.0 0.167 0.003 0.500 0.008 0.4998 0.0167 0.9992 0.0417
1.5 0.563 0.032 1.122 0.063 1.1186 0.1264 1.4865 0.2065
2.0 1.318 0.177 1.974 0.266 1.9368 0.5292 1.8988 0.6588
2.5 2.59 0.678 2.97 0.797 2.7503 1.5799 2.0223 1.5544
2.63 2.87 0.901 3.24 1.022 2.90 2.02 1.95 1.88
3.0 3.97 1.94 3.86 1.91 2.9208 3.6995 1.3354 2.9269
3.5 5.07 4.55 4.02 3.72 0.9512 6.8796 -1.0850 4.2113
4.0 4.20 8.70 2.35 5.77 -5.6379 9.2445 -6.1155 3.3059
4.05 3.84 9.13 2.00 5.89 -6.62 9.20 -6.76 2.90
antisy. sy. Sy. antisy. ay. antisy. anti y. sy.



F Fr F F 2r F 21 F Fli" F 2r"' F 21
r ^ ^ 21 ^Ir "11 ^ 21

0 0 0 1 0 1 0 0 0
0.5 0.5000 0.0052 0.9999 0.0208 0.9998 0.0417 -0.0010 0.1250
1.0 0.9980 0.0833 0.9944 0.1666 0.9861 0.3331 -0.0333 0.4999
1.5 1.4661 0.4206 0.9368 0.5595 0.8442 1.1165 -0.2525 1.1072
2.0 1.7472 1.3108 0.6469 1.2939 0.1187 2.5541 -1.0533 1.8229
2.5 1.3100 3.0463 -0.3215 2.3124 -2.2818 4.3761 -3.0889 2.0809
2.63 0.941 3.64 -0.776 2.58
3.0 -1.1161 5.4735 -2.7429 3.0209 -8.1612 4.7949 -6.8785 0.1417
3.5 -7.6314 6.7189 -7.2934 1.4618 -18.5245 -1.7400 -11.0222 -7.8189
4.0 -19.4902 0.6805 -12.5621 -6.6661 -27.2703 -26.6992 -7.4580 -26.7961
4.05 -20.94 -0.806 -12.92 -8.10
antsy. sy. sy. antisy. sy. antisy. antisy. sy.






NACA TM 1265


TABLE 4


THE FRICTION SOLUTION 9, AS FUNCTION

OF y/6 AND 71

First Neutrkl Vibration


nl mt m' I ^ 1
/1 -3ru- 31i 3r 31 3r 31
___o' _0' _30o' 3o 13 3o'
30 30 30 93 930
-2.63 o -6.40 7.65 1 0 -0.111 -0.0159
-2 .050 -5.67 .794 .665 .188 -.069 -.0099
-1 .130 -3.04 -1.801 .329 .106 -.029 .0029
0 .209 -1.92 -.932 .136 -.012 -.009 .0061
1 .288 -.995 .113 .019 -.040 -.001 .0035
2 .368 -.202 .290 -.025 -.018 0 .0010
3 .447 .063 .076 -.027 -.003 0 .0005
4 .577 -.076 -.012 -.020 .012 0 .0006


Second Neutral Vibration

-4.05 0 -17.7 34 1 0 -0.0204 -0.0639
-3.5 .029 -17.6 8.22 .460 .519 .0001 -.0554
-3 .054 -9.71 -3.08 .106 .534 .0070 -.0415
-2.63 .074 -4.91 -4.67 -.030 .455 .0076 -.0322
-2 .105 -.272 -3.88 -.103 .309 .0051 -.0202
-1 .157 1.378 -2 -.058 .166 .0005 -.0091
0 .209 .717 -1.260 0 .084 -.0008 -.0035
1 .261 -.019 -.679 .016 .035 -.0002 -.0012
2 .313 -.194 -.155 .007 .013 0 0
3 .361 -.058 .019 0 .012 0 0
4 .415 .039 0 -.005 .014 0 0






NACA TM 1265


TABLE 5


THE FRICTION CORRECTION p3


IN THE


INTERMEDIATE LAYER AS FUNCTION OF i

T1 G (TI) 1"(1n) G'(TO) S'(T)

0 0 1.285 -0.774 -1.570
.5 .443 1.165 -.659 -.953
1 .746 .857 -.354 -.448
1.5 .839 .483 .051 -.118
2 .767 .160 .458 .035
2.5 .589 -.018 .798 .073
3 .438 -.072 1.056 .048
3.5 .325 -.055 1.243 .013
4 .250 0 1.386 0


G"(-n) = -G"(TI); H =(-~) = H"(n)
G'(Tj) = G'(n); H'(-i) = -it -H'(,)






NACA TM 1265


TABLE 6


DISTRIBUTION OF AMPLITUDES (Pr3 ,Pi 'i P (r i

AS FUNCTION OF y/b
First Neutral Vibration


y/6 Pr 9 i r' ci' (Prn TI

0 0 0 0 0 7.885 -6.615
.050 .011 -.0040 .411 -.135 6.235 -.003
.090 .032 -.0080 .620 -.115 4.580 .770
.130 .059 -.0112 .782 -.025 3.200 1.717
.170 .090 -.0111 .896 .042 2.355 1.240
.209 .127 -.0088 .976 .069 1.778 .434
.250 .166 -.0063 .986 .070 -.112 -.220
.290 .203 -.0041 .975 .058 -.699 -.576
.370 .276 -.0010 .831 .018 -1.543 -.347
.451 .335 0 .670 .005 -1.778 -.015
.531 .380 0 .510 -.003 -1.641 -.016
.612 .414 0 .361 0 -1.606 0
.693 .438 0 .237 0 -1.514 0
.774 .453 0 .118 0 -1.431 0
.854 .458 0 .007 0 -1.364 0
.935 .455 0 -.101 0 -1.303 0
1.015 .445 0 -.205 0 -1.252 0






NACA TM 1265


TABLE 6


DISTRIBUTION OF AMPLITUDES rp ~' (r ', i' rM, ( i",


AS FUNCTION OF


y/5 Concluded


Second Neutral Vibration

y/5 CPr
0 0 0 0 0 28.54 -27.89
.029 .010 -.0038 .720 -.337 20.27 -2.30
.054 .033 -.0114 1.104 -.237 9.16 6.35
.074 .057 -.0140 1.245 -.113 4.25 6.28
.105 .096 -.0148 1.333 .052 .478 3.60
.157 .168 -.0086 1.350 .129 .558 -.27
.209 .236 -.0041 1.306 .076 -1.218 -1.54
.250 .287 .0005 1.203 .014 -2.967 -1.31
.290 .333 .0002 1.091 -.016 -2.961 -.334
.370 .409 0 .827 -.016 -2.86 .144
.451 .469 0 .607 0 -2.58 0
.531 .507 0 .409 0 -2.21 0
.612 .532 0 .247 0 -1.93 0
.693 .547 0 .105 0 -1.72 0
.774 .550 0 -.026 0 -1.56 0
.854 .544 0 -.148 0 -1.44 0
.935 .527 0 -.260 0 -1.32 0
1.015 .508 0 -.367 0 -1.23 0






NACA TM 1265


TABLE 7

THE MEAN FLUCTUATION VELOCITIES u'2, \42, THE KINETIC

ENERGY OF TEE DISTURBANCE MOTION u'2 + v'2, AND THE

CORRELATION COEFFICIENT k AS FUNCTION

OF y/5. [EQUATIONs (31), (32), (33fl


First Neutral Vibration


y/b 10 L 102 o 1022 + v2' k
UM UM UM
_________ __ ___ _______m__ m______
0 0 0 0 0
.050 .445 .0546 .198 .032
.090 .647 .158 .420 .061
.130 .804 .287 .648 .155
.170 .922 .434 .854 .169
.o09 1.005 .608 1.019 .140
.250 1.016 .795 1.040 .109
.290 1.003 .973 1.019 .064
.370 .854 1.322 .748 .022
.451 .689 1.605 .501 .0075
.531 .524 1.820 .309 0
.612 .371 1.984 .177 0
.693 .244 2.098 .104 0
.774 .121 2.170 .062 0
.854 .007 2.193 .048 0
.935 .103 2.180 .059 0
1.015 .211 2.108 .090 0
1.1 .203 2.027 .082 0
1.2 .193 1.935 .075 0
1.3 .185 1.849 .068 0
1.4 .176 1.763 .062 0
1.5 .168 1.681 .057 0






NACA TM 1265 37




TABLE 7

THE MEAN FLUCTUATING VELOCITIES u'\ v

THE KINETIC ENERGY OF THE DISTURBANCE MOTION u'2 + v'2

AND THE CORRELATION COEFFICIENT k AS FUNCTION OF y/5.

E&UATIONS (31), (32), (33) Concluded

Second Neutral Vibration

y/6 10 102 102 2 + -
U U U2
m m Um2

0 0 0 0 0
.029 .661 .0655 .433 o
.054 .937 .214 .869 .121
.074 1.038 .359 1.068 .170
.105 1.108 .595 1.218 .191
.157 1.127 1.028 1.262 .146
.209 1.086 1.445 1.190 .076
.250 .998 1.755 1.019 .010
.290 .906 2.04 .854 -.015
.370 .686 2.50 .528 -.019
.451 .504 2.87 .332 0
.531 .340 3.10 .208 0
.612 .205 3.26 .146 0
.693 .087 3.35 .118 0
.774 .022 3.37 .117 0
.854 .123 3.33 .124 0
.935 .216 3.22 .149 0
1.015 .305 3.11 .187 0
1.1 .292 2.925 .171 0
1.2 .272 2.72 .146 0
1.3 .252 2.53 .126 0
1.4 .234 2.35 .109 0
1.5 .218 2.18 .093 0






NACA TM 1265


TABLE 8


TEE LOCAL ENERGY CONVERSION 1) = TRANSFER FROM

PRIMARY TO SECONDARY MOTION, 2) = DISSIPATION

EQUATIONS (41) AND (43)

First Neutral Vibration


y/ el x 103 e2' x 103 E dE)1 (E)2

0 0 86.9 0 -6.78
.050 .268 31.8 .021 -2.48
.090 2.150 17.6 .168 -1.37
.130 12.25 10.8 .955 -.84
.170 23.25 5.73 1.814 -.45
.209 28 2.64 2.18 -.21
.250 27.30 .06 2.13 0
.290 22.85 .72 1.78 -.05
.370 7.49 2.20 .584 -.17
.451 1.885 2.80 .147 -.22
.531 -1.102 2.43 -.086 -.19
.612 0 2.36 0 -.18
.(93 0 2.12 0 -.17
.774 0 1.92 0 -.15
., 0 1.75 0 -.14
35 0 1.61 0 -.13
i.01: 0 1.50 0 -.12






NACA TM 2L65


TABLE 8


THE LOCAL ENERGY CONVERSION 1) = TRANSFER FROM

FRTMARY TO SECONDARY MOTION, 2) = DISSIPATION

[EQUATIONS (41) AID (43)] Conclud-ed

Second Neutral Vibration


y/6 eI x 103 e x 103 1 E)I )
1 2E dy 15 E Edy 2
0 0 320 0 -13.31
.029 1 83.6 -.044 -3.48
.054 7.84 24.3 .326 -1.01
.074 18.45 11.15 .768 -.464
.105 41.5 2.61 1.726 -.109
.157 55.6 .04 2.310 -.002
.209 37.5 .840 1.560 -.035
.250 5.22 2.31 .217 -.096
.290 7.97 2.01 -.331 -.084
.370 -8.44 1.91 -.351 -.079
.451 0 1.8o 0 -.075
.531 0 1.38 0 -.057
.612 0 1.01 0 -.042
.693 0 .819 0 -.034
.774 0 .695 0 -.029
.854 0 .608 0 -.025
.935 0 .520 0 -.022
1.01- 0 .458 0 -.019






NACA TM 1265


Figure 1.- Laminar flow past the plate.


Figure 2.- The zone of the stable and unstable disturbances of plate flow.
I = first neutral vibration. U = second neutral vibration.


aS

I






NACA TM 1265


Figure 3.- Real and imaginary part pr, (P, fpr' qi' of the amplitude of
disturbance motion plotted against wall distance for the first neutral
vibration.


Figure 4.- Real and imaginary part )r, q, ', y ir" of the amplitude of
disturbance motion plotted against wall distance for the second neutral
vibration.






NACA TM 1265


0.125

I aw


Figure 5.- The mean fluctuating velocity in the x direction U
plotted against the wall distance for both neutral vibrations.




0.04 i

0.09 "-




0.0/


61 0.4 0s oa 1.0 2 4 1i.


Figure 6.- The mean fluctuating velocity in the y direction rU m
plotted against the wall distance for both neutral vibrations.







NACA TM 1265


Figure 7.- The mean kinetic energy of the disturbance motion u'2 + v2/ m2

plotted against the wall distance for both neutral vibrations.


0.20


0.15
-k





0.06






-a05


I





/,I


a' 0.2 s5 ofl


u'v1
Figure 8.- The correlation coefficient k -= plotted against Y/6


for both neutral vibrations.






NACA TM 1265


15.0 -- ------------



10.0
I (2)

12




024 '^ '^ a6 a0 \
-25 ---II

Figure 9.- The local energy conversion of the secondary motion for the
first and second neutral vibration. I (1), U (1) = energy transfer from
primary to secondary flow; I (2), II (2) = dissipation.


NACA-Langley 4-18-50 -950



















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UNIVERSITY OF FLORIDA


3 1262 08105 023 8