On the three-dimensional instability of laminar boundary layers on concave walls

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Title:
On the three-dimensional instability of laminar boundary layers on concave walls
Series Title:
National Advisory Committee for Aeronautics technical memorandum ;
Physical Description:
32 p. : ill. ; 27 cm.
Language:
English
Creator:
Görtler, Henry, 1909-
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington
Publication Date:

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

Notes

Bibliography:
Includes bibliographic references (p. 24).
Statement of Responsibility:
By H. Görtler.
General Note:
"Translation of 'Über eine dreidimensionale Instabilität laminarer Grenzschichten an konkaven Wänden.' Ges. d. Wiss. Göttingen, Nachr. a. d. Math., Bd. 2, Nr. 1, 1940."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
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oclc - 38198621
sobekcm - AA00006152_00001
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Full Text
-' 31j~q~T







-;' 5/ .--


NATIONAL ADVISORY COrMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1575


ON THE THREE-DIIMESIONAL INSTABILITY OF LAMINAR

BOUNDARY LAYERS ON CONCAVE WALLS*

By H. Cbrtler


SUMM ARY


The present report is a study of the stability of laminar boundary-
layer profiles on slightly curved walls relative to small disturbances,
in the shape of vortices, whose axes are parallel to the principal direc-
tion of flow. The result in an eigenvalue problem by which, for a given
undisturbed flow at a prescribed wall, the amplification or decay is com-
puted for each Reynolds number and each vortex thickness. For neutral
disturbances (amplification null) a critical Reynolds number is determined
for each vortex distribution. The nfrimerical calculation produces ampli-
fied disturbances on concave walls only. The variation of the dimension-

less U0- with respect to at is only slightly dependent on the
V HR
shape of the boundary-layer profile. The numerical results yield informa-
tion about stability limit, range of wave length of vortices that can be
amplified, and about the most dangerous vortices with regard to the tran-
sition from laminar to turbulent flow. At the very first appearance of
amplified vortices the flow still is entirely regular; transition to tur-
bulent flow may not be expected until the Reynolds numbers are higher.


1. INTRODUCTION


Until now the stability calculations of laminar two-dimensional fluid
flows on straight walls had usually been based upon disturbances in the
shape of plane wave motions which travel in the direction of the flow.
After some initial failures (see Noether's comprehensive report, 1921
(ref. 2)), the researches by Prandtl, Tietjens, Tollmien, and Schlichting

*"Uber eine dreidimensionale Instabilit'it liminarer Grenzschichten
an konkaven Wnden." Ges. d. Wiss. Gottingen, Nachr. a. d. Math., Bd. 2,
Nr. 1, 1940.
presents part of a thesis by H. Gortler submitted in partial ful-
fillment of the requirements for the degree of Dr. Phil. in Math :Natur-tl
Sciences Faculty of the University of Gdttingen.






2 NACA TM 1575


have, since 1921, produced results which compared well with observations
and to a certain extent yielded information about the important question
of the origin of turbulence from small disturbances. Schlichting, 195h
(ref. 7), gave a report on the results of these investigations. A brief
glance at the method is indicated.

In these calculations, a velocity distribution U(y) that depends
only on the coordinate y at right angles to the plane of the wall is
assumed as the basic flow. The omission of variations of the laminar
basic flow in the x-direction (= principal flow direction parallel to
wall) was dictated by mathematical reasons; and the results of the calcu-
lations enabled valuable deductions to be made as long as the variations
in x-direction were not excessive. To the basic flow U(y) were added
disturbances of assumedly sufficient smallness to permit linearization of
the hydrodynamic equations with regard to the components of the disturb-
ance. This way the problem could be narrowed down to an expression for
the stream function of the disturbance in the form


j(x,y,t) = p(y)ei(ax-t) (1.1)


A particular disturbance can then be built up by the Fourier method as a
disturbance of a general kind by a linear combination of such partial
oscillations. While a is assumed as real, the prefix of the imaginary
part of p determines whether there is amplification or damping with
increasing time t.

The more general expression of three-dimensional disturbances in the
form of traveling waves, which are parallel to the flat wall but oblique
to the base flow direction, hence, for which the velocity compo-
nents ui(i = 1,2,5) are given by


ui =fi(y)ei(alX+a2z-t) (1.2)


(z coordinate parallel to wall and perpendicular to principal flow direc-
tion), was analyzed by H. B. Squire (ref. 6). By comparison with the
aforementioned special case (1.1) to be treated independently, he was able
to show that, in the case of the disturbances (1.2) with al 0, 2 0,
amplification always occurrs at higher Reynolds numbers than in the case
of the disturbances (1.1) with a2 = a12 + a22. Therefore, the investi-
gation can be limited to two-dimensional disturbances of the form (1.1).

The stability investigation of laminar boundary layers relative to
these disturbances was also applied to curved walls (x is then the arc
length of the wall). Tollmien's claculations for the flat plate with
allowance for friction were applied by Schlichting (ref. 5) to the case






MNACA TM 15375


of flow within a rotating circular cylinder. The stabilizing effect of
the wall curvature is such that the critical Reynolds number, formed with
the displacement thickness 6* of the boundary layer, increases with
increasing 6*/R (R = radius of circular cylinder). This stabilizing
effect corresponds likewise with the concepts associated with the action
of the centrifugal force (compare Prandtl, ref. 4).

Boundary-layer flows on slightly curved stationary walls were inves-
tigated by the writer (ref. 10) for stability against two-dimensional dis-
turbances of the form (1.1). Tollmien's result for flat walls, with fric-
tion neglected and, hence, with the evaluation of the critical Reynolds
number disregarded, was the well-known stability criterion which states
that boundary-layer profiles with inflection point are unstable. Such
profiles are characterized by a pressure rise from the outside of the
boundary layer in the direction of flow. For curved walls, this criterion
is modified to the extent that, instead of the stipulated change of the
sign of U"(y), a change of sign at U" + 1 U' is necessary (curvature
radius R of wall positive on walls convex to flow, negative on walls
concave to flow). This Tollmien instability occurs, therefore, on concave
walls only after the minimum of the pressure impressed on the boundary
layer from without, and on convex walls already before the pressure mini-
mum. However, the effect of the wall curvature is extremely small.

It is surprising that convex stationary walls in this sense act
amplifying, but concave walls stabilizing, hence, that the effect of the
centrifugal force does not appear. A confirmation of the criterion fol-
lows from the fact that the same can be applied also to Schlichting's
'case of a rotating cylinder, as explained in detail in the aforementioned
report. There the criterion, in accord with Schlichting's results, yields
a stabilizing effect of the rotating concave wall. In unpublished calcu-
lations, Schlichting investigated the case of the stationary curved wall
in analogy to his own and Tollmien's calculations for the flat wall, with
allowance for friction, for the purpose of observing the wall-curvature
effect on the critical Reynolds number. In a personal conversation,
Mr. Schlichting told me that these calculations also proved the stabi-
lizing effect of concave walls and amplifying effect of convex walls.

In the present report, it will be shown that boundary-layer profiles
on concave walls can become unstable relative to certain three-dimensional
disturbances. It involves an instability that does not occur on flat or
convex walls. The friction is duly allowed for in the calculations and
even the impediment due to the now more complicated type of disturbance
can be overcome in a relatively simple manner. As to the type of these
disturbances, they are similar to those investigated by Taylor in 1923
(ref. 3) in the flow between rotating cylinders and which led to the well-
known instability (excellently confirmed by experiments Taylor made at the
same time) in the form of appearance of sharply defined vortices distrib-
uted boxlike in rectangular zones (compare fig. 1) taken from Taylor's
report.






IIACA TM 1375


The expression of corresponding equidistant vortices in the boundary
layer at a curved wall,'in which the axes of the vortices are parallel to
the principal flow direction (see representation in fig. 2), leads through
the lav-ier-Stokes equations and the equation of continuity to an eigen-
value problem: The amplification of the vortex disturbance for a pre-
scribed basic flow on a given wall must be computed for each vortex dis-
turbance and each Reynolds number of the basic flow; in particular for the
neutral disturbances (zero amplification) a "critical Reynolds number" for
each vortex distance must be determined. It is interesting to know how
these results tie in with the type of basic flow and the wall curvature,
and also the size of the vortices which are amplified first at increasing
Reynolds number, as well as the question of the most dangerous vortices
from the point of view of transition from laminar to turbulent flow.

At the present state of the experimental investigations, only the
order of magnitude of the effect is of interest. All the calculations
are centered on these claims, hence do not aim at an exhaustive mathemat-
ical treatment of the present disturbance problem but rather to a reply
to the questions of interest in practice with an expenditure justifiable
to the claim.


2. DEVELOPMENT OF THE DIFFEELI1TIAL EU,'ATIONS OF DISTURBANCE


Consider the case of two-dimensional flow of a viscous fluid on a
slightly curved stationary wall. The finite curvature radius R of the
wall is, for the sake of simplicity, assumed as constant, and R is
assumed great compared to the boundary-layer thickness & on the wall
formed under the influence of the viscosity; R is chosen positive for
walls concave to the flow since the instability to be explored occurs
only on concave walls and negative for walls convex to the flow.

The basic flow is along the x-direction (x = arc length along the
wall), y( O) is the vertical wall distance, and z is the coordinate
at right angle to both in the direction of the cylinder axis out of whose
surface portion the wall is formed.





NACA TM 1375 5

In these coordinates, the first Navier-Stokes equation, for example,
reads in full rigor and generality

aU + R u u + v uv 8_+
6t R y 6x 6y R y 6z

R 1 P R2 2u 82u 2u
+v + + _
R y P0 x (R y)2x2 6x 2 z2

1 6u 2R av u
R- yy (R y)2x (R -y)2

where u, v, w are the velocity components of the total flow in x-, y-,
and z-directions, p the pressure, p the density, and v the kinematic
viscosity of the flowing medium. All flow variations in x-direction are
disregarded as customary, and R is assumed great with respect to 5 by
binomial development of 1- and R The Navier-Stokes equa-
R y (E y)2
tions and the continuity equation, up to the terms of the order read
then


5+ v( "-R"-+) 6z= 2" ;3 z2 R ly
+ v + + w z 1 + v + 2
ot by R 6z P2 62 R y y




;3w + 1 6p +2 v 1 Ew2 + ( 11w
-+v-+W +v -+
6t 6y oz P 7z 6y2 az2 R oy

Sv w
ay R 6z






6 NACA TM 1575


Th-i undisturbed flow u = u0(y,t), v = O, w = O, p = PO, which
itself is to be a solution of the hydrodynamic equations, for which,
therefore,


6t 6 y2 R cy


u 1 0o
R P y


are applicable, is to change very little during the interval in which the
aUo
disturbances are to be observed. Therefore, -uo and, hence, its equiva-
ot
lent viscosity term is deliberately disregarded hereafter and uO is
put = uO(y).

This basic flow uO(y) is a laminar boundary-layer flow formed by
some previous history based on the viscosity effect. Use is made occa-
sionally of the conventional idealization of such a boundary layer, which
consists in assuming instead of the asymptotic transition in the outer
flow uO = U0 = const., an increase of uo(O) = 0 at the wall up to the
value u0(5) = UO at a certain point y = 6 = "boundary-layer thickness,"
while putting u0 = UO for y 5. The minor effect of the assumedly
slight wall curvature on the outside flow is ignored, since it plays no
part within the framework o4 our theory of a first approximation.

On the assumption that R >> the term R relative to u and

the term with respect to 2u can be disregarded in the equa-
R ay y2
tions (2.1) on account of u uu a2 1 _u. h The same applies to the
by 6' y2 6 sy
two other velocity components. The essential effect of the wall curvature
becomes evident in the term of the second equation (2.1). Moreover,
no systematic difficulties are encountered if the cited small terms are
carried along in the subsequent calculation. But, since they only hamper
the task and contribute nothing to the effect involved, they are
discounted.






NACA IM 1575


So, in conformity with the arrangements at the beginning, the fol-
lowing disturbance equation is used:


u = u0(y) + ul(y) cos azePt

v = vl(y) cos azePt

w = wl(y) sin azePt

p = pO(y) + pl(y) cos azet


(2.2)


a is to be real and the calculation for p itself is to result in real
values; a = where 7 is the wave length of the disturbance. The
quantity p governs the amplification or damping of the flow, depending
upon whether it is greater or smaller than zero. The equation (2.2) cor-
responds to a vortex distribution at the curved wall, the axes of which
coincide with the direction of the principal flow. Figure ) represents
the streamline pattern in a section normal to the principal flow direction.

Introduction of equation (2.2) in the equations (2.1) following the
omissions arising from R >> results in the linearized equations with
respect to the disturbance


pu1 + v dy


S 2u0
Pv1 + u1 -


wl 1 P


- dUl a2u)


p1 dPl v
P iy ( dy 2


- a2)


S(l 2 a.2w


W 1 dvl
1 = a dy


They apply as long as the disturbance velocities are small with
respect to the basic-flow velocity.


To treat this system of
unknown functions ul, vl,
(2.5.5). The result is pl


ordinary differential equations for the
Wl, and pl, we insert wl from (2.5.4) in
as a differential expression of the third


(2.3.1)



(2.5.2)



(2.5.5)


(2.5.4)






8 NACA TM 1575


order in v1. On substituting this expression for pl in (2.5.2), ul
appears as differential expression of the fourth order in v1. Combined
with (2.3.1), the following system of coupled differential equations is
obtained for u1 and vl:


d2ul
v- (P + va2)
dy2


duo
.1 ( 2dy


d2vl
(p + 2va2) + a2(p + va2)v1
dv'


2C2uo
R


u1 (2.4.2)


When u1 and v1 are known, wl and pl are computed from (2.5.4)
and (2.3.3).

It is not recommended to set up a differential equation of the sixth
order for ul or for vI alone by further elimination. The subsequent
calculations are rather based direct on the systems (2.4.1) and (2.4.2) and
merely produce a simplified mode of writing. With 5 denoting a suitably
chosen measure for the boundary-layer thickness, the following dimension-
less factors are utilized:


T-*g


a =









T2
-%


T : V/2 2 + 052


(2.5)


For neutral disturbances, that is, that state of
the disturbances are neither amplified nor damped, p


transition in which
= 0, hence T = a.


d4v
dvy


(2.4.1)






NACA TM 1575


It further is appropriate to use the quantities


UO -1
u' = u/
S1 ) (2.6)

v' =


instead of ul and v1. The prime is also omitted in the following with-
out running a chance of causing a mixup with u and v defined by (2.2).

The differential equations (2.4.1) and (2.4.2) can be written briefly
as differential equations for u and v, as follows:

dU
Lu = v
dl.

LOLv = 02Uu (2.7)


by utilizing the differential operators



d2
L d2 2

O d2
0 d2


In conformity with the order of this system, six boundary conditions
can be prescribed. It is especially stipulated that ul(0) = v1(0) =
wl(0) = 0, i.e., that the fluid hugs the wall. So with consideration to
(2.5.4), it is required that u(O) = v(O) = v'(O) = 0. With the other
three conditions, the decay of the disturbance at n -- oo is attainable,
or when the boundary layer at y = 5, that is, q = 1 is permitted to
change to the constant outside flow. Thus the smooth junction of three
disturbance components with the respective values decaying with T -> oo
outside the boundary layer is assured. The symbol "c" signifies
"sufficiently great."

The homogeneous system of differential equations (2.7) together with
six homogeneous boundary conditions produce an eignevalue problem for the






NACA TM 1575


proposed values of U(n) and R/5: The magnitude of amplification P
UO,
for every given wave length A and every given Reynolds number Re --
V
must be determined (i.e., the relationship existing between the parameters
T, a, and 4, required for solving the homogeneous boundary value prob-
lem, must be calculated). The neutral disturbances (p = O, that is,
T = a) especially, call for the determination of a "critical" Reynolds
number of every wave length of disturbance h, at which the particular
disturbance is exactly maintained without amplifying or decaying.

In the subsequent analysis of the eigenvalue, the practical aspect
is the primary object-namely, at what Reynolds number does amplification
appear (stability limit)? What is the range of the wave lengths of dis-
turbances that can be amplified at all? At what wave lengths does ampli-
fication appear first when Re increases? What disturbances are ampli-
fied most and are therefore most dangerous from the point of view of
turbulence? What effect has the amount of the wall curvature on these
data? Are there appreciable differences when different boundary-layer
profiles U(j) are used as basis? The question of calculating the eigen-
function is disregarded in the present report, although it may be stated
that the method developed enables an approximate representation of it.

It is readily apparent from (2.7) that the Reynolds number and the
wall curvature appear only in the form of the dimensionless Uo/j
v RE
(namely, in parameter p).


5. CONVERSION OF THE DIFFERENTIAL EQUATIONS OF DISTURBANJCE

TO AN EQUIVALENT SYSTEM OF INTE.CRAL EQUATIONS


Green's function G(TI; 10) is identified by the following postulates:

(1) G (r; 10) in 0 < !5 o at n J 10 is twice differentiable with
respect to n.

(2) LG d- T2 G = O at n / nO in 0 5 T .
dn2

(3) G(r; nO) is continuous at the point n = no, but has in its
first derivative the discontinuity defined by


lim (0O + (0) )o 1E; ) = 1


(4) G(O; n) = 0 and G also disappears at n-- ,
(0 D





NACA TM 1375


Green's function H(T; n0) is to have the following quality:
(1) H(q; T0) is four times continuously differentiable with
respect to n in 0 14 oo at T / To.

(2) LOLH = 0 at q io0 in 0 1 5= o.

(5) At the point T = o, H(TI; T0) is continuous including first
and second derivatives, but the third derivative has the
discontinuity


-im 3 (dIO
E-- 0 LdTi


+ E; BO) dH3
drI


(4) H(0; T0)= (0; 0) = 0 and H

By these requirements, G and H at
identified. The calculation gives


disappears at n--> .

0 < I oo are clearly


G(n; lo)

1


H(T; ) =


L e-TIO sinh TTI

Se-T sinh TTI


for T) T0

for rTO TI


1 2- e-aOO e-TO )(cosh
(oa + T)(o T) 2


o -, cosh TTI)


- Te~" 0 ae TI (T sinh crq a sinh TTI) for TI
-TT = 0


1 2 e-_Ol
(a + T)(a T)2


- e-TTI


(cosh aOQ cosh


- I Te'- ae-T sinh cOTIQ a sinh TTIO


7TO)


for T0 = TI


(3.1)





NACA TM 1375


In the event that 0 = 0, H(T; 10) becomes

Se- O + TI) +.L2,, + aG(O + TI) +1


e-( T),(TI 0 n) + 1

Se-a + o) {2T2no + (I +
;7 -


for T <


10) +


e- TIO)a(TI T10) + for


The differential equation system (2.7) is equivalent
equation system


u(n) = J

v) = 02p


G(n; 'o) d- v(no)dTb0
dTIo
H(T; TIO U00) U()U()dTIO


(35.1a)




to the

to the integral


(3.2)


4. METHOD OF DEFINING THE EIGENVALUES

To begin with, the integration interval is divided into partial
intervals of the same length d, and 110(k) and q(k) signify points
of the k-th partial interval: (k 1) d < _(k) kd (k = 1, 2, 3...),
and where, for simplicity sake, I(k) = n (k). The subscript k added
to a function symbol indicates that the particular function is to be
formed at a point of the k-th partial interval, say about Uk = U 1O (k)
furthermore, Gik = G(n(i); I\(k)), ik = Hk (i); TIO(k)), By reason of
the symmetry of the Green functions, Gik = Gki and Hik = Hki.


H(1; o0)=






NACA TM 1575


Patterned after the Fredholm theory the integral in (3.2) is
replaced by summation


ui = dZGikUk Vk (4.1a)
k=l

(i = 1,2,5,...)
00
vi= c2LdZ HikUkuk (4.lb)
k=l


Uk' disappears for sufficiently great arguments TI(k). Letting, as
approximation to the asymptotic transition, the boundary layer change to
U = Constant at Tr = 1 (that is, y = 6), it can be stated more accu-
rately that Uk' = 0 at qO(k) > 1. Thus at finite d a finite sum is
involved in the summation (4.1a). If nd = 1, the summation along k
must be extended from 1 to n. As a result, only the vk with k : n
appear on the right-hand side of the equation (4.1a).

Correspondingly, considering only the equations with the vi at
which i < n in (4.1b), the infinite sum on the right-hand side can be
approximately replaced by a finite sum of k 1 up to a sufficiently
great k = N, because the values Hik decrease rapidly with increasing
Sk owing to the upwardly restricted i < n (the uk themselves decay
with increasing k). The homogeneous system of the n + N equations is
therefore investigated

n
ui + d GikUk 'vk (i = 1,2,...,N)
k=l
(4.2)
N
a2Pd~IHikUkuk vi = 0 (i = 1,2,...,n)
k=l


for the N unknown ui(i = 1,2,...N) and the n unknown
vi(i = 1,2,...n).






14 NACA TM 1575


The vanishing of the determinants


1 0 ... 0 GllU'd G12U2'd ... GnU 'd
0 1 ... 0 G U 'd G U 'd ... G U 'd
21 1 22 2 2n n
0 0 ... 1 GNIUl'd GN2Ud ... GnU'd

C2pH11Uld oa2H12U2d ... a2CHIUNd 1 0 0
21iH2H1Uld a2.H22U2d ... j~pH21,P1jd 0 1 ... 0

G2IiH~1U1d a2p.H2U2d ... 2pnNUNd 0 0 ... 1
(4.3)

postulated for the existence of a nontrivial solution system leaves an
algebraic equation for i at given U, a, and T and, especially, the

critical value of the dimensionless 0--5 for T = a.

Every point q(k) within the boundary layer contributes two non-
trivial series to the determinant-that is, two series (or gaps) in which
not only the terms of the principal diagonal are different from zero; a
point at the border or outside of the boundary layer supplies only a non-
trivial series; the wall point (say, chosen as n(1)) produces only
trivial series because there the Green functions (and U also) disappear.


5. CALCULATION OF CRITICAL REYNOLDS NUMBER VARIATION

IN ITS MINIMUM WAVE LENGTH RANGE


A few words concerning the choice of basic flow for the proposed
numerical calculations are indicated. Theoretically, the basic flow U
represents any boundary-layer flow formed at a wall due to friction and
some earlier history. The present calculations are based on the data of
the Blasius boundary layer of the flat plate (ref. 8). Several other
profile forms are included for comparison.

As regards the profile U of the plate boundary layer, the wall
distance at which the boundary layer in its asymptotic transition to the
outside flow diverges only 1 percent from this flow (curve 1, fig. 5)
serves as measure 6 for the boundary-layer thickness. This is the wall

distance at which the variable E U in Blasius's report assumes the
2 vx
value 3.






NACA T1 1375


As a practical check on the quality of convergence of the calcula-
tion method developed in section 4, 1(k) = T (k) = k--1 and d =
were selected and the following three approximations calculated: for
q(k) and 1O(k), the points 0, 1/2, 1; O, 1/5, 2/5, 1; 0, 1/4, 1/2,
5/4, 1 were taken. According to the remarks made at the conclusion of
the preceding section 4, the calculation of three-, five-, or seven-row
determinants is involved, which result in & linear or quadratic or cubic
equation for jp with respect to a and T. Evaluation for neutral
disturbances (T = a) showed that, to each value of the paranmeter a = ab,
that is, to each wave length of disturbance there corresponds the related
value of [i as smallest root. Since the equations exhibit, on the whole,
coefficients with alternating prefix, only positive roots 4 are obtained
by this calculation, that is, positive values of the critical dimension-

less (U 2vo- hence an instability of the assumed type only on concave
walls (R > 0) result.

The results of this preliminary calculation are shown in figure 4.
The convergence for the parts of the curve above greater or smaller a5
values, where the curves continue to rise, was not quite satisfactory.
But the range of the minimum, which is of chief interest here, emerges
sufficiently accurate.

Beyond these approximations, other points q(k) outside the boundary
layer were assumed for individual ca values as a check that the
approximations achieved in figure 4 are not subjected to appreciable
changes. The minimum becomes a few percent less and shifts slightly
toward smaller ab values.

Incidentally, it should be noted that the order of magnitude of
these numerical values had been checked by special calculations. Origi-
nally it had been attempted to solve (2.7) by expanding T in power
series. The convergence for u and v from n = 0 on was very slow.
Therefore, series from I = 1 on were resorted to. Corresponding to
the order of the differential-equation system and the number of boundary
conditions, three coefficients each had to be determined. In consequence,
u and v had to be joined continuously with continuous first and second
derivatives within the boundary layer. This gave six linear homogeneous
equations for the six still indeterminate coefficients and the stipulated
disappearance of the six-row determinant of this equation system produced
the conditional equation between p, a, and T. But these calculations
failed at the evaluation of the determinants. The values of u and v to
be gained from the series and their derivatives could still be determined
with an accuracy of 1 percent at T = 0.5, but on account of the unavoid-
able large figures appearing in the solution of the determinants, the






NACA TM 1375


results could no longer be regarded as reliable. On the other hand,
near the minimum on the curve of U0- against a, they yielded
results which in order of magnitude agreed with the previous calculations.
Because of the surprisingly small values of the critical Reynolds number,
the new calculations explained above were carried out.

The next step was to find the extent of the change in the results
by a different choice of basic flow U(i). To this end the calculations
with the q(k) places 0, 1/4, 1/2, 5/4, 1 were repeated for the boundary-
layer profiles

TT. \TA + it E it _
U() = (sin inn- s in 1 sinn E 1 -1
2 1 + E 21+ E 1 s + E

for 0 < T 1 (5.1)

1 for N 1 1


with --_ = l, that is, E = 1.752, E = 0.3890. The first pro-
21+ 1 1
file (e = 1) has negative curvature throughout, the second, (E = E2) has
an inversion point. (See fig. 5, curves (2) and (3).)

To assure a physically logical comparison of the results for the
several boundary-layer profiles, it was postulated that all profiles have
the same momentum thickness



S=1 (UO uO)uody = 5 (1 U)Ud (5.2)
UO 0 0


which is a measure for the loss of momentum in the boundary layer. This
condition is met when between the individual boundary-layer thicknesses
the relation 4 = 0.1116 = 0.15261 = 0.13752 exists. Here 6 denotes
the previously defined thickness of the Blasius plate boundary layer,
61 and 62 the boundary-layer thickness (n = 1) for the sine profiles
(5.1) with El and E2.

The result of the comparison is shown in figure 6. It was found
that, when r is plotted against ac, the individual curves within
v Rli






NACA TM 1575


the scope of our approximation do not differ appreciably from one another.
(A corresponding comparison based on the displacement thickness 5*
instead of a produces curves which differ from one another considerably.)

A final calculation, as the roughest approximation to an actual
boundary-layer profile, was made on the section profile

fl for 0 1
u = (5.3)
Sfor n 1

(compare curve 4, fig. 5). If 83 is the boundary-layer thickness of

this profile (y = 63 at T = 1), then '8 = 1 5. At identical momentum
6 ?
thickness with that of the profiles used so far, the difference is
slightly greater, but, considering the rough approximation (5.5), the
departure from the results so far is not very great. (Compare curve 4,
fig. 6.)

The amplifications in the explored wave-length range, at least in
vicinity of the critical Reynolds numbers, can be determined by'the same
approximate method. These calculations were made on the Blasius plate
boundary-layer profile. Instead of the extreme case (3.la) of Green's
function H(T; qO), the more general expression is obtained from (5.1).
To each ab and L-, that is, to each pair of parameters a, T, there

corresponds a particular value of the dimensionless U6 /. The curves
v YR
8d = Constant are obtained by graphical interpolation after conversion
V
of 6 to 3. (Compare fig. 8.) For greater parameter values 92, the
quality of the approximate calculation decreases quickly.


6. ASYMPTOTIC STATEMENTS


Supplemental to these results for great and small values of ab,
a few statements are indicated. A differential equation of the sixth
order for u alone can be obtained from (2.7) by elimination of v. Its
form is disagreeable for the general calculation, but it enables a pre-
diction to be made for the extreme cases of great and small values of a
and T. In this differential equation the coefficients relative to a
and T represent polynomials up to the sixth degree. Considering only
the two highest powers on the assumption of sufficiently great values of
a and T inside the boundary layer, the problem reduces to the second






NACA TM 1575


order differential equation


T2(2a2 + T2)U'2u" 2T2(a2 + T2)U'U"u' +

{T2(o2 + T2)(2U",2 U'U'') a2T4U2 u = iUU'u (6.1)

Integration of this equation across the boundary layer gives the
relation


S2 1 T4U'-2 + T2)(2U"2 U'U' )' + 2T2(c2 +T 2)U'U" r2(2a2 + T2) 2u 1
..:, 2 5 0(6.2
v R R1
2a2 / UU'3udi
"0

The integrals still contain the unknown function u and its first and
second derivative, but the derivatives multiplied by polynomials of lower
degree in a, T, so that the essential contributions to the integral are
already included in the estimation


0 2 5ur2udT
S-)- u 2 (6.2a)
UU'5udT
0

Equation (6.2a) is evaluated by an approximation expression for u by
means of a polynomial of the fourth degree in T, taking into considera-
tion the boundary conditions u(0) = 0, u"(0) = 0 (hence v(0) = 0),
u"'(0) T2u'(0) = 0 (hence w(O) = 0) and u'(l) + Tu(l) = 0 (constant
connection with constant tangent to the solution for u outside the bound-
ary layer, which according to (2.7) is given by u = Constant e-Tn on
account of U'(1) = 0 for il > 1 and hence u" v2u = 0). As a result
u is closely approximated in wall proximity and the postulated decay
toward the outside is attained. Minor errors in u near the outer edge
of the boundary layer are of no consequence in view of the rapidly
decreasing U'; errors of u in the numerator and denominator act in the
same direction (positive integrands throughout), thus affecting the result
very little. Again the asymptotic relation (6.2a) manifests the existence
of instability at concave walls only (R > 0).

The defined polynomial for u reads rigorously


u = constant + 4) + T2(T + 4) 1 + T + -+ !) (6-
+ 3! 2 3






NACA TM 1375


but in practice only the highest powers of T are effective for great
T. Figure 7 represents this approximate function u for several values
of T.

The evaluation of the above appraisal for very great a and T
gives for the Blasius plate profile the asymptotic formula


S2, 32 + (6.4)


corrected for 3.

The same calculation for the section profile (5.5) gives the factor
2.1 instead of 2.3, hence, a slight difference only. The tie-in with the
results obtained for average a values is readily accomplished with the
asymptotic formula (6.4). Figure 8 represents the variation of the crit-
ical factor --k-J plotted against ac in the double logarithmic net
v \J
p 2
(curve =0). The first amplification curves c = Constant > 0
V V
are also shown. The variation of these curves at high a~ values is
obtained by addition of 2.3c to the critical values of at equal
ma, as is readily apparent from (6.4). Moreover, by (6.4)


S0.43 2 043- (6.5)
70


at great am, which constitutes an upper limit for the dimensionless
amplification quantity a solely dependent on -.
UO R
An asymptotic prediction for small a is obtained also by an appro-
priate analysis. It is found that the critical factor --u- increases

proportional to (c~)-l with decreasing ca, as expressed in figure 8.
(For a more accurate prediction, data about the sixth derivative of u
and v are necessary.)


7. DISCUSSION OF THE RESULTS

On the basis of the data collected in the foregoing, the questions
formulated above can now be answered in some detail. As regards the






NACA TM 1375


stability limit, that is, the Reynolds number at which vortices of the
particular type can exist for the first time without decaying again, is

0 5 16 (71)



It involves vortices at which am 0.14, that is, whose wave length
\ is given by


h 45(= 5.06) (7.2)


For the Taylor vortices between stationary outside and rotating
inside cylinder (ref. 5), the vortex appearing at the stability limit has
a wave length of about double the distance of the two cylinders, the vor-
tices thus filling quadratic cells (fig. 1). In the present case they
fill cells with a width of about 2-1/2 times the boundary-layer thickness;
they even extend beyond the boundary layer. The stability limit for the
Taylor vortices is given by = 41.5 where is the cylinder
spacing and UO the velocity of the rotating inside cylinder while the
outside cylinder is at rest. At -8 = d, U- = 2.81
6 v1 -

The appearance of the first vortices in the boundary layer does in
no way indicate incipient turbulence of the flow. On the contrary, it
should be emphasized that the flow will be regular in every way, just the
same as before. (Naturally, it does not include the case in which ordi-
nary plate turbulence already occurs at very great ..) No incipient
turbulence can be produced until the Reynolds numbers become considerably
higher so that the disturbances of an entire range of wave lengths expe-
rience sufficient amplification. The same holds true for the Taylor vor-
tices between fixed outside and rotating inside cylinder; the vortices
first appear as predicted, but the flow does not become turbulent until
the velocities are higher.

The theory developed in the present report postulates that the vari-
ation of the flow in principal flow direction is small enough to be disre-
garded. When the variation in x-direction is small, the results obtained
retain their validity as good approximations. In consequence it is justi-
fied, under this hypothesis, to inquire into the fate of a vortex of given
wave length in its wandering in flow direction through a boundary-layer
thickening up at constant outside velocity. The momentary shape of the
boundary layer has no appreciable effect on the results, as already seen,
when it is referred to the momentum thickness as characteristic length.






NACA TM 1375


In the (- ad) diagram the vortices of constant wave length
A describe, by virtue of the identity

U.w (a=)-/2 U0 ( (7.3)



curves of the configuration

UO-()/2
UV = C (aR)5/2 (7.4)


with (2i)3/2 C = -- = Constant. These curves cross the system of
curves of constant amplification and are reproduced for some values of
the parameter -6-- in figure 8. Tierec are curves in this series which
v VR
cross the zone of unstable disturbances when they enter it they always
cross it since the curves of constant amplification, and especially the
curve 2 = 0 at great a values, vary proportional to (a5)2 (see
V
equation (6.4)) and there are curves in the series that never reach the
instability range. Thus, the vortices corresponding to the latter are
never amplified but always swallowed by the viscosity effect. In the
extreme case there is a curve which is tangent to the neutral curve
"2 = 0. This is the case for the curve with the parameter UN =50
V V I P
shown in figure 8. Therefore, if the disturbance of the wave length
in wandering through the thickening boundary layer ever is to reach an
uinLdired state, --VR must be 50; that is, the inequality
v 1R
1-2/3
> 13.6t -2 (7.5)
R = vY)

must be fulfilled, which affords a measure for the smallest vortices which
are able to experience amplification at all. At the instant where it
reaches its solitary neutral state, the particular boundary disturbance
has a specific wave length referred to the momentum thickness prevailing
at that point. According to figure 8, the contact of the aforementioned
curves occurs at about ab 1.1, where, therefore,


(7.6)


X 5.78 = 0.635






NACA TM 1575


Therefore, the wave length A of the disturbance must have a certain
magnitude characterized by (7.5) if it ever is to get in a critical situ-
ation with the increasing boundary-layer thickness. If equality exists
in (7.5), this instant is given by the fulfillment of (7.6); damping
occurs before and after. If inequality exists in (7.5), then the critical
ratio of wave length to boundary-layer thickness is already reached at a
certain stage, where as yet > 0.63, after which the disturbance is
amplified until a certain second ratio < 0.65 is reached; from then
on the disturbance is damped again. However, the prediction about the
second critical ratio is applicable only when the disturbance on the pre-
viously transversed path of amplification does not exceed the theoreti-
cally specified range of "small" disturbances.

The last question to be answered concerns the most dangerous disturb-
ances, that is, disturbances in the whole range of wave lengths which in
traveling through the boundary layer -at equal Reynolds number -- expe-
v
rience the highest amplification, or in figure 8, the curves
U--H = Constant, which prevail at the start of their amplification path
v yR
before transition to turbulent flow in the range of minimums of the curves
2- = Constant.
v

In an article by M. and F. Clauser (ref. 9), the appearance of tur-
bulent flow at the concave wall was observed for Rex = Ux =2.6 x 10
Sv
at point X = 0.75 and for Rex = 3.1 x 105 at point x = 0.45. Using
R R
the Blasius law of growth of the boundary layer at the flat plate

8 = 7 ) as basis, the values of the critical dimensionless factor

fl~- are 10.6 and 8.6. They are indicated by the dashed markings in
figure 8. A rough extrapolation indicates that, in the vicinity of these
values, the amplification curves = Constant (or the curves
V
x= Constant = which, based upon the law = 2 /, is the same)
UO 4 v 5 5Uo
are minimum at about a = 0.6, hence aS = 5.5. Assuming that the turbu-
lence is caused by vortices of the type investigated here, at reversal the
boundary-layer thickness would, roughly speaking, have increased up to the
order of magnitude of the width of the highest amplified vortices. The
latter, in turn, would have a wave length A of about h 50R-21.






NACA TM 1375 23


For a more accurate prediction about the most dangerous vortices, the par-
ticular total amplification throughout the unstable range would have to
be determined for different vortices a = Constant by integration; but
for this, the few amplification curves, which at greater values of
v
become unreliable, is insufficient.


Translated by J. Vanier
National Advisory Committee
for Aeronautics






NACA TM 1575


REFERENCES


1. Blasius, H.: 3renzschichten in FlUssigkeiten mit kleiner Reibung.
Diss. Gottingen 1907, erschienen in Z. f. Math. u. Physik, Bd. 56,
1908, pp. 1-57. (Available as NACA TM 12.6.)

2. Noether, F.: Das Turbulenzproblem. ZAMM 1, 1921, pp. 125-158.

5. Taylor, G. I.: Stability of Viscous Liquid Contained Between Two
Rotating Cylinders. Phil. Trans. Roy. Soc. (London), vol. 225,
Feb. 8, 1925, pp. 289-545.

4. Prandtl, L.: Einfluss stabilisierender Krafte auf die Turbulenz.
Vortrage aus dem Gebiet der Aerodynamik und verwandter Gebiete.
Aachen 1929, pp. 1-7. (Available as NACA TM 625.)

5. Schlichting, H.: iber die Entstehung der Turbulenz in einem rotie-
renden Zylinder. Nachr. Ges. Wiss. Gottingen, Math.-Phys. Kl. 1952,
pp. 160-198.

6. Squire, H. B.: Stability for Three-Dimensional Disturbances of Viscous
Fluid Flow Between Parallel Walls. Proc. Roy. Soc., (London), ser. A,
vol. 142, Nov. 1, 1955, pp. 621-628.

7. Schlichting, H.: Neuere Untersuchungen Uber die Turbulenzentstehung.
Naturwissenschaften, 22. Jahrg, 1954, pp. 376-581.

8. Prandtl, L.: The Mechanics of Viscous Fluids. Vol. III of Aerodynamic
Theory, div. G, W. F. Durand, ed., Julius Springer (Berlin), 1935,
pp. 34-208.

9. Clauser, Milton, and Clauser, Francis: The Effect of Curvature on the
Transition From Laminar to Turbulent Boundary Layer. NACA TN 615,
1957.

10. Gortler, H.: Ober den Einfluss der WandkrUmmung auf die Entstehung
der Turbulenz. Z.f.a.M.M., Bd. 20, Heft 3, June 1940, pp. 158-147.






NACA TM 1575


Colored fluid


Inside cylinder


cylinder


Figure 1.- Vortex between the walls of two concentric rotating
cylinders according to G. I. Taylor, streamline pattern following
incipient instability (inside and outside cylinder rotate in same
direction).






NACA TM 1575


Figure 2.- Vortex disturbances in the flow of a fluid on a concave wall,
axes of vortices parallel to principal flow direction.






NACA TM 1375


y









I-~-- x --


Figure 3.- Scheme of streamline pattern in a section at right angle to
the principal flow direction.





NACA TM 1375


25
Uo0 8
ujs


20




15














I 2 3
,5 ~ ~ ~ ~ ~ 0 ---(F-----^--









Figure 4.- The critical factor -- for Blasius's flat plate bound-
ary layer plotted against as computed by three increasing
approximations.





NACA TM 1575









(2)
(3)

(4)







a S81 2 83













Figure 5.- The boundary-layer profiles of equal momentum thick-
ness b used as basis of the calculation.






NACA TM 1375


1.5



(4)


1.0 \












------ .. a 0

02 0.4 0.6



Figure 6.- The critical factor -- 4T- plotted against a% for the

boundary-layer profiles of figure 5.


2.0






NACA TM 1575


1.25
77



1.00




0.75




0.50




0.25


Figure 7.- Approximate function for u (according to equation (6.3)).






NACA TM 1375


0.1 / I I V I I I I I I I I
0.01 0.025 0.050,075 0.10 0.25 0.5 0.75 1.0 2.5 5.0 7.5 10



Figure 8.- Total variation of the critical factor U-0 plotted

against as and the first amplification curves = const. The
parallel lines in this diagram represent individual vortices
(h = const.) in a slowly thickened boundary layer at constant out-
side flow velocity U0.


NACA-Langley 6-29-54 1000
























































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