As a contribution to the problem of turbulence on a surface of rotation, the method of small oscillation is applied to the flow past a sphere. It was found that the method developed for two-dimensional flow is applicable without modifications. The frictional layer in the vicinity of the stagnation point of a surface of rotation is less stable against small two-dimensional disturbances than in the stagnation point itself, as proved from an analysis of the velocity distribution made by Homann

Pretsch, J

English

NASA Technical Reports Server

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EAT IONAL ADVISORY COMMITTEE B’OE AERONAUTICS
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\ No. 1017
.,
THE STABILITY Ox LAMIIU,R YLOW PAST A SPHERE .
-k .
::-~y J. Pretsch “ - “ “-
tiuftfahrtforschung
vol. 18, Xo. 10, October 27, 1941
Verlag von R. Oldenbourg, Mnohen und Berlfn
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Washington
June 1942
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TECHNICAII &EllORANikJ.
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FOR AERONAUTICS
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No. 1017
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.: .,
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‘ THE STABILITY’ .OF.~AMINAR FLOW, PAST A SPHERE*
.
4 By J.’.Pretsch ‘“
.,
.....
SUMMARY”
As a contribution to. the problem of turbulence on a
surface of rotation, the methodof small oscillations is
applied to the flow past a sphere. It was found that’ the
method developed for two-dimensional flow is applicable
vithout modifications. The frictional layer in the vicin-
fty of the stagnation point of a surface of rotation is
less stable against small two-dimensional disturbances
than in the stagnation point itself, as proved from an
analysis of the velocity distribution made by Homann.
INTRODUCTION
..’.
The prediction of the resistance of mo”ving bodies
is predicated on the knowledge of the conditions under
which the .laminar flow in the frictional. layer adjacent
to the wall becomes turbulent. The analytical treatments
by Tollmien (references 1, 2), Schlichtung (references 3,
4; 5, 6), and G’6rtler (references 7, 8) dealt with the
two-dimensional problem. These. investigations were based
upon the method of small oscillations by ascertaining
whether the small oscillations of various frequencies.
superposed on t-be basic flow are amplified or clapped,at
given Reynolds numbers. A“critical Reynolds number, the “
so-called “stability limit,” can be considered as ‘a first
orientating measure of the stability of laminar flowt
below which the oscillations of,all frequencies for the
velocity profile concerned are exactly damped. This
stability limit can be indicated for the veloc=y-””profiles
accompanyiilg pressure gradients and pressure rise ih a
two-dimen,slonal “frictional layer and the data of these
..,.
—— — .4—--—--_______________=_____________
*fiU.berdie Stabilit~t der ‘L~mina”rst”r6mung urn e~ne Kugel.H
Luftfahrtforschung., VOL. 18, no., 10, October 27, 1941,
2P l 341-344..
.’
II ---- —
II.
2 NACA Technical Memo.~an,dum I@. 1017;T .’
,. .
.
calculations used for an approximate prediction of the
transitional inception on’ aircraft wings.
. . .
An analysis of a surface of rotation in aflow along
its longitudinal axis, such as is approximately represented
by the airplane fuselage the resistance of which plays an
increasingly important part in the power absorption of
aircraft, would be desirable but for the following difficulty:
The stability study requires the curves of the laminar
velocity distributions to be known so accurately that even
their curvatures can be reliably indicated; but at the
present timet here is little prospect of attaining this
degree of accuracy for a surface Of rotation of ran~om
s<ection.
For this reason, the stability theory of the laminar
frictional layer is to be explored on the simplest surface
of rotation, the sphere, for which sufficiently exact ve-
locity distributions in the frictional layer are likely
to be available in the near future.
In the concluding chapter:, the only reliable distri-
bution known up to now, Homann~s stagnation point profile
(reference 9), is subjected to a stability analysis.
II. GEI?ERAL DIFFERE1{TIAL 3QIJATION OF THE DISTURBANCE
The derivation of the stability theory for the sphere
flow proceeds from the general Navier-Stokes equations,
expressed .in spherical polar coordinates “R, $, and @
(fig. 1) w-ith u, v, and w denoting the velocities in
directio~,’of increasing a, R, and Q, “it affords for
for. the,.axially, symmetrical problem
,-
W 3=o;~-~ =0,,’ :,; (1)
The equation of coxat~nuity reads,” :
,.
.,
. .
R—;;:-;~2@‘in 4) +
,.
\ ‘:r”+z(<v) = ~. , ,(2)
.,, ,
.: ,.:,., .’, .;, ,’
.
. ,.
,. “ .. ) .:, . ;“ :,,:,
and the equations ‘of ‘motion fhr’ V“ ha~et:he formand,v
.—.
NACA Technical.MernorandumNo.1017 3
Cotgo b v+v(#*++*+*~+—~* a~——
2V 2 au 2uCotg8
.—— ———R1 fl,~~ )R1. ..”. f4)f4)
The arc length s starting
from stagnation point and the
distanoe n from the sphere
surface are now introduced as
coordinates
S= R019,
n= R—Ra,
with Ii. as sphere radius.
Restricted to the flow region
wherein n = Ro, the elimina-
tion of pressure from (3) and
(4) gives the equation of
motion
Let U and V designate the
tangent ial and normal componente
of the axially symmetrical base
flow about the sphere having
stability to small vlbrat lone Se
to be analysed. The superposed
interference motion has the
stream function
~(s,n,t)=p(s,n)eia($—et), . . . . . . (6)
a “indicating the spatial natu-
ral frequency and the real part
of c = Cr + lc~ tts phase ve-
locity. The interference ve-
locities are given by
where .~ ~lgdifies dlfferenti-”
at ion with’respect” to n.
!Ehe fundamental motion equa-
tion of the interference prob-
lem then followe” the introduc-
tion of the flow with velocity
component
14”=U+**;’U=V+V*
producedby sup”orposition of
basic flow and interference
motion, in equation (5).
Restricted to the terms linear
in q and with, those of subordi~
nate order of magnitude discount-
ed, the general differential
equation reads for the present:
+~60tgs/R0 [(3,U—c)”a*p— 2( U#’+Uf#)] (3)
by referring the velocities to
the velocity of undisturbed flow
U& the arc length to the sphere
radius Ro and the wall distance
as well as the wave length of the
vibration to-a reference length
Of the friction layer, say, the
displacement thickness 6* tn a
fixed point of the sphere surface
equation (8) can also be written
the form
E-kpressed nondime”n”s%onally
(U–c)(!p’’-.’qp+ g’’p+g-(uql’– U’p)
-.
+ -g cotgs/RO [(3 U—c)n%q —2(U#+U’qf)] (9)
This lin”ear-partial ‘d~fferen-
tial equation of the fourth or-
der represents the most general
motion equation of the disturbed
fricti&n layer flow on the ephere.
It differs from the corresponding
equation of two-dimensional--flow
by the addttion of the underlined
terms which have the same order
of magnitude ‘as the other terms
on the right-hand Aide of equation
(9).
The interference equation ap-
plies equally in direct proximity
to the stagnation point ,=0
4“ l?AOA.Te”chnicalMemorandumMo.1O17
because U and c decreaee a; The normal component V of
sin e/Ro, while cotg 8/Ro~ =; the velocity is obtained from (2),
for RO-OB it becomes, as expect- wh~c~ after Omi SSiOII Of the term
ed, the two-dimensional interfer- ?.1 simplifies to
ence equation. Ro
The next problem is to prove
that-the interference equation
for the flow past a sphere has
exactly the same particular solu-
tions as the two-dimensional,
despite the additive underlined
terms.
111. SOLUTIONS
Assuming a R~* as very large
and bearing in mind that
~* (j
–- . . . . . . . ..(10)
=— R&
the partiai dif-ferential e~ua-
tion (9),reduces to Tollmien~s
frictionless interference equa-
tion for two-dimensional flow
(U—c)(p’’–q)qu–q=o,=o, . . . (11)
in which ‘Ifrictionless solu-
tionsii ~I,Qe are identical with
those .of the two-dimensional
problem by reason of the same
boundary conditions; hence all
statements made elsewhere apply
here also. In the construction
of the solution CPe in the crit-
ical layer U = c the friction
must be taken into account in
form of a transitional function,
ae explained in the following.
The laminar basic velocity in
the critical layer Is approxi-
mate to a parabola
U—c=u/{n-nJ+~(n-nJ1 . . . (12)
with the new variables
n.— neq=
—; e=(LXRe&UO’) -’/’; #=s, . . (13)
define~ as in the two-dlmensioa-
al problem amd
:+~+;;co!gd~o=o . . . . (W
whence
‘=-+J(%9%-J”
o
{
,
=—* 2~(n6+e7j)— —duo n2_e2q2)~f(.
1 duo”
+ ~ ~ (%* + esr)
—2U:~(nO+~7j}-tU/@ ~g (n02—&2v’)
[+~cotgs/R, 2c(n, fSq)’— U:(n7—e’q2)
1}+~(n6s+’&q7 . . . . . . .. . . . . (17)
and
asv 1
[
dUo’ ~u,,—. ___L#q — qf, L’?e2~=— R=% at “+ dc dl
+*cotgs/Ro(u;&’+u/&’q)] (18)
Limited to the terms linear in c
,becauseof the assumably large
Reynolds number Re* equation
(1S) affords; after=multiplica-
tion by
~/u. 1
as.~ ~o,, asP ~o,t
~iGji+2uo’—s?’~—~~~
+
cc av” sc
)“
——+~&%=Re&Ue’ bq
.o~tp . *p
[
1 dUO’nO=
—~~+~&~ —dffh+~~
1 dl?c” , dne UO” dne
—— ——
6, Tnea+uQmna — 2 dl ‘“%
Ccdne
[
Uo’ Uo.,,
)1
__-_!_ cOtgs/RO cnc—~ncz+ ~nea
de R,
. . . . . [19)
But then
*%+%. [“%-’)
‘cc”’$%+u’”c’’(”%-~)$ ’20)
eo that the underlined terms in
(19), are ,disregarded.
The friction- correction for the
frictionless flow 9* in the
critical layer follows from the
‘ .,
“Ww Technical”Merno&xium”No.1017
-,
5
solution Cp= in powers of c ab The veloc Sty dfatribution in
....
pa,=@”+ep,l-f-eg~l-....,,.. .4,(21) the stagnation point of the
which j-oin”s”’thkf’ri”c~ionle$e’so- .sphere .io to be analyzed asregard8 ite ltabilfty againet
.lut.ion..?it..,eo,mdi”titiancne’ef-r.o,m..~: ~mall ~lbrat~ona.
—U=c.
Posting (21) In (20) we get “; This’ velocity profile, which
with q)eo = 1 and multiplying
-by i--: ,.
. . . . . (22)
Again disregarding ’the terms with
e the differential equation for
the transitional function ‘?8~
is the same as in the two-dimen-
sl’onal problem: namely,
(23)
Finally it can be readily proved
that the fr%ction-induced solu-
tions in wall proximity V.,*4
themselves are as in the t~o-&i-
mensional problem. For 6+0
equation (19) afforde the same “
differential equation
ha; the same fbrm in the stagna-
tion point of any other surface
Of ro.tationwith ether than-~an-
ishing curvature. radius, was
.dbrivad b#,,Hqrnanp,(ref,orence 9) .
from the hydr,o’dy,namic”.equations
of motion expressed for this
purpose in cylindrical polar
coordinates:
where u’ the velocity along the
increasing distance r from tho
stagnation point in the tangen-
tial plane at the stagnation
potnt and v the velocit~ along
the increasing distance % from
the stagnation point past, the
body axis. “
Outside of the frtction layer
the”p.otential velocity in the
et-agnat$on point ie the same as
that toward the plate
Ua = ~r; Va=-2~Z (26)
And since thti limlting eond%~’
tions of the differential inter- ,.with t~ho’~irnensionles;”walldistantim ‘ ;.,ference ’equation for flow past
the sphere are also the same as
i.nthe two-dimensi,onal .p~oblem, t =fi”.z “’ (27)
it proves that the study of sta-
... . ,.
billty of laminar flow past the and the formula! for the stream
sphere follows the very same function
method ae In the two-dimensional $ = ~ r=: f (!,).problem. (28),,
This result suggeets the sub
piclon that the method of the Hornanri~:stlifferent:ial:’equation
two-dimensional problem is”not for the stagnation-,$o’f.xitpro-
merely applicable to the sphere file on the surface of rotation
but generally to any surface of reads:
rotation. ~or rigorous proof of
thie suspicion we would have to
fnt + 2 f fll _ fta +, 1 = 0 (29)
U (1) the dimen-proceed from the equations of
motion in general orthogonal
with fI (~) = _
coordinates. a
IVo THE STABILITY Or LAHINAR YLOW
IN THE STAGNATION POINT OF A s!.onless velocity in the fric-
SUR~ACE-O~ ROTATION ti,on layer.
l----- .
6 NACA Technical Memorandum No. 1017
It differs from Eiemenzls differential equation for two-
dimensional stagnation-point flow merely by the number 2
in the second term; it has the same limiting conditions
f =ft=o for c = O, and f~ = 1 for ~ -Q. The
solution is illustrated in figure 2.
For the stability study it further requires an ac-
ceptable approximation of the velocity distribution, for
which in support of previous calculations (reference 10)
an expression of the form
u
c; =
1- (1
- Y)n (n =.2S394 l=.) (30)
is employed, in conjunction with the equation illustrated
in figure 2:
u
--
ua=l - (1 - 0.4391 ~)3 (31)
Figure 3 contains the polar diagram for determining the
neutral stability curve (fig. 4) which separates stable
from unstable interference attitudes in the plane spanned
by Reynolds number and interference frequency. The neu-
tral stability curve for the t~o-dimensional stagnatiou-
point profile according to Hiemenz is included for con-
trast . It is seen that the stability limit Re&itical
on the three-dimensional stagnation-point profile is
about a third of that of the two-dimensional profile.
As a consequence, the laminar flow near the stagnation
point of a surface of rotation is obviously more unstable
in the case af small vibrations than the two-dimensional
stagnation-point flow.
Translation by J. Vanier,
National Advisory Committee
for Aeronautics.
n.
. . . .—. . .,- .————. .-. —.— —. .— . — — .A,
NACA Technical Memorandum No. 1017 7
~
REFERENCES
A
*= ,. ,.,. ,,. .. -,.,, . .. . .
-~
.,
i T
,,,
:~
\
1. !l!ollmien, Walter: TheProduction of Turbulence.
19!.. T.M. NO: 609, NACA, 1931.
(;
~
] 2. Tollmien, W.: Ein allgemeines Kriterium der Instabil-
itgt laminarer (leschwindigkeitsverteilungen. Nachr.
‘[
1
Ges. Wiss. (G6ttingen), Math.-Phys. Klasse,1935, p. 79.
I 3. $chlichting, H.: ~ber die Stahilit~t der Couettestr~m-
ung. Ann. d. Phys. V. 3’olge, 3d. 14, 1932, p. 905.
4. Schlichting, H.: Uber die Entstehung der Turbulenz in
einem rotierenden Zylinder. Nachr. Ges. Wiss.
(G6ttingen) , Math. -Phys. Klasse,1932, p. 160.
5. Schlichting, H.: Zur Entstehung der Turbulenz bei der
Plattenstr?fmung. Nachr. Ges. Wiss. (Gb’ttingen),
Math.-Phys. Klasse, 1933, p. 181.
6. Schlichting, H.: 4mplitudenverteilung und S3nergie-
bilanz der Icleinen Std’rungen bei der Flattenstrdmung.
Nachr. Ges. Wiss. (G6’ttingen) , Math. -Phys. Klasse,
1935, p. 47.
7. G6rtler, H.: Uber den Einfluss der Wandkriimmung auf
die Entstehung der Turbulenz. Z.f.a.M.M. , Bd. 20,
1940, p. 138.
8. G3rtler, Ii.: Uber eine dreidimensionale Instabilit&”t
laminarer Grenzschichten an konkaven Wanden. Na chr .
Ges. T?iss. (Gottingen), Math.-Phys. Klasse, 1940, p. 1.
9. Homanil, F.: Der Einfluss grosser Z&higkeit bei der
Strdmung urn den Zylinder und um die Kugel. Z.f.a.M.M.,
Bd. 16, 1936, p.1~.
10. Fretsch, J,: Uber die Stabilit~t ~$ner Lamiparstr6’mung
durch ein Roh$ snit lcreisfd~m$gezn Querschnitt.
Z.f=aAf.M.,3d. 21, 1941, p. 204.
?“1. - “’ ‘-(/$
,,
NACA Technical Memorandum I?o.1O17’
/- —-----------
R
v
,.”/
,.
\\ ‘-’”
-. . . .. . ,.
+-
l
\
\\
‘\
\,
Figs. 1,2,3,4
~i
‘(II /‘,1//’.‘ I ,,/ )/
‘\
‘--=. ““
‘—----------”
I?igurel.- Definition of spherical polar coordinates.
1.0 -—
[ f---y ““
I - ‘–-
g .
Ua /’”
[.5 - –
—
/’ –-llomanwPr ofile
-- -Approximation
~/
~=1-(1-o.4391g).3
u~
—
.5 1 1.5 2 2
—-+-t
5
Figure 2.- Velocity distribution in the friction layer at the stagnation
point of a surface of rotation.
.——
.1 ----Id
I criti
‘r;10 104 . @ 106 1(Figure 3.- Polar diagram for
determining the neutral Figure 4.- Neutral stability curve for
I stability curve. the stagnationpoint profile.
7
. . .. .—.—-—.
.,,
2
pl
!.. ‘;
,.-. . .
%., .
—


The stability of laminar flow past a sphere

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