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P rin te d b y th e
G e o r g e B a n t a P u b l i s h i n g .C om pany M e n a sh a, W isconsin
C O N T E N T S
T. A lfrey: M eth o d s of represen tin g the p ro p e rtie s of v iscoelastic m ateria ls . . 143 H . J. B a rte n : On the deflection of a ca n tilev e r b e a m ... 275
K. E. B issh o p p : T h e inverse of a stiffness m a trix . 82
K. E. B isshopp an d D. C. D ru c k e r: Large deflection of c a n tilev e r beam s . . 272 D. R. B la sk e tt an d H . S c h w e rd tfe g e r: A form ula for th e solutio n of an a r b itra ry
a n a ly tic e q u a t i o n ... 266 M . H . B lew ett: (See II. Poriisky)
E. B ro m berg a n d J. J. S to k e r: N o n-linear th eo ry of curved elastic sheets . . 246 G. F. C a rrie r: On th e:n o n -lin ear v ib ra tio n problem of th e elastic strin g . . 157 G. F. C a rrie r: On th e v ib ra tio n s of th e ro ta tin g r i n g ... 235 P . Y. C h ou: On velocity correlatio n s and th e solu tio ns of th e eq u a tio n s of
tu rb u le n t f l u c t u a t i o n ... 38 P. Y. C hou: P ressu re flow of a tu rb u le n t fluid betw een tw o infinite parallel
planes ...198 N. C o b u rn : T h e K arm an -T sien pressure-volum e relatio n in th e tw o -dim en
sional supersonic flow of com pressible f l u i d s ... 106 J. W. C raggs an d C. J. T ra n te r: T h e ca p acity of tw in cable . . . . 268, 380 C. H . Dix, C. Y. Fu and E. W . M c L e m o re : R ayleigh w aves an d free surface
r e f le c tio n s ... 151 D. C. D ru c k e r: (See K . E . Bisshopp)
C. M . F o w ler: A nalysis of num erical solution s of tra n s ie n t heat-flow problem s . 361 T. C. F ry : Som e num erical m eth o d s for locating ro o ts of polynom ials . . . 89 C. Y. F u : (See C. II. D ix)
W. H o re n s te in : On ce rtain integrals in th e th eo ry of h e a t co nd u ctio n s . . 183 J . C. J a e g e r: D iffusion in tu rb u le n t flow betw een parallel planes . . . . 210 T h. v. K a rm an an d H . S. T s ie n : L ifting-line th eo ry for a wing in non-uniform
f l o w ...• . ... 1 R. K ing an d D. M id d le to n : T h e cylindrical a n te n n a ; c u rre n t an d im pedance . 302 E. G. K o g b e tlian tz: Q u a n tita tiv e in te rp re ta tio n of m aps of m ag n etic a n d
g ra v ita tio n a l anom alies by m a th em atica l m e t h o d s ...55 W. K o h n : T h e spherical g y r o c o m p a s s ... 87 C. C. L in : On th e s ta b ility of tw o-dim ensional parallel flows.
I. G eneral t h e o r y ... 117 11. S ta b ility in an inviscid f l u i d ... 218 III. S ta b ility in a viscous f l u i d ...277 A. N. L o w an: On th e problem of h ea t cond uction in a sem i-infinite ra d ia tin g
w i r e ... 84 E. W . M c L e m o re : (See C. II. D ix)
D. M id d le to n : (See R. K ing)
I. O patow ski: C an tile v er beam s of uniform s t r e n g t h ... 76 A. A. P o p o ff: A new m ethod of in teg ra tio n by m eans of o rth o g o n a lity foci . . 166 H . P o ritsk y an d M . H . B lew ett: A m ethod of solution of field problem s by
m eans of o v erlap p in g r e g i o n s ... ... 336 W . P r a g e r : On plane elastic s tra in in doubly-con nected do m ains . . . . 377
A. P r e is m a n : G rap hical analyses of nonlinear c i r c u i t s ... 185 M . R ic h a rd so n : T h e pressure d istrib u tio n on a body in shear flow . . . . 175 H . A. R o b in so n : On A. A. Popoflf’s m ethod of in teg ra tio n by o rth o g o n a lity foci . 383 S. A. S c h a a f: A cy lin d er cooling p r o b l e m ... 356 S. A. S chelkunoff: S olution of lin ear an d slig h tly n o nlinear differential e q u a
tions ...348 H . S c h w e rd tfe g e r: (See D. R. Blaskett)
C. H . W . S ed g ew ic k : On p lastic bodies w ith ro ta tio n a l sy m m e try . . . . 178 J . J. S to k e r: (See E. Bromberg)
S. T im o sh e n k o : On th e tre a tm e n t of d isco n tin u ities in beam deflection p ro b lem s ...182 C. J. T r a n te r : (See J. W. Craggs)
H . S. T s ie n : (See Th. v. K drmdn)
A. V a zso n y i: On ro ta tio n a l gas f l o w s ...29 S. E. W a rs c h a w s k i: On T h eo d o rsen ’s m ethod of conform al m app ing of nearly
circu lar r e g i o n s ... 12 B ook R e v i e w s ...384 B ibliographical l i s t s ... 184, 276,-386
277
Q U A R T E R L Y OF A P P L I E D M A T H E M A T I C S
V ol. I l l J A N U A R Y , 1946 N o . 4
O N T H E S T A B IL IT Y O F T W O -D I M E N S IO N A L P A R A L L E L F L O W S
P A R T II I.— STA B ILITY IN A V ISC O U S FLU ID *
BY C. C. L IN **
Guggenheim Laboratory, California Institute of Technology
11. G e n e ra l co n sid e ra tio n s. T h e in v e stig a tio n s in P a r t II* of th e s ta b ility c h a r
ac teristics of a p arallel flow in an inviscid fluid led to v ery useful in fo rm atio n . In th e first place, th e y enable us to visualize th e effect of pressu re forces v e ry clearly.
In th e second place, th e resu lts can be used as a guide for stu d y in g th e s ta b ility p ro b lem in a real fluid, since in sta b ility is expected to occur only for sufficiently large R eynolds n u m bers. T h u s, if we know th e general ch a rac te ristic s of “inviscid s ta b ility ” for a given v elo city d istrib u tio n , som e s ta b ility ch a rac te ristic s in a real fluid can be o b tain ed b y considering a modification of th ese re su lts b y th e effect of viscosity. Such a consideration w as first m ade b y H eisenberg, [14]f w ho d e m o n stra te d t h a t th e ef
fect of v iscosity is g enerally destabilizing a t v ery large R eynolds nu m b ers. T h e re are, how ever, a few p d in ts to be su p p lem en ted in his discussion. W e shall th ere fo re s tu d y th is problem in som e d e ta il in §12.
T o g e t a good u n d e rsta n d in g of th e s ta b ility problem , we w a n t to know th e fol
lowing p o in ts for a n y given class of v elo city d istrib u tio n . F ir s t of all, w e w a n t to know w h e th e r th is class of flows is s ta b le for all R eynolds num bers. S econdly, if it is sta b le for ce rtain R eynolds n u m b ers w ith re sp ect to d istu rb an ce s of ce rtain w a v e
lengths, b u t u n sta b le u n d e r o th e r conditions, we w a n t to know th e gen eral n a tu re of th e cu rv e cfia, R ) — 0 w hich sep a rates regions of s ta b ility an d in s ta b ility in th e a, R plane. T h ird ly , such a curve will be expected to show a m inim um in R , below w hich all sm all d istu rb a n c e s a re d am p ed o u t. I t is th erefo re desirab le to be ab le to ca lcu late th is m inim um critical R eynolds n u m b er rapidly.
In th e n ex t section, w e shall solve these problem s for tw o classes of flows; nam ely, (a) v elo city d istrib u tio n s of th e sy m m etrical ty p e, an d (b) v elocity d is trib u tio n s of th e b o u n d a ry -la y e r ty p e. Indeed, it will be show n t h a t in these cases the flow is always unstable fo r sufficiently large Reynolds numbers, w h e th e r th e v elo city curv e has a p o in t of inflection or n o t. T h e curve of n e u tra l s ta b ility cfia, R ) = 0 w ill be show n to belong to e ith e r of th e ty p e s show n in F ig. 9. W hen th e v elo city curv e has no p o in t of inflec
tio n , th e tw o a sy m p to tic bran ch es of th e curv e h av e th e com m on a sy m p to te a = 0
* R eceiv ed J u ly 18, 1945. P a r ts I a n d II o f th is p a p e r a p p e a re d in th is Q u a rte rly 3, 117-142, a n d 2 1 8 -2 3 4 (1945).
** N ow a t B row n U n iv e rsity .
f T h e figures in b ra c k e ts re fe r to title s in th e B ib lio g ra p h y a t th e end of P a r t I.
278 C. C. LIN [Vol. I l l , No. 4
(Fig. 9a). W hen th e re is a p o in t of inflection, one b ra n ch h as th e a sy m p to te a = 0, w hile th e o th e r h as th e a sy m p to te a = a s > 0 (Fig. 9b). In e ith e r case, i t will be show n t h a t th e region inside th e loop is th e region of in sta b ility . I t should be observed how th ese re su lts fit in w ith th o se of R ayleigh an d T ollm ien for th e inviscid case, an d w ith th e d estab ilizin g effect of visco sity n oted b y H eisenberg for large R ey no lds n u m b ers. Sim ple form ulae will be derived to express th e a sy m p to tic b ran ches of th e curves in Fig. 9. In fact, it is b y m eans of these a sy m p to tic b eh avio urs an d a criterion of s ta b ility of S ynge [63], t h a t th e .re s u lts m entioned ab o v e are established . Sim ple rules will also be given, b y w hich th e m inim um critical R eynolds n u m b er can be v ery easily o b tain ed from q u a n titie s involving v ery sim ple in teg ral an d differential ex
pressions of th e v elo city d istrib u tio n w{y). V ery little nu m erical lab o r is involved for th e ca lc u latio n in a n y p a rtic u la r case.
H eisenberg also discussed th e general sh ap e of th e cu rv e c,(a, R ) = 0. H ow ever, his a rg u m e n t does n o t a p p e a r to be v e ry decisive, an d som e of his resu lts are n o t well s ta te d . H e did n o t o b ta in th e a sy m p to tic form s of th e a (i? ) curv e as given below. H e also trie d to e stim a te th e o rd e r of m ag n itu d e of th e critical R ey no lds n u m b er, b u t d id n o t tr y to m ak e an a p p ro x im a te calcu latio n in term s of sim ple d ifferential and in teg ral expressions of w (y) (loc. cit., p. 600).
In o rd e r to o b ta in definite num erical results, w hich m ay be su b jecte d to experi
m en tal, verification, we shall ap p ly o u r th e o ry to th e s ta b ility problem of special v elo city d istrib u tio n s. In §13, w e shall give th e resu lts of ca lcu latio n of th e n e u tra l curves in (a) th e B lasius case an d (b) the.P o iseu ille case. T h e m eth o d of calcu latio n and its num erical accu racy will be given in th e A ppendix. F re q u e n t reference to th e eq u a tio n s in it will therefore be m ade in th e following sections.
12. H eisen b erg ’s criterion and the general characteristics of the curve of n eu tral stability. W e shall now proceed to s tu d y th e general s ta b ility c h a ra c te ristic s in a viscous fluid as in d icated above. In th e first place, we shall develop H eisen b erg ’s criterio n in a slig h tly im proved form . W e shall th en re s tric t ourselves to v elo city d is
trib u tio n s of th e sym m etrical ty p e an d of th e b o u n d ary -lay er ty p e. F o r th ese cases, w e shall prov e th e results sum m arized in th e nex t section
a ) Heisenberg’s criterion. Along th e n eu tra l curve d (a , R) = 0
(if it exists), all th e p a ra m e te rs a, c, a n d R a re fu n ctio n s of one of th e m , say , R . L e t us re s tric t ourselves to cases w here a an d c do n o t ap p ro ach zero along th e n e u tra l curv e as R becom es infinite. T h e n , th e ap p ro x im a tio n s (6.27) are v alid for sufficiently large values of R . B y using (6.26), (6.24) an d (6.27), we can tran sfo rm (6.13) in to th e fol
low ing:
K in(c) +■ w{ (1 — c) £ a inK2n+i(c) |
n-0 '
:n(c) + w i i f ] a 2”2r2„+ i ( i ; ) l •
n - 0 '
If th e re is an inviscid n eu tra l d istu rb a n c e w ith c = c„ a —a s, we h av e T . Kin+l(c,) = 0.
n—0
cc g— ri/A / «
£ *-"Kin+1 (c) = — = = = = ] £ « 2-
n—0 V \ C t R \ l C) V n—0
+ V (
px\H r n M c T 6 1
£
a i n H ,1946] ST A B IL IT Y OF PARALLEL FLOWS 279
H ence, w hen a = a s, an d c = c„+Ac, differing b u t slig h tly from cs, w e have
... ¿ a , nK2n{Ct) T , a , nH 2n(c,)
e~*,/4 „_o e*'14 „_o ■
Ac = - . . . ... h — = — ---— --- (12.1)
•\/a»22(l cs) 5 “ 2n / \ / a sRc\ A-, 2n , / y 2n-f li^«/ / ^ 2n+lV>^*j
n=0 n=0
Sim ilar con siderations of (6.14), (6.15), an d (6.17) give resp ectively
X “ « 7?2n+l(c,) z>irt/4
Ac = — == — --- , (12.2)
V « . & i A 2n , X a . A 2n+2(C.) n=0
C” ' 4 „_02 a3 I I2n(c,)
Ac = —7= ---> (12.3)
V a 37?cj “ 2n , X <*« 0-2n+l(C3)
7 1 = 0
X ] a , I I ‘2n—l ( C s ) + ( 1 — C . ) X ) a > +
o n=l n=0
Ac = —j = —— (12.4)
\ Z a sRca 2us^' r \ 1 /, n2 V'' 2n+1;r ' t \ X “ * A 2»(c,) + (1 — c,) X TsX+ifc.)
n-0
In general, it is n o t v ery easy to d eterm in e w h e th e r Ac,->0 or < 0 . H ow ever, w hen c, an d a,, are b o th sm all, b u t n o t zero, all th e abov e eq u a tio n s will reduce to
Ac = erili/ \ / a , R c * K { (c.), (12.5)
a fte r we m ake use of th e red u ctio n s corresponding to (1) an d (2) of th e A ppendix.
N ow, w hen c is sm all,
Cv2 1 w "
Ki(c) = dy(w - c)~ 2 = — — (log c + in) + 0 (1 );
J y , w { C W1 3
hence, it can be easily verified t h a t K ( (c) is a p p ro x im a te ly real an d p o sitiv e fo r sm all real values of c. H ence, (12.5) show s t h a t A c i> 0 in ev e ry case. T h e d istu rb a n c e w ith w ave length 2i r /a s, n e u tra lly sta b le in th e inviscid case, is u n stab le w hen viscous forces are considered. T h is re su lt w as first o b tain ed b y H eisenberg, an d m ay be fo rm u lated as follows:
H e i s e n b e r g ’s c r i t e r i o n . I f a velocity profile has an “inviscid,” neutral disturbance with non-vanishing wave number and phase velocity, the disturbance with the same wave number is unstable i n the real flu id when the Reynolds number is sufficiently large.
In H eisenberg’s original discussion, only th e first ty p e of m otion is considered.
T h e la st e q u a tio n on p. 597 of his p ap e r is esse n tially -o u r e q u a tio n (12.1) w ith all th e te rm s in a2 d ropped. E v id e n tly , th e ab o v e a rg u m e n ts hold only for a„ cs5^0. I t will be seen la te r from F ig. 9 t h a t one n e u tra l d istu rb a n c e w ith a = 0 a n d c = 0 for infinite R , is a c tu a lly stab le for finite values of R.
280 C. C. LIN [Vol. I l l , No. 4 b) Asym ptotic behavior of the neutral a{R ) curve. W e shall now s tu d y th e general a sy m p to tic b eh av io r of th e n e u tra l curve, assum ing t h a t it exists. T h e answ er to th e existence problem will be e v id e n t d u rin g th e course of th e in vestig atio n. F o r large values of a R , w e w ould g enerally expect z of (6.28) to be m uch g re a te r th a n 1, b u t it is also possible for z to ap p ro ach a finite v alu e or zero. W e shall th erefo re discuss b o th possibilities.
F o r large v alu es of z, we h av e ap p ro x im ately ,
J r = 1, J t = 1 /V 2 ir = w f / V ^ R ? , (12.6) w here J i is sm all. If we refer to (5) a n d (7) of A ppendix, we see t h a t th e im ag in ary p a r t v of th e rig h t-h a n d side m em ber of those eq u a tio n s d epends on t h a t of w{ cKi(c) and th o se of th e in teg rals H i, H i , M 's and N 's . If a an d c are sm all, w hich will be verified a posteriori, we h av e only th e c o n trib u tio n from th e first te rm ; thus,*
ww"
v — — irwi --- for w = c. (12.7)
w' 3
B y using (12.6) an d (12.7), we see th a t th e eq u atio n s (8) in th e A ppendix can be a p p ro x im a te d b y
u = 1, (12.8)
and
v = cw 0 = w{ / \ / 2 a R c 3. (12.9)
wl
T h ese are th e eq u a tio n s for d eterm in in g a relatio n a ( R ) , if we elim in ate c betw een them .
In th e case w here a R c3 appro ach es a finite lim it as a R —* <*>, c m u s t ap p ro ach zero.
H ence, v m u st ap p ro ach zero, an d from (8) of A ppen dix , J i m u st also ap p ro ach zero.
F rom th e curve for J ,( z ) , we see t h a t th is h ap p e n s for z = 2.294, for w hich y r = 2.292.
T h e n , using (9) of A ppendix, (12.6), and (12.7), we h ave
a R = w { h3/ c 3, z = 2.294, (12.10)
an d
« = J r = 2.292. (12.11)
T h e tw o ty p e s of relatio n s (12.8), (12.9) an d (12.10), (12.11) ev id en tly correspond to tw o different b ran ches of th e a ( R ) curve (cf. Fig. 9). T h ese cond ition s can be satisfied in cases (2a) an d (3) of section 6 (cf. (11.5), (11.7) of A pp end ix), b u t it ap p e a rs to be difficult in case (2b) (cf. (11.6) of A ppendix).
C a s e (2a). Symmetrical velocity distribution with symmetrical <f>(y) . W e consider th e cases w here b o th a an d c are sm all. B y using (12.5) a n d n o tin g t h a t u ta k e s on a finite v alu e in e ith e r (12.8) or (12.11), we see t h a t we m u st h ave
u , where Z710 = # i(0 ) = f * w2dy, (12.12)
H i0a 2 J y,
* I n fa c t, th e o th e r te rm s n e v e r g iv e c o n sid era b le c o n trib u tio n s to th e im a g in a ry p a r t even fo r only m o d e ra te ly sm all v a lu e s o f a a n d c. T h is p o in t w ill b e d iscussed in th e A p p e n d ix . T h e a p p ro x im a tio n (12.7) will b e used fo r all la te r c a lc u la d o n s.
i.e., c m u st ap p ro ach zero as fa st as a 2. T h e a sy m p to tic beh av io r of th e a ( R ) curves as given by (12.8)—(12.11) are as follows:
R = { w i1/ 2 ^ H u w ' 72) a ~ 11, c = {H w/ w [ ) a \ (first branch), (12.13) R = w'iS( z / J THio)3a~ 7, c = (H wJ T/ w f ) a 2, (second branch), (12.14) w here J T = 2.292, z = 2.294 ap p ro x im a te ly .
C a s e (3 ). Boundary-layer profile. H ere, th e e q u a tio n corresponding to (1 2 .1 2 ) is (cf. (7) of A ppendix)
u — W\ c/a, (12.15)
i.e., c m u st ap p ro ach zero as fa st as a. N o te t h a t in th e previous case, th e relation (12.12) depends b o th on w{ an d on f vv\w"dy. H ere, it d epends only on th e in itial slope of th e velo city cu rv e w { . T h e tw o bran ch es of th e a ( R ) curv e for larg e valu es of R are
R — (wi n /2ir2w o '2)a:~6, c = a/vv{, (first branch), (12.16) and
R = w{ { z / J ry a- \ c = a j r/ w i (second branch), (12.17) w here J r = 2.292, z — 2.294 ap p ro x im a te ly .
Effect of varying curvature i n the curve of velocity distribution. In e ith e r case, th e second b ra n ch of o u r a s y m p to tic cu rv e d epend s v e ry little up o n th e sh ap e of th e velo city profile, w hile th e first b ra n ch dep en d s v ery m uch upo n it th ro u g h th e term W o'. T h is fa c t will enable us to co rrelate o u r p re sen t resu lts w ith th e inviscid inves
tig a tio n s of R ayleigh an d T ollm ien, as discussed in P a r t I I .
In all th e cases considered, we h av e w " < 0 for y < y i b u t sufficiently n e a r to it.
If w " (y ) n ev e r v anishes for y i < y < y i , we can replace w i ' b y w {' in th e expressions (12.13) an d (12.16). In general,
w n / „ „ ¡ v
1946] ST A B IL IT Y OF PARALLEL FLOWS 281
wo' = w i -1 c + [ - — — — ) c2 + • • • . Wi \ 2w i2 2w i3 /
N ow , for a flow w hich is essentially parallel, th e b o u n d a ry co nditio n AA\p = 0, w hich holds on th e solid w all for all tw o-dim ensional la m in a r flows, can be easily verified to be eq u iv ale n t to w { " = 0 . H ence, we have
w iv
W = w i + — - C2 + • • • . 2 w iz
T h u s, if w {' = 0 , b u t w " does n o t van ish for y \ < y < y 2, we h av e
R = {2w '^ / n 'iH\o(wu f \ a - l\ for case (2a), (12.18) and
R r = { l w [ l i / % i { w iv) 2 ) a ~ i \ for case (3). (12.19) In case th e v elo city profile shows a p o in t of inflection,
w i = 0 for c = c„
282 C. C. LIN [Vol. I l l, No. 4
T h en , w e h av e ap p ro x im a te ly
w 'o ' = (wlv/2w(2)(c~ — c]). ( 12. 20)
I t can now be seen from (12.13) an d (12.16) t h a t R becom es infinite as c ap proaches c,. L e t th e corresponding v alu e of a b e d enoted b y a,. T h en in stead of (12.18) an d
(12.19), th e following re la tio n s hold:
T h u s, for e ith e r a sy m m etrical or a b o u n d a ry -la y e r d istrib u tio n w ith a flex, we h av e
as R —>eo, a —>a„ c—>c„. In all th ese appro x im atio n s, w e assume a s and c, to be so small t h a t th e p rev io u s ap p ro x im a tio n s still hold, b u t th e q u a lita tiv e n a tu re of o u r conclu
sions c a n n o t be changed for m o d era te values of a, and c,.
T h e general ch a rac te ristic s o b tain ed from o u r foregoing discussions are su m m a
rized in T a b le I I , an d a re in d icated b y th e a sy m p to tic bran ch es of th e solid lines in F ig. 9. L e t us proceed to discuss th e ir significance.
i) I t m ay be expected t h a t th e region betw een th e tw o a sy m p to tic b ran ch es of th e curves re p resen ts a region of in sta b ility . T h u s, every symmetrical or boundary-layer profile is unstable fo r sufficiently large values of the Reynolds number. T h is p o in t will be su b s ta n tia te d below.
ii) In th e cases w here w {' > 0 , th e tw o b ran ch es of curves ap p ro ach th e tw o differ
e n t a sy m p to te s a = 0 and a = a „ leaving a. finite instable region for infinite R eynolds n u m b er. In th e o th e r cases, th e tw o bran ch es ap p ro ach th e sam e a s y m p to te a = 0, leaving only th e p ossibility of a n e u tra l d istu rb a n c e of infinite w ave-len gth a t infinite R eynolds n u m b er. T h ese re su lts agree w ith H eisen berg’s criterio n and th e re su lts o b ta in e d from th e considerations of an inviscid fluid in P a r t I I of th is w ork. I t th u s a p pears t h a t th e inviscid d istu rb a n c e w ith a = 0, c = 0 is a c tu a lly n o t as triv ia l as it m ay first a p p e a r to be, for it is a c tu a lly th e lim iting case of n e u tra l d istu rb an ce s in a real fluid.
c) Existence of self-excited disturbances.* T o estab lish th e a c tu a l existence of self
excited d istu rb an ce s, w e tr y to show th a t c¿ > 0 a t least in th e neighborhood of th e n e u tra l cu rve. In d eed , we m a y regard c as a fun ctio n of th e tw o in d ep en d e n t p a ra m eters a an d R ' = a R , a n d show t h a t (dCi/dR')a < 0 for th e first b ra n ch of th e curv e. T h is is analogous to H eisen b erg ’s criterion , an d d e m o n stra te s th e sam e general conclusion t h a t th e effect of v iscosity is d estab ilizin g a t large R eynolds n u m b ers. T o fix o u r ideas, we shall c a rry o u t th e analy sis for th e case of sy m m etrical profiles. T h e o th e r case can be carried o u t in a sim ilar m anner.
W e begin w ith th e eq u a tio n
(6.14) /■l(a i C) $31
w h e r e /2(a, c) a n d / 4(a, c) are given b y (6.26) an d fin/fiii is given b y (6.28). B y using those relations, we can tran sfo rm (6.14) into
R ~ ( a — a 3) 2 (12.23)
* T h is se c tio n w as in se rte d la te in 1944 a f te r discu ssio n s w ith P ro f. C . L. P e k e ris. H e m e n tio n e d th e p o ss ib ility t h a t th e n e u tra l c u rv e m ig h t b e a c u rv e o f m in im u m d a m p in g w ith s ta b le reg io n s on b o th sides of it. See also S c h lic h tin g ’s c alc u la tio n s [52].
J(z) = 4>22y / ^<¿>22 H---- or, by further using (6.30) and (6.24),
7 ( a ) - 1 = w i e ' E a " K in(.c) / E a2" tf2n_i(c).
n«“ l / n<»l
We now regard a as fixed and consider the variation of c w ith R' or z, which is a known function of c and R'. Taking logarithmic derivatives on both sides, we have
1946} S T A B IL IT Y O F P A R A L L E L FLO W S 283
r , ) ) . 5 - * ^ 5
7 ( S ) _ 1 ' C K U c ) E «2" / Î2n - ! ( c ) 1 d Z
(12.24)
So far, a, c, and z are arbitrary. On the neutral curve, c and z are real, and we m ay use the relation (12.13) if we are on the first branch of the neutral curve with large values of R'. Indeed, for large values of z, (12.6) gives
7 ( s )
700 - i
3
2 s
and
3dz 1 dR' 3 dc
2z 2 R ’ 2 c
B y using these relations, it can be easily verified th at (12.24) leads to
dR' 3dc 2 R' 2c
E C X ^ K L (c) E
= dc <---1-
E <X2”K în(c) E «2n^2n- l( i) Remembering- th at c = 0 ( a 2) and noting th at
Ki (c) =
1 Wo
— (log c + iV) + 0 (1),
Ml C Mo
where 0 (1) is real, we can reduce the above equation to
' 3dc ( w'o ) 1 d [ w ic w o )
- + — • = dc 4 1 + M l c— — (log c + 17r) > — 4---— (log C + l i t ) > ,
2c I Mo3 ; dc (. M0 3 ;
dR' 3dc 2 R
or
1 dR' 3dc ( d
= --- 41 + c —
2 R' 2c I dc
_ f t t W1CUf0
— - (log c + iV)
L w o 3 }■ (12.25)
For small values of c, th e expression in th e brackets has a positive real part and a negative imaginary part. H ence, (d c / d R ' ) a has a negative imaginary part. T his com pletes the proof. Indeed, if w{ ' does not vanish, we have
284 C. C. L IN [Vol. III, N o. 4
(;
d c r \ C
) --- R ' ~ 6' 6, d R ' J a 3R'
<dCi \ 2 ■KWy C-
) = ---— ---i ? ' - 7' 6.
\ d R ' J a 9 w{ R'
(12.26)
In th e above derivation, it should be noted th at all approxim ations are made by neglecting small terms of higher orders in comparison with some terms which have been retained. T hus, the conclusion of stab ility or instability would n ot be altered by those terms of higher orders.
d) The m i n i m u m critical Reynolds number and the m i n i m u m critical wave-length.
H aving dem onstrated the instability of th e sym m etrical and th e boundary-layer pro
files, we w ant to answer the following questions. First, does there exist a minimum critical Reynolds number, below which the flow is stable for disturbances of all w ave
lengths? If so, can w e get an approxim ate estim ate of its magnitude? Secondly, does there exist a minimum w ave-length of the disturbance (maximum a) below which the flow is stable at all Reynolds numbers? If so, can w e get an approxim ate estim ate of its m agnitude? W e shall see th at in trying to answer these questions, w e can also roughly depict the com plete a ( R ) curve, which separates stab ility from instability.
T he existence of these minimum values can be m ost conveniently inferred from a condition of stab ility derived by Synge* from energy considerations. H is condition reads
(qR) 2 < (2a- + l)(4od + l ) / a 2, q = max | w' | . (12.27) T his condition insures the existence of a minimum critical R eynolds number. It per
m its a to become infinite only for R —>=o. B u t w e know from our previous considera
tions th at a— or 0 as R —> oo. H ence, we would expect th a t there exists a maximum value of a, above which there is always stability. T he neutral curve m ust therefore take the general shape shown in Fig. 9. T he asym ptotic behaviors of th e solid curves are drawn in qualitative accordance with (12.6), (12.7), and (12.23); the other parts of the solid curves are arbitrarily drawn to indicate the general shape of the curve.
It is evident th at the region outside the curve is the region of stability, and the en
closed region is the region of instability. Similar conclusions have been reached by H eisenbergf but his arguments and results appear to be som ewhat obscure.
H aving established the existence of the minimum critical value of R and the maximum critical value of a, we proceed to make an estim ate of their m agnitude.
W e shall see th at our theory perm its us to give a quite good approxim ation to the minimum value of a R (cf. (12.30)). Since this roughly corresponds to the minimum value of R and also to th e maximum value of a (as w ill be clear from th e individual exam ples given below), we can get a rough estim ate of these values by making a rough estim ate of a corresponding to the minimum value of a R .
Using th e second equation of (8) of Appendix and the approxim ation (12.7) for v, we have approxim ately
ww"
. y ,(2) = v(c) = - irWi — • (12.28)
* S yn ge, [63], eq. (11.23), p. 258. H is X is our a. T h e con d ition is originally sta ted for plane C ou ette and plane P oiseu ille m otion; b u t it is e asily seen th a t it hold s for a general v e lo c ity distribu tion w ith 2 = m ax| w ’\ .
f Heisenberg, loc. cit., p. 601.
1946] S T A B IL IT Y O F P A R A L L E L FLO W S 285
R
Fi g. 9 . General nature of the curve of neutral sta b ility . T h e d otted curve is curve of stab ility given b y Syn ge.
If w e recall th at z is proportional to c(ai?)1/3, this equation determines (a i? ) 1/3 as a function of c. It can then be easily verified th at the minimum value of (ad? ) 1/3 occurs when
z j l { z ) = cv'(c). (12.29)
If the point where this holds is denoted by z = z 0, we have approxim ately from(11.28)*
ccR = w { * ( — (12.30)
* Cf. H eisenberg, loc. cit., eq. (29a), p. 602. H e p u t z0~ l .
286 C. C. L IN [Vol. I l l , N o . 4
T he point Zo is roughly th e value where Ji (z) takes its maximum value, because (12.29) is approxim ately J i ( z ) = 0, when c is small. T he corresponding value of a can be approxim ately obtained by taking
u = J r{z o), (12.31)
in accordance with the first equation of (8) of Appendix, where u is given by the real part of (5) or (7) of Appendix, as the case m ay be.
e) Ap pr ox i ma t e rules. W e now proceed to make some rough approxim ations in order to obtain sim ple rules, which are convenient for estim ating the minimum criti
cal Reynolds number. T he condition J t (z0) = 0 gives z0 = 3.21, where ^ (zo ) = 1.49 and J i ( z 0)= 0 .5 8 . T he corresponding value of ccan be obtained from the second equation of (8) of Appendix. P u ttin g m = 1.5 in accordance w ith (12.31) and neglecting second order terms of X, we have
®(1 - 2X) = J i ( z 0) = 0.58,
- itivI (1 - 2 \ ) ( w w " / w n) = 0.58, (12. 32) where X is defined by (4) of the Appendix. To find the value of c from this equation, it is convenient to plot its left-hand side together with w { y ) against y, and read the value off the latter curve where the former curve gives th e value 0.58. T he value of c so obtained turns out to be very close to its maximum value along the neutral curve, and is approxim ately the value where R is a minimum.
T he values of a and R m ust be obtained from more rough approxim ations. W ith a consultation of th e values of the integrals H( c ), K ( c), M [ c) and N ( c ) given in the Appendix, we m ay derive th e following reasonable estim ates of a:
/
> VIw2dy, for symmetrical profiles, (12.33)1/1
a = w { c , for boundary layer profiles. (12.34)
T hese values turn out to be som ewhat lower than th e accurate values. W ith an ap
proximate allowance for these inaccuracies and taking round numbers, w e get the following approxim ate rules for the minimum critical Reynolds number:
30^1 /HioWi
R = ---\ / ---1 for symmetrical profiles, (12.35)
c3 V o
R = 25w/ > for boundary layer profiles. (12.36)
c 4
T he calculations have been carried out for the Blasius case and the plane Poiseuille case. In the first case, the thickness of the boundary layer is taken so th at the initial slope is wi =2. It is found that
R = 5906 for Poiseuille case, i?5i = 502 for Blasius case T he quantity 5i is the displacem ent thickness
Lse, |
(12.37)
/
I oo0 ( 1 — w) dy = 0.28673,1946] S T A B IL I T Y O F P A R A L L E L FLO W S 287
where y is measured from th e solid wall. T hese values for the minimum critical R ey nolds number agree fairly well with those obtained below from more elaborate numer
ical calculations.
W hen these estim ates of the minimum values of R and the corresponding values of a (eqs. (12.32)—(12.36)) are combined with the asym p totic behavior of the a ( R ) curves (eqs. (12.16)—(12.17)), th e curve of neutral stab ility in any case can be sketched w ith fair accuracy w ith very little labor.
T he maximum value of a on the neutral curve cannot be very well estim ated. It is usually som ewhat higher than the values of a given by (12.33) and (12.34).
13. Stability characteristics of special velocity distributions. We shall now apply our theory to some special cases in order to obtain numerical results comparable with experiments. W e take (a) the Blasius case as a typical boundary-layer profile, and (b) the plane Poiseuille motion as a typical sym m etrical profile. In any case, the resultant curve of stab ility lim it should have the general shape discussed in the last two sections. Only the results will be given here; the method of calculation and its accuracy will be discussed in the Appendix.
a) Stability of plane Poiseuille flow. T he velocity distribution of plane Poiseuille motion is given by
w ( y) = 2 y — y2, with w { = 2, HflO) = 8/15. (13.1)
T a b l e II. B ehavior of R ( a ) Curve for Large V alues of R for V elocity D istributions w ith w " < 0 for th e M ain P art of th e Profile.
Second branch t e ," < 0 Wi" = 0 W i " > 0
Sym m etrical profile a - u a ~ l> ( a - a . ) " 2 a - i
B oun dary-layer profile a " 6 a -10 ( a - a . ) " 2 a 4
T he tw o branches of the a ( R ) curve are given by (cf. (12.13), (12.14)) R Ui = 8.44(a2)-u /6 , c = 4a2/1 5 ; )
} (13.2)
R1' 3 = 5.96 (a2) “7'6, c = 0.611a2. j
T he numerical results are shown in T able III and Fig. 10. T he significance of the column 5 in the table will be explained in the next section. From th e figure, we see th at the minimum critical R eynolds number occurs a t Rll3 = 17,.45, or I? = 5314, agree
ing very well with our previous estim ation.*
Earlier results. T he stab ility of plane Poiseuille flow has been attem pted by m any authors. Com paratively recent papers are those of Heisenberg, [14], N oether [36], Goldstein [6 ], Pekeris [39, 40], Synge [64], and Langer [25]. T he papers of Goldstein and Synge and one of th e papers of Pekeris [39] give definite indication of stability a t sufficiently low Reynolds numbers. Heisenberg’s paper is in general agreem ent
* T h e v a lu es giv en here are som ew h at different from th ose published before [27], because the com pu tation of T ietje n ’s fun ction has been revised.
w ith the present investigations. He gave only a very rough calculation, whose result is reproduced in the figure. It seems that his curve is
R1 ' 3 = 13.4(a2)-u /6 .
T his is different from our present result (13.2) by a numerical factor. It m ay be noted
288 C. C. L IN [Vol. I l l , N o. 4
T a b l e II I. S ta b ility of P lan e Poiseuille F low .
c g a R a 2 R W
0 2 . 2 9 4 0 00 . 7 2 1 4 0 CO
0 . 0 5 2 . 3 6 3 0 . 3 0 5 6 1 3 . 6 4 X 1 0 5 . 6901 0 . 0 9 3 4 1 1 0 . 9 1
0 . 1 0 2 . 4 4 8 0 . 4 6 0 3 1 . 2 4 3 X 1 0 s . 65 4 4 0 . 2 1 1 9 4 9 . 9 0
0 . 1 5 2 . 5 4 0 0 . 6 0 2 4 3 1 0 4 8 . 6 1 9 2 0 . 3 6 2 9 3 1 . 4 3
0 . 2 0 2 . 6 6 8 0 . 7 5 0 6 12024 . 5 7 5 2 0 . 5 6 3 4 2 2 . 9 1
0 . 2 5 2 . 8 6 8 0 . 9 2 6 3 6 1 0 8 . 5161 0 . 8 5 8 0 1 8 . 2 8
0 . 2 6 6 3 . 0 1 2 1 . 0 1 0 1 5 3 6 9 . 4795 1 . 0 2 0 3 1 7 . 5 1
0 . 2 7 0 3 . 0 8 0 1 . 0 4 1 4 5 3 1 4 . 4 6 3 7 1 . 0 8 4 5 1 7 . 4 5
0 . 2 7 2 3 . 2 1 1 . 0 8 3 6 5 6 5 9 . 43 5 8 1 . 1 7 4 1 1 7 . 8 2
0 . 2 7 0 3 . 2 4 0 1 . 0 8 8 8 5 9 2 0 . 42 9 8 1 . 1 8 5 4 1 8 . 0 9
0 . 2 6 6 3 . 3 2 0 1 . 1 0 0 7 6 6 0 2 . 41 4 4 1 . 2 1 1 5 1 8 . 7 6
0 . 2 5 3 . 4 9 5 1 . 1 0 3 3 9 2 8 7 . 38 3 6 1 . 2 1 7 3 2 1 . 0 2
0 . 2 0 3 . 8 5 7 1 . 0 2 5 4 265 9 7 . 3 3 0 9 1 . 0 5 1 4 2 9 . 8 5
0 . 1 5 4 . 1 5 2 0 . 8 8 2 4 9 2 5 2 9 . 2 9 6 3 0 . 7 7 8 7 4 5 . 2 3
0 . 1 0 4 . 4 5 8 0 . 6 9 9 0 4 . 9 4 3 5 X 105 . 26 6 3 0 . 4 8 8 6 7 9 . 0 7
Fi g. 10. C urve of neutral sta b ility for the plan e P oiseu ille case.
th a t for the values of a for which his curve is drawn, the approximation used in de
riving (13.2) is no longer legitim ate. N oether’s work is based upon a very good m athe
m atical approach, which seem s to promise further developm ents. H owever, in apply
1946] S T A B IL IT Y O F P A R A L L E L FLO W S 289
ing his method to particular examples, he neglected the terms in a2 in th e invisd d solutions. As is evident from previous discussions, this is bound to lead to the wrong conclusion th a t the plane Poiseuille flow is stable (as N oether actually did). T he m athem atical analysis in Langer’s work shows th at the region of the c-plane for which ct > 0 m ust go to zero as a R becomes infinite. T his is in agreement w ith present results. Langer, however, concluded from his analysis th at the motion is probably stable in general. T his would be a natural deduction if the effect of viscosity were only stabilizing. T he instability of the plane Poiseuille flow m ust therefore be attributed to the destabilizing effect of viscosity. T his is a very significant fact and w ill be discussed in greater detail in §14.
Pekeris’ second paper [40] is a numerical treatm ent of (4.1), replacing a deriva
tive by a ratio of two finite differences. U nfortunately, his method is not suitable for the purpose, because he has virtually neglected the inner friction layer. In his approxi
m ation, he divided the half-width of the channel into (at m ost) four equal parts cor
responding to w = 0, 7/16, 3 /4 , 15/16, 1. From the present work, w e know th at the inner friction layer occurs definitely for c < 6 /1 6 . W e know also from our previous in
vestigations th at th e function <£ varies very rapidly in the neighborhood of the inner friction layer. H ence, it is not legitim ate to replace cj>' by A<f>/Ay for the interval (0, 1 /4 ), y being measured from the solid wall here. Also, m ost of the com binations of values (a, R ) he selected do not correspond to a strong instability. T hese values are marked with crosses in Fig. 10.
b) Stability of Bl asi us flow. For this case, we choose the boundary-layer thickness to be defined by
y = 5 = 6x / \ / R x, R x = i hx/ v, (13.3) where x, y are th e dimensional distances from the leading edge and th e wall respec
tively, and «i, v are the dimensional free stream velocity and the kinem atic viscosity respectively.* W ith this definition, the dim ensional displacem ent thickness is
ii = 0.286735. (13.4)
Such a choice has the convenience that th e initial part of the velocity curve can be very accurately represented by
w(y) = 2 y — 3 y A, (13.5)
y being measured from th e wall. Also, since the edge of the boundary layer is farther from the solid wall than th at set by Tollm ien and Schlichting, greater accuracy can be expected. T o make it easy to compare with other results, all final values are pre
sented in terms of
« 1 = aSi, R i = R5i. (13.6)
T he tw o.asym p totic branches of the a ( R ) curve are given by the following for
mulae (cf. (12.16) and (12.17)):
J?i = 2.21(10)“ a7 10, c = 1.74«!, (13.7)
i ? ! = 0.0622«74, c = 4.00«!. (13.8)
* Cf. G oldstein [7], vol. I, p. 135.
290 C. C. L IN [Vol. I l l , N o. 4
T hese formulae m ay be compared w ith those given by Tollm ien.* T he com plete numerical result is shown in T able IV and Fig. 11. T he minimum critical Reynolds number occurs at i? i= 4 2 0 , agreeing fairly well w ith our previous estim ation and the earlier results of Tollm ien and Schlichting.
T a b l e I V . S ta b ility o f B lasius F lo w .
c z a R s C d R i
0 2.294 0 » .7214 0 00
0.05 2.294 0.0473 81.45 X 10s .7214 0.0136 2 3 .353X105
0.10 2.296 0.1040 4.655X10* .7205 0.0298 1.335X10*
0.15 2.311 0.1730 84555 .7135 0.0496 24244
0.20 2.341 0.2588 24783 .6998 0.0742 7106
0.25 2.396 0.3693 9536 .6759 0.1059 2734
0.30 2.481 0.5156 4388 .6414 0.1478 1258
0.35 2.624 0.7149 2358 .5897 0.2050 676
0.40 2.942 1.0778 1477 .4967 0.3090 423
0.411 3.21 1.2968 1470 .3459 0.3718 421
0.40 3.540 1.4264 1944 .3763 0.4090 557
0.35 4.219 1.2992 5392 .2893 0.3725 1546
0.30 4.382 1.0055 12399 .2733 0.2883 3555
0.25 4.685 0.7578 34739 .2472 0.2173 9961
0.6
<X,
0.5
0 2 0 0 4 0 0 600 8 0 0 1000 1200 1400 1600 1800 2 0 0 0 2 2 0 0
R.
F i g . 11. Curve o f neutral sta b ility for the B lasius case:
--- present c a lc u la tio n ,--- S ch lich tin g ’s calculation.
Earlier results. T he stab ility of the boundary layer has been calculated by T o ll
mien and later by Schlichting, approxim ating the velocity distribution by linear and parabolic distributions. For the evaluation of the imaginary part corresponding to the inviscid solutions, th ey used the exact profile. T he calculation of Schlichting is shown dotted in the figure. T ollm ien’s curve agrees fairly well w ith th e present calculations,
* Loc. cit. [73], first paper, p. 42.
1946] S T A B IL I T Y OF P A R A L L E L FLO W S 291
except for a som ewhat lower peak. Schlichting also calculated the amplification of the unstable disturbances [52], and the am plitude distribution and energy balance of the neutral disturbances [54]. Since the neutral curve in his calculation is inexact, it m ight be desirable to repeat some of his work if experimental results were available.
For those calculations, the present schem e promises less numerical labor than Schlichting’s original work.
14. Physical significance of the results. Prospect of further developm ents. L et us now summarize all the results which have been obtained and discuss their physical significance. In the first place, we m ay conclude th at all the inertia forces controlling the stab ility of two-dim ensional parallel flows can be considered in terms of the dis
tribution of vorticity. If the gradient of vorticity of th e main flow does n ot vanish inside the fluid, then self-excited disturbances cannot exist except through the effect of viscosity.
In fact, the effect of viscosity is in general destabilizing for very large Reynolds numbers. T hus, if a w avy disturbance of finite wave-length can exist neutrally for an inviscid fluid, it will be amplified through the effect of viscosity. Indeed,* if the Reynolds number of a flow is continually decreased, a disturbance of finite w ave-length, which is damped at very large Reynolds numbers, becomes amplified, unless the w ave-length is so small as to cause excessive dissipation at any Reynolds number. For still smaller Reynolds numbers, the damping effect becom es predominant, and we have again a decay of the disturbance. However, for the particular disturbance of infinite w ave
length (essentially a steady deviation), the effect of viscosity m ay be said to be al
w ays of the nature of a damping.
T he effect of viscosity is essentially one of diffusion of vorticity. It can be seen more clearly from the following considerations. Let us imagine a disturbance originat
ing from the inner friction layer where the phase velo city of the disturapce is equal to th e velocity of the main flow. During one period 2 i r l / a c U of th e disturbance, the viscous forces w ill propagate it side-wise through a distance of th e order -v/2irvl / acU
= l \ / 2 i r f a R c . It is significant to compare this distance with the distance between the inner friction layer and the solid boundary. For if th ey are nearly equal, it means th at the effect of viscosity is dom inant a t least from the solid surface to that layer. T his ratio is approxim ately
s = %/2 x /z3 (14.1)
where z is defined by (6.28). T his quantity m ay be regarded as a measure of th e ef
fect of viscosity. Its value is included in Tables III and IV. W e notice th at the value of 5 decreases from 0.7 to 0.5 as we follow th e lower branch of the neutral curve of stab ility from infinite Reynolds number to th e minimum critical Reynolds number.
Then, as s decreases to zero, we are following the other branch of th e neutral curve to infinite R eynolds numbers. Thus (see Figs. 9, 10, 11), the lower branch is essen
tially controlled by the effect of viscosity. T he effect here is stabilizing, since an in
crease of Reynolds number gives instability. On the other branch, the effect of v is
cosity on diffusion of vorticity is predominant in comparison w ith the effect o f dis
sipation. Here, an increase of R eynolds number gives stability; i.e., the effect of viscosity is destabilizing. T his destabilizing mechanism is essentially to shift the phase difference between the u and v com ponents of the disturbance. It has been ex
* See F ig. 9.
