AIM-85-1 696 On the Three Dimensional Instability Modes in Boundary Layers T. Yan-Ping and C. Mao-Zhang, Beijing Institute of Aeronautics and Astronautics, Beijing, China

AIM 18th Fluid Dynamics and Plasmadynamics and lasers Conference

July 16-18, 1985 / Cincinatti Ohio 4

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ON TU3 TNRZE D I W S I O N A L IIISTABILITY f&OOI)ZS I N BOUNDARY LAYERS

Tang Yan-Ping* and C h e n Xao-Zhang** Bei j ing I n s t i t u t e of Aeronautics and Astronautics

Bei j i n g

Abstract The purFose of t h i s paper i s t o sub-

s t a n t i a t e our guess t h a t between l i n e a r and nonlinear s tages i n a t r a n s i t i o n pro- c e s s there i s a weak nonlinear s tage f o r which some important flow phenomena can be described by l i n e a r theory. To sub- s t a n t i a t e t h i s observation we car r ied out extensive c a l c u l a t i o n s and compared the r e s u l t s with the av ia lab le experimental da ta involving t h r e e dimensional i n s t a b i - l i t y modes. The comparison i s encourag- ing. The inf luences of adverse pressure gradient on t h e three dimensional wave s t r u c t u r e and i t s consequences have been studied.

1. Introduct ion

It i s known t h a t i n a boundary l a y e r t he t r a n s i t i o n process caused by an i n f i - ni tes imal dis turbance may cons is t of t he following stages. The first is a l i n e a r stage. The main c h a r a c t e r i s t i c s of t he s tage a re t h a t t h e amplitude of dis tur- bance waves i s very small and. therefore , can be well described by t h e l i n e a r theo- ry. The development of t he two dimensio- na l TS waves i s p r io r t o t h a t of the t h r e e d imens iona l one. The second i s a nonli- near stage. I n t h i s s tage. the amplitudes of dis turbance waves have exceeded a thre- shold value, and high o rde r harmonics ap- pear , meanwhile a streamwise vortex system i s developing. The t h i r d i s the s tage i n which the wave s t r u c t u r e breaks down and

* Lecturer. J e t Propulsion Department **Prefessor. J e t Propulsion Cepartment

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the turbulent spots OCCUI' a t random. The l i n e a r s t a b i l i t y theory based on

t h e Orr-Sommerf eld equation has achieved great successes i n pred ic t ing the develop mentof the first stage. However. i t cannot provide any useful knowledge involvingnom l i n e a r phenomena. Even so l i n e a r theorymay

play an important r o l e i n descr ibing the phenomena of ear ly s tage because experi- ments have shown t h a t i n t h i s s tage the disturbance waves still keep some l i n e a r behavior. For example, Klebanoff e t a l ' s

sults. The t o t a l phase of the dis turbance waves i s i n good agreement w i t h t h e l i n e a r theory even extended i n t o ear ly nonlinear stage. I n addi t ion, Wortmann i n h i s expe- rimen@ used a high amplitude value,

E -0.04, which i s usually thought t o be Ue - l a r g e r than t h e nonlinear threshold value. but t he harmonic d i s t o r t i o n was still s m a l l . Based on these r e s u l t s and i n view

1 of the f a c t t h a t Chen Id.2. and Bradshaw have successful ly simulated some charac- t e r i s t i c s of the th ree dimensional waves with t h e l i n e a r theory, we guess t h a t from l i n e a r t o non-linear s tage, t he re i s a weak nonlinear s tage i n which some impor- tant f l o w phenomena can be described by the l i n e a r theory. In the weak nonlinear s tage t h e amplitudes of the disturbance waves a re still small, the high order har- monics do not a f f e c t the f l o w ser ious ly , and the wave shape apparently is n o t d i s - tor ted. One of t h e main c h a r a c t e r i s t i c s i n t h i s s tage i s the appearence and rapid development of th ree dimensional i n s t a b i - l i t y modes. I n order t o demonstrate t he

experiment 7 has given the following re-

guess mentioned above, w e have carr ied out extensive ca l cu la t ions about t h ree dimensional waves and compared the re- sults with t h e av ia l ab le experimental data7 The comparison shows encou- raging agreement between them. This is the first subjec t of t h e present paper. Another problem to be discussed i s the r e l a t i o n s between two-and t h r e e dimensio- na l waves, f o r example, whether t h e phase v e l o c i t i e s and the wave numbers of two and t h r e e dimeneional waves would be res - peczively equal o r not. To t h i s problem the re seemed to be d i f f e r e n t po in ts of view 3*4*9g10. We expect t o f i n d some c lues he lp fu l t o c l a r i f y t h e quest ion wi th in t h e limit of l i n e a r theory. Through the ca l cu la t ion f o r t he f luc tua t ion flow f i e l d composed of a two- and 8 three-di- mensional waves. w e found t h a t the r e s u l t s ca lcu la ted with ne i the r d,, =d3D nor C2, = C3D agreed w i t h experiments. However.

when we assumed t h a t f2D;?f3D, but C2fC3,. d2Dfd3D. t h e r e s u l t s compared w e l l w i t h experiment 8.

Although the re have been i n t e r e s t i n g works on t h e e f f e c t s of pressure gradient on t h e i n s t a b i l i t y and t r a n s i t i o n of a boundary l a y e r 12s13*14, w e need further inves t iga t ion on the e f f e c t s of t h i s f a c t o r on t h e flow s t r u c t u r e of three-di- mentional waves i n de t a i l . This i s another subjec t of the present paper. To do t h i s . t he e f f e c t s of adverse pressure gradient on Wortmann's experiment a r e considered. It makes t h e ca l cu la t ion condi t ions much more c lose t o those of the experiments. This is an improvement over t he work . The amplitude c h a r a c t e r i s t i c s achieved a r e i n good agreement with Wortmann's ex- periments. We found t h a t under t h e e f f e c t s of adverse pressure grad ien t , some new b ehav iour appeared . have published aome interesting papers on three-dimensional subharmonics. i t Can be very fruitful t o study along their way.

1

2 1 Recently, Herbert", Sa r i c and Thorns

Sect ion 2 descr ibes the fundamental

theory and models. Sect ion 3 explains the ca lcu la t ion methods fo r the three-dimen- s iona l waves. Sect ion 4 presents t he nume-

periments f o r t h e ea r ly s tage of tran- s i t i on . Sect ion 5 d iscusses the e f f e c t s of pressure gradient.

r i c a l r e s u l t s and t h e comparison w i t h ex- "ae-.

2. The fundamental theory end models

For incompressible f l o w w i t h i n f i n i - tes imal d i s turbance we can get t h e fo l - lowing equation -

Ki F - vp (1 1 7 E =

- M -* - u s u - u - -

U+ i s t h e instantaneous ve loc i ty , %(u.v. w) . Using equation (1). w e study the three-dimensional dis turbance waves i n a boundary l aye r over a f l a t p l a t e under the p a r a l l e l flow assumption. We assume t h a t the d is turbance wave i s

\ae/

- u=Alc2D(y)exp ( id2,(x-C2,t$

+X2~3D(y)exp(id3,(x-C3Dt)) 0 0 s ~ ~ ~

where b o t h k l andX2 a r e supposed t o be r e a l numbers which specify the proportion of two- and three-dimensional wave;dZD. C2D, O(3D, C3D and @3D are a l l complex numbers whose real p a r t s are the wave numbers and t h e phase v e l o c i t i e s , respec- t i v e l y ; C2D=($D,v2D), u ~ ~ = ( u ~ ~ , v ~ ~ ) .

A - A , .

We regard equations(2) a s t h e d is -

turbance models which cons is t of a two- \avl

and a three-dimensional wave. O u r d i s tu r -

2

bance model i s d i f f e r e n t from t h a t of Benney's i n t h a t ne i the r C2,,=CjD nor d2D=d3D has been appl ied and they grow spa t i a l ly .

4 Subs t i t u t ing (2) i n t o (l), we can get tne following equations according t o the superposi t ion p r inc ip l e f o r l i n e a r sys- tems.

-D U J-- VZD (3 .1)

2 - dl 'oJV3D

(4.1)

(4.3)

t h e f luc tua t ion v o r t i c i t y . Ue i s t h e ex- ternal stream veloc i ty and & i s the th ic - kness of t h e boundary layer .

The boundary condi t ions a r e ... (5.1)

form a boundary value and eigenvalue pro- blem f o r t he ordinary d i f f e r e n t i e l equa- t i o n system. I n a range of appropriate Reynolds numbers, the mathematic problems

may de.termine a r b i t r a r y number of unsta- b l e so lu t ions f o r equation (1). Reference descr ibes the method of solving equation ( 3 ) i n d e t a i l .

17

2. Methods of solving t h e s p a t i a l growth problem f o r t h r e e dimensional wBye9

The key t o solving equations (4) , (6) i s to solve the problems presented by (4.1),(6;1) and (6.2) which cen determine all eigenvalues of three dimensional waves. There a r e two kind o f methods t o solve the problem. The f i rs t i s t o solve equation (4.1) d i r ec t ly . Zquation (4.1) contains seven e i g e n v a l u e s r r , r i ' c 3Dr. c 3?1, . d3Dr' d3Di and t h e s p a t i a l gro-wth condi t ion can determine t h r e e of them, and o t h e r s can Only be determined by coxparing with ex- perimental data. Yne second method i s based on Squire 's theorem. Our ca l cu la t ion has shown t h a t t h e three-dimensional waves obtained by applyin2 Squire 's t!ieorem a r e i n agreement with experiments. Therefore, we s h a l l only d iscuss t h e second method hereaf te r .

and E. The equation i t s e l f

Applying Squi re ' s theorem, 1 obta ins the three-dimensional waves on neu t r a l curves. The present paper app l i e s t h e theorem i n the nonneutral region f o r spa- t i a l growth condition. The key t o g e t t i n g three-dinensional waves through applying Squi re ' s theorem is t o f ind such a two- dimensional "equivalent" mode

which should be of the same form a s equa-

3

t i o n (4.11, and submit t o

dgDe=r (8.1)

d2DeR2De= d 3DR3D (8.2)

'2De = '3D (a. 3)

A .-. If so, it is obvious t h a t v~~~ =

Thus we can obta in three-dimensional waves f r o m t h e equivalent 7rode.

The d i r e c t numerical s i n u l a t i o n fo r the f l u c t u a t i o n f i e l d should be based on the f a c t t 3 a t t he amplified dis turbance waves measured a r e not determined by the loca l condi t ions of the measuring pos i - t i o n but t h e upstream condi t ions, because the diturbance waves perceivable a t the measuring pos i t ion a re the development of the upstream disturbance. Therefore we should seek the most unstable dis turbance wave through i n t e g r a t i n g the m p l i f i c a - t i o n r a t e of d i f f e r e n t didurbances and comparing t h e i r t o t a l amplif icat ion r a t i o s with the experimental data. However, a s me have appl ied the p a r a l l e l flow assumption, i t makes it impossible t o obta in the accurate value of to ta l am- p l i f i c a t i o n r a t i o s . A s a r e s u l t , t he e i - genvalues determined by the i n t e g r a t i o n method were c l o s e t o those determined by t h e local. condi t ions, $0 t h e latter has been used.

A t first, under the s p a t i a l growth con- d i t i o n and t h e known Reynolds number cor- responding to the experiment'. solving equation (3.l) ,ons can ge t the eigenvalues f o r two-dimensional TS waves. Then accord- i n g t o t h e soheme spec i f ied , such a s d2,=d3D. or c ~ ~ = c ~ ~ , or f2D=f3DD(notice

t h a t when R2D=R3D, no two of them can be

met simultaneously), solving equation (4.1) and t h e o thers of equation (4) through applying Squire 's theorem, one can obtain the eigenvalues f o r t h r e e dimensional waves. The programs used are taken from 1 and 17.

The computation s t eps a r e a s follows.

4. Numerical r e s u l t s and comparison w i t h experiments

b stage of t r a n s i t i o n

sis for t h e t o t a l phase of the disturbance waves. The r e s u l t s a r e showa i n f i g . 4.

The discrepancy between computed r e s u l t s and experimental data? i s within 3% which may come from the assumption of p a r a l l e l flow and the e r r o r s caused by measuring. Ross pointed out t h a t t h e e r r o r s i n measuring the parameters of f l u c t u a t i o n f i e l d might be i n a raqge of 2% - 3% i n h i s experiment . I n f ig . 4, point A is the end of l i n e a r s tage and D i s the place where the wave s t r u c t u r e breaks down. and the region between A and B i s the weak nonl inear stage. Fig. 4 shows that the computed r e s u l t s are very c l o s e t o those of experiment b e f o r B o i n t C. The ca lcu la ted d i s t r i b u t i o n of at

point A is shown i n f ig . 1 'We se t h2-0 i n c a l c u l a t i o n , i. e. s i m ? l y a two-dimensio- nal wave. Fig. 1 showa t h a t the two-dimen- s iona l wave i s a dominant f ac to r until the end of l i n e a r stage. k comparison f o r @ d i s t r i b u t i o n a t point B is shown i n Pig.2, where curve "a" represents the r e s u l t com- puted w i t h h2=0, and Curve "b" t h e r e s u l t withkl=O. It shows t h a t n e i t h e r pure two- dimensional wave nor pure three-dimeosio- nal wave can compare well w i t h t h e expe- riment at point B. It implies a rapid growth of three-dimnsional wave and then t h e formation of a composite wave. Fig.3 shows the r e s u l t of composite wave com- puted with the smplitude and frequency of two dimensional waves equal t o those Of t h ree dimensional waves, respectively. The r e s u l t so obtained agrees very w e l l with the experimental data.

From f ig . 1-4. and t h e discussion above, we can &e the following observations. In the weak nonl inear s tage the three-dimen- s iona l waves grow a t a rate much g r e a t e r than t h a t of two-dimensional waves. while

A t f irst we made a numerical analy-

5

ke

b/

4

before t h i s s t age , t h e growth r a t e of two-dimensional waves i s greater. It is an important behavior i n t r a n s i t i o n pro- cess. Kle iser ' s numerical analysis18 has

4 shorn the same r e s u l t though h i s method i s d i f f e r e n t fron ours and m r e compli- cated t h a n ours.

Calculation of instantaneous spanwise v o r t i c i t y i n the ea r ly s tage of transi- t i o n + --

Kovasznay e t a l p s measurments'l show the typ ica l d i s t r i b u t i o n of spanwise vor- t i c i t y a t t h e spanmise peak posit ion. The present paper dee l s with the same problem using numerical method. Since 11 has not provided t h e experimental condi t ion in d e t a i l , our c a l c u l a t i o n condi t ions m i g h t be d i f f e ren t f r o m those of experiments t o some exteEt. Z g . 5 anu fig- 6 show t h e r e s u l t s from :.:IT c a l c u l a t i o n and Kovasznay e t al's experiment, respectively. Fig. 6. showing the ca lcu la t ion r e s u l t . demon- s t r a t e s as tonishing s imular i ty with fig.5. It shows that our t h e o r i t i c a l method can predic t t h e development of spanwise vor- t i c i t y even i n t h e ea r ly nonl inear stage. We took u ~ ~ , % ~ I =25 in o u r Calculation.

It implies t h a t the th ree dimensional waver heve f u l l y formed. T h i s i s an addi t iona l evidence t o show t h e r a p i d growth of t h r e e dimensional waves.

Study on t h e r e l a t i o n s between two- and - three-dimensional waves

for the two- and th ree dimensional waves with e i t h e r equal wave number or equal

frequency under t h e condition of equal Reynolds number. It s u b e t a n t i a t e s that it i s impossible t h a t both C2D=C3D and dZD=D(jD hold siinultaneously.

ween two- and three-dimensional waves

I

T a b l e 1 shows t h e calculated r e s u l t s

There a r e th ree possible cases bet-

---rzD =f3D. but D ( ~ D fg3Ds C 2 ~ f C 3 ~

*This work w a s s u e s t e d by Dr. R.S.Hirsh

The calculated r e s u l t s r e l a t ed t o point B i n f ig .3 and fig.4 correspond t o t h e first case. ;:either the second case nor t he t h i r d case could produce a d i s t r i b u - t i o n curve a s w e l l comparable w i t h expe- r i n e n t s a s f ig . 3. The first case i s Fhy- s i c a l l y possible i n t h a t t he three-dimen- s ional waves a r e developing on the b a d 6

of ready-developed t w o dimensional waves, therefore the developing ones w i l l beet a t the saae r a t e a s the developed ones.

r e s u l t of d i f f e r e n t phase ve loc i t i e s , the s p a t i a l development i n streamwise should have exhibi ted the d i f fe rence between two- and three-dimensional waves. but Mebanoff e t a l ' s experiments did not show t h i s d i f -

ference. would l i k e t o disc.iss t h i s .problem. According t o t h e i r experiment, t he range f r o m point A t o the middle way between B and C i n f ig .4 covers an inter- va l of roughly two wave lengths. Assuming t h a t the anpl i txces of two- and three- dimensional waves a r e of the s m e order and no1;icing t h a t fZD=f3@ for the first case. one g e t s

If the first case is t rue . a s t he

~ i n 2 m d ; ~ ~ - f ~ ~ t + ~ ~ ~ ) + S i n z ~ ( d ~ ~ = - f t 3Dt+q3d

t t in ( (C12D+ti3D)x-2ft+ ~q,,+l~,,y~

* * where o(2D and d 3D a r e dinensiocal wave

number. Accordin& t o our c a l c u l a t i o n within t he range of two wave lengths. without l o s i n g genera l i ty , simply as- suming thatv2D='#3D for the i n i t i a l phases,

one obtains

It ind ica t e s tbat the cosinusoidal term does not a f f e c t the wave shape obviously.

5

t he re fo re t h e dis turbance w i l l behave s inusoida l ly i n a form of e composite wave. t he s p a t i a l d i s t r i b u t i o n of which depends on t h e composite wave number

d!&??b 2 agrees well w i t h t h a t of experiments. As

a r e s u l t of t hese f ac t s . i t i s d i f f i c u l t t o de tec t t h e d i s t o r t i o n of t h e composite wave within the range of two wave lengths. I n addi t ion , i n doing experiments. only t h e wave number, frequency and wave shape of t h e composite wave can be d i r e c t l y measured along t h e ztreamwise while t h e same q u a t i t i e s f o r t h e two- and three- dimensional waves cannot be measured separately. These may be the reasons why Klebanoff e t a l ' s experiment showed a preference tha t both " c 2 D = ~ 3 D end C2D=CJD

held simultaneously.

which i s given i n t a b l e 1 and

5. Numerical ana lys i s f o r the^ t h ree dimensional waves under adverse pressure grad ien t

Applying Squi re ' s theorem Chen e t a l l obtained three- dimensional waves numerically and compared wel l with Wort- mann's data. However, they d id not take i n t o account t he e f f e c t s of adverse pres- sure gradient which ex is ted i n Wortmann's experiment. The present paper has improved Chen e t a l ' s work i n t h a t i n addi t ion to the e f f e c t s of adverse pressure gradient t o be consid.ered, a s p a t i a l growth model. r a t h e r than a neu t r a l one, has been used.

The ca l cu la t ion s t e p s a r e as follows: (1) As 'Wortmanna d id not descr ibe

the experiment condi t ion i n d e t a i l , w e had t o t e s t numerically severa l schemes t o f ind out the streamwise pressure d is - t r i b u t i o n s imi l a r t o t h a t used by Wortmann. We found t h a t t he following r e l a t i o n fitt- ed the experimental d a t a best .

where according t o \Vortmann's da t a , U,=0.1047m/s, a=O. 1282, %0.755 ?=x/L,

L=5m at t h e ca l cu la t ion point.

t r i b u t i o n of f r e e stream ve loc i ty , t h e ve loc i ty p r o f i l e ac ross t h e boundary

( 2 ) With the known streamwise d is -

l a y e r can be obtained. k w

( 3 ) iipplying Squi re ' s theorem, solving equation (4 .1) . we can get t he eigenvalues and one eigenfunction f o r t he three-dimen- s iona l wave,

equation ( 4 , 1 ) , solving t h e r e s t of equa- t i o n s ( 4 ) , we can de tern ine e l l the eigen- func t ions of the three- dimensional wave.

1 6 and 17.

( 4 ) Using t h e r e s u l t s obtained from

The programs used a r e taken from 1,

The r e s u l t s obtained a r e a s follows: Numerical r e s u l t s f o r the mean end f luc- t u a t i o n flow f i e l d

Table 2 lists t h e re levant ca lcu la ted r e s u l t s and experimental data. Table 2 shows t h a t the e r r o r s a r e higher than those i n t a b l e 1. There may be two ree- sons causing the e r rors . The first is t h a t t h e e r r o r f o r t h e mean flow i s about 2% which e f f e c t s t h e r e s u l t s of f l uc tua t ron flow. The second i s t h a t the p a r a l l e l flow

l a r g e r under t h e adverse pressure gradient then those wi th zero pressure grad ien t , The d i s t r i b u t i o n of spanwise f luc tua t ion veloci ty .

Chen e t el1 pointed out that the spanwise f luc tua t ion ve loc i ty w i s a very s e n s i t i v e ind ica to r of three-dimensional wave motion. Fig.8 descr ibes the v a r i a t i o n of w with time f o r d i f f e r e n t he ights from t h e wall . The r e s u l t s e r e i n good egree- m e n t with those of t he experiments qual i - t a t i ve ly .

An important improvement over work' is t h a t t he height obtained i n the present paper corresponding to the maximum am- p l i t u d e of f luc tua t ion ve loc i ty w i s much c l o s e r t o t h a t of Wortmann's da t a (fig.8). To demonstrate t h i s p o i n t , we l ist t h e re- s u l t s as follows:

assumption w i l l g ive r i s e to e r r o r s much be,

6

Sources The he ights of

Klebanoff e t a l ' s expt (Zero pressure gradient)

WmaX

0.13 s C&c.'(Zero pressure gradient ) 0.194s

0.274s Wortmann's expt (adverse pressure grad ien t ) The present paper (adverse pressure grad ien t )

The r e s u l t s above show t h a t t h e height where the amplitude of w a t t a i n s its maxi- mun increases a s t h e adverse pressure gradient increases. It makes t h e flow more unstable (see l e t e r ) . From f ig . 8 of reference 1. we can see some disagree- ment in phase sh i f t between theory and experiment, i n t h i s respect nothing has been improved i n the present paper.

W f e c t s of adverse Dressure a rad ien t on the d i s t r i b u t i o n of f l u c t u a t i o n auan- ties -

Fig.7shows the eigenfunctions of spanwise f l u c t u a t i o n v e l o c i t y f o r d i f - f e r en t pressure gradients. With known o ther eigenfunctions from equations (4). we can get the d i s t r i b u t i o n of f l u c t u a t i o n energy. %q2 , and i ts production term P (fig.9). Fron fig.7 and fig.9 we can see tha t under t h e e f f e c t s of adverse pres- sure gradient a l l t h e regions of i n t e r e s t - ing f ea tu res r i s e away from the wall. This i s an important r e su l t . Since i t is well known t h a t t he r i s i n g of such regions is a necessary process from becoming unstable

15 t o wave breakdown (see. say. Hama", Tam and the f i l m of wortmann3. The e f f e c t s of adverse pressure gradient w i l l c e r t a in ly a c c e l e r a t e t h i s process and t h u s t he whole t r a n s i t i o n process. T h i s argument is help- ful t o explain Knapp's experimental re- suits . 13

6. Conclusions

(1) The present paper s u b s t a n t i a t e s our guess t h a t from l i n e a r t o nonlinear s tage there i s a weak nonlinear s tage , i n which

some fundamental flow phenomene can be described by the l i n e a r theory. ilsing a disturbance model which i s composed of a two- and a t h r e e diaensional waves, t he present paper has obtained t h e d i s t r i b u - t i o n of rms of t h e s t r e a m i s e f l u c t u a t i o n ve loc i ty and spanmise vo r t i c i ty . These q u a n t i t i e s compare well with experiments. ( 2 ) O u r ca lcu la t ion s u b s t a n t i a t e s t he ar- gument tha t it i s impossible t h a t both the r e l a t i o n s C20=C3D and d2n=C8'3D hold

simultaneously. According to o u r calcula- t i o n s the r e s u l t s besed on the r e l a t i o n

f2D=f 3D 2D nor C2D is equal t o i ts three dimen-

s iona l counterpart. (3) The numerical ana lys i s has shown t h a t a f t e r the s tage of two-dimensional wave growth the re is a s tage of a rapid growth of t h ree dimensional waves, t h u s forming a s t rong streamwise vortex system inducing an upward motion. (4) Taking in to account t h e e f f e c t s of adverse pressure grad ien t , we have pre- d ic ted the height where the amrlitude of spanwise f l u c t u a t i o n ve loc i ty a t t a i n s i t s maximum. This i s an improvement over 1. Vi3ile i n another respect . the predict ion o f the phase s h i f t , nothing has been done. (5) A n important r e s u l t of the present paper i s t h a t w i t h t h e illcrease of adverse pressure gradiect t h e regions of i n t e r e s t - i n g f ea tu res rise upwards. This e f fec t w i l l c e r t a in ly acce lera te the whole tran- s i t i o n prpcess.

a r e the best . i n t h i s case n e i t h e r

Acknowledgement: The authors a r e gra- t e f u l t o Prof. P. Bradshaw of Imperial College fo r he lpfu l advice.

References 1. Chen M.Z. and P.Bradshaw A I A A J. 22 -

P301 (1984) 2. J.T. S tuar t J. Fluid Mech. 2 p352

3. D.J. BeMey Phy. Fluids 7 p319 (1964) (1960)

4. D.J. Benney J. Fluid Uech. p209 (1961)

7

5. ?toss et al. J. Fluid Tdxh. Q p819

6. J.B.Arders e t el. Trans i t ion from

7. P.S. Kleba.:ofP e t a l . J. Fluid Xech.

8. F.X. Woitmann AGARD Cpp-224 (1978)

(1970)

Loininar t o Turbulent (1979)

- 1 2 p 1 (1962)

9.

10.

11.

12.

B.N. Antar e t al . Phys. F lu ids a P289 (1975) G.Nelson e t al. Phys. F lu ids 20 p698 (1977) L.S.G. Kovasznay e t el. Proceedings of t h e 1962 Heat t r a n s f e r and Flu id Xech. I n s t i t u t e p 1 (1962) G.B. Shubauer e t a l . NACA TR 909 (1947)

13. 14.

15.

16.

17.

18.

19.

20. 21.

C.F. Knapp A I A A J. 6 p29 (1968) P.Schlichting Boundary Layer Theory (1979) I. Tani Trans i t ion from Laminar t o "c?/

Trubulent p213 (1979) He L i Trans. of BIAA 30 Aniversal-g (1982) Tang Y.P. and Chen 1d.Z. Chinese 2nd Compu. Fluid Mech. Conf. (1984) L.Kleiser 8 t h Int. Conf. on IJum. Eeth. i n F lu id Dyn. p280 (1982) F.2. H a m a et a l . Proc. of 1963 Heat Transfer and Fluid Nech Inst .

T. Herbert A I A A 84-0009 W.S. Sar i c and A.S.5'. thornas, Proc. IUTAX symp. "Turbulence and Chaotic Phenomena i n Fluids'' Kyoto, Tapan,1983

P 77 (1963)

Table I Comparison o f ca lcu la ted 3D. 3D and composite waves with experiments from 7

'r 4 xx(cm) hz(cm) r( l /s) S(m) A Calc. 2D 1693 .a31 .277 -333 3.66 148 4.837

X(in) Cases Rs* d, e

37.5 C a l c . 3D 1693 -831 .294 .354 1.247 3.66 2.44 157 4.837

'sr/ 2Dr= 3Dr Calc. 3D 1693 .79 .277 -351 1.185 3.82 2.55 148 4.837

Calc. corn- 1693 .81 pos i t e wave 2xpt. (7) 1693

.341 3.75 2.50 148 4.837

.34 3.81 2.54 145 4.837

39 Calo. 2D 1727 .846 .282 .333 3.66 147.8 4.934 Calc. 3 D 1727 .846 . 3 O l .355 1.270 3.66 2.44 157 4.934 Calc. 3D 1727 .8O5 .282 .351 1.208 3.85 2.56 148 4.934 f2D'f 3D

&+=%,)

Calc. corn- 1727 .825 .341 3.75 2.50 148 4.934 pos i t e wave

Expt. (7) 1727 .340 3.81 2.54 145 4.934

B Calc. 2D 1733 .852 .284 .333 3.65 148.6 4.949 39.2 $gala 3D 1733 .850 .302 .355 1.275 3.65 2.43 157.8 4.949

ca1;.% 1733 .808 .284 .351 1.212 3.85 2.56 148.6 4.949 2Dsf 3D

C a l c . corn- 1733 .830 .342 3.74 2.50 148.6 4.949 pos i t e wave Expt. (7 ) 1733 .340 3.81 2.54 145 4.949

W

8

Table 2 Comparison between ca lcu la t ion and experiment (8)

Calo. 3D 1235 -395 .1460 .4147 -2013 -3155 -2010 4.9%

h p t . (8) 1230 -185 "29 .1821 5.2%

Repative .4% 8.85% 8.85% 8.77% 5.76% e r r o r s

&

Fig.1 D i s t r i b u t i o n of streamwise fluc- t u a t i o n ve loc i ty u'rms

- x - calc. - - Expt. (7)

Fig. 2 Dis t r ibu t ion of streamwise Velo- c i t y u'rms -.- calc. 3D wave - x- talc. 2D wave -- . -- expt (7)

Fig. 3 Dis t r ibu t ion of streamwise fluctuation veloci ty u' rms -- . -- exot. (7) . ... ~~~ - x - calc. composite wave

with f2D=f3D

0.0 f.0 0.8 0.6 0.4 0 2 Fig. 5 Contours of spanwise v o r t i c i t y i n

0

t h e ea r ly stage from KKV's expe- rimantll

- - Fig. 6 Same kind of conto& as Fig.5

from our ca lcu la t ion

9

Fig. 4 Total phase v a r i a t i o n - calc.

expt. (7) --.- -

Fig.7 Spanwise f l u c t u a t i o n v e l o c i t y e i - genf unc t i o n

} zero pressure --9- -1- 0-- gradient

I 7 9 11 13 I 3 5

Fig. 8 Variat ion of spanwise f l u c t u a t i o n v e l o c i t y w with time t fo r d i f f e - ren t d i s tances from t h e wall

-.- expt. ( 8 ) -0- talc*

-X- adverse pressure ---- .------ f gragient

Fig. 9 Dis t r ibu t ions of f l u c t u a t i o n energy and pmduotion term p f o r d i f f e r e n t pressure gradients. Scheme C i s of t he most unfavorable pressure gra- dient.

- . __ production term p f l u c t u a t i o n e n e r a X- -

I%y

10