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62 J O U R N A L O F T H E A E R O N A U T I C A L S C I E N C E S J A N U A R Y , 1 9 5 4

F I G . 2 P R O C E S S R E P R E S E N T A T I O N

t o n ' ( Y M ? )

F I G . 3 GRAPHICAL SOLUTION OF ACTUATORD I S K D I S C O N T I N U I T Y

7 Vi -\- Vz- : (P2V2 - P1V1) = - (P2 ~ Pi) ( 6 )

E q . (6) rep resen ts the locus of possible s ta tes 2 for given state 1a n d may be recognized as the R a n k i n e - H u g o n i o t r e l a t i o n s for ap lane shock wave . The cycle of processes can be rep resen tedtherefore in a fo rm shown in Fig . 2. It is of in te res t to n o t e t h a tthe foregoing formulation of the p rob lem leads to a locus of s t a t e sequa t ion wh ich is i n d e p e n d e n t of the t h r u s t and power load ingsof the disc.

H a v i n g o b t a i n e d the locus of s ta te s equa t ion for the disc, aspecific solution yielding state 2 for a g iven s ta te 1 can be obt a ined by a s imple g raph ica l me thod shown in Fig . 3. The construction follows directly from the m o m e n t u m e q u a t i o n , Eq.(2 ) , which , for the purpose of the nond imens iona l coo rd ina te s ofthis figure, can be w r i t t e n as

(Pt/Pi) " (1 + r)1 - (v 2/vi) = yM i

2 ( 7 )where r = T/pi.

Since the processes 0-1 and 2-3 are isentropic and the conditionof equal pressure p % po is i n h e r e n t in the definit ion of theboundar ie s of the f low system, the change of s t a t e 1-2 d e t e r mines explicit ly the overa l l dens i ty change

p . / p . - e - K - r - D / y ] [ < - * > / ! (8)where 5 is the un i t en t ropy and R is the gas cons tan t .

The flow problem is thus so lved wi th re spec t to the p a r a m e t e r sof s ta te 1. H o w e v e r , a prope l le r p rob lem is usually given int e r m s of t h e p a r a m e t e r of s t a t e 0 (po, VQ, VQ) and the disc loadingT. H e n c e , one more re la t ionsh ip is needed in orde r to r e l a t eth is s ta te wi th the d iscon t inu i ty . T h is is given by the m o m e n t u m e q u a t i o n for the control surface 0-3 of Fig . 1i.e.,

T = m(V, - Fo) (9)A computa t iona l so lu t ion migh t p roceed as fo l lows : With

k n o w n p0t v0, and Mo, a v a l u e of Mi is a s s u m e d and the s t a t ep roper t ie s at 1 are calculated from the i sen t rop ic re la t ions . Theprocedure desc r ibed in Fig. 3 is used to ob ta in s ta te 2, and thecycle is comple ted wi th an i sen t rop ic expans ion back to p % = po.T he va lue for F3 is thus ob ta ined . T he com puta t ion is r e p e a t e dwith different assumed M\ u n t i l the ob ta ined va lue of Vz sa t i s fies Eq. (9). Calcu la t ions for several cases of in te res t are nowbeing carried out by J. D. S t e w a r t and will be presen ted sho r t lyas pa r t of a Univers i ty of Cal i fo rn ia M.S . thes i s .

I t is in te res t ing to no te tha t , a l though all the d iag rams havebeen d rawn for the case of a prope l le r advanc ing at subson icspeed through sti l l air, the d iscon t inu i ty ana lys is p resen ted isequa l ly va l id for supersonic f low at s t a t e 1. Indeed , ac tua lc o m p u t a t i o n for a prope l le r advanc ing at supersonic speed isgreatly s implif ied in t h a t s t a t e 1 is the s a m e as s t a t e 0, and s t a t e s2 and 3 can be ca lcu la ted d i rec t ly wi thou t the recourse to thei t e ra t ive p rocedure desc r ibed above .

A s a f inal remark, it shou ld be s t a t e d t h a t all the a r g u m e n t sp resen ted he re are re s t r ic ted by the a s s u m p t i o n t h a t the flowca n be t r e a t e d as one-d imens iona l . T h is imposes a definitelimit on the app l icab i l i ty of the results , s ince with excessive discloadings the sp read of the d is tu rbances in a p l a n e n o r m a l to thedirection of motion and the genera t ion of vor t ices canno t beneg lec ted .

R E F E R E N C E S1 Krz ywobloc k i , M. Z., Elementary Propeller Theories in CompressibleInviscid Fluids, Applied Scientif ic Research, Series A, Vol. 2, No. 3, pp. 2 0 5 -224, 1950.2 Vogeley, Arthur W., Axial Momentum Theory for Propeller in Compressible Flow, N A C A TN No. 2164, July, 1951.3 Kiic he ma nn , D., and W e be r , J., Aerodynamics of Propulsion, 1st Ed.,

pp . 25-27 ; M c Gra w-Hi l l Book Compa ny , Inc . , New York , 1953.4 La i tone , E. V., Actuator Disc Theory for Compressible Flow and a Subsonic Correction for Propellers, Re a de r s ' Forum, Journa l of the Ae rona u t ic a lSciences, Vol. 20, N o . 5, p. 365, Ma y, 1953.5 La i tone , E. V., and T a l b o t , L., Subsonic Compressibility Correction forPropellers and Helicopter Rotors, J o u r n a l of the Ae rona u t ic a l Sc ie nc e s , Vol.20 , No. 10, pp. 683-690, October , 1953.

Calculated Ampl i f ied Osci l la t ions in the PlanePoiseuille and Blasius Flows*S . F. S h e nDepartment of Aeronautical Engineering, University of Maryland,College Park, Md.O c t o b e r 5, 1953" C ^ O R ACTUAL CAL CUL AT ION of hydrodynamic stability problems,

t h e r e are now two somewha t d i f fe ren t p rocedures , one dueto He isenberg 1 and L in 2 and the o t h e r due to T o l l m i e n 3 andSch l ich t ing .4 The main difference lies in the h a n d l i n g of the so-ca l led ' ' inv isc id so lu t ions . " In the classical case of Blas iusflow ; L in ' s neu t ra l cu rve ag reed we l l wi th tha t of T ol lmien butno t wi th Sch l ich t ing ' s ca lcu la t ion fo l lowing T o l lmien ' s p rocedure .F r o m the compar ison in S c h u b a u e r and S k r a m s t a d ' s r e p o r t , 5 it

* This work was done at the suggestion of, and under the supervision of,Prof. C. G. Lin while the author was at M.I.T. during the summer of 1953.It was supported by project N5 ori-07872.

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R E A D E R S ' F O R U M 63

a Po i n t s co mp u te d by Th o ma s,Re f .6 n u mb e rs b e i n g va l u e sC J X I O 3

5 8 10 2 0 3 0 4 0 5 0R = I O "3

8 0 1 0 0F IG . 1. Am plificatio n ra tes of Poiseu ille flow.

- x i o 3

PRESENT RESUL TS C H L I C H T I N G ( R E F . 4 )o A E X P E R I M E N T ( R E E 5 )

0.1 0.2 0.3 0.4F I G . 3 . Compar ison o f neu t ra l cu rves (Blas ius case ) .

0 . 4 0

0.35

0 . 3 0

0 . 2 5

0 .20

0.15

0.10

5 0 0 1 0 0 0 1 5 0 0 2 0 0 0R.

F I G . 2. Amp lif ication rat es of Blasius f low.2 5 0 0 _ L3 0 0 0

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6 4 J O U R N A L O F T H E A E R O N A U T I C A L S C I E N C E S J A N U A R Y , 1 9 5 44 0 0

F I G . 4. Com parison of amplif ication rate s (Blasi us case) .appea rs tha t the exper imen ta l po in ts a re somewha t c lose r to L in ' sneu t ra l cu rve tha n to Sch l ich t ing ' s . Fu r th e rm ore , in o the r de t a i l s , such as the ampl i f ica t ion ra te , Sch l ich t ing ' s theo re t ica l re su l t cou ld on ly be qua l i ta t iv e ly ver if ied by the exper imen t . Bu tbecause of the approximate nature of the theory and the diff icultyo f rep roduc ing the theo re t ica l mode l in ac tua l expe r imen t , oneis in no posit ion to draw hasty conclusions on only a s ingle evidence . Recen t ly , T hom as 6 computed , wi th the he lp o f the h ighspeed IBM mach ine , seve ra l e igen -va lues o f the Sommerfe ld -Orrequation for the case of plane Poiseuille flow, using finite differences wi tho u t re so r t ing to o the r approx ima t ions . T hu s wenow have a l so a l imi ted num ber o f " ex ac t" re su l t s to check aga ins tthe a sympto t ic me tho ds . T ho ma s ' va lues a re a l l complex ; i t i stherefore desirable to compute points off the neutral curve fordirect comparis on. The points tha t l ie in the unstab le region oft h e (a , i)-plane give, meanwhile , a measure of the amplif icationof th e corres pond ing dist urba nce. Aside from possible verificat ion by ava i lab le exper imen ta l da ta , the ampl i f ica t ion ra te s a l somight serve as a basis for the development of some criteria to exp la in the lamina r - tu rbu len t t ran s i t ion , such a s the d if ferent suggestions of Schlichting, 4 L i e p m a n n , 7 and Lees.8

We have completed calculations of the amplif ied oscil la tionsfor both the plane Poiseuil le and Blasius f lows by perturbing theneu t r a l cu rve ob ta in ed f rom L in ' s p rocedure . T he pe r t u rba t i onscheme is the same one previously used by Schlichting in a s imilarca lcu la t ion . 4 The rates of change of the complex wave velocity(c = cr + ici) in bo th and a- ( w a v e n u m b e r ) a n d R- ( R e y n o l d sNum ber ) d i rec tions a t the neu t ra l cu rve a re de te rmine d . Atc o n s t a n t R, a cubic in a is the n fi t ted as an appro xim ation for th ein te rmed ia te po in ts in the uns tab le reg ion . Con to urs of con s t a n t Ci ( rep resen t ing the ampl i f ica t ion ra te ) a re in te rpo la ted .Need less to say , the neu t ra l cu rve co r responds to a = 0. Th eresu l t s a re p resen ted a s F igs . 1 and 2 .

For the Po iseu i l le case , compar ison wi th T homas ' re su l t s a remad e . Rem embe r ing the h igh ly compressed sca le in the ind i rec t ion (F ig . 1 ) , we no te tha t , excep t nea r the c r i t ica l Reyno ld sNum ber , the e r ro r p robab ly shou ld be e s t ima ted a t the same va lue

of R. I t is the n seen th at a shif t in the a-dire ction of roughly 2or 3 per cent is suffic ient to account for the discrepancy betweenthe values of c%. T he c r i t ica l Reyno lds Number by L in ' s p ro cedur e, howev er, is roug hly 10 per cent too low. Gen erallyspeaking, the agreement seems to be as good as can be expected.T he upper b ranch o f L in ' s neu t ra l cu rve migh t be s l igh t ly tooh igh ( in the a -d i rec t ion ) and the min imum Reyno lds Number al i t t l e too low. S ince the me th od o f so lu t ion is on ly a sym pto t i cally correct for large R and since Lin 's procedure leaves outterms of higher orders in a, the accu racy na tu ra l ly shou ld besomewha t poore r in the lower R- and /o r h ighe r a - reg ions o f the(a, i?) -plane. Nev erthele ss , from the com pariso n i t does lendconf idence to the adequacy o f the a sympto t ic me thod .

For the Blas ius case , we aga in make a compar ison wi th Schu-bauer and Skra mst ad ' s expe r ime n ta l re su l t . T o b r ing ou t morecon t ra s t , th e neu t ra l cu rve i s rep lo t ted in the (t3 rv/Uo2, i ? i ) -p lane ,where (3 rv/Uo2 is the dimension less frequenc y of the dist urba ncea n d Rx i s the Reyno lds Nu mbe r based upon the d isp lacemen tth ickness . Bo th L in ' s and Sch l ich t ing ' s cu rves a re inc luded inF ig . 3 , toge the r wi th the exper im en ta l po in ts . T o comp are theamplif ication, the dimensionless amplif ication rates a t Ri =630, 1,840, and 2,200 are plo tted in Fig. 4* versu s the di mens ionle ss wave num ber i , aga in based on the d isp lacemen t th ic kness .In bo th F igs . 3 and 4 , the p resen t re su l t s a re in good ag reemen twi th exper imen t , ce r ta in ly much c lose r than Sch l ich t ing ' s ca lcu la t ion .

A de ta i led accoun t o f the work i s now under p repa ra t ion .R E F E R E N C E S

1 Heisenberg, W., Uber Stabilitdt und Turbulenz von Fliissigkeitsstromen,Ann. d. Phys. , Vol. 74, pp. 577-627, 1924.2 Lin , C. C , On the Stability of Two-Dimensional Parallel Flows, Qua r t .Appl. Ma th. , Vol. 3, pp. 117-142, 218-234 , 277-30 1, 1945.3 Tol lmie n , W . , Uber die Entstehung der Turbulenz, Ges. d. Wiss. Gottin gen,Ma th . Ph ys . K la sse , Na c hr . , pp . 21-44 , 1929 .4 Schlichting, H., Zur Entstehung der Turbulenz bei der Plattenstromung,Ges. d. Wiss. Gottingen, Math. Phys. Klasse , Nachr . , pp. 181-208, 1933.5 Sc huba ue r , G . B. , a nd Skra msta d , H . K., Laminar-Boundary-LayerOscillations and Transition on a Flat Plate, NACA TR No. 909, 1948.6 Thoma s , L . H . , The Stability of Plane Poiseuille Flow, Phys. Rev., Vol. 86,pp. 812-813, 1952.7 Lie pma nn , H . W . , Investigation of Boundary Layer T ransition on Con~cave Walls, NACA A CR No. 4J28 , 1945 .8 Lees, L. , Instability of Laminar Flows and Transition to Turbulence,Conso l ida te d Vul te e Re por t ZA-7-006 , Fe brua ry 25 , 1952 .

* I t migh t be po in te d ou t tha t , in r e duc ing the i r obse rve d da ta , Sc huba ue ra nd Skra msta d use d the wa ve ve loc i ty a s the downs t r e a m propa ga t ionvelocity of the disturban ce inste ad of the more logical "group velo city."The correction is to add approxim ately 10 to 20 per cent on the experim entalamplif ications of Fig. 4, there by br ing ing a c loser check with our com puta t ion a t the two h ighe r Re ynolds Num be r s . Howe ve r , the c ompute d c urve sare obtained by interpolatio n, which migh t easily err by as much as 20 per centexcept in the imm edia te neighborhoo d of the neutra l curve. Henc e, we donot p ut too much emp hasis on this correction.

Comments on Location of Detached Shocksa n d T r u i t t ' s D i l e m m aE. V . La i toneAssociate Professor, University of California at BerkeleyO c t o b e r 5 , 1 9 5 3T N REFERENCES 1 AND 2 , T ru i t t mis taken ly be l ieves he has de -

rived a new relation for predicting the effect of nose angle onthe location of a detached shock wave ahead of a plane wedge.Fur t he rm ore , in re fe rence 1 , T ru i t t c la ims Moecke l ' s 3 result iswrong by s ta t ing : " . . . o the r ( i .e . , Moecke l ' s 3 ) s imple express ions have been de r ived fo r shock de tachmen t d is tance wh ichpredict no direct effect due to nose shape or varying nose angle .