On the Transonic Flow Past Thin Airfoils Citation Cole, Julian David (1947) On the Transonic Flow Past Thin Airfoils. Engineer's thesis, California Institute of Technology. doi:10.7907/7YQN-9C32. https://resolver.caltech.edu/CaltechETD:etd-04202004-105048 Abstract No Abstract Submitted Item Type: Thesis (Engineer's thesis) Subject Keywords: Aeronautical Engineering Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Minor Option: Mathematics Awards: Caltech Distinguished Alumni Award, 1971 Thesis Availability: Public (worldwide access) Research Advisor(s): Liepmann, Hans Wolfgang Group: GALCIT, Caltech Distinguished Alumni Award Thesis Committee: Unknown, Unknown Defense Date: 9 June 1946 Record Number: CaltechETD:etd-04202004-105048 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-04202004-105048 DOI: 10.7907/7YQN-9C32 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1419 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 23 Apr 2004 Last Modified: 22 Jul 2021 20:37 Thesis Files Preview PDF (Cole_jd_1946.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

ON THE TRANSONIC FLOW PAST THIN ATRFOILS Thesis by Julian D. Cole iter J Ge In partial Fulfillment of the Requirements for the Degree of Aeronautical Engineer California Institute of Technology Pasedena, California June 9, 1946 ACKNOWLEDGMENTS The author wishes to thank Dr. Hans W, Liepmann for directing the research and giving essential assistence, Dr. Paco Lagerstrom for valuable discussions, and Harry Ashkenas for cooperation especially in obtaining experimental results. Part It III TABLE OF CONTENTS Title Page Introduction / Application of Fourier Integral to the = General Problems Some Results and Comparison with Experiments 48 INTRODUCTION This research was undertaken as part of a general program of the investigation of the formation of shock waves on airfoils. Consequently, it was thought advisable to study potential solutions for transonic flow to obtain information concerning the occurrence, size, and shape of supersonic zones. From this point of view it is especially desirable to include the effect of wind tunnel walls in modifying the flow field since the most reliable data can be obtained in wind tunnels. Until recently the main methods of investigating transonic flow were: (1) Expansion of the velocity potential in a double power series of space coordinates, This method, as applied by Taylor (Ref. 1) and Gortler (Ref. 2), solves an exact equation for the flow with aoproxima te boundary conditims, Since the area of convergence is very limited the use of this method is restricted to local problems such as the flow in the throat of a nozzle, ete. (2) Solution of the Hodograph Equation. The equation for the two-dimensional flow of a compressible fluid may be linearized by a transformation of the independent variables. (See Ref, 3) Then, in general, the simplified equation must be solved with correspondingly more difficult boundary conditions. Tsien and Kuo (Ref. 4) have applied this method to the flow past bodies of nearly circular shape, but even here the computations were lengthy. ee (3) Development of velocity potential in power series of thick- ness ratio. In the first approximation this is the so-called "small perturbation" or Prandtl-Glauert method and has been used by many authors to obtain information about compressible flows. The first approximation is relatively simple and where carefully considered yields much information, even in transonic flow. Gortler obtained supersonic regions in the flow past a wavy wall in Ref. 8, The se- cond approximation is more complex and has only been worked out for special cases, e.g. Hantzche and Wendt (Ref. 5). (For a revision of these methods see Ref. 3) (4) Solution of an Approximate Transonic Equation. If in the two-dimensional, irrotational, isentropic flow of a compressible fluid, & = velocity of sound at a point x,y in space u = velocity of flow in x-direction at that point v = velocity of flow in y-direction at that point, (a 158 200 BY sav = ° (1) 4 Then, u is considered to be only slightly different from a velocity of sound and v is considered to be small, i.e., u= a¥y 2 ood. - # ag ab ; where 5, and 3 are much less than u, so that squares of their $ ratio may be neglected and where a* = velocity of sound when it equals the flow velocity on a*® = u® + v’, Eg. (1) may be written wt of 8 , 2 26 36 2% (2) q*® 2X ax” q* ay XY -— =0 a7 oe Solution of this equation should apply to transonic flow. Recently, E, Graham and Th. von Karmen have used Eq. (2) as a starting point and have, by similarity methods, obtained some results for the drag of thése bodies in transonic flow. & consideration of the methods available made a logical first choice the application of the small perturbation theory to the tran~ sonic flow past airfoils. The small perturbation theory makes the solution of the flow of a compressible fluid an affine transformation of the solution of the flow of an incompressible fluid. The result- ing problem in potential theory is solved by means of a Fourier in-« tegral. It can then be expected that as long as the assumptions made in the theory are not violated, the results will approximate the true flow, In this way, supersonic regions over an airfoil and wind tunnel wall effects can be considered, When exact equations, like (1); are used, supersonic flow is governed by a hyperbolic equation and subsonic flow by an elliptic equation, or one can say that a supersonic zone is an hyperbolic region and a subsonic zone is an elliptic regim. The linearized equation (see Eq. 5) used in the solution is, however, elliptic for all space, This implies the analytic continuation of an elliptic solution through a hyperbolic region, (The problem in general was devised by H, Bateman, Ref. 7). However, this is not too strange, for the limited hyperbolic region of a finite supersonic zme behaves elliptically, i.e., every point on it influences every other point and in fact, every point in space, This occurs in the following wey: & Pig. 1, the influence of a disturbance or pressure pulse at B is felt in the domain downstream through B, Thus, the influence — ~ of B is flet in the boundary of / \ b \ the elliptic region and affects the F Yas lp j 8 \ rest of the elliptic region and << especially that vart of the boundary in whose domain .of:influence C lies, Fig. 1 Physically, the small perturbation approximations are interesting for it will be noticed thet stream tubes have no throat at sonic speed nor do they diverge when the velocity becomes supersonic. Eowever, this is a natural consequence of replacing the exact equation for “ a stream tube, (1 - Me) de --gA where M = Mach number = a A area of stream tube by an approximate one 2 Aa _ A (- Ma ) HF = - 24 where M disturbance, as, in effect, the theory does. Bach number far from I APPLICATION OF SMALL PERTURBATION METEOD TO TRANSONIC FLOW By considering the conservation of mass, energy, and momentum in a perfect compressible fluid which is disturbed only slightly from steady rectilinear flow, it is possible to describe the motion by the following linearized equation: (1-mg ) Ee rd + SB 0 ) where | M = Mech number in the undisturbed streem = ns if V = velocity of flow in free stream, taken parallel to x-axis and a, = varied velocity in free stream Alos, if u, vs w are the velocity components at a point x, y, z in the x, y, 2 direction respectively, then the induced velocities 3 u 2 In the diameter of B it is assumed that the induced velocities are small compared to the free stream velocity so that squares of this a? ratio may be neglected and that V = is also small, where t = & 4 a characteristic length of the body causing the disturbance, “, is any induced velocity, end oS) is any coordinate, (See Ref. 1) These assumptions must be later checked. It can also be shown, using the same approximation that the pressure coefficient va’ when p denotes pressure andp (4) _— — 4 -/¥# = tf V V density. A If the free stream has subsonic velocity and the entire motion can be considered two dimensional, z a¢ vs ao _ Vi ox ay" =? where p is a perturbation (5) a | 7 2 potential such that a” 3K a 2 ys oF Some solutior of (3) in special cases are now given. A, Flow past a wall of almost arbitrary shape, the medium extending to infinity y, as in Fig, 1, x,y are the coordinates in a physical plane and y = f(x) gives the shape of the wall, then (3) must be solved wi 1. and 2. th certain boundary conditions, These are: No disturbance in y-direction far from the wall on (6) y's Howe F: —e 00 Flow tangent to the wall. within the approximation of the theory this can be expressed as: ¥ V Fog (7) K(ho) = 69] » i.e., the boundary condition is applied on the x-axis. Y qv Now let ¥7% | so that IF (s) Ig - fy =? ' with these boundary conditions la. # —+o 7 0? ne JIG) 97? Then the solution may be written (if f'(t) is suitably restricted). __Y 7, _ / (9) BIT /+ s fe dn f (¢) 060 a tor 279 IP _o BF fa? Va a Ay blest 2M $b 2§* 4° 7 24 that 2A —y oO as a and at a=? The pressure distribution may Khon Be found from ~ f . Bfe Tz "¢ fl) win 7 Ub €) de changing the Be integration,yields, fe Liou Leo om fe- $e re and integrating, (¢- §) , yy Cs UZ J L008. At » or in physical coordinates / (tx) Lb) ra <4 Foyt The result can now be applied to the flow past a thin airfoil at zero angle of attack. 1, Circular arc airfoil = ad If the flow is past the circular arc airfoil shown in Tig, 3, —- x FU) = = opp 4 2 RK (11) = for JX) >a to the order of the theory. Then, from (9), [oI Va -§)d we Gg (12) Po Hig a (¢- 5) ey? Let a = 1, so that integrating by parts —_ tp~ sie [9 Y? & SL Suess or in ee coordinates 14” + (i+x)* (13) p(y) = fre [4 tly 52 44 (1-x)* _A “He te bx eg ay)! Then —-4 (14) Ce (0, 0) = Cp = ATR approximating the surface by y = 0, and above the maximum thickness as) % (4g/> Psu L'74 tm ag | “On the surface of the airfoil, or approximately at y = 0 Co (x, b) = Cys /- Jt x ’ ’ Ponay L 7 4 Jxle/ /-X The method of solution can now be extended to flow between boundaries, A, Flow in a chamel past a wall of arbitrary shape. For closed channels, only the case of flow between a smooth wall and a disturbing wall will be considered but it will be obvious how flow between two disturbing walls can be built up. For the case shown in Fig, 4 the differential Eq. (8) still holds as does boundary condi- tion 2a of (7) but boundary condition la must be modified so that (4%) V=22 2 2% 4 y °¢ a9 LLLELIIL ISIE MISSI LLL LL ALLL Vv 4-6 a = = /4h = A sey Pa) y Then the solution is ob MTree «O° ae fed fetter imi ya Fy =-24 at 0A pink dh for V?g=9 ena _7F hadlh-w) 4 ale Last ul PO) tte dll - 5) dt so that od. =O at. =4 and at “Sey J. ad J £1 op dle- S)dd- ft (3) 10, Then (a8) Lesh ah changing the a of integration, ye P por | AEE pi 5 bepydd (7 - This.may be integrated with respect to for Av [grr irate’! plch MF 7 Fae tor, 9) . , (bbs Hence % = = Zz S12) dé south (4 ,) (0 hl Cha) A a) or eo pf ah (EB) Sb (19) Guy) = gif [ ai CCE eb Sie)-ol fe) For the case where the disturbance is restricted to -a x a, the pressure coefficient at the surface (y = 0) is given by This is identically the result obtained by Tsien and Lees in a different way (See Ref. 10), The results can now be applied to a special case, li. ‘1, Circular are airfoil at O° angle of attack in a wind tunnel f* 7° # for /t/ £2 For this case, as before, . = o for //74& (See Fig. 5) Then from (18) ¥ ‘ é - LLLILLLLLALMLLLLLLLL LLL Yah (Lt -L 7, (2) 94 “9349 a) dt Now let as/ - 4 Opa bp J Then ur rie IY “@) - on 17 or poh py) ee 7 a a This can easily be evaluated at the surface of the airfoil (y = 0), gan 5 ze [react 4 (22) = 9 ‘for (23) ion ek (EH) = al , GO" ar FF ‘bey? Per “where the Be are Bernoulli numbers, This converges everywhere for a worst case of b = 1. Fora 2p narrower channel other forms of expansion may be used, as for example in Ref, 3, but this is sufficient for very practical cases, / ml ‘si f oqo nee gave al * ha ib Ze ES |¢0 +h or / Sr [vs Yb (24) gp (he) = i /- swlby- #4 ey” Gul! rs 4 It is seen thet the first term of (24) corresponds to a surface dis- tribution of pressure Ce, ) without a tunnel wall (Eq. 16), and the fe) second term gives the interference effects, . K i.e. (24)¢,, 14, 0) x Ge ¥ 44% » o L awn/ 2 fa" toe I pp” "49 ed ms/ where SY = Cau, 0! 4 ye . f, aa ge), E a # But Ap p/ d ad gerplp ef - ant al Nps aa (30044) Hence, (26) eg = ign nop eel me 13. and for x = 0 Gee = Prey DOM tei tne” arf ne/ or - ‘ oo Sau- jn | any Donen! ig “/ (7) 09 * pee [14 FO Gmaiyt (al maf The ¢, can also be easily obtained at the tunnel wall y= 1. From (22) ga" ae [gril It (28) 7 f Bat fat pe Fel 228) (bs), (bt Je F (300)! ees Then proceeding as before we get (29) end = a4 eZ we) fet fe tama) tem “Na) gp cb) = V mnéy patomei dé and at x = 0 or above the maximum thickness on 29, wile) Sms / 2} —/) — 7 zs Gy PAC (anes la (30) Va % 4) 7 mor mes Another place where c, may be easily found is y = 1/2 or half way between the wall and airfoil. Then from (22) ~ ih Lup-¢ 4 (31) gx, w/™ mt yf ip so that “(3 -/ 1) Ry at J pe )lren) te reyeteH) ? Bs “)” 7% 2 autonrs) ¢ Ma/ 14. an _— gute ae jo SA) 6 (8) er )/ (32) ( +) Vai . “id mo “1,) ae me} This last solution may be extended by a Taylor expansion, by putting - bls )™ ” vee (33) Gote—) * Gotes bh) + Yy- bs )g'eo, bh) + Ph. % "te ) + a? when y's #4 ete, - From (21) we have, C, (ey) == Pe Ap (34) PY” The ¢ mp - a Ss ae mone gp> aed hjoege WA 9 F 4p h 2 at siti) Zea [GI taf | f (35) _ ed (P/- As Le LB/ so - Zz Gute bh) = spi Lorian! “cay I@ (36) Then - BOT ah [G/ (31) gig ble Fe [EY [awk 5 Zi / Then from (32) and (33) uw aud} Gayl = Fe ran 7)" (y-)” a - a yen) nm ae _ ~ fo ) of fea Joe (3!) eae Vile “/ _ a (faflt- - if fook 2 - fina’ fe} For the case when the disturbing wall is in a jet, or the airfoil (38) 7 (ug! = Pesuns is in an open wind tunnel, the same method can be applied. For the flow as in Fig, 5a the boundary condition 2'a must be changed so that (2" a g = By a constant at y = b or y = h), but otherwise the P=, problem is the same. Hence, the solution can be written, i6, Ge) Hi af ete ty wn ath-$) “oad a for then Y, ap o / a” a fh af yeh, ye 2¢ . a S10 ym ttf) bb = LSB). apes Le / Then, in the same way as before, 2 fa / bk Weed fey pus LR) ya a) or changing the order of integration a Piet AL: ? seri Act-pdl a . (ae pone des Za ff Ap is jt ole io toot 7 * co See Kef, 2 tow so that _# poh Cab =z f a op These results can be applied to the same disturbance as before, 2 Circular are airfoil in open wind tunnel, Then in the same notation as before je re y (42) oo ar LEH! 4 put agen [2 js =. ht a p. and a special case of some interest isx=y= 0. Then Ets - + i poh ¢ 4 p Coe? © w woh pp -! (43) | = oug [gu ty Expanding =/ Fan-! ™ sacl, & “3 [v Ze 1 So (4) ‘/ and integrating yields « Pe [ir Zor Cs <1) Fors ()"/ (_f, an pon): me} a C so that “ GO) Em fie (44) g ee Ye, / Zo oan wl nes 17. 18, JL (RESULTS AND COMPARISON WITH EXPERIMENTS Some results and comparison with experiments are shown in Figs. 6-183. The experiments were made in the 2" x 20" transonic wind tunnel of the GALCIT (see Ref, 11). Figs, 6 and 7 show the pressure distribution at the airfoil surface. Co ori tical denotes pressure at which M= 1. As wald be expected, the agreement is better in the leading half of the airfoil and at lower Mach numbers. There is intrinsically less error in the theory at lower Mach numbers, Also, boundary layer effects in dis- torting the shape of the surface are less at smaller Mach numbers and on the leading part of the airfoil. The effect of boundary layer and occurrence of shock waves on the aft part of the airfoil have been noted in schlieren photographs (see Ref. 13). The parameters used in the computation were half chord of airfoil = 1.5" =1 a= R = radius of curvature of airfoil surface = 6,785 = 4,51 b = 1/2 height of tunnel = 10" = 6.67 a) : T o= 7 tunnel height parameter = wW2. = .471. 6. 67 It was interesting to note that the wall interference effect was approximately constant along the surface of the airfoil when com- puted with these parameters. Fig. 8 makes some comparison of the wind tunnel wall effects for this tunnel and shows especially the increasing effect at higher M . it is noted thet the interference effect of an open wind tunnel is less than that of a closed me. Fig. 9 shows how strongly 19, the ratio of tunnel height. to model chord affects the interference. Fig. 10 gives some indication of the decey of airfoil disturbance above the surface. It also indicates that the supersonic region predicted would be too small, In order to get the shape of some typical supersonic regions Pigs, ll, 12, 13 were prepared and for simplicity wind tunnel well effects were neglected, To show how mach the locel supersonic zme computed differs from the supersonic flow, the distribution of velocity on the supersonic zone at the surface was used as a starting point and the flow field extended by the method of characteristics . (See Ref. 3). The result is shown as the dotted line in Fig. ll. Actually, the supersonic zones in this airfoil become more asymmetric at higher M and hence the agreement with potential theory would be worse, An interesting theoretical method to determine conditions with- in the supersoni ¢ zone would be to successively alter the surface distribution of pressure in the supersonic zone and make the M = 1 boundary of the characteristic extension agree with that predicted by the small perturbation theory. The small perturbation theory has then yielded some interesting results for transonic flow over airfoils, but for more information more accurate theories are needed, It is felt that the best attacks in the future will be through the second ap»roximation of this theory or through the approximate transonic equation, il. 12. REFERENCE Taylor, G, I,, "The Flow of Air at High Speeds Past Curved Surfaces", R, & M, 1381 January 1930. Gortler, H., "Zum Ubergang von Unterschall-zu Uberschallgeschmindig- keiten in Dusen", Z.A.M.M. Vol. 19, 1939, pp. 325-337, Liepmann, H, W., and Puckett, A., "The Aerodynamics of Compressible Fluids", John Wiley and Sons, 1946, Tsien, H.S,, and Kuo, Y. H,, “Two-Dimensional Irrotational Mixed Subsonic and Supersonic Flow of a Compressible Fluid end the Upper Critical Mach umber", GALCIT Report to NACA under contract No, NAw 2793, Dec. 15, 1944 = Restricted, Hentsche, W., and Wendt, H,, "Die Kompressible Potential Stroemang um eine Schar von nicht angestellten symmetrische Zylinder im Kanal", LFF, Vol. 18, pp. 311-316, 1941, also NACA Tech, Memo. 1030, 1942, Lamb, H,, "Hydrodynamics", Cambridge University Press 1932, Bateman, H,, "Notes on a Differential Equation which occure in the Two-Dimensional Motion of a Compressible Fluid and the Associated Variational Problems", Proc. Koy. Soc, A Vol. 125, pp. 598-616 19264, Gortler, H., Zeit, f, Angew, Math. u. Meck. 20 (1940) pp. 254- 262, Haan, Bierens de, Table of Definite Integrals, Table 265. Tsien, H, S., and Lees, L., "The Glauert-Prandtl Approximation for Subsonic Flows of a Compressible Fluid", J. Ae, Sci., Vol. 12, 1945, Liepmann, H, W., "The 2" x 20" Transonic ‘iind Tunnel”, GALCIT Report to AAF Materiel Command under Contract No, W 33-038 acl717 (11592) Liepmann, H. W., "Investigations of the Interaction of 3oundary Layer and Shock Waves in Transonic Flow". GALCIT Report to AAF Materiel Command under Contract No. W 33-038 ac-1717 (11592) toe rp yea fad a sieeueesess Tet akess ac. osue tasers ran +44 + pees epauares pria it + + iv neers rot et if Ae ttt tHtite cannes tt ip FO ett wt? DB IV FOV ESS YD OY Oo” S Zo om y ra & oC f | | | { } i | 1 ‘ en a free —_ aod _ wa spy | -. ~ 4 seve . — + — we ee od _ . ok — cee af - _ MOIEN ILI V5 SL IVAW D-H - af | VOCL WED EAI LAD Fe 7 PAE 4 7 ae ~ ; | ~ “4 ! a fe 4 pape Seg OP BP > OW bb VOs eM Oddie | ~ i WIP De ee try SACOG bY | PAO OST by SQ VOLO SLOG ; . ' i 1 if : oh _ + ng t we 4 — —-4 r - ~ —+- coe ee | | | | | : i i ' L an a a Ho — L as a _ I kL. I eee {? rep ey reas rege ¢ on = St pues ie ceactey L are Gomen ae Za? 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