AZ/Vc/t- L^lll ACE No. LUC2lf NATIONAL ADVISORy COMMITTEE FOR AERONAUTICS WAKTIME REPORT ORIGINALLy ISSUED March 191^1^ as Advance Confidential Report L4C24 OH THE FLOW OF A COMPRESSIBLE FLUID BY THE HODOGKAPH METHOD I - UNIFTCATIOK AMD EXTENSION OF PRESENT -DAT RESULTS By I. E. Garrlck: and Carl Kaplan Langlej Memorial Aeronautical Laboratory Langley Field, Va. NACA WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. 127 DOCUMENTS DEPARTMENT Digitized by the Internet Archive in 2011 with funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/onflowofcompressOOIang n/1 00b ^1 NATIONAL ADVISORY COMMITTEE FOR AERON^lyTICS ADVANCE CONFIDENTIAL REPORT NO. L4CC4 ON TH:^ flow OF A COk'PRESSIBLE FLUID 3Y TPZ: IIOEOGRAPH i.IETHOD I - UNIFICATION AND T^ITENSION OF PRESENT-DAY RESULTS By I. E. Crarrick and Carl P^aplan SUrfi'[ARY Elerr'.entary basic solutions of the equations of motion of a compressible fluid in the hodograph variables are developed and used to provide a basis for comparison, in the form of velocity correction formulas, of corre- sponding^ compressible and incompressible flo\"s. The known approximate results of Chaplygin, von Karman and Tsien, Temple and Yarv/ood, and Prandtl ard Glauert are uiiified by means of the analysis of th" present paper. Two ne\7 types of approxir.ations, obtained from the basic solutions, are introduced; they possess certain desirable features of the other approximations and appear prefera- ble as a basis for extrapolation into the range of high stream ilach nunberr' and large disturbances to the main stream. Tables and figures giving velocity and pressure- coefficient correction factors are included in order to facilitate the practical application of the results. [NTRODUCTION The present paper is concerned vith a theoretical study of the hydro dynamical equations of a perfect com- pressible fluid in two dimensions, in vmich the so-called hodograph variables are used as the independent variables. It is hoped to achieve herein a unification of the present-day results obtained in this field and also to provide a working basis for further developinents . The earliest contributors to the hodograph m-ethod for treating compressible fluids were Molenbroek (reference 1) and Chaplygin (reference 2) . The remarkable work of Chaplygin on gas jets appeared in Russian in 1904 but re- mained relatively unnoticed. In recent years contribu- tions to the hodograph method have been miade chiefly by 2 CO!TIDSrITTAL NACA ACR IIo . 1AC2A: Demtclicnico (reference S), von Ivariran (reference 4), Tr-ien (reference 5), Ringleb (reference 5), and Ter.ple and Yar\vood (reference 7) . The c'nief recson, and perhaps tlie onl;.' reason, for preferrin.'^ tiie hodograph vfi'iable;s to the physical plane coordinates is that the equations of motion in the hodoei:raph variables are linear. This simplification Is achieved, however, at the cost of /nore difficult boundary conditions and at a loss of physical Insif^ht. The .''reat simplification in the r.iathenatics due to linearity never- theless makes it desirable to pursue this line of attack as long as it appears i:rofi table to do so. The mathematics for handllnp; the flov; equations re- ceived a substantial ir.petu.s by the v.'ork of Bers rnd Gelbart (reference 8), who developed a nev/ function theory aralo.f-"ous to ordinary anal7tic fur-.ction theory. The presont paper utilizes the r-.ethods of this new function theory to develop certain fu.nctions essential to the ooir.pressible-f low problem.. It is of historical interest that ideas similai' to those of 3crs and Gelbart v;ere o::- plcred by the renowned r!at}:iematician Ililbert (reference 9) in the early part of t}\is century but do not appear to have been further devolcped at the tine. The material to be treated is conveniently separated into two parts. In part I, the present paper, basic particular solutions of the hodograph flow equations are developed and employed in unifyiiif' and extending the re- sults obtained by Chaplypin, von Karman, and Temple and Yarwood. Tlie results obtained in part I are of iirjnediate practical application and are given in the form of tables and graphs of velocity and pressure-coefficient correction factors. In part II, v;hlch will appear later, f^eneral particular solutions of the hodograph flcv equations are developed and discussed. The material in part II, It is hoped, v/ill lead to a method for handling the actual boundary problem of the flovr of a compressibl.e fluid past a prescribed body. AKAIYSIl Flow Equations of an Incompressible Fluid It is v;rll known that ij-ie roD.ations bet\.eexa the velocity potential and the stream function \i; for COMt'IDEfTTIrvL IIACA ACR No. L4C24 COIjPIDENTIAL the steady irrotational two-dimensional motion of a perfect incompressJble fluid are bj6 ^ dii_ ox dj "^ (1) These equations ai-e the Canchy-Riemann equations and therefore 0+1^ is an analj^tic function i'{z) of the complex variable z = x + iy. The complex velocity or reflected velocity vector u - iv is obtained from the complex potential f(z) by differentiation. Thus, u - IV = dfiz) d2 = qe -19 ^-i (9 + 1 logo) (2) where q is the magnitude of the velocity vector and 9 is the angle the vector makes vrlth the positive direction of the X-axis. The variables 9 and q are sometimes referred to as "the hodograph variables." The flow equations in the variables 9 and q can be readily derived by intro- ducing 9 + i lop q as the independent complex variable in place of x + iy. Then, in analog:/ with equation (1), 60 _ dj! o6 6 log q 60 6 log q 69 (5) COIJF'IDEWTIAL CONFIDEIITIAI; IIACA ACR No. L4C24 or d9 6q I £i I q 69 J (4) These equation? aro knov;n as the holograph equations for the flow of an incoipresnible fluid. F3 o"vv liquations o:^ a Coripressible r'luid The equation.? oorrG"r:onding to equation (1) are, for a oiTip r e s r ib 1 e flu 1 -"l , 6x p 6y (5) 6y " p" 3x Vifhere p is t'.-r denpity of the fluid at any pon'nt (:;',y) and p ir. a constant density, vvhlch for convenience i3 refeT'red to a stagnation point, A s'lort vay to derive the hodo,3r&ph equations for a coTipres'^ible fluid, attributed to f-.ol.enbroek, is as fol- lows : Accordinp; to equation (5), vith u = -<-^ and v . Po d;5 + i — - di' = (u d.x + v dy) + i(-v dx + u dy) = (u - iv) (dx + i dy) ,-i3 = qe ds or dz e--^ (djc^ + i ~ d>|/ p. (6) COr^FIDENTIAL NACA ACR IIo. L4C24 COTIFTDENTTAL It follows from equation (6), 'bj considering 9 and q as independent variables, that de q ^ \.6e P 66/ V^ and dz oq 1 e"(^6i^ 1 £°M) q \tq p 6q/ Then, by assioininf; that p is a function of onlj q (equivalent to ansiuninp that the pressure is a function of only the density), 6"z 6q 69 and 62z a9 1 6^ ^ . '^(pp/pq) 6-^1 , 1 ie/l6fil. .£o_^fi_^ _"q2 6G ^ ^q 6eJ ^ ® \6q 69 ■^'- P 6q 66/ 66 6q q V6q p 6qy q ® V^9 6^ Po 62^1/ q "^ ^ P 69 cq Since, by continuity, these two expressions are identical, it follows that q \on. p 6q y "qii 69 ^ dq 69_ Hence, by equating real and iraaginary parts, 69 " P 6q 6q TTT = q ^'(Po/Pq) 6'!; I f dq 6e J (V) co:it^ide:itial C0I:PIDEI'TIAL I:ACA ACR Uo, L4C24 These are the hodof",raph equations, first obtajned. hy Molenbi'oek, for the flov; of a compressible fluid and are independent of the form of the prc?sure-denr,ity relation, It is o'l^served that, when p = p = Conrtant, equa- tions (7) reduce to equations (4). Equations (7), in contrast with equations (5), are linear in the dependent variables . Bernoulli's Equation and Equation of I^tate In the present section there is listed a collection of fomulas and definitions necessary in the analysis. Bernoulli's equation for a conipressible fluid is ^"^ ^ -^ ^ q2 - (8) Po whore p static pressure in fluid p static pressure at stapnatlon poiiit (q = 0) p density of fluid q nagnitude o.'^ velocity of fluid The adiabatic relation betv/eon the pre?s\ire and th^e density is V where Y adiabatic index (approx. 1,4 for air) pQ density of fluid at stagnation point (q = 0) The local velocity of sound a is obtained from .2 „ dp - - dp COITPIDENTIAL NACA ACR No. L4G24 CONFIDENTIAL For the adiabatlc case. a2 Y R 10) From Bernoulli's equation (8) and from equations. (0) and (10), the folloving relations nay be obtained: a- = a,." - -i (y - l)q^ 'o - 2 1 ■-C Y-1 i P = P, 1 - rr (Y - 1) -~ Y Y-1 (11) where a^ Is the velocity of sound at stat.^naticn point (q =0). From equations (11), for y > 1, a vmxlmurr. velocity q = Qjji is obtained for the liiiiltln^; conditionp- P = p = a = 0. Thu-, 2 _ d , 2 ^^ra ~ V - V ^o wh ere •pa-o' Y - 1 (12) The funda^iental nondinenslonal speed vn-iable, in f-eneral, is q/a^ bnt It 1? found U'^eful in the analysis to eriplo^r a nondimensional speed variable t defined as T = q^Ara^ (13) For Y > 1, th.e range of the variable t is £ t ^ 1, The value t = has 'a dual raeanlnc:'. T - in the case CONFIDENTIAL 8 COIITIDENTIAL NACA ACK Ho. L4G24 of a compressible fluid corresponds to a stagnation point (q = 0), or T = may pean the ll'^lting case of an in- compressi'ble fluid (a^ = ») . V/ith the definitions of t' and p, equetions (11) bee one p = p^fl - T)P !> p = Pq(1 - t)'- 3+1 (14) >• Tlie local Marli number J.I = q/s. may be expressed in terms of the speed vaj-lable' T in the follov/lng way: 2 0^ 2 ,^^ 2 2 „ q -ra a2 2pT 1 - T (15) or, by solviuf for t in terms of M, T =T 2p + M^ (16) The value of t I'or viiich the local velocity of the fluid equals the local velocity of sound (M = 1) given by T„ = 2[i + 1 (17) In the case of uniform, flow past a fixed boundary, the pressure coefficient is defined as _ p - Pi Cp,Ill - -^ -• 2 ^-1^1 CONT'IDENTIi-.L FACA ACR IIo. L4C24 COITIDENTIAL 9 virhere the subscript 1 refers to the undisturbed ptroam, The pressure coefficient for the incompressible case (!I = 0) is The pressure coefficient for the comprossiblr case is ■ 1 \ For q = q^ (sonic) l^J^*h,.-^U^^\.(^)'\r) }-(mrj '-' r r '•\-^ '1 ! P'or q = qj^ (vacuuri), (^?,:'i) m yHt "^ (ISd) Basic Solutions of Hodopraph Equations Consider the Inconpresnlble case represented by equations (3) or (4) . It is clear that ^ = 9 and \j/ = log q satisfy these equations. In fact, any con- vergent power series in w = 9 + 1 log q represents an analytic function of v/hich the real and iina,f.;inary parts satisfy equations (3) or (4) . The class of analytic functions in w (and the concept of analytic continua- tion) then yields all the partlpular solutions of these equations . The particular solution vic = 9 + i log q can. be obtained by rr^eans of an integration that is Instructive in the generalisation to the compressible case. It is v;ell known that COIJPIDEHTIAL 10 CO%TIDE".TTIAL N^CA ACR No. L4C24 r P(w) = ; f (w) dw can be represented as the sum of tv/o line inte,f;^rals where f(v.O P + Thus, given a pair of fund Ions F and Q that satir-fy equations (3) or (4), thin pr-ooe^s yields another pair of solutions, namely, the real and the imap-inary parts of T7(v;). For example, if P = 1 and Q = 0, F{vi) = G + log q (19) Afain, if P = and Q = 1, F(v^) = iw -lo,^' q + i9 (no) The phypical interpretation of equation? (19) and (20), considered as flov patterns, is of sone intei'est in con- nection with later developne nts . It is clear that equations (19) and (20) represent a vortex and a source located at the origin, respectively. The ftonoralizatior to the coinpressihle case of the fore;'i;oing elementary results was acconplished by Bers and Gelbart (reference 8) by means of simple yet fertile ideas. Bers and Gelbart treat equations of the form 60 60 = \(q) -< 6'^ 5q -Xr(q) ce (21) and show as 5,3 readily verified that, if P and Q, are a pair of solutions, the real anr] imaginary parts of the follov/ing sum of line integrals CGITFIDEFTIi.L NACA ACR I:o, L4C24 CONFIDSSiTIAL 11 / fp d9 - \,,(q) Q clq] + i / U d9 + —^r P dq I (22) J '- - -^ J L ^1^^^ J are also solution?' of equations (21), In particular, corresponding to the pair of solu- tions P = 1 and ^ = 0, there is obtained and, for P = and ''^' - "' iv; = ije + i / ^2^esult is that, if powers of q/ao higher than the third are neglected, f(T) = g(T) ^-f-^ (30) and does not involve explicitly the adiabatic index v. This circujtnstr.nce ujnderlies the present-day approximate methods for obtaining velocity and pressure-coefficient correction factors; in the »fo3 lowing sections, this point is brought out more clearly. CO:^TDENTI.lL NACA ACR IIo. L4C24 CONFIDENTIAL 17 Application of Baric Fujictions I, and L In this section, the basic fiinction? L and L are enployed to ^et up relations between velocities in "corresponding" compressible and incoT;pressible flows. These relations are of the nature of "stretching factors" 01' velocity correction formulas and contain the results of Chaplyrin, von Karman, Temple and Yary-ood^ and Glauert and Prandtl. It is important to recognise at the outset that no single velocity correction form.ula can represent in an exact v.'ay the correspondence of flow patterns past a prescribed body in a compressible and an incompressible fluid. A single velocity correction formula is actually feasible in only tv/o cases: (1) The stream Ilach number is small (even though tlie disturbance to the main stream due to the presence of the body may be large) so that the compressible-flow pattern differs only slightly from the incompressible-flow pattern or (2) the disturbance to the main stream is vanishingly small (even though the stream Mach number nay be high) so that the effect of the shape of the solid boundary is small. The various velocity correction formulas discussed in the present paper differ essentially only in the degree to which the requirements of these two cases are satisfied. Despite their limi- tations, single velocity correction formulas are extra- polated, in view of the lack of more rigorous solutions. Into the range of large distvirbances to the m.ain stream and high Mach numbers. This extrapolation can be justified by further theoretical investigations and by com_parison with experimental results. Consider again the corresponding paj.rs of functions w = 9 + i log q ! V; = + iL and (31) iw - 1(9 + i log q) 1 *> (32) i"^ = i(9 + iL) J It has previously been noted that the pairs of functions in equations (31) and (32) denote respectively a vortex and a source in an incompressible and a compressible COI'JPIDENTIAL la COITIDEFTIAL IIACA ACR No. L4C24 fluid. Each pair of functions can be employed a correspondence of flow patterns in which correspondiaf? points are identifjed Ly the sa^ne -values (j^,\J/) . Thus, In the case of the vortex (equations (31)), I-' •\ f-'c •it - \[f = log q^ = L where the su.b?cripts i and c refer to the inconprea- sible and to the compressible case, respectively. It folloivs that q.i ^ e" - q.e'^^T) (33) Simllai'ly, In the case of the soi^rce (equations (32)), ^^ = ^^ = -log q^ = -I, and M'l = M/c - 9 qi = e^ = qce^-^^ (34) At the end of the precedin,^ section it was pointed out that, to a first approxination, the functions f(T) and g(T) are equal. This fact implies that, to a fjrst appro.riration, a single velocity correction formula is feasible. The assumption is now n.ade that either equa- tion (33) or equation (34) can be adopted to provide a correspondence of flo"; patterns In the case of uniform flow past a body in an incompressible and a compressible COIIPIDEKTIAL r^ACA ACR No. L4C24 COHFIDEI^TIAL 19 fluid. V.'ith the vuidis turbed ?treans as convenient refer- ences, the following nondimsnsional forns of equations (33) and (34) can be written: ^1 = ^\ ft ana (• 4^(tift where the subscript 1 refers to the undisturbed stream. The use of the undisturbed streain ap reference in the nondinencional forri of the velocity correction formula was introduced by Tsien in reference 5, v/here also the details of the von Karman approximation are developed. It is shov.Ti in the follov,'in~ section that either of equations (35) or (36) contains the results of Chaplygin, von Karrian, and Tenple and Yarwood. As has been previ- ously pointed out, the concept of a single velocity cor- rection fornula is feasible in only two cases, nanely, small stream Ilach numbers and vanishingly small disturb- ances to the main stream. It is desirable then to seek a single velocity' correction formula that combines the features of these tv/o cases. From this point of view, equation (35) or equation (3S) is not the best choice. A better choice of a single velocity correction formula appears to be the following combination of equations (35) and (36), based on the arithmetic mean of f(T) and g(T); ~rf(T)4-g(T)l _e^^ ■ |[f(Ti)+g(Ti)] (37) In a later section, still another combination refer- red to as "the geom.etric-m.ean type of approximation" is introduced; in the section dealing with the G] auert-Prandtl approximation, certain features of the foregoing arithmetic- mean type of approrimiation and of the geometric-mean type are discussed. COI^IFIDENTIAL 20 COI^IFIDENTIiuL HACA ACR No. L4C24 At this point it is de5!irabl6 to discus.^ the practical application of equation (37). According to equation (16), T — M" 2,3 + 1 ,^2 ■^1 — Mi2 O O J. 1 ,f 2 and ^^a \2 3 + M (33) obtaired frop. eaua- Eqiiation (37) then yield?, for a .^iven set of values of the stream Mach numter !!]_ and the local !.iach number M, a value for the ratio (q/'1i )• o- ^^^^ local velocity q and the stream velo'^ity q^ in an incompressible fluid. Table 2 shows corresponding values of (q/q]_ )„ ai'id \q/q.^\ for various values of the stream Mach n'jmber I.i^. with Y = 1.4 (p = 2.5). This tabulation is perform.ed, for the purpose of comxparison, for the three cases repre- sented by equations (35), (^c) , and (37). Values of (q/qi)i, (q/oi),, and -^;^, tions (37) and (3B), are plotted arainst the local Fach number M in figure 2 for various values of the stream T.lach nu::iber M]_. Table 2 alco shov/s values of the pressure coefficients Cp q and Gp y[^ calculated by equations (13a) and (13b) for these correspondinfj values of (q/^i)' ^'^^'^ (q/qx) • Figure 3 shcvs the curves of pres3\ire coefficients corresponding to the curves of velocities of fir^ure 2. Useful cross plots of the curves in fip;ure 3 are Ehc-;vn in figure 4, in which Cp iji^ i? plotted against I'l for various values of C-. ^. 7n ■^ - P^ o COi:FIDHiTTIAL NACA ACR .10. L4C24 CONPIDETriAL 21 addition, ciTves are shown in figure 4 for (C^ Mi V ^^'^ and (C-Q ^'h) calculated bv equations (18c) and (18d), respectively. The curve for (Cp T,^-I^ corresponds to the sonic value T! = 1 or " - Tg = l/o and in effect divides the rep;icn of flov; into a subsonic and a super- sonic part. The curve of (c^ y.-, \ corresponds to the maxiriun! value M = '» or T = 1 and represents the cuter liinit of the supersonic region (or a perfect vacuum) . In order to exhibit the main differences between the various correction formulas (35), (36), and (37), the ratios of the ^'onic values (Gr^ t/t )„ and the correspond- ing incoHDresslble values Cr-, o 3-^3 plotted a'^-alnst the stream Mach nuriber M]_ in figure 5 = Observe in figure 2 that th.e ('i/^i ), -curves have raxiniim points. This fact r.ieans that the value of (q/q-^) associated vith a value of (^/li)- is not unique. Analytically, the criterion for the raax iniuni point is equivalent ^o -^: ^^-^ = (39) dT or, from velocity correction formula (37), (1 _ T)"^P"^^ - (2^ + 1)T + 1 :. For j3 = 2.5 this equation has only cne positive root, T = 5/24 or ?! -1.15. It is interesting to note that velocity correction formula (36) yields as the criterion for the naxiniiri point 1 - (2p + 1)t = The root of this equation is t = Tg = -.t-- , , ■ and, for P = 2,5, is T = i/g or I' =1, Velocity correction formula (35) yields no naximuin value of t or !', CONFIDENTIAL COKFIDENTIAL NACA ACR No. 'U^-CZli !-'eaning can bs civen to the value T = l/6 (M = 1) in the ca.se of equation {3l|.) with reference to the origi- nal interpretation of the flov; pattern as thtt of a source. It C3n be 3hov;n that the acceleration fq -r^ \ alone; a strefinline is Infinite at all points for y/hlch the loccl Mach number ic unitj- (t = l/6) and that a flov; discontinuity er.ists there. In the case cf the vortex flow p:..ttern (equation (55) )» no flov; discontinuity occurs for M< co . The velocity correc- tion fcrmulr. (57) sugr.ests a "liriiiting." value M ~ I.I5 for a Gpiral flov;, since equation (59) is anslogous to a condition of infinite acceleration. Tiius , the existence of a r.iixed c^-.-bsonic .ind supersonic region of flov/ ■.•/ithout oiHcontinultlec is indicated. Since the occurrence of tliis limiting- value of V. is a consequence of the sim}/le form assumed for -'-^le velocity correction formula, no undue significance should be at-^ached to any particular vslue at the presevit time. Tiie Chaplj'-gin ApproxLaation From the ooint of viev.- cf the nreseni ^ r; r> o ■)■■> .^ ^./ «^ J. I Chaplygin's approximation for subsonic speeds assumes a simple and lucid form. Chaplygin introduces in place cf q a new independent speed variable ^ equivalent to the quantity .c-iven on the right-hand side of equa- tion (55), namely. T) = q. .f(T) The hodograph flov; equationc; (7) th.^n assume the form '.vhere F(T) T] tr- - 1 - {2^i + 1) t (1 - T)2P+i > (iiO) 1 - p:Pp + 1;t- - t p(2ii + 1) (2p H- 2)t- CCNPIDENTIAL WACA ACR No. L4C24 COK^IDEI'TIAL 23 Values of the function 'F(t), for several values of y (or p) , are given in table 3 and are plotted in figure G against the local Mach number K, Chaplygin noted that, in the case of air (p = 2.5), F(t) differs but little from unity over about one-half the subsonic range = T - l/e. His approximation in the rsnge of lov; sub- sonic speeds consists in neglecting powers of t higher than the first or in replacing F(t) by unity. Equa- tions (40) can then be written in the Cauchy-riiemann form 69 b log T) 60 _ 6x1/ 6 log Tj 59 and + iUf therefore is an analytic function of the com- plex variable 6 + i log t). Chaplygin 's approximation thus leads to the velocity correction fo'^^'mula ^lA V^'l/e 1 - I Ti (41) where powers of t higher than the first are neglected throughout. The use of equation (34) instead of equa- tion (33) also leads to this result to the same order of approximation. The von Earnan Approximation Von Raman's approximation corresponds to the case Y = -1 (or p = — i) . It follows at once from the integral expressions for f(T) and g(T) given by equa- tions (25) and (27), respectively, that for this case ^/ ^ / ^ n 1 + (1 - t)^/^ f(T) = g(T) = - log ^ — '■ COIIPIDENTIAL 24 COHPIDENTIAL NACA ACR No. L4C24 or, with the use of equation (16), f(^) = s(t) = - lo,q -Ft 1 + (l - ^2)^/2 This function, plotted against M, Is included in fig- ure 1(b). Corresponding to equations (35) and (36), there is a single equation (^ iJi " Vqi/, I +(i - T )i/2 ''r / q \ "^ 1 Replacing t bv T-,f-3- ; and t-, bv — ■• according ■ n"i/c ^ ■" :;ii"-l to equation (16) ^ield?; q A - (^^' ^1/1 q- 1 + (l - M-]_2^' 1/2 (l - Mi^y/' Ji _ Mi2 + Mi2/JL\"! 2-]!/' (42) .^1/ c J Then, by solvin.f; for (qAi ) i^^ terns of (q/li "). s-^"^ the stream Ilach number Mt, -fcL ^^'^i ^ - u^2-^' (45) v/here t^ •1 I !i + u COIIFIDSI^TIAL NACA ACR Ko. L4C24 COl^IDEFTIAL The prersure ccefficient C^ jj., , expressed in terns of the inconpressible pi-essure coefficient Cp q, is easily obtained from the general forrnula (18b) by putting Y = -1 and making use of equations (45) and (18a). TIius , S,Wl = ^P,o : ^^ (44) (^ 1 + 1 1 - Ml ^ }' V -1- V Observe that for this case the function F(t) intro- duced by Chap].y,frin and given in equation (40) is exactly equal to unity. ^Froin the point of view of the present paper then, von Karrran's approximation appears to be equivalent to that of Chaplygin, who approximates F(t) by unity. It follows that the range of validity of In a :ed out supersonic region. Von Karmsn's choice of y - -1 has the advantage, however, of yielding slmj-le explicit ex- pressions for (i/QiV -^ terms of ('l/li^r ^nd for Cp y~ in term.s of Cr, o* Several values of G,-, ji-, calculated by equation (44) are included in figure 4. For the purpose of comparison with the other approximations, there is plotted in figure 5 the ratio of (C-n, ii-,\ to Cp Q against the stream Mach n->ir.-iber :!]_ in the case of von Karman's apnroxination. The values of C^ ^ are p , u obtained with the use of velocity correction formula (42) for the local Mach number ;I = 1, but the values of ^Cp ;j-, ) are calculated v/lth = 1.4. The Temple-Yarviood Approximation The functions and U' related by the first-order simultaneous equations (21) separately/' satisfy the second- order eciuatlons COI^IDEMTIriL 26 coit?id./:;nti-.l itaCh acr No. ni.C2Li. + A., (q) aTT ! A .- -, ■ t^- i - ^ 4. • ^ (i;-5) + r- d&2 • /,^(q) Cq i 1 - ; In topjis of the nondl-riensionai 3;_-eec' v.;\piable t and with the values of XWq) nrid X-^io) j"o^" -he adi£.TDatic case given by equations (25), thecc eq-'j.ationG tal:e ths foni ^ ^ "do ^ cT 1 - Kdf. + Lj-r or = 1 1 - (2R + I)t ^^'' ''■ T(l - T )' +1 ^p.2 CT — ( ^^ (1;6) L(i - T)'' ^■ = ; Porraal s.. lutions of these equations v/ere given by aha:DlT-in in the foria of two infinite scrios ) =-B0,, (T) - 2]^ E,,, i2t,(T) co3(nO + c^) ni=l ■> ikV I ) where the functions -J.^^Cr) and j^ (t ) are obtainod fron hyperseor.etric series and B, r^i/and e are arbitrary constants. ' A disadvantage cf the for::.al solution, as re^^arked by Te.T.ple and Yarv/ood, is that it is unsuitable for COiJPIDilTTIiiL NACA ACR No. L4C24 CONFIDENTIAL 27 nimerical computation because the hypergeometric functions involved are complicated and are not tabulated. Temple- and Yarv.'ood therefore looked for approximations that are of practical valwe in calculations of compressible flov;s. By ineans of a sliillful analysis, they found such apT)roxl- Tristion? and shov/ed fnat the simplest are of the type A m ^orns for \|/, m and -•li^i-^) ~ r-n(T):] m im 4(T) ~ [Kt)] 4(t) ~ loz lir) ^ (48) where r\{-x) and ^(t}, independent of the index m, are r\ = ^ (49) Significantly, fromi the point of viev/ of the analysis of the present paper, the functions and approxi- mated by e - t^y are none other than the functions defined on the right-hand sides of equations (35) and (24). ?he approximation of Temple and Yarv/ood then leads to the same velocity correction relation as v/as obtained bv r.eans of Clianlvgin's a-oproximation (equa- tion (41)). The velocity and pressure-coefficient correction formulas obtained by Temple and Yarwood are more involved than the explicit expressions (43) and (44) obtained by von l^.o.TrAn, Replacing ^m. thus 7-rlelds V^i/i v^i/^c in equation (41) by ^ - 4 ^ly-ll (50) conpid:^i'tial 28 COITFIDEIiTIAI NACA ^CR No. L4C24 vvhere v,2 O + : .1 The .solution of this cubic equation for (q/q]_^^ io / r- \ cos -J (it + o) JL\ = ,M-\ 3^1 - I T.^l :2 (51) vrtiere ,1/2/ COS a and < a ^ v^. The pressure ccefi'icient C,-, i\,,- is ~ ^ • j-'j-'l then calculated by equation (1-%). Eomo values of the pressure coefi^icient C,^ fr calculated with the aid of i- > ' - 1 equation i^l) are r;hown in fi,cure 4; a curye of (Cp^TT-,) /Crj o plotted against I/;^ is included in fig- ure 5. It is remarked that, with the use of equa- tion (39), the velocity correction formula. (50) ^'"lelds a llralting value VI = lo35. Approxination Based on Geometric Ilean of dL and dL V«'lthout pox'nr, into its deep significance in the present p<^per, it is of interest to introduce another function i-elated to L and L and to t>je general particul.'ir solutions. This xunctTon, which like L and L I'e duces to log q for t = 0, is defined by R(-^ - ' '•'•'- -■-~^-'■^ (T) = J (dL df)^-^^ (52) It is remarked that H(t) is clo^'cly related to a function K(t) employed by Temple and Garwood (refer- ence 7) in the determination of their approximation. In the next section, it vili be seen th:..t the function COiU'IDEMTIAL llACk ACR IIo. L4C24 COIJf'IDENTIAL 29 H(t) plays an inportant role in connection v/ith the Prandtl-Glauert appro:cirriation. Froin equations (26) and (27), and dL = ,i- dq = (1 - T)-^^ '^1 ^ dL = ^odq = 1 - (S^3 + 1)t dq (1 - T) ;+l 4 Then, (dL dL) fA lA ( .1 /p ^1/ 1 - (23 + 1)T i^/^ dq 1 - T ~! q (53) and, from equation (52), n(T) = loG q + li(T) (54) where r^ fT- h(T) .3 1 1 _ -.1/2 (23 + DtP^"^ 1 - T '}^ The function h(T) can be obtalnsd in a closed form for anv value of y (or 3) and is h(T) |(l-T)^/^.fl-fW2li;,.T)l^.(T,.T)l/^|^'^ -lor -= : '- = (55a) 2(1 - v'Tg)' ^ COI^IDIINTIAL )0 COiTI'IDET^TIAL NAGA ACR No. L4C24 w?iere t„ = 7- and where tiiis expression is valid. jp + 1 in the .^vibsonic range ^ t ^ t,,. V.ith t replaced M' by 2p + W becomes an d ^ M ^ 1, the expression for h(T) V2 !_,/" h(T) = -log 1-:=^ log 2 2v" ■^^l/2 Vt- 1 + v't^ 1 + \^;(i - i;'') + :r=: log 1/2 2yfr s 1 + ^/T (55b) It is observed that, for the supersonic region Tg ^ T ^ 1 or }\ > 1, K(t) as defined by equation (52) becomes a coriplex function; but, for present purposes, onl-^T the real function of the subsonic range is utilized. The function H(t) may be utilized to obtain a velocity cor'rection formula in the same manner as the functions L(t) and L(t). Thus, analogous to equa- tion (35) , (35), or (37) , (■4). J"(T ) (56) It is instructive to comnare equation (55) vith the approximation p-lven by equation (37). Equation (37) nay be written as ^1 J^(dL+dL) J|-(dL+dL) _7=r^ COIIFIDSNTIAL NACA AGR No. L4G24 CONFIDENTIAL 31 and eqiiatlon (56) Fiaj be written as /q Thus, the pover of the exponential is in one case the intecrral of the arithnetic mean ■ ' "" • . , — •— and. in the other case the intepral of the creoiretric mean (clL dL; Z'^. Table 1 shows values of the furictions f(T) + r(y) , , _J_L.-^i £,-1^ l,(r) in the case of air (y = 1.4, P = 2c5, and 13 = i/g) and firures 1(a) and 1(b) shov/ these functions plotted ajjainst t and II, respec- tively. Observe that these functions, and consequently the velocity correction forrulas (27) and (5G) , differ" only slightly in the subsonic range < M < 1. Fig- ure 5 exhib-its graphically a comparison of the velocity correction fornalas (37) and (56) for I! = 1. The limiting value of M (defined by equation (59)) is M =1 in the case of equation (56) as compared v/ith M ~ 1.15 in the rase of equation (37). Comparison of Results of Present Paper with Prandtl-Glauert Approximation The v;ell-knov,Ti Prandtl~GlauGrt approximation is based on the assumption of vanishingly smsl.l disturb- ances to the main stream. The F,t'andtl--&lauei't velocity correction form.uia m.ay be expressed, as Ti^ M.,^y-^' 1 /l ■ where q - q-. is vanishingly small . The left-hand side COITF'IDEFTIAL C0!JFIDEI:TTAL IIACA ACR Ko. L4C24 of this equation is actuallj the differential coeffi- ^(q/qi )e c3ent — ; — 7 — r— eval'iated at the main stream velccit- (i(q/qi)i q = q-, (or t = T-, ) . An ezact for:n of the Prandtl- Glauert approximation then Is ^C^/^iI \ •'"Jt=Tji_ \^1 - <-'h ) (55) The differential coefficient in equation (58) in nom' evaluated for the various approximations treated in the present paper. For the arithmetic-'nean approximation of the present caper given by equation (37) (y or [3 arbi- trary) , j cl(q/q lA di (q/qi)i T=T ■ (^ - ^d' 1 - '2(' + 1)ti (1 - r,y^^ it/ 1 * (i - .,^)(1 . 1^ -1 + 2 ^-^1 + 8 =^1 ■" TG -1 ^l?>8 llp^ + 4p + 1 M^B (69) COIIFIDEIITIAL ITACA ACE No. L4C24 CONFIDENTIAL >5 Per the Chaplygln or the Temple-Yarv;ood approxi- mation g-iven by equation (41) (y = 1.4 or p = 2.5), d(q/qi) T=T "'5 1 - 20 ^"1 = 1 + i Mi^ + li l\^ + ... (50) /■ / • For the von Karnan ap-:-)rcxlrri.ation given bv eaua- tion (42) (y = -1 or p -' -^), ^-(yqi)c = (1 " T) , 1/2 Ci - :^i o 'X/'^ ) (SI) For the georaetric-riean appro:-:irnation of the present paper given by equation (56) (y or p arbitrary). d (lAi), r=T' - T -il/2 1 - (^2p + i;,-^ tJ (1 - r,2 ) r/2 (62) Equation (62) Is indepfr..oent of -clie value of the adiabatic index y ^'-V\ incluc.='3 the von Karrian approxi- mationo Observe that ^jbe goori'istric. -':n''!ar approxiriP ti on yields the Pi sndtl-Glaii.ert i-eca],: '^xaotly, v4.erea£ the arithrnetlc-nean approxiratxon yialds blie Prandtl-Glauert COITFIDENTIAL 34 COIIFIDENTIAL NACA ACR No. L4G24 result insofar a.? terns iiiclusive of rf]_° are concerned. The Chaplyr;in or the T emple-Yarv/ood approxiiiiatlon con- ^ tains the' Prandtl-Glauert result only insofar as the M]_^ term is concerned. RESUI'E AIID CONCLijDII'G RLMARKS 1. Basic elementary solutions of the hodograph equa- tions have been employed to provide a basis for comparison, in the forra of velocity correction formulas ;, of corre- spondinp compressible and incompressible flews. 2. The velocity correction formiulas obtained, by Chaplyf;in, by von Karnan, and by Temple and Yarvfood have been ' minified by m.eans of these basic solutions and shown tc be essentially equivalent. 5, In the present paper tvjo types of spproxim.ations have been introduced by means of the basic elementary solutions, nam.ely, the "arithmetic-mean" type and the "geometric-mean** t^/pe. These approximations include those obtained by Chaplygin, by von Karrian, and by Temple and Yarwood. 4..- The approximations discussed in the present paper have been compared with the well-known results of Prandtl and Glauert. For this purpose, it has been em.phasized that tlie Prandtl-Glauert result is valid for vanishingly small disturbances and, in a strict sense, is the slope term in a Taylor expansion in a quantity v/hich m.easures the disturbance. It was found that tlie arithmetic-mean type yields the Prandtl-Glauert result to a higher order of approximation than the Ghaplyrin or the Temple -Yarwocd. type and that the geomietric-mean type contains the Prandtl-Glauert result exactly. The tv/o types of approxi- riations introduced in the present paper then appear to be preferable to the others as a basis fci'' extrapolation into the range of high stream iiach n\ii'.:bers and large disturb- ances to the m.ain stream. 5. The results of the present paper have been ob- tained without consideration of any particular boundary. The actual boundary problem of determ.inlng the flow past a prescribed body is of a high order of difficulty and involves in general all the' particular solutions of the hodograph equations. NACA ACR No. L4C24 CONFIDEl'ITIAL 55 6. The particular solutions di'^cussed in the present paper are well-behaved functions in both the subsonic and the supersonic refions. The hodograph equations give no reason, in general, to suppose that a discontinuity neces- sarily occurs in the solution when local sound speed is attained. Rather, it appears that the first brealrdcwn of the solution is associated with the vanishing of the Jacobian of the transfomation from the physical to the hodograph variables. Indeed, von Karnan has nade an equi- valent suggestion in that the appearance of infinite accel- erations In the flow solution is a condition for flow dis- continuities. Interesting speculations on this natter are suggested by the results of the present paper since the "limiting" curves discussed in the present paper are defined by a condition that is equivalent to the condition for infinite acceleration. The arithrie tic-mean type of approximation thus yields a limiting value of the local Mach number V ~ 1.15, and the geoinetric-riean type of approximation yields a limiting value of the local Mach ni^nber M = 1. The value M = 1 appears to be exact for vanlshingly small disturbances; that is, local r'ach number M = strear;: Mach number !i]_ = 1 (Prandtl-Glauert approximation) . However, for finite disturbances to the main flow due to the presence of a body in the fluid, infinite accelerations nay occur, for stream Ilach numbers less than unity, in regions v^here the local Ilach nijnber is greater than unity. In this regard, the arithiuetlc- mean t^.'pe of approximation, considered as an extension of the Prandtl-Glauert relation to finite disturbances, indi- cates the possibility of a mixed subsonic- and supersonic flow without discontinuities. It is inportant, however, to recognize that in general the limiting value of the local Mach number }l is a function of shape parameters and is a result of the blending of many particular solu- tions of the hodograph flow equations aeeording to the boundary conditions. Langley Memorial Aeronautical Laboratory, National Advisory Com-nittee for Aeronautics, Langley Field, Va. CCIIFIDENTTAL 36 coitt'^tdeittial naca acr tio . l4c24 repek^:jce3 1. Molenbroek, P.: Vher einige Bev/egunpen eines Gaser. rait Annalime elnes GeschT/vlndipkeitepotentialp. Arrhiv d. Math. u. Phyr , (2), vol,' 9, 1890, p. 157. 2. Chaplygln, S. ^i. : On Gas Jets, (7'f:zt in Ru^siari.) Scl. Ann., Mofcow Iraporial UniVc, Math.-Phys. Sec, vol. 21, 1904, pp. 1--121. 3. Dentchenko> B. r ^•I'^'lque^ proble'nes d'hydrodynamique biclTnonslenelle des flnldes compresslbles . Pub. ITOo 144, Pub. F^cl. et Tech. du i'llnistere do I'air (Paris), 19;-9. 4. vo'-i V&rrav.n, Tli. : Gor-iprascibllity ^;i;fff:ct3 in Aero- dvramic? r Jour. Aero. Scl., vol, 8, no. 0, July 1941, pp. o6r-c56. 5. Tsien, iirue-ilhen: ?v"c-Dinensional Subsonic Plow of ConprecFiblc Fluids:. Jour. Acre, cci., vol. 6, no, 10, Au:r. 1939, pp. 599-407 .. 6. Pinp;leb, Triedrich: Exalcte Lbw^ungen d,er Dif Terential- gleich'mfen einer adl abati schen Gasst".:'6raung . Z,fca.,noM», B5. 20, licit 4, Aug, 1Q40, pp. 185-198. 7. Templej.. G,, and Yai-v,'cod_, J,; The Approximate Solution of the riodograph Equacions for Compressible Flc-v. Rep. IJc, S.McTl. 5201, R,A.H=, June^l942. 8. Fers, Lipnan. and Gelbart, Abe; On a Cla?? o-^ Dif- fe7'entj.al EquLticns in Mechynics of Continua. ^uart^.i^ly Aopl^ T'L\th,. vol, Z, no. 2, Julv 1945, pp. 168-188 « 9. Eilbert, Davidj Grund^.l^!;''; einer allgemeiiLen Tlieorie der linearen Integra] giLoich-ung en, B. G, Teubner (Leipzig and Berlin), 19;:'4; p, 73. C0:!riD3?iTlAL NACA ACR No. L4C24 37 >- OS o tiC + to h 5 O + «0 « to ■f o*o>ot^r^c\llO^D lOOiHrH t-oc-ooDoir-cvjio 8O>C»c-tOtOr-)O>C0C3>O>O>O>00c000 0'»'<0(M'*(M«50»ON ^tOCMiHOOlCOt-lOlO ot)00o>a>a>o>i/}tot« ■<»"ION>HOO>00a3C--«) CD OD 0000 CO o m lo to f-i to o> t--«^ t~ ioc-o>'*iooQ •H iH O 3 lOC5 UJ & 0> 'I' 00 O) O iH rH CVJ lO 00 lO lO CM 00 «0 00 CO * O O CDOiHrHCVJIOODlOIOCM U5lO*IOCMf-IO>0pt^CV» C~C^C-t-t" t-t~C-t-C-C^<0'C<0<0 <0 1-1 lO to oo o lO lO OJ o oo o C»0JrHO>IOOrH00tO O00c~0>GOOOCO t-cocDrH'«>ior->ffl« 0>OlOO)C--i-lrH^OOO eOCM<7>lOK)O00r-tO'*'*CsJr-( cococ-t~c-r-t->t-t» OOCOC-IOWUJOt^lOlO O > lO CD ♦O o (J>(Or-(^'*lOlOCOO o> OQ o 00>00 Q QO ^ to o o oo o o t~«><0«0(OI0U]'OlOlO o O(O00>H>Hor-m« lOHM "•'CM * t- > ♦ 8t-OOON«>HO)t-l(5 0>0>>t0'4>01OC7>C0 0>0>0>(}>0>0>0>OOCD lO'*CM00^Ot~inr-lt~ S^^IO CM t-«5tOU5'*'<>"lOlOCMCM COOOCOCOOOOOOOCOCOCO (O <0 rH ■>»' CM CM CO CM OCOOrHlAlOCMi-l CMOCMC~IOC»t-IO O<-II£>OHiH mot'tot-'Dtoiocit- 0*CMt~CDt-lOC--0>lO «-i*C0rtO>CMt^ lOt-OOOrHCM^'iCtOt^ rHiHi-tCMCMCMCMCMCMCM iHrHIOO'VIOOPJOl 100lOCMOCM*0(H CMOCMO«0iHt-K)^ O>HCM'OCM'« OOOOOOrHi-lrH t-ioino>coo ^lAC-bOlOIO>HCM« «oe-aooi>HCM(0'HC3CMa»CMiocDooa>'4i« r-co(>i-ic4ioiof-c»r- CMCMCMWWlOlOlOlO* O I I I I I I I I I I I I I I I I I I I till • I I I I 00 •♦ CO 00 10 OWIOIOOODODO tO(0«CQlOOC-0 r :a ' - -- ' rHCM in lOO I I I iH ^ I I rHIOOCOOlOOCM^IO OOOOOrH<-lr-t»-l iocj'*^CMioQr-'*rH (OCMtniCIOCMaO^^CMIO rOOOlOOXD'fCMr 0> O CM 10 lOI ~ •-I CM CM CM CM I I VM Whi ^r v>4 >4/ > ■* CM 6 «) CM ) t- 0> O »HO)lO 10 "C •-' t~ «o •* CM O t- O "-I CM C- rH 0> 0> CM 0> C^ rH CO Oi 10 "H 0> CM "~ I I I I I I I I I I I I I I I I I I I I I I I CM m 0> rHO I I I to 00 I I »? ifloowto^comooto CMOiHOOCOrH'^C^rH oocMtommcjiocM OOOOOOOi-tr-l 00lOlOlOOrH«Ocy>(Oaj tO'fiomujc-oDcoooo -iiomio lOp-C7>O>HCM-*m<0C~ rHr-IOO'>l'lOO>0 rH'^tOmCMO*^ lOio^ioior-t^co I I I I I I I I I I I I I I I I I I I I r I O^COfHCMlCCMmCM OtJXOOtOrHWCMiH CMC-£~rHC~t-C^O>iH OOr-ltO'^'OC~(X>0 OOOOOOOOrH CDCM«)0>CMCMOlOlOC~ ^OOCMt^frHOtOi-tlO lOAlOCMCniOCMlOrHtO rHiHCMIOIOI-UJlOtOUJ '« in 0> lO lO <0 C~ CM O CM »-l CM 00 * r-l Ca lO iH CD (O CM t- to <» ^ O O t- to CM ^-^-oOcDc7>oo^^c\lm .-lrMr-lrt.HCMCMCMCMCM <0 ID O to <£> 10 ««■ m CO 01 to iH O •* lO CM rH to OOrH^lO-^IOIQO CMio«rio-ICMtO'*lO<0cnc»o>o>o CM ^ to 00 O CM in CD O O OOOOOOO- o o o o iH iH iH r-i CM to •«• m o lo o o o Q rt r-( rS M .-1 i-l M rH r^ rH 1-4 r-t CM CM tO ^ lO 6h z; Q I— I Cr. 25 o NACA ACR No. L4C24 CONFIDENTIAL 38 TABLE 2.- VALIIB3 OF (q/qj),., (q/ljlj. (l/Sllc (q/qi)i' ^^ FOR T» 1.4 AND PORfARIODS VALDB3 OP '-P.lll "I NMIONAL AOVISUKY CUMMII1EE FOR A£R0NAUI1CS If 0.8 0.3 0.4 0.6 0.S6 0,6 0.7 0.8 0,9 1,0 1.1 1.2 T 0.00794 0.01768 0.03101 0.04762 0.05705 0.06716 0.0S926 0.11S4S 0.13942 0,16667 0,19486 0.223*0 Ml « 0.2 (q/)) 0.63973 -0,69286 -1.48366 -1.89880 -2,31113 -9.11290 -3.82*09 •4,40996 -4,81466 -6.01804 -6,00869 =P.»1 (Bd. (18*)) 0,55794 -0.74111 -1.18984 -2.11685 -2,62611 -3.69298 -4.78921 -5,86302 -6.9474* -7.86968 -8,90*71 Ml > 0.4 (q/qi)« 0.90692 0.78614 1.00 1.23924 1,35640 1.47174 1.6*66* 1.91300 2.12040 1,31*40 ■ .•(M74 1,***** (q/qi)i Sq. (36) 0.6E061 0.76759 1,00 1.(1461 1.51438 1.40905 1.882*0 1.73619 1,8*764 1.9*106 8,07744 2 . 168*6 ■q. (S«) 0.62236 0.78967 1,00 I.IOSOI 1,30190 1.3BS11 1,63603 1.64401 1,71116 1.73987 1,71116 i.«44aa Bq. (37) 0.62143 0.76897 1,00 1.21124 1,30813 1.59863 i.iiaM 1.68897 1.7(7*8 1.86336 1,86642 1.66I** (q/qi)c iq/qjii Bq. (37) 0,97026 0.98253 1.00 1.0S312 1,03690 1.06236 1.06869 1.13264 1.18614 1,26092 1,92964 1.4t»40 "p.o (Bq. (18>)) 0.72811 0.40930 -0.46710 -0.71120 -0.96589 -1.4mM -l.*6262 -2,19169 -2,43494 -2,59481 -2.94929 =P.»1 (Iq. (IBb)) 0.76643 0.43714 -0.62420 -0,81188 .1.11299 •1.74192 -2.38866 -3.03429 -3.66206 -4.26938 •4.61760 Ml = o.e (q/qj). 0.40825 0.60936 0.80695 1.00 1.09454 1.18763 1,36906 1.64370 1,7110* l.>7064 2.02281 2.16694 (q/qiU ■q. (38) 0.42857 0.63202 0.82338 IwOO 1.08222 1.16017 1.30307 1.42871 1.537*9 1.63116 1.71061 1.77797 Bq. (3«) 0.43240 0.63706 0.B2781 1.00 1.07773 1.14909 1.27069 1.36092 1.41662 1.43631 1.41661 1.36110 Bq. (37) 0,43049 0.63464 0.82660 1.00 1.07998 1.16462 1.88*7* 1.39440 1.47688 1.53012 l.»*868 1.66639 (q/qi)c fq/qi), »q. (37) 0.94834 0.96032 0.97741 1.00 1.01548 1.02859 1.06393 1.10r707 1.16936 1.228*7 1.89*41 1.1931* «P,0 (Sq. (IS*)) 0.61468 0.69736 0.31838 s -0,16656 -0,53516 -0.66683 -0.94439 -1.17822 -1.34127 -1.4230T •1,41924 (Bq.''h4b)) 0. 87771 0.66366 0.56646 -0.19654 -0.40000 -0.82766 -1.26784 -1,70657 -2.13343 -2.99960 •2.91*03 Ml « 0.66 (q/qj). 0.37299 0.66672 0,73726 0.91363 1.00 1.08604 1.26081 1.41096 1.66926 1.70923 r 1.84810 1.9797* (q/qi)i Bq. (38) 0.39601 0.68399 0.76083 0.92404 1,00 1.07202 1.20408 1.32017 1.42086 1.50722 1.68087 1.64896 Bq. (36) 0.40122 0.69110 0.76B11 0.92788 1.00 1.06621 1,17906 1.26278 1.31454 1.33178 1.31436 1.26294 Bq. (37) 0.39861 0.58765 0,76446 0.92896 1.00 1.06911 1.19160 1,29114 1.36*67 1,41680 1.44199 1.4401* (q/qi)e fq/qiU Bq. (37) 0.93573 0.94766 0,96441 0.98669 1.00 1.01490 1.04978 1.09234 1.14393 1.20640 1.28222 1.97467 =P.o (Sq. (ie«)) 0.84111 0.65481 0,41560 0.14262 -0.U500 -0,41967 -0.66704 -0.86761 -1.00732 -1.07743 -1.07416 =p.«l (Sq. (IBb)) 0.91835 0.72686 0.47249 0.16T87 -0.17497 -0.6407* -0.91731 -1,29299 -1.6682* -2.006*0 •2.33068 CONFIDENTIAL NACA ACR No. L4C24 CONFIDENTIAL 39 TABLE 2.- Continued N»IIUN»L AOVISUHV CUMMITUE FOR AERONAUTICS M 0.4 0.6 0.65 0.7 0.75 O.B 0.86 0.9 1.0 1.1 1.2 1.3 T 0.03101 0.06716^ 0.07792 0.08926 0.10112 0.11348 0.12626 0.139421 0.16687 0.19485 0.22360 0.26262 1 ■l = 0.6 1 (q/qi)c 0.67947 1.00 1.07707 1.16277 1.22704 1.29981 1.37107 1.44093 1.57627 1.70324 1.82460 1.93939 («/qi)l Bq. (56) 0.70971 1.00 1.06548 I.ISSIB 1.17914 1.23146 1.28018 1.32689 1.40696 1.47438 1.63201 1.68014 Bq. (58) 0.72041 1.00 1.06612 1.10583 1.14869 1.18436 1.21246 1.23289 1.24908 1.23272 1.18451 1.10691 Eq. (37) 0.71804 1.00 1.06979 1.11447 1.16382 1.20767 1.24686 1.27841 1.32521 1.34816 1.34708 1.32263 (q/qilc (q/qi)i Eq. (37) 0.96025 1.00 1.01631 1.03437 1.06432 1.07630 1.10061 1.12713 1.18869 1.26339 1.35449 1.46642 (&1. (18«)) 0.48872 -0.12316 -0.24204 -0.38448 -0.45847 -0.68214 -0.63433 -0.76618 -0.81761 -0.81462 -0.74909 <:p,"i («a. llSb)) 0. 86492 -0.18786 -0.31929 •0.48310 -0.84774 -0.81238 -0.97599 -1.29437 -1.69762 -1.88099 -2.14181 ■l = 0.6S (q/qi)c 0.63084 0.92845 1.00 1.07029 1.13924 1.20681 1.27296 1.33766 1.46266 1.68138 1.69406 1.80060 (q/qi)i Bq. (36) 0.66734 0.94030 1.00 1.08*16 1.10878 1.16797 1.20S77 1.24830 1.32204 1.38638 1.44067 1.46682 Bq. (se) 0.68212 0.94688 1.00 1.04708 1.08766 1.12142 1.14802 1.16723 1.18271 1.16723 1.12168 1.04808 «q. (57) 0.67469 0.94356 1.00 1.06189 1.09816 1.13963 1.17666 1.20612 1.26044 1.27209 1.27110 1.24789 (il/qi)e (q/qi)i Eq. (37) 0.93601 0.98396 1.00 1.01778 1.03741 1.08904 1.08286 1.10908 1.16963 1.24314 1.33274 1.44292 =P,o (Sq. (IB.)) 0.84479 0.10969 -0.10884 -0.20696 -0.29853 •0.38192 -0.48473 -0.66580 -0.81821 -0.61670 -0.66723 Cp.Ml (Eq. lien,)) 0.64074 0. 14002 -0.14333 -0.28862 -0.4347* -0.88086 -0.72671 -1.00B82 -1.27767 -1.52913 •1.76024 Mj = 0.7 (q/qi)c 0.68942 0.86747 0.954SS 1.00 1.06442 1.12786 1.1893* 1.24960 1.38680 1.47782 1.68280 1.6S238 (q/qi)i Bq. (56) 0.65186 0.89032 0.94683 1.00 1.04983 1.09641 1.13978 1.18008 1.26178 1.3126* 1.56398 1.40681 Eq. (3<) 0.66146 0.90429 0.95604 1.00 1.03875 1.07099 1.09*40 1.11475 1.12953 1.11476 1.07114 1.00096 Bq. (37) 0.64169 0.89728 0.96094 1.00 1.04429 1.08362 1.11788 1.14696 1.18909 1.20969 1.20874 1.18666 (q/qi)c (q/qiii ai. (37) 0.91869 0.96678 0.98263 1.00 1.01928 1.04088 1.0*394 1.08967 1.14920 1.22140 1.30946 1.41772 =P,o (Bq. (IB.)) 0.88836 0.19489 0.09571 -0.09064 -0.17423 -0.2496* -0.31649 -0.41394 -0.46338 -0.46108 -0.40816 Cp.Ml (Eq. (IBb)) 0.70580 0.26616 0.12904 -0.13082 •0.26284 -0.39397 -0.62440 •0.77907 -1.02140 -1.24781 -1.46689 Hi = 0.75 1 (q/qi)c 0.55574 0.81497 0.87778 0.93947 1.00 1.06931 1.11737 1.17416 1.28380 1.38810 1.48701 1.68064 (q/qili Bq. (35) 0.80187 0.6480^ 0.90191 0.98283 1.00 1.04436 1.00868 1.12404 1 . 19238 1.28038 1.2992* 1.34006 Bq. (36) 0.62718 0.87066 0.91942 0.96868 1.00 1.03106 1.05660 1.07317 1.08740 1.07317 1.03120 0.9*562 Bq. (37) 0.61439 0. 86 924 0.91062 0.98789 1.00 1.037*8 1.07047 1.09831 1.13688 1.16B40 1.15749 1.13636 (q/qi)e (q/qj), Eq. (37) 0.90 IBB 0.94846 0.96394 0.98108 1.00 1.02084 1.04381 1.08906 1.12748 1.19829 1.28468 1.39088 (Bq. (18.)) 0.62252 0.26171 0.17077 0.08302 -0.07678 -0.14591 -0.20628 -0.29669 -0.34189 -0.33978 j-0. 29131 •^P.Xl (Bq. (IBb)) 0.78361 0.35197 0.23700 0.11937 -0.12006 -0.23997 -0.35893 -0.69124 -0.81227 -1.01872 i-1. 20883 1 Ml = 0.6 j (q/qi)c 0.52274 0.76934 0.82864 0.88687 0.94402 1.00 1.05482 1.10842 1.21193 1.31038 1.40376 , 1.49204 (q/qi)i Bq. (36) 0.67651 0.81205 0.86369 0.91207 0.95753 1.00 1.03956 1.07629 1.14171 1.19726 1.24406 1 1.28312 Eq. (36) 0.60627 0.84456 0.89172 0.93371 0.96990 1.00 1.02372 1.04065 1.06467 1.04086 1.00014 0.93460 Eq. (37) 0.59209 0.82805 0.87766 0.92283 0.96370 1.00 1.03162 1.05843 1.09734 1.11633 1.11646 1.09810 (q/qii= (q/qi)i Kq. (37) 0.88287 0.92910 0.94425 0.96103 0.97968 1.00 1.02249 1.04723 1.10443 1.17583 1.25846 1.36247 Cp.o (Sq. (18.)) 0.64943 0.31433 0.22989 0.14838 0.07128 -0.06424 -0.12027 -0.20416 -0.24*19 -0.24423 -0.19»e4 =P.«1 (Bq. (186)) 0.81620 0.43547 0.52940 0.22083 0.11074 -0.11068 -0.22036 -0.43464 -0.63867 -0.62904 -1.00411 CONFIDENTIAL NACA ACR No. L4C24 CONFIDENTIAL 40 TABLE 2 - Concluded NAIIONAL AOVISURY CUMMITltt FOR AtRONAUTICS M 0.4 0.6 0.825 0.85 0.876 0.9 0.926 0.96 1.0 1.1 1.2 1.3 -^ 0.03101 0.06716 0.11982 0.12626 0.13279 0.13942 0.14612 0.15563 0.16667 0.19485 0.22360 0.26262 Ml = 0.B25 «./qi). 0.60878 0.74871 1.00 1.02665 1.05276 1.07670 1.10434 1.13972 1.17942 1.27524 1.36610 1.45203 (q/<.i)i Bq. (36) 0.66494 0.79602 1.00 1.01905 1.03738 1.06606 1.07206 1.09476 1.11917 1.17363 1.21960 1.25761 K)) 0,67936 0.37286^0.08508 0.02660 -0.02465 -0.04716 -0.07504 -0.10139 -0.13986 -0.13602 -0.09690 Cp.Ml (Bq. (18b)) 0.88664 0.54277 0.09960 0.04971 -0.04943 -0.09846 -0.16642 -0.24286 -0.42701 -0.59898 -0.75712 Ml = 0.9 (q/qi)c 0.47160 0.69409 0.92705 0.96164 0.97596 1.00 1.02376 1.06657 1.09338 1.18220 1.26644 1.34609 (q/qiU Bq. (35) 0.6S546 0.76449 0.94783 0.96687 0.96326 1.00 1.01610 1.03764 1.06079 1.11240 1.18B88 1.19218 Eq. (36) 0.68439 0.81121 0.97291 0.98366 0.99257 1.00 1.00679 1.01113 1.01327 1.00000 0.96089 0.69792 Eq. (37) 0.65939 0.78233 0.96029 0.97466 0.98788 1.00 1.01091 1.02429 1.03676 1.05470 1.06367 1.03463 (q/qi)c (q/qi)i Eq. (37) 0.84306 0.88721 0.96539 0.97636 0.90792 1.00 1.01271 1.03151 1.06461 1.12069 1.20170 1.30104 Cp,o (Bq. (18.)) 0.68708 0.38796 0.07784 0.06004 0.02409 -0.02194 -0.04917 -0.07467 -0.11239 -0.11064 -0.07046 «P,>ll (Sq. (18b)) 0.90787 0.67490 0.14460 0.09617 0.04799 -0.04762 -0.11365 -0.18788 -0.36660 •0.53362 -0.66711 Ml = 0.925 1 (q/qi)c 0.46066 0.67798 0.90562 0.92965 0.96331 0.97678 1.00 1.03206 1.06800 1.16476 1.23704 1.31486 (q/qi)i Eq. (36) 0.62697 0.74263 0.93279 0.96057 0.96768 0.90415 1.00 1.02119 1.04397 1.09476 1.13766 1.17329 Eq. (36) 0.58103 0.80664 0.96729 0.97787 0.98686 0.99423 1.00 1.00532 1.00743 0.99424 0.95634 0.89276 Eq. (37) 0.65336 0.77387 0.94991 0.96413 0.97721 0.98919 1.00 1.01323 1.02556 1.04332 1.04249 1.02347 (q/qi)c (q/qi)i Bq. (37) 0.83249 0.87609 0.95327 0.96413 0.97664 0.96745 1.00 1 1.01857 1.04138 1.10681 1.18662 1.26471 Cp,o (Eq. (18.)) 0.69380 0.40113 0.09767 0.07046 0.04506 0.02160 -0.02664 -0.05177 -0.08862 -0.08679 -0.04749 1 Cp.Mi (Eq. (18b)) 0.92963 0.60574 0.18708 0.13991 0.09301 0.04633 -0.06426 -0.13642 -0.31038 -0.47285 -0.62P22 CON FIDEN TIAL NACA ACR No. L4C24 COKPIDEIJTIAL 41 TABLE 3.- VALUES OF F(t) FOR SEVERAL VALUES OF y F(T) = 1 - (2p -f 1)T (1 - T) 2 13+1 1 - LI' 1 + -2- 2P/ _ F Y = 1.4 Y = 1 Y = 2 Y = c M ({3 = 2.5) (p-^o°) (P = 1) (P— ^0) Adlabatic Isothernal (1) Hydraulic analogy Limiting inconpressible (2) 1.00 1.00 1.00 1.00 .2 .99901 .99918 .99879 .9600 .4 .98328 .98575 .97977 • .8400 .6 .90606 .91733 .89113 .6400 .65 .86634 .88113 .84726 .5775 .70 .81594 .83248 .79050 .5100 .75 .74558 .76763 .71822 .4375 .80 .65738 .63273 .62728 .3600 .85 .54489 .57153 .51421 .2775 .90 .40258 .42710 .37504 .1900 .95 .22355 .24041 .20534 .0975 1.00 1.05 -.27752 -.30870 -.24667 -.1025 1.10 -.62069 -.70423 -.54102 -.2100 1.20 -1.55960 -1.85711 -1.30158 -.4400 1.30 -2.95915 -3.73944 -2.34862 -.6900 1.50 -8.01227 -11.8597 -5.64466 -1.2500 2.00- -56.6884 -163.79 -26.9981 -3.0000 Y = 1, P :^ (1 - Y = ", F = 1 - M' 2 m2 COJIFIDENTIAL NACA ACR No. L4C24 Fig. la o ■ss. o f;i-: f'-M ::■' r -\ ::'i' ^-v' ytt i^: ^-f jLirc tt-C Xiit .L*: . -~t\ TTT rny TT- TT-T ir^r ■■V" t:-;| - J ' ■ « t- 1 ^ C ) "^ 2 -' : 1 ■ , 1 ■■i^ ■ ^ s . \ o T • ^ "(t "V ^t -Xi Cji QC > — 1 ^ i > VJ iT ■ . .1 M ■•^ "* \ ■s ^ v • •1 1,1 \ V, ' ■ \-\. V § •< ) 'S ^ c; 1 \ ^' s-^ c: S ^ >^ X ^ ~ \ s ^ r ^ u 1' ■ ;^ \ >»., \ ^ j- . S, <. X \ <5 "> c^ s. '.:■-■' ^ ; V, ^ \, \ > .'^ . .ii[ ^ 1 1 C> — Y - ' ^S > r \ \ <■ \i f >> > ■f r M t M "■ -ta ^ -^ s • ■ NACA ACR No. L4C24 Fig. lb a I' V -; t ■"■■■ j ;. ;; -'';'tc ^=3 < Pr n 1 "^ o J k, r, \ :\ '^ • u ; a x. ■ o \ s 0^ \ n ^ ^ \ 'H^ \ s 1- \ r \ \ \ c3 - k. o \ \ y ^ "~ . -'l \ \ V \ ^ - \ \ \ ^ \, \, i C i '■■".r !,l \, \\ \ •n 't S s. ,:S \ \ \ . \ -- \ \ \ \ \ ' v\ \ '•f{ \ k \ y\ • ^i : t'- \ \ Ov '-\- — \ \\ \ n--; -\ \\ tv ■'\ ^t-'I- \ \\ < '-il ''; ^ \^ 'i -_- V i \ Ifl 4 > -^ 1 -^ii u t — 1. '■'._u % ~- ■ "14 i' -;- \ ^ ^•"-: i- ■j:". -_• ]: v::^ 4- '■ 1 j'll K--'_ ^A ■"-; . --I * t '^ -r \ i? -':- ?:y In ■T> M fv S» ^ :iii ' ' <3 i 1 ;3 <0 -,-; ■ . 1 ;-;t ^ ,^ -: -j- • r^ "o "■'^ — . , 1 :S -< ^ ^- r \ ^1 "' 1 • -) r ^ ^ i J; 1 • ^ ^ ^" -\ ; / '_t NACA ACR No. L4C24 Figs. 2a. b rr '^ i ! 'i ' '. 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