Two-Dimensional Irrotational Mixed Subsonic and Supersonic Flow of a Compressible Fluid and the Upper Critical Mach Number Citation Kuo, Yung-Huai (1945) Two-Dimensional Irrotational Mixed Subsonic and Supersonic Flow of a Compressible Fluid and the Upper Critical Mach Number. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/D9ME-DD61. https://resolver.caltech.edu/CaltechTHESIS:08172010-111715951 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Aeronautics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): von-Kármán, Theodore Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 December 1944 Record Number: CaltechTHESIS:08172010-111715951 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:08172010-111715951 DOI: 10.7907/D9ME-DD61 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 5990 Collection: CaltechTHESIS Deposited By: INVALID USER Deposited On: 18 Aug 2010 15:50 Last Modified: 10 Nov 2023 19:50 Thesis Files Preview PDF - Final Version See Usage Policy. 46MB Repository Staff Only: item control page

THO<DIESNGTORAL IRROTATIONAL MIXED SUBSONIC AW SU2SkeOGnIC PLOY OF A Comer ess ibis FLUID AND TI UPPoP UPRIPICAL MACH NUMBER Thesis bv Yung-Huai Kuo In Partial Fulfillment of the Feqtiirements for the Degree of Doctor of Philosophy Salifornia Institute of Technology Pasadena, valifornia December, 1944 Acknowledgements The author wishes to take this opportunity to express his sincere appreciation and gratitude to Ur. Theodore von Karman for acqusinting the author with the subject and for his continued encouragement fie also wishes to express his indebtedness to Dr. Ulark 5. willikan for tha interest shown in the problem. end he would like particularly to thank Dr. Hsue-Shen Tsien for his stimuleting discussions since it is only his genuine desire for perfection and great ability of discernment that made this work possible. Thenks ere also due to Mrs. Richard Bs Canright, Miss Virginia Prettyman and irs. Lewis =. Kimball for their efficient help in the numerical calcuFations; and to Miss Alberta Pompeyen for her care in the typing of the manu- script. wi- Tabis of Contents introduction and Summery Notations Part I la Ce yw de 4. Part Ti 12. 13. ah o 150 16. L7o Part IV 18. 19. 20. Part V Zl. eo Zo 24 o Differenticl Tqustions of Compressible Flow and Propertios of thair Particular Solutions Fquations of Motion Transformation of the Differentinl Equations Particular Solutions of the Differsntial Equations Properties of the Hypargcomatric Functions of Large Order Construction of the Solutions for Compressible Flow around a Bedy Chapivrin's #rocedure Ths Functions of Incompressible Flos Conformal Mapping of Incompressible Flow on the Hodogragh Plane construction of tha Solution sxbout tre Origin of Hodogreph Plane ‘nalytic Jontinuation of the: Solution--iranch point of Order One Analytic Continuation. of the Solution--Logarithmic Singulsrity Transition to Physical Plene Improverent of the Convergence of the Solution by the Asymptotic Yroperties of Hypergeometric Functions. Goneral voncepts ksymptotic Solutions of the Hypergeometric souetions Asymptotic Tepresantation of FC a,b, ; cy; et ) end FC tp byrpj3 cy; T) Transformation of W Cw3@) <-- broanch-Point of order I Transformation of W(w;4) «= Logerithmic Singularity The Coordinate Functions x(4,9) and y(4.9) Uriterion for the Upper Critical Mach Number Limiting line and the Sreakdewn of Ieentropic Flow The Condition for the Limiting Lins The Approximete Determination of the Upper critical Mach Number Applicetion: Dlliptic Cylinder eraliminary Discussions The Functions Zo(w), Wo(w) and A,(w) The Expansions of W,(w) end A.(w) The Stream Punction (9,9) Ths Coordinate Functions «(9,9 ) and (4,3 ) Conclusion 39 43 48 5d 55 57 E2 66 72 74 81 102 _RESTRICTSD Introduction and Summary vinen @ flow of non-viscous incompressible fluid is irrotational, 4t is well-known that the problem can be reduced to either the problem of Dirichlet or that of Newmann, and thet there exists a unlau. solution for any given boundary conditions, When the fluid is non-viscous but compressible, the variation of density mkes the mthematical problem very difficult end complex, In this as ,a pure potential flow throughout the region is not always possible for a given body; this depends very much upon the condition at infinity. If a certain speed of the flow at infinity is reached, regions within the ficld of flow will be created in which the irrotational flow does not exist due to the appearance of "limiting lines", : Such regions were picturesquely designated as "forbidden regions" by Th. von Kérman (Ref. 1) and they appear when the local speed of the Plow considorably exceeds the local speed of sound, It hss bean shown that the occurrence of Limitine tines is directly connected with the breakdown of irrotational flow «nd with the resultant increese in drag of the body due to shock waves, In other words, if there is a limiting Line in the field of flow, the isentropic irrotational flow mst break down. However, the irrotational flow may break down before the appearance of limiting line due to the instability of the velocity field. On the other hand, shock waves: can only occur in supersonic flow. Therefore, there is no danger of breakdown of isentropic flow if the whole field of flow is subsonic. Consequently, the Mach number corresponding to the first appearance of local speed equal to that of sound can be designated as the “lower RESTRICTED RESTRICTED Z. critical Mach number"; and the Mach number corresponding to the first appearance of limiting lines can be designated as tho “upper critical Mach number", The actual eritical Mach number for a piven body will be influenced by the boundary layor and hence the Reynolds! number, However, it must lie between these two Limiting critical wnlues. (Ref. 2) Thus knowledge of these critical speeds of the flow are essential for the design of efficient aerodynamic bodies, To determine the critical Mach numbers, one has to solve the general problem of flow of a compressible fluid about a piven body. The often used methods treating such a problem aro Janzen-layloirh's method of successive approximations and Ulauert-Prandtl 's mothod of sm2ll perturbation. The latter method has been extended recently by both Hantzsche and Wendt (Ref. 3) and C. Kaplan (Ref. 4). Indeed, both methods yield valuable informatim regarding the offects of oom- pressibility, and are useful for many practical design probloms, particularly the determination of the lower critical Mach number of a given body. But so far as the general problem of limiting line and upper critical number is concerned, none seems to be adequate, due to the doubtful convergence of such successive approximations at the re= quired high Mach numbers, An entirely different approach was first made by Molonbraek (Ref. 5) and Chaplygin (Ref. 6) by introducing the velocity components instead of the usual space coordinates as independent variables, The advantage of the method is that, instead of a non-linear differential equation as is the case in the physical plane, it leads. to a linear RESTRICTED REST aI CTED Bo one in the velocity or hodograph plane, The particular solutims of this linear equation are found to be products of trigonometric functions of the angle of inclination of velocity vector and hyper= geonetric functi ms of the mapnitude of the velocity vector, It is then possible to construct a general solution from the particular solutions of the di°ferential equation. The difficulty, however, is that the character of the field in the physical plane to which the solution in the hodogravh plane corresponds can ot be determined be= forehand. This difficulty prevents the exact formulation of the boundary value problem in the hodograph plane. Chaplygin has overcome this handicay by first choosing a “suitable solution" in the hodograph plane and then proceeding to find the corresponding flow in ‘the physical. plane, The “suitable solution” is one which,in the limiting case of zero Mach number at infinity, becomes identical with the incompressible flow over @ body similar to the body concerned, This will ensure the satisfaction of the proper boundary conditions in the physical plane. Furthermore, such a solution would be exact ‘both for the subsonic and for the supersonic regions, as no approximation is introduced, There~ fore, 1t is particularly suitable for the problem of determining the upper critical Mach number for a given body,as limiting lines occur ort: in mixed subsonic and supersonic flows. This method is followed in the present Revort, except for the introduction of th. tra:sPormed potential function %, for easy calculation of the space coordinates. For the flow around a body, Chaplygin's procedure will Lead to a solution in the form of an infinite series, each term of which is a product of a trigonometric function and a hypergeometric function. RESTHRI ‘TED eal RESTRICTED 4. fo put the method on a firm foundation, it is necessary to establish the convergence of the infinite series, Chaplygin himself has done this for the subsonic regim., Thus, only the extension to include the supersonic region remains to be completed. In Part I of the Re= port, the general properties of hypergeometric functions of large order are investigated in preparation for the proof of the convergence given in Part Il. The essential point in these parts is to establish the upper and lower bounds for the hypergeometric functions so that the sum of the infinite series can be discussed. It is appropriate to mention here that for the proper representation of the general solu= tion in the hodograph plane, both fundamental solutions of the hyper= geometric differential equation are required, This fact seems to hrve ss- capedthe notice of all the previous investigators in this field, who use only the first solution and forget about the second one. Some of the investigators, as a result, even conclude that Chaplygin's procedure is unsuitable for the problem of the flow around a body. (Ref. 7) Tie general solution constructed by the Chaplygin method is really an existence theorem, The extremely slow convergance of the series makes nunerical calculation very difficult, if not impossible, This, in fact, constitutes the main difficulty of the method. In Part ITI of the present. Teport, this difficulty is overcomes by using the asymptotic properties of the hypergeometric functions. The result is the separation of the solution in the hedograph plane into two parts. One part is of closed form and consists of the same solution es for. ine RESTRICTED RUSTRICTE De compressible flow but with a velocity distortion, or velocity correc# tion, The other partis an infinite series which converges rapidly everywhere except in a smll region om both sides of a critical circle with a radius corresponding to gze in the hodograph plane. In practice, by using only a few terms of 4.5 infinite series, one can limit this zone of slow convergence to such a small interval that it is of no consequence. Thus the Chaplyzin procedure is improved to a point where actual numerical calculations can be made without diffie culty. As a result of this part of tho stud: it becomes clans thet ba ths mere substitution of a different speed scale, or valocity distortion , in the solution for a. incompressible fluid, one cannot obtein an accurate enough solution for the compressible flow. For if this were the case, then the second part of the solution, the rapidly convergent series given by the present method, would be negligible. But the value of the second part of the solution is not small compared with that of the first part for & speed nes. tick of sourd, In other words, the usual so-called hodopraph method (nef, 8) cannot in generel yield satisfactory results, for mixed subsonic and supersonic flow. On the other hand, the present method does show that the second part of the solution SS Zero, if the isentropic ex- ponent is equal to -1, This means that for this particular case, a simple speed distortion is sufficient. This is, of course, in accord= ence with the previous investigation of von Karman (Ref. 1) and Tsien (Ref. 9). RESTRICTED RUSTRISTED 6. Furthermore, the present method also shows that the rules of speed distortion for the first part of the solution can be used only for subsonic flow, snd thet there is a singulerity nt the local sonic speed. For regions of supersonic flow, the first cart of the solution in= volves both the incompressible stream function and the incompressible potential function, Thus even without considering the second part of the solutim, there is no possibility of making the compressible stream Lincs coincide with those for incompressible flow in the hodograph plane by a simple stretching of the speed scale. The mathematical basis of this fact is the change in character of the ‘differential, equation from elliptic to hyperbolic in the transition wtb sonic to supersonic flow. For the supersonic refions, one cannot use a real transformation of the velocity varisole ‘to convert the aifferential equation of flow to the Laplace equation, and thus make a@ Simple connection between the compressible and - incompress=- ible flows. This is one of the difficulties of the previously pro= posed hodograph method. In fact, writers using this method must generally limit ticles calculstion to cubsonis spogds (Tefs 9, 10). Now this limit is removed «nd the whole fisld of wlxed subsor.ie and supersonic flows can be trested eat once with BASS o Yor the purely subsonic flow, the second mrt of the solution is small compared with the first part and may be neglected. In other words, for this case, a simple speed distortion from the solution of incomoressible flow is sufficient to give accurate enough results, However, the subject of the "best" velocity distortion rule in subsonic regions has been the subject of men discussions (Ref.l1 and 8). The present RESTRICTED RESTRICTED 7. analysis also settles this question once and for all. This is due to the fact that the present velocity dist :*tion rie is obtained from the asymptotic properties of the hypergeometric functims, and that such properties are definite and unique. Therefore, the resultant ve- locity distortion sule is not the result of uncertain speculation. Furthermore, it is also the "best" rule, because the analysis implies that this rule will make the second nart of the solution, or the correction terms, the smallest. For the vurely supersonic flow, the second wrt of the solution is again small compared with the first part and may be neglected, In fact, the solution cam then be reduced to that of the simple wave equation with the inclination of the velocity vector and the distorted velocity as independent variables, This is, of course, the comter= part of the fact that by a sirple distortion in velocity, the differ ential equation for subsonic flows can be reduced to the Laplace equation. The usefulness of this new result for purely supersonic flows has yet to be exploited. Once the general problem of mixed subsonic and supersonic flow around a body is solved, the determination of the upper critical Mach number or the Mach number for the first appearance of the limjt= jing lines is a simple matter. This problem is discussed in Part IV of the Report. A simple method is developed, based on the properties of ube limiting line as given by von Karman (Ref. 1), Ringleb (Ref. 11), Tollmien (Ref. 12) and Tsien (Ref. 2) RESTRICTED RESTRICTED 8. To test the practicability of the method developed; two numerical examples are worked out in detail. However, in order to reduce the anount of computational work and in view of the limited time available, a slightly different procedure is actually used. This procedure is only approximate. but is believed to be sufficiently accurate in the supersonic region to give a satisfactory description of the most interesting feature of such flows. The examples chosen are derived from the incompressible solution of an elliptic eylinder of thickness ratio 0.6. The free stream Mach numbers of the compressible flow are 0.6 and 0.7 for these two examples. The first casé gives a smooth flow over an "elliptic™ cylinder of thickmess ratio 0.42. The maxi= mum local Mach number is approximately 1,25. Thus a considerable supersonic region exists, The second case gives a flow with limiting line, Finally, it must be said that due to the limitation of time, only the case of. flow without circulatiom is investigated in detail. The explicit formulae for numerical calculation are given for two cases; a) Flow aroumd a body such as an ollipse; b) periodic flow pattern such as that over a wavy surface, However, it is believed that more general cases can be studied by a slight extension of the present results and using the same method of approach, RESTRICTED RESTRICTED 96 Notations Ths symbols used in this report are clessified according to the following groups: he Physical quantities xaV u,v g =x © DT fo WM eo QO : Cartesian coordinates : the velocity components ; the absolute value of the velocity vector ; the inclination of the velocity vector with x-axis 3 density of the fluid ; density of the fluid et q = 0. ; pressure within the fluid corresponding to ~, : pressure at a = 0 > retio of the specific hents : the local speed of sound : the speed of sound at ¢ = 0 : the value of q at infinity, sssuming parallel to the x-axis. With subscript, howevor, it may be a function of t » B. Hydrodynamic functions in the physical plane. Zz xr ey, Wie) = Poway) + iW(y): complex potentiel for incompressible flow g, ¥, ; 9 ¥ in Ze : velocity potentinl for incompressible flow stream function for incompressible flow : velocity potential for compressible flow : stream function for compressible flow. Mt C. Hydrodynamic functions in the hodograph plane. Weu=-47V. We(w) = Polusv) +i Yoav): complex potential for incompressible flow in wo De POSTPICST ID 10. wb Sa DD Quy); vslocity poterntirl for incompressible flow. Vu): Stronm function for incompressible flow Aglw> = Ew- Ww) = Xow) - ev): transformed complex potentisl function Xobuiv) = uxt ve ~ Pocy) j x= dhe, y= Boe, transformed potential function Wiw;2): the complex potentirl function for conoressible flow, Wiuivy= Im { Wiwjt)) > stream function for compressible flow. \\w;9: transformed complex potentiel function for comoressible flow. Xiu) = Ux+ vy - Pry)= RY {awn} treusformed votentiel function for compressible flow @,(av) ae Qluiv) = 2B. f Wig 8) = Wigayt yf 49) : the velocity distorsion. yy” (4.9) stands for the a Wi4:9) represents the contribution by transformed infinite series, where the superscript J may n " t either mean L : the inner or "0" tha outer solution. In the case of coordinates, tho notation is exactly the same. (A) Gio= ZeaB, + P2240 $(%) Tt) Gy te) = Ftc) AB + BAF &) " fied 7%) ~, (0) yu >. ~~ (ot) Go) = JF, was. Parameters and variables. y : positive rstional nunbers m,n: positive integers RESTRICTED RSTRICTED 11. A : it denotes 1 or 2 when used as superscript with a bracket or of = yt ° p : it denotes the dependence on B when used as’ subscript —= lL s or BR \ = aiaey> |__| . the ratio of the distorted speed to (itay Jagr, Ti that at infinity. If, 28 ce 7]: With supereor subscript they denote some functions of & tt as i} or stard for the two families of the characteristic para» meters 9+Wtr) , 9- Wile) of the partial differential equations for Wig) or Xb) ° J :. complex variable or {(t) a function of t » M=:the Mach number at infinity. C 2 -+tt. T= 36 Ce é : geometrical parameter of the body BB: Laplacian or difference between the exact and approximate values of a function or a constant, Hyper geometric functions QsbsCy = parameters of the hypergeometric functions. In parti- sc cular, Bye dD are defined by (3-5) 9 yoy FO= Fla bjejet). first integral of the hypergeometric equation associated with the stream function, RESTRICTED RESTRICTED Fie) =F (Citar ty s bye cy j atv 5) -V v _ rt Ta Tlby) Ye They L Fe = (2qe2)’ Tle Tecy) b Thtayrey)Pisbrcy) 5, Tiare) Hv) integral of the same equation . Ge) = q” F, ) Ft, = F lie ay, itby jt4eys T) {rd Zt) = F,\) / $0) (y) . Fy, = F,,,) /3,,) _*K Ft) = Fi) + t Fy tr) Ree ld es| a ee + second If any function or a constant is associrted with !98)» it will be ~ marked on top by a symbol ~ , such as F, (er) ° Ronan omy POU PPT egy Part I Differentiel oquations of sompressible Flow and Properties of Their Particular Solutions 1. Hauations of Motion It is proposed to study the irrotational steady motion of on inviscid nonecondusting compressible fluid in an infinitely extended domain containing a cylindrical body with its exis perpendiculer to the constant velocity at infinity. The flow is then two-dimensional, Let x and y bo the Vartesion coordinates erd u and v be the velocity components parallel to the x-and the yenxis. The dvnemical equations governing such a motion, in the absence of body force, are Pu = + Pv oy 3x ( ) x< il uw ov vy ou ~ OF ’ (1.2) Here p is the pressure and P the density of the fluid, both being continuous functions of x and y. In addition, the following equation of continuity must be satisfied; 2 oy Furthermore, since the velocity is constant at infinity, the fiow is Hwlg4) + szlYv) =o. (1.8) Lrrotational there. Then, according to Thomson's theorem, the flow will remain irrotationel, i.e@., Dy ri ; ) if the pressure is a function of the density alone. In the case of flow of an inviscid non-conducting gas, the thermodynamic change of states of the gas is adiabatic. If the flow is assumed to be continuous, excluding shock waves, then the relation between p and P must be that of an isentropic process: = Cons ant r p= fant f (1.5) where Y is the ratio of the specific heats. As in the case of incompressible flow, there are more equations than the number of the variables, However, by virtue of Eqs. (1.4) and (1.5), the dynamical equations (1.1) and (1.2) reduce to a single differential eavation and can be integrated easily to give a relation between the pressure and the magnitude 4 of the velocity, namely XY = be { 1 4, a , with eu + we (1.6) Here Pp and c, are respectively the pressure and the speed of sound at the stagnation point 4=°. and c 21% . One can obtain a similar relation between ¢ and 4 by means of Eq. (1.5): oe PR, oe R, {i- Et \ my (1.7) where {, denotes the value of FP at Zz Oe ‘After integrating the aynanical equations the velocities u and v can be determined from the kinematic conditions. By eliminating ° from Eq. (1.3) one has 2 SV u oui ow 1s oy i. -~¥) Dy = 0 1.8 where ¢ = tbe and thus can be caloulated in terms of the speed by Eqs. (1.6) and (1.7). Jt is of interest to note that the equation of continuity (1.8) now, umlike the case of incompressible flow, be= comes dependent on the dynamical equations and consequently is non- linear, This change in the character of the fundamental equation RESTRICTED THOTT VATS TTS Par ian en ne Po 15. makes the direct soluticn of the vwroblem in space coordinates very difficult. 2, Transform: "itcn of the Differential Equations The assumotion of irrotatiorality implies the existence of a velocity-potential for such a flow. If this function is introduced to eliminate «and v, the "qs. (1.4) and (1.8) would give rise to a nonelinear vartial differential equetion of the second order. The problem is further complicated by the possible appearance of supersonic regions, or regions where the speed of flow is larger than the local sonic speed, This means that for some part of the domains the equation is of the elliptic type; while in the other part, it is of the hyperbolic type, The equation is thus of mixed type and there is as yet no successful method to deal with it directly in the physical plane. Molenbroek (Ref, 5) and Chaplygin (Ref, 6) made some progress in solving the problem by transforming the equations from the physical to the hodogravh plane in which wu and v are taken as the independent variables, If this is done the diffrrantiel equations bacome linear end thus can be solved by welleknovm methods. ist the transformatio be defined by u = ulx,y) (2.1) v= v(x,y) , (2.2) If u and v are continuous functions of x and y ‘vith continue — is finite D(heW) ous partial derivative-, and if the Jacobian /( and nonevenishing, a unique inverse tre:sformetion exists. Under these conditions, Faqs. (1.8) end (1.4) are easily transformed into RESTRICTED RESTRI TED 16. owe y OE Luv dX uy Ox We) oy t er . oy +0 sy =o (2.3) ox _ ot so abv BU (2.4) Corresponding to 9 (x,y) in the physical plane one introduces here a function (u,v) defined b x _ 2 — ok (2.5) x= 2A ) {= 2A , While Eq. (2.4) is satisfied identically, Eg. (2.3) becomes ~ uty o> vie 2 ww yy OK 2.6 As ¢ is a function of q alone, the equation for x (u,v) is then linear, One recognizes from Eq. (2.5) that if X (v7) is knowmm, a oneeto-me correspondence between the space coordimtes and the velocity com- ponents can be easily established, However, it is also clear that this function is 4neonvenient for obtaining the stream lines and the flow in the physical plane, To solve this part of the problem one may adopt a plan similar to Chaplygin'’s by introducing both the potential functia P (xy) and the stream function Y (x,y) defined bys = we Be, vee (2.7) fu=fShor pra-pBh. (2.8) RESTRICTED ROT ULCTED 17, From these definitions one has immediately the following equivalent relations des udx + vdy, (2.9) PaW = -Pvdx + pudy . (2,10) For the subsequent calculations, it was found convenient to intro= duce the polar coordirates in the hodograph plane defined by: u=qeesd , vx 4am & (2.11) where 9 is the inclination of the velocity vector to the x-axis. dx and dy can be solved for from Eqs; (2.9) and (2,10). As ax end ay are exact differentials, the conditions of integrability then give: F) , 2. sy to PRG +29 _ £f 9 13g FOE (2.18) By eliminating P between Eqs, (2.12) and (2.13), an equation for (2.12) VY is obtained: xy oy 2, 2 2 - £ vy =0, + (it & 3) 434 + WB) 352 (2,14) (2,6) can also be transformed in polar coordinates, The procedure is straightforward and yields q z _ . g tH — o, (2.15) sat thy + FS) Eqs, (2.14) and ong are the two fundemental equations in the present problem dealing with the flow of a compressible fluid. 3. The particular solutions of the differential equations As the differential equations for W(49) and X(4:%) are linear, one can certainly build a general solution by superimposing the par« RESTRICTED ee omnnntante ie oTRICTSD 18. ticular integrals of the equations. To obtain the particular inte= grals, let W493) and (4.9) be of the following forms: iva Vaoy= eye. Aq) = Fy, er” where Y is any real number, By substituting ir “qs. (2,14) and (2.15), the equaticns satisfied by V4) and Xv 4) are’ ook + + (amit & 4) £4 4B + y(y+i) ca =O, (3,1) 2 te * ae & % ri von 8 Ay = (3.2) Now each of these equations ean be further reduced by changing the independent variable. The apvpropriate transformation is By expanding the gas to zero pressure, or vacuum, the maximum velocity is obtained, Eq. (1.6) shows that the maximum speed is Vmax = Cy Therefore, the maximum value of t is unity. Similarly, it is .found that for the speed of the flow eyual to the local sonic speed t= aa Eqs. (3.1) and (3.2) then become TU t)V te) + ls ~ (a+b H1)T} Ye) = ay by Vic) = 0, (3.8) tli-*) %,@) + [ cy - tarprbepri)t |y'r0 ~ (4,48) (bp) A= 0; (3,4) whe re Oy+ by = v- B ? yb, =-+t pv(y4) and CL =v], (3 5) RESTRI “TED RESTRICTED 19, These are the hypergeometric equations, of which Eq. (3.3) was first obtained by Chaplygin in 1904. The differential equation of this type has three regular singularities at 0, 1 and +t oo, If the exponent-differences c-1, a <= bs, a tb-=-c are not integers or zero, the two fundamental independent solutions are -(4a,b,C)T) and t F (i+ a-¢ , j+b-¢ 52-37), They are single-valued and regular in the whole plane with a cut from +1 to + oo. The function F(ab;c;7) knowm as the hypergeometric function of general parameters a, b, and c, is defined by the hypergeometric series which is absolutely and uniformly convergent when |t[< |. provided RY (c-a-b) zo, and by continuation when» |t][> }] -. Furthermore, it is normalized so thatat T=o0o Fla bsc,9)=]., (3.6) Hence, the particular solutions of Eq. (3.3) are Fla, byity;e) , TF Uitarty Hb ey j2-Cy jE) 5 (3.7) The varticular solutions of Eqs. (3.4) are F aya tp cy) t) LF litayteety tebtp Cy j 2G 3b) + (3,8) Here a, , by and ¢, are parameters defined by Eq. (3.5). When yis a oositive integer while a, and b, remain as they are, the second integral will reduce to a constant multiple of the first one, This case was first studied by Gauss (Ref. 13) who found a -second integral involving a logarithmic term by considering the limiting value of the integrals given above as y teénds to an inte-= gral value. The method has been further developed by Tamery (Ref, 13) and Goursat (Ref. 14). However, the form regarded as conventional nowadays was that obtained by Frobenius's general method. According to the latter. method, the pair of fumdamental solutions. of a hyper RESTRICTED PESTRICTED 20. geonetric equation are -n nn ; y) n tr} ; ti) Fla,b; mhjt) > KyTt it Flarbsartjt)doget t QlabjatR ey (3,9) when C,= n+|,n being a positive integer; and i) 20 _™ Q (a,bjc) = oie Tlarm) TC btm) Vilo.bjm)T , “ Ttaytlb) or imei) p (aria Mm) ! Pe) - ae Tir Few ot 1" Tla-ntm) T(b-ntm)T nem) -" (3.10) * ™:! a =o ysl Here a, b may be either 4&, , by or a,* 6 » byt 8 defined in Eq. (3.5) according to whethor the system (3.9) is referred to as solution s of Eq. (3.3) or Eq. (3.4). Km can be determined so that, the product of the second integral and 7" satisfies the conditim (3,6), in view of the fact that the second integral in (3,9) does not constitute a family of solutions with the second integral given in (3.7) or (3.8), it is very desirable to define a new function as second integral which will be continuous,in y as well as in c ,. Let & te) denote the first integral F(a,b;¢,;t) + As ® second integral one may take the linear combination of the solutions: Bitd= ky \ Thi-cyray T(b) See) HT (ley) (Hac, )T Ute bee Z ey} (3.11) where hy, (t) = F (ita-cy sit b-Cy;2-Cy;t) » This is evidently a solution and valid for all values of y . The constant (iy is de= termined subject to the following conditim: ¥ . 4 Fit) = | fer tT =O, (3.12) RESTRICTED Reo PirlVrep Zl. The value’ of Ky is then found to be “l K = (apes) Than!) Pita ~Cy)Titb-ty) , Using thea ralation TlgyTli- cy) = Wcse “a7 , (311), when multipliad by qv » Will define a new function Gtx) 3 (t) . gic) = Terrie) t Fle) - Ave ] (3-13) ~ Annet | PV) Tlg-P Tlita-c)TUrb-ty) Tla-T Q-ev) When y takes integral values, the expression in the bracket vanishes; however, the limit of the ratio exists; Git) = Lorene | wy, (3.14) The usual definition of the limit of a quotient zives v 4 i) _ i" > Yortb t Fit) J Fyw n : 3y TlyenT(v) Pla-VIT Co-V) ov Fira | , anh By considering seperately the first ” terms in ay i) >» 2s Tdi-v) has poles at yan » a straightforward reduction yields: 2, (a) . 4 ir) = bm c log eZ, tc) + 2 Qe) + P.., it) (3.15) ? where (2) cn” S _ Tlatm)T (b+) _™ Q.w= TiotenayTenns) < mh Zs | yi (atm) + (hem) - Y (ent m) 7 Yoomeo Tle,rwt batt) ; ww { nl ™ . = Tla-nemy Pi (b-ntm)Pinem) mH Fon (t) Tn) Tla-n) fT (b-n) >. E+) nu-m a me f (meat) . n+! b —_ t-t) Tlast le) ; * Tim) Tl) Paw) Plow) RESTRICTED RESTRICTED Bao and Wis) denotes the derivative of ley T(z), It can be seen that the difference between (3.9) and (3. 15) lies only in a constant multiple of the first integral which has been absorbed in Qe . In the following discussions, the two fundamental solutions of the hypergeometric differential equation will be taken as fe, ) and 4G, (t) - The normalizatio conditions given by (3.6) and (3,12) are. chosen for the continuous passage of a compressible to an incompressible flow, Ultim-tely, these functions are again defined in terms of cower series which are absolutely and uniformly convergent within the domain jrj<|- However, since the maximun value of — attainable by the fluid is unity, the continuation of the solutions beyond the unit circle will not be discussed here, Let F(t) and 4 i) denote the two independent integrals of Eqe (3.3) where Y is any positive rational number, The particular solutions of Eq. (2.14) are then: 4 dy re) ay cos)9 + Ayainy§] 7 G co) By eosy9 +By any] (3.16) yw w where Ay, Ay, By and BY are constants, Similarly, those of Eq. (2.15) are ¢ Fic) | Avesver iy aim v9), 4 G,10){ Brees 94 Berets] (3.17) where Ee) and 4°50) are the two independent integrals of Eq. (3,4) and AY A’ BY and $% are constants, In addition to these solutions, there are two other integrals each of which is a function of one variable mly. Assuming Ve Vig) or V(%) , then Eq. (2.14) and Eq. (2.15) yield respectively: RESTRICTED RESTRICTED 236 c, t and Cr fue) (3.18) Co and Ca {qe} (3.19) which correspond to the fundamental solution of the Laplace's equation. As ¢C, epproaches infinity, all these particular solutions reduce to the familiar harmonic funetions, viz. -Vv cio > Ca 409 E (3.21) This property which is the consequence of (3.6) and (3.12) is essential in the method presented in this Report for connecting a compressible flow with the incompressible flow of similar configura= tion. In the subsequent calculations, one will encounter another integral for the function x 9) which corresponds to the imaginary part of wr bog wt or ¢ bog G aon - Ginr-9 eas) of the incompressible flow, Suppose the solution possesses the form: Xp 9)= Yaad = ig) -9) cos (3,22) By substituting the expression in Eqs, (2.15), x and %e are fod to satisfy simultaneously the following differential equations: u , 2. i; _ . ce FAQ + O- EGA) = Xl wd)e (3.23) th r (EY 4° An) =? (3,24) RESTRICTED RESTRICTED 24, Eq. (3.24) can be easily integrated by putting 22 = + k,\4) : The condition that 7,74 as ¢?7eo requires ky) to be a constant. The second integral of Eq. (3.24) is just the second of (3,19) which, in the limit, tends to log q. Thus X= 4, is the appropriate solution. With this solution ome can proceed to solve Bq. (3,23) by assuming (= 4 Ry) « The equation for R.') ig again integrable by quadrature and the result is tv fy) = gts [plage & + K, furV4s|+K, , (528) where K, ana K, are the constants of integration. Hence, the desired particular integral is 5 y= 4 Rit)on 9 - ql ®) tn D, (3, 26) 4, The Proverties of the Hypergeometric Functions of large Order The behavior of Flay,bvicy;t) for large positive values of ¥ has been discussed by Chaplygin in connection with the question of convergence of his scries solution for the flow of a gas jet. However, his discussions are limited to the subsonic flow and for this reason, the value of tt is restricted to the interval i . oft £ In the more general problem where both subsonic and “ 26+1 ° supersonic flow may exist, the whole interval of & < | has to be considered, Furthermore, both integrals of the hypergeometric equation are involved as will be shown in Part II. As a preparation for the proof of the convergence of the solutions, the properties of the hypergeometric functions of large order in the extended interval will be discussed presently. RESTRICTED "Y AES Tal oTe 25.6 enn vonemeeene Chaplygin (Ref. 6) introduced a new function % Bie ¥ defined as the logarithmic derivative of T* f(t) » namely: VB it) zat gr 2 Log ag (tT) , pH (4.1) where &, 't) denotes the first integral of the hypergeometric equation (3.3) or (3.4) and V can be either an integer or not an integer, Then in the place of Bq. (3.3) or (3.4), the corresponding differential equation for by is a Riccati equation: = [2 Jotapei) +1) (4.2) AG) = ty tt a Sut ae at [x - cep |= o where the lower sign corresponds to Hq, (3.4). As shown by Chaplygin, $0) » although an oscillatory function, can have no root in o4T Soa and, consequently, ty (ct) is finite and continuous in the same interval. Moreover, it can also be deduced that Fle | and “$y(e)a-B and that File) does not change signin oft apr) 30) is monotomic decreasing and eventually vanishes at T,4 T* » t® being the first root of the hypergeometric function for Vo o Since coll is a decreasing function of y 4 when Y becomes large, cv and consequently T will differ from. ape by a small quantity. , Chaplygin's theorem: In of T * apt s if a monotomic continuous function W(t) satisfies (1) Jley={ and (41) Kiy)Z0 , then Wit) Ze ie) 1 Y | (4.3) The proof is given in Chaplygin's paper (ref. 6) In the case of the second integral Fie) » the theorem is still true with the sims of inequalities reversed because it can be verified that X(Z,)=o 3 RESTRICTED RESTRICTED 26, where Lyle) corresponds to the case of F, te) instead of tw) in (4.1), and Kyl=-| » therefore By !T) is negative in oft Ase Corollary 4.1 In okt tas the fmetions Fig) and G te} fall respectively between the limits: (i) Te) < Fic) < Bie) 4) (43) Ts) > 4 ir) 7 Tie), 2] (4.5) where Tilt) = axp 7 fiw u-ty ie se (4.7) This can easily be verified by choosing 2 to be [eam or (ity » As, evidently, in o4t4 apr h ; ‘de tag+t _ty 4.8 when yv>| a << Qt ct) , ( ) and _ | J= (26+ _()- P (4.9) eh 7 Sey 7 CI tc) } and furthermore, X(%,) Z0 are satisfied; consequently, it follows the results. Corollary 4.2 In of t# eT the absolute value of the logarithmic derivative of TE(ayit;c,;t) divided by VY , is bounded both above and below, that is M(t) 4 Flavtts bth jGtlit) gy 2It) (4.10) Flay by jy 5 e) . where M,(t) snd M,lt) are independent of Y . This really a conse- quence of (4.8) and (4.9). RESTRICTED RESTRICTED 27 0 Sn a eee It shall be noted that the results established above are appli cable to Gic)=F (arp, tp jc; t ) provided y is large, because then the two Eqs. (3.3) and (3.4) tend to be the same. Obviously, Chaplygin's theorem ceases to be true when t? —— , For in the interval + <t <l , the solutions of the 26+ Apr hypergeometric equation are oscillatory and, hence, within any closed interval in sam At S | the number of roots of fy) will be pro portional to y (Ref. 16). Vhen Y is large, there will be a large number. of roots in the interval, As a consequence, the funetion ¥'*) will have there an ever increasing number of simple poles and a finite interval in which ic) remains finite for all y does not exist. To carry the investigation over into rey 424} , the method is modified. Let Fie) and Fit) be two independent solu» tions of Eq. (3.3) or (3.4); and let the linear combination be denoted by a . F, iv) = F(t) + LF tv) , (4,11) fhe complex function is, of course, a solution of the same differential equation. In terms of its modulus Re) and argument pic) » the function may also be expressed as ek. i F wy= Kit) 27, (4.12) where both ‘,‘t) and Pic) are continuous functions with continuous derivatives, By comparing with (4,11), it is necessary to have Ftv) = Ry tt) am Pte) , (4.14) RES TRIV TED RESTRICTED 28. According to the Sturm's separation theorem $i) and Fi) never vanish simultaneously in any closed interval and Ryit) never vanishes in —e ap 42 4| and remains positive in the whole interval, Then corres= * ponding to (4.1), ome ean define a complex function ty ft ) * _ d zr* VR it) = 20 Fe bog oF, tr) (4.15) which satisfies the same equation (4.2), On separating into real and imaginary parts, the Riccati equation for pe) becomes O) we) = fu) 6 (i) y lw) 2 ~ SUBS 8 + ey tel} FP +e =0, (4,16) KalZy, By )2 +he EGP =e, (4.17) a) where %, (a) . . . ~ are real continuous functions of tr defined and %y a8 12)” * i) . Bey = Filey + 6 Be (4,18) Their connection with f,\t) and ptt) separately are given by means of (4.15), Namely, y Ww d z aa (c) = At 4, log t Ry 't) (4.19) i 2¢ Pic) ; (4.20) Now Eq. (4.17) can be integrated in terms of Fito and whence ¥ ) By) can be eliminated from Eq. (4.16). Then the equations for a Y and {> are fl -2¥ v 2 - “ 4, rae Kiwoe T+ Wes Z [gfe e PE epnets, (2) fy ae is (it) =~ Fb to 7 © 2 , (4.22) a_i (22 Ryue)) RESTRICTED , sTRIVTED 29. ¢ . bet & tr Eq. (4021) together with the condition 5 l= -I determines uniquely thea solution 5) ) » The actual value of By te) can be, of course, expressed in terms of the knownhypergeometric functions. But the problem on hand is to detarmine the properties of 5) for large v . To facilitate the discussion, “%q. (4.21) is first written in the form: 9] = uy 7) _ . (4.23) Ay = oy ‘t) + Qc if if a) where = 0) t 5, (c) is itt Ja + Ye) ; _ ww at. _ it , 3, tc) = %y wr V(i-T) y ) and yy v }-7 yri-vy tY Rie) when Y is large, the character of the functions $, and ¥,, can be easily studied in the ci plane (Fig. 1) by neglecting the third term under the radical sign. This can be justified in the following manner: Consider the case when Y is positive and large but not an integer. In the interval o<7 < spel 1 Fosk because = Fiisy~ OT te) by Eqs. (3.11) and (4.5). Then ee Rithea TET,» Therefore, ~% Ryie)27 1 when Y is large. But both J) and Fit) are continuous with respect to y » 80 the above result applies equally to the case of integral Y » Hence, the third term in the radical for Ye) can be neglected for large y ° | Due to the manner in which x is defined, corresponding to each 2 . i . y there is a line v= tT, > Sri such thet Yi Z0 when t?T.. As a consequence 5, and % are real or complex conjugate according RESTRICTED STRICTED 50, as tS, In of7 € To) fzeand J,7° will give two one- parameter families radiating respectively from ( 0, -f ) and (o./ ) and joining together ata point where 7, =o , If eft £t,, the product J,1, my be negative or positive according as the point lies to the left or right of the curve J,=o end 7,=° . the other hand, if tT >t , %,3, is alweys positive. Now Oo) = B » while the initial slope of J, =o is p(i-+) » the integral-curve must lie above Y¥,=9 4, and below 4,=9 If it were not, the integral curve would cross the curve ja) 4a : -~ =o 5, =o, §, 29 where yt) and yy it) would be negative . / . ; somewhere in oO = T&T * 2 pr » ‘This is not possible, for ¥ ~ Ty by an argument similar to that used for determining the magnitude y. of t* Ky) and according to (4.5) - [cgerus 7 Fy 7- le xy J A : 22 z fs) 0 : z | 1) : ino i+] >» Henca ie > in oS t Sop and yy continues to increase until it intersects with 3, =o , After it crosses ft} the curve. J. =O 5 %, <o and never changes sigas J,3,>9 in ) 2 : . Tei TS { » Consequently, $y) is monotonic and decreasing in the interval +t,<7t4|. When Y is sufficiently large, T, will approach very rapidly to apt and °T,= 2p when Y becomes infinite. ti} Theorem 4.2 rf Hye) is continuous and monotonic in tT.<7T < | and satisfies Xz fe) » then for all VON Ww 7} 4,24) Ht) ZF, It). ( RESTRICTED RESTRICT2Y Sle Proof: Form the follqwing identity: it BME Vs EWES 2 Ay] Tt (0) ae (4; 25) It can be showmm that the differential expression possesses an integra« tion factor ti} (Pe eyo” Gee PRE st (4, 26) where Ry = Rylto) exp hy joy " ngef S, = Seo oxpfy] $e. (2,27) It will be noticed that the sign of (4.26) is determined by the first ) ) : factor (mre) only. On multiplying (4.25) by (4.26) and inte« grating the resulting total differential from T, to T 4, with a suitably chosen initial value x) =. Zt) » it is found that 1) 1) LUE CUS RS + Losi [onk ya} ze . . ys : : of) which is positive if and only if Ty" y zo everywhere in 7,474 | F wy) i) Since both %y and Ty are continuous and monotonic, ‘the condi“ tion is both necessary and sufficient, Furthermore, it should be . vy) _ ell noticed that the condition 7y't0) = %y ) 4s purely a convenience, () a If Ny 0) F %, '%) , the validity of the theorem is not in the least impaired, RsSTRICTED RESTRIC TIPD 320 Corollary 4.35 In 7,414 | «, the following inequality holds * for the modulus of &, (tc) 3 B')/gny 4 ye y>N (4.28) where (2 A+1)7,- |2- For in T<t<| , er L0 and hence ayn) = © satisfies the condition o> e) which gives (4,28) by integration. Now, since ec) is bounded by zero for all Y#0 in 1,43 <|, it is also implied that Kit) < 50) (4,29) where T(t) = ao » Here the constant to can be detormined by joining T at T=T, with J" or J, defined by 2qs. (4.6) and (4.7). Then from (4.13) and (4.14) it follows that for y > N | | 27 ) fe) Tz)» (4.30 | 4, | < Tye). t Kt | (4,32) RESTRICTED RESTELOTED 356 Part II Construction of the Solutions for Compressible Flow Around a Body 55 Chaplygin's Procedure In the vrevious sections, the particular solutions of the differ- ential equations in hodograph plene are obtained, Since the differen=- tial equations in tha hodograph plane sre linear, sudervosition of solu= sions is allowed. In other words, if these particular solutions are multiplied by different constants and then added together, the sum is again a solution of the differmtial equations, By this procedure, general solutions can be constructed from the varticular solutions, However, there is a difficulty in such a method of constructing the general solution==- the difficulty of mking a proper choice of the multiplying constants for the particular solutions so that the resultant solution will give a flow satisfying the boundary conditions specified in the physical plane. This can be seen from the fact that the space coordinates x and y are obtained from X which is not expli- citly connected with v 9 the stream function. In fact, to obtain the coordinate x and y directly from V would involve an integration in the hodograph plane. Thus the linearization of differential equa tions in the hodograph plane is obtained at the expense of the simplicity in boundary value problem. Chaplygin (Ref, 6) suggested an ingenious method of solving this difficulty by using the well-known solutions of the incompressible flow. The first step in this method is to find the incompressible flow around a body "similar" to the body concermed, (The meaning of the word "similar" will be made clear in the following paragraph). RESTRICTED RESTRICTED 346 The stream function V, for inetance, is then expressed in terms of the speed 4 and the inclination 9. The function Vy (p-8) can be expanded into an infinite series, each term of which is of the form 4 ces or Gain nd > ‘or the flow around a body with constant velocity U at infinity, the function Vt, 9) has a singularity at the point @ = U, @ = 0 in the hodograph plane, since there all the stream lines, or lines of constant Vs originate, Thus there are tvo forms of the series expansion of ¥ : one is convergent within the circle | =u, while thé other is convergent outside of the circle q=U. The first series, or the "inside" series, must be regular at the origin of the hodograph plane. Therefore mly positive values of the integers MY can occur, The second series, or the “outside” series, can have both positive y and negative Y . Chaplygin's method is to use the "inside" series for ¥, as the starting point for obtaining the de- sired solution VY for the compressible fluid. _He suggested choosing the multiplying coefficient of the particular solutions for the come pressible flow by the condition that for the limiting case of infinite sonic speed, or incompressible fluid, the series will degenerate to the “inside” serics of the incompressible flow already obtained, The series for the compressible stream function VY so constructed can be called as the "inside" series of WV o the “outside” series for W can then be obtained by the method of analytical continuation with the aid of the “outside series" of the incompressible flow, These solutions so constructed for the compressible flow contain the Mach number of the undisturbed flow as a parameter. Shey constitute a family of singly infinite solutions, Included in this family of solutions is the limiting case of zero Mach number of the RESTRICT.) RESTRICTED 350 free stream, This limiting case will give the incompressible flow around a body used as the starting point of this method, For other values of the free stream Mach number, the body contour is generally different from that corresponding to zero Mach number, Thus, if the compressible flow around a given body is desired, the body shape for the initial incompressible flow must be slightly different from the given body shape. However, if a geometric parameter is. included in the solution, such an adjustment is not difficult to make. It may be stated here that due to the regularity of the solution at the origin of the hodograph plane, only the first solution of the hypergeometric differential equation appears in the “inside” series. For the "outside" series, both the first and the second solution of the hypergeometric differential equation are necessaryo This is in direct analogy with the appearance of both positive and negative exponents of 4 in the incompressible “outside” series, This fact is particularly importent,.since previous investigators seem to be unaware of it. Chaplygin himself did not use the second solution of the hypergeometric differential equation, but that is. simply because, for his problem, there is no singularity in the hodograph plane and hence only the "inside" series is needed, 6. The Functions for Incompressible Plow Following the procedure outlined in the previous section, the analysis starts with the functions required in defining an irrotational incompressible flow, For this case, the sonic speed Cc, tends to infinity, and the equations for the velocity potential Y (x,4) and RESTRICTED PRSTRICTSD S66 the stream function V. (x, 4) all became harmonic’ o LF, =o , (s.1) = (6.2) AY, =o: | where A stands for the Laplacian operator. If Wiz) is the comlex potential, it can be shown that Wiad= Bris (6.3) wnere z= xr ey, If w denotes the complex velocity u-‘v, it is connectsd with Wiz) by we dW Ewe). (6,4) dz If w'\2)#aone can always solve for z in terms of w, namely Zz = 2,1) (6.5) In general, this solution is not single=valued and will be discussed later, By introducing this relation into Eq. (6.3), one obtains the complex potential function in tha hodogrash plane. Wow = Give ¥ tne) (6.6) In case Sa. (6.5) is many valued, this would correspond to one branch of the function, It is clear that in this case Ao ivy) is also a harmonic function. Let Giuw) be the conjuzate function defined by Ko 26, one 2 88s, 6.7 OU OV (6 ) Die = 2% (808) av OU Hence Now) 2% 6 (6.9) where Wm u-iVv R STRICTED PRSTRICTED S57. Thus /\ Jw) is on analytic function of w. From lq. (2.5), the deri~ vative of A,\w) with respect to w must be z, That is, oh = Zlw) , But z,(w) has already been found from 5q. (6.5). Therefore, A.W) = J z.1w) Aw + Conk. (6.10) The real part of AY) gives A. wv) as reguired, according to (6.9), 7, Conformal Mapping of Incomoressible Flow on the Hodograph Plane, Before the construction of solutions for the compressible flow, the general character of the solutions in the hodograph plane should be examined. This can be easily done by investigating the behavior of the transition function Z,(w) for an incompressible fluid. To start with the simplest case first; consider a steady irrotational flow in an infinite simply~comected domain D bounded by a curve C in the z-plane, with a parallel flow at infinity (Fig. 2). At every point z2 of D there is one and only one velocity vector rt o if the curve C is mapped into 6 and infinity corresponds to a point P on the axis of reals of w within @ , then the domain D is mopped into® by a mapping function w = wiz) defined in (6.4), where W(z) is an analytic function of 2 . The inverse function Z = Zw) will set up a continuous one=to-one correspondence between w= and z= plane provided the mapping is conformal. This requires that wiz) is RESTRICTED RESTRICTED 38 » analytic, simple withinD, and w'(2)#o. However, for most problems these conditions cannot be satisfied throughout the field of flow. In the first place, the function w (z) is generally non-simple, for example, in the case of &® uniform flow, w(z) ¢ const., thus w’(z)z0 and the whole z-plan@ corresponds only to a point in the w-plane. Furthermore, the complex velocity for a twoedimgnsional boundary-value problem can generally be put in the following form: Wot We + wit) , where Wo is a constant. The boundary condition requires that wW(2)=0 and, as @ consequence, | w* (2) =0 as zZ becomes infinite, Therefore, in all cases, the point @ in the weplane, is a singular point. It is a branch=point at we if zZ{w) is many=vealued; or a pole, if otherwise. In practice, there are two kinds of singularities that plsy a dominant role in the problem of two-dimensional flow. These singularities will be investigated prosehtly. Ae Branchepoint of order one* One may recall] that when a closed body is present in « uniform flow, there always exist two stagnation points which both correspond to the origin of the w-plane. If one follows a stream line PS, (Fig. 3) for instance, from + oo to § 9 the portion SMS and then to-o , *The function Z,(w) is said to have a branch point of order Rat we: Wy, if its inverse w (2) contains the part w* which has a zero of order k+/{ at Z=ze o FESTRIVTED RESTRIC TAD 396 one viould describe twice along a curve on in weplane, This indicates that the function #,(w) possesses two branches of Riamann surfaces joining together about the branch=point @ . In order to make the domain 9 single-valued, a cut is put along the axis of reals from the branch-point to +0° , Then one vortion of the zeplane is mapped into a definite branch of the Riemannsurfaces in the w-plane and this will be defined as the domain 2D . If the body is symmetrical with respect to the coordinate axes with parallel flow at infinity, then the domain D:RLQZ 42°C will be mapped conformally into 9 on me branch of the Riemannsurfaces and :D >RLZ7OC om the other, where the region within C is excluded, B, Logarithmic Singularity The flow over a wavy surface, for instance, placed parallel toa uniform stream has a p2riodic nature. For such flows there are infinitely many points in the physical plane that have the same velocity. Hence, there are infinite number of branches in the weplane, each of which corres-= ponds to a definite portion of the z~plane.The function@ol”) must have a term beg (i- @) and the point PP now is a logarithmic singularity. if, however, a cut is introduced from the branchepoint to + and -7T 4 ang (in) TT , then the domain g is again mide single- valued, 8, Gonstruction of a solution about the origin Prom the considerations of the last section, the domain within a circle with radius |wl=(@S=U, where U is the absolute value of w at infinity in z-plane, is in all cases single-valued. If a function Woi~) is associated with a definite flow in zeplane, from Section 6 PESTLL ITED neem & STRICTED AO, it is an analytic function of wand regular within the circle jw heU, Consequently, the following Taylor's expansion exists: Ww) = > Aun Ww jw\au (8.1) nso / eet) where Ams are in general complex, Since wage and by (6.3) the imaginary vart of W.\w) is equal to incompressible stream funetion vy, » if can be written as Weg, 9)= Im {Wawrt = x a" ; An con d+ Ayainn 9 (8.2) According to Chaplygin's vrocedure, the corresponding compressible " solution can be obtained by simply replacins the funstion 7 in Bq. . n ind (8,2) by the corresponginy 4 & (t) &S shown by (3.16), The second mn integral is excluded by the regularity requirement at q= o , Jilwever, in order to preserve the proper singularity at the point (U, 0) in the hodograoh plane, the compressible stream function Vv is written as 08 nN ty) Ww) tz) Va 2ST Lisa cesndt framndh , (003) where ze" Sy) FE ans bi Cn ©) a4 vf a)= a = a, mE FF laerbeyenie) } (804) J and Tt, = 2B “Cz 8 the value of T, corresponding to the free strean velocity VY. It is seen thet if Cor » thn tT#=#T,7°9° , and Ze | due to the normalizing cordition (3.6). Thus the solu- tion is reduced to the incompressible form, Furthermore, if 47 U the character of the soletion is exactly like that of the incompressible solution. Hence, all the specified conditions are satisficd. Of enurse, for the mixed subse.ie and supersonic flow, the free stream . ome ote ao batho tk. RESTRI“ TED 41. Mach number is always less than unity. Thus T, < apr ' For later analysis as given in Part III, it is convenient to write iy) . a in a different form, . Since &, (t) is a purely real auantity, a complex function Ww) can be constructed as (8.5) Ww) = 5 An Foto w" | yw Zu Taen,similar to the relation between Eqs. (8.1)’and (8.2), W (9,8) can be taken as the imaginary part of the new function Wiw;) « Thus Wigi9) = Im 4 Wiwier} , (8.6) Similarly, one can construct another function Alwje) defined (<] ~ _ ~ \y) Ww Awin = 2A, Ew, geu (8.7) In this expression, the coefficients An are obtained from the ex= pansion of /\,'~) for the incompressible flow (6.10): = Ry" (8.8) Aji) = 2 Am we, In[e« U and a _ F,, \t) ” (1) (8.9) Then the function x (4,9) for the compressible flow is given by AG) = ALL Awiny- (8.10) The functions Wiw 52) and Abs;t) are actually regular and satisfy the condition that Wiwir) = Welw) and Awe Adw) on |w|=U . However, the following question may be raised: Do the series (8,5) and (8.7) converge and represent the functions PUSTRIC TED PUSPPICTRD 42, Wiq.9) and X(4)9) in the douain of validity? To settle this question, one has to »rove the following theorem: Theorem 8.1 If the coxstants Ay ene An are given in Eqs. (8.1) and \3,8), while Kite and Fiey are defined respectively by Eqs. (8.4) and (8.9), the series (8.5) and (8.7) are uniformly and ab= solutely convergent in the same domain as those of (8,1) and (8.8). Consider the first series: Multiplying throughout the inequality (4.8), namely 3,00) ? | a (spre ' of TS _l. \- 2pr! by = and integrating both sides from t to T, , it can be deduced that ) n £, lt) < T («) where = t (1) = Tit) |, lc) 2 {. Then it follows that tr) ™~ ” | An Hey w" | < | An Cw) | Now 2s | Aw (tw) | converges when |t+,w]<U due to Hq. (8.1). By Weirstrass's theorem, the series (8.5) is uniformly and absolutely convergent if It, | =tg<U . Now t,(t,) = / , thus t14 is equal to Uwhen 9= Uand @=T, .- t.4 is zero if 9 = 0 and remains positive for o<9g U, By the definition of T,it) given by Eq. (4.6), it can be easily show that d for o< tT< 7, , Thus t)4 increases ronotomically from zero to U in the interval oét7z< T, 6 Therefore, the series (8.5) is uniformly ard absolutely convergent in eny closed domain in Jw {<VU - moor RICTao RUSTTIC IED 43e Similarly the convergence of the series (8.7) can be established, 9, Analytic continuatim of the solution «= Branch point of Order One. As proved in the last section, the series (8,5) is absolutely and uniformly convergent and does represent a regular functim W (wr52) for every tin °St£7T, and on the circle of convergence it agrees “19 : m . . . with W.( Uz ) » of which the Fourier expansion exists? “ng? Weve’) = SZ An Ue | (9.1) In the oresent section, it is proposed to continue the solution (8.5) analytically outside the domain |w/4U with the initial value given by Eq. (9.1). Tho domain outside |w|#U is generally many valued, To fix ideas, discuss first the case of a branch-=po int of order one. Generally the function Wow) has other singularities besides the one at w=U . However, such singularities lie outside the region of interest and thus need not be investigated. Let the nearest singularity be given by |w|= V>U. ‘Then, the domain to be considered outside |w|=[ is an annulus with a cut joining the two singularities, The proper representation of Ww) in such a region which has a branch point of order one at w=U , is * Wrtwr = & wi? Wels), (9,2) where Waw) is single-valued and regular within the open annulus ye Jw] 4 Vo. Hence, in any closed domain Urs < lw |< V-3 ; § being a srall number, there exists a wiiformly and absolutely convergent series: Welw) = sf. wit Caw], (9.3) 20 RESTRIC TEL ROOSTRICTRD 44 » which, on substituting in (9.2), will give the. continuation of the Taylor series (8.1). The solution for a compressible fluid, which has the same character of singularities as Wiw) and valid in the annulus V4 lw f</V, can be obtained from (9.3) by gintroducing the proper hypergeometric func- tions corresponding to each exponent of w . That is:. {o) _ 4 ¥ + _7¥ W (wot) = & >. [ By Ste) wit Cy yu w | ; (9.4) nTY : . P . i) L which is the continuation of WwW (wit) . Here yentyT yn “a being a positive integer; te) and 4 G it) are the first and second integrals of the hypergeometric equation; and By and an are constants, It.should be noticed that the coefficients Bu and a in the outside series for the compressible flow are not equal to By and Cy in Eg. (9,3) for the outside series of the incompressible flow, The outside series of the incompressible flow is only used to give the proper form of “Tw t) » While the exact determination of W" (wt) has to be made by the conditions of continuity. Since the partial differential equation considered here is of the je) second order, to ensure that twjr) dis the analytic continuation (0) of (wjt) two conditions have to be inposed at the boundary of the respective regions of convergence, i.e. the circle 420 » These two conditions are the following: vat yet) 4) _ Wve’, t1) (9,5) RESTRICTED PSSTRIC TED 45. [ d te) ere) ro (wjT) =ilo We re) > W } , ' 9.6 if t 24 Tat ( ) On account of Eqs, (9.5) and (9,6), there are two relations which have the imaginary parts: ow v + -y oo n >. [ 8, fit) V + Cy Gv) U | cos v9 =- Zi hy Usinn 9 n= ° oo n / >. |B BV YF wr aUJe))+ C, Ul (-vfys)eeni eye ~Yhad (40 Se von § nro Here the orime denotes differentiation with respect to tt Evidently, the coefficients on the left hand side can be solved for in terms of the know emstants Am . They ares * -Vv | BL + Cu Gin =- Ls Am U mb dy) | (9.7) x oY J ev J | & ™ i Pr U (YZ ta) tae, Fee) + Cy UV (-v4 J) 42% G00) =-a¢ 57 A Ann 5, {t,) Nia" m-v mo), (9. 8) From these two equations, the constants B.. and C can be uniquely determined provided the determinant A (J, B) does not vanish. These results are: y Gyn, B. o=- Tron Am Ue (se +sts) (mz n- ¥E sn) ; (9.9) ad By CU's Seed FU "ogi + cv) (MF, 10-¥E ts) (9,10) as the Wronskien A (4,,F F\= _ am "u-ty¢o and 37) is defined in (4.1). RESTRICTED RESTRICTED 45. The solution is again formal. To prove that the function Wow 32) is a regular function in the annulus region, the truth of the following theorem must be first demonstrated. * * Theorem 9.1 If the constants §&, and C,, are determined according to (9.5) and (9.6) and if the seriss (9.4) converges uniformly and absolutely in a closed domain U+$ 4 jwlé V-$ then the series (9.4) will converge uniformly and absolutely in the 4 domain Urs ¢ bwié V- 3 ; 5S 20 It is observed that the following idontities exist among the constants involved in (9.1) and (2.2) ; Vv hind ™ | Ba” == eS AU Uae taal), (91) “Vv | — ™ ; CoV = ayaa Am UO sy my) | (9.12) Now, by the inequalities (4.8) and (4.9), the functions F'n) 3 it,) “Vv can be bounded both above and below for all V#o , when o£ lara 9 And if a smaller value of A\T,,F-) is taken, it can be deduced that 2 ic, | < M, Loot. ; where M, and M » are constants independent of n. On the other hand, from the inequality (4.8) 3 it) < 1 ? o= ~ 2pr\ it follows that ¥ WO 5 ,) < t, (e) F tT e084 ape RESTRICTED AT. Consequently, the first part of (9.4) can be dominated: | B, Fa) w | < |Ba Ltyw” | whe re ty.lt) = hit) /tit,) © The continuation of this inequality for TT > rot can be easily done by defining a new t,(~) in conformity with (4.29). By hypothesis, =, | Ba (tw) | converges if |taw]< VY. Since t,t) < doit) for T,24<] , the inequality | tow }< V is uniformly bounded, Similarly, it can be shown that [erg mw" | = | Ca yw)’ |, Lad - Bub 3 | Co tw) | converges if |t,w/ > U © Since. on neo Iwl=U t, t= | and fy fog!" 7 O° when 047 “agri I sha or fi, ltw| = when ipl <t< |, the condition It) | >U holds for all t in t, £74 <{f{ © -Hence, by Weierstrass's theorem the series (9.4) converges uniformly and absolutely in u+S <\iwl<V-8 . By similar procedure, the continuation of (8.7) is Niwey 2S | Be Zc) w's ce Gin) wr ] (9.13) where f,') and G et) are the first and second integrals of Ease (3.4) and the constants B. and Ce can be similarly determined, ay ay 7) & Ba U = Bry = 7,5? =, fie Uo (59 = sa), Ii) 8 aay ee 27Y (iT +> Mak, ° J) (mF taI-¥F oy} (9.26) RESTRIVTSD 48, The solution so fer determined is understood to be the: principal branch of the function Ww; 2) It was assumed thet tho flow at infinity is parallel to the x-axis. If in addition, the body is symmetrical with respect to the coordinate axes, the expression for to) the second branch of W tw;) will be identical. However, in a more general case where asymmetry comes in, the two branches will reauire separate considerstion, 10. Vontinuation--Logerithmic sinrulerity Consider now the second important type of singularity: it is assumed here thet the only singularity possessed by the function Wo (wr) in the finite part of the w=plane is a logarithmic branch=point eat w=U ebout which infinitely many Riemann surfaces are joined. By analogy with (9.2), Wow) can now be conveniently written as ~ « Woiw) = Wolw) + Wow) , (10.1) where Win) is a regular function in the entire domain with possibly ar essential singulsrity at infinity, and hence generally is given by a Teylor series or a polynomial in wW, and Welw) = 9 14,9) + ivig.9) is an analytic function which characterizes the singulsrity of Wa(’) ° Thus, aside from a constant factor Woy = + bg li-#) (10.2) If a cut is laid from +f) fp) +00 and the argument of \\- &) is restricted in -T Zag F270 , then the function Ww) will be Single-valued in the whole cut plane. The question of constructing a solution for the compressible x ~ fluid consists, therefore, of two parts; Wav) and Wow) . However, RESTRICTED PESTRIIT: 49a + the constructicn for Wow) is, in principle, exactly the same as thet of (8.6) and hence only Wiw) will be considered. First let Waow) be developed into power series in the respective domains of validity. The imaginary parts are: Ww omit) Vip) = Es * (4) cone, 4<eU (10.3) wv" 9) = - q = 4 (4) come ,(4.9) = Meg + 2 mG 47U (10.4) The corresponding expression for Wig. 9) will accordingly be; ~i) wm Wig.9) = >. Am ty (3) coon 9 } gay (10.5) nat 4, f Pat = 4)" Viq.3)=- B jules + 25 Cy Geer | G) cons 4 7V (10.6) where SK) stand for Flay, by 3 em 3t) and &.,, (v) is defined by (4.15). The function Wow) may be regarded as the complex potential of a complex source situated at w= WU . It is known that in this case the normal derivative of W, (419) on fwi-U is a constant, except at w=U where it becomes infinite. This boundary value can be expanded uniquely; , & — - ¥ toond = 1 ’ 9 #0 (10.7) The. sorrégponding problem in the case of compressible flow can be put in an analogous menner: to find a function Wiq, ®) which is continuous together with continuous partial derivatives and whose normal derivative on |w{[ =U is constant. Thus, the conditions PESTRICGTLD POETRICTED 50, (9.5) and (9.6) in conjunction with (10.7) demand: F&A. 7 a (10.8) y) / = 9 [ Frey rat Foren] Ant Lm Gatto ati Gated] Co =4BU-T, — (19,9) where’ the constant B can be determined when the normal derivative Vy (4:9) on Iwl=-U is assigned. By solving (10.8) and (10.9) these yield An = % BG ls) , (10.20) Cw \t KBR tt) » (20.31) by the relation of the Wronskinan of the two independent integrals of Eqe (305). Thus the function W(4,9) is completely determined. The associated function. X'9:9) Ab) can ba similarly constructed. As is derived from (6.10) by integration of the inverse mapping function, it must involve a term (1-4) bog (t--#) which represents the singularity of the function A, lw) > As in Eqe (10.1) No\w) is again split into two parts; * ~ Ao (wy) = Now) + A.\w) ? (10.12) where Nui) is an entire function and Aoi) is iN 4 w ~ Ww Awd =e lie ) bog Cl“ # ) (10.13) Now the solution corresponding fo hog (ie can be determined in p ¢ U exactly the same manner except thet the hvpergeometric functions in» RESTRICTSD RESTEICTa&y Bl. rhe volved are icc) and G ie) instead of fic} and & it) . n The pert thet will require speciel consideration is the term ep Z) ° ji . ~ ~ ‘ Let it be denoted by - A,iw)= X,- + Rew sob WM deg in BE), (10.14) This function is also multiple-valued. Let the argument of (1- 4) again be restricted in ~ TW Lang l-F 4 a , then in the cut plane one will have Vet St (2) jwi<sU (10.15) ney So Ra t[- legato, >i) ] \wl>U (1016) ° L According to (6.9), the function %.4(4:%) is defined as the real part of Aolw) . That part represented by (10.15) and (10.16) is then a a ” x'q.9) — 2 a (3) we (10.17) “ww He.) = flog d ain 9 Ln 9 to +2) Plarend (10.18) nas The particular solution corresponding to — 4 log 4 aim v- Sn. 9) ewe & has already been given in Zq. (3.26). Hence the solution for the compressible flow is ao zw ~w x Xqne - SAE (A )ainn d. (10.19) K qe LR amg- Lind) wd +5, 0.410) [asian (10,20) neh RESTRICTED wnere sy Reed = ha [ tape bog E -(4-$) +k, fo xy az] (10.21) ys t The conditions (9.5) end (9.6) together with an expansion vin 3+ S| nti? Rel | tinny = (1-9) cw 9 , os 2 an nel require thst; ~ Fy Ay + Gree bok. (10.22) n+| neo} ati net [w Fey tat Fle) ] Aw [eGo eag@ia] = +b | et (10,28) and ~ ~ Fe Cet, (10.24) vi- NE [-Gmrrar Fey] C, + 2, Kee 1 =I (10.25) ~ By solving (10.22) and (10.23) for A, and C, , one obtains Re = ASB Lie Ese) tar 0.28) . +)" (, oz Lee? (| - 4%, \t NF la) 7) (10,27) by using the Wronskian of the independent intepre1s of Bae (S04). With Cy given by (10.24), tha constant Ky can be solved for from (10.26), it is ry ~y A, =7 bt) | ve pt, + bps 4, 5/5 te) | (10.28) The soluticns Wiq.8) and H (4:8) in the whole domain under consideraticn sre uniquely determined. Since the dominant properties EGTRICTED of the hypergeometric functions discussed in section 4 hold in general, consequently the equation of convergence can be similarly settled, 11. Transition to Physical Plane «= In the previous sectims, it has been proved that for a given incompressible flow for which two asso= clated functions W(4.9) and A(49) are defined, there exist two associated fmotions W439) and X (4%) for the corresponding compress= ible flow, depending upon two parameters Y end T, ° The question is whether the associated functions W(4.9) and X(4. 5) belong to the same flow pattern in the physical plene. To answer this question one has to fall back once more to the fundamental equetions (2.5) and (2,10). On substituting dx and dy from Eqs. (2.5) in Eq . (2.10), the relation between W(4.9) and X (49) so obtained leads to qs. (2.12) and (2.18) if ; tp KEEP. This is, of course, the Legendre transformation and can be regarded ag an equation of compatibility. Since when V (4.9) is given, ? (4,9) is known by solving Eqs. (2.12) and (2.13), Hence if X19: 9)» satisfying Eqs (2,15) and approaching to Xe as C,-~ oo is to be associated with Wig.9) for the same flow, then it is necessary thet the equation of compatibility be satisfied, lIxcept in the case of section 10 where the complete W(9.3) function was not discussed, this condition has been properly considered. Once the relatimship between W(4,9) and X(4,%) 4s established, the next object is to caloulate the flow pattern W¥) = const, in the physical plane corresponding to W(4.9) and (49) - In the first place, te must recall the fact that the transformation defined by Eqs. (2.1) and (2.2) is generally one-to-one, Suppose RESTRICTED RSUPTC TOD 5 that in the hodograph plane there is a line defined by Wi4.9)= Const. = K (11.1) which will correspond to a definite, or a definite part of, a strean~ dine in the physical plane, The stream=-line oan be obtained by eliminating one of the two variables in x(4,0) and y(4,@). To do this, first the Eq, (11.1) is solved for 9 , namely J=9 (4K) (11.2) provided that Wo (4,9) #0 . Introducing this relation into Eq. (2.5) which, when transformed into polar coordinates, are ac ne (11.3) 4 x sing + “23 A (1104) one obteins a parametric representation of this varticular stream= line corresponding to V (4.9) = MK in the hodograph plene, PESTFICTED Bae Part III Improvement of the Convergence of Solution by the Asymptotio Properties of Hvpergeometric Functions 12. General Concepts - The significance of the general solutions cone structed in Part Il of the present Report when viewed from the practical point, rests in the fact thet thay constitute an oxistance theorem. It shows thet an irrotational isentropic flow about a body can be ob-= tained from the corresponding problem of an incompressible fluid, if the free stream Mach number is not too hirh. However, the solution in the form of a slowly convorgent infinite series cannot be conveniently used to obtain numerical vnlues, as the labor of computation would be prohibitive. By examining the infinite series obtained in Part II, the essential difference between the compressible flow solution and the incompressible flow solution is seen to be associated with the fact that,while in incompressible flow solution the individual terms of the series are of the forms q cos po q cosy F any 3 ay & } ? in compressible flow solution the individual terms of the series are of the forms v Cosy > a cosy 7 4 Jy) anys: q 4 Ainy § If it were possible to writs ~V q” F,'v) = raw] q g oF [aw] then thers would be no difference between the incompressible flow solution RESTRICTED end the compressible flow solution except the "distortion of the speed" q by the new scale . . In fact, this possibility is realized under the special condition of /Y=-1 as shown by von Karman (Ref. 1) and Tsien (Fofs 9)e For the case of isentropic flow with the general exponent 7 there is no such scale factor %.» However, if Vv is assumed to ba vory large, then there is such a function 9, at least to a first approximation. In othur words, the leading term in the asymptotic representations of Git) and Y (v) does give vthe desired form, Qn the other hend, the use of asymptotic representntion necessarily implies 2n approximation. But this defect is not difficult to remedy as the difference between an exact hypergeometric function end its asymptotic form can be added to the approximate solution as a cor= rection term . Since there are an infinite number of terms in the series form of the solution and each gives a correction term, the correction terms also constitute an infinite series. fhoerefore, the original infinite. series is now transformed into a closed function plus another infinite series of correction terms. At first sight, such a transforma- _tion seems unable to give a result thet will avoid the difficulty of prohibitive computetional work. But actually, due to the good epproxi- mation given by the asymototic representation even for moderate values of y , the correction series converges yery rapidly. A few terms seam to be all that are necessary. Thus for all practical purposes, the original infinite series is now converted into a closed function with "speed distortion” plus a few correction terms. The fundementally interesting point is that for the case of a general exponent / , the simplo method of speed distortion will not give an accurate enough solution (Cf. ref. 8). PESTRICTED RBSTRI STE, 57, The chenge in type of the differentisl eountion at the sonic speed niso introduces 5s singulrrity in tre speed distortion function Q. However, by using the correction terms, the effect of the singularity ean be limited [Oo @ Very nerrow range in the neigrborhoods of sonic speed and no practical difficulty is experienced, This will be made clear by the numerical exemplo giver in Part V of this Report, 4 13, Asymptotic solutions of the Hypergeometric Zquations ret Ute) end Vie) be two new dependent variatles defined by - ye! “a Yous oF wot thee, (en) : -£. ~ vet 2 The differential equations (504) and (4.4) reduce respectively to " 5.8 Wen- [ ¥ gms Sw) =o, (15.4) where = Jc leprOt = PT (pteay- (ity? Pie) = 4c" (i-T) fegh™) eT (J-td* Both Eqs. (15.6) and (13.4) involve a constant parameter VY which is real and positive but otherwise arbitrery for any fixed constant PB . In the intorvel o 4 t< | in which the flow tekes place, the functions Q(t) end fet) are finite end continuous except at the extremities T = 0 and T2 1. To avoid the repetition, let Fas. (15.5) and (14.4) be ree placed by a“. : Ver y™ ~ | V grey ual Uy = o. (15.5) where Up vit = UY) when a= fh 5 and Up) = Vw) when «4=-B8 » In the interval $<t < a7 1205 9(t) is bounded from zero and is positive. Horn (Ref. 17). showed that when Y is a large positive number, @ pair of solutions of the following forms exist in the interval RSOTTICTED FSCTPICT!) 58, concerned; y ad a i ¥ y ‘ a “VK -% Jove 2 | g" + bys! + ee + oct bat | , (14.7) v where Kie) = ) ¢erde ' OK TK Tex (13.8) A constant in (13.8) was left out as it can be absorbed in the constant factor in (15.6) and (15.7). This representation can be shown to bo unique so long as V remeins greater than a large positive numbor N. By substituting Vay) and say WD in Sg. (14.5) and choosing the coefficients f,, ,&) (rel snd 2; and 8 = 1,2,3, «+-) to make the individual terms vonish, Eq... (13.6) reduces to ‘ “ _ u aK $F set + K Fis ~ fa tis ~ fi , (135.9) wf i 4 mW fey TK Sse 2-R t Fars + geezer (18610) where fer ef,soeg™ The coefficients f,,(0) are then given suc- cessively by a first order ordinary differenti] ecuntion and their determination dovs not involve any difficulty. The problem is thus formally sclved. Obviously, the solution is of the exponential type when. Pit) is positive in the range concerned and of an oscillatory type when Yi) is nogative. Now in the interval 8424-3 , 8 2° where Y(t) ZO when tsEé Spit ,» both types of solution exist. It is evident thst in the neighborhood of Tay a change of cheracter of the solutions must take place; but the manner in which the transition occurs cennot he deduced from Eas. (15.6) and (14.7) due to the frilure of the repre- | apt| ° sentetion of tha solutions in the neighborhood T= This is closely relisted to the Stokes phenomenon. REST? ISTSD SUnTPRIVTED 59, The method w%s extended by Jeffrevs (Ref. 18) to include the case where (t) has a simple root in en interval under consideration and can be applied suitably to the first order of approximation. The general problem has been treated by Lenger (ef. 19) in e series of papers, considering both the case where VY and t are real and that where y and @T ara complex. Attention wa: especially given to the Stokes phenomenon and » law of connection of the solution valid on each side of the critical point was explicitly stated. In the present casa, however, only the first approximation is used and Jeffrey's method is edosted for convenience. It is seon from Eas. (13.6) and (15.7) thet the first approxi- mation depends only on Y/t,and the effect of f(t} is felt only by the higher order terms. Hence, for the first approximation only, the Eq. (13.5) can be written as Unie) = eee) =o, (13.11) ‘where Vay = Uny = J, . Thus, when y>N , tho dominant terms of the asymptotic solutions are \ -b o¥K Vive) ~ ot 2 [ ++ Oy) ] (15.12) 3) wb “VK | os td | Ulm ~ go” rs | + Ol) | * (15.13) Here O(+) , in each case, denotes the fact thet the term is uni fornly of the order. y when Y is sufficiently large in an interval , sets apri~ 8 » S370 and is a function of y” ° On the other hand; in the intervel it et <i-d » where Pit) <0. and K is a pure imaginery quantity iw where W is real, the dominant terms of the asymptotic solutions must be a linear combinstion of (14.6) and (13.7) and must be of the forms; Ua) ~ Gi coo (vwte,) | (13.14) RESTPIUTED RESTRICTED 60. (2) c . ae ge . Uy ~ qe Aun (VWs by) Geer®™*! (18.16) where c, p ¢, and €, are constants to be determined. The question of determination of these constants is actuslly the same as that of determining the mode of continuation of the asymatotic representation of the solutions in the range ie tft * i-$ , This can be done, according to Jeffreys, by considering the solutions valid in the neighborhood of t= - let x= SFr > When % is sufficiently small and Y is large, Eq. (13.11) can be written approximately as f 2 Uy wy) +y gio ge Ug) =0 (15.16) provided fe ~~} » This is mown as Stokes equation. The inde= ni P10 pendent intsgrals are 4,0) 4 > ; . _ 4, SPH, AS) 3 His . with y= Fy ing? (13.17) Consider as two independent solutions the following linear combinstions: Uri = FH dt FHP), 09-78) + 4 4 Uae Hy $0- Be Hy (3) (13.19) As Hy, 9) and Hy (5) are analytic functions in the whole < plane, this immediately provides a means of identifving the asvmptotic forms that represent the same function. Suppose first that for ang % =o » the solutions are given in Eq. (18.18) and (13.19). The same solutions for which ag %= 7 ond , + me 4s) ant me gate Uy = Fre Hytye*) + a 2* Heiser ), (18.20) i) 2 8 sm 4 ay a} are Uy (3) = Fe EH (5%?) - Fo e Hy (422) (13.21) RESTRICTED REGEN ISTERED 61. Now a me : ria (2) = TTL _ a ae Hy, (see )=- e* Hy (se) th te Hy, se*.e =2 cost Bis) + Hy 4e) : and when { is large and = - mk mg tote » the dominant terms of w 8 al aie the asymptotic expansions of Hy, (42%) and Hy, (+e?) are owe 4ee Zz) Higety w | 2 {is ogy t 45 ay % te Tw. -il{ 2? ~ ) Hy se) ~ [2 2 iso t. By substituting in Eqs. (13.20) and (13.21) and neglecting the term of lower order in <{ , one obtains by expanding at the same time Bq. (13.18) and (13.19): 2-4" cw (¥~ gre: (13.22) FO 4 8 3 aw (4- %)—> z4 2. (15.23) Here the arrow is used to indicate the trrsition of the asymptotic representation of the same function from the left hand to the right wb ct hand.member. For small 3 Fy ies gt and f~ -v¥W 3 (13.14) and (13.15) finally become Uw ~~ »% wl(vwor- | i+ Ol) b (13.24) hy ea apt (2) j Uli ~ “gh co (ywt TP) \ 1+ oy). (13.25) with C= 2% 5 Cyz-} and €,=- 2 » Under the hypothesis made above, the pair of expressions (13.12), (13.24) and (13.13), (13.25) actually represent respectively the dominant terms of the two asymptotic (a) expansions of the solutions Uy (ty and U, [t) for a Y which may be any positive but large number, EESTRICTED RESTRICTED 626 14. The Asymptotic Representation of F( ay, by; cy ; t ) and F(arp.bB,0; c ) The dominant terms of the asymptotic expansion of Ue) (23 . end U, (+t) are given respectively by (13.12), (13.24) and (13.13), (13.25). By evaluating the simple integrals in (13.12) end (13.24), i 2 the explicit expresions for the first approximation of UV {t) and Uirix) are rt) ~ Via) Vrs. Upan~ (ap) 2 pave suey 1" 2 THe) | (14.1) % oct eel. U yt) —™~ (2p) 7 4 ued 1” Tit). (14.2) 0) He yy + Up it) ~ a { tone Hee i T* tuo (yo~ HE), (14.3) gre] 2ptl 42) ab u (1-7) a). Uyo~ jae ’ t* co (VWs % ) } (14.4) © ate tye (east y 2 (et )y% + (1-ee)* r| 1 Ee eel Wits Xa tan et) The values of the function Wit) are given in Fig. 4 together with the function Kh?) » defined by Cor mM = W ° In the respective ranges of validity, each pair of expressions differs from the exact solution only by a constant factor which can be determined to satisfy the normalization conditions (3.6) ‘and (3.12). By substituting Eqe (14.1) into £&q. (13.1), these were found to be a-i zy Coy = Fg Wp) = A aah RESTRICTED eEa Te Loeb ose Thus, the expressions fer tre desired asymototic forms, when yoy , ara for the interval of tT < —L. > 2a+l yv f, v) ~ fixed) Tir) | (14.6) -y Gye) —~ FO) Tt) | (14.7) fet %. (tee where fit) = mala on > Tt) = +, [x (1% )+ (-feY"] . (14.8) (ieee y4 tay a tp ey Ue key For the interval. ~~ <t < + , they are y (14.9) Sy it) ~ Mt) Tix) Gol vyw-E) , g ~ + fic) af twa (yo+ (14.10) y(t) 2 T ty) =) : . where ry, - 2° & fey eg WD = 22h? 1, (14.11) The values of Tit) are given Fig. 5 as a function of +t together with the local Mach number M. Similarly, as from (13.11) Uyiet ~ V, [t) » corresponding expressions for F(_ at pr bt By cy) Tt ) are; ~ Y Tied ~ Gy Tw) (14.12) o¢ rel ~ -v *prl ; Git) ~ gic) Tit). (14,13) Bit "=—+ where Git) = (i- +)” * . (14,14) (i-de , RESTRICTED Ke STRICTED ee nee 642 ~ end 4 Fw) ~ q) Tis) too (Yw-Z), (14.16) at arc @ ix) Lait) —™ wa (Yw+ 2 r 4 ~ FIT w) , (14.16) ae - (i- t) 2 where Jity er (eo- 1) | (14.17) Here fit) denotes invariably the first integral F( ay.by; Cyjt ) while ,'t) , when multiplied by 9°” » denotes the second inte~ gral F, \t), defined by (3.15) when V is not an integer or by (3015) when Y is an integer, since the asymptotic oxpansions are valid for both integral snd nonintegral values of y provided y>/# In the domains of validity, the asymptotic expansions mey be differentiated with respect to + with the seme order of approxe imation. Hense, for VFA , it can de shown thet for oft <a ~ ¥ yi ~ hie) TH) Sit oly)y. (14.18) Gt) ~ tie) Tx) 4 it OL vy (14.19) where _<* -L % ~ yal Riv) = 2 (ie t)F(edey* [tery seer? 7]; (14.20) and for tents] oe 4 | (14.21) Fe) ~ thle) Th) cor ( yom mH) h+ OFF} G,,,) ~ $4) 0) to (por mre)dito(p)t » (14,22) ~a® wt where it) = 4 (i- 1j4 (42- aia (apt)* , pAit)z tw ape (14,23) The values of the functions 7 and 4,10) are given in Figs 6. REDPRIGTED seared dara 65. It is intaresting to note that when Y=-!| the constait & vanishes and onl+ the exsonential types of solutions exist . In the case of Vt) the solution is exact, namely for =- 7 y : = 2 Fi) [ earcae , (14.24) “VY — 2 Fy) = [ 1+ rope | , (14.25) of which the first is in agreenent with tho result obtained by Tsien (Ref.9), while for X,\) the solutions which are not exact reduce to ~ y ; v Fix) ~ [i+ | "| singe | Sar ols) , (14.26) is) y2N ~ rhe, ~v 4, it) ~ li+ 2, | Laas. yi ol) (14.27) This may be the cause that destroys the analogy between the coordinates of the corresponding compressible and the incompressible flows. For Y=})4o5and y=n++ ,nbeing a positive integer the three . - ~~ ~ ~ ~ groups of functions Ff, tt) ZY Ft)» Fy) and Hy, , Ty, together with their asymptotic expressions werg calculeted for t varying from o to o5 and n from oto |O0 , The rdsults are presented in the Tables 1 - le. The behaviour of the approximation is illustrated in Figs. 7 = 12. It can be observed thet the degree of approximation of the functions increases, on the one hand, with y for any fixed T e On the other hand, for any fixed », the spproxe } imation becomes worse as t approaches the critical point T= Ter > t corresponding to the local sonic speed.. On the whole, if ths RESTRIVTED vw oO ¢ F a in gy . ‘ : ’ critical point % = is not reached, the agreement cnn t aor! 3 ; generally be regarded as excellent, es7 pecially for larger values of we 15. Transformation of the panipmeil © ): Branch-point of order 1 The recs Hier ) for a Plow t E PORrenens & branch=Joint of order 1 was given in Sections 8 and 9. Ths forms of representetion, as can be seen, ara not in general suitable for practical calculetion. The saalliliatal is two-fold: First, the series involves infinitely many 2 — functions which themselves ara in turn definsd as my inite series. The convergence of hypergeometric serias generally oe fee reeses with an inerease of the parameter Yo. This means that it is io) i+) very difficult to compute the value of the later terms of the series for Wow, }e Secondly, the convergence of the power series defining the function Wi wz ) itself is, as expected, very slow in the neighborhood of the singularity. To increase the convergence one resorts to the following method: Observe that the corresponding function for the incompressible flow thet has the ‘same character of singularity is fi) Wow = BAnw wis which is absolutely one ankiormny convergent in any closed domain in jwi < U . Now, if in (8.5) Hie) is replaced by (i?) me Foie = ) Et 04th (26-2) ei r hw thas where 4igys Tid/fiay as by hypothesis, aks eal clear that () ; W iwyo) = zo a Aw (tw) ‘ {tw|<U (15.2) nie a Sean Lt ¥ aa | oS oC e which is also absolutely and uniformly convergent in the ssme domein . Wow) — ‘ tt al $09 Wy es 5 and consecuently (15.2) will be denoted by la) (tw) « In i { doing this, however, one violetes the restriction thet (15.1) holds only when n is greater then ® large number N. The error can be removed by adding and subtracting from (8.4) the quentity given in (15.2), then it follows immediately thet (0) wit 2 Wf — i ——_— (15.3) where . : fit) se) = Lt Wetwie) = res We ltw , = i) ad n We ity = S AaGata ww", twice (ase) nio with oe Galt) = — fw +" G F,, t) te Kere n is a positive integer. The function W wie) is then represented . . . © by the sum of two functions W(w;2) which is of closed form and W, (wie) which is the difference of two convergent power series and hence is also convergent. But, according to the theory of asymptotic expansion, G,(\t) tends to zero as n approsches to infinity. In fect, Git) is of order wn , therefore, the convergence of Wiw32) is increased by the order of yw. This actually is the gist of the whole problen. As the form of the representation of the hypergeometric function given in Zq. (15.1) is valid for all tin of ore P W tw;~) given by Ea. (15.4) holds automatically even outside the circle jw/=U. For this resson, W, tw ix) should be idanticel in form with that darived fror, Eas (9.4). That this is the case can be seen from the following aomsideretion. For, in addition to Zq. (15.1), if it is essumed that . -v Gy — fic) T (tc) J (15.6) RPOTP PRD FESTRICTED packeuestin laa on sliascencid ao ($<) o it follows that : =- ( = 1- 4°, Fe (15.7) 3a) Tym) ja : then Eqe (9.9) and (9.10) yield, by Sq. (9.11) and (9.12). * is Bu ff * By using these sets of approximete coefficients and replacing Ft) and G kx) by their respective asymptotic expression, one obtains the following relation with the aid of Eq. (9.3) >) {0} (wj)t) = Wow; + We: x), (15.9) where o _ & (i) a ey 2 wit) = + Ps 4 G, ) m «© Gs (t) W ‘4 (15.10) + * In this case the coefficients 5. and Cu a8 well es the functions 4) and G J) used in deriving W, 15%) are approximete. W) (2) Hence, if both are corrected Gy tu) and Gy (t) should be of the forms Q@ Ge (t) \t 5 ABu Fy tt) + PAT HAE ; (15.11) ( _ * o. , Gi = AC Gi) + Fen) Th) 4 ke) » where e> * ¥ At. = Ba.” ps am u) , Af tr)= Fw- fie) Tex); (15.12) ~~ * a Y : _ Gi -v A Cm = Ce = Fu “Te ’ A4 = iO) ~ Fle) Te) { * ¥- : Here the differences A Bu and A Cx _ depend upon the con- \ dition at infinity for any sets of constants B,, and Cu while those of AFjxy end AG fx) are functions of t only and, for this reason, can be tabulsted once for all. It can be also shown thet the order met eeTricrs 35.4 alae = a ae. arn £ 4 bosaense +h es ee 2 of ABw if at least of WwW and therefore the convergence of -t (15.10) is again increesed by * .« (4 Consequently, if Weq.t) = VW, (4,9) + Wi44.9) where the supsrscript (() denotes either (i) or (0), and if the coefficients avs real the stream function for the subsonic flow is according to (8.6), given by . tw Ek tees V,14,8) = Fe, (24.9) ; ete é ge (18618) ) = w , n>o ond in USGS VY ao (0) iis 7) v a) ~Vv ; 2 (9,9)= 2: [ Gy (w) 4 + G, ce) g Jews | (15.15) with 9 restricted by of 9 < aw ~. This result is striking in thst for: T=, » Wiu, Fz Vieu,9) as G,'%|)=0, i.e., the function VY, (q,9) represents the correct singulerity of the exact function. (x) Far eway from the singulerity the term V,«4. 9) gradually comes into prominance especially neer = ptt but the convergence there is already so rapid thet a small number of terms is enough to secure © high accuracy in W 4. 3) . On the other hund, if in (9.4) F, @) and G ic) ere replaced by Fyit) = fie) T te) Co (yw-B) , (15.16) -” 4c <]| 2ptl ~¥ ee of - = Ywr Gc) = F fey Te) co (yor), ba 88) where fir) ? T(x) and Wet) are given in (14.11) and (14.6), then Li vs hs et pe pio ta] ilyw-F) -i(yw-t Fay ezterie + 2 al and substituting as before in Eq. (9.4), we are lead again to the sum of W (w;t) = oe id [a1 Bi, ware Cattwery' + & ay { Buwe y” * Ge “we” ) Tl & re) t i2) = ee bn <i gre) = L >, 4 Gpoww G, *)w ; , 2pri -c) can also be summed: W wire 4 2 | 2° = W, (twee ‘y+ i Wotwe> I]: (15218) Furthermore, from (14.11) it can be seen thet |twl=AU , \ being a — _acopy* | | (tay ( (2p, Y2 Tit,) # which is a function of the Mech number and the characteristic constant 6 of the gas but independent of the shepe of the bound*rye The velue of this function \ is given in Table 13 and Fig. 14 for Y=] 405. As a consequence, the functions constituting the streem function for the supersonic flow e & “4, Y, (4. 9) * sh [ V ,( 40) + Wi) + Fl aewl- Fle-O} , (15.20) §-W 20 a oo -y VW, (4.8) = > } Gy mq + G. ‘ex) q }cove U<g<V_ (15421) Here the functions WV. and }, are defined, on ace mnt of (15-19) 5 VW, (940) = V(yu, 924) , GO, (9) = Pau, st) . (15422) PPRAHGID Testa fat par ore 5, 5 Oe GU 2 Tle where Q and VY, are the velocity potential and the stream function respectively of the corresponding in-compressible flow. The functions (2) 5 Gy't) and Gy,l<) are the seme as defined in (15.11) except that AF, (c) and O4§,!t) ere now given by AF, tt) = Z, 2} £2 feowa-Z) , BG a)e Ge fe) Feiner) , 416-25) Unlike ‘th previous calculations, Gute) in (15.21) is not of the order of v! due to the presences of 1/2 in front of fie) Tcalv-Z). This, however, does not offer 2 serious oe es the series in which it aospears already converges with (t9) » t being less than It is worth noting, moreover, thet in the hyperbolic domain the function V9.9) depends, aside from a factor fit) , only on the two independent families of characteristics defined by 75 S+w } 4 = 9-H , (15.24) This result is most striking as it shows that the main part of the solution satisfies the simple wave equation and thus clearly demonstrates its hyperbolic character. With both the incompressible streem function NA end the incompressible potentiel function appearing in the solution, it 258 LapoeerbTe to esteblish « simple relation between the incompressible stream lines and the compressible stream lines. Since such 2 simple reletion is the foundstion of the so-celled speed correction formula for e quick estimetion of velocity distribution in compressible flow from thet of incompressible flow over the same body, this idea cannot be extended to supersonic regions. ta & sae va beg Cw Le | st & Be to o ~ Qn the other hend, this also indicates thet although the differenticl ecunticn r Wiq,0) is hye gerbolic in the supersonic range, it cannot be reduced to the simple wave ecurtion by ® mere distortion of the speed scale es given by the function @(t). For if this were the case then a (9.9) would constitute an exact solution without the additional (ONG ae This fact is all the more important as th additional Ye: 4) igs not emall in comparison with V4, 9) for the mixed subsonic and supersonic flows, especially for the trersitionel region near sonic velocity. However, in the case of pure supersonic (9) ” fiow, Vv (9,9) might be small; then W (4, 9) alone mey be used as a nt satisfactory approximation. Of course, when Y=-! ; thetis ag in Cal the corressonding ¢*se in subsonic flow, the exact differentiel equ“tion for ths streem function can be reduced to the simple weve oO f a. & y a ct pe Q pest . oe ts is] co Pe pes s cease, the appropriate form for the spesd function’ ~{ —— wig) = - tan _ (15.25) 7 on where the subscript 1 denotes the conditions 2t the point of tsngency of the trus isentrooic curve end the aporoximating tangent. This ue ° agrees with the result dhtained “by wy Coburn (Ref. 20 16. Jontinuation: Logarithmic Singulisrity In the case of the logarithmic singularity the function W tw;7) was broken up into two parts of which only the one thet characterizes the singulerity wes given in Eq. (10-6) and (10.6). As an exemple, “it is preposed to show chet this srobl em can be treated by the same thod. If the same appreximaticn is introduced as in Eq. (15.1) [om and (15.7), then the coefficients defined in Eq. (10.11) end (10.12) pe ag ee a oa +. ean panies: once q ae m3come Apprceximately: — ~ . Te a es 616.1) A <=: eit ' c = exes ‘ ( 164) * “ fen) “en O™ teayy face x —_ i % , , with Bfiy=t; so chosen thet the form of Eq. (15.14) is agsin preserved. With these coefficients and if one writes for the function inside the circle 4 = Teed _ val 3) wi By 4 ) reduces to the sum of td Ae] ° om a) re) ° en V (4.9) = = Wi {t9,9) , pa a (16.2) ' \ oe) oo Ww 2199 = SHG I(t) cons | geu (16.3) ven here Git) = Fiz) AG ie) + re ant | iti (T%)) 7 (<3) with ay = a Af, '&) = F<) ~ fey Tex) » AG ia)\= fo) fey Tie) - (16.5) Similarly, in the case of Ev. (10.6) it reduces to Vga: = Viae) + W%CG8) + 7 (8) Here Wig9.8) is again the same as (16.2) while VY, (9,9) is t Were, = (u. «4h + ES Log 28 ‘2% &. (4 9) cosn 9 (16.6). $tT) fi Ty where ~(o) a n Gat= Fe AF a) + fasta) AG te); (16.7) . a, 1) i. with A Ie) = n — Tite) - Th, Fis i) #4) fiz) ' AG tx) Git) - f(x) Tit) . (16.8) ww ~~ Unlike the previous casa, Wiq.9) = ¥,09 8) when and only when GC, tends to infinity. Dus to (16.2), however, the singul-rity of W418) vemsins unchanged. Again, if in (10.6) ae = ty gee ac Ge) = ase) Try wo (ww+F) is substituted for @ ic) , it can similerly be shown that. ; - - Me 0) = YF2[ Dowie Kew) ~ Brows Bls-v)] (a8. yi | f, fae , 3 to a id ae Tt. Y,~ (9 9)=-F ar, feore +2? Fay agh= tt) + Ge (A esn®, (16-10) 1 an =i where NY, (sw) and B,(0tw) are defined anslogousiy to (15.22) and ry \e) AG.) in w(t) is now given by AG we) = Fit) -4 fey Tey co(nwet) , (16.11) This scems to indicate thet the results obteined so fer for W(q,9) ulte general: It may differ for different cases, at most, by a constant factor. The general property, however, is not shared by V4.8) whose character changes radicelly with the nature of tho singulerity end the shaps of the boundary. Hence its import»nce in the present problem is evident 17. The voordinrte Functions x(94.8 ) and y(4.2 ) Whenever the function (49) for a boundary problem is determined, the coordinate function x(q, 8) and 4(9,9) can be calculated according to Hqs. (11.s) and (11.4). Suppose, for instance, © boundary od a i) 2. *4 he ct ee signec . the property that the function AlWjt) is truly described by (8.7) and (9.14) of which the resl pert X64 9), defined within the circle |w([=U ,is oe) ia Tumor -PeD 8 ~ j = = urd) Me (17.1) ae) =, A. F,. it) | Cond , 94U Bae: nro where the constents A iv ee Lt > (8.7) sre a / Fi) . lutely and ; mid Sin res] and He are regserded = ; wW 2s known and 4 it)= As the serie > is ab Qo tf uniformly convergent in q4<U., it cen be difforentieted partially term by term with respec - % ‘hen the diffe coefficients X9'4s 9) and Aah) are substituted in Eqs. (11.3) and (11.4), there results: calculated and are - r “4 (G5) = zs + Aw a @ fet (n-1)9 - ptm ie Fe gteondend , (17.2) 4< U oo a 4(4,8) =-S nf wt. 7 FF asinine ae a + Fig Conddin 9 23) nai " we barptl > Cy tli where J, (t) = Fhawrpel horert 7 Ow FIED F l@ars os Bae” | d Cn 5 T,) a “a . Ba It is interesting to note that the stsgnation point which corresponds to J=0 remains fixed in the ashysical plane for both the incompressible and the compressi cen b ible flows. For it ye verified that F(t) = Ss) BN Hy rence Z(w)s Zlwjt) when | wr | =0 Now, since KelG, 9) = a in Cea (n-1) 9 Yoiq9)=- 5 0A rR) Ss ray x t! nrIh oo 5 n G.9)= SAF ain nd - N=I by introducing the aporoximetion given by tqs. (14.12) and (14.18), i.e. dy a 4 {a la a ct a d Ft) = fie) 4") ae eres ° sr da fi Line DY Geraning K(IQ.9)= Xj 9,9) + x6 709,9) , (17.5) yiQ9d= YQ) FHC Gay | (17.6) it can be shown by the seme menner that (cv) , % (4,9) = 22 tee) x, (4q.9)- BE WO (29,9) C009 | (17.7) 4 fm) of BSG $6419) = Fey ti We. (9) - = oe ot Sain 9 (17.8) and Ps) wi x4, $y= Suh 7. 4 — peoanh Gu Jeornd end, cl 7 a2) Heh 4<U rer ‘) vy yg,9) = F wh. Glo) Yai tw-i9- peaanA li condains, (17.10) vnere Coo eee. (17311) F lan 1 Bm } Ca 75) ‘f) . G deja Beh Eldar bel boots lit) tye, (r7.12) in) n+| AT a ay rt) pt . . ‘ Rie 9 = OTe | (17.78% 5 ? DQ , Bra) On the other hand, the exorgssion for XM 9) valid outside the circle of convergence is = ce = Vio eG ‘a we ; AG8) = > [BLE 4 - Cy Ger [eens , Uiteda? n= : ” 2 * 2 “ Ps provided the coefficients BY end CX in (9.13) are real. The functions X(q.9) and “4(q4,9) corresponding to (17-14) can bs found similarly. These are: oo -v-! Alay = 5 | vB. Za 4 ainwwiyg ry Ch G,10) 4 aia iveiy 9} nro _ -~ is * (17.15) ~ > (vant F, OF + VOT HLG, eZ fatnvs co08 , V<4<V 4 (4 ,9)= = {vBx Ft) 4 cosiven$-vCx G sx) q coven} (17.16) o as ~Vv-{ - ; g* y-\ we (t Pe AF bn ar dd Esq + Ch HG, 109 pommvsaing | Here the constants Be and Cu satisfy the relations (9.14) and (9.15) and can be reduced to Br = = Tie) ' oe icy Tt) (17.17) 1 provided the same approximation is made as in (15.8). Furthermore, g oo = y-t ~ vel eto) >, \v Bu | <n Ain \v-1) 9+ V Cn Gain (yei)9h , nao os 4, (4.8) = a Lv By q ‘wee 9-¥ Cy q" co WHIT n= and if Fi) and Z, ir) for the high order terms sre substituted by the asymptotic forms, viz. ~ -v ~ 235 Vv ess { FiryEgeTe) ; F,,, = hc) Tit) / 22% <aeri shsn in like manner (17.15) and (17.16) can be trensformad and can a@ach be represented by tre sum of two functions X,{(4, 8) 5 4, (4. $) » X3(4.9) 5 4,@.8) > where x, and y, are the same as (17.7) and LLT.2], while x, and ¥e are: Xo (409) = >, Gy eq ainva* Gy eagiatnioe (17.18) pe Ev Gueod a0 | Yornys 08 , RESTRICTED es mv. . ar it Na os v3 G24 cos(v- —_ CG: eq ‘caved 8 (17.19) nio - 7 and =~ W y-l Pe -v-i Sc “oer - et Divi G4 + Gy) 4 J Ainy 9 aims nMro re cs and os are defined br: mes a Ov = . Gye = SBA Foe Got TAF, Gyley Cn Gert See 27. 20) >W) aay Bu -Y 2 CRG 4 CAD Gy, = ABs =n vl + Tay > Fi, Gui =A On var de, 1 > v-1 . oo AF, oO ae a hit) “T te) 3 ‘ (17.22) AG,,'9) = Yet G,o- BOTH) , Y- and AF,\t) are defined just the same as those given 4 $ 5: : Loe 4 a Similarly, if the hypergeometric functions involved in Vs - v ~ . v Fit) S eT eo (ve-E) , F, w= tit) T coo ( yw-m-E J -V : ~ -v ee ‘tile oot ia Gey Sb get coolvorhy, WIS ZAOT co YOer a); 2 sim (v-1) 8 Coo (yw-T ) = AM [ (y-1)(G+w)+ (w-2 I+ Arm | ( ¥-1)(9-0)- w-B)] ; 2 Atay (v4l)9 Con (var B) = Avr [ (vst) (9+) - (w-3)] + am [tv 8-5 \w-E)| a brief reduction gi 5 OI Pe Queuis ae X(4,9) = tbe | [ Kol sewe Kea-oi}cos tf -a)-[ Ye (8+0)-Yie-ey} eral -us)} “e mal [ Qasr) idolfeolargy] Os) OF) ain Feu)} cos. , , t+ felu, Yq) = He) +, {[ Yseore Ys 0-01] cm( F-a)+ | Rates -K 8-9j] oi tg -w)} ~ BE His} {[ Lorre Lye-wy]em[ur)-Leusrl-QG-w)] ini g.w) aim 9 44 by the fact that at = AU in the interval under consideration. Here Yl 9EW) = Xo(AU, 94H), Yu 82w)= Yo(AU, 94H); @,(9£0) = @,(AU, StwW) 1 SHLo(8tW)= Nolay, 240) 5 whera @.(49)= Be 0a ’ lo) Pel ala rq =i. %, 4,8) = D5 v4 Gi) 9 Ma lv-)o4 Gy tt) 4 Ain (v4i) 3 i “a6 C17. 25) oe. ~~ a v-l See ) . t 3 me -6t Dv { Gr + Gy) 4 ere = f shes Le 2 | to) Lad vel v-| 7p ~ G47 ss Ve 41,4872 >, v} G, 9 cas (v-1)9- Gt) 4 cortvei) | (17.26). Ware po =) Vel lai ~v=| . - ar Sv) Gilt 4 + Git lanvd eins nie ~ ) ~ ia)” ‘1 where Gi i) and Gur © retain the definitions given in (17.20) and (17.21) except AG le) 1 BFy,)'*) DY, t&) and dG,,,'®) are replaced by AF, it) = AG x\ = G se) ~ oy Y FZ, (2) - 4 Jie) T cw (vw-Z ) i 12) + eo (Yw+e) AG,,; be noted egsin thet the seme ac those of Ad, t+) A Fy, {t) = ae x) = V+! im v BOT cw (yw-m-E) ae - ew ney 5 orders of Gi! and and eB Part IV Criteria for the Upper Critical Mach Number 18. Limiting Line and the Breakdown of Isentropic Flow The solutions constructed in the previous sections ere known to be regular in the hodograph plane except at a few singular points. It is also known that for the limiting case of infinite sonic speed, or tg-—F &* » the solution will give the desired flow pattern in the physical plane. When the sonic speed is finite or when the Mach munber of the free steam is different from zero, one hes no guerantee as to the behavior of the solution in the physical plane except the probable continuity of the flow pattern with respect to the free stream Mach number. It is found that such continuity in the flow pattern actually exists up to a certain Mech number. In other words, the pettern of the compressible flow is only slightly different from that of the incompressible flaw up to a certain Mach number at which the so-called limiting lines appear. At the limiting line, the acceleration of the floy is infinite and the flow is reversed. It was shown by — (Ref. 12) and Tsien (Ref. 2) that, without considering viscosity, the flow cannot be continued across the ma limiting lines,and a forbidden region is created in the space where no fluid can enter. In othér words, eontinuity of flow pattern fe 1 e3) exists up to a critical Mach number beyond which no isentropic flow is possible with the imposed physical boundary conditions. The breakdown of isentropic flow, or the compressibility burble, can be effected in two ways. First of all, the acceleration in the TRICTED fi wa 4 neignborhood of the limiting line is very large. Thus each one of the following factors gives apprecieble sltsrstions in the dynamic relations: a) Viscous stress due to ordinery viscosity of the fluid (Ref. 21); b) Stress due to expansion or compression of the fluid, or viscous stress due to the second viscosity coefficient (Ref. 22); ¢) Snall but appreciable relaxation time required for the vibrationel modes of the molecules to reach equibibrium state {Ref. 23); a) Heat conduction from fluid element to fluid element. Secondly, the isentropic flow can elso break down — tae appearance of + steal waves. The breakdown of isentropic flow is associeted with the introduction of vorticity to the flow. Thus the flow becomes rotational with part of the mechanical energy of the fluid converted into heat energy. All these factors tend to increase the entropy of the fluid and finally to inereese the drag of the body. Thus the critical Mech number so defined is of great physical importance to the aerodynamic characteristics of the body concerned. | — Of course, the isentropic flow might break down due to the instability of flow fluid with the final appearance of shock weves. Furthermore, the action of boundary layer and possible eondensation of one component of the fluid” on the flow might also lead to the premature destruction of the isentropic flow. On the other hand, shock wevescan appear only in supersonic flow; thus, if the speed of the fluid is everywhere subsonic, tee aA tream Mech, oy there is no danger of ihe compressibility burble. Hence the free * . — ; ‘ The phenomenon of condensation shocxs due to water vapor in the air flow around an airfoil was first brougn to tne attention of the authors by Kate Liepmenn who observed them in her wind-tunnel experiments. RESTRICTED RESTRICTED oO ta number for the first aosearence of sonic speed in the field is called the "lower critical Mach number" while the free stream Mach number for the first appearance of limiting lines is called the “upper critical Mach number®. (Ref. 2) The latter is always higher than the former due to the fact that limiting lines appear only in supersonic flow. The actual critical Mach number for the compressibility burble must lie between these two limits and depends, among other parameters, upon the Reynold's number of the flow. 19. The Condition for the Limiting Line. At the limiting hodograph, or the hodograph of the limiting line, it was shown (Ref. 1, 2, 11 and 12) that d(x.4) ~_ (| RY 2 t-Lyy le ; du) Lea) lv (a ae) Y% - (19.1) Since the factor before the term Vv, is positive for supersonic regions only, ¢ <4, where P#O , the limiting line cen appear only when the local speed exceeds that of sound. It should be noted that the vanishing of the Jacobian is the condi tion for the failure of the hodograph method,as the trensformation (2.1) and (2.2) wowld no long- er be one-to-one and continuous. Thus, the appearance of the limiting lines is taen the physical counterpart of the singularity of the trans- formation. a . As Vir) is known, Eq. (19.1) defines two lines in the hodograph planes y | 1- T — at ee Ve- Ve =e? ae ) y ac [Et | + We =o) e8 (19.3) 2t-| RESTRICTED re Geometrically, thir exoresses the fact that the straem line (9.9) = const. and & ecnaracteristie curve belonging to either family heve €& common tangent (Ref. 1). The problem can then be fornulated based on this property: the necessary and sufficient condition for the existence of a limiting line is that there exists a solution between the tro simultaneous equations 4, ¥ 39.5) It fo) (19.4) \\ fe) =e ‘ or at { Fels Bae) (19.6) Wor; (19.7) where W(t.9) is a definite branch associated with the largest possible t for 6 given boundary end a free stream Mach number. The zero stream line is chosen as it generally gives the highest velocity and is the place for the eerliest appearance of the limiting line. 2? Generally, these equations may no 23 possess & solution for & known function W(r,8) when the peraneter M, is assigmed. This means that there will be a system of boundaries corresponding to a sequence of values of Ms for which the limiting line does not oeceur. The first Méeh number for which Equus. (29.4! 4nd (19.5; ol heave a solution wil r be defined as the upper critical Mac. number and the corresponding boundary fo s the critical boundary he actusl solution of the equation is, in general, difficult owing to the fact that V(e.9} is, in most cases, represented by an infinite series. ) CA x an b TRLCTED However, if the stream lines are determined in the hnodograph plane for the calculetion of the shape of the body, a simple graphical test of whether there is s point of tangency between the zero streem line and the characteristic can be casily made. On the other hend, if the form (15.30) and (15.21), for instance, is used, an approximéete analytic solution can be obtained without involving much labor. 20, The Approximate Determination of the Upper Critical Mech Number. As can be seen from Section 15, the importance of VWe'C218) relative to Vv (t, 4) will decrease as — recedes from the eritical circle to pe towards the supersonic region. For the first appearence of the limiting line, + is almost always high, especially when the boundary is a slender closed body. Let this be vex 2 (01 x ; * : I oe eglect } t the case, then W, (2,9) can be neglected in comparison with ¥ (2,9) and a great simplification is possible. The zero stream line cen then Wits) = Yix,9)=0 Furthernore, & simple reduction shows thet the two pairs of equations, (19.4), (19.5) and (19.6), (9.7) reduce respectively to Bly) + Yly) =o» (20.1) By) + Wiy= $.i1)- Um) (20.2) or Biz) + Wiz) =° + (20.3) $.l7) - Woy) = BUG ; (20.4) waere 5 and 4 are the characteristic parameters defined in RESTrICTED RESTRICTED Eq. (15.24). This reduction is made possible by the fact that f(t) never vanishes in the interval A £et</ Whenever the stream function YW and the potential function & of the incompressible flow are given, the functions WV, and §, can be easily obtained by substituting AV for q according to Eq. (15.22). Then, since \ decreases with an increase in the free stream Mach number M, as shown by Table 13 and Fig. 9, the upper eritical Mach number will be given by the lergest value of that gives a solution either of the Fqs. (20.1) and (20.2) or the Eqs. (20.3) and (20.4). An analytical solution can be made as the functions W, end G. are quite simple. There is, however, en interesting direct geometrical inter- pretation of these sets of equations in the physical plane of the incompressible flow as shown by Fig. 10. According to Eq. (15.22), the functions W, and ©, sare the stream function Y, and the potential function F, at the constant value of the speed AY . Since DI 9 for the body shown in Fig. 14, the constent speed MU curve G forms a loop symmetrical with respect to the y-axis. The variables are really the angle of inclination of the incompressible velocity wensbor, Along the constant speed curve Gy from the point So to P, the angle of inclinetion of the velocity vector is monotonically decreasing. -There- fore, the parameter of the angle of inclinetion can be replaced by the distances elong the _— CG, « Let Bg. (20.3) be satisfied at the point S$ = So 5 then | BUS )=- Wels) . | (20.5) This means that, ¢t the point §& = 8, , the rate of chenge of the otential function @ elong CG,is equal to the negative of the rate of change of the streazr fimction V, e Since potential lines end BS TRICTED Qremg@rersan a: stream lines in incompressibie flow forman infinitesimal sauare mesh, this eondition requires that the angle between the tangent to the curve Gat & = &, be 45°, as shown in Fig. 10. This is easily seen 2 by remembering that from 59 to P , the value of the stream function increases while the value of the potential function decresses, due to the indicated flow direction. Thus the point S, can be easily 2 determined by this graphical condition, Eq. (20.4) can then be written as Bi (s)- Wats) = $15.) + Vals.) (20.6) If this condition is satisfied at point S,, then the condition for a limiting line is completely satisfied, A similar graphical interpretation for the Eqs, (20.1) and (20.2) can be worked out for the side of the constant speed curve lying to the right of the yeaxis., From these considerations, it is clear that the upper critical Mach number is the lowest free stream Mach number which gives a constant speed G containing two points, Sy and Se, defined by Eqs.(20.5) and (20.6), RESTRICTED Application: Elliptic Cylinders ey 2le Preliminary Discussions. This part of the Report is devoted to the application of the general method, developed in Part II], to the study of the flow of a compressible fluid around an elliptic cylinder. According to Sections 8 and 9, if a solution was constructed about the stagnation point, the continuation of this solution would reouire that the eonditions (9.5) and (9.6) and hence (9.9) and @10) be s etis fied. These equations involve two sets of hypergeometric functions with parameters m and mn + #, as. well 98 their derivatives. Te shorten the lengthy calculations, in view of the limited smount of time avellable, the following approximate procedures was adovted. Given the domain@Y, the solution valid in tha anulu: Ui “y Go oR fe © (2) » retrer than thet about the stagnetion point, was first constructed. s The constants whish determine the Leurent expansion of tha solution, for example, are now assigned and, consequently, the sat ar hypargeomesric functions with interral parameters is nst “ 3 ‘gsquired.s The difficuity, however, is the q: Q convergence. This ‘ stringent continuity conditions given by Eas. (9.5) and (9.6), snd to the ontinuation may not be possible owing to the requirement that the function must be regular within the circie c 5 U. This, however, does not offer a serious objection from tha ractical point of view. In the first place, the summed function, P p 3 for instance etually holds even within the circle of ‘ 4,9 3 EUSTELITED RESTRICT UD convergence 44U, and the correction function, ¥(4.9) , is generally small compared with W(q.9) due to the close asymptotic approximation of the hypergeometric functions in the elliptic domain, In other —" although the solution within the circle of convergence strictly represents a different flow, numerically it approximates vory closely to that defined in the annulus region, In the second piace, since this region 4<U is relabively unimportant in the case of mixed flow, where 7, is very much less than Seri g BeGs Tor free stream Mach number considerably less than unity, the inaccuracy of the solution is limited to a small region in. the hodograph plane. Furthermore, the most interesting phenomena of such a flow, such as the appearance of limiting lines. always take place in the annulus region, Therefore, this modified procedure, although unsatisfactory from the general view point, is an expedient capable of yielding an interesting result and furnishing a test of the practicability of the proposed solution, . The situation may also be considered from another angle, The procedure _ in this section can be derived by replacing the functions 3) «and 3_i«) with the approximate velues given in Eq (15.7) in the expressions for the coefficients involved in the solution within ‘thes annulus region, eeZ. (9.9) and (8.10). Thus the procedure may be regarded as an appropriate method of approximation, The error introduced is generally negligible if t, << 51 « This is indicated by the fact that the correction funetion Wylq.t) for instance, is very small in comparison with W(q.3) when 9<U., Another simplification is made by using the elementary integral =2.V ay Fie) instead of 4 Gc) in the continued solution, as, inthis case, JF (+) is a well-defined function, Im doing so, the PESTRICTED BSSTRIGTSD oUs pam a eo asymptotic behaviour of the second solution remains unehanged because the first term in (t) is always small in comparison with the second, If, however, all the required hypergeometric functions are computed, there is no difficulty in carrying out the exact method developed in Part III of the Report for any accurate study of two-dimensional flow, For this reason, the expressions for the hydrodynamic functions derivec for bot: the exact and approximate procedures for the problem at hand evo ziven. In the numerical exemple, detailed calculations are made for the flow of air about a cylindrical body derived from the incompressible flow about an elliptic section with a ratio of the minor and major axes equal to 0,6. The calculations were carried out for two different free stream Mach numbers, 0,6 and 0.7. 22, The Functions Z.iw) , Welw) and Aw), The irrotational flow of an incompressible fluid about an elliptic cylinder with the velocity at infinity parallel to the major axis is represented by the complex potential [7 (2) : Wa) = t+ a! (22,1) with a> 5 7 5 (22,2) For convenience i. practical calculation, all the physical quantities Zo 4Y and Pf , will be normalized consistently throughout the present Part, The major and minor axes of the Section are respectively Ite and j-€* » Where €<!/, q 21 at infinity and Pp =| when q #0, This will automatically render the hydrodynamic functions dimensionless and the constants [[ and f, will be eliminsted #rov ths formulae in the succeeding sactions, RESTRICTED nmemoreror By differentiating (22.1) with respect to 2 , the dimensionless complex velocity of the flow is a s ~ oat ‘ 3 ~ €, Thus 4 t=- [ey , fieewlto | (22,3) 1- WwW This function is two-valued with two branch-pointe at we 1 and we € e in order to make Z.lw) a ninete~va tued function of w, the expression (22.5) is supposed to be the principal value so that {ang u-w) {<r and = (<lwile Ee . The condition Jewl< | must be satisfied, for w= é corresponds to J = 0 which is another singularity, ‘With the principal value so defined, if the negative sign in (22,3) is taken, then the domain © corresponds to the half plane fh ~< 0 amd |4|2{ . On the other hand, since the transformation (22.2) is one-to-one when |{/2/, then the domain Dwhich is RiZ,< 0 with the region inside the section excluded, corresponds uniquely to 9. Consequently, the inverse mapping function zZ,(w) is ~ gt % My Z,(0) = > { [ Mee) 4 ree] i (22,4) l-w \-ew which will be singleevalued provided a cut is introduced to join the branchepoints in such a way that the argument of (l-w) is restricted to -W2Z magnon) w and few/4| » Separating into real and imaginary parts, it is found that as of 9277 X.(4,9) =-3 {Lig.aegiaat+ é {16q0)4 514-99 | (2248) Crp Tam rT May Lot % zs - 4 414.9) = ral $-1(q.9)4 Jag} - € PP ACOESICE (22,6) -19 with w=q4@ , where the functions 114.9), 1, (4.3) , and J3(4:9) stand for: z Sok 1(4.3) = 1- (te) gen9t € 4 (22.7) » f[~12q cor 9+ 9° . 22 I, (4.9)= }- Ute*)Qer9+ E49 ; (22,8) |- 2674 coo8 + e 9% 2 Hoe T(48)= | J- 2€ gcwrS+eE 4 ‘y (22.9) 1-29 e078 + i ; On the other hand, substituting Eq. (22.2) in Eq. (22.1), the function W. (z.) is carried over into 9, namely —~ _ \- ews 2 j-w + » Wot) = - { | =] +( £5] } . (22.10) Now Wiiw) = 9 (q.9) +e (94.9) “ sun similarly h “1 - 9,(4+8) =-4,| { 114.8) + (4.995 rs 41,449) +J cart | 4 (22,11) ¥,, -\ Ja, W.i4.8) = rr | 2g.) + C409) f - 41. (4,9) +J 4-8) | : (22,12) By integrating Zo.(w) according to (6,10) the trans- formed potential function Ay l) », aside from a constant, Ro Sy rT adh ti a takes the form: \ 5 vf Aww) = 2 U-wh Un ew)? , (22,13) The principal value of this function is again defined by restricting the argument of (lew) to -1< eee er and {wi< a e Within this domain #, the real and imaginary parts’ are: , / Aolqa) = 2 [ Kegs) + Lege] ? (22,14) 04 8420 ¥ ee ; 3140) =-2> [-Kiq.d) + Liq.) | (22,15) as Agi) = Xel49) -i5 (4,9) » where the functions K (4.9) and La.) . are defined by: K(4.9) = 1- Ureyqend +e pemad (22,16) 2. h L $2 Ia Liq.) = [1-290 9 +4 ] [\-acqens+2ee] (22,17) PMeEPMT imopy 4 r rey etal ow wee ~ te ce Expansions of Welw) and A.(w). Tha function Wytw) dofined in (22.10) is single-valuad and no ex e reguler everywhere in |w]< | and hence possesses the following expansion: tw) =- >” Bux “ae. Jurf< | (2854) nW=0 2 where the coefficients A» are real and given by Ww Mi) An = 280 + lites, u 2 | (24.2) fee agneg , with ‘ se) ae St _Thwmee) Tmeg) am mo T(n-m+i/)T( mtt) However, in the region outside jwi(<J the function W.') is double valued; and when e cut is put between the branch-points w ={ -h ’ § cgeal i ied and w=e , the principal velue is discontinuous along the positive axis of reals within the annulus region. To obtain the desired expansion, the function is written in the following form ‘ Q- (i+w B atts ¢ = (23.3) * ay, 24 ‘ ‘ how (y-w JY? Ge ew) is Single-valued and continuous within the annulus region, its Leurent expansion is ~ nat ’ oo A bey 2, ) (0) in nm baa 5 = (i-w 72 ew)? = a 2 > S$ [ Ee wWw+tew 1<|wh< @ (23.4) where Ge = =i ‘= Plaeepeny Ties) 2 . (23.5) vo T (m+meriyT Omi) \ es . ah Ee 4 23 2 7 r buting (i-w J# (i-ew yt from (23.4) in (28.3) , the expansion w ho sy “a co foto my or Wheto) in the annulus region is o a Wo\“) = > [ Ba E wal % Ca. W we | ; [Zjwlcé- (23.6) n=O RESTRICTED FS ’ cr) woe when the constants » Cy, and the exponsnt Y ars defined by: n n v 8 B= 26S - ee) se, 2as7) n+ (0) 5 {o) a 2, 5B [ik Er? Saas , y=aune+t Similarly, the transformed potential function A,\w) can be oXx~ panded and is ae o n Aww) = 2 >. Aww, jw} c | (25.8) Niro when the constants A, are ~ i 4) (9 A, = Sa -teUS ee oe | (25.9) » A - pet Ore) j Aaz! ; m1 ; Pe TEE od and SS, is given in (23.2). Un the other hand, in ths annulus region the exprnsion is a 7s ae SF = 25.10 Aw=-20 > [Bie wi i+ C,w |. i |wlLe ( ) no with the constants B, and Cy defined as ~ ins (6) 2 lo 25 0t4 Be Se -theog ¢ © Ba ; uz | ( ) ~ (oe) > Bo = 2¢7S, = “ites , ~ to) (o) 2 Aly Cy = in ee we * © Saxy fo) ¥: , where §, (e’) is defined in (25.5). 24. The Stream Function Wi9.9), The relationship between the domain Q and D is thus fully established and the functions corresponding to such domains are also given. From the general schemes developed in Sections 8 and 9 the or the similar motion of a compressible fluid can be constructed. First of all, the stream function Wiq.e) governing the RESTRICTED rdin D fe u ' co @ = <o — oO ie | [oF << ar to ~~ ped : 8 >] ct QO a. ai 4 aml [- 1 (44.9) 47 (49.9)] - [-1(44.9)4Tieq.9) it (24.1) xh 2 <0 aA a 2 where the functions 1444,9) , 1,498). and J(t4:®) are obtained from I, T¢ and J in (22,7) to (22.9) by replacing q by t4 , t being ue * 2 mes, ce + defined in (15-1). For q4<l , the f unction “Y, (4:9) is vy, (4,3) = > An Gait) 4 Ann nd i (24.2) Nr=o where A, is defined in (26.2) and Gt) in (15.5). For 47! and in : as . = " {0} subsonic region the function Wa: (4.9) io) bl w) an y (2) -v 29<27 24 nt Pigsy= D, [Gime 94+ G4 |cove , oF (24.5) n=o0 Ww where Gy(t) and Gite) are defined by (15.11) with the constants a sippy, 5 Bw and Cy defined in (23.7). When the motion becomes supersonic, the continuation of Wtq.9) = ' . % V4.8) = tte L Lia Reap). [-1.A.g)4 JAD] % = Yn + L(A) eT Ae] ~ [-Tethepe TU) ~ [+TiAig) tygy]? [rag sang]? + +7 2 [riamesarn]*s [yap Fay’ be (eee) according to (15.20). Here 3 and 7% ere the cheracteristic parameters defined in (15.24). The upper sign in the last two terms corresdonds to 47° while the lower one, to 7<O «| The accompanying function Wie(qs9) as & Aygo) = A [ Gye) é "q° + Gri) 4 "Jeo igi! (24.5) nN=o0 RESTEICTSD * : f oa Soe ° 5 me a + tinuetion of (24.5). It should also be noticed thet the varisble : : { : "id is restricted to jay, at 4\ instead of nn 8 STE , as Fé is generally greater than unity which is impossible for the actual gas. It should be remembered that We 24,9) is always negligible compared with W(9.9) within and on the unit circle ge | when T, is small in comparison with ea 5 Yq: 8) can be approximately . represented by Wig. 8) alone throughout the interior of the unit circle. As a consequence, the calcounltion can he simplified considerably by constructing first a solution for the annulus region by using Jy) instead of 9) and making an aporoximats connection across the unit circle, In thet event, the stream function will be reduced to Wigay = Wiad), (24.6) when of 4 <¢ | » hore W(q,9) is agein defined in (24.1). o7 the other hand, when 4,4 T< ne 2p 9) ; (2427) W949) = Yigd) + Ylqes) , where the function W093, 9) which is small.on g={ is given by (24.8) 402 R un - Y, (9,6) = os [ B.. Gy te) é q+ bw Gute) 4” ]eosvs Here tha functions Git) and G yit) can be shown to be (2469) v ry) ~-y - 7 fx) a, f(t) (t) = = w= F wy- Le ; andthe coefficients B, and C, are defined in (28.9 js The continuation of VW 4.8) is naturally the Pas xpress pao Q 3 ™ re <q @ Fe eon =) (24.4) while that of (24.8) differs only in the definition of G,(t) and Gy't) which are 98s G y= 2" 4 i{@ + coo(vw- 2) , vt) = Fie) - 2 Fay 4 (24.10) Zaet es, EE | — Ain 1 fo cy YV@Out,. 7 25. The Coordinates Functions x(4.9) and y9(4.9), With the functions Zo) and A.lw) defined in Sections 22 and 23 Hwy; the corresponding functions Altwjt) and consequently Z(“jt) f the motion of a compressible fluid can be constructed. The coordinete functions derived from A(wit) are given respectively by the sum of two functions X,(4,9) and Y4/( 4:9) (17.7) to ( which, according to Ea. bh 7.8) are 4 % st + X, 14.9) =~ Fl 1(t4.9)+ J(t4.9)] + é[zeqales eal, f _ Bt he) hw) tam29 t9,9)+ € 25.1 2 fa) aut [4 we egend-E4 Tegal Tee» (2642) ty 4 5A Jr, ¥,(4.9) = Te Hl [-Lieq.9)+Jitq,0)] - € [-1pleq.8)+Fteq.0)) a at) - PU Fe pee ih tue tg end- ee Jueg.9e OT (4 oh a e “ eS ° nm ~~ wheres 0 (14.9) is obtained from (4:9) in (22.15) by replecing q , ms . «i ‘ , - by t9 » The functions X, (q.8) and 4ig.8) » according to Ease (oy si 7 (25.3 Xi4g8)= 2 > nAyGyte “4 — 18 ape Fh = (94 con necas ‘6 ) n=| 441 eo i eo & nel é 464.9) 2-1 Fn Aue) admin -ape 5 nh Gn, 4 cond and (25.4) ne=| che functions G.(t) and Gylt) sre defined by Eqs. (17.11) and The same functions velid in the annulus region are again represented by the sums X,(4,8)+X;%q.8) and Ylqi8)+ F>qel where X,(q.9) . 4 oo Pose \ we ¥ —— Yiq.9) are dofined by Eqs. (25.1) and (25.2) respectively ye When o) ° 48 2 tg , xq 9) » and 4 (4,9) are ° 2” -y-! . 5M 8) =-15.y[ Gey eG aimty-94 Gorey aime 9| -y-l tape Ss v [ Gvlere” q+ Glu] Joiwv9 coo , (25-8) (0) co v-l apy . ds 690) =-2 5 vl Gyre 4 cov-9+ Grier g env] Niro r an Y-! (25.6) +29t 5 vi om The 4 i c. ) q "ates Am fhe functions Gite) : ~ (+) involved above are defined in: Eqse (17.20) and (17.21) together with Eq. (17.22) with the constants Bu and Cy defined in Bas. (25.11). On the other hand, when = Lee { the continued expressions 3 - see — { of X,(9.8) 5 4,049.9) across the critical circle tari are, necording to Eqs. (17.23) and (17.24), -e ) a =f Ye igo) e- 8 He Fo { [2a3)+ Tag] + € [Ag+ Jag) + [rameTaopt + @ [Lamia] | co (a) = te) (x) qiv) %, a 4 rs HS} [aq dog] - € [-JXg}+ Tay] - Eraneraant's @ brace sam) foie a) Bt tir) cod Hq ft) iI ming e4e Acwoe-€ + Jing) + Fy) RESTRIC OTR PREP Tram | + Acted (teu eacage &+ Tams ET) ] ant mtz) ~| Asin’ | isthe ge e+ TOF és ug) ; =} . haimt (| ye ures C4 TINT) FET AN)) etn (met oo or i »)] Th, (oe) 2 \ = ele at 2 2 -| ¥, q.(4a)= =e fey {ELA +7 09)] E(-Tikg+ yong] 4 oy 4 wh My +t vA + io te [LiMgy+ JOR) - € [Ze(Ae)t T(A.3)] ,-, es . +[1aApezan] + é ft (ae) Op] {ain (G-0) = ptheaind (7 hat » aif oi 4 Asn (+4 +ueA UD "> E+ Jum ETA) eoo( pee y TA) ae ain ood 7; & on Ting (ee A Cogs E+ TOMGI# & HedG)) ss ae ( pipe h cory + e+ J (A. )4 ET AM) faint co 5) While X,(q4.9) and 4,(4,9) remain to be defined by Eas. (25.5) and CS) Pe) (25.6) except the functions G, ce) and Gy,,'« are replaced by those given in Zqs. (17.20), (17.21) together with Eqs. (17.27). By the same argument as used for the stream function, the practical TP TAT ie thud OV HES . . : , on eee er re) calculation of xX(94,8) and 4 (4:9) can be simplified by neglecting x 148) and 4 (4,9) when eet ,» namsly ¥(94:9) S Hlq9) + (25.9) 4(q,8) = 4.19.9), s< qc i (25.10) where X,\q:9) and y,(4.9) ere defined in Eqs. (25.1) and (25.2); and in the ennulus region KEqr@)= Kelas) + KG 4.9) | TAT <( 4(q.9)= ylqd) + 4c qo) (25.1 do ~~ Here x,(4.9) and 4 (498) are either given by Eqs. (2501), (25.2) or (25.7), (25.8). x5” (4,9) and 4g. 9) » on the other hand, become -y-1 fn 4) xq ay=-2 21 v [BG tc) E "gi caiatvni9 4 Geo!) 4 acmivei) 3] ne an (25.13) +26t 5, v EG q “eles (t) q” emvs end , n-o veaer a) = 28 YEG q. — Evin) 9 “eosin ] nso nz=o For TST % <7 1 the functions G,'t) : Ge) are defined as Ge) = Fa . cad . Gi = ‘% “ey 9) iy (25615) fu) ; ~ we Yl > (v) _ fic) ¥ A ( y+l tr) hic) Gy.1) rat ey, 4a , oi - id al | fi e (25.16) RESTRICTED baat ie D idee a tg | For apr aA cay OED L Gio) v = _ oe . . Got By ey Fag CODE) «Gull Brey - 142 Leong 1; (25.19) has (t= mT fe) LHL coweru-g) Gay t= 4 1 Te) - 82 Leary ®).(0 ‘ins Pa 254158) As an example the motion of air past a cylindricel body was Sea AEeA ae telat = 2 » fae * in Heth +h considerad by teking € = 3. Ths flor patterns in both the +, 9- and x, y = plene for two free stream reach numbers M, = 0.6 and 0.7 gs. (15) to (18). It should Fs ey have been calculsted and were given in Fi bo noticed thet there is considerable distortion in the £ the bodies in the compressible flow from thet in the incompressible flow. if the compressible flow around given | body is desired, & sesries of computations should be made with various gsorstric parn 6 € 4 @ fae the practicability ‘of uv ths proposed method. They also show thet, in the case of two dimensio ioticn of a compressible fluid ,the mixed subsonic and uy 94 i 3 23 oO ben} i Qa Hy +“ ¢ fi 2) wa ra ee m cr pus cr * poe a] tr oo ray) a) hr tJ ' pay oO FS e) tsa fee ie ba} QO ct % ct pe oO | ei isentropic : tion from one te tha other ua pe flow about © suitabis body and the tren is continuous and reversible. Fufthermore the breakdown of the irrotational 2 the occurance of limiting lines ‘ "S Oo et urn, is determined by the condition at infinity or the ber ta tn Pee IOTAD | 103, shape of the boundery, while = cr es 33 id 3 % e ie ¢ Qu oO o By ct a ct 2d oO o ine) pay aw ; ¢ @ fu sey] ct et so a) | @ jor (ep) ct ie et te " g) ch 2.) je QO pes! Fe) hed a] i : 3 at © gS 2 Ved wal Mech number M = 1.25 whereas ne = ey z by the eppeserance of 2 fluid. Of course, the chsnge in bety contour in the two cases ta a ce} S ro) fu i & ws \ | ct tee] Qu e > Bs* cr) ase @ s G Ls J he © 2 rs Pi) Q er AS 3 hy co) si ee ce + hy oO > ey t th Par ‘a +} similar except that in tho compressible case the pressure is lower due ’ , a snfinite at M = 1.22 (Fic. 2c) eg im-ensely augunented rnd becomes infinite at ¥ lect (Figs > This appeerance of a limiting line implies the break down of isentropic rrotationsl flow. RESTRICTED tt te pe of ° po U ? : The von Karman, "Compressibility Effects in Aerodynamics” Journel of Aeronautical Sciences, Vol. 8, pp. 447-356 (1941) He Ss Taisen, = Rp anit, Line' in Mixed Subsonic and Supersonic Flow Compressible Fluids" NACA Technical Note No. 961 ana ; . " tt We Hantzsche and H. Wendt, "Der K Kompio ewibi li tetesinfluss fur dunne wenig gekrummte Profile bei Unterschallgeschwindigkeit”. Ze & M. Me Vol. 22, pp. 72-86 (1942) Oe Keplan, "The Flow of a Compressible Fluid past a Curved Surface” NACA Advenced Restricted Report No. 3KO2 (1943) tt P. Molenbroek, "Uber einige Bewegungon eines Gases mit Annahme eines Geschwindigkeitspotentials" Arch. d. Mathem. u. Physe, Grunert Hoppe, (2) Vol. 9, pe. 157 (1890) S. A, Uhaplygin, "Gas Jets" Sci. Mem. Moscow Univ. (1902); Trans. by Slud, Brown University (1944) Se Bergman, "On Two-Dimensional Flows of Vompressible Fluids" Report submitted to NACA (1944) I. Ee Garrick and C. Kaplen, "On the Flow of a Compressible Fluid by Hodograph Method" NACA Advenced Confidential Peport | No. L4024 (1944) + 4 S. Tsien, “Two-Dimensional Subsonic Flow of Compressible Fluids" Journal of Aeronautical Sciences, Vol. 6, p. 399, (1939) C ° L. Bers, "On a tethod of Vonstructing Two-Dimensional Subsonic Compressible Flows sround Closed Profiles” Report sub- mitted to NACA (1941) bi F. Ringleb, "Uber,die Differenhalgleichungen einer adiebahschen Gasstromung und den Stromungstoss" Deutsch Methemetik Vol. 5; pp. 357-884 (1941) . : tt We Tollmien, "Grenzlienien adiabahschen Potsnhalstromungen" Ze &. Me Me Vol. 21, pp 140-152, (1941) F. Gauss, "Werke", Vol. 5, p. 207-229 ca ° Je Tannery, "Propriétes des Intégraloes des figquetions airrSrentielles lin@siras." Ann. Sc. Ho. Norm. (2) Vol. 4, pp. 113-182 (1875) ‘ af = a7 : 3. Goursat, "Sur l'écuetions différentielle lineaire qui admet pour Intigrale hypergéométricue." Ann Sc. Kee Korm. (2) Vols 19, pp. SS-bi4e (1881) ke “4 are i eo o Fe He’ ‘gy _ 2 Le Ince, “Ordinary Lifferential Equations", p. 227 (Longmans & ‘ [9] oe @ O re ~~ Ps 1 L ¢ = oo Horn, “Uber eine linesre re Diftoraentielpleichimg zweiter Ordnung mit einem wielkurlich =~ en Parereter."” Meth. Anne Vol. 52, pp. 271-292 (1899) Jeffreys, "On certain Approximate Solutions of linear bifferential Equations cf the Second Urder." Proc. London Meth. Soc. (2) Vow, 23 pp. 428-56 5 ferentirt Fountians" 1942) "Asymptotic Solutions r Phil. Meg. (7) Vol, 35 pp. 46 bj ey ee os hy Re Fy t. Langer, "On the asymptotic solutions of ordinery differentiel equator” Trans. Americen Meth. Soc. Vol. o3, pps eae (1981) Bh ec} ans. Mmerican Meth. Soc. vole 34, poo 447-480 (1942) eo te at 4 , . , ‘ Soburn, "The Harmen-Tsien Pressure = volume Reistion i dimensions] Supersonic Flow of Compredsihle Fluids Metheneticnl Quarterly (1945) Kraft and Cc. G. Dibtle, "Some Two#Dimensional Adiabrtic Cor pressible Flow Petterns" Journal of Aeroneutical Se sciences, Vol. tly pe 296 (1944) Busenann, "Gasdynamik", Section 1, Part IV, Handbuch der axpe srimentalphysik, p. S51 and p. Sb& Akademische weipzie (1981) < rat + (ew id v9 a Kantrowitz, "Sffects of Heat-Capacity Lag in Gas. Dynamics NAGA Adverce Restricted Report No. 4A2¢ (1944) (1,0) (-1,0)} Fig.1: The behaviour of the | (1) integral-curve 3 it) Y GQ auojd ajoym ayy $0 Buiddow ay n ou auol Q auojid joy ayy yo Buiddow ay, :¢ B14 audjd-M aupjd-°z pou Riera ere See ' Hi i i ; ts ' ppemeuneyit nana tin ones ae & } Om 4 i Spree Guarentee Oreo i ' ' ; Riel $4 oh] Te wee: . t } } it i a) red ' oo Fige5 t 1.405 —“ . Fig. 6.9 Siecnekgh ee rn Tease } $ % f : i . i j eee eee 5 : : i i Ou Cie | j be j i i | ! ; i i i | j i i‘ f j te + i i i ‘ H { N - line denotes the The thin i “3 Jues—of- - approximate va | i chaneainenemme nape neues untae iene 6 : 30 5: i | ! { { ! i ; i 5 EES So 3 : aaa 20 os eon ray comnran ts Sap urnE econ ye! Wars ig sia oan Bi wD oe 114, Fig 9 FR age wonthoe ares Gre taae Bea Seana th rte 30 00 : - eae eet Sane ! meee on arene me ve ersenes a rs fos ds Dagaeen, «| ASS i i ‘ pew Gwe be dhe tage docks: ce Pees gal ee i encore od : i : fax i { i | i ’ * ' " i : i if i t 4 t t H F i i i ! H i i a4 nk H t : i i : i eee eee ee ee : i : Po RearS OR ib mee: CRE eee a dsesteeeentnmen.niee } i ! i i : 40 20 115. [ : 3 ' + ‘ ot ’ ' H i i i - " een ee ~ i ' ‘ . t ‘ 4 ' + oc Ly ‘ t i i : i : foe \ ‘ i ' ' ' ti | ' i 4 ' i ; (ig vt ‘ ‘ i TEM Ra Bes Boer ee, ee ee ay oe ee ee ere iE gal i Bn AOS As en te Peet Gea ag ree he ee . { . } iF } me 1 LIés ifs i ean ee H so tence -500- 118. : " F— etmmeipetY boro “ “ el ; H i : ' i — ; ¢ H . : ; Hi t ' * + $ { ' : ; o : : (@) “AO AY, vv VW = Constant o = Constant \ l _.~ Curve CK / of constant speed Fig. 14: Geometrical condition of Eq. (20.5) ~ <, “Buald MO|} ajqissaidwioo ay) ‘ 2 -@'2 UI : Gl bi4 Ws a ‘ *% \. \ a ine acer LO=W' $29 “guoid- 9g‘ ul MOj}d ajqissaidwod ay, : 9] Bi4 o0r- iP sch $2. ae | #3 auojd- A*x ul MO|} JIGISSaidwod ayy <f| Hig sO - SO- LO=w‘#=) ‘eubjd ~A‘x ul MO|} S1qiSsasdwod ayy - Bl ‘Big SO i24, “Fig, 19: ‘Prassuc distribution a along #he- er “Oxis- — Sgt a - ae eee peer ee Beis a eee: eee ‘ | along. the mojor_ axis 5 he ‘ i i 4 cnertearameiiay entmenn diaes Ramat a i i oe: Tables of th e hypergeomstric Functicns The values of the hypergsorstric funetions given in Tables 1 = 5 are ciloulated from power series for TY: 40405, The function. (c) “Vif in Table 6 is connécted with (tS (t) and Fy ey E,,, &) through the fellowing aquavicn: five) s,s 7 > Aly-1) tH t= Fie) fic) ~ ae c <- fee 2(y-!) a Ens ) : F, ) zy!) ean gh it)— (i “J This is simply the Wronskian of the two independent integrals of the hypergeonetric equaticn and it helds everywhere except 2b the corresponding apprcximete functions as indicated. a na of approe ira The numbers in these tables ere expressed in term é priatepowers of 10. However, a nctetion was devised in which only the indicated by tha sign fT on the heading, ecidental errors were detected and s inated by the difference method. a os » ee ae oe ed wt? CO DNQO Mm om pO om 2 o s es C7 eo ° e 7 - ro Loe O Soe or wn en o Sot Oe AC COME COME cv Co oe 1 of. ON o e e * row ool 0 Cy OF ° mc Py ik UF * ® er) ta © CO On We BO * we ce ° 2 e hog uy 12m Ooh & OF 2 ery Table i 4 Fy (t) $50) Fi, 10) at (t) Fa, (t) Fi') Fs, vt) Fa) Fu, \t) Fay) 1.00000, fe) 1.00000, o 1.00000, 0 1.00600, 0 1.00000, © 1,000, c 1.cocoe, 0 1.00000, 9 1.00000, 6 1.00600, -0 1.66050, §.40592, <1 8.26748, -1 7.23508, <1 6.31726, <1 5.50840, -1 4.79886, <1 4.17817, <1 2.63614, -1 5016339, -1 2e751S7, = -1 2.39525, 9.29261, -1l 7.94894, <1 6.75546, =l 572109, ~-1 4.55687, -1 4.08015, <1 3.44878, -1 2.83778, -2 2.43980, -1 2.05352, -1 1.72739, §.18112, -1l 7.64945, -1 -6.298€C, <1 6.166923, -1 4.22612, -1 36447€1, -1 2.80875, -1 2.28561, -1 1.85821, <1 1.50967, -1 1.22578, 9.12595, -1 7048807, 1 6.07873, -1 4.90498, -1 3.94290, <1 Ee1612C, 9-1 2e5c9ut, <1 2002124, 1 l.613ce, 1 1.28641, -1 1.02495, 9.07143, 1 7.35892, -1 5.86410, <1 4.65282, -1, 67698, <1 2.89241, <1 2.27271, -1 1.7°196, -1 1.79515, -1 1.09110, -} 8.52437, 9.04429, #h 7.2b516, =i 5.758910, -l 4.53033, wy, 3.54621, <1 2.7621, el 2.15219, -1 1.67109, =-1 1s Z9589; -1 1.00303, <1 7.75697, 9.01726, -1 7.19196, -1 5.65503, =1 4.41018, <2 642087, Ll 2.64328, 1 2.04672, <1 1.F6681, -1 1.20162, =-1 9.20855, <2 7.04764, @.99065, -1 7211930, -1l 5.65249, -1 4.29235, <1 3.29881, -1 2.52468, -l 1.92611, -i 1.46585, -1 1.13435, =-1 8.44275, --2 6.39285, 8.96358, <1 7.06719, <1 6.45180, =] 4.17682, <1 %.18004, <1 2.41012, +1. 1.82027, 1 1.387102, <1 1.G30%F, -1 7.72969, -2 5.78904, 6.93684, <1 6.97563, -1l 5.36140, -1l 4.06256, -) 3.06243, ol 2.29956,° 1 1.71898, -1 223111, -1 9.62408, -2 7.06636 -2 6.24290, 8.91217, -1 6.90461, <1 5.25281, -1l 3.95262, <1 2.S529R, =) 2.19288, -1 1.62218, -1 1.19592, <1 &.78255, =-2 6.44991, +2 4.72128, 8.8838C, <2 S.83411, <1 5.15553, -1 3.84370, <1 2094274, @l 2.09003, -l 1.53955, -1 1.11523, -1 8.10647, <2 5.96499, -2 4.25122,, 8.35742, -1 6.76417, 1 5.05952, <1 2.72707, <-k 2.7862, <1 1.99083, <1 1.44109, -1 1.05850, -1 7.46665, +2 5.34692, -2 wee1sco, 8.85718, <-1 6.62589, <1 4.87854, <1 4.F3020, <1 2.58.68, -1 1.80316, = +2 1.27601, -1 ag.9°454, <2 6.29906, =-2 4.40045, << 5206307, ®.7F312, <1 6.48974, <1 4.68817, -1 5, C6152 -1 2.84025, =i 1.62911, = 1.12582,, <1 7.7é2Z25, +2 5.28225, -2 3.59219, 2 2543131, 8.70161, -1 Ge 5AT1, -1 4.50995, 2 7.14141, <1 2.16882, -1 1.46839, =1 1.08852, <i 6.€1895, -2 4.39813, -2 2.90544, -2 1.90736, 8.65054, -1 6.22378, <1 4.35662, <-1 2.95901, <1 1.99792, <1 1.31893, <1 B.UbSE9, == 2 5.63245, <2 B.G3284, <2 2.32524, -2 1.47590, A.59991, -1 6.09396, -1 4.16807, =] 2.78426, <1 1.82708, <1 1.18145, -i1 7.54263, <2 4.70140, -2 2.87302, <2 1.88786, ° “o2 1.12607, 6.54973, <1 5.96618, -1 4.90426, -1 2.61721, <1 1.67851, <1 1.05460, -1 6.83858, ~2 3.99519, -2 2.40766, -2 1.42122, ~-2 8.38046, @,49998, -1 6.84048, <1 Z.845C07, 1 2045734, -1 1.53401, <2 9.38418, -2 5.64524, <2 BeSLSSo, Ek L.92562, -2 1.09583, <£ 6.09334, 8.45066, -1 5.71682, <i $.68046, 1 2.30457, 1 1.40084, -l 8.31600, <2. 4.82627, <2 2.75864, 2 L.El7hS, <2 8.20153, -% 4.28202, 8.40177, -1 £.59519, -1 3.54034, -1 2.15869, <1 1.27656, 1 7e33811, <2. 4.10444, -2 BitE0CT, 2 1,174C6, <-2 6.96281, ~-3 2.66731, 8.35332, -1 5.47557, -l 3.39463, -1 2.01951, <1 1.16959, <1 6.44495, <2 B.46251, -2 1.79223, -2 8.87595, -3 4.16579, =3 1.78066, 8.25771, -1 S.24252, 1 Z.1l1619, -1 1.76065, <1 $.49754, <2 4.89140, <2 2.35154, 2 1.09549, -2 4.56947, -3 1.63451, “<8 3.63955, 8.16381, 1 5.01692, <i 2.25451, -1 1.52558, <1 7.67739, <2 B.61599, <2 1.56622, -2 6.37055, <3 1.75646, <3 1.61765, ~4 <-3.4%649, 8.07162, 1 4.79526, -l 2.60899, -l 1.51370, <1 Sellis4, <2 2.68187, <2 G.48050, <3 2.62652, -3 3.65867. -B -5.94075, --4 <6.15156, 7.S8111, <1 4.58923, -1 2.37908, -1 1.12320, <1 4.7755¢, <2 1.755C8, =2 4.85885, <-2 2.80912, 94 -G.C84E2 he B8UET4, «= $k BAGG, 7.89228, <1 4.38670, ‘-1 2.16418, <1 9362680, <2 3.64602, 2 1.10641, 9-2 1.62280, -& 1.06571, <4 -1.31917, -3 -8.99eC%, <4 -5.31185, 7.80512, -1 4.19514, -21 1.96359, <1 8.90764, <2 2.70018, -2 6.06927, 3 8.84278, <4 77069, 9S 1 S89SE, = 57662512, a = 3.82053, 7.71962, -1 4.00362, <1 1.77706, 1 6.65119, <-2 1.91178, <2 2.36621, <3 -1.90872, -3 <-8.02854, -3 -1.25658, -3 -6.63137, “4 -2.54660, 7.68877, 1 3.82284, i 2.60868, <1 547462, <2 1.27926, <2 1.50059, 9-4 2.65549, -e A SO7ERT, «80-3 0 L.OZ127, 9-3 -3.56238°, -4 -1.12718, 7.55368, -1 2664895, -1 1.44304, -=1 4.43554, <2 7.07404, <3 -2.12546, <3 2.90284, -3 -1.74815, -3 -7.52550, -4 +1.73337?, -@ -2.48122, 7.47296, -1 Ge48262, -1 1.29454, <1 $.55206, <2 3.65270, <3 “4.82352, -=8 <-2586455, -3 -1.42551, -3 <-4.98152, -4 <-2.9237%, <5 +2.98827, «lO 212 014 215 »16 «185 ol? ei 75 0155 aa eae 2195 220 eel 222 023 "ee4 «28° 26 eet 028 29 200 ote 2 o4 236 058 042 ‘oth 246 248 250 Z, 37/2 1.08839, 1.06928, 1,079¢3, 1.08515, 1.05C32, 1.09289, 1.09543, 1.09796, 1.10046, 1.10298, 1.108465, 1.11039, 1.11525, 1.12006, 1.12481, 1.12960, 1.1s414, 1.13871, 1.14323, 1.14776, 1.16211, 1.15646, 1.16500, 1.17532, 1.18145 1.18935 1.19706 5 2 1.20456, 1.21187, . 1621899 s 1.22691, 1.23286, a ~~ & 2 Ch eye Cl. <3 Kae ho Oo Se Co gerd ogoes ooo eaecoe oe 6 5 4 a) eile 1.09060 1.120 eit Teldicl, lelz “RY i4, »L6050, LabtiOns de l42Cl, 1iazis, Lel&e £35 1.14199, 1.14162, 1.14108, 1.14036, 1.15949, 1.13724, 1.15438, is S095, 1612695; 1.12244, 1.11743, 1.11196, 1.10605, 1.09973, £ sOO soe » 1.07861, 1.05296, 1.Cé¢30, 1.02881, “ore 9.91968, © Les ° J ~J 6 e s On " ~ > & ie Co 0 8 6 ° en pes DD us c2 @ st io te a oO | S ie Oo OD OS Oo a voy A MIOoOCooa OM Oe oO Dm a «x La) > ¢ 2 Lo e e “nm omen tn © ® ta 6A ta Pom Ft Co Ww =a Oe eh pe mo © wo ch O pet COON SD ke OD OO es a Se tnoen . « ~~ wo wo oO OO thm ft pe 8 Co «7D ° + C2 &) Cal 9 cf] mm e » &© & & od e e th on e we J " @ ON on t . e CG) po en ee Be be TN OT cn om oD ch tO ew & we ~~ * & BO be CFT ed ° we > OO ht aD et 3 e NOeY NACH MA AHONM ON AH oo wm tn ° e ch lop of @ wad & 0 960797 8.316238, 6.89872, 5.44847, 2.08639, 2.58286, $.10727, CO a a Co co a ee er Co eC eC eee a vw & © LODO OO Ow o o oOnmnoeoa Si) OO 6 Oo = c 2 1 ed Se ca - -1 nH a a at o4 BOG 6 * ke ad * = * € 2 * a > CA OO “I Go wh Goa © a ww NY OO ° COA NM HME Ow N & ~j & te) ONION OO © O rhs . Be (or) ~4 » oe CAP CO A te pw on Ag CO OD ..& cn @ MY 69 OO H FH) FO OO PO FO PO FE Fe Oo ca 2.66430, 2.37172, 2.37625, 2.36292, 2.33411, 2.28816, lid NW tA «J Oh (0 eo vw oe & 2 it , Oro OCOD New HeY OOO Os feokoeeroncyoverehegs no onan meron egal «| 3.03264, CA ys iS Re ca (> GF 0D es OF CA de CO me ono Po om %® e = a me Ny - we eo e “FO et Ww cg ote 6 oD OF et AR bo os i ee ed eX ran ol A) if Oe oo Om CO Ole @9 Go e « © . . a o DO i+ +3 Nw OOPS OF OA OF On CA CO Ga Oe Ca ote ° e ~t OD om or 6 e ° e C2 pro ~y TO Go Ad Re OT we OG “ Of DY oo © wd CD eo ty ifs tO ibs w we we aT ® oh UD Re 2 3 an 0 0 OD GH Pot fhe © swap C3) OND oy cr o ew & ww» & * He pet met > re ate & ag ot o 4 oe e wo O are od oN CO ow No AD é oD pie pat ped CTF bee co “hy YOO rr OOO oa2oov90 Oo @ wWwOoOooodo oo oO & a, aie mw Oo ° ° ° o AEE ae Ome © Re Ao OD O en OD . tn ts on On ~ ©& e } ce, 4 ~ ” Ph wh A OI 1 ND EY ° 4 re 4.86368, 4.73892, i 4. ms 83784, 4.85625, 4.78045, 4.50764, 4.29654, Se8SE85, $.50152, 2.59526, 1.74570, 8.53485, 3.77554, -3.02435, “6.09800, -9,.45145, 1.28041, -1.62038, -1.91%85, -2.14584, =2.29447, “2.27261, iVBoooopaoco oo OO 9d VGOoOo°o 5.50301, 5.73721, 5.96026, 6.16844, 6.55782, 662426, 6.66343, 6.77088, 6.84198, 6.67186, 6.24957, 5.58798, 4.56782, 3.25617, 1.60767, -3«86535, 3.82195, =5 28566, ~1.15868, ‘wl R6550, =2 60588, -3.52151, ~3.91243, =4.32781, @4,43151, 4.31290, 3.79201, @2.91157, 5 ey ey whi PMO Oe- OOOO OOO OO DOO OC OC mas pe = a rare lare) oS wat Ws O00 Oo e ie oa .e ° CH te OO te oO ws ex o we ® we ® w ° onc wo) © oo eh Om OL om ce cs ec we w& © on . ey ° Om ® re st Gi cn wd on nO 2 ° #2 0 MQ —™ Ft tA fe 2 ko C2 oe eh ‘A ¢ C PO Oe Ae Tw OM Oo: ABAnMwWQOWMWONAM tA Hh we we nm bh OO GO uD IO ON “ ww vw © w& A toe oe OO OO tO 6 1 6 80 6 00 OD OO we AE On te GH ae ” -1,72976 3.07618, wh, 65705, -6 00662, -7.21257, -7.96883, =7, 09367, -7,17273, =f, 59505 <2 “67117, 8.7286 ’ — OOD Oa 6 @ Oa O:6'a © 2} co go kom © * sat Or eH ee eee ee OO a nm ren 6D © & . * Om dr rh ND He 8 be oH yet a DO © go 4p ef 61 dow te HM Aw Oo ) * e e 7D Web OW CA GW OA Po DD RY be en - & » Ce en Se ee ee ee oo oe oe oS os I FO @ @ e ve ww © Me On & ob OM kt tO we 6D CO AD Om w we -1,06 ce ~4.2875 ~7.02522, -9.86607, ~2 626174, -1.: S7846, -1,.56028, CM Ce oc ce Sac ee oO a OO OO) ' Pe a a e FOP aH ON Pow OY rw rw HRne mh 3d M oO oO wwe * ® e ® e * cA WO et NS [9] WOO 8H OO e HO om 8 (Oo & we we 1 Gy sg Ort COMO rE we ol ee ee oo oe oe OM ee Sa * wd ae ee) we = w O22 tc Ww MH Ss load % ~~ e& & & & -4.85917, ~7 01848, “3.42783, ee ee DD MH BN ee Re tO oo vt © 2.51778, 2.58766, 2e61858, 2062419, 2014934, 1.46274, 1.90368, -1.5872°5, ~3.95195, 26 6 98167, to? 5466, =t.47011, -1,92579, w2.36512, wS5eH3051, 2$-32623, -5.15871, tel. . 1.708 6.6710: 9.9417: Ls Bass 5355 > § & » © w& © et ga 02 2 AS RO RD SD RD ORD DOD et et eet tet bt ee et e e . « m e rf NM 0 ete RS bt be be oT an ey Log) REO 0 6 OD SQ AD e 3 1249 ise | _ F i i Tigh T T T a T Tt Fy ic} 3{) ts \t) Fys) 79, yl) Zig.) gry, (0) Fy ®) Fy) fait) OOO 1.90000, 3 1.0000C, @) 1.00000, C 1.00000, 0 1.05000, 0 1.00000, fe) 1.00000, 3) 1.00900, C 1.00000, Q 1.00000, 9 0CD0, 0 1.02255, 9) 9.60632, <1 B.65656, -1 7.68544, -] 6.77456, 1 5.94750, -1 Be207c6, -1 4.55215, -1 3.97426, <1 Z.4.662, -1 3.02178, -1 1.02756, a) G.52156, -1 38224, -1 7.24588, <1 6.20821, -1 be2esce, -1. 4.48602, -1 3.79276, <i 3.21218, <1 2.71583, <1. 2.28954, -i 1.06281, e 42406, -1 8.10616, -1 6.80974, <1 B.E5E1T, <1 4.6676, -1 B.O2e76, ol 3.14008, =-1 2.56617, <-1 2.C9E74, <1 1.7C610 -1 1.08351, 5 9.38948, <1 7.96735, i 6.59661, -1 5.39172, <2 4.37484, -1 B.6cead, = = 1 2.84246, <1 2.28146, -1 1.82770, -2 1.46199, -1 1.fSe27, 6 9.8427, =i 7.82803, <-l (6688517, <1 B.lggez, <i 4.1977¢, -2 3.24641, <2 2.56517, <i 2.02045, -1 L.6@815, <1 1.24604, <1 1.02967, 0G $.42141, -1 T.7561€, lL 6.27775, 1 B.COBSE, =i 3.95746, -1- &.10998, -1 2043885, “1 1.89850, -) 1.67768, -1 1.14781, <i 1.94105, C 9.22839, -l 7.68817, -1 6.7276, <1 4.€8035, <i 3.82570, -1 2.97082, <1 2.40726, <1 1.78196, -1 1.87209, -i 1.C55€8, <1 1.C4252, 9 9.27520, <1 7.61804, -1 6.0682C, -1 4.75625, -1 3.9268, -1 2.84729, <1 2.15527, -1 2.67068, -1 1.274is, <-1 9.69363, -2 1.C4387, ) 9.25183, <1 T.B4777, -1 5.96432, -1 4.62456, -L 3eE6874, -2 2.72164, -1 2.06781, <1 1.66450, -1 1.18062, -1 8.86560, <2 1.04543, 0 9.22829, -1 TAT7TSE, m1 5.86082, <1 4.51260, <-1 2.45822, <1 2.69949, <2 1.95475, -1 1.46524, <2 1.09222, -1 8.13007 “2 1.04691, C 9.20457, -i 7.40681, <1 Be75764, -l 4.29416, -1 *.314S1, <1 2.45005, +) 1.84598, -1 1.26674, =] 1.0080, <i Tat «2 1.64841, 0 9.18068, <1 TS2511, -1 5.€8555, -% 4.27615, -1 3.18416, <-1 2.26578, <1 1.74140, -1 1.27454, =i 9.30451, 2° 6.76613, <2 1.04992, 6 G.lss69, -1 7.26528, -i B.5f505, “2 4.18972, -1 3.O07659R, <1 2.26412, +1 L.640s1, -l 1.12727, -1 8.56585, <2 - > 6.46277, <2 1.08250, ) 9.10786, -1 71227, = 5.25087, =i B.oeice, =) 2.84726, -1 2.04085, -1l 1.45177, -1 1.02521, <-1 7.21522, 2 BeO6858, +2 1.05614, G 9.05835, -1 6.28047, =1 5.15041, =i 3.7067, <i 2.628035, <1 VeARACKH, =) L277, 1 8.79050, <-2 6.02599, <2. (4.630822, <2 1.06986, re] 9.00806, <-l G.82718, -1 4.95206, <1 3.4uz208, 1 2.41870, <1 1.85504, -1 lsii8i., *1 7.47776, <2 AsteeCT, -% Seceléc, <2 1.06286, 0 8.95694, -1 6.69522, -1 4.76571, =2 3.28125, 1 2.21889, -1 1.47759, <1 9,72075, <2 6.30301, <2 A,CTSES, -2 -2.60114, <2 1.06604, @° 6.90496, -1 6.54864, -1 4.56150, =i E.CTEHE, =e 2.02841, <2 1.21586, -1l 8.38876, -2 5.25589, <2 3,294G9, <2 eslges, ~2 1.06950, 9) 8.86210, -1 6.40542, <1 4.86942, <1 S27. «J 1.84708, 1 1.36140, -1 7.17813, ~2 4.32651, -2 2.8C40Z, -2 “1.65124, <2 1.07805, © 8.79833, -1l 6.25762, -2 4.17949, 1 2.°8442, <1 1.67472, -1 1.0167%, <1 6.08189, <2 3.50558, <2 2.02219, <2 1.12740, -2 1.07668, 0 8.74562, -1 6.11072, 1 B.O8]74, =r 249714, 1 1.51115, <-1 &.88598, <2 5.09535, <2 2.78547, <2 1.62862, 2 7.S6572, -2 ‘1.08042, 2) 8.88792, -1 8.96314, -1 3.P0nZ1, =1 2.271565, <1 1.25618, -1 7.87415, <2 4.20588, 2 2.15Z1&, «2 1.11658, -2 bef9LG1l, +¢ 1.08425, 9) 8.64120, -1 6.815¢c2, -i SeE22R8, =k 2.159 7, <1 ° 1.2093, -1 @.55817, <2 B.413¢1, -2 1.50620, <2 7.58658, 3 2.16844, <2 1.09221, Q 8.51457, -l 5.51734, “i 2026210, -1 1.8C586, -2 9.41052, -2 4.59760, <2 2.08796, <-2 7.21412, -¢ 2.57268, 3% 2.49754, <4 1.10068, G 8.59358, <1 5.21591, <-1 2.21262, =L 1.4947C, 1 7.02861, =2 2.27227, -2 1.07105, <2 7.98990, <4 -S.29683, <4 -1.31670, -é 1.10954, re) 8.26726, <-1 4.91116, -1 2.67170, «1 1.20628, -1 4.°6611, <2 1.65104, <2 3.19266, 8 6.7292, <2 <8.40270, 95> -1.98991, -2 1.11898, o 8.15792, -1 4.60288, -1l 2024064, ll 9.40421, <-2 Bel7771, “2 <6.03769, <-4 -2.07404, -3 8.87287, <2 -S.14715, <3 WE.008'7, 898 1sA28C2, % 7.92850, <1 4.22064, -1 1.9194, <i 6.05546, 2 1.65772, =2 <2L.987cl, -S <5.45776, -8 <9.02991, -2 -Se1f032, -8 “1.71556, <2 1.18872, o 7.85822, <-1 3.97480, -1l 3060924, ih 4.75666, <2 3.90922, =3 7.84725, -3 -7.29514, <3 -1.05881, <2 2.79384, -% -1.26203, <2 1.15114, C 7.70425, -l 3.66460, ol 1.30976, <1 2.76346, -2 6.33482, “8 <3F.18i31, -2 <7.89179, -% 1.19892, <2 -2.]el¢e, 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TICiG, 4) 1.85205, fe) 2.0577, 9 2.17225, 0 2.35255, 1,C9629, c 1,22335, a 1.564 G LeAiceet, Q 1.693809, 3) 1.89435, 9 211326, 0 4.35756, 9 2o830Cc, 0 2.93400, o S.27515, 1.148582, C S2SCS 3) 1152410, o 1.75493, a 2 020%, a) 2eA2E75, a 2.87980, S 3.08499 0 3.55224, c 4.09025, 9) 4.70975, 1.23522, G 1.46925, Oo 1, 76068, 6) 2.08594, 9) 2.48545, 0 2.98146, 9 3.52862, G 4.20441, a) 5.00962 @) 5.96905, 4) TLk22i, 1.38760, fe 1.71659, o 2.12278, G 2.62744, CG 0 4.02161, a) 4.975846, 0 6.15555, 9) 7.61553 fe) 9.42179, Oo 1.16565, 1.3566, o 1.93944, 3 2.44959, S 3.09343, 2) 4) 4.92446, 0 6.23139, O 7.86986, 9) 9.93913 a) 1.25825, 1 1.58530, 1.85639, 9) 2.49764, CG 3.09669, 9) 3.99955, & 0) 6.67174, 0 8.61692, G Lal F2CR, L 1.43741, 1 1.85648, 1 2.59777, 2.52115, G 2.08489, 0 3.968C7, 8 5.18822, 9) 6 0 8.86942, 0 1.15967, i 1.51626, 3 1.98249, ] 2.59208, 1 3.38915, 1.97764, 0 2,62022, % 3.47696, C 4.60227, G c ° R,O8400, 5) LL, O7MO, ) 1.41995, i 1.88189, 1 2.49412, 1 3.30561, 1.68331, 0 2.85506, G 2.48904, e Be8E870, 0 é ¢) £00585, o 8.05695, 8 1.08085, 1 1.44944, 1 1.94403, i 2.60737, 1.19687, o 1.H22e,, % 2.19864, 0 2.27957, ¢) 4 0 5.470€5 G 7.41185, e 1.00410, i 1.5014, 1 184275, i 2.49598, T.C895i, G 1.49048, G 2 USSSC,, o 2.78751, ¢) Ba Gb 8 G &. 20310, c 7611618, 4] 9.72065, 5 1.5¢748, 1 1.81286, i 2.47362, 1.01502, 0 1.89982, 6 1.92961, G 2.65822, 5 S.e80l4, 8) 5.03680, 0 6.92715, 9) 9.52115, ¢) 1s30782, 1 1.79621, 1 2.46255, 9.58226, <1 L, SAGE, G 1.84661, a) 2, 58947, Q Subd tot, 9) 4.9917C, OQ 677165, 2) 9.34345, 6 1.28762, 1 1.77174, i 2.446445, 9.1208), <1 1.27494, 0 L.77869, o 2 ATIAG « o Ss44e05, 5 4.77857, QO : 6.61655, o 9.14090, c 1.25976, 1 1.75162, l 2. 20k, 8.40755 wi 1.18658, C 1.66788, 4) Silt. 2) 2.25925, fs) 4.50221, O° | 6.24245, a) &.66135, fe) 1.16569, 1 1.57871, i PW«LGS1is me s7016, ~j 1 L16h), Z LahFAlS, G 2,20397, G 8 ESTE, 9 4.19696, 9 5.68605, 0 7.58025, a) 2.9042¢, 0 1.25725, 1 1.67558, “93S 1 1.05441, 9 Lw48sts, eo 2.C6459, c Z.A19E2, C 3.78S01, re) 4.88319, 9 6.06990, 9) 7.06475, 3) 7.50395, 9) 7.10740, 7 eeean “1 9.98825, 1 1.%9893, o 1.91322, a Ga S85 2, ° 3.21218, 379553, 6) 3..92921L, a) 2.95415, CO 4.99969, -1 2.81451, 6.85402, -1 9647280, =] 1.109%, 0 1.7482, a 2.19821, © 2.820755 9 2.88206, G -10702, C 2.61475, O 1.12276, 1 2.91659, 6.34158, -1 8.98065, <1 1.22048, 9 1.56128, Q 1.20448, Qo 1.€80¢89, 6 6.28765, QO 2.48611, OQ #-9.77445, GO 2.51642, lL #8.8587C, 6.05684, 2 8.50527, <1 1.12678, ) 1.36793, 9) I 88ht 7,5 fe) 6.89815, <1 1.49096, O 6.86430, G 1.85750, 1 4.25645, Ll 8.84745, 5.78147, -1 8.0599%, <1 1.02928, ¢) La Lees, }. R.ALECG, el 4.52902, <1 23.96352, QO 1.20262, Ll -2,89943, lL +-6.27515, L 0) .27225, 5.54475, <1 7.58697, <1 9.28625, <-1 8.96665, <1 2.74121, <1 +1.7889¢, O 6.77225, QO 1.79149, 1 pipet, 1 = 8.58950, Y 1.75776, 6.41244, -1 7.13734, «2 8.22800, -1 6.38575, <1 6.42859, -i 8.16472, 0 =9,90920, QO 2.44750, l 540835, 1 oli2e2 2 «#f,17516, 4.88404, <1 6.26916, -1 6.04795, -1 7.79700, =-2 1.72409, 9 96.5791 9 -1.69445, 1 -%,90549, 1 MA. 20014, 1 2 8.09189, 4.49410, -1 5.48827, -1 Be74912, -1 -6.34045, <1 -%.28521, O 9-9,93145, 0 2.34784, 1 5.43885, 1 1.10463, 2 2 2.735929, 4.14515, <1 4.54018, -<-1 1.48204, -1 1.17927, O 4.7107, C «1.88061, “1 3.21255, Lo #6 .807%0, I wL,Zlses, 2 G «#8, "LF, 3.80195, -l Be71285, <1 -21.01642, -1 -1,82946, O 8.49980, QO 1.71474, 1 + 1 -7.80826, 1 --1.40383, 2 2 2.72873, 3.49107, -2 2.90807, -1 8.40012, -1 2.49811, GC 8.0560, Q 2.02670, 1 136045, 1 8.16840, 1 2.29698, z 2 94,95521, 3.19986, -1 2.15267, -1 =5.72770, -1 -3,11942, G 9.46850, O 2.26740, ° ¥ rie TST 1 6.45765, 1 +8.52640, tL 5 1 2.93140, 2.92665, -1 1.39091, -1 7.98420, -1 -3,€9668, 0 1.06211, L 2.40989, “lL 4.45007, 1 6.18929, 1 366233, QO -2L.48$21, e T<lGSTE, 2.66912, <1 6.87472, ~e “1, O0sR2, GC 94. 20472, 3) ~1.14€89, i. -2542071, ‘DT 3.93544, L =-%.64254, i 4.64526, i 3.82650, 2 1.14148, 2.42713, <-1 2.69075, -3 1.55858, O 4.62520, QO 21,1977, L «2.21584, 1 -2,08625, 1 1.65651, Q 1.43491, 2 5. 619€0, Z 1.45880, 2.19955, <1 6.86493, -2 -1.67510, O 4.94366, G -1.195495, 1 #2, CETE4, 1 1.72385, Ll -4,00740, 1 2.43137, Zz 7.29325, 2 1,€7658, AAR NM HY WMO RM Meee OOM mM YP HEE ee eee OD OOOO ta) P oy oe) = qe) Tx) > q(t) Fit) cos(yw-Z) 3 3 of ee ee oO ie] el ” Fi ey “ 7) mio saa (ey x 1O ~ 0 ° 7s ~ i © ici Fae ae Tye) Fate! Js (c) Jogi) Fig cx) + oon it) Ce 1.03963, 0 1.01%693, G 9.88402, «1 9.65742, -1 2.29697, <-1 9.16251, -1 8.93390, <1 8.71101, <1 8.49366, -1 8.281768, <1 8. 07511, -1 04 1,08556, © 1,03073, © 9.78762, -1 9.29368, -1 ©,82469, -1 8.37935, -1 7.95648, -1 7.55496, -1 7.17370, -1 6.81168, -1 6.48793, = 006 1414018, = 9 1605279, = C8 .72109, -1 = 8. 9760S, 2 8468818, ol 7.65294, -1 7,06642, -1 6.52485, -1 6.02479, -1 5.56295, -1 5413669, -1 008 = «1.20746, 9 1.08257, 0 9.70250, 1 8 9ETIZ, 1 = 7.79807, l= 98838, -1 «= 6 26436, -1 «= 5.61535, -1 5.03359, -1 4.51209, -1 4.04462, =2 710 -1.29495, 0 1.12461, 0 76686, = -1 848217, 2 PEGE, ==) = 639750, -1 8.58500, -l 4.82518, -1 4.19050, -1 +.83829, -1 3.16060, 1 a — nen > 5.90855, -L . 4.96895, -1 4.16198, -1 3. 4 “1 2.93160, -1 2.46031, <1 714 «1662754, OG 1,31555, O 1.06835, 0 8.59475, -1 6.04705, -1 5.61522, -1 4.58872, -1 3.68860, -1 2.96529, -1 2.59681, -1 1.98752, <1 115 21,81680, 0 = 1.43828, 91415883, 0 8.C]Ge0, =1— 7413959, -1 = 865301, =1 «4.47608, -1 «5.54418, 1 «2.80680, -1 2.22204, -1 «1.75948, i — ioe - ge a . es e 7.94448, 1 6.13694, -1 4.76253, -1 3.68743, -1 2.85503, -1 2.21053, -1 1.70998, <1 — ee ee Eee vr ll lr CF 7.25168, -1 5.54625, -l 4.24181, -l 3.24452, -1 2.48185, -1 1.89778, -1 17 «4672637, 890 SERGE, = BAGSOBL, = GSC, LaewdRe, = 1.15587, © 8.72144, -1 6.58217, -1 4.98533, -1 3.74529, ~-1 2.82683, <1 “ATG BBG, = RA, 6 LR, LA, 7.63403, <1 6.69059, -1 4.05090, -1 3.16183, -1 2.35670, -1 1.75859, -1 «18 -2,88255, 0 = 2.12836, 90 «= 1456840, CG 1.15677, © 8.63116, -2 6.29112, -1 4.63889, -1 3.42036, -1 2.52177, -1 1.95905, -1 2037041, 1 185 2.635894, 9) 1.92812, ) 1.40845, Q 1,02863, 0 7.51075, -1 5.48505, =} 4.00188, =] 2.92025, «] 2.13061, a 1.58415, = 1.128465, “1 -19 2.47544, © 1.79811, 0 1.29937, OG 9.359665, -1 6.79759, <1 4.915534, -2 3.55283, -1 2.56687, -1 1.85351, -1 1.30835, -1 9.65840, -2 -1SE 2.85554, G 1.60287, — 0 1.21601, O 8.72516, -1 6.25428, -1 4.48075, -1 %.20796, -1 2.29496, -1 1.64078, -1 1.17245, -1 8.6970, -2 -20 2.26219, 0 1.61418, © 1.15030, © 8.18491, -1 5.81743, -1 4.12985, -1 2.92841, -1 2.07429, -1 1.48774, -1 1.08749, -1 7.82630, <2 21 2.12529, C — 1.49857, 9108058, = ESB, «= 15415841, -1 3458072, “-1 2.48968, = «1.72742, -2 «119610, -1 BLESS, «2 BL TCIBE, ==? 02k 2.02900, © 1.41171, © 9.77366, -1 6.78587, 1 4.92278, 1 B.18Cl4, <1 2.15231, -1 1.46078, -1 9.86121, 2 G.6uzia, 2 4.47652, 2 22300 1695765, 0 1.54782, =O S20208, 9-1 6.24195, 9-2 42055E, =) = 2.81732, 1 1.87689, <1 1.243384, -1 8.20170, -2 6.58186, -2 %.81008, =2 24 1,90304, © 1.29661, 9 8.73569, -1 5.82184, -1 3.65225, =1 2.52556, -1 1.64294, -1 1.96085, -1 6.80140, -2 4.52992, -2 2.7873%, -2 25 = 1.86046, 1 2SS4e, = 884265, -2 BLA TCOS, «= 1554353, -1 «= «227014, 9 -2Ss1642920, -2 9.08252, -2 5.61801, -2 S.485dl, -2 Z.4045Z, 2 -26 1.82695, 9 1.22157, C 8.00298, -1 5.15365, ~-1 3.26748, <1 2.042¢1, -1 1.25878, -1 7.65623, -2 4.59427, -2 2.71871, -2 1.58516, -2 -27 1.80155, 01618842, 0770751, -1 4687004, 2 S51787, -i=—1,8368C, -1 «1.09784, -1 6.44850, -2 3.71902, -2 2.10283, -2 liiezsz, -2 +28 1.77972, iS) 1.195832, Q a ae | 4.60400, -1 2.78408, -1 1.646590, -1 2.51444, -é 5.37404, -2 2.96016, -2 1.58448, -2 8.19256, 5 229° 1.70372, 9 1.10828, 0 6.94167, -1 4.21076, -1 2.48080, -1 1.42144, -1 7.91678, -2 4.27697, -2 2.23121, -2 1.11498, -2 §.25865, <3 250 «1.75144, 0 = 1.1e8SE, 0 695719, = -1 «= 412747, =k = 2.88495, -1 1.30984, -1 6.99637, -2 3.58860, -2 1.75078, -2 7.97632, -3 © 3.26045, -8 <2, 1.73675, © 1.0980, O 6.54014, -1 3.69786, -2 1.99152, -2 1.01946, +1 4.92078, -2 2.19926, -2 8.72767, -3 2.71864, -3 2,80349, <3 254 1.75244, 0 = 1.07452, 0 6.16085, -1 3.30061, -1 1.65368, -1 7.67616, -2 3.22470, -2 1.18647, -2 2.57564, -3 -5.74978, -% 1.32608, 3 86 1.73656, © = 1.05485, 0 6.30435, -i = 2.02719, -1 = 424599, -1 = 49BZR, 9 32Ss«LB3B19, -2 «= 3.48102, -3 1.48561, -3 -2.40783, -S 9.99155, -3 8 1A7A7E7, «408842, 48362, 1 «= 2457259, 1 1.08489, -1 3.61771, “<2 7242911, 3 =2.11660, -3 -3.91145, -3 -5.19504, -7 -2.7455, -3 040 1.76586, ) 1.92434, ° Sreearaerar “1 2o24307, -1 8.08441, - 2.01980, -2 -1,00608, 2 -5 81232, 3 -5.08977, = LaESTS, See ~~ a 61578, 6 GE, 6 BO, wt LOTS, «0-1, A TEORE, «=e 6.79837, -3 7.248656, -3 9.52393, -3 -5.54867, -3 -2.8421, -% -1.41176, -3 044 1.81833, o 1.00118, 0 4.48CEl, -1 1.59312, -1 5,64048, <2 -4.19461, -3 1.11778, -2 -8.86186, -S 4.96352, -% -2.25519, -3 8.37482, <4 046 1,85505, © 9.91200, -2 4.15481, -1 1.29063, -1 1.74711, -2 1.29465, -2 1.41896, -2 -8.80350, -3 -4.16712, -3 -1.53572, -3 <%.59576, 4 -48 1.89087, 0 481845, = 3.38806, -1 9.99156, -2 6.63943, -4 1.96124, --2 -1.67499, -2 -8.03295, -3 -3.14823, -3 -8.51431, -4 <1.26787, -5 60 1.99927, © 9.72785,° -2 5.00787, -1 7.18620, -2 1.40378, -2 2.43514, <2 1.54161, -2 -6.76831, -S 2.05775, -3 -2.25031, -4 <2.s19421, 4 -v Fw) al Gable 10: Jott) = gic) Tir) —> $4°*) Tit) cos(yw+ F) = € yi) = “ti fi ie) sed ‘ith pple + (0) ~ (0) = (9) ~{0) >to) F-4 tx ) + § F-1't) Fy (t) Fu (t) 3 te) F.15 (t) Fg (t) 19 (x) Aa tt) 202 1.06623, 5 1.09252, G 1.12150, a} 1.15802¢ a — ; ee = “Od dLsd4B2E,«=it«‘aCROMOL«S(isid2 ROBO, «= s«iC BDL rt ° — ° 1.240775 C 1.27262, 0 1.80808, = 9 BBBA7g = OL RVATAZ OO 206 1.23481, Q 1.43730, o 1.44830, } 1.56861, a are : ase og ———y 6 1.64274, 0 1.75004, 0 1.82123, C 1.91822, C 08 1.34702, G 250270, © 1.67634, 5 1.87014, 6 o. LoREDT, 5 a. 5 een 0 2.15777, O 2.45986, O 2.53082, oO 62.74089, G to «1sa8108, ««)sCTueSL, «=«0-«ALDTBE, ©«BaateR7] «6g ersuiss «LOLS; «oo eaTSeT, «Oo acomes, «0 LEOTTE, «}BlAOESR,«S1OSIE, "i2,«1069082, «= Gti«i AHS, «= s—“‘ié‘iw OOS, «=O, «=i LOTGOR, «©—s—i‘i;SOBOS, 4B ply 4 Sees C Bei, 8 feniibly f Eeitiity 2 “le -2LO1SB7,««G:C*«‘i CN DTIE] «© BLORZOO, «SB LBIRS, §«60ksgs7nas, 6B eB3e2e, «= Tezz0as, beTeans, 0 BERS, Be BAEC, «= BeTEIAL, 15 2.2988, C 2.89702, G 3.65875, Oo 4.62078, D5 ,88574. 0 eon siege ° me ‘ aoereis i 1.36751, 2 1.69161, 1 16 2.85518, 2) BoG68764, Cc 4.76280, 5 6.15142, 0 7. 9ae92. e 1 O2E1S _— “ 0 ag ro 3 al 1 1.87502, 1 2.36802, l "165 3.82301, 0 4.72708, rs 6.19864, i a cafe - secar et ° — 4 ~——y 1 Le7L172, 1 2.21078, i 2.85532, 1 8.69754, 1 Pe 3.13201, G 4.15093, ) 5.50817, * 7.29103, G 5 +r a0r 5 ete, : See 4 e-s6608, LL B.0Sd41, 1 = 404592, 1 5.29999, l «175 2022438, QO 2.98372, 9 4,00z4C, OC «BBR BEE, . Les wos - es ; goitig L etpoll, 1 8.95118, 1 - Be BRITE, 1 .18 =-1495367, © 2.64805, © 8.58855 @ 4.86261, © €.r9082, 6 Lo if Eee | Le72771, Ll 2.55071, 1 Bel2604, 2 4419268, 1 -185 1.80549, Q 2246998, 0 AvSThee, c 4.61935, n 6.71491, 0 eptriry, ee : i, 2 manna 1 §.00721, 1° ‘#07277, l 19 «1e70780, «0's=« BHAA, «= si ALAEKE, «= G4 TZES, «= sO ABBSC, «= ss—(“(«é‘éaC TAT, «=Os«O: DESI, ae of Ebest : Gam: 4.09818, (4195 -1.68709, 0 2427364, 0 ‘aa, tf nee 5 auc co ame eo geet «¢ geese sigue << eee i-amee’ § 20 1.58294, 6 2.21194, & ~ &.0B592 x 4.29701, é apetng - 5 chery ola eer s : —_——s So ASOD y i oel2625, Le ooh 18914, bi “21: 1.50468, 6 2.12355, = 298495, 41821, 8 g.8Bte 6 cite #|S a | 1.58588, 2 | 2.29561, 1 3.00808, 1 4.11712, 1 22 1.45552, 9g 2.06036, re) D go73l, : 4.07042, D 6.04724 0 “tts 5 10 015, 7 apetie . @eOBE20, 3 2.81642, 1 30 TEZE7, 2 «23 1.40978, C 2.01020, 8) 2.83288, & %. 05645, fa 5.37584. 0 7.1arAe, - op . ots i seBebaby i 2.52196, 1 2, 83565, i 24 «1.37862, © 1.96707, © 2.78477, 0 3.78751, 0 4.99251, 0 6638842, 0 ae 7022, yy L 2.88700, 11689261, i 1 85E1, = o25 1.35886, re) 1.92738, Q 2.06741, 6 3.85274, 6 4.47275. “ acs a 4. BenAe e Lg hag U SeB17S1, 0 -9,84541, <1 -& 54253, Q 226 1.83378, C 1.88884, C 2.66694, a) E.eRe7Z, ' 279528, 4 S Gaaeae a a ae ~ iy GO -5.e2010, O -2,28444, = 75695406, 1 ~27 1031797, © 1.85100, © 2.45219, G 2.95652, : cee ¢ erry a -kMaee . 7 —e Te 1 5.29048, i 21.17085, 2 "28 «1.30383, 9 e81464, © 2.31654, © 2.85843, 0 L.s9eee, 6 -2.c1922, 0 -8.91958, 0 ~1.49e90, 42 -4.04255, 1 -9.21906, 1 -1.92580, 2 129 1629167, © 176740, © 2.16825, © 2.08880, 0 6.472883. <1 4.05081, 0 a eee 1 WEW70640, 4G 95652494, 2 241217, 2k BSEIC, =? nod 1.28157, a 1.72134, 6 1.98745, S 1.5413E, 5 nd, BCs $1. “i ehace 6 ire pee eg pins lL -9.52252, 1 “* O0066, ed 3.37822, 2 -32 1.26572, Q 1.62467, fs) 1.56727, 8 2.02062, -=1 -4,46807, , ; ie iy 1 “6.90489, 1 “1.80865, 2 -68352, 2 =5.2476°, 2 34 1.25376, G 1.50524, fe) 1.04593, © 1.48985, 6 Py 5 4. Saee : eure oe . Md ol iy z 72,1406, Z ny 2 -%.°1279, 2 036 1.24468, c 1.26660, o 4, 15995, ~l wt, H48B4,, 3 -1. weciite, 7 ahr 4 - pens . ns _ -$.081 73, 2 -5 24329, Pa -1 eUéSis, 3 238 «(23773 : + 20838, 0 -.308S2, -1 2B .S883. . eee . i -2. eel, 1 - hes @ -3.97017, 2 ee 2 1.12870, 3 240 = 1.23235, i) 1.02655, 9 1.20024, C =8,80068, O “eb.Razer. o7.16425, 2 a pee “ —, & 98 5701S, é +-7,18680 2 8, 89230, 2 42 = 1.22815, © 8.18566, -1l 2.19844, © 1.19736, 3 "sa.escan, 1 “AORTA, : "See : ane 2 =4.57237, S =H 4800S 2 1.75978, 2 (044 1.22484, © 6.82381, -1 -3.32952, C +1,.54728, i u4,aan5n i "5 ofabee a’ 9 seeps = * seemet 2 5.85840, 2 2.20798, 2 1,12514, 3 46 1.22215, re} 3.14753, A “4.59645, 0 1. 92528 "| es, 7 = "4 EES Vy * i BETS 3. a 2.5 =i & 1.49106, ai &, 0238-7, z Se IUSTey 3 48 1.21989, 0 1.38241, -2 <7.72500, © 2, 22465 1 =8.$9986. 1 pen, . rr ° oa GOT, 2 g.1Rt00, @ 1.61487, S 622668, 8 50 =: 1421792, Q 3.24763, 1 <-%.27525, 0 22.93727 Began, : es : - SE, z “BWPZETZ, Q 7.20082, 2 2.Azaad, & Tetnedd, 3 een, wi.1deis, -0,54518, 1 ~of1SlS5 2 1.a¢828, 3 4.02726, 3 9.29452, 3 137 v (ct) Tit) cos ( vw- e-F) 4 y ) “Try —> a to) sa aor oF or Fed , ft) > io) als feu £ & ele (Hy e (a) en alee? | (eo) i (t) BY Es Zi ) to) $ ({t) .e) Laie ety os i) 3 AA coy redoet oedt a @ 4 a 5 Om ty Co oD CO if 0 Ww or wp eed TD eo © 4 a> t WO eit ety | ~ & & rdw Wd ND Ga & © fd Us * * « OO be a ot drt rn a 5 ed xh Co tf ow dt RO wr Om cs eo @ Oo gs Wiican Ms a ob Ft a & a O st at aN OM OD et Ou . e « Ra > r4o rmdir of 6 a a aw 9) wo CH xt tad G7 ON a h- oO ® ° e hs a oo mt et el #3 oo ¢3 uid a eed eet OG a «a CO ke ras wa ON I a e s yet ed ~~ 8 a >) we Or ° e et oe 23 2 34 Oo iO a a) ot wd Qo + 274, 1.e is) edt rt ed ed et et 6 £ # 6 t Fk Cr ee er ee, ee. ey C2 92 22 Me 80 2D el ge ee a wy “2 uD OM WH Om on VN et OO OD oy © os 6 ° oe 73) bh BD oD ry WO Ww oF UDO wD: be th Dd YD Kw) 42 wo oat CO <t 1) Gwe a CON ord a e cy ca s ° e Wot ow be 8 ef 1 Ce ee. 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