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I N A 1 \13i8 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1581 ON THE DETERMINATION OF CERTAIN BASIC TYPES OF SUPERSONIC FLOW FIELDS* By Carlo Ferrari SUMMARY A quite universal mode of attack on problems which arise in super sonic flow, whether connected with flow over wings or over bodies of revolution, is explained, first, in great generality, and then in more detail, as specific applications to concrete cases are illustrated. The method depends on the use of Fourier series in the formal definition of the potential governing the flow and in the setting up of the boundary conditions. This new formulation of the many problems met in supersonic flow is really an extension of the doublet type of "fundamental solution" to higher order types of singularity. The limitations and, in contrast, the wide field of applicability of such a means of handling these prob lems with complex boundary conditions is discussed in some detail, and a specific example of a wingbody interference problem is cited as proof of the versatility of the method, because the esults obtained by applying the techniques expounded herein agree well with experimentally determined data, even for the quite complex configurrtior .srd to exemplify the kind of problem amenable to such treatment. 1. INTRODUCTION For purposes of analytic treatment of the flow problem to be con sidered here the usual rectangular Cartesian coordinate system is employed with the xaxis taken to lie in the direction of and having the same sense as the uniform (undisturbed) freestream velocity, V,. This freestream __> velocity, V,,, is taken to be supersonic in the discussion that follows; i.e., V. > C, where C, denotes the velocity of sound in the undis turbed stream. The flow of the gaseous fluid to be investigated is to "Sulla determinazione di alcuni tipi di campi di corrente iper sonora," Rendiconti dell'Accademia Nazionale dei Lincei, Classe di Scienze fisiche, matematiche e natural, series VIII, vol. VII, no. 6; read at the meeting held on December 10, 1949. NACA TM 1581 be considered as resulting from the superposition upon the freestream velocity of a nonuniform flow, having velocity components that are des ignated as Vx, Vy, and Vz, and lying in the direction of the respec tive axes (x,y,z) of the coordinate system. This nonuniform super imposed flow is supposed to be small enough, in comparison with the speed of sound, C,, that it is permissible to neglect the ratios Vx/Coo V/ C), etc. in the equations governing the flow. It is taken for granted that, under the conditions stated above, there exists a velocity potential describing the flow in question, and in practically all cases which are of any interest for actual designs it will really be true that this assumption can be made legitimately. If it is then agreed that the nonuniform superimposed part of the flow is to be denoted by the potential 0, it will be recognized that this potential will have to satisfy the relationship: ( M2 =O (1) 1x2 ay2 0z2 where the freestream Mach number, MN, is defined as M, = Vg/C and, of course, the potential 0 will also have to obey the boundary condi tions which are peculiar to each stated problem. A way of handling the determination of the function 0, so that it will satisfy equation (1) and so that it will obey the imposed boundary conditions, will now be explained, and its usefulness illustrated by consideration of problems which can be attacked by this means, both in the case of lifting surfaces (that is, wings) as well as in the case of bodies of revolution. The proposed method is based on the use of Fourier series. Although this technique does not afford complete universality in treatment of all the posed problems, as will be more clearly pointed out in what follows, it can be used to fine advantage in a.goodly number of situations by replacing the procedures which are based on the Fourier or the Laplace transforms (which, for that matter, have just as restricted limits of applicability as the analogous ones which arise in connection with the approach being discussed herein) or by being substituted in place of the techniques which stem from use of the "fundamental" (source, sink, doublet) solutions to equation (1), or from use of transformations carried out in the complex plane. 2. PROBLEMS HAVING TO DO WITH FLOW OVER WINGS As usual, the wings are imagined to be very slender and so placed that the wing span lies along the yaxis; i.e., the long dimension is NACA TM 1381 out the yaxis (see fig. 1). Let the equations which define the ventral and dorsal surfaces of the wing surface be, in fact, given in the form: z = Zv(x,y) and z = zd(x,y) and then the slenderness of the wing is supposed to be slight enough that the abovedefined values of z will be so small at all locations on these surfaces as to make it possible to accept the fact that the derivative is, to all intents and purposes, equal to the direction 6x cosine, with respect to the freestream xaxis, of the normal to the surface. It is further assumed that the wing is immersed in a stream of supersonic flow which has a constant value for its component lying in the direction of the xaxis, of magnitude Vo. The component in the direction of the zaxis, meanwhile, is assumed to be known, but of rela tively small size in comparison with the V, velocity, and it may take on various values, which will be denoted by Vz'. If, now, the potential describing the flow perturbed by the wing is denoted by 0 this poten tial will have to satisfy equation (1), and it will also have to conform to the conditions which are imposed at the boundaries. These further (boundary) conditions may be stated as follows: (2) Upstream of a certain surface, which may be immediately defined just as soon as the winglike body is specified which is to invest the impinging stream, the value of is zero; i.e., the basic condition is = 0 (2) (5) On the wing surface, it must be true that = V 8 Vz cos (n,z) = H(x,y) (5) wherein the value of z to be employed is either the zv or zd quanti ties, depending on whether one is concerned with a point which is lying on the under ventral surface or on the upper dorsal surface, respectively. The notation cos (n,) signifies the cosine of the angle between the zaxis and the unit vector taken in the direction of the exterior normal to the wing surface in question; i.e., this vector is represented by the vector n, and under the present hypothesis cos ,z = il. NACA TM 1581 It is convenient to distinguish between two basic types of problem which come under this kind of analysis, and to make the differentiation on the basis of the sort of boundary conditions met with in each type; that is, Symmetric Types of Configuration In this case, the boundary conditions to be satisfied on the wing may be expressed in the form =( H(x,y) (5') and \z=O= H(x,y) Asymmetric Types of Configuration In this case, the boundary conditions are expressed as ()z=o+ z=0 ,y) (") The first type of problem corresponds to a configuration for which the wing has a zero angle of attack with respect to the freestream undisturbed flow, V,, and which possesses a symmetric profile. The sec ond type of problem corresponds to a configuration for which the wing is a flat plate, but which has any local angle of attack whatsoever, 4 with respect to the freestream vector, V,, so long as it is small. 5. DEVELOPMENT OF THE CASE OF THE SYMMETRIC TYPE OF CONFIGURATION In this case it will suffice to examine the flow solely in the upper halfplane, where z > 0. If 0(l)(x,y,z) stands for the flow which takes place in this upper region, and if 0(2)(x,y,z) represents the flow in the nether region, then, of course, (2)(x.yz) = #()(x,y,z) NACA TM 1581 5 The boundary conditions in this case are composed of equations (5'), together with the restriction that ( = 0 (for locations lying beyond the region \ /z /z=0 occupied by the wing surface) (2') Now let the definition of the function describing the velocity com ponent at the wing surface, and also the potential function itself, be cast into the convenient forms H*(x,y) = V Hm (1) cos y = V. Hm() cos mi m m and 0(l)(x,y,z) = O(l) = V. b m(e ,) cos I m1 (4) m wherein = x/b, y = y/b, and = z/b, while b is a suitable length used for purposes of nondimensionalization. The value used for b will be equal to the semispan of the wing in the case where the leading edge of the wing is supersonic everywhere, and provided that the wing tips are cut off in such a way that the wing surface remains outside of the tip Mach cones emanating from either one of the wingtip extremities out at the farthest reaches of the wing span. The value used for b will be larger than this semispan just defined, if, in contrast, these geo metrical relationships do not hold; the magnitude employed for b in this latter case is illustrated in figure 2. Finally, it should be observed that H* is a periodic function of y, which is equal to the values taken on by the function H at the wing's surface and it is zero for points lying out of this region, and this definition is to hold throughout the spanwise interval for which b < y < b. The fact that it is possible to write H*(x,y) in the form given as equation (4) (i.e., the possibility of expressing the component velocity field describing the normal velocities to the wing surface by means of a Fourier series instead of in terms of a Fourier integral) stems from the property already noted to the effect that the perturba tions, which are created at any arbitrary point P(x,y) whatsoever, do not make themselves felt anywhere outside of the Mach cone emanating from P. As a result of this situation, therefore, as far as the NACA TM 1581 determination of the field of flow about the given wing is concerned, it makes no difference to this flow whether one considers the wing to be operating by itself as an isolated entity within the impinging stream or whether, instead, one imagines it to be accompanied by an infinite number of reflections of this primary wing in the planes y = tmb. If one now inserts the second of the expressions given as equa tion (4) into equation (1), it will be seen that this differential equa tion reduces to  B2 2 = k2m wherein B2 = Ma2 1 and where k replaces the constant 1M. 2 Meanwhile, it is also evident that, on the basis of the first of the formal developments given as equation (4), the boundary condition reduces to (1m\ The expression given as equation (5) above is formally analogous to the socalled "telegraph equation," and its solution, which is suitable for applying the type of boundary condition exemplified in equation (6), is SB B0 ^=%j hm(') Jol (l7) ETB2tj dti where JO is the cylindrical Bessel function of zeroth order. Consequently, the vertical derivative turns out to be a0M = (6B +) j B h(') 1 BE, B22')2 B~ Q g 2);B2 = Bm(S) NACA TM 1581 and, because of the boundary condition (6), it follows that so that th sought potential must have the frm so that the sought potential must have the form (7') B1 4. DEVELOPMENT OF THE CASE OF THE ASYMMETRIC TYPE OF CONFIGURATION The possibility of being able to find solutions to such asymmetric problems by means of the method being propounded here is restricted in this case to those configurations for which the leading edge as well as the trailing edge of the wing are supersonic, and where the wing tips are cut off in such a way that the wing surface lies outside of the Mach cone emanating from the very tip of the leading edge where the maximum span occurs. Under these circumstances the boundary conditions are constituted from the restrictions given as equations (3"), and of equation (2') once again. If one then follows the same procedure as was utilized in section 5, it follows that the expression for the sought potential is formally given as (ref. 1) 0M = nhmE B + k f B I 1 ^ )2 B2 (gg)2B22 m\~ ''L~t t where h, is, a priori, an undetermined function, and where it should be recognized that the + sign is to be employed for the lower halfplane where t < 0, and where the sign is to be employed for the upper half plane where t > 0. It is evident, therefore, that the derivative of m with respect to will be continuous along the plane 0 = 0, but the Bl(S ')Jo dq' NACA TM 1581 derivative of m with respect to t will be discontinuous, and the "jump" will be of such size that holds true. It is clearly permissible here again to concentrate attention solely upon the disturbed flow in the upper halfplane where t > 0, therefore, because the observation just made above will tell one how to compute what the flow will be in the other lower halfplane, once the former is obtained. The boundary conditions in this instance may now be recast into the IBh. () + k2 /F it T h(t ') Jo nk (t') Jil k ''0 I BL) d'  S') d' = Gm(d) provided, as in the previous section, one sets up the convenient con vention that G*(x,y) is to represent a periodic function in y that is to be equal to the values taken on by the function G(x,y) at the wing's surface, and it is to be zero for points lying out of this region. This definition is to hold throughout the spanwise interval for which b S y < b. In addition, the form of G*(x,y) is to be assumed, specif ically, to have the appearance G*(x,y) = Vc G~ m() cos mT m while it has also been assumed that the derivative of a function by the sole parameter upon which it depends is to be denoted by a dot over the function, that is, d1h de form =O+0 /H  W6t /t~o NACA TM 1581 The integrodifferential equation defining bm may also be immedi ately simplified to the compressed expression B(hm() + j hm( ') (Jo + J2)d' = Gm( ) (9) Now apply a Laplace transformation to this integrodifferential equation (i.e., multiply through by the factor ePE and integrate from 0 to m). Thus, one obtains 2 +2 k2 1 p2 +2 CBfmp + Gm B p 2 2 k2 where a bar over a symbol serves to indicate that this quantity stands for the Laplace transform of the function so designated. Standard tables of Laplace transforms could be consulted to check these results, which may now be simplified by noting that 1 + 2 k2 + 1 4(5B2 p + k2 k2 +2 k2 SB2 B2_ TB2 k2 + p2 k2 p2 2p /p2 + k2 k2 p B2 B2 T B2 = 2 B p2 + p k2B2 Thus the Laplace transforms of equation (9) simplifies to tBpbm + xB p2 + p m = Om or the explicit expression for the Laplace transform of the unknown func tion hm is given in the form 1 2 k2 p B2 xB NACA T' 1381 so that finally one may invert the transformation to obtain hmtf 3xBJ GCm(')Jo ( ')] d' Once having obtained the value of hm, it is easy to write down the expression for the component of velocity lying in the xdirection and located at the wingsurface, because one has simply that this component is given by the partial derivative of the potential 0, taken with respect to t, and evaluated at the plane of the wing; i.e., one finds that m cos m2 = V Y ]m(w) cos m 2 m Furthermore, the formula giving the lift on the wing is just L = 8pV. 2b2 m+l1 S(1) 2 hm/b)m m (11) for m = 1,,5,5,. ., etc. where the symbol, 2, is used to denote the distance along the xaxis measured back from the leading edge of the root chord to the projection into the plane of symmetry of the trailing edge of the wingtip profile. In regard to the moment taken about the yaxis, it is apparent that it may be computed from the relation: My = 2npV.2b5 m = 2zpm b5V2 f cos (on Wl ) mtl ()(l) 2 for m = 1,3,5,. ., etc. (10) dI f/b 0 bm ) 0Z/b 0 hm() d) (12) a ( =0 S(5cO=0 NACA TM 1381 5. PROBLEMS HAVING TO DO WITH FLOW PAST BODIES OF REVOLUTION The procedure discussed in the preceding sections can be extended at once to apply also to the solution of problems which are concerned with the flow over bodies of revolution. For this purpose let a cylindrical coordinate system (x,Y,8) be set up, and then the equation which governs the potential, 0, being sought will have the form 20 L B2 02 (13) 6y2 aY y2 6e2 ,x2 Now let R = R(x) be the equation of the meridian line of the body, and let it be assumed that R is sufficiently small at all locations along the body so that the direction cosine of the normal to this merid ian line, measured from the Yaxis, may be taken to be equal to unity. Furthermore, let Vnf' stand for the component, taken in a direc tion perpendicular to the circular crosssection of the body, which arises from the impinging flow which invests the body. Then the boundary condition which must be satisfied at the surface of the body may be expressed mathematically by the relation = Vnf For sake of simplicity, it is also now assumed that the treatment to be developed is to be restricted to the case where symmetry with respect to the semiplanes 8 = exists in the incident flow. Under 2 this hypothesis it is convenient to write the normal velocity components and the potential being sought in the following explicit formulations: r S= Vm Fm () sin m o m (14) 0 = V. m(x,Y)Ym sin m a m 12 NACA TM 1581 If one now inserts the second of the expressions given as equa tion (14) into the differential equation governing the flow (13), it will be found that the defining equation for the potential will have the form a&2 Y2m ay2 + 2m + 1 m Y Ny (15) = B2 m 6X2 and the boundary condition turns out to be a(YY) Y Y=R = F (15') A suitable solution to equation (15), which can be made to satisfy the boundary condition being imposed as equation (15'), will be found to be ~m (lm 0 Y Y) 00aY 0a are ~ cosh BY (16) fo(xBY cosh u)du where NACA TM 1581 Thus, the successive individual potentials are given by the expressionsI Yo1 = B f arc Y02 = B2 f arc cosh  BY cosh XA BY fl(xBY cosh u) cosh u du f2(xBY cosh u) cosh2 u du etc. Upon imposition of the requirement that the boundary condition (15') is to be satisfied, one obtains a set of integral equations which serve iTranslator's note: It was pointed out on page 650 of an article by R. H. Cramer in the Journal of the Aeronautical Sciences, vol. 18, no. 9, September 1951, entitled "Interference Between Wing and Body at Supersonic Speeds Theoretical and Experimental Determination of Pressures on the Body," that the result given here for O, for m > 1, is incorrect; the correct formula is, for m = 2, Y2 = B2 r x f2(xBY cosh u)(2 cosh2 u 1) du arc cosh By while, in general, the use of hyperbolic functions of multiples of the argument u gives a more compact form, which is easy to work with; i.e., in general it is true that Y = Bm()m Sco fm(xBY cosh u) (cosh m u) du are cosh BY (17) NACA TM 1381 to determine the arbitrary functions fm, which are a priori unknown. Thus, applying these conditions, one finds that2 Bm+l (_)m+l f fm(xBR cosh u) (cosh u)m+l du = Fm (18) arc cosh x BR The determination of the values of the fm's appearing in formula (18) may be carried out by using a stepbystep procedure which is entirely analogous to the one employed by Von KArmAn in his work on determining the flow about a body of revolution at zero angle of attack. It is important to point out that if one only has in mind to calcu late the force distribution along the axis of the body and the corre sponding moment, and if one is not interested in knowing the local veloci ties or pressures around the body, then it is merely necessary to calculate 0 and 02. 6. APPLICATIONS The procedure that has been propounded above has been applied (ref. 2) to the situation arising in the study of the question of wingbody inter ference. The wingbody configuration considered in this particular appli cation of the method is depicted in the appended figure 5. The wing used in this configuration is a flat plate, whose semispan is equal to 4R0, where RO is the radius of the circular crosssection taken through the body at the location where the body is widest. The leading edge of this wing is located 5RO downstream from the tip of the nose of the body. The freestream flow is impinging on the body at a speed which is twice the speed of sound in the undisturbed stream. 2Translator's note: In view of the correction pointed out in Note 1 above, it will be seen that this formula for determining the fm functions is also incorrect, except for m = 1; for higher integral values of m, the correct formula is: Bm+1 (i)m+l1 m(xBR cosh u) [cosh m u cosh u] du = Fm arc cosh BR NACA TM 1381 The curves shown in figure 5 give the value of the pressure coeffi cient, P P, out along the span of the wing, in the midchord loca 2 tion (i.e., along the wing axis), for points on the upper (dorsal) side of the wing. These coefficients have been calculated by the method out lined in section 4, and there are shown results for various angles of attack, which apply to such points on the upper side of the wing pro files at their midchord positions. In addition, some experimental test points obtained by R. H. Cramer (see ref. 2) are also plotted on these curves. These results were obtained from experiments carried out in the supersonic tunnel of the Daingerfield Aeronautical Laboratory. The agreement between the computed and experimentally determined results is very good from a qualitative viewpoint. In regard to the more precise details of the quantitative comparison between the results it is worthy of note that the experimental results exhibit a certain amount of dissymmetry as one passes from positive angles of attack to negative angles of attack. Such a dissymmetry cannot be predicted, or should not be expected, from the type of theoretical treatment being considered here. In order to bring about a more valid comparison of these results, it would appear logical, in face of such evident dissymmetry, to take for the representative experimental value, at a given value of the angle of attack, 0, the one which is obtained from averaging the result obtained for an angle of attack equal to +p with the result obtained at 0. Such average values have been computed and are designated in the plots of figure 3 by means of solid circles. These adjusted values lie much closer to the theoretically derived curves at almost all locations. Translated by R. H. Cramer Cornell Aeronautical Laboratory, Inc. Buffalo, New York REFERENCES 1. Ferrari, C.: Interference Between Wing and Body at Supersonic Speeds  Theory and Numerical Application. Jour. Aero. Sci., vol. 15, no. 6, June 1948, pp. 317556. 2. Ferrari, C.: Interference Between Wing and Body at Supersonic Speeds  Note on Wind Tunnel Results and Addendum to Calculations. Jour. Aero. Sci., vol. 16, no. 9, Sept. 1949, pp. 542546. NACA TM 1381 Voo Figure 1. Orientation of coordinate axes and location of typical wing plan form therein. /I F I8 F7 LI Figure 2. Definition of the interval of periodicity required for application of the Fourier series technique when leading edges are subsonic. NACA TM 1581 Cp Poo 2 0.00 1/2p, Voo 0.00 e Upper 0.04 surface 4 0.08 0 =40 o I X 1 I 0oB 012 0.160 4Ro=b Q20 2R QuWing juncture 0.20 0. .10 .20 .30 .40 .50 .60 .70 .80 .90 100 77 y/b Figure 3. Pressure distribution along the wing axis: Comparison of experimental results with predictions based on the method expounded in section 4. NACALangley 11354 1000 9 *) 43, Q a S < < . ra 0   4M u a cc V C4 dL CL L ^s st Ca, ON~y CL o *a, a S C a, 6 ....  w1a mo '4 a E Za, C 0 CL *OmE gC O.C m 0 V Co0.~ 0 C =a, ., L, w ss !:i^ a L.z.: ;CL cc 0 w co asw l a a,^^J  S a, .^  a c a s c c~r c U0 zzo  .." I ,'* a, 0 C4 4 SC1 Z En a U 0 0 OL U  bL0o.. o "J,3  C M C. .. C~z V;; __~ g * !1 * S < as *gi I . S * C.) C~a C s0lw4 Is .5!ok p o 0 U 0 t 'Q4) E 4>, ' 4 04 4 r*sPE! ~ i oa CT *w ii a u m S.. : d I CCc z* O*. 9 trO 0. l ~0.L Is ;E;^s a0 0 V5*g* 0. 11 oa u i i ss ^ to 60 ,>.t>.z *&4 o Oh XF ~ O ')C')~ Qa E &^g~g faS^I  c a ; co i tiH < C 4 .u' . C ,) 0 .6 0. OcU 0 W 3 ^r; 81 i I. o 8 cw' 1 .2 0< 0 * . 0 "I I Ci 0 w M, P U 0 a m1 4  U)i^  s OK o o' s ~ m t i ^ .j^'ml,< A0 0 10, 0 IdC cN  m hS a t~'.'S S /. 1^: sl~: Li 's 0g Eli cc ; 0s V 5 UVmDO .4 o 04 0ca 0 Zd 0 IDH ~ ~ o P4 id >0 .2 W g ls L) u r,4 . I I1 v o t sB vs s S^s^ g s 6I m M5 0 b o 0 ZZNZ 0 fi "a 3015 0 i u 9o a o 0 u 0 M._ C > D r E~f a~a coE ^ E 5L  0 q 0 W 0 0 w0 0 ou bo ar E a E z .0 W w w o~ I UNIVfRSITY UlF LUHIUA 3 1262 o 06 523 6 
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