/hf 4r^ '}fO(y' B. M. LE ADON 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1100 
 
 INFINITESIMAL CONICAL SUPERSONIC FLOW 
 By Adolf Busemaim 
 
 Translation 
 Infinitesimalo kogelige Uberschallstromung 
 
 30 
 
 Washington 
 
 UNIVERSITY OF FLORIDA 
 March 1947 DOCUMENTS DEPARTMENT 
 
 1 20 MARSTON SCIENCE LIBRARY 
 
 P.O. BOX 117011 
 
 GAINESVILLE, FL 32611-7011 USA 
 
127, Of 1 ^ "^ 
 
 NATIONAL ADVISORY COMMITTEE FOR iffiRONAUTIGS 
 
 TECHNICAL MEMORAl^lDITM NO, 1100 
 
 INPINITESIMi^i CONICAL SUPERSONIC PLOW--- 
 By Adolf B^iseiiiaim 
 
 Conic a l^ flov/ f l.e lds , - Real flov;s alv/ays occur In 
 three-dimensional space. In calculating a riov.r, hov/ever, 
 one will greatly appreciate it if there are onl^f two 
 essential coordinates to deal v/ith. plows of this I:lnd, 
 limited to two coordinates ^ forr.i the plane flov/ and the 
 flow of a.::ial syiiinietry. The space '.vhich is filled out 
 by the streamlines Is represented in planes parallel to 
 these lines; they contain certain streajiilines to their 
 whole extent. In conical flow fields, however, the 
 streaxiillnes are cut through slantingly so that each 
 streaLTillne is contained in the plana but appears there 
 as a point only. These relations are made cleaz-^ in fig- 
 ure 1, If the friction is neglected the shape of the 
 body leads owe to expect a pattern that can be increased 
 or decreased geometrically. The fixed point P and the 
 direction of the three spatial a:;:es, x, y, and a remain 
 the same. All essential characteristics of the flow and 
 the shape of the body can "be inl'erred from the plane 
 z = 1. A piano z =; 2 \!fould, if diatarxccs vicro doublod, . 
 show ident.,lcal values for gas conditions and velocities. 
 The isobax' planes in the space x, y, a are of conical 
 shape and have the cone vertex p| therefore these flows 
 shall be called abbreviatedly conical flow fields. 
 
 Infinitesi mal differe nces in pressure .- In figxire 1 
 there iihrelT^be one' more limltat'fon for the general conical 
 flow fields nan'iely that the body disturbs the parallel 
 flow only to a slight degree. So the conical isobars 
 reflect bvor- and under-pressures differing infinltesl- 
 mally from the pressure of the parallel flov/. This two- 
 fold" limitation, to conical and Infinitesimal, is not 
 actually very stringent Insofar as in the class of poten- 
 tial flows there are present only the conical fields of 
 a:'cial syimiietry and the infinite sim.al conical fields. All 
 other conical" flo?/s are affected by rotation. The infini- 
 tesimal supersonic flows, however, also excel in another 
 way: the superposition of fields with different fixed 
 points P is permitted in spite of the fact that the dif- 
 ferential equations ordinarily are not linear; thus the 
 applicability is broadened most gratifyingl^'-. 
 
 -"-"Ini'initesimalo kcgo'llge ijberschallstromung. " 
 Deutschen i"jiademie dei- Luf tf alirtf orshung, 19lj-2-Ii.5* P. -'-l-f'S 
 
NAG A TM No. 1100 
 
 Dlf 
 the dUT 
 fields t 
 Unit th 
 spatial 
 torical 
 Therefor 
 known, t 
 space X 
 basic ve 
 if the 
 
 ferential eqi\aticn.- There are two ways to limit 
 
 potential in conical 
 u, V, w, and to 
 
 erential equation for th 
 
 o small additional velocities. ^, v, -.. , 
 
 e differencial equation for nearly ^Darallel 
 
 flov/s to conical fields; the first is the his- 
 
 one, the second, however, the simpler one. 
 
 here the second one is chosen. as is well 
 he line noised differential equation in the 
 , y, z, for the additional potential cp o-srev a 
 locity W in the direction of the a_vis z reads 
 2; as has the sonic velocity a: 
 
 + o. 
 
 + ? 
 
 .= {- 
 
 (1) 
 
 The coordinates & anc 
 respond to the spatial 
 plane z = 1 : 
 
 T) of the conical current cor- 
 co ordinate 3 x and y in the 
 
 X 
 
 z 
 
 and 
 
 •n 
 
 V 
 
 •J 
 
 z 
 
 (2) 
 
 The additional potential cp increases on each ray through 
 the fixed oomt P in proportion to the distance. There- 
 
 Pore , the potential divided by 
 
 z 
 
 is invariant on 
 
 single ray, 
 flow : 
 
 ind provides the potential of the conical 
 
 yS-l^-^.) ^ T ^o(x,y,z) 
 
 (5) 
 
 The additional velocities 
 of the f ^m o r tj o t ont i a 1 , 
 
 U, Vj 
 
 'c tho derivatives 
 
 ^ = 9x = ^ 
 
 •- = CD = ' 
 
 _ ;^ 
 
 
 > 
 
 ik) 
 
 The differential equation for the new potential "^ 
 determined from the old differential equatior, a^-' 
 
 obtains 
 
 d 
 
 13 
 
 one 
 
NAG A TM IJo. 1100 
 
 •X A -siV-/ fi-^^-y sin 
 
 mi'ith 
 
 1 
 
 A 
 
 1 
 
 tg a 
 
 ^ 
 
 (5) 
 
 It certainly is gratirylng to roco 
 this differential equation an old 
 plane gas flow; for thostroaiii f'anc 
 transformed according to Legendre 
 the components of the current dons 
 differential eauation. I 
 
 gnize in the type of 
 acquaintance froiu the 
 tion of the plane flow 
 and superimposed over 
 ity prodi 
 n or dinar'' 
 
 ices exactly 
 
 eve] 
 
 tL 
 
 aenominator 
 
 A^ IS 
 
 gasf 
 
 a ±oc 
 
 is a special gas with rectilinear 
 pressure -volume -diagram in which, 
 denominator also remains constaiat, 
 favorite v/here it is a mere quest! 
 calculations ♦ 
 
 hov/- 
 
 al function, but there 
 adiabatics in the 
 as i^e quired, this 
 
 This gas is a special 
 on of numerical 
 
 Regions 
 
 of influence.- The spatial differential 
 as flow at supersonic velocity is of 
 
 equation of the gj 
 
 hyperbolic character, as shown in equation (1). That 
 means: each point of the flow dominates a conical range 
 opening dovYnstrearaj each locus, on the other hand, is 
 dominated solely by those points v/hich are situated in 
 the cone prolonged bacla^/ard and opening upstreaiii. Here- 
 with the relationships are divided definitely araong the 
 three possibilities : superior, subordinate, and independent, 
 Mach's cones in the supersonic flov/ considered as regions 
 of disturbance of a small trial bodv make this fullv com- 
 
 a physical sense. It must seem odd at 
 dependencies of the general spatial flow 
 soon as one proceeds to a more lim-ited 
 But the above-mentioned differential equa- 
 
 the radxus A 
 
 prehensible in 
 first that the 
 are widened as 
 spatial i'lov/. 
 
 tlon shows that inside of the circle v/iti 
 there prevails the elliptic character. 
 
 This behavior is easily explained hj the fact tlis.t 
 all points of a ray stai^tlng from^ P are cojuprehended 
 as a whole. The relation of dependencies of two rays 
 results from, the dependencies of the single points. Only 
 the characteristic "independent'' appears uniformly in 
 certain cases for all x^airs of points (P itself is 
 excluded). The combination suiDerior and independent 
 
k 
 
 NAG A TM No. 1100 
 
 becomes superior; subordinate aiid independent become 
 subordinate. But i" there are pairs of points of all 
 kinds on the rays^ then the rays are subject to the new 
 characteristic "reciprocally dependent." Rays of this 
 kind fill out the interior of Mach's cone starting fi-on 
 point P. 
 
 Characteristics.- Mach's cone starting from P 
 Intersects the'^'plane 2=1 on the circle having the 
 radius A. In the field outside of this cone, i, e., 
 outside of the circle in the intersecting plane, one 
 gets rectilinear characteristics of the differential 
 equation (5) which are tangents of the circle. In fig- 
 ure 2 this is demonstrated by t\/o wires, a and b. 
 The wire b is bent slightly upstream in order not to 
 exclude cases of this kind. The range of disturbance 
 results from, the svm of all of Mach's cones starting from 
 all points of the v-:iro. It is iiiimediately obvious that 
 only the cii'cle with the radius A and its tangents can 
 form the boundaries of the area of disturbance. Outside 
 of Mach's cone starting from point P these character- 
 istics settle all questions; they can be traced back to 
 the plane case with a transverse component of the velecity. 
 The essential and different part of the conical fields, 
 therefore, is concerned with the convex surface of Mach's 
 cone starting from poijit P, &iid with its interior. 
 
 Tschapligin' s illustra tion .- In the plane of inter- 
 section ^""=""1 we' find inside of the circle with the 
 radius A the elliptic character of the differential 
 equation (5)« Near the center the differential eqiiation 
 of the potential theory is valid; in plane cases, this 
 equation can be satisfied by analytic functions of the 
 complex variable. In this circle, therefore, there only 
 exists a mutual dependence but not yet a full equivalence 
 of all loci. This is not surprising, because the analytic 
 continuation of the plane reaches to the outer range of 
 the circle. Tschapligin, hov/ever, has devised a geomet- 
 rical construction which so distorts the field inside the 
 circle that equivalence regarding the differential 'equa- 
 tion will result. As figure 5 shows this distortion is 
 attained by transferring the plane z = 1, with the 
 com.plex variable t = £, + it], through parallel projec- 
 tion to a sphere with the radius A, and by then pro- 
 jecting it fro!-ii a pole of the sphere on to a plane in 
 the distance. One v/ill easily recognize that only tlie 
 interior of the circle with the radius A v;ill be . 
 
NACA TM No, 1100 
 
 depicted; fi.rst it will be delineated from the lower 
 half-sphere on the Interior of the unit circle of the 
 new plane v;ith the new complex variable e; a second 
 time it vi/ili go from the upper half -sphere on to the 
 outer field of the imit circle. In these cooi'dinates 
 one can use analytic functions for the solutions.. 
 
 SOLUTIOH OF TilS DIPFSREHTIAG EQUATION 
 
 For each of tlie velocity coiiroonent; 
 
 u, V, ana w 
 
 one can oqxiate the real pai't of an analytic function f(€). 
 It ?;ill serve the purpose best to set up the equation 
 for the coniporient w, because then the more closely 
 related components u and v can be calculated jointly; 
 
 w = A . Re (f (e )) or w + is = A • f (e) 
 
 (6) 
 
 The completion represented here by s is for the time 
 being completely meaningless. According to Tschapligin 
 there then results the complex velocity: 
 
 0) 
 
 u + 
 
 - = - |J(f ^ -) 
 
 (7) 
 
 The pressure in the current, with the aid of the den- 
 sity p, results from the velocity components as follows 
 
 P = -P 
 
 W 
 
 w + 
 
 2 , 2 , ; 
 
 U + V + w 
 
 \ 
 
 K -|-p|AtV (f + f ) + ww] 
 
 (8) 
 
 The right fianction f(e) is to be selected with the aid 
 of the boundary conditions. 
 
 Bound ary conditions .- The outside of Mach's cone is 
 superior to Mach's cone itself. Therefore, first, those 
 velocities u, v, and w on the circle of the t -plane 
 (and therefore on the uniform, circle of the e -plane) 
 that I'esult from the outer field must be ascertained. 
 
NACA TM No. 1100 
 
 If the body does not protrude anywhere out of Mach's 
 cone, the values u = v = vv = on the uniform circle 
 are given. If on the contrary no part of the body is 
 Inside of Mach's cone, the values of vif are to be 
 represented by an analytic function f(G) free of 
 singularities with the given boundary values on the 
 circle. If f ( e) does not produce a stationai'y value 
 df = 0, then u and v according to equation (7) v/ill 
 have a logarithmic singularity at zero. The many-leaved 
 f tmction can be selected in a unique way by using radial 
 intersections with the boundai^y values of u and v on 
 the uniform circle. The radial intersections produce 
 rotationcJ. layers, as is 'phjslcs^llj to be expected from 
 a lifting surface, 
 
 Imperm.eable boundaries of the body can be trans- 
 ferred into the e-pl£uie at the same time. They nus t be 
 streamlines in the field of the relative velocity: 
 
 ^rel = ^'^ " ^^^ - -^'^^ (9) 
 
 This condition is not alv/ays easy to comply with. How- 
 ever, if the body possesses rectilinear surface elements 
 passing near zero, the other'wise meaningless imaginary 
 part s of the function f(^) will remain constant on 
 these elements. If the straight part goes over zero, a 
 stationary value for f is to be stipulated at zero. 
 Conditions of this kind are especially agreeable. From 
 the pressure equation (3) conditions applicable to cases 
 of given pressures or ox" given lifts are to be understood. 
 
 The disappearance of the real or of the imaginary 
 part of f on certain lines because of syfiimetry can be 
 attained in the well knovifn way by reflexion, as the 
 examples will show. 
 
 Examples 
 
 1, The circular cone in the straight flow 
 
 For the only axial-s;>/riirnetrical case, 1. e,, the 
 circular cone with an Infinitesimal apex angle, the 
 right solution is, of couj^se, given by the statement 
 
 w + is = C • Inc 
 
WAGA TM LTo. 1100 
 
 The pressure on the convex surface of the cone resiilts 
 in the known way (fig. h,.) and conforms vi^lth v. Karman's 
 values and rnlne . 
 
 2. The c1,rculai' cone in oblique flow 
 
 Oiie succeeds, with the aid of the relative velocity 
 according to equation (9)> in solving the circular cone 
 in oblique flow. Herein the apex angle and the angle of 
 attack may, though infinitesimal, yet bear a relation to 
 each other. The solution is shown in figure 5* W one 
 makes the angle of obliquity y zero, one gets again the 
 circular cone in straight flow. If one makes the apex 
 angle 2p disappear, one gets the pressure distribution 
 of a circuleT' cone in an incompressible current. The 
 comparison witli Ferrari is rendered somewhat difficult 
 by the fact that Fei^rari measures the volocity field per- 
 pendicular to the Gone axis while it is here perpendicular 
 to the wind direction. If the system of coordinates is 
 rotated adequately, the conformity is complete. 
 
 5. Tip of a rectangular plate 
 
 If a plane rectangular plate of infinitesimal thick- 
 ness is placed in a flow perpendicular to the front edge 
 with an infinitesimal angle of attack, and if the velocity 
 field is needed only up to the rear edge of the plate, 
 one can place the fixed point P at the right corner 
 point of the front edge. On the supposition of an 
 infinitesimal angle of attack y ('■''^ith the x-a:cis 
 forming the axis of rotation) the pressure distribution 
 will be represented on the quax'ter plane betv/oen the 
 positive axis (z) and the negative a:cis (x). For the 
 plane z = 1 the section of the body, except for infini- 
 tesim.al distances, is then rendered by the negative real 
 axis. Let the reduced pressure above the plate and the 
 increased pressure below the plate be adjusted to a unit 
 value outside of Mach's cone. These values hold on the 
 boundary circle. On the left half of the unit circle of 
 the e -plane corresponding values for w are then to be 
 assigned. On the right semici3:^cle the outer field is 
 undisturbed; here v/ = 0. For reasons of syimnetry the 
 value w = must also result on the positive real axis. 
 y\long the negative real axis, on the contrary, 
 s = Im(f(e)) must be fixed because of the fixed radial 
 boundary. Since s is given only up to one constant, 
 one can demand here s = 0. All conditions can be 
 
8 NAG A TM No. 1100 
 
 attained by reflexion if one undertakers a preliminary 
 conTormal mapping on the plane V = yc* 
 
 The solution is represented in figure 7« Figure 8 
 shows the pressure distribution on both edges of a rec- 
 t angul ar plate. 
 
 14.. Supporting triangle 
 
 Every t /o radii starting from P form a triangular 
 plane as far as the piano z = 1, v/hen all points ai-e 
 connected. Because of the required infinites i:aal dis- 
 turbance of the parallel current, hov/ever, the angle of 
 attack must be inf initesim.al, so that the plane of the 
 two rays will ne&vlj pass through the z-a^cis. Such 
 triangles are possible completely inside of Mach's cone, 
 completely outside, and uni- and bl-laterally protrv.din.g. 
 Here v;e shall only consider the simplest case of the 
 supporting trlaiigle outside o£ Mach'iJ cone, although a.11 
 other cases can be easily integrated. 
 
 Pigtu-'O 9 shows this supporting triangle. The 
 velocity component w which predominantly influences 
 the pressTxre is different from, zero only on the short 
 arcs between S^ and ^3 as also t^ and ^g. The 
 value zero results froia the undisturbed state on the 
 right, and also en the lex't because of the pressure 
 adjustment behind the triangle, when consideration is 
 given to the symmetry v/ith a positive and with a nega- 
 tive angle of attack. Plgui^e 10 shov/s the relations in 
 the ^ -plane. If one intends to let the rear edge of 
 the triangle travel while the front edge lies fixed, one 
 will at first tr.nsfei' only the points t^ and t^ into 
 the € -plane. Withi suitable regulation there must result 
 an increase of w from. to +Tr at ^j_, and from 
 -rr to at t^ (If one rnoves on the circle in the 
 direction of increasing angles). One can treat this 
 part of the solution independently if one assumes a 
 further singularity at zero. Physically spealving, one 
 then has a uniformly loaded triangle bctv/een the front 
 edge at t,^^ and the s-axis. Inside of Mach's cone, 
 
 however, t is triangle is not flat, but is twisted to 
 uniform or load,- /Is 3oon as on:-: suporimpoaos at 
 the rear edge a negatively loaded triangle and its influ- 
 ence between t^^ ^^'id the z-axis, the part behind the 
 rear edge u'ill no longer be supporting, and the singu- 
 larity in w at the point zero of the e-plane disappears. 
 
NACA TM No. 1100 
 
 However, a vortex layer in the field u, v is left. 
 The pai'tial solution in the e-plane is represented in 
 figure 11. 
 
 5. Superposition of tv/o conical flows 
 
 The infinitesimal conical flows can be superposed 
 without having the fixed point in coramon as in f igrire l\.. 
 Therefore the relations in the plane plate can also be 
 represented v/hen the plate has more depth. Figure 12 
 shows the isobars of the edge of the plate and also their 
 superposition after Mach's cones have overlapped. Dif- 
 ferent boundary coudltions for the partial solutions 
 need be considered onlj v;hen the cones reach the other 
 edge of the plate. The disappearance of pressure along 
 a straight line in just the distance at which the cones 
 arrive at the other edge of the plate is remai^kable. 
 The positively loaded part of the plate ends here. Fig- 
 ure 15 shows the lift distribution of the positively 
 loaded part in perspective representation. 
 
 To find the velocity field behind a rectangular 
 plate of finite depth one can annul the svipporting pres- 
 sure differences of a plate of infinite depth by conical 
 fields having the apices on the rear edge of the plate. 
 If one superposes a negatively loaded plate shifted 
 inf initesimally in the direction of the z-ajils one 
 obtains a supporting line as a limiting case of the 
 supporting strip. The cases calculated by Schlichtlng 
 according to Prandtl's method are obtained in this v/ay. 
 Here, too, the conformity is perfect, except for an 
 error of sign in the calculation of the Integral equation. 
 
 SUm^ARY 
 
 The calculation of infinitesimal conical supersonic 
 flows has been applied fir-st to the sim.plest examples 
 that have also been calculated in another way. Except 
 for the discovery of a miscalculation in an older report 
 there was found the expected conformity, Tlie new method 
 of calculation is limited more definitely to the conical 
 case; but, as a compensation, it is m,uch m.ore convenient 
 because the solution is obtained by analytic fiinctions . 
 The fundsanental i-ecognition that there the hyperbolic 
 character is replaced by the elliptic one will lead to 
 
10 NAGA TM No. 1100 
 
 more thorough Investigation of conical fields as special 
 cases in supersonic flov/s. Of course, one ■«ill be 
 teivipted to call the elliptic character seemingly elliptic 
 onlji for if one notches an indentation into a cone, the 
 real hxpcrbolic character of the field of the flo^v down- 
 streatn from this indentation immediately becomes obvious 
 again. P^iowevt-r , if one traces the flow to a point very 
 far behind the indentation there i/ill appear inside of 
 Mach's cone a reciprocal relation betiveen every two rays 
 v/hich will gradually restore the conical course of the 
 flov;. Instead of trying to produce the final flow hj an 
 infinite succession of hyperbolic dependencies it will 
 be more expedient to consider special elliptic singu- 
 larities at the points of disturbance. In these rela- 
 tions I see the significance of the conical field; the 
 irrfinitesim.al case represents only a first approximation 
 to it. 
 
 Translation by Mary L. Maiiler 
 and Robert T. Jones, 
 National Advisory Conuiiittoe 
 for Aeronautics . 
 
NACA TM No. 1100 
 
 Figs. 1,2 
 
 Figure 1. Coordinates in conical field 
 
 Figure 2. Disturbance field of elements a and b 
 
Figs. 3,4 
 
 NACA TM No. 1100 
 
 C " 2 
 
 Figure 3. Chaplygin's transformation 
 
 / = Cine 
 
 p = ^W^ f/32 In 
 
 A = tgOc 
 
 / 
 
 1 
 1 
 1 
 
 I 
 
 \ 
 
 \ 
 
 1 
 
 
 B~\ 1 
 
 
 \ 
 
 © ; 
 
 z 
 
 1 
 
 I 
 \ 
 
 / 
 / 
 / 
 
 
 \ 
 
 / 
 
 
 Figure 4. Circular cone in axial flow. 
 
NACA TM No. 1100 
 
 Figs. 5,6 
 
 / = 4V AR^ iln 
 
 e-e, 
 
 e-- 
 
 1-- 
 
 3^0 ^^« 
 
 
 i-e„ e-e„ i-ee 
 
 ^e-^o^^ ^^-e^o^ 
 
 2.1 p^ y^ 
 
 P 
 V 
 
 = Qn'^ ( 13'^ Iv -^-^ - -i— ~ + 2/3ycos S+y^ cos 21 
 
 t^a 
 
 fi 2 2 
 
 ) 
 
 Figure 5. Circular cone in yawed flow 
 
 — (pd-ps) 
 a 
 
 A'.t 
 
 Figure 6. Edge of a rectangular plate 
 
Figs. 7,8,9 
 
 NACA TM No. 1100 
 
 
 V - )f€~ 
 
 r 
 
 f(v) =--i4ln(v-v^) -ZnCv-VgJ 
 -ln(v-v )+ ln(v-v )\ 
 
 Figure 7. Conformal representation at edge of a 
 rectangular plate 
 
 Lift 
 
 Direction of motion of 
 plate 
 
 Figure 8. Pressure distribution on a flat plate- 
 
 W 
 
 Figure 9. The 
 lifting triangle 
 
NACA TM No. 1100 
 
 Figs. 10,11 
 
 i = 
 
 sAe 
 
 !+€€ 
 
 Figure 10. Cross section of the lifting triangle 
 
 / 
 
 f^) = i{^n(e-e,JHn(e-e^)-Lnej 
 
 Figure 11. Representation of the lifting triangle 
 
 in the e plane 
 
Figs. 12,13 
 
 NACA TM No. 1100 
 
 Figure 12. Superposition of edge influences for 
 the rectangular plate at supersonic velocities 
 
 Figure 13. Pressure distribution on the rectangular 
 plate at supersonic velocities 
 
fill 111 III ™ ™'iii;i,^L^'i9"'DA — 
 .iiiiff 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 120 MARSTON SCIENCE LIBRARY 
 P.O. BOX 117011 
 GAINESVILLE. FL 32611-7011 USA