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NATIONAL ADVISORY OMIMITTZE FOR AERONAUTICS TECHNICAL M EMORANDTUmI INO. 1100 II;FIMITESIMAL CONICAL SUPERSONIC FLO':" Ey Adolf Buseiaaruni Conical flov' fields. Real flows always occur in threedii;innsionlal space. In calculating a flow, hx.'cver, one will greatly appreciate it if there are onl. t.wo essential coord.inate;e to deal with. Flo.'s of this hind, limited co tvo coordinates, rorn te plane flow and the flow of aial .:,,''i try. The space hiclh is filled o,.t by the stream.'ines is repr,sent e;. in planes parallel to these lines; the.: contain certain s,;r.eo.nines to t!eir. whole extent. In, conical float fields, ho'.rever, the stre a,:lines are cut t:hroughi sl ntin.l y so that e.a;ch streamline is contained in the ,lan, buit appe Sari the as a pcint c.nl. Tihte relati.:; ar. made clea: in fig ure 1. If ths friction is ne l'cted the s.:e [ V body r I 1.;2 one to e:'pect a pa 'tern t!i'it can be i nc r ase .; or decr ased Zoie trically. mTW f ixedi point P and the direction of " the three spa'ial a:.ces, x, Y, and z ren,,ain the sare. All esEaenti charCactr..LS tics of the lo' alnd the share ,:, thi; bod.G ci bLe iLfterr~ed. fr: 'i the opline S = . ...'E. .' ,t b 'c. u :1., if ..z t ' 2 ,cr ,ou.bl. d, show ice.nt'.l va ].i.us fcr gJS' c:ondi ions and velocities. The isotar plans in ;lhe spe s, ,e cf are of conical shape and Ihave tbe cone vr t :: P; therefore th'ee .r'lows shall be called abltrviatedi cojnicl flow fields. ILfx.ni esimal diflfeences il prressure. In 'iu?.re 1 there .shll be one more 1li.iLct:on for the general .onical flow field, nr.!.ean ly th.ai: the body distLUtbs the parall .i flow onl ta a liht decree. 3o the cornical isobar s. reflect ov;r and underpresoures diffCrin ifinitesi mally from the rre ss .re of th par allel flo'... This t3',o fold l.i.itation, to conical a: iLfini si.ial, is not actually. vo.ry s tringeritC insof ri as ia the class :. pote n tial flno.vs ti ere are present cnly the co iical fields of axial sym!.,etry and the infinitesimal conic:.al fiild.. .'ill other conical flowy. are affected by r't nation. TJ infini tesinal supersonic ?lo.,s, however, also excel in an."ther way: the supeerpositio) of fields with d.fferent fi'ed points P is permitte,~ in s pite of the fact that the dif ferential equations ordinarily are not linear" thus the applicability is broadened most gratif'yingly. ::"Infinitesimalo kc'ogli ie be rschal itrorLung." Deutschen Akaclemie dex LuftfauLrtforslhung, 19u2145 p. ).155. 2 NACA TM No. 1100 Differential equation. There are two ways to limit the differential equation for the potential in conical fields to small additional velocities. u, v, w, and to limit the differential equation for nearly parallel spatial flows to conical fields; the first is the his torical one, the second, however, the simpler one. Therefore here the second one is chosen. As is well knovtn, the linearized differential equation in the space x, y, z, for the additional potential over a basic velocity W in the direction of the axis z reads if the gas has the sonic velocity a: SI + e + z z I 0 (1) The coordinates & and n of the conical current cor respond to the spatial coordinates x and y in the plane z = 1: : = and (2) z z The additional potential c increases on each ray through the fixed point P in proportion to the distance. There fore, the potential divided by z is invariant on a single ray, and provides the potential of the conical flow: = p(x, y,z) (5) z The additional velocities u, v, w are the d. rivatives of the former potential. (4) = =. x T The differential equation for the new potential ) is determined from the old differential equation, and one obtains: NACA TM To. 1100 7 ) + 2 = 0 1 1 I with =  a tg a It certainly is gratfying to recognize in the type of this dilffe.rential equation an oic acquaintance fro: the plane gas ilow: for th; steac;.' function of the plane flow transformi.d accodin, to Legendre and supevimose.d over the coi:.ponents of the current dn.:ity proc.duces exactly the s ric d'i'frcntial equation. in rdiinar gaz;., how ever, tt.e d. Crorinatoi L is a Ical f.uncLu .on; buLt. there is a special gas tditi 'rectilin;ear a.dabatics in the pressujleolumc;diagr.i iin whi.lch,, as required, thl denominrt,,r also re.aains cnEst&iu. Thns as is a special favorite w.ere it is a rmere :ues :;on of numn.rical calculations. Regions of influence. The spatial differential equation: of the as flox at supersonic velocity i: of hyperbolic character, as shov.'n in equation (i). That means: eash point of the flocw dominates a conical rane opening dlownsistrea..i; eaach locus, on the other hand, is dominated solely by those points which are situaced in the cone porolonged bacnkwvard and opening ups3trean. Ier?e with tlie relationships are divided definitely among the three possibo'.lities:superior, subordinate, ant. independenI. Machts cones in the supersonic flow consdei.'ed as regiorns of dis Ltrtance of a small trial body nmal:e this fuller coi. prehensible i a phyisal sence. It must seei d od at first th .t th e deoondencies of the general spatial flow' are videlned as soon as one p'oces is to a more limited spatial 'low. But the abcvenentio.ned differential e _ia tion shows thdt inside of the circle witli the rad.is A there rEtv_ ilr the clliptic character. This 1beavior is easily explained by the fact that all points of a. ray starting froi P are comiprehended as a whole. The relation of dependencies of two rays results from the dependencies of the single points. Only the characteristic "indendendent"; appears uniformly in certain cases for all pairs of points (P itself is excluded). The combination superior and independent NACA TM No. 1100 becomes superior; subordinate and independent become subordinate. But if 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 Ilachts cone starting from point P. Characteristics. Miacl's cone starting from P intersects 'the ane z = 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 two 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 sum of all of Mach's ccnes starting from all points of the wire. It is immediately obvious that only the circle with the radius A and its tangents can form the boundaries of the area of disturbance. Outside of :ach'is 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 velocity. The essential and different part of the conical fields, therefore, is concerned with the convex surface of ITaccts cone starting from point P, and with its interior. Tschapligin's illustration. In the plane of inter section = 1 iv. i_ nd .nsi,:r e f the circle with the radius A the elliptic character of the differential equation (5). HTear the center the differential equation 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, however, 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 complex variable L = + iT, through parallel projec tion to a sphere with the radius A, and by then pro jecting it from a pole of the sphere on to a plane in the distance. One will easily recognize that only the interior of the circle with the radius A will be. NACA TLM No. 1100 5 depicted; first it will be delineated from the lower halfsphere on the interior of the unit circle of the new plan i wvith1 the new complex variable c; a second tire it will go from the upper halfsphere on to the outer field of the unit circle. In these coordinates one can use analytic functions for the solutions. SCLUTIOIN :P T~ DIFFEREITTIAL EQUATION For each of the velocity components u, v, and w one can eqcuacc the real part uf an analytic function f(c). It will serve the purpose best to set up the equation for the component w, because tnen the more closely., related ccrmipnents u and v can be calculated jointiya w = A Re (f(c)) or w + iz = A f (E) (6) The completion represented here by s is for the time being completely .ne aningless. According to Tschapligin there then results the complex : velocity: 0 = u + i = I L + edf (7) 2j E The pressure in the current, with the aid of the den sity p, results from the velocity components as follows: S1.12 + v + w P = Pp + 2 w w=p "+ 2 f (E) = p (f +f) + The right function f(e) is to be selected with the aid of the boundary conditions. Doundary conditions. The outside of T.ch's cone is superior to I'[ach's cone itself. Therefore, first, those velocities u, v, and w on the circle of the bplane (and therefore on the uniform circle of the eplane) that result from the outer field must be ascertained. NACA T1M No. 1100 If the body does not protrude anywhere out of Machts cone, the values u = v = w = 0 on the uniform circle are given. If on the contrary no part of the body is inside of Mach's cone, the values of w are to be represented by an analytic function f(C) free of singularities with the given boundary values on the circle. If f(c) does not produce a stationary value df = 0, then u and v according to equation (7) i'll have a logarithmic singularity at zero. The manyleaved function can be selected in a unique way by using radial intersections with the boundary values of u and v on the uniform circle. The radial intersections produce rotational layers, as is physically to be expected from a lifting surface. Impermeable boundaries of the body can be trans ferred into the Eplane at the sane time. They must be streamlines in the field of the relative velocity: rel = + E 0W 2A (9) This condition is not always easy to comply with. How evor, if the body possesses rectilinear surface elements passing near zero, the otherwise meaningless imaginary part s of the function f(C) 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 (8) conditions applicable to cases of given pressures or of given lifts are to be understood. The disappearance of the real or of the imaginary part of f on certain lines because of symmetry can be attained in the well known way by reflexion, as the examples will show. Examples 1. The circular cone in the straight flow For the only axialsymmetrical case, i. e., the circular cone with an infinitesimal apex angle, the right solution is, of course, given by the statement w + is = C Inc NACA Ti': No. 1100 The pressure on the convex surface of the cone results in the k:no'wn way (fig. L.) and conforms with v. Karman's values and mine. 2. The cir1culavi cone in oblique flow One succeeds, with the aid o?: the relative velocity according to equation (9), in solving the circular core in oblique flao. Herein the apex angle and the .:ngle of attack miy;,, though infinitesimal, yet bear a reltiCon to each otler. The solution is bhc.wn in figure 5. It' one makes thr. ancle; of obliquity y scro, one gets again the circular cone in straight flow. If' onea makes the apex angle 2P disappear, one ge'u the pressure distribution of a circ.ulcr cor in an irnoj:':;i.ssible current. The comparison v.iti] Ferrari. is rendered somewhat diffici'.lt by the fac.t that 'Ferr ;Pari m easures Lhe v:locitv fieldI. par pendicular to ti.e .core axi!s wirle it is ,here perpendicular to the wind direction. If the system of coor.dirabes is rotated adequately, the confom.idty is cor. plete. 5. Tip of n rectangular plate If a plane ractaniguler plate of ii'finitesimnal thick: ness is placed in a flo' perpendicular to the front eo.:e with an ird;finites i1nt1 antle of attack, and if the velocity field is needed only up to the rear edge of the plat, one can pl.ce the fi'ed point P at the right corner point of the front edge. On the supposition of an infinitesimal anls of attack y ('withi the xais forming the axis of rotation) the pressure disZLribution will be represented on the quarter plane between tne positive axis:3 (z) and the negative ax:is (v). For the plane z = 1 tih section of the body, except for infini tesimal distances, is then rendered by the negative real axis. Let the reduced pressure above the plate ani 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 Eplane corresponding values for w arc then to be assigned. On the right semicircle the outer field is undisturbed; here w = 0. For reasons o.C s.yr ietry the value w = 0 must also result on the positive real axis. Along 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 NACA TM No. 1100 attained by reflexion if one undertakes a preliminary conformal mapping on the plane V = Ve. The solution is represented in figure 7. Figure 8 shows the pressure distribution on both edges of a rec tangular plate. 4. Supporting triangle Every two radii starting from P form a triangular plane as far as the plano z = 1, when all points are connected. Because of the required infinitestual dis turbance of the parallel current, however, the angle of attack must be infinitesimal, so that the plane of the two rays will nearly pass through the zaxis. Such triangles are possible completely inside of ,.!a.i's cone, completely outside, and uni and bilaterally protr"ding. Here we shall only consider the simplest case of the supporting; triangle outside of i.ach1s cone, although all other cases can be easily integrated. Figure 9 shows this supporting triangle. The velocity component w which predominantly influences the pressure is different from zero only on the short arcs between t a:LId t3 as also 4 and t, The value zero results from the undisturbed state on the right, and also on the left because of the pressure adjustment behind the triangle, when consideration is given to the syrmetry with a positive and with a ne.ga tive angle of attack. Figure 10 shows the relations in the tplane. If one intends to let the rear edge *f' the triangle travel while the front edge lies fixed, one will at first transfer only the points and ts into the eplane. With suitable regulation there must rcul.t an increase of w from 0 to +TI at t, and from r to 0 at t2 (if one moves 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 speaking, one then has a uniformly loaded triangle between the front edge at am a.i the zaxis. Inside of lacts conr, however, t is triangle is not flat, but is twisted to uniform or loT..d. As soon as onr supor"r.;oaoscs at the rear edge a negatively loaded triangle and its influ ence between ,3 and the zaxis, the part behind the rear edge will no longer be supporting, and the singu larity in w at the point zero of the Cplane disangears. NACA TiM No. 1100 However, a vortex layer in the field u, v is left. The partial solution in the eplane is represented in figur 11. 5. Superposition of two conical flows The infinitesimal conical flows can be superposed without having the fixed point in coiLrmCon as in figure 4. Therefore the relations in the plane late can also be represented whv.en the pl.tte ha. mr)ri depth. Figure 12 shov:s the isobars of the edge cC the plate and also their superposition ~i'ter M.ach's coins 'havi overlapped. Dif ferent boundary .c j.itions for the partial solutions need be cosiered osinly vwhen thL c'oes reach the other edge of the plate. The ,.iaa ,c, irance of ipreuro alon. a straight line in just the ist tnce ai which the cones arrive at th,; ct ..r edge of tle plate :,s zerm!:ablc. The posjtivv..ly loaded prt of the p ate nids here. Fij ure 1i shc.o s' the 1ft distribution of the positively loaded part in perspective repr aesentation. To find the velocity field behind a. rectcarulaL plate of finite dcpth one can E:en.pul the supporting; : pres sure differiEinces of a pletas of infinite' 'crith by con,ical fields having t the apices on the r:ar cdgc of the plate. If one superposes a negatively. loaded ol.ate shifted infinritosimall in the di'rcotion of the za supporting sirip. The cases calculated by Silichting according to Prandtl's :,imetliod a e obtained in this way. Here, too, the conformitv is perfict, except for an error of in in thealculatn these Integral equation. SUMNl WRY The calculatlon of irfinitesimal conical supuersc.nic flows has been applied firvt to the simplest xmir.,lcs that have iso been calculated in anotlt.her way. Except for the discovery of a miscalculation in an older rencrt there was found the expected conforriity. The taew Tmithod of calcullation is limited more definitsi.y to the conical case." tub, as a compensation, it is much more convenient because tlh solution is obtained by analytic functions. The fundamental recognition that; there the hyperbolic character is replaced by the elliptic one will lead to NACA Ti No. 1100 more thorough investigation of conical fields as special cases in supersonic flows. Of course, one will be tempted to call the elliptic character seemingly elliptic only; for if one notches an indentation into a cone, the real hi. rbdlic character of the field of the flow down stream from this indentation immediately becomes obvious again. However, if one traces the flowr to a point very far behind the indentation there will appear inside of liacl ts cone a reciprocal relation between every two rays which will gradually restore the donical course of the flow. Instead of trying to produce the final flovw by an infinite succession of hyperbolic dependencies it will be more expedient to consider special elliptic singu larities at the points of disturbance. In those rela tions I see the significance of the conical field; the infinitesimal case represents only a first approximation to it. Translation by :,.y L. :ibler and Robert T. Jones, National Advissory Committoee for Aeronautics. NACA TM No. 1100 P Figure 1. Coordinates in conical field Figure 2. Disturbance field of elements a and b Figs. 1,2 NACA TM No. 1100 Figure 3. Chaplygin's transformation W 'gg L? f d f = Clne A = tg A = tgQ: Figure 4. Circular cone in axial flow. Figs. 3,4 A 2 (EI + T= z~e NACA TMNo. 1100 Figs. 5,6 y I  p o f = 4WAR2 k n p =Qe ,2 (8 I S= ita 1 1 CE 0 E 2A _ S  r, /3 2 .0 oE 1EE a 2 ro y + 2 ycos i+Y r0c 2) Figure 5. Circular cone in yawed flow W 1 apr. tga.y tgaeatt  ( d#s) = arc cos 2 77 tga A.tfi Figure 6. Edge of a rectangular Tlate 2 : + '0 lE60)21 Figs. 7,8,9 Figure 7. NACA TM No. 1100 v = r, f(v) =i { In (v ) In(vv2) In(vv3)+ In(vv4) Conformal representation at edge of a rectangular plate Direction of motion of plate Figure 8. Pressure distribution on a flat plate Figure 9. The lifting triangle. NACA TM No. 1100 2Ae 1+e" Figure 10. Cross section of the lifting triangle ffe) = i {n(e2)+tin(e)ine pdps = const. Figure 11. Representation of the lifting triangle in the e plane Figs. 10,11 NACA TM No. 1100 Figure 12. Superposition of edge influences for the rectangular plate at supersonic velocities LIFT Figure 13. Pressure distribution on the rectangular plate at supersonic velocities Figs. 12,13 3 1262 08106 273 8 
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