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c, / 7 r/ L 2. 2.. ;. NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1333 ON ROTATIONAL CONICAL FLOW By Carlo Ferrari SUMMARY The author determines some general properties of isoenergetic rota tional conical fields. For such fields, provided the physical parameters of the fluid flow are known on a conical reference surface E, it being understood that they satisfy certain imposed conditions, it is shown how to construct the hodographs in the various meridional semiplanes, as the envelope of either the tangents to the hodographs or of the osculatory circles. ANALYSIS 1. A method for determining the field of flow about a cone of revo lution, the axis of which is aligned with the direction of the impinging supersonic stream, which is taken to have a uniform velocity distribution sufficiently far ahead of the body before the conical field is created, was developed by Busemann (reference 1) at an early date. Several years later the present author (reference 2) extended Busemann's procedure to cover the case of a cone of any shape whatsoever situated in the flow, so that its axis was at any arbitrary finite angle of attack; that inves tigation was confined, however, solely to irrotational conditions. With proper alterations, nevertheless, this treatment of the yawed arbitrary shaped conical surface can be applied to the case of rotational flows. The purpose of the present investigation is just to give the relationships which permit one to draw the hodographs for the flow, in reference to the various meridional planes through the body axis, in the case where the motion is defined by a rotational field of conical flow with arbitrary specification of the cone shape. 2. Upon the mere assumption of isoenergetic flow in a perfect fluid, one may write the equations of motion in the form (reference 3): *Original Italian Report appeared as Sui Moti Conici Rotazionali in Onore di Modesto Panetti, published by L'Aerotecnica, Associazione Tecnica Automobile, and La Termotecnica, Turin, Italy, November 25, 1950. NACA TM 1333 4 > C2 curl V x V = grad S 7R 1 div V2 v2 = 0 (1) wherein R is the universal gas constant, 7 is the adiabatic exponent, V2 is the limiting velocity obtained when the flow expands to a vacuum, C is the velocity of sound, S is the entropy, and V is the fluid velocity. Upon employment of a spherical coordinate system (r,9,(), as depicted curl V in figure 1, the components of l V are taken as wr, Wp, and we; V1 where wr is the radial component, w(p is the component lying in the meridional plane and normal to the radius vector, while we is the com ponent that is perpendicular to the meridional plane. Likewise, the corresponding components of V/V1 are denoted by vr, vm, and ve. Between these components there subsist the following relationships 1 'v O(vo sin cp) = rwr sin p 0 =T 6(rvo) 1 vr (2) 6r sin (p 0 rw &vr ( rvqp) r = rve Based on the assumption that the flow is conical, the following scalar equations are derived in a straightforward way from the first of the equations (1): NACA TM 1333 wevp Wqv = 0 c2 1 as  Vr + W(Vr = 7R r sinq) e 1 c2 s VoWr VrW r yR a wherein c2 = C2/V2. The set of equations (3) are not independent of each other as is evident from consideration of equation (1) directly, but they are inter related through the expression V grad S = 0 () V x grad S = 0 (4) which, for a conical field, becomes s vye 1 as vcp ) V sin cp e (4') From the second of equations (1), upon use of the hypothesis the flow is conical, and by taking into account the relationships by equations (2) and (4'), one then obtains that c2 ( c'/ v2 ) c2 2 V(PV qdV _V sin cot 9 v sin c c2 0 v + 1 sin (p V~P ds YR d'V ao / that expressed c2 ^> vcpVr crw c2 This expression differs only by the presence of the rotationality terms from the analogous relationship derived in reference 2 previously mentioned. NACA TM 1333 3. By means of equations (3), (4), and (4') one may deduce some interesting properties of conical fields. Let it be assumed that. one of the stream surfaces of the flow is conical (this will be the case for a field of flow arising by the action of a uniform supersonic stream impinging from any direction whatsoever upon a conicalshaped obstacle); it will be convenient to designate this surface as Ec. Let the versor of any arbitrary general on eof the generatrices of the conical sur face Zc be denoted by then grad S x r = 0. On the other hand, if the versor of the tangent to any streamline whatsoever that is traced upon the surface of the cone Ec is denoted by r then it is true, in addition, that grad S x T = 0. It is evident, therefore, that grad S is perpendicular to the surface Lc; that is, the abovedescribed conical surface is a surface of constant entropy. Besides, let it be assumed that the conical flow is symmetric with respect to the meridional plane e = +900 (this will be the case already mentioned for the field of flow about a conicalshaped obstacle). At all the points of this plane it is true that vg = 0. One then deduces, upon the basis of the first of equations (3), that we = 0 provided that v~, is not zero everywhere. On account of this, and through utilizing the third of equations (3) it follows that _S = 0. Thus even the meridional plane of symmetry for the conical field is itself a con stant entropy surface, and at this plane the flow is irrotational as is easily deduced upon taking cognizance of equations (2). In conjunction with the result obtained above one can derive from this latter fact that, for the case of flow about a conical obstacle, the shock wave in the two semiplanes e = 900 and 0 = 900 must produce the same change in direction of the stream velocities; and so the tangents to the trace of the shock wave in these semiplanes are symmetrically inclined with respect to the undisturbed stream velocity vector. Now let us consider an obstacle in the form of a right circular cone. Upon the surface of this cone it is true that __S = vqp = 0, and therefore one Ce gets that vw = 0. Thus the following relationship results ve =  r (6) sin p 6@ NACA TM 1333 5 On the cone one can always express the vr values as a periodic function of 0, and thus vr = Bo + L Bn sin nO. It follows that on the cone's surface the peripherial velocity component is given by vQ = 1 n Bnn cos ne sin cp just as in the case of irrotational flow. Now, if we let the angle of incidence of the axis of the cone be denoted by P, then the expression for vr becomes simply vr = Bo + B1 sin 8 if only terms of the order of magnitude of 0 are taken into account. Since this is true, then because B1 is proportional to 0, the Bn coefficients have to be at least as small as 32. The relationship given by equation (6) may be generalized for the case of a cone of any shape whatsoever. It is assumed for this purpose that the cone's surface is divided up by a network of orthogonal coordi nate lines 0a and a2 (r = const. and c = const., respectively). The former of which are the intersections of the spheres with radius r upon the cone under consideration, while the latter are the generatrices of the cone itself. At an arbitrary general point P on the cone the length of the linear element dol can be written as: dol = rh1 (E) dO while the length of the linear element do2 along the line a2 is given by: do2 = dr. The component, in the direction of the normal to the cone at the point P, of the curl is 1 v \' 1 3V2 wn = (vihr)  rhl 6r Y lh1 where v1 and v2 are now the velocity components in the direction of al and 02, respectively. On the other hand, upon referring to the first of equations (1), it is still true that wn = 0, and on account of this it is evident that: NACA TM 1333 (6') If the component of velocity in the expressed as a periodic function of equation (6') immediately furnishes spending expression for vl. direction 0, as is the means of the radius vector is still permissible, then of obtaining the corre 4. It is now easy to determine how to continue the construction of the flow field downstream of a given conical reference surface, E, upon which the physical conditions of the flow are assumed known. Let the equation of the conical reference surface, Z, be given by (P = (p(e) From the relationship /dS\ = 6S ~S \d0^ 60 ac wherein q:p one obtains ( dS)  = ve s CCP 4vJ vp sin q provided p is expressed as a function of 0 above relationship allows one to calculate LS surface E. In like manner, by use of the equation as in equation (7). The at the points of the 36( vp\ l60 , 1 v2 vl = h1i e hl 60 + CP C N NACA TM 1333 and by setting S+Vr = RI C)CP it is found that 3vqP e \de /z  pR1 + vr = A1 RR1 (10) wherein the Al are calculated at the points of the conical surface through means of the relation A = ( v (10') Finally, it is found (the intermediate steps are omitted, and just the final result presented) that: cv 9 ? i2 in Ri sin (p OVr V82 vs 1 re = R2 Vp =) s An byc V9 p sin 'P where the quantities A2 and A3 are the expressions: \de v\ A2 = WE V( E9G7 P sn ve cos \ sin \ c2 1 as + Al sin cp 7R ve p ve sin c + (11) Vr V(P (11') NACA TM 1333 T (2 + vy e) I sin CP Vq( sin q( ( ve and their values are known on the reference conical surface, so also is the value of R2. E, and thus By means of equation (5), therefore, one obtains c2 cL 2 sin rp s2 V c2V sin p c2 c2 Av c2 A1  )2 c 1 sin2 cp v 2 / 1 + 2) cot (P  c2 ] R1 = v 2) C2 c A2 sin v{ 3S VcVr  c2 rwe RR5 c2 2 (12) Thus it is possible to calculate the values of R1 at the points of the reference conical surface, Z. This formula is the natural extension of the analogous relationship already derived in reference 2 in the case of an irrotational flow. The complete solution of the problem as to how to continue constructing the flow field downstream of the reference surface Z is thus presented by the formulations given as equations (8), (11), (12), and the additional equation (13), since it is also evident that 1 i A c2 1 v A Rt I vO cos (  6p sin Cp 7R v sin (] fdv\ vv wk Avr A3 = o9 (ill") (13) NACA TM 1333 5. The graph of the hodograph corresponding to an arbitrary gen eral semiplane, 0 = const., is represented in figure 2. Let P be any point whatsoever on this hodograph, and let R signify the vec @ I tor R= Rlr R2T. In addition let r again denote the versor that has the sense and direction of the radius vector in the semiplane in question, and let T now be the versor normal to r in the semiplane and oriented in the sense of an increasing cp. With these conventions dP it follows that R x = 0 and therefore it becomes clear that the dp tangent to the hodograph at the point P is perpendicular to the above 9 defined vector R. Thus this tangent is inclined to the direction of r by an angle 1 R2 X = tan (14) If the linear element of length along the hodograph is denoted by ds then the absolute value of ds/dcp is given by H s= R12 + R22 Thanks to the formulae developed in the preceding sections the drawing in of the tangent to the hodograph at the point P, situated on this hodograph, is therefore made possible, since everything is known about the point P. If the element of length along the hodograph is calculated as ds = /R2 + R22 dp then it is easy to find another neighboring point P'. When used repe titiously in the various semiplanes, this procedure allows one to determine the respective hodographs as the envelope of their tangent lines. The angle of intersection da with respect to the direction of the linear element of the hodograph ds is thus determined by NACA TM 1333 ( dR2 dR / Rd W1 ~  da = d (1 = dp 1 2 S12 + R2 / The radius of curvature of the hodograph at the point P just mentioned turns out to be consequently ( 12 + 3/2 R12 + R22 Rc = (15) / dR2\ / dRi1 R (3 + R2 R2 +  dp dp When the point P' is determined in the manner that was described above, in consequence of having available all information about the known point P, it follows that R1 and R2 are also known at the point P', dR1 dR2 and thus the values of  and are calculable. Through means of dip dcp equation (15) the radius of curvature Rc is then determined, using dR1 dR2 these values of  and . The direction of the principal normal d4p dCp as defined by equation (14) is also known, and thus the hodograph can be obtained in every meridional semiplane as the envelope of the respec tive osculatory circles. 6. The results obtained here for the determination of the conical field of flow in the case of rotational motion are employed in an exactly analogous way as described in reference 2 when making a numerical appli cation. In general it will be convenient to assume as the reference surface, Z, upon which the initial values are taken to be known, the conical surface of the shock wave. The required information about the shock wave may be obtained in first approximation by utilization of the hypothesis that the flow is irrotational. Translated by R. H. Cramer Cornell Aeronautical Laboratory, Inc. Buffalo, New York NACA TM 1333 REFERENCES 1. Busemann, A.: Drucke auf kegelf6rmige Spitzen bei Bewegung mit Uberschallgeschwindigkeit. Z.f.a.M.M., Bd. 9, Heft 6, Dec. 1929, pp. 496498. 2. Ferrari, C.: Atti R. Accad. Sci. Torino, vol. 72, Nov.Dec. 1936, pp. 140163. (Available as R.T.P. Translation No. 1105, British Ministry of Aircraft Production) and Ferrari, Carlo: Interference between Wing and Body at Supersonic Speeds Analysis by the Method of Characteristics. Jour. Aero. Sci., vol. 16, no. 7, July 1949, pp. 411434. 3. Crocco, L.: Una nuova funzione di corrente per lo studio del moto rotazionale dei gas. Rend. della R. Accad. dei Lincei. Vol. XXIII, ser. 6, fasc. 2, Feb. 1936. NACA TM 1333 x Figure 1. Coordinate and vectorial orientation in Lhe spherical coordinate system. Figure 2. Construction in the hodograph plane for finding P' when they are known at P. velocities at NACALangl.r 3111 1000 a:. A I e u ,im a Zd 0 14 4 f4 ; 0 d mm i. . 0 LL. .., S0 s '5 g C 1. 0 U U 0 zt rI z. 7 M 2 o 0 * co I mii % o~ o  b aa v 6 . * i a . 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