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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1333 
 
 ON ROTATIONAL CONICAL FLOW 
 By Carlo Ferrari 
 
 Translation of "Sui Moti Conici Rotazionali" in "Onore di 
 Modesto Panetti" published by L'Aerotecnica, 
 Associazione Tecnica Automobile., and La 
 Termotecnica, Turin, Italy, 
 November 25, 1950 
 
 Washington 
 February 1952 ' ^ITY OF FLORIDA 
 
 LIBRARY] 
 
 
 ' 1 us v 
 
^c/7w'/^~ 
 
 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 semip lanes, 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 l) 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. 
 
curl V x V = ii- grad S 
 7R B 
 
 NACA TM 1333 
 
 div 
 
 -/) 
 
 1 
 
 Mi 
 v 
 
 = 
 
 (1) 
 
 wherein R is the universal gas constant, 7 is the adiabatic exponent, 
 V^ 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,0,cp), as depicted 
 
 -* 
 curl V 
 in figure 1, the components of are taken as w r , w©, and vq; 
 
 where w r is the radial component, Wm is the component lying in the 
 meridional plane and normal to the radius vector, while w@ is the com- 
 ponent that is perpendicular to the meridional plane. Likewise, the 
 corresponding components of v/Vj are denoted by v r , Vm, and Vg. 
 
 Between these components there subsist the following relationships 
 
 sin cp 
 
 cVm b(v e sin cp) 
 
 be 
 
 o? 
 
 = rw r 
 
 Sr 
 
 sin 
 
 5v T 
 
 cp be 
 
 = rw, 
 
 cp 
 
 (2) 
 
 bv-r S(rv ( 
 
 Sep 
 
 cp, 
 
 rw 
 
 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 (l): 
 
NACA TM 1333 
 
 w 9 v cp " w cp v " ° 
 
 c 2 
 
 V m W„ + WmV,, = 
 
 *cp»r "cp v r " 7R r sin ^ bQ 
 
 (3) 
 
 v Q w„ - vv a = ■? 
 
 1 c 2 d 
 
 s 
 
 wherein c 2 = C 2 /Vj 2 . 
 
 The set of equations (3) are not independent of each other as is 
 evident from consideration of equation (l) directly, but they are inter- 
 related through the expression 
 
 V x grad S = (h) 
 
 which, for a conical field, becomes 
 
 SS = _ V0 1 SS t^,\ 
 
 Sep v cp sin cp $e 
 
 From the second of equations ( l) , upon use of the hypothesis that 
 the flow is conical, and by taking into account the relationships expressed 
 by equations (2) and (k 1 ), one then obtains that 
 
 VV V ,^A = ^L_V^_ cotq) v m L 
 
 c*p / " sin cp c 2 S9 * \ c 2 
 
 \ . *£)L + -i- *£\ - 31 » „ ZBBlr ^ (5 ) 
 
 , c 2 / V sin cp d6> / 7R ctp c 2 
 
 This expression differs only by the presence of the rotationality terms 
 from the analogous relationship derived in reference 2 previously mentioned. 
 
k NACA TM 1333 
 
 3. By means of equations (3)> (M, and (*+') 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 conical-shaped obstacle); 
 it will be convenient to designate this surface as Z c . Let the versor 
 
 of any arbitrary general one of the generatrices of the conical sur- 
 face Z c be denoted by r, 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 L c is 1 denoted by t then it is true, in 
 
 addition, that grad S x t = 0. It is evident, therefore, that grad S 
 is perpendicular to the surface Z c ; that is, the above-described 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 9 = +90° (this will be the case already 
 mentioned for the field of flow about a conical-shaped 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 wg = provided 
 that v,p is not zero everywhere. On account of this, and through 
 
 utilizing the third of equations (3) it follows that — = 0. Thus even 
 
 dp 
 
 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 (2J~ I 
 
 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 9 = 90° and 9 - -90° 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 — = Vm =0, and therefore one 
 
 ~b9 
 gets that Wm = . Thus the following relationship results 
 
 1 Sv r ic\ 
 
 v e = - (6) 
 
 sin cp de 
 
NACA TM 1333 ! 
 
 On the cone one can always express the v r values as a periodic 
 function of 6, and thus v r = B + £ B n sin n0. It follows that on 
 the cone's surface the peripherial velocity component is given by 
 
 L B n n cos n© 
 
 sin cp 
 just as in the case of irrotational flow. 
 
 Now, if we let the angle of incidence of the axis of the cone he 
 denoted by B, then the expression for v r becomes simply 
 
 v = B + B-, sin 9 
 
 if only terms of the order of magnitude of 3 are taken into account. 
 
 Since this is true, then because B-^ is proportional to 3> the B n 
 
 o 
 coefficients have to be at least as small as 3 • 
 
 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 a^ and Op (r = const, and cp = 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 dcr-^ can be written as: da^ = rh-^ ( 0") &6 
 
 while the length of the linear element dap along the line <7p is 
 
 given by: dap = dr. 
 
 The component, in the direction of the normal to the cone at the 
 point P, of the curl is 
 
 n rh x c>r V 1 1 J 
 
 dvp 
 
 where v-^ and vp are now the velocity components in the direction 
 of U]_ and Op, respectively. On the other hand, upon referring to 
 the first of equations (l), it is still true that w n = , and on account 
 of this it is evident that: 
 
NACA TM 1333 
 
 1 hi 50 
 
 If the component of velocity in the direction of the radius vector is 
 
 expressed as a periodic function of 0, as is still permissible, then 
 
 equation (6 1 ) immediately furnishes the means of obtaining the corre- 
 sponding expression for v^. 
 
 k. It is now easy to determine how to continue the construction of 
 the flow field downstream of a given conical reference surface, Z, upon 
 which the physical conditions of the flow are assumed known. Let the 
 equation of the conical reference surface, £ , be given by 
 
 From the relationship 
 
 <P = 9(0) (7) 
 
 &e) L be " &p 
 
 wherein 
 
 cp = 
 
 dtp 
 d0 
 
 one obtains 
 
 'dS^ 
 Vd6 
 
 . = v — : 
 
 °P vgfp - Vj, sin cp 
 
 Ss \aej z 
 
 = v fi — : (o) 
 
 v 9 
 
 provided cp is expressed as a function of as in equation (7). The 
 
 above relationship allows one to calculate ^ at the points of the 
 
 cYp 
 
 surface E. 
 
 In like manner, by use of the equation 
 
 /Svcp\ = ^ + ^ H 
 
 \be l z be Sep 
 
NACA TM 1333 
 and by setting 
 
 ^ + v r = % (9) 
 
 it is found that 
 
 — ■ I— 1 " <P R l + * y T " A l " $*1 (1 °) 
 
 Be \dJ0 Jz 
 
 wherein the A-j_ are calculated at the points of the conical surface 
 through means of the relation 
 
 H3* 4 *' (10,) 
 
 Finally, it is found (the intermediate steps are omitted, and just 
 the final result presented) that: 
 
 a- 
 
 v fl CD 
 
 = h. + ~7Z R i 
 
 de sin cp 
 
 and 
 
 rVe -_ E2= ^r. v l£.Zl_^_ (11) 
 
 y ^ Sep * Vq, vq, sin cp J 
 
 where the quantities Ag and Ao are the expressions: 
 
 A P = — £ 1 — _ Vfl cos cp+An - i= ± sincp 
 
 d \&Q J L sin cp ^ e 1 7 R v Sep 
 
 -E v fl sin cp + IE A. (11') 
 
 V «P V <P I 
 
8 
 
 NACA TM 1333 
 
 'dv T 
 
 Ao = 
 
 
 (vcp 2 + v 2 ) 
 
 v^ sin cp - cp v 
 
 sin 9 
 
 (11") 
 
 and their values are known on the reference conical surface, Z, and thus 
 so also is the value of R2- 
 
 By means of equation (5), therefore, one obtains 
 
 Ri = 
 
 sin cp c 
 
 Vcp 
 
 — A l 
 
 v 9 
 
 1 + 
 
 cot cp 
 
 i v r + 
 
 sin cp/ 
 
 ;v<£ SS v cp v r 
 yE ^P c 2 
 
 rwe 
 
 (12) 
 
 Thus it is possible to calculate the values of R]_ 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), (ll), (12), and the additional 
 equation (13), since it is also evident that 
 
 SV£ 
 
 Sep 
 
 sin cp 
 
 A ^,-r, _ c 1 <JS . _ 
 At - cpRi - v fl cos cp — — sin cp 
 
 11 ° 7E v 0cto 
 
 (13) 
 
NACA TM 1333 
 
 5. The graph of the hodograph corresponding to an arbitrary gen- 
 eral semiplane, 6 = const., is represented in figure 2. Let P be 
 
 any point whatsoever on this hodograph, and let R signify the vec- 
 tor R = RjV - P^t. 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 r 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 Rx — = and therefore it becomes clear that the 
 
 dcp 
 tangent to the hodograph at the point P is perpendicular to the above- 
 defined vector R . Thus this tangent is inclined to the direction of 
 r by an angle 
 
 - 1 R 2 / , x 
 
 X = tan =F (lk) 
 
 R l 
 
 If the linear element of length along the hodograph is denoted by ds 
 then the absolute value of ds/dcp is given by 
 
 ds 
 
 dp 
 
 = l/ Rl 2 + Rg 2 
 
 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 = l/R-, 2 + Rg 2 dcp 
 
 then it is easy to find another neighboring point P'. When used repe- 
 titious ly 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 
 
10 NACA TM 1333 
 
 dX\ 
 
 = dtp 1 l • 
 
 dR 2 dR^ 
 R l W ~ % dcp 
 
 R l 2 + ^2 2 
 
 da = dcp 1 
 
 The radius of curvature of the hodograph at the point P just 
 mentioned turns out to be consequently 
 
 ( 2 2\ 3/2 
 
 Re = - (15) 
 
 R i( R i - w) + ** t* 2 + 3 
 
 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 R^ and R2 are also known at the point P', 
 
 d -Rl dR? 
 and thus the values of and — - are calculable. Through means of 
 
 dcp dcp 
 
 equation (15) the radius of curvature R^, is then determined, using 
 
 dRn dRo 
 
 these values of and . The direction of the principal normal 
 
 dcp dcp 
 
 as defined by equation (ik) 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 11 
 
 REFERENCES 
 
 1. Busemann, A. : Driicke auf kegelformige Spitzen bei Bewegung mit 
 
 Uberschallgeschwindigkeit. Z.f.a.M.M. , Bd. 9, Heft 6, Dec. 1929, 
 pp. k96-h9&. 
 
 2. Ferrari, C: Atti R. Accad. Sci. Torino, vol. 72, Nov. -Dec. 1936, 
 
 pp. lUO-163. (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 19^9, pp. Ull-^34. 
 
 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. 
 
12 
 
 NACA TM 1333 
 
 Figure 1.- Coordinate and vectorial orientation in the spherical coordinate 
 
 system. 
 
 Figure 2.- Construction in the hodograph plane for finding velocities at 
 
 P' when they are known at P. 
 
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