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NATIONAL ADjVISORY COM~I~TTEE FOR AERONAUTICS TECHNICAL MEMORANSDUM 1415 LAMINAR FLOW ABOUT A ROTATINJG BODY OF REVOLUTION IN AN AX.IAL AIR.STREAM*C By H. Schlichting 1. INPR~ODUCTIONI The flow about a body of revolution rotating about its axis and simultaneously subjected to an airstream in the direction of the axis of rotation is of importance for the ballistics of projectiles with spin. In jet engines of all kinds, too, an important role is played by the flow phenomena on a body which is situated in a flow and which at the same time performs a rotary motion. Investigations of C. Wieselsbergerl regarding the air drag of slender bodies of revolution which rotate about their axis and are at the same time subjected to a flow in the direction of the axis of rotation showed a considerable increase of the drag with the ratio of the circumferential velocity to the freestream velocity increasing more and more, the slenderer the body. Similar results were obtained by S. Luthander and A. Rydberg2 in tests on rotating spheres which are subjected to a flow in the direc tion of the axis of rotation. These authors observed, in particular, a considerable shifting of the critical Reynolds number of the sphere dependent on the ratio of the circurmferential velocity to the freestream velocity. The physical reason for these phenomena may be found in the processes in the friction layer where, due to the rotary motion, the fluid corotates in the neighborhood of the wall and, consequently, is subjected to the influence of a strong centrifugal force. It is clear that the process of separation and also the transition from lamilnar to turbulent conditions are strongly affected thereby, and that, therefore, the rotary motion must exert a strong influence on the dlrag of the body. *"DiMe laminare Stro~mung um einen axial angestrojmten rotierenden Drehkijrper." IngenieurArchiv, vol. XXI, no. 4, 1955, PP. 227i244. An abstract from ths report was read on the VIII International Mechanics Congress in Istanbul on August 27, 1952. IC. Wieselsberger, Phys. Z. 28, 1927, p. 84. 2S. Luthander, A. Rydberg, Phys. Z. 36, 1935, p. 552. NACA TM 14c15 In the flow processes in the corotating layer of the fluid, one deals with complicated threedimensional boundarylayer flows which so far have been 1 ttle investigated, experimentally as well as theoretically. Th. v. Karmin~ treated at an early date the special case of a disk rotating in a stationary liquid, for laminar and turbulent flow, as a boundarylayer problem, according to an approximation method. Later, W. G. Cochran4 also solved this problem for the laminar case as an exact solution of the NavierStokes equations. A generalization of this case, namely the flow about a rotating disk in a flow approaching in the direction of the axis of rotation, for laminar flow, has been treated recently by H. Schlichting and E. Truckenbrodt The result most important for practical purposes are the formulas for the torque of the rotating disk; it is highly depend ent on the ratio of the circumferential velocity to the freestream veloc ity of the disk. For the general case of a rotating body simultaneously subjected to a flow, J. M. Burgers6 gave a few general formulations. We have set our selves the problem of calculating the laminar flow on a body of revolution in an axial flow which simultaneously rotates about its axis7. The prob lem mentioned above, the flow about a rotating disk in a flow, which we solved some time ago, represents the first step in the calculation of the flow on the rotating body of revolution in a flow insofar as, in the case of a round nose, a small region about the front stagnation point of the body of revolution may be replaced by its tangential plane. In our problem regarding the rotating: body of revolution in a flow, for luminar flow, one of the limiting cases is known: that of the body which is in an axial approach flow but does not rotate. The solution of XTh. v. Karmin, Z. angew. Math. Mech. 1, 1921, p. 255. W. G. Cochran, Proc. Cambridge Philos. Soc. 3O, 1934, p. 365. H. Schlichting, E. Truckenbrodt, Z. angew. Math. Mech. 32, 1952, p. 97; abstract in Journal Aeron. Sciences 18, 1951, p. 658. J5. M. Burgers, Kon. Akad. van Wetenschappen, Amsterdaml 45, 1941, p. 13. I1t is pointed out that the turbulent case, for the rotating disk in a flow as well as for the rotating body of revolution in a flow, mean while has been solved, in continuation of the present investigations, by E. Truckeabrodt. Publication will take place later. E. Truckenbrodt, "Die Strojmung an einer angeblasenen rotierenden Scheibe bel turbulenter Strajmung," will be published in Z. angew. Math. Mech. E. Truckenbrodt "Ein Quadraturverfahren zur Berechnung der Reibungsschicht an axial angestrtjmten rotierenden Drkehkijrpern." Report 52/20 of the Institut fur Strojmungsmechanik der T. H. Bra~unschweig, 1952. NACA TIM 1415 this case was given by S. Tomotikai, by means of transfer of the well known approximation method of K. Pohlasen9g to the rotationally symmt rical. case. The other limiting case, namely the flow in the neighborhood of a body which rotates but is not subjected to a flow is known only for the rotating circular cylinderlO, aside~ from t~he rotating disk. In, the case of the cylinder one deals with, a distribution of the circuimferential velocity according to the law v = ER2/r where R signi fies th ecyli nder radius, r th~e distance from. the center, and a the angular velocity of the rotation. The velocity distribution, as it is produced here by the friction effect is therefore the same as in the neighborhood of a poten tial vortex. In contrast to the first limiting case (nonrotating body subjected to a flow), the flow in, the case of slender bodies which rotate about their longitudinal axis in a stationary fluid does not hav "boundarylayer character," that is, the friction effect is not limited to a thin layer in the proximnityr of the wall but takes effect in the entire environment of the rotating body. Very recently, L. Howarthll also made an attempt at solution for a sphere rotating in a stationaary fluid. This flow is of such a type that in the friction layer the fluid is transported by the centrifugal forces from the poles to the equator, and in the equator plane flows off toward the outside. When we treat, in what follows, the general case of the rotating: body of revolution in a flow according to the calculation methods of Prandtl's boundarylayer theory, we must keep in mind that this solution cannot con tain the limiting case of the body of revolution which only rotates but is not subjected to a flow. However, this is no essential limitation since this case is not of particular importance for practical purposes. The dominant dimensionless quantity for our problem is the ratio Circumferential velocity _V, Rmu Freestream velocity U, U, where R, is to denote the radius of the maximum cross section of the body of revolution. The calculations must aim at determining for a prescribed body of revolution the torque, the drag, and beyond that, the entire boundarylayer variation as a function of V,/Um. The 8S. Tomotika, "Laminar Boundary Layer on the Surface of a Sphere in a Uniform Stream." ARC Rep. 1678, 135). 4K. Pohlhausen, Z. angew. Math. Mech. 1, 1921, p. 253. 10H. Schlichting, "GrenzschichtTheorie," p. 63. Karlsruhe 1951. lL. Howarth, Philos. Mag. VII Ser. 42, 1951, p. 1,j08. NACA TM 14c15 particular case V,/Um = O is already known from the boundarylayer theory established so far. Considering what has been said above, we must not expect our solution to be valid for arbitrarily large Vjm*um The upper limit of the value of V,/Um for which our calculations holds true, still remains to be determined. Presumably, it will lie considerably above Vz U, = . 2. THE FUNDjAEDWEAL EQUATIIONS We take the coordinate system indicated in figure I as a basis for the calculation of the flow. Let (x,y,z) be a rectangular curvilinear fixed coordinate system. Let the x.axis be measured along a meridional section, and the yaxis along a circular cross section so that the xyplane is the tangential plane. The zaxis is at right angles to the tangential plane. Let ut, v, w be the velocity components in the direc tion of these three coordinate axes. Furthermore, let R(x) be the radius of the circular cross section, a, the angular velocity of rota tion, U(x) the potentialtheoretical velocity distribution, and v =ppthe kinematic viscosity. The equations of motion simplified according to the calculation methods of the boundarylayer theory are for this coordinate system +u u dR aw (continuity) (1) ox R dx ae u ,  w U+ v (momentum, meridional) (2) Ax R dx ae dx iz2 u21 uv dR avy & 1 (momentum, azimuthal) (5) ax R dx az az2 The cioundary conditions are z = 0: u = 0, v = v0 = Rmu, w = 0; z = m: u = U(x), v = 0 (4) NACA TIM 1415 A solution of this system of differential equations for an arbitrarily prescribed body shape R(x) with the pertaining potential. theoretical velocity distribution U(x) leads to insurmountable mathematical. diffi culties. We use therefore the more convenient approximation method which makes use of the momentum theorem. We obtain the two momentum equations for the meridional and the azimuthal direction by Lntegration of the corresponding equations of motion over z from the wall a = O to a. dis tance z = h > 6 which lies outside of the friction layer. For the meridional direction there results by integration of (2) over z, with consideration of the continuity equation (1) and after introduction of the wallshear stress for the xdirection (bul~l the momentum equation for the meridional direction 7O U2, ,, x + 2,+ + ~U 4x+ v '24 ) Therein, as is well known, Bx 1 i  uds is the displacement thickness whereas 9, 8 x Oul u dz may be denoted as momentumloss thickcnesses for the x or ydirection. 9 1O6~62 NACA TM 1415 In an analogous manner there results for the animuthal direction by integration of (3) over z with consideration of the continuity equa tion (1) and introduction of the wallshear stress for the ydirection ato = 1 (10) as the momentum theorem for the circumferential direction m=d _{URd 39Y= R2 70 (ll) dx\ xyP Therein 6,y =hU1 u ZOd (12) has been introduced as the "momentum loss thickness due to spin." 3. APPROXDIATION6 METHOjDS (a) The Velocity Distributions According to the approximation method of the boundarylayer theory as given first by Th. v. Karmain and K,. Pohlhausen, the momentum equa tions (6) and (ll) are satisfied by setting up suitable formulations for the velocity distributions u and v which satisfy the most important boundary conditions. For the present case, two parameters may still be left undetermined in these equations for the determination of which the two momentum equations are then available. As expressions for the veloc ity distribution, polynomials in the distance from the wall have proved to be suitable, with the property that the boundary layer joins at a finite wall distance z = 8 the frictionless outer flow. The boundary layer thickness may be different for the meridional and the azimuthal velocity component. Let these boundarylayer thicknesses be by and 6 respectively; we introduce the dimensionless wall distances formed with them z t and z t' (15) NACA TM 1415 For the velocity distributions u and v, we select polynomials of the fourth degree in t and t', respectively. These contain five coeffi clents each so that, for determination of these coefficients, we can satisfy five boundary conditions each for u and v. We choose the following 10 boundary conditions: VO2 dR R ax z22 2u z2 dz dU t = 0: u = 0 t= 1: u =U (lka,b,c,d,e) 2 az2 '  t' = 0: v =v0 Ru t' = 1: v = 0 (15a,b,c,d,e) The boundary conditions (14a, b, c) and (10a, b, c) result immediately from the fundamental equations (2) and (3) with (4) for z = O and z 8 or S The remaining boundary conditions provide a gentle transition of the boundary layer into the outer flow. Taking these boundary conditions into consideration, one obtains the following poly nomials as expressions for the velocity distributions u= 2t 2ti + t + K1 t t2 + 3tS t U6 1 2t' + 2t'3 t' 8x2 dU'01U dRI Sdx \U R dxJ (16) (17) (18) Therein NACA TIM 1415 is a form parameter of the uvelocity profile, which is analogous to the form paramter h of the Poh~hausen meth~odl2. The velocity distributions ulU and .vlvO are represented in figure 2. Let t;he point of separation be given by the beginning of the return flow of the meridional velocity component u(z) in the proximity of the \a z=0 This yields K = 12 (separation) (19) The expression for the ucomponent is the same as in the Pohlhausen method for the plane and rotationally symm~etrical. case. This guarantees that our solution in the case without rotation, w = O, will be trans formd inrto the solution of S. Tomotika and F. W. Scholkemeyerli for the nourotating body of revolution. Introduction of the expressions (16) and (1'1) into the momentum equations (6;) and (ll) yields two differential equations for the still unknown boundarylayer thicknesses by(x) and 6 (x) or the quantities derived from them. (b) The Momentum Equation for the Circumferential Direction We present first the further calculation for the momentum equation of the3 circumferential direction. With TY0 lv = 2'Rm.v (20) p dqJ s there results from (ll), after division by w, 1 ~R5Us = 2'R3 (21) 120f. H. Schlichting, "GrenzschichtTheorie ," p. 195 13F. W. Scholkemeyer, "Die laminare Reibungsschicht an rotations syrmmetrischen Kojrpern." Dissertation Braunschweig 1945, Of. H. Schlichting, GrenzschichtTheorie, p. 204. NACA TlM 1415 With introduction of the further parameters gO = and n = (2 as well as 8_xyr (23) and vx R dxl one obtains from (21_) the following differential equation for O(x) =~E G(K,AL) (5 ax U Therein is G(K,a) = gO 2a (26) a universal function of the two parameters K and C. This function has been determined already by W. Dienemanal4 in the calculation of the temperature boundary layer on a cylinder (two dimensional problem).1) For the temperature distribution in the boundary layer there we chose the same polynomial of the fourth degree as we did for the azimuthal velocity distribution according to (17). According to (12) we have 9 1l x~Y u v~t a, J U YO Because of x, y x, 1.Dienemann, "Berechnung des Warme~berganges nliarutote Korpern mit konstanter und ortsveranderlicher Wandtemperatur." Disserta tion Braunschweig, 1951, Z. angew. Math. Mech. 55, 1955, p. 89. 15with the symbols according to W. Dienemann there apply the identities Ht a gO and A K. 10 MACA TM 1415 one obtains after calculation of the integral with the velocity distri butions (16) and (17) the quantity gO as a function of K and n. According to W. Dienemann, there results g0(K,6) = gl~a + 2(LA) 15 140 180 90 84 5b0 1,080 (27) (28) A 2 : gl( )= + 3 10 10 G 15 02 ) 140 1 1 1 S+   4 180 5 g2(n)120 a 180 a2 840 ,4 3,024 p g0(a) as a function of a for various values of The function K is represented in figure 3. Table I gives a few numerical values of the functions gl(A) and g2 A)* AND g2 () TABLE I. THIE UN~IVERSAL, FUNCTIONS gl(n) ACCORDING TO EQUATION (28) 16Cf. H. Schiliehting, GrentschichtTheorie, p. 195 NIACA TIM 141j 11 (c) The Momentum Equation for the Meridional Direction FrL~ther transformation of the mometum equaion for the meridjional direction yields, if one introduces, according to IiolsteinBohlenl6 and analogous to (25) v dx (29) the following differential equation for Z(x) dZ= F(K n dx U F(K~,.d = 2 f (2Xi + Y:) g gRI h R dx Uf0~ ( 50) Therein (3l) G(K,C) in (26j) a universal. function of the two Individually, the following relationships apply: exactly as K and 12. parameters f0(K) 9x K_ K2 8x 315 945 9,072 fl(K) x_ K 6K 10 120 8x* fl(K) 2(K) 4, f0(K) ($2) 6x s x s x 6x f0(K) (35) f3(K) =_0 9Ux~~ a~l~ O )f(K) (36) 6, 126 The above functions of K are already known from the calculation of the boundary layer of the twodimensional case.17 The connection between Z and. K results from (18), with considera tion of (29) and (32), and is Kf02(K) =~ Z 6 + '0 dR g Taking X = ZU', from (29), into consideration, one may write this because of (32) also in the form xE +  P' =X* =K7 K_~ ~\g0 (39) U R U' Sl 945 9,072~ In figure 4 the universal functions f0> f2> f, and K are represented as functions of X+. At the point of separation, for arbi trary rotational velocity, one will have, because of K = LB, the parameter X* = 0.1567. At the stagnation point, without rotation, K = 4.716 and :Ir* = 0.05708, whereas with rotation the values at the stagnation point are dependent on the spin parameter v0 U (cf. the following section). From (58) the form parameter K can be determined when Z is given. Furthermore, for the later calculation a connection between the parameters 6,gO, 13 Z, and K is needed. There results according to (22), (25), (29), and (32) as follows OgO(K,n) = f0g(K) (40) The two differential equations (25) for 93(x) and (30) for Z(x) are two simultaneous differential equations coupled by the universal functions G(k,a) and F(K,n). In the case of the nonrotating body, VO = O, the coupling is eliminated since then, according to (31), the function F becomes independent of C and remains dependent only on K. 17Cf. Hi. Schlichting, GrenzschichtThe~orie, Chapter XII. NACA TJM 1415  O20" L 0 2 = __ NACA TIM 141'j In this case, one can first determine Z(x) from (50), and sulbsequently S~x) from (25). This solution for 8 has it is true no physical significance. It serves merely for giving the: limiting value for vanishing speed of rotation., (d) The Initial Values at thez Stagnaion Point At the stagnation point where~ U = 0, the two differential equa tions (25) and (50) have a singularr vaue since in. both equations on the right side thei denominator vanshes. In order to obtain at the stagnation point initial slopes of finite magnitude, d6/ldx and dZ /dx finite, the numerators also mut disappear iLn these two equations for the stagnation point. This requirement yields the initial values of the parameters KO and 60O at the stagnation point. For the~ potential flow there applies at the stagnation point R= 1 dx x4: U(x) = U0'R = aR (41) The initial values of the meridional equation are F = 0 according to (31) obtained from f30 2XD With according to (38)J with KO 002 D 2c . according to (36), and brief calculation KO . 1 /,2 1 + hO according to ( 7), there results after a 2 + l gK (42) 157 29 1 2 210 ,2 ,4 25p NACA TM 1415 For a given speed of rotation m/a, this is the first equation between the initial values KO, and 60. For the case without rotation, a = , the boundarylayer thickness ratio 13C drops out from this equation, and an1 equation for the initial value KO only remains which reads 2 + Kg_ + 29 K2+ __ 24O = 0 (45) The physically useful solution of this equation is =~ KOO= .716 (44) as known according to S. Tomotika. For the initial values of the azimuthal equation, one obtains from G = 0 according to (26) 280 O,60 ~ a KO*0g = Because of kaO2 0~2 2 a KO,60= 4;ae0 according to (24) and according; to (18) and because of (27) mediate calculation KO 1 +o 2 gl 0) one obtains after a short inter 1 + K(OE2 0 02 (4S) For a given speed of rotation m/a, this is the second equation between the initial values KO and nO* (KO m=0 ,fi 2 1 Kg aB, 1+I v \ai] M~ACA ITM 1415 For the case without rotation, a = O, one obtains from (45) with KO = KOO = 4.716 according to (44) for the initial value of 6~0 m 0= 00 the equation 1 9.432C002 Pcl i OO + 4.716g2 00 (46) Bence results with gl(n) and g2(a) according to (27) and (28) n00 = 0.915 (47) The ratio of the boundarylayer thicknesses A = by/6x for the azimuthal and meridional velocity distribution therefore lies near I which is physically plausible. The two equations (42) and (4;) now represent, for prescribed angular velocity m/~a, two equations for the initial values KO and GO* A solution was obtained by determining from both equations the values of KO T1 + 2 as a function of 60 for various fixed values KO' Rence, the initial values indicated in table 2 result. These values are presented in figure 5 as a function of m/a. It was found that for values of m/a > 0'.815, no usable initial values of KO and SO exist; that is, our method fails for these larger values of o/a. The limit beyond which our calculation method fails coincides with the value K = 12 of the form parameterl8 The initial values XO, O*, and g00 deter mined fromn the initial values KOj and 60, are represented in figure 5 and table 2, as a function of oj/a. 18For K( > 12, because of the effect of the centrifugal forces, it is entirely possible in the present case to obtain velocity profiles with ufU > 1. a TABLE 2. ~INITIAL VALUES AT THE STAGNATION POINT 0 4.716 0.915 5.71 5.71 0.0629 .221 5 .908 5.71 5.99 .0652 .454 6 .882 B.T 6.89 .0640 .679 8 .858 5.69 8.52 .0651 .785 10 .781 5.69 9.19 .0661 .815 12 .726 5.69 9.49 .0664 19See footnote 5 on page 2. NTACA T1M 1415 2 100 XO+ 100 XO Finally we obtain the initial manner with U0' = a value for Z simply in the following zo = 5 a ( 8) The initial value for 8 results with GO = 2gOO according to (24) as S=1 g00 26= a (49) The expression for the velocity distribution used here (parabola of the fourth degree for u and v) is different from that of our former calculationl9 for the rotating disk in a flow. It must be expected, however, that the boundarylayer parameters of the rotating; disk in a flow should agree approximately with those at the stagnation point of the rotating body of revolution if both methods are to yield usable results. We give this comparison for the mcomentumloss thickness in xdirection (8) at the stagnation point and for the meridional component of the wall shear stress at the stagnation point. The dimensionless momentumloss thickness at the stagnation point is according to (29) with Ux=0' = a 4x0I~ 0 ~ (5O) NACA TM 141', The meridional component of the wallshear stress at the stagnation point iTO x=0 = TPO is according to (5), (16), and (j2) "u;oI2 +Y~i (51) The values calculated accordingly are compared with those of the rotating disk in figure 6.20 The agreement up to the validity limit of our cal culation (w/a = 0.815) is quite satisfactory. Hence we conclude that our present calculation yields satisfactory results in the entire range 0 I m/a 5 0.815. 4. TORQUE AN~D FRICTION~AL D1RAG (a) Torque The entire torque of the body of revolution may be easily ascer tained from the results of the boundarylayer calculation in the following manner: The contribution of an element of the btody of revolution with the radius R.(x) and the arc length d~x is (fig. 7) E = 2nR2 iygdX and thus the total torque M =2xS A YR2dx (52) where xA signifies the arc length from the stagnation point to the point of separation. Taking the momentum theorem for the circumferential direction (ll) into consideration, one obtains 20Whereas the values for the wallshear stress could be taken directly from the report referred to in footnote 5 (p. 227, table 2), the values for the momentumloss thickness were calculated subsequently with application of equation (8) with the velocity distributions indi cated there. NACA TM 1415 M =2non [R ,UG A = 2npRA IUAB ? xyA where the subscript A denotes the values at the separation point. From the boundarylayer calculation, one knlows the value of the momentum thickness due to spin at the separation point in the dimensionless form R, v where E, is assumed to denote the radius of the maximum crosssectional area. If one Lntroduces in the same manner as for the rotating disk  a dimensionless spin coefficient by one obtains where Vm = Rmm is the circumferential velocity of the maximum cross sectional area. Since, as the completely calculated examples show, the dimensionless momentum thickness due to spin B varies at the separa tion point only a little with V /Um, eg is in first approximation proportional to Um V, and inversely proportional to the Reynolds number UmR d~v. For the case of the rotating disk in a flow, with the radius R, = R, one obtains because of RA = R, UA = aR from (56) in combination with (54) NACA T1M 1415 19 and with the numerical value =~ m/Ve~2I~a=0 = 0.2$1 according to (49) in very good agreement with the former investigation where the num~eri cal value is 3.17. (b) Frictional Drag The frictional drag of the rotating body of revolution may be deter mined by integration of the wallshear stress components 7 0 A sur face ring element of the body of revolution with the radius R,(x) and the are length dx (fig. 7) yields th~e drag dW = 2nRTxd (~b) Therein TI is the coordinate measured along the body axis. Integration from the stagnation point x = 0 to the separation point xA, where TXO = 0, yields w =2n7 ,RaX (59) We shall refer the drag to the maximum crosssectional area nR 2 and define the drag coefficient 0, (60) 2 21Cf. footnote 5 on page 2, equation (49a). NACA TM 14151 Since we obtain the wallshear stress in the dimensionless form pU= T (61) we ma~y write for tlhe drag coefficient 0 = 4 (62) 5. EXAMPLES (a) Sphere As the first example, the friction layer on the rotating sphere was calculated. When, R, signifies the sphere radius, x the are length, and xlR = Q, the center angle measured starting from the stagnation point, the radius distribution is R(x) = Rm sin 9 65 and the theoretical potential velocity distribution U(x) = IU, sin rp (6 ) Thfe velocity gradient at the stagnation point is dx =0~ 2 R, U and thus Since, according to the explanations in section ), the calculation can be carried out only for m/a 5 0.815, we must limit ourselves to V, U <3 0.815 = 1.22. The solutions are obtained by numerical integration of the two sim~ul taneous differential equations (25) and (30) for the two cases V,IUm = 0 and 1. The calculation scheme is given in table 3. The results for further values of V /Um could hence be obtained conveniently by inter polation. The case Vm U, = 0 (nonrotating sphere) agrees with the case of Scholkemeyer22. The results of the calculation are represented in table 4 and figures 8 to 12. TABLE :. CALCULATIONS SCHEME FOR THE SOLittOI O THE TWO SIMULTANEOUS CIIIFERE21IAL SQUATIONS (25,) AND (J)) a prrecribea ollt.) dU K f 9 aR~) =R' U(x) =U' v0 = J dz dr To be calculatePd line by line TABLE 4. POSITION OF SEPARATION POINT AND OF THE TO~RQUE TN spin Separation Torque, parameter, point, o 108.2 9.15 .25 108.0 9.14 .50 107.5 9.06 .75 106.2 9.03 1.00 104.9 8.95 1.22 los.) 8.85 22Footnote 13 on page 8. NACA TIM 1415 Initial givn (eq. (48)) Eq. (59) (table 2) Fig. 4  Fig. 4 Fig. 4 Given P body fonn and potential flow  To be caClculted ( LinL~e by Lne ) FOR THE ROTATING SPHERE IN A FLOW DEPENDENCE ON V, Um NACA TIM 1415 Figure 8 gives the variation of the form parameter K of the merid ional component of the velocity distribution in the boundary layer. The initial values KO at the stagnation point are Lanediately given in table 2 with equation (0)). At maximum velocity, 9 = 900, K is, according to (18), equal to zero for all VIUm, because in a sphere at the point where dU/dx. = O, also dR/ldx = The value K = 12 gives the position of the separation point A. In figure 8 the variation of the boundarylayer thickness ratio a = B 8,x is also plotted; it always lies close to I and also changes only little with V~JUm. Figure 9 shows the variation of the momentum thickness due to spin 4gy. The curves for various V, Um almost coincide. The same is true for the momentumloss thickness 9, and the frictionlayer thiicknesses by and by. Figure 10 shows the variation of the meridional and aziauthal component of the wallshear stress. The meridional component TXO increases with the spin coefficient V, Um only a little whereas the azimuthal. compo nent TYO in first approximation is proportional to the spin coeffi clent V, Um. The position of the separation point as a function of the spin coefficient VUml" is given in table 4. For the nonrotating sphere cpA = 108?.20, and for V@Um = *22 the separation point shifts forward to VA = 103.5o. This displacement of the separation point because of the rotation is due to the effect of the centrifugal forces and is, clearly, immediately plausible. For the velocity profiles behind the equatorial plane ((9 > 900), the centrifugal forces have the effect of an additional pressure increase in flow direction and there fore cause the separation point to shift forward. In figure 11 the dimensionless torque coefficient formed according to equation (56) is represented as a function of the spin coefficient Vm/U,. (Cf. table 4.) One sees that the proportionality with V /Um is fulfilled with very good approximation. Finally, figure 12 shows several velocity profiles in photographic reproduction. A sphere is rather unsuitable for the comparison of the theoretical calculation with test results, because of the large deadwater zone which has the effect that even in the case of the nonrotat~ng sphere the posi tions of the separation point according to theory and to measurement do not agree when the boundarylayer calculation is based on the potential theoretical pressure distribution as we have done here. A valid compari son regarding the influence of the rotation on the behavior of the fric tion layer can be made only for a slender body where no noteworthy deadwater zone develops. Nevertheless we mention here the measured results of S. Luthander and A. Rydberg 23. In figure 13 the drag coef ficient of the sphere in dependence on the Reynolds number Re for various values of V, Um is givell according to these measurements. For the nonrotating sphere, VmlUm =0, and up to values of V,/U, to about 3, the curve c, against t Re shows the characteristic variation with 2}Footnote 2 on page 1. NACA T1M 1415 the familiar sudden drop at the socalled critical Reynnolds n~umber. It is known that for Reynojlds numbers below the critical Reynolds number the friction layer undergoes laminar separation, and for numbers above the critical Reynolds number, in contrast, a turbulent one. In the case without rotation, the laminar separation point lies at about (O = 81", the turbulent one, in contrast, at about (p = 1100 to 1200. The meas urements with rotation show for Vm/U, = 0 to 1.4 a shifting of the critical Beynolds numbers toward higher values of Re. This shifting of the critical Reynolds number to higher values for small V,/Um is probably brought about by the fact that for Vm/Um, = O the laminar separation point is shifted from (p = 810 to higher (pvalues, with the separation still remaining laminar, however. Only for higher values of V,IU,, the rotation causes the friction layer to become prematurely tur bulent, and it then has the effect of a trip wire whereby a shifting of the critical Reynolds number to lower Reynolds numbers takes place. Whereas in our theoretical calculations a forward displacement of the separation point occurs, due to the influence of the rotation, the measurements for small values of V /U, indicate a shifting of the separation point toward the rear. On the basis of the effect of the centrifugal forces, this must be expected, if one takes into considera tion that in the case without rotation the laminar separation point lies, according to theory, behind the equator, according to measurement, how ever, ahead of the equator. In both cases, the separation point is shifted toward the equator by the effect of the centrifugal forces as is to be expected, at least for small Vm Um, as long as no premature laminar/turbulent transition has been produced by the rotation. (b) Bodies With a Base (HalfBodies) As a second example we shall now treat the socalled halfbody (body of revolution I) which originates by superposition of a transla tional flow on a threedimnensional source flow. If one denotes by Rm the largest radius at infinity, the following parametric representation for the geormetrical data of the body24 is = sin (o Rm 2 24For these relationships as well as for the numerical calculations of section Sa, I am indebted to Dr. E. Truckenbrodt. The example calcu lations of sections 5b and c are taken from the thesis of K. B. Gronau, 1952. NACA TM 1415 S= tan sel 1 sin2 9E + F ,(67)25 Here is the angle measured from the forward stagnation point. F and E 3re the incomplete elliptical integrals of the first and second kind for the modulus a = OO The velocity distribution is U 2 sin 2E ,1 sin2! 2 (6g) Um 2V 2 The form of the body anrd the velocity distribution are represented in figure 14. Thiis figure shows, for various values of the spin param eter V IU,, the variation of the form parameter K with the distance along the bodly. One sees that already for VU, Um l .j only positive values of K result. This means that due to the rotation the laminar friction layer has become more stable because in the present case the centrifugal forces accelerate in the direction of the flow and thus have the effect of an additional pressure drop. We shall forego dis cussing here all the results. In figure 15 we have represented the torque coefficient rE eg against the length L R, of the half body. Moreover, the asymptotic solution was drawn in for comparison; one can derive for it the relationship Aside from the torque, the frictional drag also was determined. Figure 16 presents a compilation of the torque coefficient and of the drag coefficient in dependence on the spin parameter Vm Um for vari ous body lengths L R It should be emphasized that the drag coeffi cient is increasing about quadratically with the spin parameter which is in qualitative agreement with the test results that have become known so far. 25E and F signify FG~I) = 09 1 d sin2 6 and E~ ~ = i24d NACA TM 1415 (c) Streamline Bodies As further examples we also calculated two streamline bodies of the thickness ratio D/L = 0.2 (bodlies of revolution II: and III). The body shapes and the pertaining velocity distributions were taken from the report of A. D. Young and E. Young2i (fig. 17). The body of revolution II has as a meridional section a normal profile; the body of revolution III, in contrast, has a laminar profile withi the velocity maximum lying rela tively far downstream. Of the results, figure 18 shows the torque coef ficient and the frictional drag coefficient as a function of the spin parameter V JU,. In both cases, there are not large differences between the bodies. For the rest, the variation is similar to that in the case of the body with a base. In figure 119, the position of the separation points is shown as a function of the spin parameter V ,U,. In agreement with the values for the rotating sphere (cf. table 4), the separation point shifts forward with increasing rotational speed. This displacement is larger for the body of revolution II than for the body of revolu tion III which is made understandable by the position of the velocity maximum. Finally, we gave for the body of revolution II a graphic repre sentation of the velocity distributions in the friction layer for the spin parameters V /Um = and VG/U, =1 (fig. 20). Fr~m it one sees that ahead of the pressure minimum the meridional velocity component does not vary noticeably due to the influence of the rotation whereas between the pressure minimum and the separation point the influence of the rotation is considerable. a. SUMMZARY A calculation method is given by which the flow about a rotating body of revolution in a flow which approaches in the direction of the axis of rotation may be determined on the basis of boundarylayer theory. The investigations yield a contribution to the aerodynamics of a pro jectile with spin. The calculation is carried out for the lamrinar boundary layer with the aid of the momentum theorem which is stated for the meridional and for the circumferential. direction. The performance of the calculation requires the solution of two ordinary staultaneous differential equations of the first order. It yields, in addition to the boundarylayer parameters, the frictional drag and the torque as a function of the dimensionless spin coefficient Vm/Um = circumferential velocity freestream velocity. The displacement of the separation point 2A. D. Young, E. Young, "A family of streamline bodies of revolu tion suitable for highspeed and lowdrag requirements." ARC Report 2204, 1951. NACA T1M 141'5 with the spin coefficient also is obtained. As examples, the flow about a rotating sphere, about a body with a base, and about two streamline bodies is treated. Translated by Mary L. Mahler National Advisory Cormmittee for Aeronautics NACA TIM 1415 27 U (x) a Lux,z) ''R ( I x o' b c U(x) V (x,z) Rc, R ?x) Figure 1, Explanat~ory sketch. 8 10 0 20 0 22 C Figure 3. Tie universal function L = s7" x as a function of g~ = xy /x according to (2r7) and (28). K=19 to8 42 O 2 4 ~6 8 10 2 I ~ _ . . NACA 'IN 1415 Z. t= s Z t Figir~e 2~. Velocity distributions in meridional and in azimuthal direction. 04 0.e 00cnn / on r~ 016~ 'nf NACA TM 14c15 Figure 4. The universal functions fil = 9,5 x, f2 = Sx*Sx, f3 = rx0 x/9>,U and K as a function of :l* according to (32) to (39). NACA TM 141) O 0.2 0.4 0.6 0.8 1.0 Figure 5. The initial values at the stagnation point h0, KO' 0O, and x 0 2 NACA TM 1415 UR xo ~ 1.6 E T 4UR 4V ~ RI O.8 ~ 0~4t According t H. Schlichting and E. Truckenbrodt 0.4 0.6 0.8 1.0 Figure 6. Momentum10ss thickn~ess a, and wallshear stress r,0 at the stagnation point. Comparison with the values of the rotating disk in a flow according to H. Schlichting and E. Truckenbrodt. Figure 7. Sketch for calculation of the torque and of the frictional drag. NACA TM 14.15 Figure 8. Form parameter K: a~nd boundarylayer thickness ratio 6 y/SX for the rotating sphere in a flow for different spin parameters Vm/TJ. NACA 'IN 1415 Figure 9. Iv10menturn10ss thickness due to spin 29 x for the rotating sphere in a flow. Re = U. Rm/ * NACA TIM 1115 O~ pUm O8 = LO 'I UmD O 6 0.75 O.25 O 200 40* 600 80" 100. Figure 10. The wallshear stress components in meridional and azimuthal direction rz0 and ry0 for the rotating sphere in a flow. Re = U.R m' NACA TM 1415 vm U00 Figure 11. Torque coefficient cy, for the rotating sphere in a flow. Re = UmRm; . NACA 'lM 141'5 1 ?: '1 \ I:s u!, '~~ Figure 12. Photographic representation of the velocityr distributions in the boundary lawyer of the rotatin~g sphere in a flow, Vm .u = 1, A = separation line. Figure 13. Drag coefficients of the rotating spihere in a flow as a function of the Reynolds number, according to measurements of S. Luthander and A. Rydberg. NACA TM~ 1415 x y Figure 14. Form parameter K for the rotating body with a base in a flow (body of revolution I). NACA I1 1415 Figure 15. a flow as Torque coefficient cpJ for the rotating body with a base in a function of the length of the body. Re = U Rm/v . r ~t~t~_ ...... NACA 'IN 16 15 VcM vm U, IReCw L 6 R 4  2 Vm Um, 1.0 1.2 O 0.2 0.4 0.6 0.8 Figure 16. Torque coefficient cMn and drag coefficient cw for the rotating body with a base in a flow (body of revolution I) as a function of the spin parameter Vm/U,. H~e = U,Rm/V  NACA '1M 1415 O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Figure 17. Body shape and potential flow for tw~o streamline bodies (bodies of revolution L1 and Il); D/L = 0.2. IVm ~ec, Um, ~~t V, Um, 0.2 0.4 0.6 0.8 1.0 4 Vm Um NACA TM 12C1' 20 8 4 C Re c sp 0.2 1.0 Figure 18. Torque coefficient eM and drag coefficient ew for the two streamline bodies according to figure 17 as a function of the spin parameter Vm/U,. Re = U,R,/V. NACA TM 1415 Body of revolution II Body of revolution U. O 0.2 0.4 0.6 0.8 1.0 Figure 19., Position of the separation point A for the two streamline bodies according to figure 17 as a function of the spin parameter Vm/IU,. Position of the separation point: o for Vm/: U. = 0; for N~ACA 1I 141+ 1 54 o 0o Or O OO rD~ IS oa o od o9 Ic I a~ a o O so O .J 8 11 i I.. O L : NACA Langley FLEld. Va. ii; ~s ~0 r~ a ~a ~1 O s 2z r I Bt~. s OE II1 5 s~ B ;~ c, a a 'j r;i; bbC ib r:A o c ~a~9~ a ti38~ 1; ruc w9 ru M S Z E r b~ r (U a e o~ "~:iP s" a h g 58 ,o vl a ,P E ;ii c: cl~~t ca4: c c65 .5 r iX 5 cu m ** rvr ei r iiio Se~ s o, ;a I Oka Fj (D d 3 o cc M~ Ei EiS oe~~~ u u h~ U"3 ,O a E a 1 "e*O r o:: Bs e ., e n ~c~o, ImI a E a a ~uu, aSe d E: ... 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