iVAt/j-f/ti-ri NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1415 LAMINAR FLOW ABOUT A ROTATING BODY OF REVOLUTION IN AN AXIAL AIRSTREAM By H. Schlichting Translation of *Die laminare Stromung um einen axial angestromten rotierenden Drehkorper." Ingenieur-Archiv, vol. XXI, no. 4, 1953. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY RO. BOX 117011 GAINESVILLE. FL 32611-7011 USA Washington February 1956 I'^ipf'^) OIH 3'ili5\^>' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM lUl5 LAMINAR FLOW ABOUT A ROTATING BODY OF REVOLUTION IN AN AXIAL AIRSTREAM* By H. Schlichting 1. INTRODUCTION 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 rotaxy motion. Investigations of C. Wieselsberger-'- 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 free-stream velocity - increasing more and more, the slenderer the body. Similar results were obtained by S. Luthander and A. Rydberg^ 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 circumferential velocity to the free-stream velocity. The physical reason for these phenomena may be found in the processes in the friction layer where, due to the rotary motion, the fl\iid 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 laminar to turbulent conditions are strongly affected thereby, and that, therefore, the rotary motion must exert a strong influence on the drag of the body. *"Die lamlnare Stromung um einen axial angestromten rotierenden Drehkorper." Ingenleur-Archlv, vol. XXI, no. k, 1955, PP- 227-2^4.- An abstract from -this report was read on the VIII International Mechanics Congress in Istanbul on August 27, 1952. ■k:. Wieselsberger, Phys. Z. 28, I927, p. 8k. S. Luthander, A. Rydberg, Phys. Z. 36, 1955, P- 552. MCA IM liH5 In the flow processes in the corotating layer of the fluid, one deals with complicated three-dimensional boundary -layer flows which so far have been little investigated, experimentally as well as theoretically. Th. v. Karman^ treated at an early date the special case of a disk rotating in a stationary liquid, for laminar and turbulent flow, as a boundary -layer problem, according to an approximation method. Later, W. G. Cochran^ also solved this problem for the laminar case as an exact solution of the Navier-Stokes eqioations. 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. Truckenbrodt5 . 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 free -stream veloc- ity of the disk. For the general case of a rotating body simultaneously subjected to a flow, J. M. Burgers" gave a few general formulations. We have set o\ir- selves the problem of calculating the laminar flow on a body of revolution in an axial flow which simultaneously rotates about its axisT. 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 laminar 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 ^h. V. Karman, Z. angew. Math. Mech. 1, 1921, p. 235. \ W. G. Cochran, Proc. Cambridge Philos. Soc. 50, I95U, p. 365. H. Schlichting, E. Truckenbrodt , Z. angew. Math. Mech. 32, 1952, p. 97; abstract in Journal Aeron. Sciences I8, 1951 > P« 638. J. M. Burgers, Kon. Akad. van Wetenschappen, Amsterdam ^5, 19^1 ? p. 13. 'It 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. Truckenbrodt. Publication will take place later.- E. Truckenbrodt, "Die Strbmung an einer angeblasenen rotierenden Scheibe bei turbulenter Stromimg," will be published in Z. angew. Math. Mech.- E. Truckenbrodt "Ein Quadraturverfahren zur Berechnung der Reibungsschicht an axial angestromten rotierenden Drehkorpern." Report 52/20 of the Instltut fiir Stromungsmechanik der T. H. Braunschweig, 1952. NACA TM 1J|15 this case vas given by S. Tomotika°, by means of transfer of the well known approximation method of K. Pohlhausen9 to the rotationally symmet- 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 the rotating disk. In the case of the cylinder one deals with a distribution of the circumferential velocity according to the law v = oE^/r where R signifies the cylinder radius, r the distance from the center, and cd the angiilar 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 have "boundary-layer character," that is, the friction effect is not limited to a thin layer in the proximity of the wall but takes effect in the entire environment of the rotating body. Very recently, L. Howarth-'--'- also made an attempt at solution for a sphere rotating in a stationary 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 boundary -layer 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 Y^ Ej^o) Free-stream velocity Uqo U^ where Rj^ 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 boundary-layer veiriation as a function of Vm/Uoo. The "S. Tomotika, "Laminar Boundary Layer on the Surface of a Sphere in a Uniform Stream." ARC Rep. 1678, 1955. 9k. Pohlhausen, Z. angew. Math. Mech. 1, 1921, p. 255. -'-'-^H. Schlichting, "Grenzschicht-Theorie, " p. 63. Karlsruhe I95I. ■"■■4.. Howarth, Phllos. Mag. VII Ser. k2, I95I, p. 1,508. MCA ™ lifl5 is already known from the boundary-layer Considering what has been said above, we particular case Vjjj/U^ = theory established so far. _ must not expect our solution to be valid for arbitrarily large VjjJU^ The upper limit of the value of Vj^/Uoo for which our calculations holds true, still remains to be determined. Presumably, it will lie considerably above Vj^/Uoo = 1. 2. THE FUNDAMENTAL EQUATIONS We take the coordinate system indicated in figure 1 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 y-axis along a circular cross section so that the xy -plane is the tangential plane. The z-axis is at right angles to the tangential plane. Let u, 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, cu the angular velocity of rota- tion, U(x) the potential -theoretical velocity distribution, and V = ti/p the kinematic viscosity. The equations of motion simplified according to the calculation methods of the boundary-layer theory are for this coordinate system ^1 u dR 5w _ Q Sx R dx Sz (continuity) (l) , Su v2 dR 5u ,,dU u + w — = U — + ^ R dx bz dx f — — (momentum, meridional) (2) Sz^ u^ + MX dR + w^ = v^ 3x R dx Sz Sz^ (momentum, azimuthal) (5) The Doundary conditions are z = 0: u = 0, v = Vq = Ro), w = 0; z = <»: u = U(x), v = (k) MCA TM liH5 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 integration of the corresponding equations of motion over z from the wall z = to a distance z = h > 5 which lies outside of the friction layer. For the meridionaJ. direction there resiilts by integration of (2) over z, with consideration of the continuity equation (l) and after introduction of the wall-shear stress for the x-direction the momentum equation for the meridional direction Therein, as is well known, is the displacement thickness whereas 5 * - f^l - i^]dz (7) ^^ . r u/l _ u^^ (8) (9) may be denoted as momentum-loss thicknesses for the x- or y-dlrectlon. MCA TM l4l5 In an analogous manner there results for the azimuthal direction by- integration of (3) over z with consideration of the continuity equa- tion (1) and introduction of the wall-shear stress for the y-direction as the momentum theorem for the circumferential direction dx V "^•y / p Therein ^m^^\ = -R2ly0 (11) dx V ^J I D ^ = ^^z (12) ^ Jo U Vq has been introduced as the "momentimi loss thickness due to spin." 3. APPROXIMATION METHODS (a) The Velocity Distributions According to the approximation method of the boiJindary-layer theory as given first by Th. v. Karman and K. Pohlhausen, the momentum equa- tions (6) and (11) 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 = 6 the frictionless outer flow. The boundary- layer thickness may be different for the meridional and the azimuthal velocity component. Let these boundary-layer thicknesses be S^ and 5 , respectively; we introduce the dimensionless wall distances formed with them -Z- = t and -^ = t' (13) 5. x h NACA TO li*-15 For the velocity distributions u and v, we select polynomials of the foiu-th degree in t and t', respectively. These contain five coeffi- cients 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 : t = 0: u = t = 1: u = U v^ = -U^ - ^ dE az2 bn dx R dx dz2 (l4a,b,c,d,e) t'=0: v = v^ = RfD t' = 1: V = d^v_ az2 av = h^^ > (I5a,b,c,d,e) The boundary conditions (l^a, b, c) and (l^e., b, c) result immediately from the fundamental equations (2) and (j) with (k) for z = and z = 5 or 6y. The remaining boundary conditions pro-vide 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 U 2t - 2t5 + t^ + Kift - 3t2 + 5t5 - t^) (16) — = 1 - 2t' + 2t'3 ,k (17) Therein K = -^ V dU _^_ l"^\ U m dx U / R dx (18) 8 NACA TM l4l5 Is a form parameter of the u-velocity profile, which is analogous to the form parameter A of the Pohlhausen methodic. The velocity distributions u/u and v/vq are represented in figure 2. Let the point of separation be given by the beginning of the return flow of the meridional velocity component u(z) in the proximity of the wall This yields K = -12 (separation) (19) The expression for the u-component is the same as in the Pohlhausen method for the plajie and rotationally symmetrical case. This guarantees that our solution in the case without rotation, co = 0, will be trans- formed into the solution of S. Tcmotika and F. W. Scholkemeyer-'-5 for the nonrotating body of revolution. Introduction of the expressions (16) and (17) into the momentum equations (6) and (ll) yields two differential equations for the still unknown boundaxy-layer thicknesses 5x(x) and 6y(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 the circumferential direction. With y -ZO ^ ^/^\ ^ _2JLr^ (20) there results from (ll), after division by to, A{r5u0^} . 3^5 (31) -^Cf. H. Schlichting, "Grenzschicht-Theorie, " p. I93. 13 -^F. W. Scholkemeyer, "Die lamlnare Reibungsschicht an rotations- syrametrischen Korpern." Dissertation Braimschweig 19if3, Cf. H. Schlichting, Grenzschicht-Theorie, p. 204. NACA TOI 11+15 With introduction of the further parameters as well as and gp, = ^21 and A = -^ (22) &y &x G = -^ (25) a = ef^ + 5U dE\ i2k) Vdx R dxj one obtains from (21) the following differential equation for 0(x) de ^ G(K,A) (25) dx U Therein is G(K,A) = kgQ - 2a (26) a universal function of the two parameters K ajid A. This function has been determined already by W. Dienemann-'-^ in the calculation of the temperature boundary layer on a cylinder (two- dimensional problem). 5 por 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 xy _ / u j^^, """ 6y Jo U Vq Because of t = ^ = ^!y = t'A ^x 5y \ w. Dienemann, "Berechnung des Warmeiiberganges an laminar umstromten Korpern mit konstanter und ortsveranderlicher Wandtemperatur . " Disserta- tion Braunschweig, 1951^ Z,. angew. Math. Mech. 55^ 1953j p. 89. ^5yxth the symbols according to W. Dienemann there apply the identities Ht = &q and A = K. 10 MCA TM lill5 one obtains after calculation of the integral with the velocity distri- butions (16) and (17) the quantity gQ as a function of K and A. According to W. Dienemann, there results a1 1: A ^ 1: go(K,A) = g-L(A) + Kg2(A) gl(A) g2(A) gl(A) g2(A) 1a 15 -4^ ll+O 180 90 - 1a2 + 4-a5 - 560 1,080 5 10 5 1. 10 A 2 1 15 A^ 5 1 ^ 1J^^4 1 180 1 a5 > 1 1 1 1_ 1_ ^ ^ _!_ 120 A " 180 ^2 840 ^4 " 3,024 ^5 (27) (28) The function go (A) as a function of A for various values of K is represented in figure J. Table 1 gives a few numerical values of the functions g]_(A) and ggCA). TABLE I.- THE UNIVERSAL FUNCTIONS g2_(A) AND ACCORDING TO EQUATION (28) g2(A) A gl(A) 100g2(A) .2 .0055 .036 .4 .0208 .114 .6 .0457 .205 .7 .0606 .249 .8 .0784 .291 .9 .0970 .329 1.0 .1175 .364 1.2 .1614 .423 1.4 .2089 .471 1.6 .2589 .510 1.8 .3109 .541 2.0 .3643 .568 2.5 .5021 .618 3.0 .6437 .651 MCA TM li<-15 11 (c) The Momentum Equation for the Meridional Direction Further transformation of the momentum equation for the meridional direction yields, if one introduces, according to Holstein-Bohlenl° and analogous to (25) X = dU dx (29) the following differential equation for Z(x) dZ ^ F(K,A) dx U (30) Therein F(K,A) = 2 f. _ (2X + Xfo) - zM ^ 1 + '^0? ho u/ fo (51) exactly as G(K,A) in (26) a universal function of the two parameters K and A. Individually, the following relationships apply: f,(K) = ?^ = ^ K K^ 315 9i^5 9,072 (52) f. (K) = ^ = ^ - J^ -^ 6x 10 120 (33) ^ ^x fo(K) (3^) 6„ 6 'X 5y fp,(K) b -d ^ U^ ' XX X (35) 'Xo ^ du &„ -a. f (K) = ^i ^ = ^ rx ^ = 2 + ^ f^(K) (36) l^Cf. H. Schlichting, Grenzschicht-Theorie, p. I95. 12 MCA TM IU15 ^ s 126 •7 (37) The above functions of K are already known from the calculation of the botmdary layer of the two-dimensional case. -'-7 The connection between Z and K results from (18), with considera- tion of (29) and (52), and is Kfo^(K) = Z dU dx vof u j u dR R dx (58) Taking X= ZU', from (29), into consideration, one may write this because of (52) also in the form \ U / R U' X* K 37 K ^ 315 9^5 9,072^ (39) In figiire k the universal functions fr fz, and K are -0^ ^2> represented as functions of X*. At the point of separation, for arbi- trary rotational velocity, one will have, because of K = -12, the parameter X* = -O.I567. At the stagnation point, without rotation, K = 4.716 and X* = O.O57O8, whereas with rotation the values at the stagnation point are dependent on the spin parameter Vq/U {cf . the following section). From (38) the form parameter K can be determined when Z is given. Furthermore, for the later calculation a connection between the pajrameters A, gg^ Q , Z, and K is needed. There results according to (22), (23), (29), and (32) as follows Ag^(K,A) =^/af (K) (40) The two differential equations (25) for 9(x) and (50) for Z(x) are two simultaneoiis differential equations coupled by the universal functions G(k,A) and F(K,A). In the case of the nonrotating body, Vq = 0, the coupling is eliminated since then, according to (3I), the function F becomes independent of A and remains dependent only on K. ^7cf. H. Schlichting, Grenzschicht-Theorie, Chapter XII. NACA TM IU15 15 In this case, one can first determine Z(x) from (jO), and subsequently G(x) from (25). This solution for 9 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 the Stagnation Point At the stagnation point where U = 0, the two differential equa- tions (25) and (50) have a singular value since in both equations on the right side the denominator vanishes. In order to obtain at the stagnation point initial slopes of finite magnitude, dQ/dx and dZ/dx finite, the niimerators also must disappear in these two equations for the stagnation point. This requirement yields the initial values of the parameters Kq and Aq at the stagnation point. For the potential flow there applies at the stagnation point 0: U(x) = Uq'R = aR ^ = 1 (kl) The initial values of the meridional equation are obtained from F = according to (5I) ^50 " ^^ ~ ^^20 " \) 00 f^t = With according to (58), with ^^00 " \) ^50 - ^00(2 + f according to (56), and h^^ according to (37)^ there results after a brief calculation K, 2 + i - ^^)^ 137 210 3,02V ^'- ^ i^2) l!+ NACA TM 1^4-15 For a given speed of rotation co/a, this is the first equation between the initial values Kq and A^. For the case without rotation, cd = 0, the boundary-layer thickness ratio Aq drops out from this equation, and an equation for the initial value Kq only remains which reads 105^ 2,550^-' 3,02i^^ (^3) The physically useful solution of this equation is ('o)^o = "-0 - ^-n^ (H) as known according to S. Tomotika. For the initial values of the azimuthal equation, one obtains from G = according to (26) 2go(Ko>^) - ^(Kq^^) = Because of a(Ko,Ao) = kaeQ = 4agoo%2. "Xr according to (24) and ^ a5x, ■0 1-, i^ according to (l8) and because of (27) one obtains after a short inter- mediate calculation KO 1 + (^f 2[gi(/^) + Kog2(^)]z^2 (^5) For a given speed of rotation cu/a, this is the second equation between the initial values Kq and /Sq. NACA TM IU15 15 For the case without rotation, to = 0, one obtains from (ij-5) with Kq = Kqq = 4.716 according to (kk) for the initial value of ("^0)^^=0 ^ "^0 ^^^ equation 1 - 9.452Aoo^ si(^o) + ^ -71682 (Aqo^ = (46) Hence resiilts with gi(A) and g2(A) according to (27) and (28) Aqo = 0.915 (^7) The ratio of the boundary-layer thicknesses A = Sy/s^ ^°^ ^^'^ azimuthal and meridional velocity distribution therefore lies near 1 which is physically plausible. The two equations (42) and (45) now represent, for prescribed angular velocity ca/a, two equations for the initial values Kq and Aq- A solution was obtained by determining from both equations the values of v2' 1 + m as a function of /Sq for various fixed values K, 0- Hence, the initial values indicated in table 2 result. These values are presented in figiire 5 as a function of co/a. It was found that for values of u)/a > O.815, no usable initial values of Kq and Aq exist; that is, our method fails for these larger values of co/a. The limit beyond which our calculation method fails coincides with the value K = 12 of the form parameter . The initial values Xq, Xq*? and gQQ deter- mined from the initial values Kq and Z^q are represented in figure 5 and table 2, as a function of cu/a. l^For K > 12, because of the effect of the centrifugal forces, it is entirely possible in the present case to obtain velocity profiles with u/U > 1. 16 NACA TM 114-15 TABLE 2.- INITIAL VALUES AT THE STAGNATION POINT a ^0 ^0 100 Xq 100 Xq* |eoo i+.7l6 0.915 5.71 5.71 0.0629 .221 5 .908 5.71 5-99 .0652 .k^k 6 .882 5.70 6.89 .o6i+o .679 8 .858 5.69 8.32 .0651 .785 10 .781 5.69 9.19 .0661 .815 12 .726 5.69 9.^9 .0661+ Finally we obtain the initial value for Z simply in the following manner with Uq' = a Z -^0 ihS) The initial value for 9 results with CTq = 2goo according to (24) as % = - — ^ p a ih9) The expression for the velocity distribution used here (parabola of the fourth degree for u and v) is different from that of o\ar former calculationl9 for the rotating disk in a flow. It must be expected, however, that the boundary-layer parameters of the rotating disk in a flow shoiild agree approximately with those at the stagnation point of the rotating body of revolution if both methods are to yield usable resxolts. We give this comparison for the momentum-loss thickness in x-direction (8) at the stagnation point and for the meridional component of the wall shear stress at the stagnation point. The dlmensionless momentum-loss thickness at the stagnation point is according to (29) with U^_ x=0 ^^0 ? = ^^ (50) -^^See footnote 5 on page 2. MCA TM li+15 IT The meridional component of the wall-shear stress at the stagnation point ("^x^^ _ = Tp^ is according to (5), (16), and (32) '^)x=0 -I — = 1 _ pU^Vv Va ^^ 2^^ (51) The values calculated accordingly are compared with those of the rotating disk in figure 6.20 ^p^g agreement up to the validity limit of our cal- culation (a)/a = 0.815) is quite satisfactory. Hence we conclude that our present calculation yields satisfactory results in the entire range £ ao/a £ 0.815. k. TORQUE MD FRICTIONAL DRAG (a) Torque The entire torque of the body of revolution may be easily ascer- tained from the results of the boundary-layer calculation in the following manner: The contribution of an element of the body of revolution with the radius R(x) and the arc length dx is (fig. 7) dM = -2jtR2T ±x and thus the total torque M = -2rt / T R2dx (52) where x^ signifies the arc length from the stagnation point to the point of separation. Taking the momentima theorem for the circumferential direction (ll) into consideration, one obtains ^^ereas the values for the wall-shear stress could be taken directly from the report referred to in footnote 5 (p. 227, table 2), the values for the momentum-loss thickness were calculated subsequently with application of equation (8) with the velocity distributions indi- cated there. l8 MCA TM lit- 15 M = 2jtpm do R^U^ I - o*^R.5 xyjn 2«pR/UA^^y^ (55) where the subscript A denotes the values at the separation point. From the boundary-layer calculation, one knows the value of the momentum thickness due to spin at the separation point in the dimensionless form ^^A /UocRm ^ = B i-^k) where Rj^ is assumed to denote the radius of the maxira-um cross -sectional area. If one introduces - in the same manner as for the rotating disk - a dimensionless spin coefficient by 2 one obtains where Y^ = ILm 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 vaxies at the separa- tion point only a little with Vjjj/U^, Cy^ is in first approximation proportional to U^j^/V™ and inversely proportional to the Reynolds number l/Ucx^A For the case of the rotating disk in a flow, with the radius R^^ = R, one obtains because of R^ = R, U. = aR from (56) in combination with (54) /r2 NACA TM lij-15 19 and with the numerical value i^ - PLo - --'' according to {k9) 4^ ^M = 5-^5^ in very good agreement with the former investigation^-'- where the numeri- 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 wall-shear stress components t^ . A sur- face ring element of the body of revolution with the radius R(x) and the arc length dx (fig. 7) yields the drag dW = 2nRT^^d5c (58) Therein x is the coordinate measured along the body axis. Integration from the stagnation point x = to the separation point 5c^, where r^ = 0, yields pA W = 2n I Tj^Rdx (59) We shall refer the drag to the maximum cross -sectional area itR^ and define the drag coefficient ^w = _W (60) ^J^\' 21cf. footnote 5 on page 2, equation (49a). 20 NACA ™ li+15 Since we obtain the wall-shear stress in the dimensionless form I^S=T (61) we may write for the drag coefficient / ^'Ai 1, H^ ^E^^ (62) •Jo ^ ^ 5. EXAMPLES (a) Sphere As the first example, the friction layer on the rotating sphere was calculated. When B.-^ signifies the sphere radius, x the arc length, and x/Rjj^ = cp the center angle measured starting from the stagnation point, the radius distribution is R(x) = I^ sin cp (65) and the theoretical potential velocity distribution U(x) = |j„ sin cp {6k) The velocity gradient at the stagnation point is and thus dx L 2 Rm yUJo Uoc 2 a (65) MCA ™ l4l5 21 Since, according to the explajiations in section 5, the calcixLation can be carried out only for to/a ^ O.815, we must limit ourselves to Vj^/U„ ^ ^ 0.815 = 1.22. The solutions are obtained by numerical integration of the tvo simul- taneous differential equations (25) and (50) for the two cases Vj^/U«, = and 1. The calculation scheme is given in table 5. The results for further values of Vjjj/U^ could hence be obtained conveniently by inter- polation. The case ^jjl^oo = (nonrotating sphere) agrees with the case of SchoLkemeyer^^ . The results of the calculation are represented in table 4 and figures 8 to 12. TABLE 3.- CALCULATION SCHEME FOR THE SOLUTION OF THE TWO SIMULTANEOUS DDTEKENTIAL EQUATIONS (25) AND (jO) 0) prescribed ^0 9 X R(x) dx U(x) dU _ dx " U' "0 = oR % "^2 ^3 G Iven -^- Initial value given (eq. (W)) =ZU' Eq. (39) To be Fig. k initial value (table 2) calculatec Fig. h Fig. k Fig. 1* % 2o ecA A "So (?)> F dZ dx AZ ^VM-1 G de dx ^ Sv+1 Initial vBlue (eq. (1^9)) «^ E 1- {2k ) Eq. (4 0) ¥i -6. 5 >d A — To ^ 25 " 126 alcnla Eq. ( ted U 31) TIP I- I q. (30) line Eq. (26) Eq. (25) ^ TABLE 4.- POSITION OF SEPARATION POINT AND OF THE TORQUE IN DEPENDENCE ON Vj^ U^ FOR THE ROTATING SPHERE IN A FLOW Spin paxameter, Separation point, < Torque , u J V ^ .25 .50 .75 1.00 1.22 108.2 108.0 107.3 106.2 101^-9 105.5 9.15 9.11^ 9.06 9.05 8.95 8.85 ^^Footnote 13 on page 8 . 22 MCA TM li4-15 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 Kq at the stagnation point are Immediately given in table 2 with eqiiation (65). At maximum velocity^ cp = 90°> K is, according to (I8), equal to zero for all Vjj^/U^, because in a sphere at the point where du/dx = 0, also dR/dx = 0. The value K = -12 gives the position of the separation point A. In figure 8 the variation of the boundary-layer thickness ratio A = 5y/5x is also plotted; it always lies close to 1 ajid also changes only little with Vjjj/Uco. Figure 9 shows the variation of the momentum thickness due to spin •^xy- The ciorves for various Vj^/Uoo almost coincide. The same is true for the momentum-loss thickness ^.^ and the friction-layer thicknesses 6^ and. 6y. Figure 10 shows the variation of the meridional and azimuthal component of the wall-shear stress. The meridional component t^^ increases with the spin coefficient Vjjj/Uoo only a little whereas the azimuthal compo- nent Ty_ in first approximation is proportional to the spin coeffi- cient Vjjj/u^. The position of the separation point as a function of the spin coefficient Vj^/tJ^j^ is given in table k. For the nonrotating sphere cp. = 108.2°, and for Vjn/U^ = 1.22 the separation point shifts forward to cp. = 103.5°. 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 > 90° )j "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 Vj^/U^. (Cf. table k.) One sees that the proportionality with Vj^/Uoo is fiilfilled 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 calc\alation with test results, becaiise of the large dead-water zone which has the effect that even in the case of the nonrotating sphere the posi- tions of the separation point according to theory and to measurement do not agree when the boimdary -layer 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 dead-water zone develops. Nevertheless we mention here the measured results of S. Luthander and A. Rydberg ^5. in figure I5 the drag coef- ficient of the sphere in dependence on the Reynolds number Re for various values of Vm/Uoo is given according to these measurements. For the nonrotating sphere, Vjjj/U„ = 0, and up to values of Vm/U^o to about 5, the curve c^ against Re shows the characteristic variation with — -^Footnote 2 on page 1. NACA TM IU15 25 the familiar sudden drop at the so-called critical Reynolds number. It is known that for Reynolds 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 cp = 8l°, the turbulent one, in contrast, at about cp = 110° to 120°. The meas- \irements with rotation show for Vjji/U(„ = to 1.^4- a shifting of the critical Reynolds numbers toward higher values of Re. This shifting of the critical Rej^nolds number to higher values for small V^/Uoo is probably brought about by the fact that for Vj^/Uoo = the laminar separation point is shifted from cp = 8l° to higher cp-values, with the separation still remaining laminar, however. Only for higher values of Vjjj/Uqo, 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 calciilations a forward displacement of the separation point occurs, due to the influence of the rotation, the measurements for small values of Vjjj/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, aJiead 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 Y-^IU^, as long as no premature laminar /turbulent transition has been produced by the rotation. (b) Bodies With a Base (Half -Bodies) As a second example we shall now treat the so-called half -body (body of revolution I) which originates by superposition of a transla- tlonal flow on a three-dimensional source flow. If one denotes by Rm the largest radius at Infinity, the following parametric representation for the geometrical data of the bodyS^- is ^ = sin 2 (66) Kffl 2 ^or these relationships as well as for the numerical calculations of section ^a, I am indebted to Dr. E. Truckenbrodt . The example calcu- lations of sections 5^ ^^'^ c are taken from the thesis of K. H. Gronau, 1952. 2k MCA m l4l5 X *-lf^F^*P^^ = 8-6,F (69) ^00 y V YRm Aside from the torque, the frictional drag also was determined. Figure I6 presents a compilation of the torque coefficient and of the drag coefficient in dependence on the spin parameter Vjjj/Uoo for vari- ous body lengths L/l\r,- I"t 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 ^ sin2 i3 d-a NACA TM 1415 25 (c) Streamline Bodies As further examples we also calculated tvo streamline bodies of the thickness ratio d/l = 0.2 (bodies of revolution II and III). The body shapes and the pertaining velocity distributions were taken from the report of A. D. Young and E. Young^o (fig. I7). The body of revolution II has as a meridional section a normal profile; the body of revolution III, in contrast, has a laminar profile with the velocity maximum lying rela- tively far downstream. Of the results, figure I8 shows the torque coef- ficient and the frictional drag coefficient as a function of the spin parameter V /U^. 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 I9, the position of the separation points is shown as a function of the spin parameter Y^lu^. In agreement with the values for the rotating sphere (cf . table k) , 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 ^^Na> = and Vj^/u^j^ = 1 (fig. 20). From 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. 6. SUMMARY 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 boundary-layer theory. The investigations yield a contribution to the aerodynamics of a pro- jectile with spin. The calculation is carried out for the laminar 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 simultaneous differential equations of the first order. It yields, in addition to the boundary -layer parameters, the frictional drag and the torque as a function of the dimensionless spin coefficient V Aj^ = circumferential velocity /free -stream velocity. The displacement of the separation point 26 A. D. Young, E. Young, "A family of streamline bodies of revolu- tion suitable for high-speed and low-drag requirements . " ARC Report 2204, 1951. 26 MCA IH li^-15 with the spin coefficient stLso 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 Committee for Aeronautics NACA TM lij-15 27 U(x) Figure 1.- Explanatory sketch. 28 MCA TO lij-15 t=— ; t'= ^ 8x' "^ 0-2 0.4 0.6 0.8 1.0 Figure 2.- Velocity distributions in meridional and in azimuthal direction. Figure 3.- The universal function a = 6v/b^ as a function of gn,{A) =z/q according to (27) and (28). MCA TM li4-15 29 1 n ! 1 l.o 1 c ^^ A 1 K 4 A ^ ^ 5f3 1 1 1 I \ ^ y /^ 10fo \ 4 i — — *^ \ y y "^ — . -^ V ^ y ^ r> o /^ y / ^ U.o 0.6- 04- -^2 -2 / ' 1 1 / / 10 "^ 1 / / r\ 1 1/ /^ ^-■ 1 y ^ 1 0.0948 \ -0.12 -0.08 -0,04 1-0.1567 . 1 1 y y 0.02 0.06 i / y y / / y -0.6" -0,8 -1.0 -1 2- ,/ /' — V Figure 4.- The universal functions fg = ^x/^x» ^2 " ^x*'^'^x> ^3 " "^xo/^^ "^x/U and K as a function of x* according to (32) to (39). 30 MCA TM li+15 0.2 0.4 0.6 0.8 1.0 Figure 5.- The initial values at the stagnation point A„, K„, x^, and x^'' NACA IM li+15 51 According to H. Schlichting and E.Truckenbrodt _J I L_J \ \ \ \ Cl) a 0.2 0.4 0.6 0.8 1.0 Figure 6.- Momentum -loss thickness ^^ and wall -shear stress t^ 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. 32 WACA "M IU15 *' 12 8 -8 -12 Vm -c" IJ =^-' ?2 1.0 "^ /075 ■-^ s — tcr- ^ / 0.50 1 ^^^^^ ^ k \ 1 20*" 40*' 60" 80" " V 100° Vc >0<' 1 ,Se 1 paration 1 1

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