VOL. 2, NO. 2, MARCH-APRIL 1986 J. PROPULSION 155 An Analytical Investigation of Swirl in Annular Propulsive Nozzles Barbara T.Kornblum,* H. Doyle Thompson,! and Joe D. Hoffmanf Purdue University, West Lafayette, Indiana An analytical performance prediction methodology for annular propulsive nozzles with swirl introduced in the combustor upstream of the nozzle is presented. The application of that methodology to a specific nozzle design for a free vortex swirl distribution is discussed. Discharge coefficients, specific impulses, and wall pressure distributions are presented. These numerical studies show that the discharge coefficient, thrust, and specific im- pulse decrease as the amount of swirl increases. This methodology will enable nozzle designers to account for the effects of swirl in nozzle design. Nomenclature a - speed of sound Cj -C 4 = constants specifying tangential velocity distributions F = thrust m - mass flow rate M = Mach number P,P Q = static and stagnation pressure, respectively R =gas constant S - swirl number t =time r, T 0 =static and stagnation temperature, respectively u,v,w = velocity components x,y = axial and radial coordinates, respectively yw - swirl Yi,Y 2 = inner and outer radii of the nozzle 7 = specific heat ratio 0 = tangential coordinate p =.density Subscripts A... F = geometric stations IVL = initial-valve line B,C,P =base, cowl, and plug, respectively t,x,y = partial differentiation Introduction R ECENT studies indicate that the introduction of swirl ahead of the combustor in axisymmetric dump com- bustors can have very beneficial effects on the combustion process. Buckley et al. 1 found that swirl both reduced the reattachment length of the combustor flowfield (thereby reducing the overall combustor length needed for good per- formance), and helped eliminate destructive very-low- frequency instabilities. They further concluded that (in the range of swirl intensities of their study), "losses in thrust due to residual swirl, at least to the sonic point of the nozzle, are negligible." Presented as Paper 85-0364 al the AIAA 23rd Aerospace Sciences Meeting, Reno, NV, Jan. 14-17, 1985; received March 12, 1985; revi- sion received July 12, 1985. Copyright © American Institute of Aeronautics and Astronautics, Inc., 1985. All rights reserved, * Graduate Student; presently, Mechanical Engineer, Nuclear Test Division, Lawrence Livermore National Laboratory, Livermore, CA. Member AIAA. tProfessor of Mechanical Engineering, Thermal Sciences and Pro- pulsion Center. Associate Fellow AIAA. ^Professor of Mechanical Engineering, Thermal Sciences and Pro- pulsion Center. Member AIAA. Scharrer and Lilley 2 made five-hole pitot probe measure- ments of the effects of swirl in simulated dump combustors followed by a nozzle. They observed a significant interaction between the swirling flowfield in the simulated combustor and the nozzle flowfield. Their major objective, however, was the measurement of the confined turbulent flow in the simulated combustor, not the nozzle flowfield. Consequently, their nozzles simply were used as downstream blockage components. Both nozzles used in their studies were conven- tional converging nozzles without centerbodies. Conley et al. 3 presented an analytical and experimental in- vestigation of the performance of annular propulsive nozzles without swirl. The present work is an extension of the per- formance prediction methodology developed by Conley et al. to include the effects of swirl introduced in the combustor on the performance of annular propulsive nozzles. A similar study, performed by Dutton, 4 shows trends similar to the results obtained in the present investigation for conventional convergent-divergent nozzles without center- bodies. The present investigation is concerned with con- vergent-divergent nozzles with centerbodies. The objective of the present work is to analytically in- vestigate the effect of swirl on the transonic and supersonic flowfields in annular propulsive nozzles, and to determine the effects of swirl on mass flow rate, thrust, and specific impulse. Performance Prediction Methodology Geometric Model The geometric model considered in this investigation is il- lustrated in Fig. 1. Air enters at station A and flows through a swirler where tangential momentum is transferred to the air to give the desired tangential velocity distribution at station B. Station B is followed by a sudden expansion dump into the combustor inlet at station C. Combustion takes place between stations C and D, where the stagnation temperature T Q rises corresponding to the amount of fuel added and the stagnation pressure P 0 decreases slightly due to friction, mix- ing, and heat addition. The combustion products accelerate in the nozzle to the choked condition at the nozzle throat, station E, after which the flow continues to accelerate super- sonically from station E to F. The concern in the present study was the effect of swirl in- duced in the swirler on the performance of the nozzle (i.e., mass flow rate, thrust, and specific impulse). Consequently, the swirler, sudden expansion dump, and combustor are not modeled in detail in the present analysis. Emphasis is placed on the nozzle flowfield. However, the swirler is the source of the swirl in the flowfield. In the present analysis, the swirl