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Abstract
An analytical model has been developed and an approximate solution for the free vortex swirling supersonic flow through nozzles. Earlier work by Mager, is potential flow based. In the present model swirl number has been redefined to confine the flow properties within the valid range. Flow properties are always determined by the total conditions (P0 and a0), nozzle profile and swirl intensity as in the case of one-dimensional flow. Integral form of the continuity equation has been used in order to satisfy the boundary conditions, since radial velocity component is ignored. For free vortex case, it has been shown that there exists a minimum radius along the length of the nozzle at which the density is zero and flow seems to be void causing reasonable blockage to the flow at throat. In defining the non-dimensional swirl number (β), axial velocity at each section was employed for non-dimensionalization as compared to Mager’s definition of using a global constant to find swirl number (α). The variations of performance parameters of the nozzle such as co-efficient of discharge (Cd) thrust efficiency (ηs) and impulse efficiency (ηI) and flow field parameters such as density, pressure, velocity etc. are presented as a function of swirl numbers (α, β) and compared with the available experimental data. The swirl numbers (α, β) are also correlated.
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References
- Carpenter, P.W., "A General One-dimensional Theory of Compressible Inviscid Swirling Flows in Nozzles", Aeronautical Quarterly , Vol.27, pp.201-216.
- Mager, A., "Approximate Solution of Isentropic Swirling Flow Through a Nozzle", ARS Journal, Vol.31, No.8, August 1961, pp.1140-1148.
- Hsu, C.T., "Swirling Nozzle Flow Equations from Crocco’s Relation", AIAA Journal, Vol.9, 1971, pp.1866-1868.
- Swithenbank, J. and Scotter, G., "Vortex Generation in Solid Propellant Rockets", AIAA Journal, Vol.2, December 1964, p.1297.
- Glick, R.L. and Kilgore, M.S., "Effect of Specific Heat Ratio on Mass Flow for Swirling Nozzle Flow", Journal of Spacecraft and Rockets, Vol.4, No.8, August 1967, pp.1908-1909.
- Bussi, G., "Analisi Numerica di Flussi Vorticosi in Ugell", Instituto di Macchine e Motori per Aeromobili, Torino, Pub. No. 166, 1974.
- Batson, J.L. and Sforzini, R.h., "Swirling Flow Through Nozzle", Journal of Spacecraft and Rockets, Vol.7, February 1970, pp.159-163.
- Donaldson, C.D. and Snedeker, R.S., "Experimental Investigation of the Structure of Vortices in Simple Cylindrical Vortex Chambers", Rept.47, 1962, Aeronautical Research Association of Princeton, Princeton, New Jersy, USA.
- So, K.L., "Vortex Phenomena in a Conical Diffuser", AIAA Journal, Vol.5, No.6, June 1967, pp.10721078.
- King, W.S., "A Theoretical Investigation of Swirling Flows Through a Nozzle", PhD. Dissertation, 1967, UCLA, Los Angeles, California, USA.
- Bastress, E.K., "Interior Ballistics of Spinning SolidPropellant Rockets", Journal os Spacecraft and Rockets, Vol.2, No.3, May-June 1965, pp.455-457.
- Manda, L., "Spin Effects on Rocket Nozzle Performance", Journal of Spacecraft and Rockets, Vol.3, No.12, November 1966, pp.1695-1696.
- Carpenter, P.W. and Johannesen, N.H., "An Extension of One-Dimensional Theory to Inviscid Swirling Flow Through Chocked Nozzles", Aeronautical Quarterly Journal, Vol.26, May 1975, pp.71-87.
- Lewellen, W.S., Burns, W.J. and Strickland, H.J., "Transonic Swirling Flow", AIAA Journal, Vol.7, November 1969, pp.1290-1297.
- Gostintsev, Yu. A. et al., "On the Structure of a Supersonic Swirling Gas Jet with Underwexpanded Outflow (in Russian)", Izv Akad Nauk SSSR, Mekh Zhidk Gaza, No.5, 1969, p.158.
- Gostintsev, Yu.A., "The Output Characteristics of a Nozzle Issuing Spiralling Gas Flow (in Russian)", Izv Akad Nauk SSSR, Mekh Zhidk Gaza, No.4, 1969, p.153.
- Moore, A.W., "Transonic Flows in Nozzles of Arbitrary Shape and Swirling Flows in Nozzles and Jets", Ph.D. Thesis, University of Manchester, 1964.
- Smith, R., "An Investigation of the Effects of Swirl on Some Flows at Transonic, Supersonic and Hypersonic Speeds", Ph.D. Thesis, University of Manchester, 1971.
- Armitage, J.V., "The Lax-Wendroff Method Applied to Axi-Symmetric Swirl Flow", ITS Blanch Anniv., Vol.21, N67-39462, February 1967, pp.1-20.
- Dutton, J.C., "Swirling Supersonic Nozzle Flow", J. Propulsion and Power, Vol.3, July, 1987, pp.342-349.
- Cline, M.C., "VNAP : A Computer Program for Computation of Two-Dimensional, Time-Dependent, Compressible, Viscous, Internal Flow", Los Alamos Scientific Laboratory, NM Report LA-7326, November 1978.
- Chang., Chau-Lyan., Merkle. And Charles., "Viscous Swirling Nozzle Flow", AIAA Journal, Aerospace Sciences Meeting, 1989.
- Binnie, A.M., Hookings, G.A. and Kamel, M.Y.M., "The Flow of Swirling Water Through a Convergent Divergent Nozzle", J. Fluid Mechanics, Vol.3, December 1957, pp.261-274.
- Iserland, K., "Untersuchungen uber die Umelenkung eines freien Luftstrahis mit Hilfe von Drall", Mitteilungen aus dem Institut for Aerodynamic ETH Zurich, Verlag Leemann Zurich Nr. 25, 1958.
- Massier, P.F., "Axisymmetric Steady Flow of a Compressible Fluid Through a Convergent-Divergent Nozzle without External Heat Transfer", Jet Propulsion Laboratory Space Programs Summary, No.37-33, Vol.4, June 1965, pp.133-141.
- Massier, P.F., "Swirling Flow of Argon Through an Axisymmetric Convergent-Divergent Nozzle", Jet Propulsion Laboratory Space Programs Summary, No.37-24, Vol.4, August 1965, pp.149-163.
- Massier, P.F., "Thrust Comparisons for Swirling and Non-Swirling Flows of Argon Through a Convergent-Divergent Nozzle as Determined from Wall Static Pressure Measurements", Jet Propulsion Laboratory Space Programs Summary, No.37-35, Vol.4, October 1965, pp.161-165.
- Parkinson, R.C., "The Effect on the Mass-Flow of Swirl in a Supersonic Nozzle", Spaceflight, Vol.9, 1967, pp.360.
- Dutton, J.C., "Correlation of Nozzle Performance Degradation Due to Swirl", J. Propulsion and Power, Vol.5, No.1, January 1989, pp.126-128.
References
Carpenter, P.W., "A General One-dimensional Theory of Compressible Inviscid Swirling Flows in Nozzles", Aeronautical Quarterly , Vol.27, pp.201-216.
Mager, A., "Approximate Solution of Isentropic Swirling Flow Through a Nozzle", ARS Journal, Vol.31, No.8, August 1961, pp.1140-1148.
Hsu, C.T., "Swirling Nozzle Flow Equations from Crocco’s Relation", AIAA Journal, Vol.9, 1971, pp.1866-1868.
Swithenbank, J. and Scotter, G., "Vortex Generation in Solid Propellant Rockets", AIAA Journal, Vol.2, December 1964, p.1297.
Glick, R.L. and Kilgore, M.S., "Effect of Specific Heat Ratio on Mass Flow for Swirling Nozzle Flow", Journal of Spacecraft and Rockets, Vol.4, No.8, August 1967, pp.1908-1909.
Bussi, G., "Analisi Numerica di Flussi Vorticosi in Ugell", Instituto di Macchine e Motori per Aeromobili, Torino, Pub. No. 166, 1974.
Batson, J.L. and Sforzini, R.h., "Swirling Flow Through Nozzle", Journal of Spacecraft and Rockets, Vol.7, February 1970, pp.159-163.
Donaldson, C.D. and Snedeker, R.S., "Experimental Investigation of the Structure of Vortices in Simple Cylindrical Vortex Chambers", Rept.47, 1962, Aeronautical Research Association of Princeton, Princeton, New Jersy, USA.
So, K.L., "Vortex Phenomena in a Conical Diffuser", AIAA Journal, Vol.5, No.6, June 1967, pp.10721078.
King, W.S., "A Theoretical Investigation of Swirling Flows Through a Nozzle", PhD. Dissertation, 1967, UCLA, Los Angeles, California, USA.
Bastress, E.K., "Interior Ballistics of Spinning SolidPropellant Rockets", Journal os Spacecraft and Rockets, Vol.2, No.3, May-June 1965, pp.455-457.
Manda, L., "Spin Effects on Rocket Nozzle Performance", Journal of Spacecraft and Rockets, Vol.3, No.12, November 1966, pp.1695-1696.
Carpenter, P.W. and Johannesen, N.H., "An Extension of One-Dimensional Theory to Inviscid Swirling Flow Through Chocked Nozzles", Aeronautical Quarterly Journal, Vol.26, May 1975, pp.71-87.
Lewellen, W.S., Burns, W.J. and Strickland, H.J., "Transonic Swirling Flow", AIAA Journal, Vol.7, November 1969, pp.1290-1297.
Gostintsev, Yu. A. et al., "On the Structure of a Supersonic Swirling Gas Jet with Underwexpanded Outflow (in Russian)", Izv Akad Nauk SSSR, Mekh Zhidk Gaza, No.5, 1969, p.158.
Gostintsev, Yu.A., "The Output Characteristics of a Nozzle Issuing Spiralling Gas Flow (in Russian)", Izv Akad Nauk SSSR, Mekh Zhidk Gaza, No.4, 1969, p.153.
Moore, A.W., "Transonic Flows in Nozzles of Arbitrary Shape and Swirling Flows in Nozzles and Jets", Ph.D. Thesis, University of Manchester, 1964.
Smith, R., "An Investigation of the Effects of Swirl on Some Flows at Transonic, Supersonic and Hypersonic Speeds", Ph.D. Thesis, University of Manchester, 1971.
Armitage, J.V., "The Lax-Wendroff Method Applied to Axi-Symmetric Swirl Flow", ITS Blanch Anniv., Vol.21, N67-39462, February 1967, pp.1-20.
Dutton, J.C., "Swirling Supersonic Nozzle Flow", J. Propulsion and Power, Vol.3, July, 1987, pp.342-349.
Cline, M.C., "VNAP : A Computer Program for Computation of Two-Dimensional, Time-Dependent, Compressible, Viscous, Internal Flow", Los Alamos Scientific Laboratory, NM Report LA-7326, November 1978.
Chang., Chau-Lyan., Merkle. And Charles., "Viscous Swirling Nozzle Flow", AIAA Journal, Aerospace Sciences Meeting, 1989.
Binnie, A.M., Hookings, G.A. and Kamel, M.Y.M., "The Flow of Swirling Water Through a Convergent Divergent Nozzle", J. Fluid Mechanics, Vol.3, December 1957, pp.261-274.
Iserland, K., "Untersuchungen uber die Umelenkung eines freien Luftstrahis mit Hilfe von Drall", Mitteilungen aus dem Institut for Aerodynamic ETH Zurich, Verlag Leemann Zurich Nr. 25, 1958.
Massier, P.F., "Axisymmetric Steady Flow of a Compressible Fluid Through a Convergent-Divergent Nozzle without External Heat Transfer", Jet Propulsion Laboratory Space Programs Summary, No.37-33, Vol.4, June 1965, pp.133-141.
Massier, P.F., "Swirling Flow of Argon Through an Axisymmetric Convergent-Divergent Nozzle", Jet Propulsion Laboratory Space Programs Summary, No.37-24, Vol.4, August 1965, pp.149-163.
Massier, P.F., "Thrust Comparisons for Swirling and Non-Swirling Flows of Argon Through a Convergent-Divergent Nozzle as Determined from Wall Static Pressure Measurements", Jet Propulsion Laboratory Space Programs Summary, No.37-35, Vol.4, October 1965, pp.161-165.
Parkinson, R.C., "The Effect on the Mass-Flow of Swirl in a Supersonic Nozzle", Spaceflight, Vol.9, 1967, pp.360.
Dutton, J.C., "Correlation of Nozzle Performance Degradation Due to Swirl", J. Propulsion and Power, Vol.5, No.1, January 1989, pp.126-128.