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Entrainment waves in decelerating transient turbulent jets

  • MARK P. B. MUSCULUS (a1)


A simplified one-dimensional partial differential equation for the integral axial momentum flux during the deceleration phase of single-pulsed transient incompressible jets is derived and solved analytically. The wave speed of the derived first-order nonlinear wave equation shows that the momentum flux transient from the deceleration phase propagates downstream at twice the initial jet penetration rate. Transient-jet velocity data from the existing literature is shown to be consistent with this derivation, and an algebraic analytical solution matches the measured timing and decay of axial velocity after the deceleration transient. The solution also shows that a wave of increased entrainment accompanies the deceleration transient as it travels downstream through the jet. In the long-time limit, the peak entrainment rate at the leading edge of this ‘entrainment wave’ approaches an asymptotic value of three times that of the initial steady jet. The rate of approach to the asymptotic behaviour is controlled by the deceleration rate, which suggests that rate-shaping may be tailored to achieve a desired mixing state at a given time after the end of a single-pulsed jet. In the wake of the entrainment wave, the absolute entrainment rate eventually decays to zero. The local injected fluid concentration also decays, however, so that entrainment rate relative to the local concentration of injected fluid remains higher than in the initial steady jet. An analysis of diesel engine fuel-jets is provided as one example of a transient-jet application in which the considerable increase in the mixing rate after the deceleration phase has important implications.


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Abramovich, G. N. 1963 Chapter 5. Jet of an incompressible fluid in a coflowing external stream. In The Theory of Turbulent Jets. MIT Press.
Akagawa, H., Miyamoto, T., Harada, A., Sasaki, S., Shimazaki, N., Hashizume, T. & Tsujimura, K. 1999 Approaches to solve problems of the premixed lean diesel combustion. SAE Trans. 108 (3), 120132.
Atassi, N., Boree, J. & Charnay, G. 1993 Transient behaivor of an axisymmetric turbulent jet. Appl. Sci. Res. 51, 137142.
Borée, J., Atassi, N. & Charnay, G. 1996 Phase averaged velocity field in an axisymmetric jet subject to a sudden velocity decrease. Exp. Fluids 21, 447456.
Borée, J., Atassi, N., Charnay, G. & Taubert, L. 1997 Measurements and image analysis of the turbulent field in an axisymmetric jet subject to a sudden velocity decrease. Exp. Thermal Fluid Sci. 14, 4551.
Breidenthal, R. 1986 The turbulent exponential jet. Phys. Fluids 29, 23462347.
Breidenthal, R. E. 2008 The effect of acceleration on turbulent entrainment. Phys. Scripta T132, 15.
Bremhorst, K. 1979 Unsteady subsonic turbulent jets. In Recent Developments in Theoretical and Experimental Fluid Mechanics: Compressible and Incompressible Flows (ed. Müller, U., Roesner, K. G. & Schmidt, B.) Springer, pp. 480520.
Bremhorst, K. & Hollis, P. G. 1990 Velocity field of an axisymmetric pulsed, subsonic air jet. AIAA J. 28, 20432049.
Bruneaux, G. 2005 Mixing process in high pressure diesel jets by normalized laser induced exciplex fluorescence. Part I. Free jet. SAE Trans. 114 (3), 14441461.
Cossali, G. E., Geria, A., Coghe, A. & Brunello, G. 1996 Effect of gas density and temperature on air entrainment in a transient diesel spray. SAE Trans. 105 (3), 12931301.
Crow, S. C. & Champagne, F. H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48, 547591.
Doudou, A. 2005 Turbulent flow study of an isothermal diesel spray injected by a common rail system. Fuel 84, 287298.
Genzale, C. L., Reitz, R. D. & Musculus, M. P. B. 2008 Effects of piston bowl geometry on mixture development and late-injection low-temperature combustion in a heavy-duty diesel engine. SAE International Journal of Engines 1 (1), 913937.
Hill, P. G. & Ouellette, P. 1999 Transient turbulent gaseous fuel jets for diesel engines. J. Fluids Engng 121, 93101.
Hinze, J. O. 1975 Chapter 6. Free turbulent shear flows, In Turbulence, 2nd ed. McGraw-Hill.
Iyer, V. & Abraham, J. 2003 An evaluation of a two-fluid Eulerian-liquid Eulerian-gas model for diesel sprays. J. Fluids Engng 125, 660669.
Johari, H. & Paduano, R. 1997 Dilution and mixing in an unsteady jet. Exp. Fluids 23, 272280.
Joshi, A. & Schreiber, W. 2006 An experimental examination of an impulsively started incompressible turbulent jet. Exp. Fluids 40, 156160.
Kashdan, J. T., Mendez, S. & Bruneaux, G. 2007 On the origin of unburned hydrocarbon emissions in a wall-guided, low NO2 diesel combustion system. SAE Trans. 116 (4), 234257.
Kato, S. M., Groenewegen, B. C. & Breidenthal, R. E. 1987 Turbulent mixing in nonsteady jets. AIAA J. 25, 165168.
Kevorkian, J. 1996 Partial Differential Equations. Chapman & Hall.
Kim, T. & Ghandhi, J. B. 2001 Quantitative two-dimensional fuel vapour concentration measurements in an evaporating diesel spray using the exciplex fluorescence method. SAE Trans. 110 (3), 21652181.
Lakshminarasimhan, K., Clemens, N. T. & Ezekoye, O. A. 2006 Characteristics of strongly-forced turbulent jets and non-premixed jet flames. Exp. Fluids 41, 523542.
Levandosky, J. 2008 Stanford MA220 lecture notes.
Musculus, M. P. B. & Kattke, K. 2009 Entrainment waves in diesel jets. to appear in SAE International Journal of Engines 2.
Musculus, M. P. B., Lachaux, T., Pickett, L. M. & Idicheria, C. A. 2007 End-of-injection over-mixing and unburned hydrocarbon emissions in low-temperature-combustion diesel engines. SAE Trans. 116 (3), 515541.
Naber, J. D. & Siebers, D. L. 1996 Effects of gas density and vaporization on penetration and dispersion of diesel sprays. SAE Trans. 105 (3), 82111.
Nathan, G. J., Mi, J., Alwahabi, Z. T., Newbold, G. J. R. & Nobes, D. S. 2006 Impacts of a jet's exit flow pattern on mixing and combustion performance. Prog. Energy Combust. Sci. 32, 496538.
Pickett, L. M. 2009 Engine combustion network.
Reynolds, W. C., Parekh, D. E., Juvet, P. J. D. & Lee, M. J. D. 2003 Bifurcating and blooming jets. Annu. Rev. Fluid Mech. 35, 295315.
Ricou, F. P. & Spalding, D. B. 1961 Measurements of entrainment by axisymmetrical turbulent jets. J. Fluid Mech. 11, 2132.
Sangras, R., Kwon, O. C. & Faeth, G. M. 2002 Self-preserving properties of unsteady round nonbuoyant turbulent starting jets and puffs in still fluids. ASME J. Heat Trans. 124, 460469.
Schlichting, H. 1979 Chapter XXIV. Free turbulent flows: jets and wakes. In Boundary Layer Theory, 7th ed. McGraw-Hill.
Witze, P. O. 1980 The impulsively started incompressible turbulent jet. Energy Rep. SAND80-8617. Sandia National Laboratories.
Witze, P. O. 1983 Hot-film anemometer measurements in a starting turbulent jet. AIAA J. 21, 308309.
Wu, K.-J., Santavicca, D. A. & Bracco, F. V. 1984 LDV measurements of drop velocity in diesel-type sprays. AIAA J. 22, 12631270.
Zhang, Q. & Johari, H. 1996 Effects of acceleration on turbulent jets. Phys. Fluids 8, 21852195.
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Entrainment waves in decelerating transient turbulent jets

  • MARK P. B. MUSCULUS (a1)


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