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Non-modal stability of round viscous jets

Published online by Cambridge University Press:  30 January 2013

S. A. Boronin ,
J. J. Healey  and
S. S. Sazhin
S. A. Boronin
Affiliation:
Sir Harry Ricardo Laboratories, School of Computing, Engineering and Mathematics, University of Brighton, Brighton BN2 4GJ, UK
J. J. Healey
Affiliation:
Department of Mathematics, Keele University, Keele, Staffs ST5 5BG, UK
S. S. Sazhin
Affiliation:
Sir Harry Ricardo Laboratories, School of Computing, Engineering and Mathematics, University of Brighton, Brighton BN2 4GJ, UK
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Abstract

Hydrodynamic stability of round viscous fluid jets is considered within the framework of the non-modal approach. Both the jet fluid and surrounding gas are assumed to be incompressible and Newtonian; the effect of surface capillary pressure is taken into account. The linearized Navier–Stokes equations coupled with boundary conditions at the jet axis, interface and infinity are reduced to a system of four ordinary differential equations for the amplitudes of disturbances in the form of spatial normal modes. The eigenvalue problem is solved by using the orthonormalization method with Newton iterations and the system of least stable normal modes is found. Linear combinations of modes (optimal disturbances) leading to the maximum kinetic energy at a specified set of governing parameters are found. Parametric study of optimal disturbances is carried out for both an air jet and a liquid jet in air. For the velocity profiles under consideration, it is found that the non-modal instability mechanism is significant for non-axisymmetric disturbances. The maximum energy of the optimal disturbances to the jets at the Reynolds number of 1000 is found to be two orders of magnitude larger than that of the single mode. The largest growth is gained by the streamwise velocity component.

JFM classification

Wakes/Jets: Jets Multiphase and Particle-laden Flows: Gas/liquid flow Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 716 , 10 February 2013 , pp. 96 - 119
Copyright
©2013 Cambridge University Press

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References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Batchelor, G. K. & Gill, A. E. 1962 Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14, 529–551.Google Scholar
Butler, K. M. & Farrel, D. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (5), 1637–1650.CrossRefGoogle Scholar
Crua, C., Kennaird, D. A., Sazhin, S. S., Heikal, M. R. & Gold, M. R. 2004 Diesel autoignition at elevated in-cylinder pressures. Intl J. Engine Res. 5 (4), 365–374.CrossRefGoogle Scholar
De Luca, L. 2001 Non-modal growth of disturbances in free-surface flows. In Proceedings of International Conference RDAMM-2001, 6(2).Google Scholar
Duda, J. L. & Vrentas, J. S. 1967 Fluid mechanics of laminar liquid jets. Chem. Engng Sci. 22, 855–869.Google Scholar
Drazin, P. G. & Reid, W. H. 1983 Hydrodynamic Stability, 2nd edn. Cambridge University Press.Google Scholar
Ellingsen, T. 1975 Stability of linear flow. Phys. Fluids 18, 487–488.Google Scholar
Funada, T., Joseph, D. D. & Yamashita, S. 2004 Stability of a liquid jet into incompressible gases and liquids. Intl J. Multiphase Flow 30, 1279–1310.Google Scholar
Garg, V. K. & Rouleau, W. T. 1971 Linear spatial stability of pipe Poiseuille flow. J. Fluid Mech. 54, 113–127.Google Scholar
Gary, J. & Helgason, R. 1970 A matrix method for ordinary differential eigenvalue problems. J. Comput. Phys. 5, 169–187.Google Scholar
Godunov, S. K. 1961 Numerical solution of boundary-value problems for the systems of linear ordinary differential equations. Usp. Mat. Nauk 16 (3), 171–174.Google Scholar
Hoyt, J. W. & Taylor, J. J. 1977 Waves on water jets. J. Fluid Mech. 83, 119–127.Google Scholar
Klingmann, G. B. 1992 On transition due to three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 240, 167–195.CrossRefGoogle Scholar
Landahl, M. L. 1980 A note on algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1959 Fluid Mechanics, 2nd edn. Pergamon.Google Scholar
Lasheras, J. C. & Hopfinger, E. J. 2000 Liquid jet instability and atomization in a coaxial gas stream. Annu. Rev. Fluid Mech. 32, 275–308.Google Scholar
Levin, O., Chernoray, V. G., Löfdahl, L. & Henningson, D. S. 2005 A study of the Blasius wall jet. J. Fluid Mech. 539, 313–347.Google Scholar
Lin, S. P. & Reitz, R. D. 1998 Drop and spray formation from a liquid jet. Annu. Rev. Fluid Mech. 30, 85–105.Google Scholar
Malik, S. V. & Hooper, A. P. 2007 Three-dimensional disturbances in channel flows. Phys. Fluids 19, 052102.Google Scholar
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.Google Scholar
Mayer, W. O. H. 1994 Coaxial atomization of a round liquid jet in a high speed gas stream: A phenomenological study. Exp. Fluids 16, 401–410.Google Scholar
Morris, P. J. 1976 The spatial viscous instability of axisymmetric jets. J. Fluid Mech. 77, 511–529.Google Scholar
van Noorden, T. L., Boomkamp, P. A. M., Knaap, M. C. & Verheggen, T. M. M. 1998 Transient growth in parallel two-phase flow: analogies and differences with single-phase flow. Phys. Fluids 10 (8), 2099–2101.Google Scholar
Rayleigh, J. W. S. 1945 The Theory of Sound, vol. II. 2nd edn. Dover.Google Scholar
Renardy, Y. 1987 The thin-layer effect and interfacial stability in a two-layer Couette flow with similar liquids. Phys. Fluids 30 (6), 1627–1637.Google Scholar
Reshotko, E. & Tumin, A. 2001 Spatial theory of optimal disturbances in a circular pipe flow. Phys. Fluids 13 (4), 991–996.CrossRefGoogle Scholar
Reynolds, O. 1883 An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. 174, 935–982.Google Scholar
Sazhin, S. S., Crua, C., Kennaird, D. A. & Heikal, M. R. 2003 The initial stage of fuel spray penetration. Fuel 82 (8), 875–885.Google Scholar
Sazhin, S. S., Martynov, S. B., Kristyadi, T., Crua, C. & Heikal, M. R. 2008 Diesel fuel spray penetration, heating, evapouration and ignition: modelling versus experimentation. Intl J. Engng Syst. Modelling Simul. 1 (1), 1–19.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Schmid, P. J. 2007 Non-modal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Shkadov, V. Ya. & Sisoev, G. M. 1996 Instability of a two-layer capillary jet. Intl J. Multiphase Flow 22 (2), 363–377.Google Scholar
Söderberg, L. D. & Alfredsson, P. H. 1998 Experimental and theoretical investigations of plane liquid jets. Eur. J. Mech. B/Fluids 17 (5), 689–737.Google Scholar
South, M. J. & Hooper, A. P. 1999 Linear growth in two-fluid plane Poiseuille flow. J. Fluid Mech. 381, 121–139.Google Scholar
Tumin, A. & Reshotko, E. 2001 Spatial theory of optimal disturbances in boundary layers. Phys. Fluids 13 (7), 2097–2104.Google Scholar
Turner, M. R., Healey, J. J., Sazhin, S. S. & Piazzesi, R. 2012a Wave packet analysis and break-up length calculations for an accelerating planar liquid jet. Fluid Dyn. Res. 44, 015503 26pp.Google Scholar
Turner, M. R., Sazhin, S. S., Healey, J. J., Crua, C. & Martynov, S. B. 2012b A breakup model for transient diesel fuel sprays. Fuel 97, 288–305.Google Scholar
Wilkinson, J. H. 1988 Algebraic eigenvalue problem. Clarendon Press.Google Scholar
Yang, H. Q. 1991 Asymmetric instability of a liquid jet. Phys. Fluids 4 (4), 681–689.CrossRefGoogle Scholar
Yecko, P. A. & Zaleski, S. 2005 Transient growth in two-phase mixing layers. J. Fluid Mech. 528, 43–52.Google Scholar