Abstract
The inviscid temporal stability analysis of an unbounded shear layer of two fluids of different density is investigated. Two background velocity profiles are considered: the piecewise-linear profile and the more realistic error-function profile. The disturbance kinetic energy is analyzed to physically understand the mechanism that causes instability. The surface-tension effect is investigated extensively. Surface tension is found to destabilize the neutrally stable waves that exist when surface tension is absent. This surface-tension-induced unstable mode is generally weaker than the dominant mode and extremely less evident when the density and/or viscosity difference increases. Short-wavelength instability is observed with a background viscosity jump at the interface. A comparison between the two velocity profiles is presented. The piecewise-linear profile does not match the more realistic results obtained with the error-function profile in the short wavelength range, especially in nonhomogeneous shear-layer flows; however, the phase-speed results are in a good agreement with those of the error-function profile.
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References
Esch R.E. (1957). The instability of a shear layer between two parallel streams. J. Fluid Mech. 3: 289–303
Miles J.W. (1959). On the generation of surface waves by shear flows. Part 3. J. Fluid Mech. 7: 583–598
Lindsay K.A. (1984). The Kelvin–Helmholtz instability for a viscous interface. Acta Mech. 52: 51–61
Zalosh R.G. (1976). Discretized simulation of vortex sheet evolution with buoyancy and surface tension effects. AIAA J. 14: 1517–1523
Rangel R.H. and Sirignano W.A. (1988). Nonlinear growth of Kelvin-Helmholtz instability: Effect of surface tension and density ratio. Phys. Fluids 31: 1845–1855
Drazin P.G. and Howard L.N. (1962). The instability to long waves of unbounded parallel inviscid flow. J. Fluid Mech. 14: 257–283
Michalke A. (1964). On the inviscid instability of the hyperbolic-tangent velocity profile. J. Fluid Mech. 19: 543–556
Garcia R.V. (1956). Barotropic waves in straight parallel flow with curved velocity profile. Tellus 8: 82–93
Tatsumi T., Gotoh K. and Ayukawa K. (1964). The stability of a free boundary layer at large Reynolds numbers. J. Phys. Soc. Jpn. 19: 1966–1980
Pouliquen O., Chomaz J.M. and Huerre P. (1994). Propagating Holmboe waves at the interface between two immiscible fluids. J. Fluid Mech. 266: 277–302
L.G. Redekopp, Elements of instability theory for environmental flows. In: R. Grimshaw et al. (eds.), Environmental Stratified Flows. Kluwer Academic Publishers (2002) pp. 223–81.
Drazin P.G. and Reid W.H. (1981). Hydrodynamic Stability. 2nd ed. Cambridge University Press, New York, 525 pp
Prandtl L. (1935). The mechanics of viscous fluids. In: Durand W.F. (eds) Aerodynamic Theory. J. Springer, Berlin, pp. 34–208
Tatsumi T. and Gotoh K. (1960). The stability of free boundary layers between two uniform streams. J. Fluid Mech. 7: 433–441
Yih C.-S. (1967). Instability due to viscosity stratification. J. Fluid Mech. 27: 337–352
White F.M. (1991). Viscous Fluid Flow. 2nd ed. McGraw-Hill, New York, 614 pp
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Alabduljalil, S., Rangel, R.H. Inviscid instability of an unbounded shear layer: effect of surface tension, density and velocity profile. J Eng Math 54, 99–118 (2006). https://doi.org/10.1007/s10665-005-9017-y
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DOI: https://doi.org/10.1007/s10665-005-9017-y