Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T19:46:05.109Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous effects on Kelvin–Helmholtz instability in a channel

Published online by Cambridge University Press:  23 June 2011

H. KIM ,
J. C. PADRINO  and
D. D. JOSEPH
H. KIM*
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
J. C. PADRINO
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
D. D. JOSEPH
Affiliation:
Department of Aerospace Engineering and Mechanics, University of Minnesota, Minneapolis, MN 55455, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The effects of viscosity on Kelvin–Helmholtz instability in a channel are studied using three different theories; a purely irrotational theory based on the dissipation method, an exact rotational theory and a hybrid irrotational–rotational theory. These new results are compared with previous results from a viscous irrotational theory. An analysis of the neutral state is conducted and its predictions are compared with experimental results related to the transition from a stratified-smooth to a stratified-wavy or slug flow. For values of the gas fraction greater than about 0.20, there is an interval of velocity differences for which the flow is unstable for an interval of wavenumbers between two cutoff wavenumbers, k− and k+. For unstable flows with a velocity difference above that interval or with gas fractions less than 0.20, k− = 0. The maximum critical relative velocity that determines the onset of instability can be found when the kinematic viscosity of the gas and liquid are the same. This critical value is surprisingly achieved when both fluids are inviscid. The neutral curves from the analyses of potential flow of viscous fluids and the hybrid method, the only theories that account for the viscosity of both fluids in this work, indicate that the critical velocity does not change with the viscosity ratio when the kinematic viscosity of the liquid is greater than a critical value. For smaller liquid viscosities, the critical relative velocity decreases.

JFM classification

Instability: Instability Interfacial Flows (free surface): Interfacial Flows (free surface)
Type
Papers
Information
Journal of Fluid Mechanics , Volume 680 , 10 August 2011 , pp. 398 - 416
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

This paper is dedicated to the memory of Daniel D. Joseph (1929–2011)

References

REFERENCES

Adham-Khodaparast, K., Kawaji, M. & Antar, B. N. 1995 The Rayleigh–Taylor and Kelvin–Helmholtz stability of a viscous liquid–vapor interface with heat and mass transfer. Phys. Fluids 7 (2), 359–364.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamc and Hydromagnetic Stability. Oxford University Press. (Reprinted by Dover in 1981).Google Scholar
Funada, T. & Joseph, D. D. 2002 Viscous potential flow analysis of capillary instability. Intl J. Multiphase Flow 28, 1459–1478.CrossRefGoogle Scholar
Funada, T. & Joseph, D. D. 2001 Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel. J. Fluid Mech. 445, 263–283.CrossRefGoogle Scholar
Johns, L. & Narayanan, R. 2002 Interfacial Instability. Springer.Google Scholar
Joseph, D. D., Funada, T. & Wang, J. 2007 Potential Flows of Viscous and Viscoelastic Fluids. Cambridge University Press.CrossRefGoogle Scholar
Kordyban, E. & Ranov, T. 1970 Mechanism of slug formation in horizontal two-phase flow. Trans. ASME J. Basic Engng 92, 857–864.CrossRefGoogle Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press. (Reprinted in 1993).Google Scholar
Levich, V. 1949 The motion of bubbles at high Reynolds numbers. Zh. Eksp. Teor. Fiz. 19, 18–24.Google Scholar
Li, X. & Tankin, R. S. 1991 On the temporal instability of a two-dimensional viscous liquid sheet. J. Fluid Mech. 226, 425–443.CrossRefGoogle Scholar
Lin, P. Y. & Hanratty, T. J. 1986 Prediction of the initiation of slugs with linear stability theory. Intl J. Multiphase Flow 12, 79–98.CrossRefGoogle Scholar
Moore, D. 1959 The rise of a gas bubble in a viscous liquid. J. Fluid Mech. 6, 113–130.CrossRefGoogle Scholar
Moore, D. 1963 The boundary layer on a spherical gas bubble. J. Fluid Mech. 16, 161–176.CrossRefGoogle Scholar
Padrino, J. & Joseph, D. 2007 Correction of Lamb's dissipation calculation for the effects of viscosity on capillary–gravity waves. Phys. Fluids 19, 082105.CrossRefGoogle Scholar
Stokes, G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. IX, 8–106.Google Scholar
Wallis, G. B. & Dobson, J. 1973 The onset of slugging in horizontal stratified air–water flow. Intl J. Multiphase Flow 1, 173–193.CrossRefGoogle Scholar
Wang, J., Joseph, D. & Funada, T. 2005 a Pressure corrections for potential flow analysis of capillary instability of viscous fluids. J. Fluid Mech. 522, 383–394.CrossRefGoogle Scholar
Wang, J., Joseph, D. & Funada, T. 2005 b Viscous contributions to the pressure for potential flow analysis of capillary instability of two viscous fluids. Phys. Fluids 17, 052105.CrossRefGoogle Scholar