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Propagating Holmboe waves at the interface between two immiscible fluids

Published online by Cambridge University Press:  26 April 2006

O. Pouliquen ,
J. M. Chomaz  and
P. Huerre
O. Pouliquen
Affiliation:
Laboratoire d’hydrodynamique (LadHyX, URA 317), Ecole Polytechnique, F-91128 Palaiseau, France
J. M. Chomaz
Affiliation:
Laboratoire d’hydrodynamique (LadHyX, URA 317), Ecole Polytechnique, F-91128 Palaiseau, France
P. Huerre
Affiliation:
Laboratoire d’hydrodynamique (LadHyX, URA 317), Ecole Polytechnique, F-91128 Palaiseau, France
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Abstract

The instability of the surface of discontinuity separating two streams of immiscible constant-density fluids is studied experimentally and theoretically near onset when surface tension effects are significant. Following Thorpe's original idea, a tube filled with two immiscible fluids is tilted at an angle and returned to its horizontal position to produce a nearly constant velocity difference between both streams that can be varied continuously across threshold. In order to control the wavenumber near onset, the flow is spatially forced by periodically distributing small obstacles on the upper side of the tank. When the kinematic viscosities of each fluid are nearly equal, ones observes two counter-propagating waves of equal amplitude, which cannot be explained from a vortex sheet model. A linear stability analysis of a density discontinuity embedded within a piecewise-linear velocity profile demonstrates that such waves are Holmboe modes associated with the diffusive layers above and below the interface. Good agreement is obtained between the measured and predicted values of the critical velocity difference, propagation velocity and growth factors of the waves. The instability analysis of the asymmetric velocity profile reveals that the breaking of reflectional symmetry gives rise to a single propagating wave near onset. When the kinematic viscosities of each fluid differ, the first destabilized wave is observed to propagate in the same direction as the less-viscous fluid, in agreement with the theoretical results, and the dominant direction of propagation can be manipulated by adjusting the viscosities accordingly.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 266 , 10 May 1994 , pp. 277 - 302
Copyright
© 1994 Cambridge University Press

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References

Balsa, T. F. 1987 On the spatial instability of piecewise linear free shear layers. J. Fluid Mech. 174, 553–563.Google Scholar
Betchov, R. & Szewczyk, A. 1963 Stability of a shear layer between parallel streams. Phys. Fluids 6, 1391–1396.Google Scholar
Browand, F. K. & Winant, C. D. 1973 Laboratory observations of shear layer instability in a stratified fluid. Boundary-Layer Met. 5, 67–77.Google Scholar
Delisi, D. & Corcos, G. M. 1973 A study of internal waves in a wind tunnel. Boundary-Layer Met. 5, 121–137.Google Scholar
Drazin, P. G. & Howard, L. N. 1966 Hydrodynamic stability of parallel flow of inviscid fluids. Adv. Appl. Mech. 9, 1–89.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Hazel, P. 1972 Numerical studies of the stability of inviscid parallel shear flows. J. Fluid Mech. 51, 39–62.Google Scholar
Ho, C. M. & Huang, L. S. 1982 Subharmonics and vortex merging in mixing layers. J. Fluid Mech. 119, 443–473.Google Scholar
Ho, C. M. & Huerre, P. 1984 Perturbed free shear layers. Ann. Rev. Fluid Mech. 16, 365–424.Google Scholar
Holmboe, J. 1962 On the behavior of symmetric waves in stratified shear layers. Geofys. Publ. 24, 67–113.Google Scholar
Howard, L. N. & Maslowe, S. A. 1973 Stability of stratified shear flows. Boundary-Layer Met. 4, 511–523.Google Scholar
Kolodner, P., Passner, A., Surko, C. M. & Walden, R. W. 1986 Onset of oscillatory convection in a binary fluid mixture. Phys. Rev. Lett. 56, 2661–2624.Google Scholar
Koop, C. G. & Browand, F. K. 1979 Instability and turbulence in a stratified fluid with shear. J. Fluid Mech. 93, 135–159.Google Scholar
Lawrence, G. A., Browand, F. K. & Redekopp, L. G. 1991 The stability of a sheared density interface. Phys. Fluids A 3, 2360–2370.Google Scholar
Lawrence, G. A., Lasheras, J. C. & Browand, F. K. 1987 Shear instabilities in stratified flow. Proc. Third Intl Symp. on Stratified Flows, February 2–5, 1987, Pasadena, California.
Maslowe, S. A. & Kelly, R. E. 1971 Inviscid instability of an unbounded heterogeneous shear layer. J. Fluid Mech. 48, 405–415.Google Scholar
Maslowe, S. A. & Thompson, J. M. 1971 Stability of a stratified free shear layer. Phys. Fluids A 14, 453–458.Google Scholar
Maxworthy, T. & Browand, F. K. 1975 Experiments in rotating and stratified flows: oceanographic applications. Ann. Rev. Fluid Mech. 7, 273–305.Google Scholar
Pouliquen, O., Chomaz, J. M., Huerre, P. & Tabeling, P. 1992 Wavenumber selection and phase solitons in spatially forced temporal mixing layers. Phys. Rev. Lett. 68, 2596–2599.Google 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. R. Soc. Lond. 174, 935–985.Google Scholar
Scotti, R. S. & Corcos, G. M. 1972 An experiment on the stability of small disturbances in a stratified shear layer. J. Fluid Mech. 52, 499–528.Google Scholar
Smyth, W. D., Klaasen, G. P. & Peltier, W. R. 1988 Finite amplitude Holmboe waves. Geophys. Astrophys. Fluid Dyn. 43, 181–222.Google Scholar
Smyth, W. D. & Peltier, W. R. 1989 The transition between Kelvin–Helmholtz and Holmboe instability: an investigation of the overreflection hypothesis. J. Atmos. Sci. 46, 3698–3720.Google Scholar
Smyth, W. D. & Peltier, W. R. 1990 Three-dimensional primary instabilities of a stratified dissipative, parallel flow. Geophys. Astrophys. Fluid Dyn. 52, 249–261.Google Scholar
Smyth, W. D. & Peltier, W. R. 1991 Instability and transition in finite-amplitude Kelvin–Helmholtz and Holmboe waves. J. Fluid Mech. 228, 387–415.Google Scholar
Surko, C. M. & Kolodner, P. 1987 Oscillatory traveling-wave convection in a finite container. Phys. Rev. Lett. 58, 2055–2058.Google Scholar
Thorpe, S. A. 1968 A method of producing a shear flow in a stratified fluid. J. Fluid Mech. 32, 693–704.Google Scholar
Thorpe, S. A. 1969 Experiments on the instability of stratified shear flows: immiscible fluids. J. Fluid Mech. 39, 25–48.Google Scholar
Thorpe, S. A. 1973a Turbulence in stably stratified fluids: a review of laboratory experiments. Boundary-Layer Met. 5, 95–119.Google Scholar
Thorpe, S. A. 1973b Experiments on instability and turbulence in a stratified shear flow. J. Fluid Mech. 61, 731–751.Google Scholar
Weissman, M. A. 1979 Non linear wave packets in the Kelvin–Helmholtz instability. Phil. Trans. R. Soc. Lond. A 290, 639–685.Google Scholar