Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T23:48:40.722Z Has data issue: false hasContentIssue false
  • English
  • Français

Instability of a free-shear layer in the vicinity of a viscosity-stratified layer

Published online by Cambridge University Press:  11 July 2014

Kirti Chandra Sahu  and
Rama Govindarajan
Kirti Chandra Sahu*
Affiliation:
Department of Chemical Engineering, Indian Institute of Technology Hyderabad, Yeddumailaram 502205, India
Rama Govindarajan
Affiliation:
TIFR Centre for Interdisciplinary Sciences, Tata Institute of Fundamental Research Narsingi, Hyderabad 500075, India
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability of a mixing layer made up of two miscible fluids, with a viscosity-stratified layer between them, is studied. The two fluids are of the same density. It is shown that unlike other viscosity-stratified shear flows, where species diffusivity is a dominant factor determining stability, species diffusivity variations over orders of magnitude do not change the answer to any noticeable degree in this case. Viscosity stratification, however, does matter, and can stabilize or destabilize the flow, depending on whether the layer of varying velocity is located within the less or more viscous fluid. By making an inviscid model flow with a slope change across the ‘viscosity’ interface, we show that viscous and inviscid results are in qualitative agreement. The absolute instability of the flow can also be significantly altered by viscosity stratification.

JFM classification

Wakes/Jets: Shear layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 752 , 10 August 2014 , pp. 626 - 648
Copyright
© 2014 Cambridge University Press 

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.)

References

Bague, A., Fuster, D., Popinet, S., Scardovelli, R. & Zaleski, S. 2010 Instability growth rate of two-phase mixing layers from a linear eigenvalue problem and an initial-value problem. Phys. Fluids 22, 092104.Google Scholar
Bers, A. 1983 Space–time evolution of plasma instabilities – absolute and convective. In Handbook of Plasma Physics (ed. Rosenbluth, M. N. & Sagdeev, R. Z.), vol. 1, pp. 451–517. North-Holland Publishing Company.Google Scholar
Betchov, R. & Szewczyk, A. 1963 Stability of a shear layer between parallel streams. Phys. Fluids 6 (10), 1391–1396.CrossRefGoogle Scholar
Bhattacharya, P., Manoharan, M. P., Govindarajan, R. & Narasimha, R. 2005 The critical Reynolds number of a laminar incompressible mixing layer from minimal composite theory. J. Fluid Mech. 565, 105–114.Google Scholar
Biancofiore, L. & Gallaire, F. 2012 Counterpropagating Rossby waves in confined plane wakes. Phys. Fluids 24, 074102.Google Scholar
Bishop, C. H. & Heifetz, E. 2000 Apparent absolute instability and the continuous spectrum. J. Atmos. Sci. 57, 3592–3608.Google Scholar
Boeck, T. & Zaleski, S. 2005 Viscous versus inviscid instability of two-phase mixing layers with continuous velocity prole. Phys. Fluids 17, 032106.Google Scholar
Briggs, R. J. 1964 Electron–Stream Interaction with Plasmas, Research Monograph, vol. 29. MIT Press.Google Scholar
Burridge, D. M. & Drazin, P. G. 1969 Comments on stability of pipe Poiseuille flow. Phys. Fluids 12, 264–265.Google Scholar
Carpenter, J. R., Balmforth, N. J. & Lawrence, G. A. 2010 Indetifying unstable modes in stratified shear layers. Phys. Fluids 22, 054104.Google Scholar
Caulfield, C. P. 1994 On the behaviour of symmetric waves in stratified shear layers. J. Fluid Mech. 258, 255–285.Google Scholar
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.Google Scholar
Craik, A. D. D. 1985 Wave Interactions and Fluid Flows. Cambridge University Press.Google Scholar
Criminale, W. O., Jackson, T. L. & Roslin, R. D. 2003 Theory and Compuation of Hydrodynamic Stability. Cambridge University Press.Google Scholar
Drazin, P. G. & Reid, W. H. 1985 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Ern, P., Charru, F. & Luchini, P. 2003 Stability analysis of a shear flow with strongly stratified viscosity. J. Fluid Mech. 496, 295–312.Google Scholar
Govindarajan, R. 2004 Effect of miscibility on the linear instability of two-fluid channel flow. Intl J. Multiphase Flow 30, 1177–1192.Google Scholar
Govindarajan, R., L’vov, S. V. & Procaccia, I. 2001 Retardation of the onset of turbulence by minor viscosity contrasts. Phys. Rev. Lett. 87, 174501.CrossRefGoogle ScholarPubMed
Govindarajan, R. & Sahu, K. C. 2014 Instabilities in viscosity-stratified flows. Annu. Rev. Fluid Mech. 46, 331–353.Google Scholar
Gubbins, D., Sreenivasan, B., Mound, J. & Rost, S. 2011 Melting of the Earth’s inner core. Nature 473, 361–363.CrossRefGoogle ScholarPubMed
Harnik, N., Heifetz, E., Umurhan, O. M. & Lott, F. 2006 A buoyancy–vorticity wave interaction approach to stratified shear flow. J. Atmos. Sci. 65, 2615–2630.Google Scholar
Healey, J. J. 2009 Destabilising effects of confinement on homogeneous mixing layers. J. Fluid Mech. 623, 241–271.Google Scholar
Heifetz, E. & Methven, J. 2005 Relating optimal growth to counterpropagating Rossby waves in shear instability. Phys. Fluids 17, 064107.Google Scholar
Heifetz, E., Reuveni, Y., Gelfgat, A., Kit, E. & Methven, J. 2006 The counterpropagating rossby wave perspective on Kelvin Helmholtz instability as a limiting case of a Rayleigh shear layer with zero width. Phys. Fluids 18, 018101.Google Scholar
Hinch, E. J. 1984 A note on the mechanism of the instability at the interface between two shearing fluids. J. Fluid Mech. 144, 463–465.CrossRefGoogle Scholar
Hooper, A. P. & Boyd, W. G. C. 1983 Shear flow instability at the interface between two fluids. J. Fluid Mech. 128, 507–528.Google Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 156, 151–168.Google Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instability in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537.Google Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in openflows. In Hydrodynamics and Nonlinear Instabilities (ed. Godreche, C. & Manneville, P.), pp. 81–288. Cambridge University Press.Google Scholar
Juniper, M. P. 2006 The effect of confinement on the stability of two-dimensional shear flows. J. Fluid Mech. 565, 171–195.Google Scholar
Kaiktsis, L. & Monkewitz, P. A. 2003 Global destabilization of flow over a backward-facing step. Phys. Fluids 15 (12), 3647–3658.Google Scholar
McIntyre, M. E.2013 Rossby-wave propagation and shear instability (appendix 1). In GEFD Summer School, Cambridge.Google Scholar
Mishra, M., De Wit, A. & Sahu, K. C. 2012 Double diffusive effects on pressure-driven miscible displacement flows in a channel. J. Fluid Mech. 712, 579–597.Google Scholar
Nasr-Azadani, M. M., Hall, B. & Meiburg, E. 2013 Polydisperse turbidity currents propagating over complex topography: comparison of experimental and depth-resolved simulation results. Comput. Geosci. 53, 141–153.Google Scholar
Nasr-Azadani, M. M. & Meiburg, E. 2014 Turbidity currents interacting with three-dimensional seafloor topography. J. Fluid Mech. 745, 409–443.CrossRefGoogle Scholar
Ranganathan, B. T. & Govindarajan, R. 2001 Stabilisation and destabilisation of channel flow by location of viscosity-stratified fluid layer. Phys. Fluids 13 (1), 1–3.Google Scholar
Robinet, J. C. & Dussauge, J. C. 2001 Wall effect on the convective–absolute boundary for the compressible shear layer. Theor. Comput. Fluid Dyn. 15, 143–163.Google Scholar
Sahu, K. C., Ding, H., Valluri, P. & Matar, O. K. 2009 Linear stability analysis and numerical simulation of miscible channel flows. Phys. Fluids 21, 042104.Google Scholar
Sahu, K. C. & Govindarajan, R. 2012 Spatio-temporal linear stability of double-diffusive two-fluid channel flow. Phys. Fluids 24, 054103.Google Scholar
Sahu, K. C. & Matar, O. K. 2011 Three-dimensional convective and absolute instabilities in pressure-driven two-layer channel flow. Intl J. Multiphase Flow 37, 987–993.Google Scholar
Schaflinger, U. 1994 A short note on Squire’s theorem for interfacial instabilities in a stratified flow of two superposed fluids. Fluid Dyn. Res. 14, 223–227.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Selvam, B., Talon, L., Lesshafft, L. & Meiburg, E. 2009 Convective/absolute instability in miscible core-annular flow. Part 2. Numerical simulations and nonlinear global modes. J. Fluid Mech. 618, 323–348.Google Scholar
Usha, R., Tammisola, O. & Govindarajan, R. 2013 Linear stability of miscible two-fluid flow down an incline. Phys. Fluids 25, 104102.CrossRefGoogle Scholar
Vihinen, I., Honohan, A. M. & Lin, S. P. 1997 Image of absolute instability in a liquid jet. Phys. Fluids 9 (11), 3117–3119.Google Scholar
Wilson, H. J. & Rallison, J. M. 1999 Instability of channel flows of elastic liquids having continuously stratified properties. J. Non-Newtonian Fluid Mech. 85, 273–298.Google Scholar
Yecko, P. & Zaleski, S. 2005 Transient growth in two-phase mixing layers. J. Fluid Mech. 528, 43–52.CrossRefGoogle Scholar
Yih, C. S. 1955 Stability of two-dimensional parallel flows for three dimensional disturbances. Q. Appl. Maths 12, 434–435.Google Scholar
Yih, C. S. 1967 Instability due to viscous stratification. J. Fluid Mech. 27, 337–352.Google Scholar