Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-23T08:47:06.513Z Has data issue: false hasContentIssue false
  • English
  • Français

Inviscid linear stability analysis of two vertical columns of different densities in a gravitational acceleration field

Published online by Cambridge University Press:  09 August 2017

Aditya Heru Prathama Open the ORCID record for Aditya Heru Prathama [Opens in a new window]  and
Carlos Pantano
Aditya Heru Prathama*
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
Carlos Pantano
Affiliation:
Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA
*
†Email address for correspondence: ahp2@illinois.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the inviscid linear stability of a vertical interface separating two fluids of different densities and subject to a gravitational acceleration field parallel to the interface. In this arrangement, the two free streams are constantly accelerated, which means that the linear stability analysis is not amenable to Fourier or Laplace solution in time. Instead, we derive the equations analytically by the initial-value problem method and express the solution in terms of the well-known parabolic cylinder function. The results, which can be classified as an accelerating Kelvin–Helmholtz configuration, show that even in the presence of surface tension, the interface is unconditionally unstable at all wavemodes. This is a consequence of the ever increasing momentum of the free streams, as gravity accelerates them indefinitely. The instability can be shown to grow as the exponential of a quadratic function of time.

JFM classification

Geophysical and Geological Flows: Baroclinic flows Instability: Instability
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , R4
Check for updates
Copyright
© 2017 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Chandrasekhar, S. 1981 Hydrodynamic and Hydromagnetic Stability. Dover.Google Scholar
Drazin, P. G. & Reid, W. H. 1982 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gat, I., Matheou, G., Chung, D. & Dimotakis, P. E. 2016 Acceleration-driven variable-density turbulent flow. VIIIth International Symposium on Stratified Flows, 1 (1). Available at: http://escholarship.org/uc/item/61d722q6.Google Scholar
Gat, I., Matheou, G., Chung, D. & Dimotakis, P. E. 2017 Incompressible variable-density turbulence subject to an external acceleration field. J. Fluid Mech. (in press).Google Scholar
Olson, B. J., Larsson, J., Lele, S. K. & Cook, A. W. 2011 Nonlinear effects in the combined Rayleigh–Taylor/Kelvin–Helmholtz instability. Phys. Fluids 23, 114107.Google Scholar
Rayleigh, Lord 1883 Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177.Google Scholar
Richtmyer, R. D. 1960 Taylor instability in shock acceleration of compressible fluids. Commun. Pure Appl. Maths 13 (2), 297–319.CrossRefGoogle Scholar
Sandoval, D. L.1995 The dynamics of variable-density turbulence. PhD thesis, University of Washington, Seattle, WA.CrossRefGoogle Scholar
Sharp, D. H. 1984 An overview of Rayleigh–Taylor instability. Physica D 12, 3–18.CrossRefGoogle Scholar
Squire, H. B. 1933 On the stability of three-dimensional disturbances of viscous flow between parallel walls. Proc. R. Soc. Lond. A 142 (621‐8), 129–155.Google Scholar
Taylor, G. 1950 The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. A 201, 192–196.Google Scholar