Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-28T09:05:06.174Z Has data issue: false hasContentIssue false
  • English
  • Français

Vertically forced stably stratified cavity flow: instabilities of the basic state

Published online by Cambridge University Press:  31 July 2018

Jason Yalim Open the ORCID record for Jason Yalim [Opens in a new window] ,
Juan M. Lopez Open the ORCID record for Juan M. Lopez [Opens in a new window]  and
Bruno D. Welfert
Jason Yalim
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Juan M. Lopez*
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
Bruno D. Welfert
Affiliation:
School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, USA
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear stability of a stably stratified fluid-filled cavity subject to vertical oscillations is determined via Floquet analysis. Retaining the viscous and diffusion terms in the Navier–Stokes–Boussinesq equations, with no-slip velocity boundary conditions, no-flux temperature conditions on the sidewalls and constant temperatures on the top and bottom walls, we find that the instabilities are primarily subharmonic (as is typical in many parametrically forced systems), except for in a few low-forcing-frequency ranges where the instabilities are synchronous. When the viscosity is small, the Floquet modes resemble the inviscid eigenmodes of the unforced problem, except in boundary layers. We establish scaling laws quantifying how viscosity regularizes the degeneracy associated with the inviscid idealization, and how it scales the thickness and intensity of the boundary layers. The product of boundary layer thickness and intensity remains constant with decreasing viscosity, leading to a delta distribution of vorticity on the walls in the limit of zero viscosity. This is in contrast to the zero wall vorticity in the inviscid case.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Instability: Parametric instability Geophysical and Geological Flows: Stratified flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 851 , 25 September 2018 , R6
Check for updates
Copyright
© 2018 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

Benielli, D. & Sommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.Google Scholar
Benjamin, T. B. & Ursell, F. 1954 The stability of the plane free surface of a liquid in vertical periodic motion. Proc. R. Soc. Lond. A 225, 505–515.Google Scholar
Dauxois, T., Joubaud, S., Odier, P. & Venaille, A. 2018 Instabilities of internal gravity wave beams. Annu. Rev. Fluid Mech. 50, 1–28.Google Scholar
Edwards, W. S. & Fauve, S. 1994 Patterns and quasi-patterns in the Faraday experiment. J. Fluid Mech. 278, 125–148.Google Scholar
Kumar, K. & Tuckerman, L. S. 1994 Parametric instability of the interface between two fluids. J. Fluid Mech. 279, 49–68.Google Scholar
McEwan, A. D. 1971 Degeneration of resonantly-excited standing internal gravity waves. J. Fluid Mech. 50, 431–448.Google Scholar
Mercader, I., Batiste, O. & Alonso, A. 2010 An efficient spectral code for incompressible flows in cylindrical geometries. Comput. Fluids 39, 215–224.Google Scholar
Miles, J. & Henderson, D. M. 1990 Parametrically forced surface-waves. Annu. Rev. Fluid Mech. 22, 143–165.Google Scholar
Orlanski, I. 1972 On the breaking of standing internal gravity waves. J. Fluid Mech. 54, 577–598.Google Scholar
Staquet, C. & Sommeria, J. 2002 Internal gravity waves: from instabilities to turbulence. Annu. Rev. Fluid Mech. 34, 559–593.Google Scholar
Thorpe, S. A. 1968 On standing internal gravity waves of finite amplitude. J. Fluid Mech. 32, 489–528.Google Scholar
Yih, C.-S. 1960 Gravity waves in a stratified fluid. J. Fluid Mech. 8, 481–508.Google Scholar