Hostname: page-component-6dbcb7884d-wnk8j Total loading time: 0 Render date: 2025-02-14T08:29:38.537Z Has data issue: false hasContentIssue false
  • English
  • Français

Linear and nonlinear dynamics of a differentially heated slot under gravity modulation

Published online by Cambridge University Press:  26 April 2006

A. Farooq  and
G. M. Homsy
A. Farooq
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
G. M. Homsy
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we consider the effect of sinusoidal gravity modulation of size ε on a differentially heated infinite slot in which a vertical temperature stratification is imposed on the walls. The slot problem is characterized by a Rayleigh number, Prandtl number, and the imposed uniform stratification on the walls. When ε is small, we show by regular perturbation expansion in ε that the modulation interacts with the natural mode of the system to produce resonances, confirming the results of Farooq & Homsy (1994). For ε ∼ O(1) we show that the modulation can potentially destabilize the longwave eigenmodes of the slot problem. This is achieved by projecting the governing equations onto the least-damped eigenmode, and investigating the resulting Mathieu equation via Floquet theory. No instability was found at large values of the Prandtl number and also low stratification, when there are no travelling modes present.

To understand the nonlinear saturation mechanisms of this growth, we consider a two-mode model of the slot problem with the primary mode being the least-damped travelling parallel-flow mode as before and a secondary mode of finite wavenumber. By projecting the governing equations onto these two modes we obtained the equations for temporal evolution of the two modes. For modulation amplitudes above critical, the growth of the primary mode is saturated resulting in a stable weak nonlinear synchronous oscillation of the primary mode. An unexpected and intriguing feature of the coupling is that the secondary mode exhibits very high-frequency bursts which appear once every cycle of the forcing frequency.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 313 , 25 April 1996 , pp. 1 - 38
Copyright
© 1996 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

Alexander, J. I. D. 1990 Low gravity experiment sensitivity to residual acceleration: A review. Microgravity Sci. Technol. III 2, 52–68.Google Scholar
Alexander, J. I. D., Amiroudine, S., Ouzzani, J. & Rosenberger, F. 1991 Analysis of the low gravity tolerance of Bridgman–Stockbarger crystal growth II. Transient and periodic accelerations. J. Cryst. Growth 113, 244 21–38.Google Scholar
Amin, N. 1988 The effect of g-jitter on heat transfer. Proc. R. Soc. Lond. A419, 151–172.Google Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.Google Scholar
Batchelor, G. K. 1954 Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures. Q. Appl. Maths 12, 209–233.Google Scholar
Bergholz, R. F. 1978 Stability of natural convection in a vertical fluid layer. J. Fluid Mech. 84, 743–768.Google Scholar
Biringen, S. & Danabasoglu, G. 1990 Computation of convective flow with gravity modulation in rectangular cavities. J. Thermophys. 4, 357–365.Google Scholar
Biringen, S. & Peltier, L. J. 1990 Computational study of 3-D Benard convection with gravitational modulation. Phys. Fluids A2, 279–283.Google Scholar
Bloch, F. 1928 Uber die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555–600.Google Scholar
Davis, S. H. 1976 The stability of time periodic flows. Ann. Rev. Fluid Mech. 8, 57–74.Google Scholar
Elder, J. W. 1965 Laminar free convection in a vertical slot. J. Fluid Mech. 23, 77–98.Google Scholar
Farooq, A. & Homsy, G. M. 1994 Streaming flows due to g-jitter-induced natural convection. J. Fluid Mech. 271, 351–378.Google Scholar
Gill, A. E. 1966 The boundary layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515–536.Google Scholar
Gresho, P. M. & Sani, R. L. 1970 The effects of gravity modulation on the stability of a heated fluid layer. J. Fluid Mech. 40, 783–806.Google Scholar
Hart, J. 1971 Stability of flow in a differentially heated inclined box. J. Fluid Mech. 47, 547–576.Google Scholar
Iyer, P. A. 1973 Instabilities in buoyancy driven boundary layer flows in a stably stratified medium. Boundary-Layer Met. 5, 53–66.Google Scholar
Kerczek, C. Von & Davis, S. H. 1974 Linear stability theory of oscillatory Stokes layers. J. Fluid Mech. 62, 753–773.Google Scholar
Kerczek, C. Von & Davis, S. H. 1976 Instability of a stratified periodic boundary layer. J. Fluid Mech. 75, 287–303.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 162, 339–363.Google Scholar
Magnus, W. & Winkler, S. 1966 Hills Equation. Interscience
Mizushima, J. & Gotoh, K. 1976 The stability of natural convection in a vertical fluid layer. J. Fluid. Mech. 73, 65–75.Google Scholar
Moler, C. B. & Stewart, G. W. 1973 An algorithm for generalized eigenvalue problems. SIAM J. Numer. Anal. 10, 241–256.Google Scholar
Nelson, E. S. 1991 An examination of anticipated g-jitter on space station and its effects on materials processing. NASA Tech. Mem. 103775.Google Scholar
Paolucci, S. & Chenoweth, D. R. 1989 Transition to chaos in a differentially heated cavity. J. Fluid Mech. 201, 379–410.Google Scholar
Stoker, J. 1950 Non-linear Vibrations in Electrical and Mechanical Systems. Wiley
Vest, C. M. & Arpaci, V. S. 1969 Stability of natural convection in a vertical slot. J. Fluid. Mech. 36, 1–15.Google Scholar