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Effect of gravity modulation on the stability of a horizontal double-diffusive layer

Published online by Cambridge University Press:  08 October 2007

YOUMIN YU ,
CHO LIK CHAN  and
C. F. CHEN
YOUMIN YU
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona Tucson, AZ 85721, USA
CHO LIK CHAN
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona Tucson, AZ 85721, USA
C. F. CHEN
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Arizona Tucson, AZ 85721, USA
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Abstract

The instability characteristics of a horizontal stably stratified fluid layer being heated from below, including its subsequent nonlinear evolution under steady and modulated gravity, have been investigated by experiments and two-dimensional numerical simulations. The critical condition at instability onset is also checked using linear stability analysis. The fluid is contained in a horizontal test tank with an initial stable solute gradient and a constant-temperature gradient imposed by heating from below. Because of the non-diffusive boundaries, the vertical solute gradient slowly decreases and, eventually, the layer becomes unstable. From the time of the onset of instability, the critical solute Rayleigh number is determined. For the experiments with modulated gravity, the tank is fixed onto a platform that oscillates vertically at 1 Hz with an amplitude of 10 cm. The experiment is designed such that no internal wave mode of instability can be excited. The experimental results show that gravity modulation destabilizes the system slightly by increasing the solute Rayleigh number at onset by 8.4% and causes the oscillation frequency at onset to increase by 32.6%. Linear stability analysis and two-dimensional numerical simulations for the steady gravity case yield results that are in good agreement with the experiment. For the gravity modulation case, linear stability results do not show any effect of gravity modulation at the frequency of 1 Hz. Numerical simulation results do show increases in both the onset solute Rayleigh number and the oscillation frequency; however, their values are smaller than those obtained in the experiment. The characteristics of the internal wave mode of instability are explored by numerical simulations of a stably stratified solute fluid layer under gravity modulation. The interference effects between the internal wave mode and double-diffusive mode of instabilities are studied by imposing an adverse temperature gradient on the stratified layer.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 589 , 25 October 2007 , pp. 183 - 213
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Arakawa, A. 1966 Computational design for long-term numerical integration of the equations of fluid motion: two-dimensional incompressible flow. Part 1. J. Comput. Phys. 1, 119–143.CrossRefGoogle Scholar
Benielli, D. & S ommeria, J. 1998 Excitation and breaking of internal gravity waves by parametric instability. J. Fluid Mech. 374, 117–144.CrossRefGoogle Scholar
Chan, C. L., Yu, Y., Chen, C. F. 2004 Instability of convection of an ethanol–water solution in a vertical tank. J. Fluid Mech. 510, 243–265.CrossRefGoogle Scholar
Chen, W.-Y. & Chen, C. F. 1999 Effect of gravity modulation on the stability of convection in a vertical slot. J. Fluid Mech. 395, 327–344.CrossRefGoogle Scholar
DuFort, E. C., Frankel, S. P. 1953 Stability conditions in the numerical treatment of parabolic differential equations. Math. Tables Other Aids Comput. 7, 135–152.Google Scholar
Gershuni, G. Z., Zhukhovitskii, E. M., Iurkov, I. S. 1970 On the convective stability in the presence of periodically varying parameter. Z. Angew. Math. Mech. 34, 442–452.CrossRefGoogle Scholar
Ghorayeb, K. & Mojatabi, A. 1997 Double diffusive convection in a vertical rectangular cavity. Phys. Fluids 9, 2339–2348.CrossRefGoogle 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.CrossRefGoogle Scholar
Hayes, M. H. 1996 Statistical Digital Signal Processing and Modeling. Wiley.Google Scholar
Houstis, E. N. & Papatheodorou, T. S. 1979 High-order fast elliptic equation solvers. ACM Trans. Math. Software 5, 431–441.CrossRefGoogle Scholar
Huang, H. & Wetton, B. R. 1996 Discrete compatibility in finite difference methods for viscous incompressible fluid flow. J. Comput. Phys. 126, 468–478.CrossRefGoogle Scholar
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double-diffusive convection. J. Fluid Mech. 78, 821–854.CrossRefGoogle Scholar
Landolt, H. & Bornstein, R. 1989 Eigenschaften der Materie in ihren aggregatzustaden, part 5, p. 640. Springer.Google Scholar
Napolitano, M., Pascazio, G. & Quartapelle, L. 1999 A review of vorticity conditions in the numerical solution of the ζ–ψ equations. Computers Fluids 28, 139–185.CrossRefGoogle Scholar
Nield, D. A. 1967 The thermohaline Rayleigh–Jeffreys problem. J. Fluid Mech. 29, 545–558.CrossRefGoogle Scholar
Proakis, J. G. & Manolakis, D. G. 1996 Digital Signal Processing: Principles, Algorithms, and Applications. Prentice–Hall.Google Scholar
Roache, P. J. 1982 Computational Fluid Dynamics. Hermosa, Albuquerque, NM.Google Scholar
Rogers, J. R., Pesch, W., Brausch, O., Schatz, M. F. 2005 Complex-ordered patterns in shaken convection. Phys. Rev. E 71, 066214-1-18.Google ScholarPubMed
Saunders, B. V., Murray, B. T., McFadden, G. B., Coriell, S. R., Wheeler, A. A. 1992 The effect of gravity modulation on thermosolutal convection in an infinite layer of fluid. Phys. Fluids A 4, 1176–1189.CrossRefGoogle Scholar
Sekerzh-Zen'kovich, S. Ya. 1983 Parametric resonance in a stratified liquid in a container undergoing vertical vibration. Sov. Phys. Dokl. 28 (6), 445–446.Google Scholar
Shirtcliffe, T. G. L. 1967 Thermosolutal convection: observation of an overstable mode. Nature 213, 489–490.CrossRefGoogle Scholar
Shirtcliffe, T. G. L. 1969 An experimental investigation of thermosolutal convection at marginal stability. J. Fluid Mech. 35, 677–688.CrossRefGoogle Scholar
Sinha, S. C. & Wu, D.-H. 1991 An efficient computational scheme for the analysis of periodic systems. J. Sound Vib. 151, 91–117.CrossRefGoogle Scholar
Tanny, J., Chen, C. C. & Chen, C. F. 1995 Effects of interaction between Marangoni and double-diffusive instabilities. J. Fluid Mech. 303, 1–21.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar
Wright, J. H. & Loehrke, R. I. 1976 The onset of thermohaline convection in a linearly-stratified horizontal layer. Trans. ASME C: J. Heat Transfer 98, 558–563.CrossRefGoogle Scholar