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Numerical Experiments in Double-Diffusive Convection

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Summary

We present and discuss some numerical experiments in double-diffusive convection where a layer of fluid, stably stratified by concentration, is subject to heating. By considering first the case where the fluid is heated from below we justify the need of highly accurate numerical approximations, like spectral methods. The spectral Chebyshev method for solving the equations governing the motion of a double-diffusive fluid is then briefly described. Finally, the case where the fluid is laterally heated is discussed in more details. In particular, the formation of convective layers followed by the merging of some of them is discussed from the numerical results.

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References

  1. J.S. Turner, “Double-diffusive phenomena”, Ann. Rev. Fluid Mech., 6, 37–56 (1974).

    Article  ADS  Google Scholar 

  2. J.S. Turner, “Multicomponent convection”, Ann. Rev. Fluid Mech., 17, 11–44 (1985).

    Article  ADS  Google Scholar 

  3. J.S. Turner, “The behaviour of a stable salinity gradient heated from below”, J. Fluid Mech., 33, 168–200 (1968).

    Google Scholar 

  4. T.G.L. Shirtcliffe, “An experimental investigation of thermosolutal convection at marginal stability”, J. Fluid Mech., 35, 677–688 (1969).

    Article  ADS  Google Scholar 

  5. H.E. Huppert and P.F. Linden, “On heating a stable salinity gradient from below”, J. Fluid Mech., 95, 431–464 (1979).

    Article  ADS  Google Scholar 

  6. G. Veronis, “Effect of a stabilizing gradient of solute on thermal convection”, J. Fluid Mech., 34, 314–336 (1968).

    Article  ADS  Google Scholar 

  7. E.K. Knobloch, D.R. Moore, J. Toomre and N.O. Weiss, “Transition to chaos in two-dimensional double-diffusive convection”, J. Fluid Mech., 166, 409–448 (1986).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  8. E.K. Knobloch, M.R.E. Proctor and N.O. Weiss, “Heteroclinic bifurcations in a simple model of double-diffusive convection”, J. Fluid Mech., 239, 273–292 (1992).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. H.E. Huppert and D.R. Moore, “Nonlinear double-diffusive convection”, J. Fluid Mech., 78, 851–854 (1976).

    Article  ADS  Google Scholar 

  10. S.M. Chang, S.A. Korpela and Y. Lee, “Double-diffusive convection in the diffusive regime”, Appl. Sei. Res., 39, 301–319 (1982).

    Article  MATH  Google Scholar 

  11. U. Ehrenstein and R. Peyret, “A Chebyshev collocation method for the Navier-Stokes equations with application to double-diffusive convection”, Int. J. Numer. Methods Fluids, 9, 427–452 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. B. Roux (ed), “Numerical Simulation of Oscillatory Convection in Lower-Pr Fluids”, Proc. GAMM Workshop, Marseille ( 1988 ), Vieweg, Braunschweig, 1990.

    Google Scholar 

  13. P. Le Quéré, “Transition to unsteady natural convection in a tall water-filled cavity”, Phys. Fluids, A2, 503–515 (1990).

    Article  ADS  MATH  Google Scholar 

  14. J. Ouazzani, R. Peyret and A. Zakaria, “Stability of collocation-Chebyshev schemes with application to the Navier-Stokes equations”, in D.Rues and W. Kordulla (eds), Proc. Sixth GAMM Conf. on Numerical Methods in Fluid Mechanics, Vieweg, Braunschweig, 1986, pp. 287–294.

    Google Scholar 

  15. J. Fröhlich, T. Gerhold, J.M. Lacroix and R. Peyret, “Fully implicit spectral methods for convection”, in M. Durand and F. El Dabaghi (eds), High Performance Computing II, North-Holland, Amsterdam, 1991, pp. 585–596.

    Google Scholar 

  16. J.M. Vanel, R. Peyret and P. Bontoux, “A pseudo-spectral solution of vorticitystream function equations using the influence matrix technique”, in K.W. Morton and M.J. Baines (eds), Numerical Methods for Fluid Dynamics II, Clarendon Press, Oxford, 1986, pp. 463–475.

    Google Scholar 

  17. D.B. Haidvogel and T. Zang, “The accurate solution of Poisson’s equation by expansion in Chebyshev polynomials”, J. Comput. Phys., 30, 167–180 (1979).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. U. Ehrenstein, “Méthodes spectrales de résolution des équations de Stokes et de Navier-Stokes. Application à des écoulements de convection double-diffusive”, Thèse Doctorat, Mathématiques Appliquées, Université de Nice, 1986.

    Google Scholar 

  19. R. Bwemba and R. Pasquetti, “About the influence matrix used in the spectral solution of the 2D incompressible Navier-Stokes equations (vorticity-stream function formulation)”, to appear.

    Google Scholar 

  20. S.A. Thorpe, P.K. Hütt ans R. Soulsby, “The effect of horizontal gradients on thermohaline convection”, J. Fluid Mech., 38, 375–400 (1969).

    Article  ADS  Google Scholar 

  21. R. A. Wirtz, D.G. Briggs and C.F.Chen, “Physical and numerical experiments on layered convection in a density stratified fluid”, Geophys. Fluid Dyn., 3, 265–288 (1972).

    ADS  Google Scholar 

  22. V.N. Nekrasov, V.A. Popov and Yu.D. Chashechkin, “Formation of periodic convective-flow structure on lateral heating of a stratified liquid”, Izv. Atmosph. Ocean. Phys., 12, 1191–1200 (1976), English transi, pp.733–739.

    Google Scholar 

  23. Y. Suzukawa and U. Narusawa, “Structure of growing double-diffusive convection cells”, J.Heat Transfer, 104, 248–254 (1982).

    Article  Google Scholar 

  24. J. Tanny and A.B. Tsinober, “The dynamics and structure of double-diffusive layers in sidewall-heating experiments”, J. Fluid Mech., 196, 135–156 (1988).

    Article  ADS  Google Scholar 

  25. J. Tanny and A.B. Tsinober, “On the behavior of a system of double diffusive layers during its evolution”, Phys. Fluids, Al, 606–609 (1989).

    ADS  Google Scholar 

  26. J.E. Hart, “On sideways diffusive instability”, J. Fluid Mech., 49, 279–288 (1971).

    Article  ADS  Google Scholar 

  27. J.E. Hart, “Finite amplitude sideways diffusive convection”, J. Fluid Mech., 59, 47–64 (1973).

    Article  ADS  Google Scholar 

  28. O. Kerr, “Heating a salinity gradient from a vertical sidewall: linear theory”, J. Fluid Mech., 207, 323–352 (1989).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. O. Kerr, “Heating a salinity gradient from a vertical sidewall: nonlinear Theory”, J. Fluid Mech., 217, 529–546 (1990).

    Article  ADS  MATH  Google Scholar 

  30. C.F. Chen, “Onset of cellular convection in a salinity gradient due to a lateral temperature gradient”, J. Fluid. Mech., 63, 563–576 (1974).

    Google Scholar 

  31. S. Thangam, A. Zebib and C. F. Chen, “Double-diffusive convection in an inclined fluid layer”, J. Fluid Mech., 116, 363–378 (1982).

    Article  MathSciNet  ADS  Google Scholar 

  32. R. A. Wirtz and L. H. Liu, “Numerical experiments on the onset of layered convection in a narrow slot containing a stably stratified fluid”, Int. J. Heat Mass Transfer, 18, 1299–1305 (1975).

    Article  Google Scholar 

  33. C.S. Reddy, “Cell merging and its effect on heat transfer in thermosolutal convection”, J. Heat Transfer, 102, 172–174 (1980).

    Article  Google Scholar 

  34. Y. Demay, J.M. Lacroix, R. Peyret and J.M. Vanel, “Numerical experiments on stratified fluids subject to heating”, Proc. Third Inter. Symp. on Stratified Flows, Pasadena, 1987, E.J. List and G.H. Jirka (eds), American Society of Civil Engineers, New York, 1990, pp. 588–597.

    Google Scholar 

  35. H.C. Ku, R.S Hirsh and T.D. Taylor, “A numerical simulation of the effect of salinity on a thermally driven flow”, in M. Deville (ed), Proc. Seventh GAMM Conf. on Numerical Methods in Fluid Mechanics, Vieweg, Braunschweig, 1988, pp. 151–158.

    Google Scholar 

  36. C.E. Mendenhall and M. Mason, “The stratified subsistence of fine particles”, Proc. Nat. Acad. Sci. USA, 9, 199–207 (1923).

    Article  ADS  Google Scholar 

  37. C. Sabbah, “Etude numérique de la formation et l’évolution de cellules convectives dans un fluide stratifié chauffé latéralement”, Projet de stage de DEA “Turbulence et Systèmes Dynamiques”, Université de Nice-Sophia Antipolis, 1994.

    Google Scholar 

  38. J.P. Pulicani, “A spectral multi-domain method for the solution of 1-D Helmholtz and Stokes-type equations”, Computers and Fluids, 16, 207–215 (1988).

    Article  MATH  Google Scholar 

  39. R. Peyret, “The Chebyshev multidomain approach to stiff problem in Fluid Mechanics”, Comp. Meth. Appl. Mech. Eng., 80, 129–145 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  40. H. Guillard and R. Peyret, “ On the use of spectral methods for the numerical solution of stiff problems”, Comp. Meth. App. Mech. Eng., 66, 17–43 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  41. J.M. Lacroix, R. Peyret and J.P. Pulicani, “A pseudospectral multidomain method for the Navier-Stokes equations with application to double-diffusive convection”, in M. Deville (ed), Proc. Seventh GAMM Conf. on Numerical Methods in Fluid Mechanics, Vieweg, Braunschweig, 1988, pp. 167–174.

    Google Scholar 

  42. A. Mofid, “Application des méthodes spectrales à l’étude de l’effet d’une source de chaleur dans un fluide stratifié”, Thèse de Doctorat, Sciences de l’ingénieur, Université de Nice-Sophia Antipolis, 1992.

    Google Scholar 

  43. J. P. Pulicani, “Modélisation isotherme d’une interface diffusive en convection de double-diffusion”, Int. J. Heat Mass Transfer (to appear).

    Google Scholar 

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Peyret, R., Vanel, J.M. (1995). Numerical Experiments in Double-Diffusive Convection. In: Leutloff, D., Srivastava, R.C. (eds) Computational Fluid Dynamics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-79440-7_3

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  • DOI: https://doi.org/10.1007/978-3-642-79440-7_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-79442-1

  • Online ISBN: 978-3-642-79440-7

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