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Thermal Convection in Fluid Layers with Nonuniform Thickness or Surface Temperature

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Nonlinear Instability of Nonparallel Flows

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Abstract

A review is made of results obtained mainly during the past 15 years for the topic described in the title. Reference is also made to results obtained for the analogous problem in centrifugal instability. The aim is to use this prototype problem to understand how stability problems with spatially varying boundary conditions differ from classical problems with spatially homogeneous conditions.

This work was supported by the National Science Foundation grant CTS-9123553

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References

  • Ahlers, G., Cannell, D.S., Dominguez-Lerma, M.A. and Heinrichs, R. 1986 Wavenumber selection and Eckhaus instability in Couette-Taylor flow. Physica 23D, 202–219.

    ADS  Google Scholar 

  • Balzer, G., Coullet, P., Pomeau, Y., Berg, P., Dubois, M. and Gil, L. 1991 Brisure de la parit dans une exp rience de convection forc e. Comptes Rendus Acad. Sci. Paris 312 (Ser. II), 585–591.

    Google Scholar 

  • Behringer, R.P. and Ahlers, G., 1982 Heat transport and temporal evolution of fluid flow near the Rayleigh-B nard instability in cylindrical containers. J. Fluid Mech. 125, 219–258.

    Article  ADS  Google Scholar 

  • Bhattacharjee, J.K., 1991 Parametric resonance in Rayleigh-B nard convection with corrugated geometry. Phys. Rev. A 43, 819–821.

    Article  MathSciNet  ADS  Google Scholar 

  • Bouthier, M., 1972 Stabilit lin aire des coulements presque parall les. J. M chanique 11, 599–621.

    Google Scholar 

  • Bouthier, M., 1973 Stabilit lin aire des coulements presque parall les. Partie II. La Couche Limite de Blasius. J. Mecanique 12, 75–95.

    Google Scholar 

  • Buell, J.C. and Catton, I., 1986 Wavenumber selection in ramped Rayleigh-B nard convection. J. Fluid Mech. 171, 477–494.

    Article  ADS  MATH  Google Scholar 

  • Coullet, P., 1986 Commensurate incommensurate transition in non equilibrium systems. Phys. Rev. Lett. 56, 724–727.

    Article  MathSciNet  ADS  Google Scholar 

  • Coullet, P. and Huerre, P., 1986 Resonance and phase solitons in spatially-forced thermal convection. Physica 23D, 27–44.

    ADS  Google Scholar 

  • Crighton, D.G. and Gaster, M., 1976 Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413.

    Article  ADS  MATH  Google Scholar 

  • Daniels, P.G., 1977 The effect of distant sidewalls on the transition to finite amplitude B nard convection. Proc. Roy. Soc. A358, 173–197

    MathSciNet  ADS  Google Scholar 

  • Daniels, P.G., 1982 Effects of geometrical imperfection at the onset of convection in a shallow two-dimensional cavity. Int. J. Heat and Mass Transfer 25, 337–343.

    Article  ADS  MATH  Google Scholar 

  • Domaradzki, J.A., 1989 Heat transfer enhancement in Rayleigh-B nard convection, Int. J. Heat Mass Trans. 32, 2475–2483.

    Article  Google Scholar 

  • Drazin, P.G. and Reid, W.H., 1981 Hydrodynamic Stability. Cambridge University Press.

    MATH  Google Scholar 

  • Duck, P.W., 1979 Flow induced by a torsionally oscillating wavy cylinder. Q. J. Mech. Appl. Math. 32, 74–91.

    Article  MathSciNet  Google Scholar 

  • Eagles, P.M., 1980 A B nard convection problem with a perturbed lower wall. Proc. Roy. Soc. A371, 359–379.

    MathSciNet  ADS  Google Scholar 

  • Eagles, P.M., 1985 Ramped Taylor-Couette flow. Phys. Rev. A31, 1955–1956.

    ADS  Google Scholar 

  • Eagles, P.M. and Eames, K., 1983 Taylor vortices between almost cylindrical boundaries. J. Engr. Math. 17, 263–280.

    Article  MathSciNet  MATH  Google Scholar 

  • Eagles, P.M. and Weissman, M.A., 1975 On the stability of slowly varying flow: the divergent channel. J. Fluid Mech. 69, 241–262.

    Article  ADS  MATH  Google Scholar 

  • Fasel, H., 1976 Investigation of the stability of boundary layers by a finite-difference model of the Navier-Stokes equations. J. Fluid Mech. 78, 355–383.

    Article  ADS  MATH  Google Scholar 

  • Gaster, M., 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480.

    Article  ADS  MATH  Google Scholar 

  • Gil, L., Balzer, G., Coullet, P., Dubois, M. and Berge, P., 1991 Hopf bifurcation in a broken parity pattern. Phys. Rev. Lett. 66, 3249–3252.

    Article  ADS  Google Scholar 

  • Hall, P. and Walton, I.C., 1977 The smooth transition to a convective regime in a two-dimensional box. Proc. Roy. Soc. A358, 199–221.

    MathSciNet  ADS  Google Scholar 

  • Kelly, R.E. and Pal, D, 1976 Thermal convection induced between non-uniformly heated horizontal surfaces. Proc. 1976 Heat Transfer and Fluid Mech. Inst. pp. 1–17. Stanford University Press.

    Google Scholar 

  • Kelly, R.E. and Pal, D., 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86, 433–456.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Kelly, R.E., Cuendet, G.J. and Walton, I.C., 1983. Axisymmetric convection induced by radial and vertical temperature gradients. ASME Paper 83-WA/HT-71.

    Google Scholar 

  • Kelly, R.E., and Pearlstein, A.J., 1985 Local resonance in thermal convection. In Stability in Convective Flows, ed. by W.S. Saric and A.A. Szewczyk. ASME HTD-vol. 54.

    Google Scholar 

  • Koschmieder, E.L., 1966 On convection on a nonuniformly heated plane. Beitr. Phys. Atm. 39, 208–216.

    Google Scholar 

  • Koschmieder, E.L., 1968 Convection on a non-uniformly heated, rotating plane. J. Fluid Mech. 33, 515–527.

    Article  ADS  Google Scholar 

  • Kramer, L. and Riecke, H., 1985 Wavelength selection in Rayleigh-B nard convection. Z. Phys. B - Condensed Matter 59, 245–251.

    Article  ADS  Google Scholar 

  • Krettenauer, K. and Schumann, U., 1992 Numerical simulation of turbulent convection over wavy terrain. J. Fluid Mech. 237, 261–299.

    Article  ADS  Google Scholar 

  • Lowe, M., Albert, B.S. and Gollub, J.P., 1986 Convective flows with multiple spatial periodicities. J. Fluid Mech. 173, 253–272.

    Article  ADS  Google Scholar 

  • Lucke, M. and Roth, D., 1990 Structure and dynamics of Taylor vortex flow and the effect of subcritical driving ramps. Z. Phys. B - Condensed Matter 78, 147–158.

    Article  ADS  Google Scholar 

  • Mtiller, U., 1966 Uber Zellularkonvektionsstromiingen in horizontalen Fliissigkeitsschichten mit ungleichmassig erwarmten Bodenflache. Beitr. Phys. Atm. 39, 217–234.

    Google Scholar 

  • Pal, D. and Kelly, R.E. 1979 Three-dimensional thermal convection produced by two-dimensional thermal forcing. ASME Paper 79-HT-109.

    Google Scholar 

  • Papp, H.-G. and Riecke, H. 1991 Drifting vortices in ramped Tayler vortex flow: quantitative results from phase equation. Phys. Fluids A3, 1519–1532.

    ADS  Google Scholar 

  • Pismen, L.M., 1987 Bifurcation of quasiperiodic and nonstationary patterns under external forcing. Phys. Rev. Lett. 59, 2740–2743.

    Article  ADS  Google Scholar 

  • Powe, R.E., Warrington, R.O., and Scanlan, J.A., 1980 Natural convective flow between a body and its spherical enclosure. Int. J. Heat and Mass Transfer 23, 1337–1350.

    Article  ADS  Google Scholar 

  • Rees, D.A.S. and Riley, D.S., 1989 The effects of boundary imperfections on convection in a saturated porous layer: non-resonant wavelength excitation. Proc. Roy. Soc. A421, 303–339.

    ADS  Google Scholar 

  • Riahi, D.N., 1993 Preferred pattern of convection in a porous layer with spatially non-uniform boundary temperature. J. Fluid Mech. 246, 529–543.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Richter, F.M., 1973 Dynamical models for sea floor spreading. Revs. Geophys. Space Phys. 11, 223–287.

    Article  ADS  Google Scholar 

  • Riecke, H. and Papp, H.G., 1987 Perfect wave-number selection and drifting patterns in ramped Taylor vortex flow. Phys. Rev. Lett. 59, 2570–2573.

    Article  ADS  Google Scholar 

  • Riecke, H. and Papp, H.G., 1992 Chaotic phase diffusion through the interaction of phase slip processes in ordered and turbulent patterns in Taylor-Couette Flow. NATO ASI Series B, v. 297, ed. by C.D. Andereck and F. Hayot. Plenum Press.

    Google Scholar 

  • Smith, F.T., 1979a On the nonparallel flow stability of the Blasius boundary layer. Proc. Roy. Soc. A366, 91–109.

    ADS  Google Scholar 

  • Smith, F.T., 1979b Nonlinear stability of boundary layers for disturbances of various sizes. Proc. Roy. Soc. A368, 573–589.

    ADS  Google Scholar 

  • Somerville, R.C.J., 1967 A nonlinear spectral model of convection in a fluid unevenly heated from below. J. Atm. Sei. 24, 665–676.

    Article  ADS  Google Scholar 

  • Soward, A.M., and Jones, C.A., 1983 The linear stability of the flow in the narrow gap between two concentric rotating spheres. Q. Jl. Mech. Appl. Math. 36, 19–42.

    Article  MathSciNet  MATH  Google Scholar 

  • Srulijes, J.A., 1979 Zelluarkonvection in Beh ltern mit Horizontalen Temperaturgradienten. Dr.-Ing. thesis, University of Karlsruhe.

    Google Scholar 

  • Vozovoi, L.P., 1989 Nonlinear quasi-periodic convection regimes in a vertical wall with wavy boundaries. Heat Transfer Sov. Res. 21, 189–195.

    Google Scholar 

  • Vozovoi, L.P. and Nepomnyashchii, A.A., 1974 Convection in a horizontal layer in the presence of spatial modulation of temperature at the boundaries. Gidrodinamika 7, 105–117. (Publ. Perm University, in Russian.)

    Google Scholar 

  • Walton, I.C., 1982a The effects of slow spatial variations on Benard convection. Q. Jl. Mech. Appl. Math. 35, 33–48.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  • Walton, I.C., 1982b On the onset of Rayleigh-Benard convection in a fluid layer of slowly increasing depth. Studies Appl. Math. 67, 199–216.

    MathSciNet  ADS  MATH  Google Scholar 

  • Walton, I.C., 1983 The onset of cellular convection in a shallow two dimensional container of fluid heated non-uniformly from below. J. Fluid Mech. 131, 455–470.

    Article  ADS  MATH  Google Scholar 

  • Walton, I.C. and Kelly, R.E., 1987 Convection in a shallow vertical cylinder with nonuniform heating. In Bifurcation Phenomena in Thermal Processes and Convection, ed. by H.H. Bau, L.A. Bertram and S.A. Korpela. ASME HTD-vol. 94.

    Google Scholar 

  • Yoo, J.-S. and Kim, M.-U., 1991 Two-dimensional convection in a horizontal fluid layer with spatially periodic boundary temperatures. Fluid Dyn. Res. 7, 181–200.

    Article  ADS  Google Scholar 

  • Zierep, J., 1961 Thermokonvektive Zellularstr mungen bei inkonstanter Erw mung der Grundflche. Z. angew. Math. Mech. 3, 114–125.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Mechanical, Aerospace and Nuclear Engineering Department, University of California, Los Angeles, Los Angeles, CA, 90024-1597, USA

    R. E. Kelly

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  1. R. E. Kelly
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Editors and Affiliations

  1. Dept. of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY, 13699-5725, USA

    S. P. Lin , W. R. C. Phillips  & D. T. Valentine ,  & 

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© 1994 Springer-Verlag, Berlin Heidelberg

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Kelly, R.E. (1994). Thermal Convection in Fluid Layers with Nonuniform Thickness or Surface Temperature. In: Lin, S.P., Phillips, W.R.C., Valentine, D.T. (eds) Nonlinear Instability of Nonparallel Flows. International Union of Theoretical and Applied Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-85084-4_16

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  • DOI: https://doi.org/10.1007/978-3-642-85084-4_16

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  • Print ISBN: 978-3-642-85086-8

  • Online ISBN: 978-3-642-85084-4

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