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The effects of boundary imperfections on convection in a saturated porous layer: near-resonant wavelength excitation

  • D. A. S. Rees (a1) and D. S. Riley (a2)

Abstract

Weakly nonlinear theory is used to study the porous-medium analogue of the classical Rayleigh-Bénard problem, i.e. Lapwood convection in a saturated porous layer heated from below. Two particular aspects of the problem are focused upon: (i) the effect of thermal imperfections on the stability characteristics of steady rolls near onset; and (ii) the evolution of unstable rolls.

For Rayleigh-Bénard convection it is well known (see Busse and co-workers 1974, 1979, 1986) that the stability of steady two-dimensional rolls near onset is limited by the presence of cross-roll, zigzag and sideband disturbances; this is shown to be true also in Lapwood convection. We further determine the modifications to the stability boundaries when small-amplitude imperfections in the boundary temperatures are present. In practice imperfections would usually consist of broadband thermal noise, but it is the Fourier component with wavenumber close to the critical wavenumber for the perfect problem (i.e. in the absence of imperfections) which, when present, has the greatest effect due to resonant forcing. This particular case is the sole concern of the present paper; other resonances are considered in a complementary study (Rees & Riley 1989).

For the case when the modulations on the upper and lower boundaries are in phase, asymptotic analysis and a spectral method are used to determine the stability of roll solutions and to calculate the evolution of the unstable flows. It is shown that steady rolls with spatially deformed axes or spatially varying wavenumbers evolve. The evolution of the flow that is unstable to sideband disturbances is also calculated when the modulations are π out of phase. Again rolls with a spatially varying wavenumber result.

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Bolton, E. W., Busse, F. H. & Clever, R. M., 1986 Oscillatory instabilities of convection rolls at intermediate Prandtl numbers. J. Fluid Mech. 164, 469485.
Busse, F. H. & Clever, R. M., 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91, 319335.
Busse, F. H. & Whitehead, J. A., 1971 Instability of convective rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305320.
Busse, F. H. & Whitehead, J. A., 1974 Oscillatory and collective instabilities in large Prandtl number convection. J. Fluid Mech. 66, 6779.
Coullet, P.: 1987 Commensurate–incommensurate transitions in nonequilibrium systems. Phys. Rev. Lett. 56, 724727.
Daniels, P. G.: 1982 Effects of geometrical imperfection at the onset of convection in a shallow two-dimensional cavity. Intl J. Heat Mass Transfer 25, 337343.
Eagles, P. M.: 1980 A Bénard convection problem with a perturbed lower wall. Proc. R. Soc. Lond. A 371, 359379.
Hall, P. & Walton, I. C., 1977 The smooth transition to a convective régime in a two-dimensional box. Proc. R. Soc. Lond. A 358, 199221.
Hall, P. & Walton, I. C., 1979 Bénard convection in a finite box: secondary and imperfect bifurcations. J. Fluid Mech. 90, 377395.
Joseph, D. D.: 1976 Stability of Fluid Motions II. Springer Tracts in Natural Philosophy, vol. 28, p. 138. Springer.
Kelly, R. E. & Pal, D., 1976 Thermal convection within non-uniformly heated horizontal surfaces. In Proc. 1976 Heat Transfer and Fluid Mech. Inst., pp. 117. Stanford University Press.
Kelly, R. E. & Pal, D., 1978 Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation. J. Fluid Mech. 86, 433456.
Lapwood, E. R.: 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508521.
Lowe, M., Albert, B. S. & Gollub, J. P., 1986 Convective flows with multiple spatial periodicities. J. Fluid Mech. 173, 253272.
Lowe, M. & Gollub, J. P., 1985a Solitons and the commensurate–incommensurate transition in a convecting nematic fluid. Phys. Rev. A 31, 38933897.
Lowe, M. & Gollub, J. P., 1985b Pattern selection near the onset of convection: the Eckhaus instability. Phys. Rev. Lett. 55, 25752578.
Lowe, M., Gollub, J. P. & Lubensky, T. C., 1983 Commensurate and incommensurate structures in a nonequilibrium system. Phys. Rev. Lett. 51, 786789.
Newell, A. C. & Whitehead, J. A., 1969 Finite bandwidth, finite amplitude convection. J. Fluid Mech. 38, 279303.
Pal, D. & Kelly, R. E., 1978 Thermal convection with spatially periodic non-uniform heating: non-resonant wavelength excitation. In Proc. 6th Intl Heat Transfer Conf., Toronto, vol. 2.
Pal, D. & Kelly, R. E., 1979 Three dimensional thermal convection produced by two-dimensional thermal forcing. ASME Paper 79-HT-109.
Rees, D. A. S. & Riley, D. S. 1986 Free convection in an undulating saturated porous layer: resonant wavelength excitation. J. Fluid Mech. 166, 503530.
Rees, D. A. S. & Riley, D. S. 1989 The effects of boundary imperfections on free convection in a saturated porous layer: non-resonant wavelength excitation. Proc. R. Soc. Lond. A A421, 303339.
Straus, J. M.: 1974 Large amplitude convection in porous media. J. Fluid Mech. 64, 5163.
Tavantzis, J., Reiss, E. L. & Matkowsky, B. J., 1978 On the smooth transition to convection. SIAM J. Appl. Maths 34, 322337.
Vozovoi, L. P. & Nepomnyaschh, A. A., 1974 Convection in a horizontal layer in the presence of spatial modulation of the temperature at the boundaries. Gidrodinamika 8, 105117.
Walton, I. C.: 1982 The effects of slow spatial variations on Bénard convection. Q. J. Mech. Appl. Maths 35, 3348.
Wynne, M. C.: 1987 The effects of boundary imperfections on free convection in fluid layers. Ph.D. dissertation, University of Bristol.
Zaleski, S.: 1984 Cellular patterns with boundary forcing. J. Fluid Mech. 149, 101125.
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The effects of boundary imperfections on convection in a saturated porous layer: near-resonant wavelength excitation

  • D. A. S. Rees (a1) and D. S. Riley (a2)

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