Abstract
For a uniform saturated porous layer heated from below, the dependence of the quantity of heat transferred on the distribution of the heat source is investigated. It is found, using perturbation methods and numerical techniques, that very small nonuniformities in the heat source having the same wavelength as the preferred convection mode significantly reinforce natural convection.
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Abbreviations
- A mn , B mn :
-
Fourier series coefficients in expansions for ψ, T
- c :
-
fluid specific heat
- d :
-
porous layer thickness
- f n , g n :
-
Fourier series coefficients
- \(\hat f_{n, } \hat g_n\) :
-
‘normalized’ Fourier series coefficients
- g :
-
gravitational acceleration
- K :
-
permeability of porous layer
- k :
-
thermal conductivity of porous layer
- l :
-
system aspect ratio (dimensionless)
- m :
-
wavenumber of flow pattern at onset of convection
- Nu:
-
Nusselt number
- p :
-
pressure
- R:
-
Rayleigh number
- R c :
-
critical value of Rayleigh number
- T :
-
temperature
- T mn :
-
Fourier series coefficient in expansion for T
- ΔT :
-
temperature difference across system
- u, w :
-
nondimensional components of mass flux velocity vector
- x, z :
-
nondimensional horizontal, vertical coordinates.
- α :
-
thermal expansion coefficient for fluid
- β m :
-
power series coefficient
- δ :
-
(small) amplitude of temperature nonuniformity
- ε :
-
(small) amplitude of convection
- θ(x) :
-
nondimensional variation in temperature at base of porous layer
- θ n :
-
Fourier series coefficient in expansion of θ(x)
- ν :
-
kinematic viscosity of fluid
- ϱ :
-
calculation parameter
- ϱ a :
-
reference density of fluid
- τ :
-
calculation parameter
- ψ :
-
stream function
- ψ mn :
-
Fourier series coefficient in expansion for ψ
- (n):
-
refers to coefficient of ε n in series expansions in powers of small amplitude ε
References
Caltagirone, J. P., 1975, Thermoconvective instabilities in a horizontal porous layer, J. Fluid Mech. 72, 269–287.
Cheng, P., 1978, Heat transfer in geothermal systems, Adv. Heat Transfer 14, 1–105.
Combarnous, M. A. and Bia, P., 1971, Combined free and forced convection in porous media, Soc. Petroleum Engineers J. 11, 399–405.
Combarnous, M. A. and Bories, S. A., 1975, Hydrothermal convection in saturated porous media, Adv. Hydrosci. 10, 231–307.
Donaldson, I. G., 1962, Temperature gradients in the upper layers of the earth's crust due to convective water flows, J. Geophys. Res. 67, 3449–3459.
Horne, R. N., 1975, Transient effects in geothermal convecting systems, PhD thesis, University of Auckland.
Kelly, R. E. and Pal, D., 1978, Thermal convection with spatially periodic boundary conditions: resonant wavelength excitation, J. Fluid Mech. 86, 433–456.
McKibbin, R., 1983, Convection in an aquifer above a layer of heated impermeable bedrock, New Zealand J. Sci. 26, 49–64.
Nield, D. A., 1968, Onset of thermohaline convection in a porous medium, Water Resources Res. 4, 553–660.
O'Sullivan, M. J. and McKibbin, R., 1982, Variation of heat transfer with cell-width in a porous layer, in A. McNabb, R. A. Wooding and Mervyn Rosser (eds.), Mathematics and Models in Engineering Science, DSIR, Wellington, New Zealand, pp. 129–138.
Palm, E., Weber, J. E., and Kvernvold, O., 1972, On steady convection in a porous medium, J. Fluid Mech. 54, 153–161.
Robinson, J. L. and O'Sullivan, M. J., 1976, A boundary-layer model of flow in a porous medium at high Rayleigh number, J. Fluid Mech. 75, 459–467.
Rubin, H., 1975, On the analysis of cellular convection in porous media, Int. J. Heat Mass Transfer 18, 1483–1486.
Straus, J. M., 1972, Finite amplitude doubly diffusive convection, J. Fluid Mech. 56, 353–374.
Straus, J. M., 1974, Large amplitude convection in porous media, J. Fluid Mech. 64, 51–63.
Straus, J. M. and Schubert, G., 1978, On the existence of three dimensional convection in a rectangular box of fluid-saturated porous material, J. Fluid Mech. 87, 385–394.
Straus, J. M. and Schubert, G., 1979, Three-dimensional convection in a cubic box of fluid-saturated porous material, J. Fluid Mech. 91, 155–165.
Straus, J. M. and Schubert, G., 1981, Modes of finite-amplitude three-dimensional convection in rectangular boxes of fluid-saturated porous material, J. Fluid Mech. 103, 23–32.
Tavantzis, J., Reiss, E. L., and Matkowsky, B. J., 1978, On the smooth transition to convection, SIAM J. Appl. Math. 34, 322–337.
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O'Sullivan, M.J., McKibbin, R. Heat transfer in an unevenly heated porous layer. Transp Porous Med 1, 293–312 (1986). https://doi.org/10.1007/BF00238184
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DOI: https://doi.org/10.1007/BF00238184