Abstract
The effect of vertical heterogeneity of permeability, on the onset of convection in a horizontal layer of a saturated porous medium, uniformly heated from below but with a non-uniform basic temperature gradient resulting from vertical throughflow, is studied analytically using linear stability theory. It is found that, to first order, a linear variation of the reciprocal of permeability with depth has no effect on the critical value of the Rayleigh number Ra c based on the harmonic mean of the permeability, but a quadratic variation increasing in the upwards direction leads to a reduction in Ra c.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Abbreviations
- a :
-
Dimensionless horizontal wavenumber
- c a :
-
Acceleration coefficient
- f(z):
-
Function characterizing the basic temperature gradient, defined by Eq. 25
- g(z):
-
Function characterizing the reciprocal of the permeability, defined by Eq. 25
- g :
-
Gravitational acceleration
- H :
-
Dimensional layer depth
- k m :
-
Effective thermal conductivity of the porous medium
- K :
-
Permeability of the porous medium
- P*:
-
Pressure
- P :
-
Dimensionless pressure, P* K/μα m
- Q :
-
Péclet number defined by Eq. 14
- Ra:
-
Rayleigh–Darcy number
- t*:
-
Time
- t :
-
Dimensionless time, t*α m /σ H 2
- T*:
-
Temperature
- T :
-
Dimensionless temperature, (T* − T 0)/(T 1 − T 0)
- \({T^{\ast}_{0}}\) :
-
Temperature at the upper wall
- \({T^{\ast}_{1}}\) :
-
Temperature at the lower wall
- (u, v, w):
-
Dimensionless Darcy velocity components, (u*, v*, w*)H/α m
- V 0 :
-
Throughflow velocity
- v :
-
Dimensionless Darcy velocity, \({\frac{(\rho{c})_{\rm f} H}{k_{\rm m}}}\)
- v*:
-
Dimensional Darcy velocity, (u*, v*, w*)
- (x, y, z):
-
Dimensionless Cartesian coordinates, (x*, y*, z*)/H; z is the vertically upward coordinate
- (x*, y*, z*):
-
Cartesian coordinates
- α m :
-
Thermal diffusivity of the porous medium, \({\frac{k_{\rm m}}{(\rho{c}_P )_{\rm f}}}\)
- γ a :
-
Acceleration coefficient defined by Eq. 10
- γ :
-
Permeability linear heterogeneity parameter defined by Eq. 36
- δ :
-
Permeability quadratic heterogeneity parameter defined by Eq. 40
- μ :
-
Viscosity of the fluid
- ρ :
-
Fluid density
- (ρc)f :
-
Heat capacity of the fluid
- (ρc)m :
-
Effective heat capacity of the porous medium
- σ :
-
Thermal capacity ratio defined by Eq. 6
- *:
-
Dimensional variable
- ′:
-
Perturbation variable
- b:
-
Basic solution
References
Barletta A., di Schio E.R., Storesletten L.: Convective roll instabilities of vertical throughflow with viscous dissipation in a horizontal porous layer. Transp. Porous Med. 81, 461–477 (2010)
Braester C., Vadasz P.: The effect of weak heterogeneity of a porous medium on natural convection. J. Fluid Mech. 254, 345–362 (1993)
Brevdo L.: Three-dimensional absolute and convective instabilities at the onset of convection in a porous medium with inclined temperature gradient and vertical throughflow. J. Fluid Mech. 641, 475–487 (2009)
Brevdo L., Ruderman S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part I. Normal modes. Transp. Porous Med. 80, 137–151 (2009a)
Brevdo L., Ruderman S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part II. Absolute and convective instabilities, and spatially amplifying waves. Transp. Porous Med. 80, 153–172 (2009b)
Hill A.A.: Unconditional nonlinear stability for convection in a porous medium with vertical throughflow. Acta Mech. 193, 197–206 (2007)
Hill A.A., Rionero S., Straughan B.: Global stability for penetrative convection with throughflow in a porous material. IMA J. Appl. Math. 72, 635–643 (2007)
Kuznetsov A.V., Nield D.A., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium: periodic and localized variation. Transp. Porous Med. 81, 123–139 (2010)
Kuznetsov A.V., Nield D.A., Simmons C.T.: The onset of convection in a strongly heterogeneous porous medium with transient temperature profile. Transp. Porous Med. 86, 851–865 (2011)
Nield D.A.: Convective instability in porous media with throughflow. AIChE J. 33, 1222–1224 (1987)
Nield D.A.: General heterogeneity effects on the onset of convection in a porous medium. In: Vadasz, P. (ed.) Emerging Topics in Heat and Mass Transfer in Porous Media—from Bioengineering and Microelectronics to Nanotechnology, pp. 63–84. Springer, New York (2008)
Nield D.A., Bejan A.: Convection in Porous Media. 3rd edn. Springer, New York (2006)
Nield, D.A., Kuznetsov, A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium. Int. J. Heat Mass Transf. 50, 2361–2367; erratum 4512–4512 (2007)
Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: moderate heterogeneity. Int. J. Heat Mass Transfer 51, 2361–2367 (2008)
Nield D.A., Simmons C.T.: A discussion on the effect of heterogeneity on the onset of convection in a porous medium. Transp. Porous Med. 68, 413–421 (2007)
Nield D.A., Kuznetsov A.V., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium. Transp. Porous Med. 77, 169–186 (2009)
Nield D.A., Kuznetsov A.V., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium: 2D/3D localization and spatially correlated random permeability fields. Transp. Porous Med. 83, 465–477 (2010)
Shivakumara I.S., Nanjundappa C.E.: Effects of quadratic drag and throughflow on double diffusive convection in a porous layer. Int. Commun. Heat Mass Transf. 33, 357–363 (2006)
Simmons C.T., Kuznetsov A.V., Nield D.A.: The effect of strong heterogeneity on the onset of convection in a porous medium: Importance of spatial dimensionality and geologic controls. Water Resour. Res. 46, W09539 (2010). doi:10.1029/2009WR008606
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nield, D.A., Kuznetsov, A.V. The Onset of Convection in a Heterogeneous Porous Medium with Vertical Throughflow. Transp Porous Med 88, 347–355 (2011). https://doi.org/10.1007/s11242-011-9742-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11242-011-9742-9