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Effect of Time Periodic Boundary Conditions on Convective Flows in a Porous Square Enclosure with Non-Uniform Internal Heating

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Abstract

The problem of unsteady natural convection in a square region filled with a fluid-saturated porous medium having non-uniform internal heating and heated laterally is considered. The heated wall surface temperature varies sinusoidally with the time about fixed mean temperature. The opposite cold wall is maintained at a constant temperature. The top and bottom horizontal walls are kept adiabatic. The flow field is modelled with the Darcy model and is solved numerically using a finite difference method. The transient solutions obtained are all periodic in time. The effect of Rayleigh number, internal heating parameters, heating amplitude and oscillating frequency on the flow and temperature field as well as the total heat generated within the convective region are presented. It was found that strong internal heating can generate significant maximum fluid temperatures above the heated wall. The location of the maximum fluid temperature moves with time according to the periodically changing heated wall temperature. The augmentation of the space-averaged temperature in the cavity strongly depends on the heating amplitude and rather insensitive to the oscillating frequency.

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Abbreviations

A, a:

Amplitude, dimensionless amplitude

f :

Dimensionless oscillating frequency

g :

Gravitational acceleration

G :

Measure of the internal heat generation

ℓ :

Width of the cavity

K :

Permeability of the porous medium

p :

Local heating exponent

Q :

Space-averaged temperature

Ra :

Rayleigh number

t′, t:

Time, dimensionless time

t p :

Dimensionless period

\({\ddot{t}}\) :

Denoting a summation of time steady state and initial periods

T :

Fluid temperature

u, v:

Velocity components in the x- and y-directions

x′, y′ & x, y:

Space coordinates & dimensionless space coordinates

α m :

Effective thermal diffusivity

β :

Thermal expansion coefficient

γ :

Internal heating parameter

ψ′, ψ:

Stream function, dimensionless stream function

θ :

Dimensionless temperature

μ :

Dynamic viscosity

ν :

Kinematic viscosity

ω :

Oscillating frequency

av:

Average

c :

Cold

h :

Hot

max:

Maximum

ss:

Basic steady state

–:

denoting a quantity average over time

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Authors and Affiliations

  1. School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM, Bangi, Selangor, Malaysia

    H. Saleh & I. Hashim

  2. Department of Mechanical, Materials and Manufacturing Engineering, The University of Nottingham Malaysia Campus, 43500, Semenyih, Selangor, Malaysia

    N. Saeid

Authors
  1. H. Saleh
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  2. I. Hashim
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  3. N. Saeid
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Correspondence to I. Hashim.

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Saleh, H., Hashim, I. & Saeid, N. Effect of Time Periodic Boundary Conditions on Convective Flows in a Porous Square Enclosure with Non-Uniform Internal Heating. Transp Porous Med 85, 885–903 (2010). https://doi.org/10.1007/s11242-010-9597-5

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  • DOI: https://doi.org/10.1007/s11242-010-9597-5

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