Free convection from a heated vertical cylinder embedded in a fluid-saturated porous medium

Summary We consider the free convection boundary layer flow induced by a heated vertical cylinder which is embedded in a fluid-saturated porous medium. The surface of the cylinder is maintained at a temperature whose value above the ambient temperature of the surrounding fluid varies as then ^th power of the distance from the leading edge. Asymptotic analyses and numerical calculations are presented for the governing nonsimilar boundary layer equations and it is shown that, whenn<1, the asymptotic flowfield far from the leading edge of the cylinder takes on a multiple-layer structure. However, forn>1, only a simple single layer is present far downstream, but a multiple layer structure exists close to the cylinder leading edge. We have shown that the fully numerical and asymptotic calculations are in stisfactory agreement, especially for exponentsn close to zero. Comparisons of the present numerical solutions obtained using the Keller-box method with previous numerical solutions using local methods are also given.List of symbols a radius - scaled streamfunctions - f ₀,f ₁,f ₂ inner zone streamfunctions whenn<1 - leading order streamfunctions inn>1, 1 asymptotic solution - F ₀,F ₁ outer zone streamfunctions whenn<1 - G large parameter satisfyingG=X ² lnG - g gravitational acceleration - K permeability of the porous medium - n exponent in prescribed temperature law - r radial co-ordinate - r rescaled radial co-ordinate - R Darcy-Rayleigh number - T temperature of convective fluid - T _w temperature of cylinder at leading edge - T ambient temperature of fluid - u velocity in axial direction - v velocity in azimuthal direction - w velocity in radial direction - x axial co-ordinate - x escaled axial co-ordinate - X dimensionless axial co-ordinate - thermal diffusivity of the saturated medium - coefficient of thermal expansion - constant in the boundary conditions forF ₀ - dimensionless radial co-ordinate - co-ordinate for the outer zone in then<1 solution - scaled radial co-ordinates - scaled fluid temperature - similarity variable for then=1 problem - nondimensionalisation constant (Eq. (9)) - viscosity of fluid - scaled axial co-ordinates - density of fluid - co-ordinate for the inner zone in then<1 solution - azimuthal co-ordinate - similarity variables for then>1 problem - streamfunction     

Mixed convection boundary layer flow on a vertical surface in a saturated porous medium

Summary The flow of a uniform stream past an impermeable vertical surface embedded in a saturated porous medium and which is supplying heat to the porous medium at a constant rate is considered. The cases when the flow and the buoyancy forces are in the same direction and when they are in opposite direction are discussed. In the former case, the flow develops from mainly forced convection near the leading edge to mainly free convection far downstream. Series solutions are derived in both cases and a numerical solution of the equations is used to describe the flow in the intermediate region. In the latter case, the numerical solution indicates that the flow separates downstream of the leading edge and the nature of the solution near this separation point is discussed.     

Free convection flow past a vertical flat plate embedded in a saturated porous medium

A numerical solution for the free convection flow past a vertical semi-infinite flat plate embedded in a highly saturated porous medium by allowing the plate to have a non-uniform temperature or a non-uniform heat flux distributions has been developed. Both local heat transfer rate and excess surface temperature as a function of the distance along the plate are tabulated for a few cases of prescribed wall temperature and heat flux distributions. Such tabulations serve as a reference against which other approximate solutions can be compared in the future.     

Non-Darcy mixed convection flow with thermal dispersion on a vertical cylinder in a saturated porous medium

Summary The non-Darcy mixed convection flow on a vertical cylinder embedded in a saturated porous medium has been studied taking into account the effect of thermal dispersion. Both forced flow and buoyancy force dominated cases with constant wall temperature condition have been considered. The governing partial differential equations have been solved numerically using the Keller box method. The results are presented for the buoyancy parameter which cover the entire regime of mixed convection flow ranging from pure forced convection to pure free convection. The effect of thermal dispersion is found to be more pronounced on the heat transfer than on the skin friction and it enhances the heat transfer but reduces the skin friction.     

Transient free convection about a vertical flat plate embedded in a porous medium

The problems of transient free convection in a porous medium adjacent to a vertical semi-infinite flat plate with a step increase in wall temperature and surface heat flux are considered in this paper. By assuming a temperature profile in each case, the governing equation for the boundary layer thickness is obtained by an integral method. These governing equations are first-order partial differential equations of the hyperbolic type that can be solved exactly by the method of characteristics and approximately by the method of integral relations. The results based on the method of characteristics clearly indicate that during the initial stage when the leading edge effect is not being felt, heat is transferred as if by transient 1-dimensional heat conduction. At a later time, depending on the vertical location, the heat transfer characteristics change from transient 1-dimensional heat conduction to steady 2-dimensional convection. The thickness of the boundary layer is shown to be increasing with time until it reaches steady state where its value remains constant thereafter. The growth rate of the boundary layer thickness exhibits a discontinuity at the end of the transient period and the beginning of the steady state period. On the other hand, the results based on the method of integral relations show that the boundary layer thickness grows continuously with time and approaches the steady state value asymptotically; the growth rate of the boundary layer thickness decreases from a finite value to zero continuously as the steady state is approached. Except between the end of the transient period and the beginning of the steady state period, the results based on the method of integral relations are in good agreement with those based on the method of characteristics.     

Mixed convection flow in a porous medium bounded by two vertical walls

The aim of the investigation is to study the fully developed mixed convection flow in a porous medium bounded by two vertical walls having a linear axial temperature variation. Solution of the governing second order simultaneous differential equations, derived by non-dimensionalising the momentum and energy equations, has been obtained by converting them into a fourth order differential equation. It has been found that the fourth order differential equation has five different solutions depending upon the values of the Darcy number, Rayleigh number and ratio of the effective viscosity of the porous domain to the viscosity of the fluid. Graphical representation of the analytical solution shows that the flow is of parabolic type only for higher Darcy number whereas there is a reverse type of motion for lower Darcy numbers. The effect of the ratio of the effective viscosity of the porous matrix to the viscosity of the fluid is to decrease the velocity field.     

Free convection and radiation characteristics for a vertical plate embedded in a porous medium

Steady two‐dimensional free convection flow due to combined effect of radiation and convection through a porous medium bounded by a vertical infinite plate is considered. The behaviour of Darcy and non‐Darcy flow is investigated. The flow of water through different porous media under different environmental conditions is discussed. Effect of four non‐dimensional parameters, i.e. Prandtl number (Pr), modified Grashof number (G), permeability parameter (K) and radiation parameter (N) has been studied. Effect of these parameters on Nusselt number is analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed convection with viscous dissipation in a vertical channel filled with a porous medium

Summary The fully developed laminar mixed convection flow in a vertical plane parallel channel filled with a porous medium and subject to isoflux ÷ isothermal wall conditions is investigated assuming that (i) the Darcy law and the Boussinesq approximation hold, (ii) the effect of viscous dissipation is significant, and (iii) the average flow velocity U _m (as an experimentally accessible quantity) is prescribed. It is shown that under these conditions both upward (U _m > 0) and downward (U _m < 0) laminar flow solutions may exist as long as U _m does not exceed a maximum value U _{m, max}. The velocity field can either be unidirectional or bidirectional. Moreover, bidirectional flow configurations are possible also for U _m = 0. A remarkable feature of the problem is that for U _m < U _{m, max} even two solution branches (dual solutions) exist, which merge when U _m approaches its maximum value U _{m, max}. The mechanical and thermal characteristics of the flow configurations associated with the dual solutions are investigated in the paper analytically and numerically in detail.     

Natural convection on a thin vertical cylinder moving in a high-porosity ambient medium

An analysis has been performed to study the natural convection flow over a thin vertical cylinder which is moving with a constant velocity in a non-Darcy high-porosity ambient medium. Both constant wall temperature and constant heat flux conditions have been considered. The coupled non-linear parabolic partial differential equations have been solved numerically by using an implicit finite-difference scheme. The heat transfer is found to be significantly affected by the inertia and porosity parameters, and the Prandtl number, whereas the skin friction is weakly affected. The heat transfer for the constant heat flux case is more than that of the constant wall temperature case and this difference increases with the Prandtl number. The heat transfer increases with the buoyancy force, but the skin friction is slightly reduced.     

Mixed convection flow near an axisymmetric stagnation point on a vertical cylinder

The mixed convection flow near an axisymmetric stagnation point on a vertical cylinder is considered. The equations for the fluid flow and temperature fields reduce to similarity form that involves a Reynolds number R and a mixed convection parameter λ, as well as the Prandtl number σ. Numerical solutions are obtained for representative values of these parameters, which show the existence of a critical value λ _c  = λ _c (R, σ) for the existence of solutions in the opposing (λ < 0) case. The variation of λ _c with R is considered. In the aiding (λ > 0) case solutions are possible for all λ and the asymptotic limit λ → ∞ is obtained. The limits of large and small R are also treated and the nature of the solution in the asymptotic limit of large Prandtl number is briefly discussed.     

The interaction of convection modes in a box of a saturated porous medium

Convection in an infinite layer of a porous medium occurs if the dimensionless Rayleigh number exceeds a critical value. This is also true for a box of a porous medium; however, each discrete modal solution has its own associated critical Rayleigh number. Usually just one mode will be generated at the onset of convection, but there are many critical box dimensions for which up to four modes share the same critical Rayleigh number, and all may be generated at the onset of convection. In such circumstances there will be a slow interchange of energy between the preferred modes. A perturbation method is applied to a system where three modes are generated at onset to yield a system of ordinary differential equations which govern the evolution of the amplitudes of the viable modes. Three unique cases arise, each with a different phase space structure. Critical boxes with ‘moderate’ aspect ratios are systematically categorised into these cases. While two of the examples represent the usual case where just one mode survives in the final state, the third example is a special case where it is possible for the three modes to coexist. The initial conditions determine which mode(s) will survive.     

Unconditional nonlinear stability for convection in a porous medium with vertical throughflow

Summary Linear and nonlinear stability analyses of vertical throughflow in a fluid saturated porous layer, which is modelled using a cubic Forchheimer model, are studied. To ensure unconditional nonlinear results are obtainable, and to avoid the loss of key terms, a weighted functional is used in the energy analysis. The linear instability and nonlinear stability thresholds show considerable agreement when the vertical throughflow is small, although there is substantial deterioration of this agreement as the vertical throughflow increases.     

Free convection on a horizontal plate in a saturated porous medium with prescribed heat transfer coefficient

Summary The free convection on a horizontal plate embedded in a saturated porous medium is considered assuming that the plate is subjected to a prescribed temperature or a prescribed heat flux or a prescribed heat transfer coefficient. By similarity transformation the governing equations are reduced to identical coupled equations for all the three cases with three common boundary conditions and one boundary condition depending on the thermal boundary condition imposed. It is shown that there is no need to solve the three boundary value problems independently and that the solution for one case can be used to obtain the solution for any other case by a simple algebraic method.Notation A transition parameter used in Eq. (19) - C function ofx defined in Eq. (11.1) - f dimensionless stream function - g acceleration due to gravity - k thermal conductivity - K permeability of the porous medium - N heat transfer coefficient - p fluid pressure - p _e ambient pressure - PT prescribed temperature - PHF prescribed heat flux - PHTC prescribed heat transfer coefficient - q _w surface heat flux - R function ofx defined in Eq. (11.2) - T temperature - T _w plate temperature - T _e ambient temperature - T _w temperature difference=T _w-T _e - u velocity in thex-direction - v velocity in they-direction - x coordinate along the plate in the upward direction - y coordinate normal to the plate - equivalent thermal diffusivity - coefficient of thermal expansion - dimensionless similarity variable - dimensionless temperature - exponent inC - viscosity - kinematic viscosity - fluid density - _e ambient density - stream function