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The present numerical study deals with natural convection flow in a trapezoidal cavity when the bottom wall is uniformly heated and the vertical wall(s) are linearly heated and cooled whereas the top wall is well insulated. Non-linear coupled partial differential equations governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Parametric study for the wide range of Rayleigh number (Ra), 103 ≤ Ra ≤ 105 and Prandtl number (Pr), 0.07 ≤ Pr ≤ 100 shows consistent performance of the present numerical approach to obtain the solutions in terms of stream functions and the temperature profiles. For linearly heated side walls symmetry is observed while representing the flow patterns in terms of stream functions whereas secondary circulation is observed for the linearly heated left wall and cooled right wall. Local Nusselt number becomes negative at the side wall for linearly heated side walls and at the left wall for linearly heated left wall and cooled right wall indicating the reversal of heat flow. The effect of Prandtl number in the variation of average Nusselt numbers is more significant for Prandtl numbers in the range 0.07 to 0.7 than 10 to 100. Copyright © 2007 by ASME.

GENERAL

Department Of Chemical Engineering

Department of Mathematics

2007 Proceedings of the ASME/JSME Thermal Engineering Summer Heat Transfer Conference - HT 2007

Natural convection in a trapezoidal cavity with linearly heated side wall(S)

The influence of uniform and nonuniform heating of wall(s) on natural-convection flow in a square cavity filled with a porous matrix has been studied numerically by using the penalty finite-element method with biquadratic rectangular elements. In the present investigation, the left vertical wall and the bottom wall are uniformly and nonuniformly heated, while the right vertical wall is maintained at constant cold temperature and the top wall is well insulated. The Darcy-Forchheimer model is used to simulate the momentum transfer in the porous medium. The present numerical approach yields consistent performance over the range of parameters (Rayleigh number Ra, 103Ra106, Darcy number Da, 10-5Da10-3, and Prandtl number Pr, 0.71Pr10) in order to obtain the solutions in terms of stream functions, temperature profiles, and Nusselt numbers. Nonuniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than the uniform heating case for all Rayleigh numbers, but average Nusselt number shows overall lower heat transfer rate for the nonuniform heating case. Critical Rayleigh numbers for conduction-dominant heat transfer cases have been obtained. For convection-dominated regimes the power-law correlations between average Nusselt number and Rayleigh numbers are presented.

Numerical Heat Transfer; Part A: Applications

Effects of nonuniformly heated wall(S) on a natural-convection flow in a square cavity filled with a porous medium

The present numerical study deals with natural convection flow in a closed square cavity when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Non-linear coupled PDEs governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Numerical results are obtained for various values of Rayleigh number (Ra) (103 ≤ Ra ≤ 105) and Prandtl number (Pr) (0.7 ≤ Pr ≤ 10). Results are presented in the form of streamlines, isotherm contours, local Nusselt number and the average Nusselt as a function of Rayleigh number. © 2006.

International Journal of Heat and Mass Transfer

Steady natural convection flows in a square cavity with linearly heated side wall(s)

An analysis is performed to obtain the non-similar solution of a steady laminar forced convection boundary layer flow over a horizontal slender cylinder including the effect of non-uniform slot injection (suction). The effects of transverse curvature and viscous dissipation are also included in the analysis. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form using suitable transformations and the resulting system of nonlinear coupled partial differential equations is then solved by an implicit finite difference scheme in combination with the quasilinearization technique. Numerical results for the effect of non-uniform slot injection (suction) on skin friction coefficient and heat transfer rate are presented. The effects of transverse curvature, viscous dissipation and Prandtl number on velocity and temperature profiles and skin friction and heat transfer coefficients are also reported. © 2005 Elsevier Ltd. All rights reserved.

Effect of non-uniform slot injection (suction) on a forced flow over a slender cylinder

A general analysis has been developed to study flow and heat transfer characteristics of an unsteady laminar mixed convection on a continuously moving vertical slender cylinder with surface mass transfer where the slender cylinder is inline with the flow. The unsteadiness is introduced by the time-dependent velocity of the slender cylinder as well as that of the free stream. The calculations of momentum and heat transfer on slender cylinders considered the transverse curvature effect, especially in applications such as wire and fiber drawing, where accurate predictions are required. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a nonsimilar transformation, and the resulting system of nonlinear coupled partial differential equations is then solved by an implicit finite difference scheme in combination with the quasi-linearization technique. Numerical results are presented for the skin friction coefficient and Nusselt number. The effects of various parameters on the velocity and temperature profiles are also reported here. Copyright © 2006 by ASME.

Journal of Heat Transfer

Unsteady mixed convection from a moving vertical slender cylinder

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy-Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 103 ≤ Ra ≤ 106, Darcy number Da, 10-5 ≤ Da ≤ 10-3, and Prandtl number Pr, 0.71 ≤ Pr ≤ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ≤ 10 -5 irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ≥ 10-4. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented. © 2005 Elsevier Ltd. All rights reserved.

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 103 ≤ Ra ≤ 105 and Prandtl number Pr, 0.7 ≤ Pr ≤ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented. © 2006 Elsevier Ltd. All rights reserved.

Effects of thermal boundary conditions on natural convection flows within a square cavity

A general analysis has been developed to study the flow and heat transfer characteristics of an unsteady laminar mixed convection on a continuously moving vertical slender cylinder. The governing boundary layer equations along with the boundary conditions are first cast into a dimensionless form by a nonsimilar transformation and the resulting system of nonlinear coupled partial differential equations is then solved by an implicit finite difference scheme in combination with the quasilinearization technique. © 2005 Elsevier Ltd.

3rd M.I.T. Conference on Computational Fluid and Solid Mechanics

Nonsimilar solution of an unsteady mixed convection flow on a moving slender cylinder

A penalty finite element analysis with bi-quadratic rectangular elements is performed to investigate the influence of uniform and non-uniform heating of wall(s) on natural convection flows in a square cavity. In the present investigation, one vertical wall and the bottom wall are uniformly and non-uniformly heated while the other vertical wall is maintained at constant cold temperature and the top wall is well insulated. Parametric study for a wide range of Rayleigh number (Ra), 103 ≤ Ra ≤ 106 and Prandtl number (Pr), 0.2 ≤ Pr ≤ 100 shows consistent performance of the present numerical approach to obtain the solutions as stream functions and temperature profiles. Heat transfer rates at the heated walls are presented in terms of local Nusselt number. © 2005 Elsevier Ltd. All rights reserved.

International Journal of Engineering Science

Finite element analysis of natural convection flows in a square cavity with non-uniformly heated wall(s)

A numerical study is performed to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over the range of parameters (Rayleigh number Ra, 10 3 ≤ Ra ≤ 105 and Prandtl number Pr, 0.7 ≤ Pr ≤ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained. Copyright © 2005 by ASME.

Proceedings of the ASME Summer Heat Transfer Conference

Natural convection inside a square cavity heated from below and symmetrically cooled at both sides: Effects of various dirichlet boundary conditions at the bottom wall

This paper deals with a new self-similar solution of unsteady mixed convection flow on a rotating cone in a rotating fluid due to the combined effects of thermal and mass diffusion. It has been shown that a self-similar solution is possible when the free stream angular velocity and the angular velocity of the cone vary inversely as a linear functions of time. The system of ordinary differential equations governing the flow has been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. Both prescribe wall temperature and prescribed heat flux conditions are considered. Numerical results are reported for the skin friction coefficients, Nusselt number and Sherwood number. The effect of various parameters on the velocity, temperature and concentration profiles are also presented here. © 2003 Elsevier Inc. All rights reserved.

Applied Mathematics and Computation

Unsteady mixed convection flow on a rotating cone in a rotating fluid

An analysis is performed to present a new self-similar solution of unsteady mixed convection boundary layer flow in the forward stagnation point region of a rotating sphere where the free stream velocity and the angular velocity of the rotating sphere vary continuously with time. It is shown that a self-similar solution is possible when the free stream velocity varies inversely with time. Both constant wall temperature and constant heat flux conditions have been considered in the present study. The system of ordinary differential equations governing the flow have been solved numerically using an implicit finite difference scheme in combination with a quasilinearization technique. It is observed that the surface shear stresses and the surface heat transfer parameters increase with the acceleration and rotation parameters. For a certain value of the acceleration parameter, the surface shear stress in x-direction vanishes and due to further reduction in the value of the acceleration parameter, reverse flow occurs in the x-component of the velocity profiles. The effect of buoyancy parameter is to increase the surface heat transfer rate for buoyancy assisting flow and to decrease it for buoyancy opposing flow. For a fixed buoyancy force, heating by constant heat flux yields a higher value of surface heat transfer rate than heating by constant wall temperature. © Springer-Verlag 2003.

Heat and Mass Transfer/Waerme- und Stoffuebertragung

Self-similar solution of the unsteady mixed convection flow in the stagnation point region of a rotating sphere

Unsteady flow over an infinite permeable rotating cone in a rotating fluid in the presence of an applied magnetic field has been investigated. The unsteadiness is induced by the time-dependent angular velocity of the body, as well as that of the fluid. The partial differential equations governing the flow have been solved numerically by using an implicit finite-difference scheme in combination with the quasi-linearization technique. For large values of the magnetic parameter, analytical solutions have also been obtained for the steady-state case. It is observed that the magnetic field, surface velocity, and suction and injection strongly affect the local skin friction coefficients in the tangential and azimuthal directions. The local skin friction coefficients increase when the angular velocity of the fluid or body increases with time, but these decrease with decreasing angular velocity. The skin friction coefficients in the tangential and azimuthal directions vanish when the angular velocities of fluid and the body are equal but this does not imply separation. When the angular velocity of the fluid is greater than that of the body, the velocity profiles reach their asymptotic values at the edge of the boundary layer in an oscillatory manner, but the magnetic field or suction reduces or suppresses these oscillations.

Meccanica

Unsteady MHD flow on a rotating cone in a rotating fluid

The non-uniform slot injection (suction) into water boundary layer flow over a yawed infinite circular cylinder is analyzed including the temperature-dependent viscosity and Prandtl number. The difficulties arising at the starting point of the streamwise coordinate, at the edges of the slot and at the point of separation are overcome by applying the implicit finite difference scheme in combination with the quasilinearization technique. The results indicate that the separation can be delayed by non-uniform slot suction and also by moving the slot downstream but effect of non-uniform slot injection is just the opposite. Further, the effect of variable fluid properties is to move the point of separation downstream but yaw angle has very little affect on the location of the point of separation. © 2004 Elsevier Ltd. All rights reserved.

Non-uniform slot injection (suction) into water boundary layer flow past yawed cylinder

A semi-similar solution of an unsteady mixed convection flow over a rotating cone in a rotating viscous fluid has been obtained when the free stream angular velocity and the angular velocity of the cone vary arbitrarily with the time. Both the prescribed wall temperature and the prescribed heat flux conditions are considered in the present investigation. The non-linear coupled partial differential equations governing the mixed convection flow have been solved numerically using an implicit finite difference scheme in combination with the quasi-linearization technique. Numerical results are presented for the skin friction coefficients, Nusselt number and Sherwood number. The effects of various parameters on the velocity, temperature and concentration profiles are also reported here. © 2003 Elsevier Ltd. All rights reserved.

Unsteady mixed convection from a rotating cone in a rotating fluid due to the combined effects of thermal and mass diffusion

An analysis is performed to study the effect of non-uniform slot injection (suction) into steady two-dimensional and axi-symmetric laminar boundary layers with variable viscosity and Prandtl number. Non-similar solutions have been obtained from the starting point of the streamwise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the streamwise co-ordinate, at the edges of the slot and at the point of separation have been overcome by applying an implicit finite difference scheme with the quasi-linearization technique and an appropriate selection of finer step size along the streamwise direction. The results indicate that the separation can be delayed by non-uniform slot suction and also by moving the slot downstream but the effect of non-uniform slot injection is just the opposite. The heat transfer rate is found to depend strongly on viscous dissipation but the skin friction is little affected by it. © 2003 Elsevier Science Ltd. All rights reserved.

Non-uniform slot injection (suction) into water boundary layers over (i) a cylinder and (ii) a sphere

The unsteady free convection flow over an infinite vertical porous plate, which moves with time-dependent velocity in an ambient fluid, has been studied. The effects of the magnetic field and Hall current are included in the analysis. The buoyancy forces arise due to both the thermal and mass diffusion. The partial differential equations governing the flow have been solved numerically using both the implicit finite difference scheme and the difference-differential method. For the steady case, analytical solutions have also been obtained. The effect of time variation on the skin friction, heat transfer and mass transfer is very significant. Suction increases the skin friction coefficient in the primary flow, and also the Nusselt and Sherwood numbers, but the skin friction coefficient in the secondary flow is reduced. The effect of injection is opposite to that of suction. The buoyancy force, injection and the Hall parameter induce an overshoot in the velocity profiles in the primary flow which changes the velocity gradient from a negative to a positive value, but the magnetic field and suction reduce this velocity overshoot.

Unsteady free convection flow over an infinite vertical porous plate due to the combined effects of thermal and mass diffusion, magnetic field and Hall currents

The influence of non-uniform slot injection (suction) into steady axi-symmetric laminar incompressible boundary layer flows with temperature dependent viscosity and Prandtl number has been examined from the origin of the streamwise coordinate to the exact point of separation. The difficulties in obtaining the non-similar solutions at the origin of the streamwise coordinate, at the edges of the slot and at the point of separation have been overcome by applying an implicit finite difference scheme with the quasilinearization technique and an appropriate selection of finer step size along the streamwise coordinate. The results indicate that the separation can be delayed by non-uniform slot suction and also by moving the slot downstream but the effect of non-uniform slot injection is just the opposite. Further, the effect of variable fluid properties is to move the point of separation downstream but rotation parameter has the reverse effect. © 2003 Elsevier Science Ltd. All rights reserved.

Non-uniform slot injection (suction) into steady laminar water boundary layer flow over a rotating sphere

Non-similar solutions of steady two-dimensional and axi-symmetric compressible laminar boundary layer flows with non-uniform slot injection (or suction) have been obtained, from the starting point of the stream-wise co-ordinate to the exact point of separation. The difficulties arising at the starting point of the stream-wise co-ordinate, and at the edges of the slot and at the point of separation have been overcome by applying the method of quasi-linear implicit finite difference scheme, along with an appropriate selection of finer step size along the stream-wise direction. It is observed that the separation can be delayed by a non-uniform slot suction and also by moving the slot downstream but the effect of non-uniform slot injection is just the opposite. The increase of total enthalpy at the wall causes the separation to occur earlier while cooling delays it. The increase of Mach number shifts the point of separation upstream due to the adverse pressure gradient.

Compressible boundary layer flow with non-uniform slot injection or (suction) over i) a cylinder and ii) a sphere

An analysis is performed to study the influence of temperature-dependent viscosity and Prandtl number on the steady nonsimilar laminar forced convection flow over a rotating sphere up to the point of separation. The difficulties arising at the starting point of the streamwise coordinate and at the point of separation are overcome by applying the method of quasilinear implicit finite difference scheme with an appropriate selection of finer step size along the streamwise direction. The results indicate that the effect of variable viscosity and Prandtl number is to move the point of separation downstream, but the rotation parameter has the reverse effect. For higher wall temperature than the free stream temperature, beyond a certain critical value of the dissipation parameter, the cooler free stream is unable to cool the hot wall due to the "heat cushion" provided by the frictional heating. The heat transfer rate is found to depend strongly on viscous dissipation but the skin frictions are little affected by it. In general, the results pertaining to variable fluid properties differ significantly from those of constant fluid properties.