Thermal Boundary Layer Flow Research Articles

Theoretical analysis and numerical study of natural convection inside combined solar chimney

AbstractSolar chimneys may induce natural ventilation through solar radiation. However, sufficient theoretical studies are needed as a basis to fully exploit passive design in practical green building design. In this work, we investigate the heat transfer properties of turbulent natural convective flows in a combined solar chimney with a thermal flux at the absorption wall by means of theoretical analysis and numerical simulations. Two different flow patterns have been found, one with a clear thermal boundary layer flow pattern and the other without, based on high Rayleigh numbers. For flow development in these two flow regimes, the transient scaling analysis is performed separately and the control mechanism for each phase is presented. Some new scale relationships are established to characterize the ventilation performance of solar chimneys, including thermal boundary layer thickness δT, velocity vT, mass flow m, and so on. For the distinct thermal boundary layers, δT,s ~ HΓ2/5/Bo1/5κ2/5, vT,s ~ Bo2/5κ4/5Γ1/5/H, m ~ ρBo1/5κ2/5Γ3/5. For nonobvious thermal boundary layers, vT,f ~ Bo1/3κ/H2/3W1/3, m ~ ρBo1/3κ/A2/3. The important scale relationships are validated using corresponding numerical simulation data, such as the mass flow rate scale M ~ (Γ/κ)3/5Bo1/5 in the distinct thermal boundary layer flow state, and so on. The air changes per hour and heat exchange efficiencies are calculated for a solar chimney with a fixed height‐to‐width ratio to provide a basis for the design of a solar chimney.

Computation of hydromagnetic tangent hyperbolic non-Newtonian flow from a rotating non-isothermal cone to a non-Darcy porous medium with thermal radiative flux

A theoretical and numerical study is conducted on nonlinear, steady-state thermal convection boundary layer flow of a magnetized incompressible Tangent Hyperbolic non-Newtonian fluid from a rotating cone to a non-Darcy porous medium. Power-law variation in temperature on the cone surface is considered and thermal radiation heat transfer is also present. The Brinkman-Darcy-Forchheimer model is deployed for the porous medium. The study is motivated by rotational (spin) coating with new emerging magnetic rheological polymers, a process which often utilizes filtration media and high temperatures. The transformed non-dimensional conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite-difference Keller Box technique. The numerical code is validated with previous studies. Extensive visualization of axial, tangential velocity components and temperature distributions with variation in key parameters including Rosseland radiative number, Darcy number, Forchheimer number (non-Darcy inertial parameter), magnetic interaction parameter, tangent-hyperbolic non-Newtonian power-law index and Weissenberg (non-Newtonian) number is included. Additionally, axial and tangential (circumferential) skin friction and Nusselt number values are tabulated for variation in key control parameters. With increasing Weissenberg number, axial velocity is depleted near the cone surface, tangential velocity is suppressed throughout the boundary layer regime and temperature is strongly enhanced. Axial flow is strongly decelerated further from the cone surface with increasing tangent-hyperbolic power-law index and there is also a significant depletion in tangential (swirl) velocity. Temperature is however boosted throughout the boundary layer transverse to the cone surface with a rise in tangent-hyperbolic power-law index. Both tangential and axial velocity are suppressed with increment in magnetic interaction parameter whereas temperature and thermal boundary layer thickness are enhanced. With larger Darcy number (i.e. greater permeability), axial velocity is strongly increased near the cone surface with no tangible modification further from the cone; However tangential velocity is consistently elevated throughout the boundary layer with greater Darcy number whereas temperature is depleted. An increase in Forchheimer number substantially damps both axial and tangential velocity whereas it elevates temperature. Increasing radiative flux strongly energizes the magnetic polymer and elevates temperature but suppresses the axial and tangential velocities. With elevation in non-isothermal wall exponent, axial skin friction is suppressed whereas Nusselt number are elevated at the cone vertex. Further along the cone surface a similar response is observed but there is also a reduction in magnitudes of tangential skin friction.

Thermal Marangoni convection in two-phase quadratic convective flow of dusty MHD trihybrid nanofluid with non-linear heat source

The current study examines the effect of heat generation on thermal Marangoni convective boundary layer flow of dusty trihybrid nanofluid across a flat surface with thermal radiation and non-linear mixed convection. We consider, two phase dusty liquid model with non-linear heat generation. The fluctuation of surface tension gradients leads to the discovery of Marangoni convection. It can be used for growing crystals, drying silicon wafers, stabilising soap films, and wielding. The main objective of this study is to ascertain the trihybrid nanofluid thermal mobility. A trihybrid nanofluid consisting of magnesium oxide (Mgo), titanium oxide (TiO2), silver (Ag) and water as the base fluid is used. This model can improve the efficiency and dependability of thermal systems in a variety of applications by helping to optimize heat transfer processes in materials processing, electronics cooling, and the creation of cutting-edge cooling technologies in energy systems. The existing PDEs are converted into nonlinear ODEs via similarity variables. The nonlinear ODEs are then solved numerically using shooting technique (RKF-45th approach). When the Marangoni convection parameter rises, higher surface tension gradients lead to stronger induced flows and more efficient heat transfer inside the liquid. As the temperature profiles of the dust and fluid phases drop, the distribution of these characteristics in the liquid becomes more homogeneous.

Thermal Analysis of a Casson Boundary Layer Flow over a Penetrable Stretching Porous Wedge

This work aims to analyze the Casson thermal boundary layer flow over an expanding wedge in a porous medium with convective boundary conditions and ohmic heating. Moreover, the effects of porosity and viscous dissipation are studied in detail and included in the analysis. The importance of this study is due to its applications in biomedical engineering where the analysis of behavior of non-Newtonian blood flow in arteries and veins is desired. Within the context of blood flow, it is also applicable to many other fields, for instance, radiative therapy, MHD generators, soil machines, melt-spinning, and insulation processes. The modeled problem is a set of PDEs, which is nondimensionalized to derive a nonlinear boundary value problem (BVP). The obtained BVP is solved using the shooting technique, endowed with the order four Runge-Kutta and Newton methods. The impact of different parameters on the momentum and temperature fields is investigated along with two important parameters of physical significance, i.e., the Nusselt number and the surface drag force. Results are validated, and an excellent agreement is seen for the parameters of interest using MATLAB built-in function bvp4c. A significant finding is that by increasing the Casson liquid parameter, the velocity decreases as the wedge expands quicker than the free stream velocity at R=2. However, the velocity increases for the case when R=0.1. A decrease in the Darcy number increases the temperature profile. Furthermore, the convective parameter accelerated the heat transmission rate, and a rise in the Prandtl number thickens the thermal boundary layer. The findings of this investigation contribute to problems in fluid dynamics and heat transfer that involve studying the behavior of a non-Newtonian fluid with Casson rheological properties near a solid porous wedge surface.

Study on the steady-oscillatory transition of three-dimensional (3D) natural convection via Hopf bifurcation

The transition from steady-state flow to periodic oscillatory flow for the natural convection by Hopf bifurcation is investigated in a three-dimensional (3D) cavity. The spectral collocation method (SCM) in combination with the artificial compressibility method (ACM), which is developed by ourselves as a numerical method SCM-ACM with high accuracy, is employed to solve the governing equations directly instead of linear stability analysis method that is commonly used for the research on flow instability. The results show that the amplitude decays exponentially with time and the decay rate is linear with the Grashof number (Gr). The critical Grashof number for steady-oscillatory transition is obtained as Grcr = 3.423 × 106. The dimensionless angular frequency ωcr = 0.24 is also determined by Fourier analysis. In this work, we also examine the heat-momentum interactions within the boundary layers, visualize the periodic oscillations of temperature and velocity amplitudes, and analyze the origin of instability from multiple angles. The results show that large oscillations of velocity and temperature are observed near the isothermal walls. The oscillation is enhanced by the increase of thermal boundary layer thickness and flow velocity at both ends of isothermal walls. The maximum velocity and temperature amplitudes appear at the lower left and upper right corners of the mid-plane (Z = 0.5), where are the origin of instability, and the spanwise walls are almost independent of oscillations. The oscillatory flow of natural convection in three-dimensional cavity originates from the continuously increasing buoyancy force, and its transition occurs by Hopf bifurcation. Moreover, the temperature amplitude exhibits a wavy distribution on the mid-plane (X = 0.5) and strongly depends on the depth Z. These results provide benchmark data for future numerical studies and engineering application.

Amplitude and oscillating assessment of thermal and magnetic boundary layer flow across circular heated cylinder with heat source/sink

The effects of heat source/sink and magnetohydrodynamics on the oscillatory and periodic quantities of heat transfer and current density characteristics of viscous fluid along the magnetized and heated circular cylinder has been investigated. The governing mathematical model in terms of partial differential equations is converted into dimensionless form. The dimensionless model is again converted into steady, real and imaginary part for oscillating results. The steady, real and imaginary part is transformed into primitive form for smooth algorithm with the help of Primitive Variable Formulation (PVF). The primitive equations are reduced into system of algebraic equations by using Finite Difference Method (FDM). The heat source/sink parameter δ, the magnetic force number ξ, the buoyancy parameter λ, magneto-Prandtl factor γ, and remaining secured factors are utilized to determine computational findings of unknown quantities. Graphs are displayed for velocity, temperature distribution and magnetic profile for all pertinent parameters by FORTRAN and Teplot 360 software. The main novelty of current work is to evaluate oscillatory and periodic quantities of heat transfer and current density by using the steady solutions. It was found that prominent results in temperature distribution are deduced at each angle with the heat source effects. It was expected physically because the heat source is used as a sporting agent to compute heat transfer performance in electrically conducting fluid. The significant amplitude of oscillation in heat transfer and current density is evaluated around every position of circular cylinder. The numerical results of skin friction are compared with existing literature with excellent agreement.

Convective Boundary Layer Flow Adjacent to an Inclined and Linearly Heated Semi-Infinite Plate

Abstract The transient convective flow adjacent to an inclined semi-infinite plate which is heated by a linear temperature gradient is investigated with scaling analysis and direct numerical simulation (DNS) in this study. Both Pr < 1 and Pr > 1 fluids are considered. The initial ambient fluid is quiescent and thermally homogeneous. Important parameters characterizing the thermal boundary layer flow are thickness, characteristic velocity, and time to reach the steady stage. Scaling analysis is carried out to obtain scales for these flow parameters. Compared to previous similar studies, the obtained scale relations are more generalized and they can be utilized for different inclination angles. The derived scales are compared against the DNS results for a variety of flow parameters, e.g., Rayleigh number Ra, Prandtl number Pr, stratification factor s (s = dθw(y)/dy, where θw(y) is the local temperature at a streamwise location of y), inclination angle of the heated plate α, evolutionary time τ, and streamwise location y. The scale relations and the DNS results compare well suggesting the proposed scale laws can provide a sound description for the dynamics of the convective flow subjected to a tilted surface and a linear heating condition.

Nonlinear hydrodynamic and thermodynamic aspects of unsteady Görtler flows

Both steady and unsteady Görtler vortices may lead to an incompressible boundary layer flow to transition from laminar to turbulent flow. Görtler structures can modify hydrodynamic and thermal boundary layers nonlinearly, increasing the time and spanwise averaged velocity and temperature gradients near the surface compared with the laminar counterpart. The nonlinear aspects of unsteady Görtler vortices evolution on thermal boundary layer flows are investigated using a high-order numerical method. In the validation process, the adopted numerical method shows agreement with previously published experimental data in both linear and weakly-nonlinear regions. Nonlinear high-order analyses reveal competition between the temporal fundamental (1,±1) and the steady super-harmonic (0,2) modes. The higher the frequency of the excited mode, the lower its amplification rate. The results show that when a high-frequency three-dimensional wave (modes (1,±1)) is introduced, the transition process is dominated by the super-harmonic (0,2) mode if it is unstable. The time and spanwise average skin friction coefficient and the Stanton number are investigated. The results show that the lower the disturbance frequency, the higher the gain in these two coefficients.

Analysis of Non-Linear Radiation and Activation Energy Analysis on Hydromagnetic Reiner–Philippoff Fluid Flow with Cattaneo–Christov Double Diffusions

Using magnetohydrodynamics (MHD), the thermal energy and mass transport boundary layer flow parameters of Reiner–Philippoff fluid (non-Newtonian) are numerically investigated. In terms of energy and mass transfer, non-linear radiation, Cattaneo–Christov double diffusions, convective conditions at the surface, and the species reaction pertaining to activation energy are all addressed. The stated governing system of partial differential equations (PDEs) is drained into a non-linear differential system using appropriate similarity variables. Numerical solutions are found for the flow equations that have been determined. Two-dimensional charts are employed to demonstrate the flow field, temperature and species distributions, and rate of heat and mass transfers for the concerned parameters for both Newtonian and Reiner–Philippoff fluid examples. The stream line phenomenon is also mentioned in this paper. A table has also been utilized to illustrate the comparison with published results, which shows that the current numerical data are in good accord. The findings point to a new role for heat and mass transfer. According to the findings, increasing values of solutal and thermal relaxation time parameters diminish the associated mass and thermal energy layers. The current study has significant ramifications for chemical engineering systems.

Thermal boundary layer of laminar flow of dilute polymer solution

The thermal boundary layer flow is a canonical flow with characteristics that are present in most natural and industrial convection flows. An approximate self-similar solution is proposed for the first time for the thermal boundary layer of steady laminar flow of viscoelastic fluids, described by the finitely extensible nonlinear elastic constitutive equation with Peterlin's closure (FENE-P model). This semi-analytical thermal solution is obtained by performing an order of magnitude analysis and ensuing simplifications of the governing equations by assuming that the fluid properties are independent of temperature therefore decoupling the flow governing equations from the energy equation. The effects of viscoelasticity quantified with the Weissenberg number based on the streamwise coordinate (x) (Wix) up to Wix=1 and viscous dissipation (results are presented for Brinkman numbers between -40 and +40) on thermal boundary layer characteristics are investigated comprehensively for both constant wall temperature and constant wall heat flux. At low elasticity levels (Wix<0.01) the solution exhibits a global self-similar behavior in which flow and thermal quantities collapse on the corresponding Newtonian curves, and the polymer characteristics show a unique behavior if adequately normalized. However, by increasing flow elasticity the unique self-similar behavior of the approximate solution is lost, with the elasticity dependent results exhibiting local variations. In addition, the effects of elasticity are intensified by viscous dissipation. For the present study cases, it is observed that elasticity may change Nusselt numbers by more than 8%, and the thermal boundary layer thickens by up to 10%.

Flow felid and heat transfer enhancement investigations by using a combination of corrugated tubes with a twisted tape within 3D circular tube based on different dimple configurations

AbstractDifferent dimple geometrical configurations with a combination of corrugated tubes and twisted tape are numerically investigated. Water is used as a working fluid for constant heat flux heat transfer conditions at the pipe wall. The dimensionless diameter of the dimples (d/D) used in this study is 0.09, 0.18, 0.27, and 0.36. However, the corrugation configuration diameter is 1 mm. The numerical simulations are carried out at the Reynolds number in the range of 1500–14,000. The outcomes reveal that the friction factor (f) and Nu number are augmented as the dimple diameter increases. The Nu number ratio of 1.25 is found for a dimple pipe tube with a diameter of 4 mm. The numerical outcome presented more mixing, secondary, and vortex produced in the main flow direction and near the pipe wall to the rotating flow induced by twisted tape. Moreover, mixed, secondary vortices and rotational flow originate behind and near the dimple, twisted tape, and corrugation surfaces. These rotational and vortices can promote mixing in flow between the thermal boundary layer and velocity boundary flow layer. So, increase the heat transfer enhancement. The improved pipes with different dimple diameters produce a maximum performance evaluation factor of is more than 1.25.

MHD thermal boundary layer flow of a Casson fluid over a penetrable stretching wedge in the existence of nonlinear radiation and convective boundary condition

In this paper, we discuss the solution of the magneto hydrodynamics (MHD) thermal boundary layer flow of Casson liquid over a penetrable extending wedge with ohmic heating and convective boundary condition (BC). Important applications of such flows and condition include the MHD pump, radiation therapy, MHD generator, soil mechanics, melt spinning process, thermal insulation and many others. Practical applications of convective BC are often encountered in laser damage or laser processing. The problem is non dimensionalized and computational solutions are achieved for the governing nonlinear coupled ordinary differential equations (PDEs) by using homotopy analysis method (HAM) and influence of various emerging parameters on momentum and temperature field are examined. A table containing computational data for the wall shear stress f′′(0) and local Nusselt -θ′(0) number is also provided. Major findings/objectivesof the present study are: an increasing in Casson fluid parameter δ decreases velocity when wedge stretches faster than free stream velocity for R=2. However, velocity increases for R=0.1 case. Temperature velocity decreases for R=2 or R=0.1 cases for δ escalations. Enhancing Bi for convective heat transfer increases the momentum and thermal boundary layer thickness in the presence of suction C. Computational outcomes for velocity, drag force, temperature velocity and heat transfer rate are discussed through tables numerically. Results are verified with varied choices of suction parameter C. It is noted that the convective parameter magnified the heat transfer rate.

A convective analytical model in turbulent boundary layer on a flat plate based on the unifying heat flux formula

The two-dimensional (2D) convective heat fluxes are formulated for quasi-steady incompressible turbulent thermal boundary layer flow over a flat plate based on the unifying heat-flux formula developed previously. It is indicated that the total turbulent convection heat flux is the combination of three heat transport mechanisms: conduction due to random molecular motion, laminar advection due to time-averaged motion of fluid bulk and turbulent advection due to fluctuations of velocity and temperature. The interdependent relationship between two heat flux components in different directions is obtained for the quasi-steady turbulent boundary layer flow, which is proved to be equivalent to the turbulent differential and integral energy equations with constant physical properties, respectively. The turbulent thermal boundary layer is divided into laminar sublayer and turbulent layer, in which the linear and one-seventh-power temperature (velocity) profiles are adopted, respectively. Considering the different thicknesses of turbulent thermal and velocity boundary layers, the explicit analytical expression of convective turbulent heat transfer coefficient, as well as the simplified analytical expressions of boundary layer thickness ratio, is proposed based on the total turbulent convection heat transfer structure. The suggested analytical turbulent convective heat transfer coefficients (or Stanton numbers) for air and water are validated by comparing with those of previous theories and experimental measurements. Relevant key characteristics and physical insights of the turbulent structure on the flat plates are revealed in virtue of 2D analytical convective heat fluxes proposed.

Homotopy Simulation of Dissipative Micropolar Flow and Heat Transfer from a Two-Dimensional Body with Heat Sink Effect

Non-Newtonian flow from a wedge constitutes a fundamental problem in chemical&lt;br /&gt; engineering systems and is relevant to processing of polymers, coating systems, etc. Motivated by such applications, the homotopy analysis method (HAM) was employed to&lt;br /&gt; obtain semi-analytical solutions for thermal convection boundary layer flow of incompressible micropolar fluid from a two-dimensional body (wedge). Viscous dissipation&lt;br /&gt; and heat sink effects were included. The non-dimensional boundary value problem&lt;br /&gt; emerges as a system of nonlinear coupled ordinary differential equations, by virtue of&lt;br /&gt; suitable coordinate transformations. The so-called Falkner-Skan flow cases are elaborated. Validation of the HAM solutions was achieved with earlier simpler models, as well as with a Nakamura finite difference method for the general model. The micropolar model employed simulates certain polymeric solutions quite accurately, and features rotary motions of micro-elements. Primary and secondary shear stress, wall couple stress, Nusselt number, microrotation velocity, and temperature were computed for the effect of&lt;br /&gt; vortex viscosity parameter (micropolar rheological), Eckert number (viscous dissipation),&lt;br /&gt; Falkner-Skan (pressure gradient) parameter, micro-inertia density, and heat sink parameter. The special cases of Blasius and stagnation flow were also addressed. It was observed from the study that the temperature and thermal boundary layer thickness are both suppressed with increasing wedge parameter and wall heat sink effect, which is beneficial to temperature regulation in polymer coating dynamics. Further, strong reverse spin was generated in the microrotation with increasing vortex viscosity, which resulted in&lt;br /&gt; increase in angular momentum boundary layer thickness. Also, both primary and secondary skin friction components were reduced with increasing wedge parameter. Nusselt number was also enhanced substantially with greater wedge parameter.

The effects of zero and high shear rates viscosities on the transportation of heat and mass in boundary layer regions: A non-Newtonian fluid with Carreau model

The momentum and thermal boundary layer flow problem for shear thinning fluid with modified viscosity model over moving wedge is discussed. The mathematical model of the problem is developed by using improved Carreau model relates viscosity to shear rate and using the thermal conductivity model in the similar form of viscosity expression. The improved Carreau model contains five-parameters: zero viscosity, high shear rate viscosity, flow behavior index, consistency index and viscosity curvature which cover almost all aspects of the fluid's rheology. On the governing equations of the problem, boundary layer approximations are used and then reduced into ordinary differential equations by using similarity transformations. The results in form of velocity and temperature profiles, thickness of boundary layers regions, deflation in volume and momentum flow rates under the influence of all the parameters are calculated and displayed in graphs and tables for discussion. The effects of zero viscosity on different profiles are dominant as compared to high shear rate viscosity. The velocity is decreased but temperature profile is increased by the parameters that are involved to increase to viscosity.

Heat transfer through mixed convection boundary layer in a porous medium: LTNE analysis

This paper concerns the LTNE model for two-dimensional mixed convection boundary layer flow of a viscous fluid over a wedge embedded in the porous medium. The temperature for the fluid and solid field is governed by the two separate energy equations: one each for fluid and solid phase due to LTNE condition. Since the flow velocity is sufficiently large, the momentum and thermal boundary layers form near the wedge surface. The resulting mathematical equations in terms of partial differential equations are converted into equivalent ordinary differential equations using appropriate similarity transformations and are solved numerically. Numerical results on various physical quantities reveal that the LTNE effects are more predominant for small inter-phase heat transfer rate and porosity scaled conductivity. For mixed convection thermal boundary layer flow, LTNE effects are not as strong as for forced convection heat transfer. However, when the inter-phase interaction and porosity-scaled conductivity are increased, the LTE is recovered for which both solid and fluid phases become almost equal. Simulations on velocity profiles show that the pressure gradient and permeability have less effects on the boundary layer but cannot be neglected. The physical dynamics behind these effects are discussed in some detail.

A Unified Treatment for Local Thermal Nonequilibrium Free, Forced, and Mixed Convective Boundary Layer Flows Over an Arbitrarily Shaped Nonisothermal Body in a Fluid Saturated Porous Medium

Abstract A unified integral solution procedure has been proposed to analyze all possible Darcian local thermal nonequilibrium (LTNE) free, forced, and mixed convective boundary layer flows, commonly encountered in porous media engineering applications. The heated body may be arbitrarily shaped, and its temperature may vary over the surface. The integral energy equation for the solid phase yields an algebraic equation between the dimensionless fluid thermal boundary layer thickness and its ratio to the solid-phase counterpart, while the integral energy equation for the fluid phase reduces to a first order ordinary differential equation in terms of the dimensionless fluid thermal boundary layer thickness. This set of the equations for determining the local Nusselt number of our primary interest proved to be valid for all possible Darcian cases of LTNE free, forced, and mixed convective boundary layer flows over an arbitrarily shaped nonisothermal body in a fluid saturated porous medium. Asymptotic expressions for the cases of arbitrary shapes were also obtained analytically for both leading edge and far downstream regions. The results are found to agree well with available direct numerical integration results. Furthermore, the regime map has been constructed to show the boundary layer transition point from the LTNE to equilibrium. The proposed unified method is found quite useful when designing thermal engineering systems associated with fluid saturated porous media.

Time-Dependent Nonlinear Finite Difference Analysis of Buoyancy-Driven Convective Flow over an Oscillating Porous Moving Vertical Cylinder

The thermal and mass boundary layer flow of a time-dependent isochoric viscous fluid over an oscillating cylinder entrenched in a porous medium is investigated numerically. The governing problem contains a set of coupled nonlinear partial differential equations with significant initial and boundary conditions and are solved using an implicit finite difference scheme, namely Crank–Nicholson method. The importance of physical parameters on velocity profiles, temperature profiles, concentration profiles, local skin-friction coefficient, mean skin-friction coefficient, local Nusselt number, mean Nusselt number, local Sherwood number and mean Sherwood number is explained in detail and is depicted graphically. The validation of these results is verified with the existing results, and it is correlated well.

Two-Phase Microscopic Heat Transfer Model for Three-Dimensional Stagnation Boundary-Layer Flow in a Porous Medium

Abstract This work examines the steady three-dimensional forced convective thermal boundary-layer flow of laminar and incompressible fluid in a porous medium. In this analysis, it is assumed that the solid phase and the fluid phase, which is immersed in a porous medium are subjected to local thermal nonequilibrium (LTNE) conditions, which essentially leads to one thermal boundary-layer equation for each phase. Suitable similarity transformations are introduced to reduce the boundary-layer equations into system of nonlinear ordinary differential equations, which are analyzed numerically using an implicit finite difference-based Keller-box method. The numerical results are further confirmed by the asymptotic solution of the same system for large three-dimensionality parameter, and the corresponding results agree well. Our results show that the thickness of boundary layer is always thinner for all permeability parameters tested when compared to the nonporous case. Also, it is noticed that the temperature of solid phase is found to be higher than the corresponding fluid phase for any set of parameters. There is a visible temperature difference in the two phases when the microscopic interphase rate is quite large. The physical hydrodynamics to these parameters is studied in some detail.

Effect of three modes of linear thermal forcing on convective flow and heat transfer in rectangular cavities

The convective flow in rectangular cavities with different linear thermal forcing at the sidewalls is investigated by DNS (direct numerical simulation) in the present study. A non-finned cavity and a cavity with an adiabatic horizontal thin fin attached to the middle of both sidewalls are modelled. Three heating modes are considered and calculated, i.e. s < 0, s = 0 and s > 0 thermal forcing respectively and six Rayleigh numbers ranging from 1.15 × 108 to 3.68 × 109 are examined. It demonstrates that compared to the s = 0 case, which is the classic homogenously heating problem, the thermal boundary layer flow as well as the convective flow in the cavities is strengthened for the s < 0 heating mode and weakened for the s > 0 heating mode. It is found from the finned cavity calculation that, for all the three thermal forcing considered, significant flow oscillation is observed on top the thin fin which in turn causes strong disturbances in its downstream flow adjacent to the sidewalls. The present results suggest that compared to simply increasing the Rayleigh number of the flow, switching the heating mode to s < 0 is potentially a shortcut for the purposes of promoting the flow. However, it is further revealed that due to the existence of the so called neutral buoyant effect at the sidewalls, the heat transfer associated with the s < 0 scenario is lower than the other two heating modes.