Natural convection in a square cavity due to thermally active plates for different boundary conditions

This work deals with the study of natural convection cooling of thermally active plates placed inside an air filled cavity at the center, with two different boundary conditions imposed on the cavity walls. By an active plate we mean one that is hotter due to isothermal heating or inherent heat generation. The walls of the cavity are subjected to either an isothermal temperature or a uniform outward heat flux. The finite difference method using the alternating direction implicit method coupled with the successive over-relaxation technique is employed to solve the governing nonlinear coupled equations. The results are presented and discussed in terms of a steady state isotherm and streamline plot, and over all Nusselt numbers. This study will provide qualitative suggestions that may improve the thermal design of sealed modern electronic packages which are encountered frequently in the electronics industry.     

Parameter-free symmetry-preserving regularization modeling of a turbulent differentially heated cavity

Since direct numerical simulations of buoyancy driven flows cannot be computed at high Rayleigh numbers, a dynamically less complex mathematical formulation is sought. In the quest for such a formulation, we consider regularizations (smooth approximations) of the non-linearity: the convective term is altered to reduce the production of small scales of motion by means of vortex stretching. In doing so, we propose to preserve the symmetry and conservation properties of the convective terms exactly. This requirement yielded a novel class of regularizations [Comput Fluids 2008;37:887] that restrain the convective production of smaller and smaller scales of motion in an unconditionally stable manner, meaning that the velocity cannot blow up in the energy-norm (in 2D also: enstrophy-norm). The numerical algorithm used to solve the governing equations preserves the symmetry and conservation properties too. In the present work, a criterion to determine dynamically the regularization parameter (local filter length) is proposed: it is based on the requirement that the vortex stretching must stop at the scale set by the grid. Therefore, the proposed method constitutes a parameter-free turbulence model. The resulting regularization method is tested for a 3D natural convection flow in an air-filled (Pr = 0.71) differentially heated cavity of height aspect ratio 4. Direct comparison with DNS results at Rayleigh number 6.4 × 10⁸ ? Ra ? 10¹¹ shows fairly good agreement even for very coarse grids. Finally, the robustness of the method is tested by performing simulations with Ra up to 10¹⁷. A 2/7 scaling law of Nusselt number has been obtained for the investigated range of Ra.     

Natural convection in a porous trapezoidal enclosure with an inclined magnetic field

The effect of a magnetic field on steady convection in a trapezoidal enclosure filled with a fluid-saturated porous medium is studied numerically by the finite difference method. The inclined sloping boundaries is treated by adopting staircase-like zigzag lines. The sloping walls are maintained isothermally at different temperatures. The top and bottom horizontal straight walls are kept adiabatic. The results indicate that the heat transfer performance decreases by decreasing the angle of sloping wall. Optimum reducing of the heat transfer rate was obtained for an acute trapezoidal enclosure and large magnetic field in the horizontal direction.     

Natural convection boundary layer of a non-Newtonian fluid about a permeable vertical cone embedded in a porous medium saturated with a nanofluid

An analysis was performed to study the effect of uniform transpiration velocity on free convection boundary-layer flow of a non-Newtonian fluid over a permeable vertical cone embedded in a porous medium saturated with a nanofluid. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. The governing partial differential equations are transformed into a set of non-similar equations and solved numerically by an efficient implicit, iterative, finite-difference method. Comparisons with previously published work are performed and excellent agreement is obtained. A parametric study of the physical parameters is conducted and a representative set of numerical results for the velocity, temperature, and volume fraction profiles as well as the local Nusselt and Sherwood numbers is illustrated graphically to show interesting features of the solutions.     

Double-diffusive and Soret-induced convection of a micropolar fluid in a vertical channel

This paper reports an investigation of the fully developed natural convection heat and mass transfer of a micropolar fluid in a vertical channel. Asymmetric temperature and concentration boundary conditions are applied to the walls of the channel. The cases of double diffusion and Soret-induced convection are both considered. The governing parameters for the problem are the buoyancy ratio and the various material parameters of the micropolar fluid. The resulting non-dimensional boundary value problem is solved analytically in closed form using MAPLE software. A numerical solution of the time dependent governing equations is demonstrated to be in good agreement with the analytical model. The influence of the governing parameters on the fluid flow as well as heat and solute transfers is demonstrated to be significant.     

Effect of aspect ratio on convection in a porous enclosure with partially active thermal walls

The aim of the present numerical investigation is to understand the effect of aspect ratio and partially thermally active zones on convective flow and heat transfer in a rectangular porous enclosure. Five different heating and cooling zones are considered along the vertical walls while the remaining portions of the sidewalls and top and bottom of the enclosure are adiabatic. The Brinkman-Forchheimer extended Darcy model is used in the study. The governing equations are solved by the finite volume method with the SIMPLE algorithm. The computations are carried out for a wide range of parameters and the results are presented graphically. The results reveal that the location of heating and cooling zones has a significant influence on the flow pattern and the corresponding heat transfer in the enclosure. The rate of heat transfer approaches to a constant value for very low values of the Darcy number. The heat transfer rate is decreased on increasing the aspect ratio.     

Numerical analysis of natural convection in shed roofs with eave of buildings for cold climates

A numerical study has been carried out on partially heated triangular enclosures with eave. The eave and inclined wall has a cold temperature. The laminar, two-dimensional, steady governing equations of natural convection are solved in the streamfunction-vorticity form using a finite difference technique. Streamline, isotherm and Nusselt number are presented for different parameters such as aspect ratio AR=H/L from 0.25 to 1, ratio of eave length E=L/n where n changes from 3 to 7 and Rayleigh number from 10³ to 10⁶. It is observed that the heat transfer decreases with increasing aspect ratio for small Rayleigh number but increases for higher Rayleigh number. Heat transfer also increases with decreasing eave length.     

Effect of heat generation on mixed convection in porous cavity with sinusoidal heated moving lid and uniformly heated or cooled bottom walls

Effect of heat generation and absorption on mixed convection flows in a sinusoidal heated lid-driven square cavity filled with a porous medium is investigated numerically. Both the vertical walls of the enclosure are insulated while the bottom wall is uniformly heated or cooled. The top wall is moving at a constant speed and is heated sinusoidally. The governing equations and boundary conditions are non-dimensionalized and solved numerically by using finite volume method approach along with SIMPLE algorithm together with non-uniform grid system. The effect of Darcy and heat generation parameters are investigated in terms of the flow, heat transfer, and Nusselt number. The results for stream function and isotherm are plotted and it is found that there have significant influence with the presence of heat generation and porous medium.

     

Effect of temperature gradient orientation on the characteristics of mixed convection flow in a lid-driven square cavity

The effect of temperature gradient orientation on the fluid flow and heat transfer in a lid-driven differentially heated square cavity is investigated numerically. The transport equations are solved using the high-order compact scheme. Four cases are considered depending on the direction of temperature gradient imposed. The differentially heated top and bottom walls result in gravitationally stable and unstable temperature gradients. While the differentially heated left and right side walls lead to assisting and opposing buoyancy effects. The governing parameters are Pr = 0.7 and Ri = 0.1, 1, and 10. It is found that both Richardson number and direction of temperature gradient affect the flow patterns, heat transport processes, and heat transfer rates in the cavity. Computed average Nusselt number indicates that the heat transfer rate increases with decreasing Ri regardless the orientation of temperature gradient imposed. And the assisting buoyancy flows have best performance on heat transport over the other three cases.     

Transition in a 2-D lid-driven cavity flow

Direct numerical simulations about the transition process from laminar to chaotic flow in square lid-driven cavity flows are considered in this paper. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasi-periodic, inverse period-doubling, period-doubling, and chaotic self-sustained flow regimes. The numerical experiments are conducted by solving the 2-D incompressible Navier-Stokes equations with increasing Reynolds numbers (Re). The spatial discretization consists of a seventh-order upwind-biased method for the convection term and a sixth-order central method for the diffusive term. The numerical experiments reveal that the first Hopf bifurcation takes place at Re equal to 7402±4%, and a consequent periodic flow with the frequency equal to 0.59 is obtained. As Re is increased to 10,300, a new fundamental frequency (FF) is added to the velocity spectrum and a quasi-periodic flow regime is reached. For slightly higher Re (10,325), the new FF disappears and the flow returns to a periodic regime. Furthermore, the flow experiences an inverse period doubling at 10,325 <Re< 10,700 and a period-doubling regime at 10,600 <Re< 10,900. Eventually, for flows with Re greater than 11,000, a scenario for the onset of chaotic flow is obtained. The transition processes are illustrated by increasing Re using time-velocity histories, Fourier power spectra, and the phase-space trajectories. In view of the conducted grid independent study, the values of the critical Re presented above are estimated to be accurate within ±4%.     

Numerical study of laminar natural convection in a complicated cavity heated from top with sinusoidal temperature and cooled from other sides

Steady, laminar natural convection in a two-dimensional enclosure with three flat and one wavy walls is numerically investigated. The top wall is heated with a sinusoidal temperature profile. The other three walls, including the wavy wall, are maintained at constant low temperature. Air is considered as the working fluid. This problem is numerically solved by SIMPLE algorithm with deferred QUICK scheme in non-orthogonal curvilinear co-ordinates. The mesh generation has been done by solving the partial differential equation with grid control functions. Tests are carried out for wave amplitudes 0.0-0.10 in steps of 0.01 and Rayleigh numbers 10⁰-10⁶ while the Prandtl number is kept constant. The number of undulations considered are one, two or three.The effect of the various parameters (Rayleigh number, amplitude of undulation and number of undulations) on the flow pattern and heat transfer has been studied. The heat transfer mode remains conductive up to Ra = 10³. With increase of Ra, the mode of heat transfer changes from conduction to convection. It has been observed that the average Nusselt number remains constant for Ra up to 10³ and then starts changing when Ra is increased further. This led to the further detailed study about the flow behavior in a cavity with undulations. Because of the nature of the imposed boundary conditions, there are two large vortices formed. The left side vortex adjacent to the flat vertical wall always remains single cell and is unaffected by Ra, amplitude and number of undulations. For a single undulation, when Ra is increased to 10⁶, the right side large vortex breaks into two cells for amplitude above a certain value (0.06) giving rise to first saddle point and right side second vortex center. For two and three undulation cases, in addition to these, a second saddle point and right side third vortex appear beyond some amplitude (0.07 and 0.03, respectively).     

具有非均匀内热源的多孔介质中自然对流传热的数值研究

采用数值方法分析了填充多孔介质的竖直同心套管内,非均匀分布的内热源和内外壁面温差对自然对流的影响。考察了高宽比A、内热源分布系数m以及内外Rayleigh数之比Rai/Ra对流场、温度场和内外壁面Nusselt数的影响。结果表明：Rai/Ra较大时流场中部形成逆向环流,并出现θ＞1的高温区;内壁面Nusselt数呈现先增大后减小的趋势,大约在Z=0．8处出现转折。外壁面Nusselt数在Z＞0．8处变化加剧,表明外壁面对流传热主要集中在管上部区域。m增大流体中心逆向环流随之减小并最终消失。     

Coupled flow and reaction during natural convection PCR

Polymerase chain reaction (PCR) in a microfluidic Rayleigh–Benard convection cell represents a promising route towards portable PCR for point-of-care uses. In the present contribution, the coupled fluid mechanics and heat transport processes are solved numerically for a 2-D flow cell. The resultant velocity and temperature fields serve as the inputs to a convection-diffusion-reaction model for the DNA amplification, wherein the reaction kinetics are modeled by Gaussian distributions around the conventional bulk PCR reaction temperatures. These evolution equations are integrated to determine the exponential growth rate of the double-stranded DNA concentration. The predicted doubling time is approximately 10–25 s, increasing with the Péclet number. This effect is attributed to low velocity, slow kinetics “dead zones” located at the center of the reactor. The latter observation provides an alternative rationalization for the use of loop-based natural convection PCR systems.     

Thermocapillary convection and interface deformation in a liquid film within a micro-slot with structured walls

Thermocapillary convection in a thin liquid film inside a micro-slot with structured walls kept at different temperatures is studied. The liquid film is wetting the substrate wall and is separated from the cover wall by a gas layer. If the slot walls are structured, the temperature at the liquid–gas interface is non-uniform. The temperature variation induces thermocapillary stresses which bring the liquid into motion and lead to the interface deformation. We investigate the film flow inside the micro-slot, the heat transfer, the liquid–gas interface deformations and the film stability in the framework of the long-wave theory. We show that the amplitude of the interface deformation increases with increasing of the wall structure period. We demonstrate that the structured walls lead to the heat transfer enhancement, which effect is for the studied range of parameters stronger if the cover wall is structured. We also show that the wall structure enhances the long-wave Marangoni instability. The destabilizing effect of the substrate structure is stronger than that of the cover wall structure. This work has been originally presented at the 3rd International Conference on Microchannels and Minichannels, 13–15 June 2005, Toronto, Canada.     

Identification of the parameters of a convection diffusion system

Computational algorithms are proposed for the realization of gradient methods based on the solution of direct and conjugate problems in weak formulations for some complex inverse problems of the recovery of parameters of multicomponent parabolic distributed systems. The approach proposed makes it unnecessary to set up Lagrange functionals in explicit form and to use the Green function. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 42–63, January–February 2009.     

A parameter uniform numerical method for a system of singularly perturbed convection–diffusion equations with discontinuous convection coefficients

In this paper, a weakly coupled system of two singularly perturbed convection-diffusion equations with discontinuous convection coefficients is examined. A finite difference scheme on Shishkin mesh generating the parameter uniform convergence in the global maximum norm is constructed for solving this problem. Numerical results which are in agreement with the theoretical results are presented.     

Falkner–Skan wedge flow of a power-law fluid with mixed convection and porous medium

This article describes the mixed convection flow of a non-Newtonian fluid past a wedge. An incompressible power-law fluid occupies the porous space. The arising mathematical problem has been solved by homotopy analysis method (HAM). Convergence of the derived solution is checked. The local skin friction coefficient and Nusselt number are also discussed.     

A two-dimensional study on natural convection and heat transfer in the enclosure with heat transfer and radiation coupled in natural convection

Numerical investigation using SIMPLE algorithm with QUICK scheme for natural convection and heat transfer in the enclosure bounded by a solid wall and with heat transfer and radiation coupled in natural convection has been conducted.The various parameters are:Rayleigh number(from 103 to 105),dimensionless conductivity of bounding wall(from 0 to 100),dimensionless wall thickness(from 0 to 0.6) and radiation emissivity of all surfaces(from 0 to 1).The results suggest that flow and heat transfer are influenced...     

Study of Europa’s crust via a Stefan model with convection

We present results of a numerical simulation of the thermal convection in the subsurface mushy ice layer of Europa, one of the Jupiter’s moons. Beside fluid dynamics and heat transfer within such a layer, heat conduction in the solid crustal surface and heat exchange between the two phases – mushy ice and solid crust – are included in our model in order to follow also the evolution of the phase front.     

Entropy generation analysis and heatline visualization of free convection in nanofluid (KKL model-based)-filled cavity including internal active fins using lattice Boltzmann method

Two-dimensional natural convection and entropy generation in a square cavity filled with CuO–water nanofluid is performed. The lattice Boltzmann method is employed to solve the problem numerically. The influences of different Rayleigh numbers $\left(10^3<Ra<10^6\right)$ and solid volume fractions $\left(0<\phi <0.05\right)$ on the fluid flow, heat transfer and total/local entropy generation are presented comprehensively. Also, the heatline visualization is employed to identify the heat energy flow. To predict the thermo-physical properties, dynamic viscosity and thermal conductivity, of CuO–water nanofluid, the KKL model is applied to consider the effect of Brownian motion on nanofluid properties. It is concluded that the configurations of active fins have pronounced effect on the fluid flow, heat transfer and entropy generation. Furthermore, the Nusselt number has direct relationship with Rayleigh number and solid volume fraction, and the entropy generation has direct and reverse relationships with Rayleigh number and solid volume fraction, respectively.