Natural convection in an inclined rectangular porous cavity with uniform heat flux from the side

The problem of natural convection in an inclined rectangular porous layer enclosure is studied numerically. The enclosure is heated from one side and cooled from the other by a constant heat flux while the two other walls are insulated. The effect of aspect ratio, inclination angle and Rayleigh number on heat transfer is studied. It is found that the enclosure orientation has a considerable effect on the heat transfer. The negative orientation sharply inhibits the convection and consequently the heat transfer and a positive orientation maximizes the energy transfer. The maximum temperature within the porous medium can be considerably higher than that induced by pure conduction when the cavity is negatively oriented. The peak of the average Nusselt number depends on the Rayleigh number and the aspect ratio. The heat transfer between the two thermally active boundaries is sensitive to the effect of aspect ratio. For an enclosure at high or low aspect ratio, the convection is considerably decreased and the heat transfer depends mainly on conduction.     

Free convection of a liquid binary mixture in an inclined rectangular cavity

A study is made of the influence of an inclination on the distribution of concentrations produced by thermal diffusion in a cavity of rectangular section. It is shown that even a very weak convective motion produced in the cavity in the case of heating from above leads to significant perturbations of the concentration field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 176–179, July–August, 1979.We thank G. Z. Gershuni, Yu. K. Bratukhin, and V. I. Chernatynskii for discussing the results and for helpful comments.     

Free convection from an arbitrarily inclined plate in a porous medium

A mathematical model for the flow and heat transfer in the free convection from an arbitrary inclined isothermal flat plate embedded in a porous medium is presented, in which the Darcy–Boussinesq approximation is adopted to account for bouyancy force. A novel inclination parameter ξ is proposed such that all cases of the horizontal, inclined and vertical plates can be described by a single set of transformed boundary layer equations. Moreover, the similarity equations for the limiting cases of the horizontal and vertical plates are recovered from the transformed equations by setting ξ=0 and ξ=1, respectively. Detailed results for the skin friction coefficient and Nusselt number as well as for the dimensionless velocity and temperature profiles are presented for a wide range of the parameter ξ. A comparison with similarity solution shows excellent agreement.     

矩形倾斜腔体中的两种对流斑图和分区

为了研究矩形倾斜腔体中普朗特数Pr=0.72的流体对流斑图和分区,本文基于流体力学方程组进行了数值模拟。在相对瑞利数r=6.0的情况下,观察了倾角θ=10°和θ=60°时对流斑图随着时间的发展,发现系统存在单圈型对流和多圈型对流两种斑图。流线随着倾角的变化说明:随着倾角增加,对流圈数逐渐减少,对流波长逐渐增加,对流波数减小;然后,随着对流圈数显著地减少,系统由多圈型对流过渡到单圈型对流。根据模拟计算结果,给出了多圈型对流过渡到单圈型对流的临界倾角θc随着相对瑞利数r变化的关系曲线。对流在θ-r平面上分为两个区域:θ<θc时系统是单圈型对流,θ>θc时系统是多圈型对流。对流最大振幅A和努塞尔数Nu随着倾角θ的变化曲线被临界倾角θc分成两段,它们有不同的变化规律。因此,临界倾角也可以由对流最大振幅A或努塞尔数Nu的变化曲线来确定。     

Effect of free convection on thermodiffusion in a liquid mixture filling an inclined rectangular cavity

The convective motion which develops in an inclined cavity upon heating from above determines to a significant degree the form of the concentration field produced by thermodiffusion. The interaction of convective and thermodiffusion fluxes at small thermal Grashof numbers Gr causes the appearance of longitudinal jumps in concentration. Increase in temperature difference intensifies convection and encourages reduction in concentration gradients. The dominant role of convection for fixed Gr is determined by the angle of inclination of the liquid layer [1, 2]. A significant feature of liquid solutions is their low diffusion coefficient and thus high Schmidt number. This fact does not permit use of results obtained for gas mixtures, and greatly complicates numerical simulations. In contrast to [2], the present study will investigate thermodiffusion separation in a cavity with impermeable boundaries.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 73–76, September–October, 1986.In conclusion, the authors thank G. Z. Gershun for evaluation of the results and helpful remarks.     

Conjugate natural convection between horizontal concentric cylinders filled with a porous medium

A conjugate problem of natural convection between two horizontal concentric, isothermal cylinders filled with a fluid-saturated porous medium is studied. The flow field and energy equations are solved under the conditions of equality in temperature but with jump in heat flux at the fluid-solid interface. Numerical results by a finite-difference technique are presented for a large variation in the parameters entering the problem. A comparison of the solution with non-conjugate problem is also given.Das konjugierte Problem der freien Konvektion zwischen zwei horizontalen, konzentrischen, isothermen Zylindern, die mit einem fluid-gesättigten porösen Medium gefüllt sind, wurde hier untersucht.Das Strömungsfeld und die Energiegleichung sind unter den Bedingungen gelöst worden, daß die Temperatur gleich ist, aber ein Wärmeflußsprung am Übergang von der Flüssigkeits- zur Festphase stattfindet. Die numerischen Ergebnisse werden mit dem Finite-Differenzen-Verfahren für viele Parametervariationen dargestellt. Zudem wurde ein Lösungsvergleich mit einem nicht konjugierten Problem gegeben.     

Mixed convection adjacent to inclined flat surfaces embedded in a porous medium

An analysis is performed to study the flow and heat transfer characteristics of laminar mixed convection boundary layer flows from inclined (including horizontal and vertical) surfaces embedded in a saturated porous medium with constant aiding external flows and uniform surface temperature. Both the streamwise and normal components of the buoyancy forces are retained in the momentum equations. Nondimensionalization of the boundary layer equations results in the following three governing parameter: (1)Gr/Re, the ratio of the Grashof number to the Reynolds number; (2)Pe _x =Re _x Pr, the Peclet number; (3) φ, the angle of inclination from the horizontal. The resulting nonsimilar equations are solved by an efficient implicit finite-difference scheme. Numerical results are presented for flows with different values ofGr/Re in the range of 0 to 50, over a wide range of the Peclet numbersPe _x, and various values of φ ranging from 0 to 90 degrees. It is found that the local surface heat transfer rate increases with increasing the local Peclet number. In addition, as the plate is tilted from the horizontal to the vertical orientation, the local Nusselt number increases for a given Peclet number and the effect of the buoyancy force on the surface heat transfer rate increases.     

Effect of inclined gradients on thermohaline convection in porous medium

Onset of convection in a thermohaline fluid staturating a porous medium subjected to inclined temperature as well as salinity gradients of finite magnitude is analysed using Galerkin technique. Due to unequal horizontal gradients of heat and salt the basic state fluid density varies horizontally also. It is found that (i) when the horizontal temperature gradient is greater than salinity gradient the system becomes unstable and stationary convection is possible in the stable quadrant, (ii) when the horizontal salinity gradient is greater than temperature gradient the stable region extends to the basically unstable quadrant, (iii) in the case of compensating horizontal gradients also the region of stationary convection gets extended to the stable quadrant when horizontal gradient increases, (iv) if only one of the horizontal gradients (heat or salt) is present the point where stationary convection changes over to oscillatory pattern is shifted to the region conductive to stationary convection; the effect of salinity gradient being more than that of temperature gradient.Mit Hilfe der Galerkin-Methode wird das Einsetzen der Konvektion in einem, ein poröses Medium tränkenden Thermohaline-Fluid unter dem gleichzeitigen Einfluß von veränderlichen Temperatur- und Salzkonzentrationsgradienten endlicher Höhe untersucht. Wegen der ungleichen Horizontalgradienten von Temperatur und Konzentration ändert sich auch die Gesamtfluiddichte in Horizontalrichtung. Es zeigt sich: (1) Ist der horizontale Temperaturgradient größer als der Konzentrationsgradient, so wird das System instabil und im stabilen Quadranten ist stationäre Konvektion möglich. (2) Im umgekehrten Fall erstreckt sich das stabile Gebiet bis zum grundsätzlich instabilen Quadranten. (3) Kompensieren sich die Horizontalgradienten, so erstreckt sich auch das Gebiet stationärer Konvektion bis zum stabilen Quadranten, falls der Horizontalgradient zunimmt. (4) Ist nur einer der beiden Horizontalgradienten vorhanden, so verschiebt sich der Punkt, wo stationäre Konvektion in die oszillatorische Mode übergeht, vom Gebiet reiner Wärmeleitung in das stationärer Konvektion. Der Einfluß des Konzentrationsgradienten überwiegt dabei den des Temperaturgradienten.C. Parthiban thanks University Grants Commission for awarding a fellowship.     

Natural convection flow with solidification between two vertical plates filled with a porous medium

Quasi-steady solidification between two vertical flat plates filled with a saturated porous medium has been investigated. The medium is homogeneous and isotropic. The convection flow of liquid takes place in the porous medium in the variable space between the two walls. One of the vertical walls is set to a temperature lower than the solidification temperature of the medium and therefore a frozen crust is formed on this wall. The second wall has a high temperature then the fusion temperature of the medium. The problem has been simplified by assuming laminar flow and the Brinkman and the Oberbeck–Bousinesq’s approximations. The results are presented in terms of the velocity for different properties of the porous medium. Various velocities are displayed in dependence of the Rayleigh and Darcy numbers. The study indicates that asymmetric boundary conditions have an important effect on the temperature and flow field. In addition, the growth of the thickness of the frozen layer with time has been derived from a simple analytical solution of the interface energy equation.     

Natural convection in an inclined porous enclosure with an off-center diathermal partition

A numerical study has been made of natural convection in an inclinded porous enclosure with an off-center diathermal partition. A temperature difference is imposed between the two isothermal end walls and the other two walls are assumed to be adiabatic. Numerical results are obtained for Rayleigh numbers (Ra) in the range of 10 to 500, the dimensionless partition location ( \(\bar S\) ) ranging from 0.125 to 0.875, the aspect ratios (A) of the enclosure ranging from 0.5 to 5, and the inclination angles (φ) of ?60, ?30, 0, 30, 60 degrees. It is found that the partition location has strong influence at lowRa and relatively weaker influence at highRa. The average Nusselt number reaches the minimum value when the partition is in the middle of the vertical enclosure, and the maximum Nusselt number occurs around φ = 30 degrees.     

Vibrational thermal convection in a rectangular cavity

Vibrational thermal convection in a rectangular cavity under conditions of weightlessness is studied. Some equilibrium configurations were obtained in earlier papers of two of the authors [1, 2] and their linear stability investigated. In the present paper, a numerical investigation is made of the developed vibrational convection which arises under conditions when equilibrium is impossible. The structure of the average vibrational-convective flows and the characteristics of the heat transfer are determined. The change of regimes and the connection with the stability problem are discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 94–99, July–August, 1982.     

Double-diffusive convection in an inclined porous layer with a concentration-based internal heat source

The thermosolutal instability of double-diffusive convection in an inclined fluid-saturated porous layer with a concentration-based internal heat source is investigated. The linear instability of small-amplitude perturbations to the system is analyzed with respect to transverse and longitudinal rolls. The resultant eigenvalue problem is solved numerically utilizing the Chebyshev tau method. It is shown that an increasing inclination angle causes a strong stabilization in the transverse rolls irrespective of the internal heat source or vertical solutal Rayleigh number. Furthermore, substantial qualitative changes are demonstrated in the linear instability thresholds with variations in the inclination angle and concentration-based heat source.     

Radiation effects on mixed convection about a cone embedded in a porous medium filled with a nanofluid

The problem of steady, laminar, mixed convection boundary-layer flow over a vertical cone embedded in a porous medium saturated with a nanofluid is studied, in the presence of thermal radiation. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis with Rosseland diffusion approximation. The cone surface is maintained at a constant temperature and a constant nanoparticle volume fraction. The resulting governing equations are non-dimensionalized and transformed into a non-similar form and then solved by Keller box method. A comparison is made with the available results in the literature, and our results are in very good agreement with the known results. A parametric study of the physical parameters is made and a representative set of numerical results for the local Nusselt and Sherwood numbers are presented graphically. Also, the salient features of the results are analyzed and discussed.     

Non-Darcian effects on conjugate natural convection between horizontal concentric cylinders filled with a porous medium

An analysis is presented of steady conjugate free convection between two horizontal concentric cylinders filled with a fluid-saturated porous medium; the innermost cylinder surface is maintained at a high temperature and the outermost cylinder surface at a lower one. The velocity-pressure-gradient relation is taken to be nonlinear, with departure from the linear Darcy situation measured by a parameter F₀. The investigation is based on the numerical solution, by a finite-difference method, of the full momentum and energy equations. The streamline and isotherm patterns as well as the local and mean Nusselt numbers are plotted for several physical parameters to show some of the flow and heat transfer characteristics. It is found that all parameters play an important role in the flow and heat transfer characteristics. The model can be applied to a variety of engineering problems.     

Influence of inclination angle on buoyancy-driven convection in triangular enclosure filled with a fluid-saturated porous medium

The present investigation deals with the numerical analysis of steady-state laminar buoyancy-driven convection in an inclined triangular enclosure filled with fluid saturated porous media using the Darcy law equation. One wall of the enclosure is isothermally heated and the other is cooled, while the remaining wall is adiabatic. The effect of inclination angle on natural convection is investigated by varying the angle of inclination (φ) between 0° and 360°. The governing transformed equations are solved numerically using a finite-difference method. Obtained results are shown in the form of streamlines, isotherms, mean Nusselt numbers and dimensionless stream function for different values of the Rayleigh number Ra in the range 100 ≤ Ra ≤ 1,000. It is found that the values of the maximum and minimum mean Nusselt number are reached for φ = 330° and φ = 210° , respectively. However, the lowest flow strength is formed at φ = 240° for all values of Ra.     

Natural convection in an inclined enclosure with a fluid layer and a heat-generating porous bed

Multiple steady-state solutions of natural convection in an inclined enclosure with a fluid layer and a heat-generating porous bed is investigated numerically by the finite volume method. The conservation equations for the porous layer are based on a general flow model which includes both the effects of flow inertia and friction. The flow in fluid layer is modeled by Navier–Stokes equations. The method of pseudo arc-length continuation is adapted in studying the effects of tilt angle on flow pattern and heat transfer. It is found that, in the whole domain of tilt angle, there exist two groups of solutions with quite different flow pattern and heat transfer behavior. The effects of aspect ratio on flow pattern and heat transfer have also been studied. Received on 04 March 1997     

Mixed convection flow in a lid-driven inclined square enclosure filled with a nanofluid

This work is focused on the numerical modeling of steady laminar mixed convection flow in a lid-driven inclined square enclosure filled with water–Al₂O₃ nanofluid. The left and right walls of the enclosure are kept insulated while the bottom and top walls are maintained at constant temperatures with the top surface being the hot wall and moving at a constant speed. The developed equations are given in terms of the stream function–vorticity formulation and are non-dimensionalized and then solved numerically subject to appropriate boundary conditions by a second-order accurate finite-volume method. Comparisons with previously published work are performed and found to be in good agreement. A parametric study is conducted and a set of graphical results is presented and discussed to illustrate the effects of the presence of nanoparticles and enclosure inclination angle on the flow and heat transfer characteristics. It is found that significant heat transfer enhancement can be obtained due to the presence of nanoparticles and that this is accentuated by inclination of the enclosure at moderate and large Richardson numbers.     

Mixed convection in an inclined lid-driven cavity with non-uniform heating on both sidewalls

The present work reports a numerical simulation of mixed convection in an inclined square cavity. The vertical sidewalls are assumed to have a nonuniform temperature distribution. The finite volume method is used to solve dimensionless governing equations. Simulations are performed for different Richardson numbers, amplitude ratios, phase deviations, and cavity inclination angles. The results are presented graphically. The mean heat transfer significantly increases in the buoyancy-dominated mode on increasing cavity inclination angle if both walls have identical heating and cooling zones.