Article

Non-Darcy Mixed Convection in a Porous Square Enclosure Under Suction/Injection Effects with a Non-Isothermal Vertical Wall

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Abstract

The influence of suction/injection at bottom/top or top/bottom walls on non-Darcy mixed convection flow with a sinusoidally varying temperature on the left vertical wall in a two-dimensional square porous enclosure is analyzed. The Forchheimer extended Darcy model is considered for flow equations. The fully developed equations are nondimensionalized, and then solved numerically by the Galerkin finite element method. A parametric study is carried out and the results are obtained for various values of the parameters such as the Grashof number (Gr*), Rayleigh number (Ra), suction/injection velocity (a), suction/injection window width (D/H), and amplitude (a;) of the sinusoidally varying temperature profile. The computed flow and temperature fields are visualized through streamlines, isotherms, and loal/global cumulative heat flux plots.

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... Extensive research [10][11][12][13][14][15][16] has been done to understand the fluid flow and heat transfer process in closed cavities during mixed convection in the presence of porous media in recent years. A numerical study on laminar mixed convection in a square enclosure filled with water-saturated porous medium was done by Amiri [10]. ...
... Unsteady mixed convection heat transfer in a rectangular cavity with a horizontal strip occupied by two different porous medium was studied by Moraga et al. [12]. Murthy and Kumar [13] presented non-Darcy mixed convection model in a vertical square fluid saturated porous enclosure with multiple injection or suction. Analysis of mixed convection with interplay of various dimensionless numbers for porous media inside a square enclosure with various thermal boundary conditions was done by Basak et al. [14] using Galerkin finite element method. ...
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... Researchers observed shifts in flow patterns from unicellular to multicellular based on Rayleigh, Peclet, and vent width measurements. According to Murthy and Kumar [16], injections and suctions can lead to non-Darcian convection within porous cavities, resulting in the emergence of multicellular flows within the cavity as a result of variations in critical parameters such as Grashof number, injection/suction opening width, and injection/suction inlet velocity. ...
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Mixed convection inside a vertical square porous cavity is investigated numerically, with special attention given to left-side heating. In the study, the importance of the problem, methods used, main results, conclusions, and novelty of the work are discussed. The Brinkmann-Forchheimer-Darcy coupled momentum equations and the Local Thermal Equilibrium energy (LTE) equation are solved using the finite volume technique. Several elemental parameters are investigated, including Reynolds number (50 × Re × 300), Darcy number (0.01 × Da × 100), Richardson number (0 × Ri × 30), and porosity (0.5 × ε ≥ 0.95). The thermal conductivity ratio of solid to liquid was (1<Kr<105), and the inlet port width between (0.05<d/H<0.3). Results show that Nusselt number increases with Reynolds number, Richardson number, and pore thermal conductivity. With increasing porosity, an inverse relationship was observed. Nusselt number was maximized by finding the optimal inlet aperture size (d/H = 0.25). It provides valuable insights into how different parameters affect mixed convection in vertical square porous cavities. Heat transfer processes in porous media are improved by these results. This work goes beyond previous efforts in the literature in exploring a wide parameter space and determining the optimal inlet aperture size.
... Also Jha et al., [11] analyses the unsteady hydromagnetic natural convection flow near impulsive wall as accelerated motion of the infinite vertical porous plate. The influence of suction/injection at top/bottom wall on non-darcy mixed convection flow is analyzed by Murthy and Kumar [12]. Chamkha et al., [13] investigate the effect of internal heat generation on free convection along a vertical plate. ...
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... They also found that the rates of heat transfer are most sensitive to the alteration in the inlet opening size. Murthy and Kumar [26] tested the influence of non-isothermal heating and the suction/ injection effects at bottom/top or top/bottom walls on non-Darcian convective flow in a square porous cavity. Results were reported for various parameters such as Rayleigh number, Grashof number, the suction/injection port size, the inlet suction/injection velocity, and the amplitude of the boundary sinusoidal temperature profile. ...
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... Chin et al. [10] investigated the steady mixed convection boundary layer flow over a vertical impermeable surface embedded in a porous medium when the viscosity of the fluid varies inversely as a linear function of the temperature. A few other applications of mixed convection were also carried out by other investigators [11][12][13][14][15][16]. ...
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Mixed convection flows in a lid-driven square cavity filled with porous medium are studied numerically using penalty finite element analysis for uniform and non-uniform heating of bottom wall. The relevant parameters in the present study are Darcy number (Da = 10− 5− 10− 5), Grashof number (Gr = 103− 105), Prandtl number (Pr = 0.026−10) and Reynolds number (Re = 1−102). The influence of convection is analyzed with Peclet number (Pe = Re.Pr). It is observed that the temperature profiles are symmetric for low values of Pe or Pr even in the presence of asymmetric flow fields irrespective of Da. The flow distribution affects significantly temperature distributions at high Pe irrespective of Da. Effect of Peclet numbers have been further investigated for both natural convection and forced convection dominant regimes at high Da. Strong coupling between flow fields and temperature are observed at high Pe. It is interesting to observe that large isothermal mixing zone at Pr = 10 reduces the overall flow strength compared to Pr = 0.026 case. Local Nusselt numbers show almost uniform and low values for low Peclet numbers and localized enhanced heat transfer rates are observed for high Peclet numbers at Da = 10− 3.
... Coupled fluid flow and heat transfer problems in hybrid domains including saturated porous medium, clear fluid and pure solid have important practical applications in many fields, such as thermal management, alloy casting, nuclear reactors, chemical processing and groundwater flow, etc. In recent years, there has been a substantial interest in the use of porous medium to improve the heat transfer performance [1][2][3][4][5][6][7][8][9][10]. Several relevant mathematical models have been developed to solve these above problems. ...
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Chapter
We already have discussed one form of mixed convection in a horizontal layer, namely the onset of convection with throughflow when the heating is from below (see Sect. 6.10). In this chapter we discuss some more general aspects of mixed convection. Since we have dealt with natural convection and forced convection in some detail, our treatment of mixed convection in a porous medium [first treated by Wooding (1960b)] can be brief. It is guided by the surveys by Lai et al. (1991a) and Lai (2000). We endorse the statement by Lai (2000) that despite the increased volume of research in this field, experimental results are still very few. In particular experimental data on thermal dispersion are very scarce and this is hindering the study of the functional relationship between effective thermal conductivity and thermal dispersion.
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Convection in Porous Media, 4th Edition, provides a user-friendly introduction to the subject, covering a wide range of topics, such as fibrous insulation, geological strata, and catalytic reactors. The presentation is self-contained, requiring only routine mathematics and the basic elements of fluid mechanics and heat transfer. The book will be of use not only to researchers and practicing engineers as a review and reference, but also to graduate students and others entering the field. The new edition features approximately 1,750 new references and covers current research in nanofluids, cellular porous materials, strong heterogeneity, pulsating flow, and more. © Springer Science+Business Media New York 2013. All rights are reserved.
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Heat and fluid flow in fluid-saturated porous media has become increas­ ingly more attractive to researchers and thus it has become a very pro­ ductive field for many researchers and practical engineers in very diverse range of fields. The great interest in the topic stems from its widespread number of different practical applications in modern industries and in many environmental issues, such as nuclear waste management, build­ ing thermal insulators, geothermal power plants, grain storage, etc. In building sciences and thermal insulation engineering, an appreciable in­ sulating effect has been derived by placing porous material in the gap between the cavity walls and multishield structures of nuclear reactors between the pressure vessel and the reactor. Geophysical applications include modeling of the spread of pollutants (e. g. radioactive mater­ ial), water movements in geothermal reservoirs, enhanced recovery of petroleum reservoirs, etc. These, and many other, important practical applications have resulted in a rapid expansion of research in the general area of porous media and thus generated a vast amount of both theor­ etical and experimental research work. It has attracted the attention of industrialists, engineers and scientists from many varying disciplines, such as applied mathematics, chemical, civil, environmental, mechanical and nuclear engineering, geothermal physics, food science, medicine, etc. This book contains some of the contributions to the NATO Advanced Study Institute on Emerging Technologies and Techniques in Porous Media that was held in Neptun-Olimp, Constanta, Black Sea, Romania on 9-20 June, 2003.
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A penalty finite element analysis with bi-quadratic elements is performed to investigate the influence of uniform and non-uniform heating of bottom wall on mixed convection lid driven flows in a square cavity. In the present investigation, bottom wall is uniformly and non-uniformly heated while the two vertical walls are maintained at constant cold temperature and the top wall is well insulated and moving with uniform velocity. A complete study on the effect of Gr shows that the strength of circulation increases with the increase in the value of Gr irrespective of Re and Pr. As the value of Gr increases, there occurs a transition from conduction to convection dominated flow at and for . A detailed analysis of flow pattern shows that the natural or forced convection is based on both the parameters Ri ( ) and Pr. As the value of Re increases from 1 to 102, there occurs a transition from natural convection to forced convection depending on the value of Gr irrespective of Pr. Particularly for higher value of Grashof number ( ), the effect of natural convection is dominant upto and thereafter the forced convection is dominant with further increase in Re. As Pr increases from 0.015 to 10 for a fixed Re and Gr ( ), the inertial force gradually becomes stronger and the intensity of secondary circulation gradually weakens. The local Nusselt number ( ) plot shows that the heat transfer rate is very high at the edges of the bottom wall and then decreases at the center of the bottom wall for the uniform heating and that contrasts lower heat transfer rate at the edges for the non-uniform heating of the bottom wall. It is also observed that shows non-monotonic behavior with both uniform and non-uniform heating cases for at higher value of Pr. The average Nusselt number plot for the left or right wall shows a kink or inflexion at for highest value of Pr. Thus the overall power law correlation for average Nusselt number may not be obtained for mixed convection effects at higher Pr.
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The unsteady mixed convection boundary layer flow near the stagnation point on a heated vertical plate embedded in a fluid saturated porous medium is studied. It is assumed that the unsteadiness is caused by the impulsive motion of the free stream velocity and by sudden increase in the surface temperature. Both the buoyancy assisting and the buoyancy opposing flow situations are considered with combined effects of the first and second order resistance of solid matrix of non-Darcy porous medium, variable viscosity and radiation. The problem is reduced to a system of non-dimensional partial differential equations, which is solved numerically using the Keller-box method. The features of the flow and the heat transfer characteristics for different values of the governing parameters are analyzed and discussed. The surface shear stress and the heat transfer of the present study are compared with the available results and a good agreement is found.
Article
Buoyancy induced flow and thermal fields characteristics inside a porous wavy walled enclosure have been numerically solved and analyzed. The enclosure consists of two isothermal wavy walls. The two parallel straight walls at the top and the bottom are flat and kept adiabatic. Governing equations are discretized using the Finite Element Method. Simulation was carried out for a range of surface waviness ratios, a = 0–0.5; aspect ratio, A = 2; inverse Darcy number, Da = 0.01–∞; and Rayleigh numbers, Ra = 10°–107 for a fluid having Prandtl number equal to 1.0. Results are presented in the form of streamlines and isothermal lines for different values of surface waviness and porosity.
Article
A numerical study has been carried out in rectangular enclosures, which have a vertical active wall with all the other walls insulated. The equally divided active sidewall is heated and cooled with sinusoidal temperature profiles. Two cases have been considered: the first is the lower part is heated while the upper part is cooled and the second, the upper part is heated and lower part is cooled. Steady state heat transfer by laminar natural convection has been studied by numerically solving equations of mass, momentum and energy, to determine the thermal penetration in the enclosures and heat transfer as a function of Rayleigh number, the aspect ratio and the position of side heating with respect to side cooling. Rayleigh number was varied from 103 to 106 and the aspect ratio from 0.2 to 5, and the results are presented in the form of streamlines and isotherms, local and average Nusselt number, and heat penetration length. It is found that the penetration approaches to 100% at high Rayleigh numbers when the lower part is heated while the higher part is cooled. In the case of the higher part is heated and the lower part is cooled, the penetration is limited to 70% passing through maxima at Rayleigh number below 106.
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We examine the combined effect of spatially stationary surface waves and the presence of fluid inertia on the free convection induced by a vertical heated surface embedded in a fluid-saturated porous medium. We consider the boundary-layer regime where the Darcy-Rayleigh number, Ra, is very large, and assume that the surface waves have O(1) amplitude and wavelength. The resulting boundary-layer equations are found to be nonsimilar only when the surface is nonuniform and inertia effects are present; self-similarity results when either or both effects are absent. Detailed results for the local and global rates of heat transfer are presented for a range of values of the inertia parameter and the surface wave amplitude.
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Thermophoresis particle deposition in free convection on a vertical plate embedded in a fluid saturated non-Darcy porous medium is studied using similarity solution technique. The effect of Soret and Dufour parameters on concentration distribution, wall thermophoretic deposition velocity, heat transfer and mass transfer is discussed in detail for different values of dispersion parameters (Ra γ, Ra ξ) inertial parameter F and Lewis number Le. The result indicates that the Soret effect is more influential in increasing the concentration distribution in both aiding as well as opposing buoyancies. Also, the non-dimensional heat transfer coefficient and non-dimensional mass transfer coefficient changes according to different values of thermophoretic coefficient k.
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The Silences of the Archives, the Reknown of the Story. The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, before captivated judges and a guilty feeling Bertrande de Rols. The historian wanted to go beyond the known story in order to discover the world of the heroes. This research led to disappointments and surprizes as documents were discovered concerning the environment of Artigat’s inhabitants and bearing directly on the main characters thanks to notarial contracts. Along the way, study of the works of Coras and Lesueur took a new direction. Coming back to the affair a quarter century later did not result in finding new documents (some are perhaps still buried in Spanish archives), but by going back over her tracks, the historian could only be struck by the silences of the archives that refuse to reveal their secrets and, at the same time, by the possible openings they suggest, by the intuition that almost invisible threads link here and there characters and events.
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Natural convection heat transfer in a porous media filled and non-isothermally heated from the bottom wall of triangular enclosure is analyzed using finite difference technique. Darcy law was used to write equations of porous media. Dimensionless heatfunction was used to visualize the heat transport due to buoyancy forces. Three different boundary conditions were applied for the vertical and inclined boundaries of triangular enclosures as Case I; both vertical and inclined walls were isothermal, Case II; vertical wall was adiabatic and inclined one was isothermal, Case III; vertical wall is isothermal and inclined one is adiabatic. A cosine function was utilized to get non-isothermal wall condition. The study was performed for different aspect ratios (0.25 ⩽ AR ⩽ 1.0) and Darcy-modified Rayleigh numbers (100 ⩽ Ra ⩽ 1000). It was observed that heat transfer enhancement was formed when vertical and inclined walls were isothermal while bottom wall was at non-uniform temperature. Heat transfer from bottom wall did not vary when the value of aspect ratio was higher than 0.50. In addition, heatline visualization technique was a useful technique for non-isothermally heated and porous media filled triangular enclosures.
Article
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.
Article
A numerical study of mixed convection heat and mass transfer along a vertical wavy surface has been carried out numerically. The wavy surface is maintained at uniform wall temperature and constant wall concentration that is higher than that of the ambient. A simple coordinate transformation is employed to transform the complex wavy surface to a flat plate. A marching finite-difference scheme is used for present analysis. The buoyancy ratio N, amplitude–wavelength ratio α, and Richardson number (Gr/Re2) are important parameters for this problem. The numerical results, including the developments of skin-friction coefficient, velocity, temperature, concentration, Nusselt number as well as Sherwood number along the wavy surface are presented. The effects of the buoyancy ratio N and the dimensionless amplitude of wavy surface on the local Nusselt number and the local Sherwood number have been examined in detail.
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A series of numerical simulations were performed in order to study liquid metal MHD natural convection in a vertical cylindrical container with a sinusoidal temperature distribution at the upper wall and the other surfaces being adiabatic. Starting from the basic hydrodynamic case, the effect of vertical (axial) and horizontal magnetic fields is assessed. Depending on the magnitude of the Rayleigh and Hartmann numbers, both turbulent and laminar (azimuthally symmetric or not) flows are observed. The results show that the increase of Rayleigh number promotes heat transfer by convection while the increase of Hartmann number favors heat conduction. The vertical magnetic field reduces the Nusselt number more than the horizontal. The circulation patterns for the most convective cases are confined close to the top corner of the container with the simultaneous formation of a secondary flow pattern at the bottom corner, while for the more conductive cases only one circulation pattern exists covering the entire domain.
Article
Steady mixed convection flow in a vented enclosure with an isothermal vertical wall and filled with a fluid-saturated porous medium is investigated numerically. The forced flow conditions are imposed by providing an inlet at the bottom surface, and a vent at the top, facing the inlet. The nature and the basic characteristics of the mixed aiding as well as mixed opposing flows that arise are investigated using the Darcy law model. The governing parameters are the Rayleigh number, Péclet number, and the width of the inlet as a fraction of the height of the square enclosure. These parameters are varied over wide ranges and their effect on the heat transfer characteristics is studied in detail.
Article
This paper describes a comprehensive study of the natural convection phenomenon occurring inside a porous layer with both heat and mass transfer from the side. The natural circulation is driven by a combination of buoyancy effects due to both temperature and concentration variations. The first part of the study consists of an extensive series of numerical simulations conducted in the range 0.01 ⩽ Le ⩽ 100, 50 ⩽ RaH ⩽ 104, − 5 ⩽ N ⩽ +3 and H/L = 1, where Le, RaH, N and H/L are the Lewis number, Dareymodified Rayleigh number, buoyancy ratio and geometric aspect ratio of the porous layer. In the second part, the phenomenon is studied based on scale analysis : the chief conclusions of this part, namely, the order-ofmagnitude predictions for the overall heat and mass transfer rates and their respective domains of validity, are shown to be in agreement with the results produced by discrete numerical experiments. Furthermore, the scale analysis is used to sort out the many effects that influence the overall heat and mass transfer results of numerical experiments.
Article
The free convective boundary layer above a near-horizontal heated flat surface bounding a saturated porous medium is studied. Two configurations are considered: one where the component of the buoyancy force along the surface aids the flow, the other where it opposes the flow. Series solutions are developed: one valid near the leading edge, where the flow is driven along by an induced pressure gradient, and for the favourable case only, a second valid asymptotically far along the boundary, where the flow is driven along by the direct action of buoyancy forces. The description is completed by a numerical solution based on a scheme by Keller and Cebeci, which gives a step-by-step solution throughout the boundary layer. In the unfavourable case, the boundary layer separates and a region of reverse-flow develops.
Article
Unsteady laminar natural convection flow has been considered in a rectangular enclosure formed by non-isothermal walls, filled with a fluid-saturated porous medium and with internal heat generation. The top horizontal wall and right vertical wall of the enclosure are cold, the bottom wall is heated at a constant temperature and the left vertical wall is considered to be non-isothermal. The equations are non-dimensionalized and solved numerically by an upwind finite difference method together with a successive over-relaxation (SOR) technique. The effects of heat generation and the porosity of the medium on the streamlines and isotherms are presented, as well as on the rate of heat transfer from the walls of the enclosure. The fluid has Prandtl number Pr=0.7 while the value of the Rayleigh number is 105.
Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluid and Porous Media
  • I Pop
  • D B Ingham
I. Pop and D. B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluid and Porous Media, Pergamon, Oxford, 2001.