# Finite Element Simulation on Natural Convection Flow in a Triangular Enclosure Due to Uniform and Nonuniform Bottom Heating

01 Mar 2008-Journal of Heat Transfer-transactions of The Asme (American Society of Mechanical Engineers)-Vol. 130, Iss: 3, pp 032501

TL;DR: A penalty finite element analysis with biquadratic elements has been carried out to investigate natural convection flows within an isosceles triangular enclosure with an aspect ratio of 0.5 as discussed by the authors.

Abstract: A penalty finite element analysis with biquadratic elements has been carried out to investigate natural convection flows within an isosceles triangular enclosure with an aspect ratio of 0.5. Two cases of thermal boundary conditions are considered with uniform and nonuniform heating of bottom wall. The numerical solution of the problem is illustrated for Rayleigh numbers (Ra), 10 3 ≤Ra≤10 5 and Prandtl numbers (Pr), 0.026 ≤Pr≤1000. In general, the intensity of circulation is found to be larger for nonuniform heating at a specific Pr and Ra. Multiple circulation cells are found to occur at the central and corner regimes of the bottom wall for a small Prandtl number regime (Pr =0.026-0.07). As a result, the oscillatory distribution of the local Nusselt number or heat transfer rate is seen. In contrast, the intensity of primary circulation is found to be stronger, and secondary circulation is completely absent for a high Prandtl number regime (Pr=0. 7-1000). Based on overall heat transfer rates, it is found that the average Nusselt number for the bottom wall is √2 times that of the inclined wall, which is well, matched in two cases, verifying the thermal equilibrium of the system. The correlations are proposed for the average Nusselt number in terms of the Rayleigh number for a convection dominant region with higher Prandtl numbers (Pr=0.7 and 10).

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TL;DR: A review of studies on natural convection heat transfer in the triangular enclosure namely, in attic-shaped space is presented, for example, attics subject to localized heating and attics filled with porous media.

Abstract: This paper presents a review of studies on natural convection heat transfer in the triangular enclosure namely, in attic-shaped space Much research activity has been devoted to this topic over the last three decades with a view to providing thermal comfort to the occupants in attic-shaped buildings and to minimising the energy costs associated with heating and air-conditioning Two basic thermal boundary conditions of attic are considered to represent hot and cold climates or day and night time This paper also reports on a significant number of studies which have been performed recently on other topics related to the attic space, for example, attics subject to localized heating and attics filled with porous media

80 citations

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TL;DR: In this article, a numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was carried out, where the enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures.

Abstract: A numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was carried out. The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. The buoyancy effect was modeled by applying the Boussinesq approximation of density to the momentum equation and the governing equations were iteratively solved using the control volume approach. The effects of the Rayleigh number and the aspect ratio were examined. Flow and thermal fields were exhibited by means of streamlines and isotherms, respectively. Variations of the maximum value of the dimensionless stream function and the local and average Nusselt numbers were also presented. The average Nusselt number was correlated to the Rayleigh number based on curve-fitting for each aspect ratio. At the highest Rayleigh number studied, the effects of different inclination angles of the enclosure and various cross-section geometries of the inner cylinder were investigated. The computed results indicated that at constant aspect ratio, both the inclination angle and cross-section geometry have insignificant effects on the overall heat transfer rates though the flow patterns are significantly modified.

72 citations

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TL;DR: A comprehensive survey of the literature in the area of numerical heat transfer (NHT) published between 2000 and 2009 has been conducted by as mentioned in this paper, where the authors conducted a comprehensive survey.

Abstract: A comprehensive survey of the literature in the area of numerical heat transfer (NHT) published between 2000 and 2009 has been conducted Due to the immenseness of the literature volume, the survey

58 citations

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TL;DR: In this article, the authors deal with experimental and numerical analysis of natural convection in a right-angled triangular cavity heated from below and cooled on sidewall while its other wall, the hypotenuse, is kept adiabatic.

Abstract: This present investigation deals with experimental and numerical analysis of natural convection in a right-angled triangular cavity heated from below and cooled on sidewall while its other wall, the hypotenuse, is kept adiabatic. The enclosure is filled with water and heat transfer surfaces are maintained at constant temperature. Experimental study covers flow visualization studies involving the use of the particle tracing method. Numerical solutions are obtained using a commercial CFD package, FLUENT, using the finite volume method. Contradictory results existing in the open literature for the Nusselt number resulting from the singularity at the corners of heated surfaces and from the definition of the Nusselt number are discussed in detail. A new approach is used to overcome the singularity at the corner joining the differentially heated isothermal walls when determining Nu. Effects of Rayleigh number, Ra on the Nusselt number, Nu as well as velocity and temperature fields are investigated for the range of Ra from 103 to 107. It is shown that the experimental and numerical results agree fairly well. Finally, a correlation for Nu is developed.

50 citations

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TL;DR: In this article, a comprehensive view of the research area is sought by critically examining the ex-perimental and numerical approaches adopted in studies of this problem in the literature, and areas of further research are highlighted.

Abstract: Natural convection in triangular enclosures is an important problem. It displays well thegeneric attributes of this class of convection, with its dependence on enclosure geometry,orientation and thermal boundary conditions. It is particularly rich in its variety of ﬂowregimes and thermal ﬁelds as well as having signiﬁcant practical application. In this pa-per, a comprehensive view of the research area is sought by critically examining the ex-perimental and numerical approaches adopted in studies of this problem in the literature.Different thermal boundary conditions for the evolution of the ﬂow regimes and thermalﬁelds are considered. Effects of changes in pitch angle and the Rayleigh number on theﬂow and thermal ﬁelds are examined in detail. Although most of the past studies are inthe laminar regime, the review extends up to the recent studies of the low turbulent re-gime. Finally, areas of further research are highlighted. [DOI: 10.1115/1.4004290]Keywords: buoyancy-induced ﬂows, ﬂow ﬁelds, triangular enclosures

48 citations

### Cites methods from "Finite Element Simulation on Natura..."

...[38] employed a finite element method to analyze laminar natural convection in an isosceles triangular enclosure heated uniformly and sinusoidally at the base wall for Pr1⁄4 0....

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##### References

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01 Jan 1967TL;DR: The dynamique des : fluides Reference Record created on 2005-11-18 is updated on 2016-08-08 and shows improvements in the quality of the data over the past decade.

Abstract: Preface Conventions and notation 1. The physical properties of fluids 2. Kinematics of the flow field 3. Equations governing the motion of a fluid 4. Flow of a uniform incompressible viscous fluid 5. Flow at large Reynolds number: effects of viscosity 6. Irrotational flow theory and its applications 7. Flow of effectively inviscid liquid with vorticity Appendices.

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TL;DR: In this paper, the Navier-Stokes equation is derived for an inviscid fluid, and a finite difference method is proposed to solve the Euler's equations for a fluid flow in 3D space.

Abstract: This brief paper derives Euler’s equations for an inviscid fluid, summarizes the Cauchy momentum equation, derives the Navier-Stokes equation from that, and then talks about finite difference method approaches to solutions. Typical texts for this material are apparently Acheson, Elementary Fluid Dynamics and Landau and Lifschitz, Fluid Mechanics. 1. Basic Definitions We describe a fluid flow in three-dimensional space R as a vector field representing the velocity at all locations in the fluid. Concretely, then, a fluid flow is a function ~v : R× R → R that assigns to each point (t, ~x) in spacetime a velocity ~v(t, ~x) in space. In the special situation where ~v does not depend on t we say that the flow is steady. A trajectory or particle path is a curve ~x : R→ R such that for all t ∈ R, d dt ~x(t) = ~v(t, ~x(t)). Fix a t0 ∈ R; a streamline at time t0 is a curve ~x : R→ R such that for all t ∈ R, d dt ~x(t) = ~v(t0, ~x(t)). In the special case of steady flow the streamlines are constant across times t0 and any trajectory is a streamline. In non-steady flows, particle paths need not be streamlines. Consider the 2-dimensional example ~v = [− sin t cos t]>. At t0 = 0 the velocities all point up and the streamlines are vertical straight lines. At t0 = π/2 the velocities all point left and the streamlines are horizontal straight lines. Any trajectory is of the form ~x = [cos t + C1 sin t + C2] >; this traces out a radius-1 circle centered at [C1 C2] >. Indeed, all radius-1 circles in the plane arise as trajectories. These circles cross each other at many (in fact, all) points. If you find it counterintuitive that distinct trajectories can pass through a single point, remember that they do so at different times. 2. Acceleration Let f : R × R → R be some scalar field (such as temperature). Then ∂f/∂t is the rate of change of f at some fixed point in space. If we precompose f with a 1 Fluid Dynamics Math 211, Fall 2014, Carleton College trajectory ~x, then the chain rule gives us the rate of change of f with respect to time along that curve: D Dt f := d dt f(t, x(t), y(t), z(t)) = ∂f ∂t + ∂f ∂x dx dt + ∂f ∂y dy dt + ∂f ∂z dz dt = ( ∂ ∂t + dx dt ∂ ∂x + dy dt ∂ ∂y + dz dt ∂ ∂z ) f = ( ∂ ∂t + ~v · ∇ ) f. Intuitively, if ~x describes the trajectory of a small sensor for the quantity f (such as a thermometer), then Df/Dt gives the rate of change of the output of the sensor with respect to time. The ∂f/∂t term arises because f varies with time. The ~v ·∇f term arises because f is being measured at varying points in space. If we apply this idea to each component function of ~v, then we obtain an acceleration (or force per unit mass) vector field ~a(t, x) := D~v Dt = ∂~v ∂t + (~v · ∇)~v. That is, for any spacetime point (t, ~x), the vector ~a(t, ~x) is the acceleration of the particle whose trajectory happens to pass through ~x at time t. Let’s check that it agrees with our usual notion of acceleration. Suppose that a curve ~x describes the trajectory of a particle. The acceleration should be d dt d dt~x. By the definition of trajectory, d dt d dt ~x = d dt ~v(t, ~x(t)). The right-hand side is precisely D~v/Dt. Returning to our 2-dimensional example ~v = [− sin t cos t]>, we have ~a = [− cos t − sin t]>. Notice that ~v · ~a = 0. This is the well-known fact that in constant-speed circular motion the centripetal acceleration is perpendicular to the velocity. (In fact, the acceleration of any constant-speed trajectory is perpendicular to its velocity.) 3. Ideal Fluids An ideal fluid is one of constant density ρ, such that for any surface within the fluid the only stresses on the surface are normal. That is, there exists a scalar field p : R × R → R, called the pressure, such that for any surface element ∆S with outward-pointing unit normal vector ~n, the force exerted by the fluid inside ∆S on the fluid outside ∆S is p~n ∆S. The constant density condition implies that the fluid is incompressible, meaning ∇ · ~v = 0, as follows. For any region of space R, the rate of flow of mass out of the region is ∫∫ ∂R ρ~v · ~n dS = ∫∫∫

9,804 citations

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01 Jan 1984

TL;DR: Second-order Differential Equations in One Dimension: Finite Element Models (FEM) as discussed by the authors is a generalization of the second-order differential equation in two dimensions.

Abstract: 1 Introduction 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods 3 Second-order Differential Equations in One Dimension: Finite Element Models 4 Second-order Differential Equations in One Dimension: Applications 5 Beams and Frames 6 Eigenvalue and Time-Dependent Problems 7 Computer Implementation 8 Single-Variable Problems in Two Dimensions 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 10 Flows of Viscous Incompressible Fluids 11 Plane Elasticity 12 Bending of Elastic Plates 13 Computer Implementation of Two-Dimensional Problems 14 Prelude to Advanced Topics

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TL;DR: In this paper, a numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed.

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

297 citations

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TL;DR: In this article, the steady natural convection in an enclosure heated from below and symmetrically cooled from the sides is studied numerically, using a stream function-vorticity formulation.

Abstract: Steady natural convection in an enclosure heated from below and symmetrically cooled from the sides is studied numerically, using a streamfunction-vorticity formulation. The Allen discretization scheme is adopted and the discretized equations were solved in a line by line basis. The Rayleigh number based on the cavity height is varied from 103 to 107. Values of 0.7 and 7.0 for the Prandtl number are considered. The aspect ratio L H (length to height of the enclosure) is varied from 1 to 9. Boundary conditions are uniform wall temperature and uniform heat flux. For the range of the parameters studied, a single cell is observed to represent the flow pattern. Numerical values of the Nusselt number as a function of the Rayleigh number are reported, and the Prandtl number is found to have little influence on the Nusselt number. A scale analysis is presented in order to better understand the phenomenon.

237 citations