Effects of thermal boundary conditions on natural convection flows within a square cavity
01 Nov 2006-International Journal of Heat and Mass Transfer (Pergamon)-Vol. 49, Iss: 23, pp 4525-4535
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.
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01 Jan 1997
TL;DR: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems and discusses the main points in the application to electromagnetic design, including formulation and implementation.
Abstract: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.
1,820 citations
TL;DR: In this paper, a numerical study of the effect of the magnetic field with inclined angle on the flow and heat transfer rate of liquid gallium in a square cavity is presented, where the penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations.
Abstract: A numerical study is presented for natural convection flow of electrically conducting liquid gallium in a square cavity. In the cavity, the bottom wall is uniformly heated and the left vertical wall is linearly heated whereas the right vertical wall is linearly heated or cooled while the top wall kept thermally insulated. A uniform magnetic field inclined at an angle ϕ with respect to horizontal plane is externally imposed. Penalty finite element method with bi-quadratic rectangular elements is used to solve the non-dimensional governing equations. Numerical results are presented for wide range of Hartmann number (Ha = 0–100) with inclined angles ( ϕ = 0 , π / 2 ) for high Rayleigh number Ra = 105 in terms of stream functions and isotherm contours, local and average Nusselt number. The results shows that the magnetic field with inclined angle has effects on the flow and heat transfer rates in the cavity. It is also found that the average Nusselt number decreases non-linearly by increasing Hartmann number for any inclined angle.
210 citations
TL;DR: In this article, the Galerkin finite element method has been employed to solve momentum and energy balance as well as post processing streamfunctions and heatfunctions in the presence of hot and cold side walls.
Abstract: Natural convection of nanofluids in presence of hot and cold side walls (case 1) or uniform or non-uniform heating of bottom wall with cold side walls (case 2) have been investigated based on visualization of heat flow via heatfunctions or heatlines. Galerkin finite element method has been employed to solve momentum and energy balance as well as post processing streamfunctions and heatfunctions. Various nanofluids are considered as Copper–Water, TiO2–Water and Alumina–Water. Enhancement of heat transfer with respect to base fluid (water) has been observed for all ranges of Rayleigh number (Ra). Dominance of viscous force or buoyancy force are found to play significant roles to characterize the heat transfer rates and temperature patterns which are also established based on heatlines. In general, convective closed loop heatlines are present even at low Rayleigh number (Ra=103) within base fluid, but all nanofluids exhibit dominant conductive heat transport as the flow is also found to be weak due to dominance of viscous force for case 1. On the other hand, convective heat transport at the core of a circulation cell, typically represented by closed loop heatlines, is more intense for nanofluids compared to base fluid (water) for case 2 at Ra = 105. It is also found that heatlines with larger heatfunctions values for nanofluids coincide with heatlines with smaller heatfunction values for water at walls. Consequently, Nusselt number which is also correlated with heatfunctions show larger values of nanofluids for all ranges of Ra. Average Nusselt numbers show that larger enhancement of heat transfer rates for all nanofluids at Ra=105 and Alumina–Water and Copper–Water exhibit larger enhancement of heat transfer rates.
172 citations
Cites methods or result from "Effects of thermal boundary conditi..."
...The jump discontinuity in Dirichlet type of wall boundary conditions for temperature at the corner points is tackled by implementation of exact boundary conditions at those singular points as mentioned earlier [50]....
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...Current solution scheme produces grid invariant results as discussed in our previous articles [50,54]....
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...The detailed solution procedure may be found in earlier works [50,54],...
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TL;DR: In this paper, a review of the studies on natural convection heat transfer in triangular, trapezoidal, parallelogrammic enclosures and enclosures with curved and wavy walls filled with fluid or porous media is presented.
Abstract: Natural convection in an enclosure (internal convection) is an important problem due to its significant practical applications. In energy related applications, natural convection plays a dominant role in transport of energy for the proper design of enclosures in order to achieve higher heat transfer rates. This review summarizes the studies on natural convection heat transfer in triangular, trapezoidal, parallelogrammic enclosures and enclosures with curved and wavy walls filled with fluid or porous media. In addition, this review also summarizes the natural convection studies in the nanofluid filled enclosures. Studies have been performed for the enclosures subjected to different thermal boundary conditions. A number of the studies demonstrated that the variation of the aspect ratio and base angle of the triangular and rhombic/parallelogrammic enclosures had a wide influence on the flow distribution pattern. In the trapezoidal enclosure, the aspect ratio of the cavity as well as the presence of the baffles along the walls played a significant role in the temperature and flow distribution. The flow patterns within the complex enclosures were found to be largely dependent on the amplitude-wavelength ratio and number of undulations of the wavy walls. In addition, the researchers have also studied the effect of the various parameters such as the Rayleigh numbers, Prandtl numbers, Darcy numbers, Darcy–Rayleigh number, irreversibility distribution ratios, volume fraction of the nanoparticles, etc. Overall, the current review paper presents an useful insight into the potential strategies for enhancing the convection heat transfer performance.
168 citations
TL;DR: In this article, 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.
Abstract: 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 Gr=5×103 and Re=1 for Pr=07 A detailed analysis of flow pattern shows that the natural or forced convection is based on both the parameters Ri (GrRe2) 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 (Gr=105), the effect of natural convection is dominant upto Re=10 and thereafter the forced convection is dominant with further increase in Re As Pr increases from 0015 to 10 for a fixed Re and Gr (Gr=103), the inertial force gradually becomes stronger and the intensity of secondary circulation gradually weakens The local Nusselt number (Nub) 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 Nul shows non-monotonic behavior with both uniform and non-uniform heating cases for Re=10 at higher value of Pr The average Nusselt number plot for the left or right wall shows a kink or inflexion at Gr=104 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
137 citations
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.
11,187 citations
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
"Effects of thermal boundary conditi..." refers background in this paper
...The relationships between stream function, w (Batchelor [23]) and velocity components for two dimensional flows are: U 1⁄4 ow oY and V 1⁄4 ow oX ; ð24Þ...
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Book•
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
3,043 citations
01 Jan 1997
TL;DR: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems and discusses the main points in the application to electromagnetic design, including formulation and implementation.
Abstract: This chapter introduces the finite element method (FEM) as a tool for solution of classical electromagnetic problems. Although we discuss the main points in the application of the finite element method to electromagnetic design, including formulation and implementation, those who seek deeper understanding of the finite element method should consult some of the works listed in the bibliography section.
1,820 citations
"Effects of thermal boundary conditi..." refers background or methods in this paper
...Typical values of c that yield consistent solutions are 10(7) (Basak and Ayappa [21]; Reddy [22])....
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...A nine node bi-quadratic elements with each element mapped using iso-parametric mapping (Reddy [22]) from X Y to a unit square n g domain has been used....
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...The continuity equation (7) will be used as a constraint due to mass conservation and this constraint may be used to obtain the pressure distribution (Basak and Ayappa [21]; Reddy [22])....
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...(7) (see Reddy [22]) which results in: P ¼ c oU oX þ oV oY : ð12Þ The continuity equations (7) is automatically satisfied for large values of c. Typical values of c that yield consistent solutions are 107 (Basak and Ayappa [21]; Reddy [22])....
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...(16) and (17), the second term containing the penalty parameter (c) are evaluated with two point Gaussian quadrature (reduced integration penalty formulation, Reddy [22])....
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Book•
15 Aug 2003
TL;DR: Advances in Heat Transfer as mentioned in this paper provides in-depth review articles over a broader scope than in traditional journals or texts, which serve as a broad review for experts in the field and are also of great interest to non-specialists who need to keep up to date with the results of the latest research.
Abstract: Advances in Heat Transfer fills the information gap between regularly scheduled journals and university-level textbooks by providing in-depth review articles over a broader scope than in traditional journals or texts. The articles, which serve as a broad review for experts in the field are also of great interest to non-specialists who need to keep up-to-date with the results of the latest research. This serial is essential reading for all mechanical, chemical, and industrial engineers working in the field of heat transfer, or in graduate schools or industry. * Compiles the expert opinions of leaders in the industry* Fills the information gap between regularly scheduled journals and university-level textbooks by providing in-depth review articles over a broader scope than in traditional journals or texts* Essential reading for all mechanical, chemical, and industrial engineers working in the field of heat transfer, or in graduate schools or industry
1,591 citations