Comparative Analysis of Thermal Models in the Lattice Boltzmann Method for the Simulation of Natural Convection in a Square Cavity

In this article, a comparative analysis of thermal models in the lattice Boltzmann method for the simulation of natural convection in a square cavity is presented. A hybrid method, in which the thermal equation is solved by the Navier-Stokes equation method while the mass and momentum equations are solved by the lattice Boltzmann method (LBM), is introduced and its merits are explained. All the governing equations are discretized on a cell-centered, nonuniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with a deferred-correction method to ensure accuracy of solutions. The resulting algebraic equations are solved by a strongly implicit procedure. The hybrid method is applied to the simulation of natural convection in a square cavity and the predicted results are compared with the benchmark solutions given in the literatures. The predicted results are also compared with those by the double-population LBM and by the Navier-Stokes equation method. In general, the present hybrid method is as accurate as the Navier-Stokes equation method and the double-population LBM. The hybrid method shows better convergence and stability than the double-population LBM. These observations indicate that this hybrid method is an efficient and economic method for the simulation of incompressible fluid flow and heat transfer problems involving complex geometries.     

Natural convection in a partially porous cavity: Roads to chaos

Abstract

A numerical study of a periodic and chaotic behavior of natural convection in a square cavity, with a porous layer is presented. The cavity under study consists of two opposite vertical walls of which the lower half walls are hot (hold high temperature) and the upper half walls are cold (hold low temperature), whereas the horizontal walls are adiabatic. The porous medium is located in the lower part of the cavity. The natural heat transfer and the Darcy Brinkman equations are solved by using the finite volume method and the TDMA. The results show that thermal natural convection evolves towards a deterministic chaos by following Curry York scenario.     

Numerical Simulation of Double Diffusive Mixed Convection in a Lid-Driven Square Cavity Using Velocity-Vorticity Formulation

In this article, convection driven by combined thermal and solutal concentration buoyancy effects in a lid-driven square cavity is examined using velocity-vorticity form of Navier-Stokes equations. The governing equations consist of vorticity transport equation, velocity Poisson equations, energy equation, and concentration equation. Validation results are discussed for convection due to heat and mass transfer in a lid-driven square cavity at Re = 500, Le = 2, and G_RT  = G_RS  = 100. These results indicate that the present velocity-vorticity formulation could predict the characteristic parameters of flow, temperature, and solutal concentration fields using a much coarser mesh compared to the mesh used in a stream function-vorticity formulation. The capability of the proposed algorithm to handle complex geometry is demonstrated by application to mixed convection in a lid-driven square cavity with a square blockage. The effect of buoyancy ratio on the convection phenomenon is discussed for buoyancy ratio varying from ? 100 to 100 at Re = 100. Under opposing temperature and concentration gradients along the vertical direction, the negative buoyancy ratios give rise to aiding flows.     

Buoyancy-driven convection of water near its density maximum with partially active vertical walls

Transient natural convection of cold water around its density maximum in a square cavity is studied numerically. Nine different positions of the active zones are considered. The governing equations are solved using Control volume method with power low scheme. The results obtained for various values of parameters are presented graphically in the form of streamlines and isotherms. It is found that the average Nusselt number behaves non-linearly as a function of Grashof number. The heat transfer rate is decreased in the density maximum regions.     

Laminar natural convection in an inclined cavity with a wavy wall

In the present work, a numerical study of the effect of a hot wavy wall of a laminar natural convection in an inclined square cavity, differentially heated, was carried out. This problem is solved by using the partial differential equations, which are the vorticity transport, heat transfer and stream function in curvilinear co-ordinates. The tests were performed for different inclination angles, amplitudes and Rayleigh numbers while the Prandtl number was kept constant. Two geometrical configurations were used namely one and three undulations.The results obtained show that the hot wall undulation affects the flow and the heat transfer rate in the cavity. The mean Nusselt number decreases comparing with the square cavity. The trend of the local heat transfer is wavy. The frequency of the latter is different from the undulated wall frequency.     

Effect of baffle–cavity ratios on buoyancy convection in a cavity with mutually orthogonal heated baffles

Buoyancy driven convection in a square cavity induced by two mutually orthogonal and arbitrarily located baffles is studied numerically. The baffles are of different sizes and the flow is two-dimensional. The coupled governing equations were solved by finite difference method using Alternating Direction Implicit technique and Successive Over-Relaxation method. The steady state results are presented in the form of streamline and isotherm plots. It is found that buoyancy force plays a key role and overall heat transfer in the cavity is enhanced for higher values of both baffle–cavity ratios. Flow inhibition emerges as a deciding factor and diminishes heat transfer when the horizontal baffle is located above the center of the cavity.     

A finite volume method to solve the Navier–Stokes equations for incompressible flows on unstructured meshes

A method to solve the Navier–Stokes equations for incompressible viscous flows and the convection and diffusion of a scalar is proposed in the present paper. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. A recently proposed diffusion scheme with interesting theoretical and numerical properties is tested and integrated into the Navier–Stokes solver. Predictions of Poiseuille flows, backward-facing step flows and lid-driven cavity flows are then performed to validate the method. We finally demonstrate the versatility of the method by predicting buoyancy force driven flows of a Boussinesq fluid (natural convection of air in a square cavity with Rayleigh numbers of 10 ³ and 10 ⁶ ).     

Conjugate Heat Transfer Analysis in a Glazed Room Modeled as a Square Cavity

A numerical study of conjugate heat transfer by turbulent natural convection in a room with three different glazed configurations is presented. The room is modeled as a square closed cavity, where the lower wall is adiabatic, the right wall is semitransparent, and the upper and left walls are opaque conductive surfaces. Governing equations of mass, momentum, and energy were solved by the finite volume method with a two equation turbulence model. The results are presented in terms of streamlines, isotherms, heatlines, turbulent viscosity isolines, and thermal parameters, such as indoor temperatures and heat transfer coefficients. From the three cases considered in this study, the reflective glass window was the optimal configuration from the thermal comfort point of view for both design days. On the contrary, the glass-film configuration showed the worst indoor thermal performance inside the cavity despite of being the configuration that allows lower energy transferred into the room through the glazed surface. A set of useful heat transfer correlations are obtained for building design applications and energy codes in temperate climates.     

Numerical conjugate air mixed convection/non-Newtonian liquid solidification for various cavity configurations and rheological models

Unsteady 2D natural convection/phase change of a non-Newtonian liquid inside a square container caused by external mixed convection of a Newtonian fluid with various cavity configurations has been studied numerically. Air was chosen as external cooling fluid and modified non-Newtonian water as the internal solidifying fluid. Conjugate convective fluid and heat transport, described in terms of non-linear coupled continuity, momentum, and energy equations, were solved by using the finite volume method with the SIMPLE algorithm. Effects of four external fluid inlet/outlet locations and four non-Newtonian rheological models were studied. Results for the time evolution of streamlines, isotherms and freezing curves are analyzed. The effect of the cavity inlet/outlet configuration on streamlines of the external fluid is remarkable, near the region close to the non-Newtonian liquid filled container.     

A Reference Benchmark Solution for Free Convection in A Square Cavity Filled with A Heterogeneous Porous Medium

The Fourier-Galerkin (FG) method is used to produce a highly accurate solution for free convection in a square cavity filled with heterogeneous porous medium. To this end, the governing equations are reformulated in terms of the temperature and the stream function. These unknowns are then expanded in infinite Fourier series truncated at given orders. The accuracy of the FG solution is investigated for different truncation orders and compared to the results of an advanced finite-element numerical model using fine-mesh discretization. The obtained results represent a set of high-quality data that can be used for benchmarking numerical models.     

GDQ METHOD FOR NATURAL CONVECTION IN A SQUARE CAVITY USING VELOCITY–VORTICITY FORMULATION

ABSTRACT

This article describes a compact numerical algorithm based on the generalized differential quadrature (GDQ) method for the numerical analysis of natural convection in a differentially heated square cavity. The velocity–vorticity form of the Navier–Stokes equations and energy equation are used to represent the mass, momentum, and energy conservations of the fluid medium in the cavity. The GDQ form of the governing equations and the vorticity definition at the boundaries are solved by a coupled solution algorithm using a global matrix scheme for all the field variables. The vorticity values at the boundary are correctly imposed using the GDQ method, which approximates a given space derivative with higher-order accuracy compared to the existing schemes based on Taylor's series expansion. This has assured a divergence-free solution for the flow field by satisfying the continuity constraint, though the pressure term is not used directly in the present formulation. The proposed algorithm is validated for a lid-driven cavity flow for Reynolds number Re = 100, 400, and 1,000, and the predicted velocity profiles are in excellent agreement with the benchmark solutions. The algorithm is then used to compute the average Nusselt number and flow parameters for natural convection in a square cavity for Rayleigh number Ra = 10³, 10⁴, 10⁵, and 10⁶. These results are in better agreement with the benchmark solutions than the results obtained by other numerical schemes, which used much finer grids compared to the present scheme.     

Computation of Turbulent Natural Convection in a Rectangular Cavity with the Lattice Boltzmann Method

A numerical study of a turbulent natural convection in a rectangular cavity with the lattice Boltzmann method (LBM) is presented. The primary emphasis of the present study is placed on investigation of accuracy and numerical stability of the LBM for the turbulent natural-convection flow. A HYBRID method in which the thermal equation is solved by the conventional Reynolds-averaged Navier-Stokes equation (RANS) method while the conservation of mass and momentum equations are resolved by the LBM is employed in the present study. The elliptic-relaxation model is employed for the turbulence model and the turbulent heat fluxes are treated by the algebraic flux model. All the governing equations are discretized on a cell-centered, nonuniform grid using the finite-volume method. The convection terms are treated by a second-order central-difference scheme with the deferred correction method to ensure accuracy and stability of solutions. The present LBM is applied to the prediction of a turbulent natural convection in a rectangular cavity and the computed results are compared with the experimental data commonly used for the validation of turbulence models and those by the conventional finite-volume method. It is shown that the LBM with the present HYBRID thermal model predicts mean velocity components and turbulent quantities which are as good as those by the conventional finite-volume method. It is also found that the accuracy and stability of the solution is significantly affected by the treatment of the convection term, especially near the wall.     

Effect of Radiation on Mixed Convection Inside a Lid-Driven Square Cavity With Various Optical Thicknesses and Richardson Numbers

This work studies numerically the effect of the radiative heat transfer on the flow and thermal behaviors of the mixed convection in a lid-driven square cavity in the presence of radiatively emitting, absorbing, and isotropically scattering gray medium. The Boussinesq approximation has been used in modeling the governing equations, and the SIMPLE (semi-implicit method for pressure-linked equations) algorithm is used in coupling the velocity and pressure fields. The radiative transfer equation and the governing equations have been solved respectively by the discrete ordinates method and the finite-volume method in order to obtain the temperature, velocity, and heat flux distributions in the participating medium. The present numerical simulations are validated by comparison with several earlier studies. Then, the temperature and velocity distributions and Nusselt numbers have been analyzed in a broad range of optical thicknesses from 0 to 100 and Richardson numbers from 0.01 to 100. The results show that the radiation has a significant role on the flow and thermal behaviors in the lid-driven square cavity. As an example, we can refer to a sweep behavior that is detected in the velocity distributions of the lid-driven cavity.     

Hydro-magnetic combined convection in a lid-driven cavity with sinusoidal boundary conditions on both sidewalls

Mixed convection in a square cavity of sinusoidal boundary temperatures at the sidewalls in the presence of magnetic field is investigated numerically. The horizontal walls of the cavity are adiabatic. The governing equations are solved by finite-volume method. The results are obtained for various combinations of amplitude ratio, phase deviation, Richardson number, and Hartmann number. The heat transfer rate increases with the phase deviation up to ? = π/2 and then it decreases for further increase in the phase deviation. The heat transfer rate increases on increasing the amplitude ratio. The flow behavior and heat transfer rate inside the cavity are strongly affected by the presence of the magnetic field.     

An Improved Bubble Packing Method for Unstructured Grid Generation with Application to Computational Fluid Dynamics

An improved bubble packing method (BPM) is proposed to generate high-quality unstructured grids for prediction of the flow field in a domain with complex geometry. For a curved-boundary domain, bubble departure from the curved boundaries during the dynamic movement of bubbles can be avoided by using the mapping and the arc-length parameterization methods. Furthermore, the grid density of the whole region can be controlled effectively. Local mesh refinement is achieved by adding bubbles with different sizes to the real and artificial vertices of the domain, and vertex information is transferred to the inner nodes of the domain using the Shepard interpolation method. In order to validate the proposed BPM, a finite-volume solver on an unstructured collocated grid is developed to simulate both square and polar lid-driven cavity flows. The numerical simulation results agree well with the experimental data under different Reynolds numbers.     

NUMERICAL STUDY OF TURBULENT NATURAL CONVECTION IN A SIDE-HEATED SQUARE CAVITYAT VARIOUS ANGLES OF INCLINATION

Turbulent natural convection at a moderately high Rayleigh number (4.9 2 10 10 ) in a two-dimensional side-heated square cavity at various angles of inclination is studied numerically. Initially, the performance of the low Reynolds number k - y model of Wilcox (1994) and the low Reynolds number k m l turbulence model of Lam and Bremhorst (1981), in predicting buoyancy-driven flow in a noninclined enclosure, is evaluated against experimental measurements. The evaluation is focused on the prediction of the flow patterns and convective heat transfer in the boundary layer and corner regions. The performance of the Wilcox k m y model is found to be superior in capturing the flow physics such as the strong streamline curvature in the corner regions. The Lam and Bremhorst k m l model is not capable of predicting these features but provides reasonable predictions away from the corners. None of these models, however, is capable of predicting the boundary-layer transition from laminar to turbulent. In order to study the effect of the inclination of the square cavity on the heat transfer and flow patterns, computations are then performed using the Wilcox k m y model for a range of inclination angles from 0° through 90°, keeping other parameters fixed. The computed flow patterns, isotherms, convection strengths, variation of the local Nusselt numbers along the heated walls, and the average Nusselt number for various inclination angles of the square cavity are reported. It is noticed that the flow fields and heat transfer characteristics become significantly different for inclinations greater than45°. The computational procedure is based on finite-volume collocated mesh. The pressure-velocity coupling in the governing equations is achieved using the well-known SIMPLE method for numerical computation. The linear algebraic system of equations is solved sequentially using the strongly implicit procedure (SIP).     

Inverse boundary design of square enclosures with natural convection

An optimization technique is applied to design of heat transfer systems in which the natural convection is important. The inverse methodology is employed to estimate the unknown strengths of heaters on the heater surface of a square cavity with free convection from the knowledge of the desired temperature and heat flux distributions over a given design surface. The direct and the sensitivity problems are solved by finite volume method. The conjugate gradient method is used for minimization of an objective function, which is expressed by the sum of square residuals between estimated and desired heat fluxes over the design surface. The performance and accuracy of the present method for solving inverse convection heat transfer problems is evaluated by comparing the results with a benchmark problem and a numerical experiment.     

NUMERICAL ANALYSIS OF HEAT TRANSFER BY NATURAL CONVECTION AND RADIATION IN PARTICIPATING FLUIDS ENCLOSED IN SQUARE CAVITIES

The influence of thermal radiation on natural convection in a participating fluid contained in a square cavity is studied numerically. The radiative transfer process is solved from the PI approximation. The Navier-Stokes equations are solved by a finite difference scheme integrated over control volumes. A numerical study of the so-called window problem (thermally driven cavity) shows the influence of thermal radiation on this reference problem for Rayleigh numbers in the range of 10³-10⁷ and Planck numbers varying from 1 to 0.05. The isotherms, streamlines, and heat lines show an increase of the dynamical effects in the central part of the cavity and a significant modification of the boundary layers. Results obtained from the simulation of an isotropically scattering medium are given.     

A Pseudospectral Multidomain Method for Conjugate Conduction-Convection in Enclosures

A pseudospectral multidomain method is proposed for the solution of the two-dimensional incompressible Navier-Stokes equations and energy equation. The governing equations are spatially discretized by the Chebyshev pseudospectral method. Within each subdomain, the algebraic system is solved by a semi-implicit pseudotime method, in which the convective and source terms are explicitly marched by the Runge-Kutta method, and the diffusive terms are implicitly marched by the matrix diagonalization method. An interface/boundary-value updating algorithm is proposed to obtain the interfaces and boundaries values to satisfy both the boundary conditions and interface transmission conditions. Since the solution of the interior collocation point values and the updating of interface/boundary values are carried out independently, the multidomain method is easy to implement with an existing single-domain solver.

The pseudospectral multidomain method is validated by three benchmark heat transfer problems: natural convection in a cavity, conjugate conduction-convection in a cavity with one finite-thickness wall, and conjugate conduction-convection in a cavity with both an internal heat source and finite-thickness walls. The numerical results are in excellent agreement with the benchmark solutions; high accuracy and the ability to treat complex problems with the present pseudospectral multidomain method are confirmed. The effects of wall thermal conductivity and Rayleigh number are accurately predicted.     

Interaction Between Two Mutually Orthogonal Heat-Generating Baffles in a Square Cavity

Laminar free convection induced by two mutually orthogonal discrete heat-generating baffles in a two-dimensional square cavity is analyzed numerically. The computations were carried out for different locations and combinations of heat source strengths of the baffles for a fixed Grashof number of 10⁶and Prandtl number of 0.71. The coupled governing equations were solved bya finite-difference method using alternating direction implicit technique and successive overrelaxation methods. The obtained results clearly show that the hydrodynamic and thermal fields in the cavity depend on both the location and strength of the heat-generating baffles. Though the flow inhibition is caused by both the baffles, the baffle with higher source strength plays a decisive role in inducing the flow. The locations of baffles with unequal source strengths produce significant changes in the net heat transfer rate. This is further magnified for higher contrast in source strengths. This study provides qualitative suggestions that may improve the thermal design of sealed enclosures, which are encountered frequently in the electronics industry.