GDQ Method for Natural Convection in a Cubic Cavity Using Velocity–Vorticity Formulation

ABSTRACT

Natural convection in a differentially heated cubic enclosure is studied by solving the velocity–vorticity form of the Navier–Stokes equations by a generalized differential quadrature (GDQ) method. The governing equations in the form of velocity Poisson equations, vorticity transport equations, and energy equation are solved using a coupled numerical scheme via a single global matrix for velocities, vorticities, and temperature. Vorticity and velocity coupling at the solid boundaries is enforced through a higher-order approximation by the GDQ method, thus assuring accurate satisfaction of the continuity equation. Nusselt numbers computed for Ra = 10³, 10⁴, 10⁵, and 10⁶ show good agreement with the benchmark results. A mesh independence study indicates that the present numerical procedure requires much coarse mesh compared to other numerical schemes to produce the benchmark solutions of the flow and heat transfer problems.     

A Fully Coupled Navier-Stokes Solver for Fluid Flow at All Speeds

This article deals with the formulation and testing of a newly developed, fully coupled, pressure-based algorithm for the solution of fluid flow at all speeds. The new algorithm is an extension into compressible flows of a fully coupled algorithm developed by the authors for laminar incompressible flows. The implicit velocity–pressure–density coupling is resolved by deriving a pressure equation following a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique. The coefficients of the momentum and continuity equations are assembled into one matrix and solved simultaneously, with their convergence accelerated via an algebraic multigrid method. The performance of the coupled solver is assessed by solving a number of two-dimensional problems in the subsonic, transsonic, supersonic, and hypersonic regimes over several grid systems of increasing sizes. For a desired level of convergence, results for each problem are presented in the form of convergence history plots, tabulated values of the maximum number of required iterations, the total CPU time, and the CPU time per control volume.     

Coupled Thermoelasticity of Functionally Graded Beams

This paper presents the finite element solution of an Euler–Bernoulli beam with functionally graded material (FGM) subjected to lateral thermal shock loads. The FGM beam is assumed to be graded across the thickness. The material properties across the thickness direction follow the volume fraction of the constitutive materials in power law form. The solution is obtained under coupled thermoelastic assumption. The equation of motion and the conventional coupled energy equation are simultaneously solved to obtain the transverse deflection and temperature distribution in the beam. The governing partial differential equations of the problem are solved simultaneously using the Galerkin finite element method with the C ¹-continuous shape function leading to fast convergence of the solution. Results are presented for different power law indexes and coupling coefficients for simply supported boundary conditions. The results are verified with those reported in the literature.     

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.     

A PRESSURE CORRECTION SCHEME USING COUPLED MOMENTUM EQUATIONS

Abstract

This article presents a method for the numerical solution of the incompressible 2D Navier-Stokes equations, based on the coupled solution of the momentum equations and a fully compatible pressure correction equation. With in each iteration, the linearized momentum equations are simultaneously, though inexactly, solved through a two-step, noniterative scheme. Their solution employs the appropriate factorization of the diagonal coefficient matrices into upper and lower triangular ones. The continuity equation is satisfied by means of the SIMPLE concept in a particular form that arises from the coupled solution of the momentum equations. The algorithm is more efficient in terms of convergence rates when compared to a segregated algorithm, given that identical discretization schemes are used.     

Velocity–vorticity formulation for 3D natural convection in an inclined cavity by DQ method

The present work proposes a novel numerical solution algorithm based on a differential quadrature (DQ) method to simulate natural convection in an inclined cubic cavity using velocity–vorticity form of the Navier–Stokes equations. Since the DQ method employs a higher-order polynomial to approximate any given differential operator, the vorticity values at the boundaries can be computed more accurately than the conventionally followed second-order accurate Taylor’s series expansion scheme. The numerical capability of the present algorithm is demonstrated by the application to natural convection in an inclined cubic cavity. The velocity Poisson equations, the continuity equation, the vorticity transport equations and the energy equation are all solved as a coupled system of equations for the seven field variables consisting of three velocities, three vorticities and temperature. Thus coupling the velocity and the vorticity transport equations allows the determination of the vorticity boundary values implicitly without requiring the explicit specification of the vorticity boundary conditions. The present algorithm is proved to be an efficient method to resolve the non-linearity involved with the vorticity transport equations and the energy equation. Test results obtained for an inclined cubic cavity with different angle of inclinations for Rayleigh number equal to 10³, 10⁴, 10⁵ and 10⁶ indicate that the present coupled solution algorithm could predict the benchmark results for temperature and flow fields using a much coarse computational grid compared to other numerical schemes.     

Coupling Kinetic and Continuum Equations for Micro Scale Flow Computations

A hybrid method, coupling the direct numerical solution of the Bhatnagar-Gross-Krook (BGK) kinetic equation and hydrodynamic (Navier-Stokes) equations is presented. The computational physical domain is decomposed into kinetic and continuum sub-domains using an appropriate criterion based on the local Knudsen number and proper gradients of macro-parameters, computed via a preliminary Navier-Stokes solution throughout the whole physical domain. The coupling is achieved by matching half fluxes at the interface of the kinetic and Navier-Stokes domains, thus taking care of the conservation of momentum, energy, and mass through the interface. The advantage of the presented hybrid algorithm is that it easily allows the coupling of existing codes for the numerical solution of the BGK and Navier-Stokes equations. To validate and estimate the efficiency of the proposed method the simulation of the monatomic gas flow through a slit has been considered for outlet to inlet pressure ratio of 0.1, 0.5, and 0.9, and a wide range of Knudsen number. The comparison of local parameters (density, velocity, and temperature) with pure kinetic solutions shows satisfactory agreement with those computed by the hybrid solver.     

UNSTEADY THREE-DIMENSIONAL NATURAL CONVECTION OF WATER COOLED INSIDE A CUBIC ENCLOSURE

Three-dimensional unsteady natural convection of cooling water inside a cubical cavity at Ra = 10⁵ is investigated. The finite-volume method with the SIMPLE algorithm is used to solve the nonlinear coupled continuity, momentum, and energy equations. All physical water properties—density, viscosity, thermal conductivity, and specific heat—are allowed to change with temperature. The numerical results for the 3-D geometry show that the side effects are relevant in fluid mechanics and in heat transfer, with larger differences than with the 2-D predictions in the region where the density anomaly of water is important.     

Low Rayleigh number flow in a heat generating porous media

A numerical analysis has been performed for the steady-state temperature and stream function distributions in a short cylinder, having an isothermal side and top, an insulated bottom, for a uniform heat generating porous medium. The analysis uses the stream function formulation of Darcy's equation in cylindrical coordinates and the Boussinesq approximation. A single energy equation was used for the fluid and solid, since conduction was the expected mode of heat transfer at low heat generation rates for a lead sphere air porous media system. The solution of the non-dimensionalized momentum and energy equations resulted in small Rayleigh numbers (2×10⁻⁶ to 0.2) indicating the heat transfer is by conduction. Solutions for the stream lines and isotherms were obtained using a transient explicit finite-difference approximation using a mean bed thermal conductivity.     

A note on the thermal stability of a reactive non-Newtonian flow in a cylindrical pipe

The effect of variable viscosity and viscous dissipation on the thermal stability of a one-step exothermic, reactive non-Newtonian flow in a cylindrical pipe was considered assuming negligible reactant consumption. The governing equations; momentum and energy equations, which are coupled due to the dissipative term in the energy equation, were solved by closed-form and approximate techniques respectively. We obtained closed-form solutions for the momentum equation via the Homotopy analysis method(HAM), while the resulting energy equation was analysed for thermal stability via the variational technique. Parametric analysis of the solutions were conducted on the dimensionless velocity (v(r)), the critical Frank-Kamenetskii parameter(δ_cr) and the critical temperature(θ_cr) and expressed graphically and in tabular forms.     

An unstructured SMAC algorithm for thermal non-equilibrium two-phase flows

The SMAC (Simplified Marker And Cell) algorithm is extended for an application to thermal non-equilibrium two-phase flows in light water nuclear reactors (LWRs). A two-fluid three-field model is adopted and a multi-dimensional unstructured grid is used for complicated geometries. The phase change and the time derivative terms appearing in the continuity equations are implemented implicitly in the pressure correction step. The energy equations are decoupled from the momentum equations for faster convergence. The verification of the present numerical method was carried out against a set of test problems which includes the single and the two-phase flows. The results are also compared to those of the semi-implicit ICE method, where the energy equations are coupled with the momentum equation for pressure correction.     

Analysis of wind flow around a parabolic collector (2) heat transfer from receiver tube

Parabolic collectors of commercial solar thermal power plants are subject to variable convection heat transfer from the receiver tube. In the present study heat transfer from a receiver tube of the parabolic trough collector of the 250 kW solar power plants in Shiraz, Iran, is studied taking into account the effects of variation of collector angel of attack, wind velocity and its distribution with respect to height from the ground.The governing equations for the two-dimensional steady state wind flow include continuity, momentum and energy equations and RNG-based k–ε model for turbulence scheme. Finite volume discretization method is used to solve the governing equations with wall function boundary condition and the SIMPLE approach is employed to iterate for the pressure correction and convergence of the velocity field. The momentum equation contains buoyancy force when the buoyancy effect is high and force convection effect is low.Computation is carried out for various wind velocities and different collector orientations with respect to wind direction. For solution of the energy equation, temperature of the receiver tube is taken as 350 K and ambient temperature is assumed to be 300 K. Various recirculation and temperature fields were observed around the receiver tube for different flow conditions. Effect of collector orientation on the average Nu number for the receiver tube was found negligible when the wind speed is low (Re⩽4.5×10⁵ based on the collector aperture). But when the wind velocity is high (Re>4.5×10⁵), the collector effect on the variation of Nu around the glass cover of the absorber tube is considerable.     

Effect of Radiative Heat Transfer on Three-Dimensional Double Diffusive Natural Convection

This work presents a numerical study of the effect of the radiative heat transfer on the three-dimensional double diffusive convection in a differentially heated cubic cavity for different optical parameters of the medium. This numerical study is conducted for fixed Prandtl, Rayleigh, and Lewis numbers, Pr = 13.6, Ra = 10⁵, Le = 2, and buoyancy ratio N in the range [–2, 0]. The natural convection equations, using the Boussinesq approximation for the treatment of buoyancy term in the momentum equation, are expressed using the vorticity–stream function formulation. These equations and the radiative transfer equation are discretized, respectively, with the control volume finite difference method and the FTn finite volume method. The influences of the optical thickness and the conduction–radiation parameter of the semitransparent fluid on heat and mass transfer are depicted. Results show different transitions of the structure of the main flow when varying the conduction–radiation parameter and the optical thickness.     

Numerical modeling of dynamics of steam superheaters

The paper presents a numerical solution of transient flow in superheaters in the state of parallel- and counter flow. The time and space temperature distributions of the working media and separating wall were evaluated. The partial differential equations describing the mentioned heat transfer were also solved using the method of lines coupled with Gear's method. Furthermore, the above temperature distributions were obtained by solving the mathematical model describing the conservation laws of mass, momentum and energy. Obtained results were compared to the suggested approximate solution.     

Non-thermal equilibrium model of the coupled heat and mass transfer in strong endothermic chemical reaction system of porous media

A two dimension mathematical model has been developed to simulate the coupled heat and mass transfer in a porous medium undergoing a strong endothermic chemical reaction. Differing from the traditional two phase equation model, just the temperature field of bulk flow is known from the solution of energy equation. The temperature distribution of the solid matrix is solved according to the reaction kinetics of the decomposition of calcium carbonate. The coupling of these two equations is given by the item of chemical reaction. The fluid flow is modeled by the Ergun–Forchheimer–Brinkman equation. This model is solved numerically by the alternate dimension implicit method, and the numerical results are validated by comparing with the experimental data in literature. The influence of the strongly endothermic chemical reaction on the heat and mass transfers in the porous medium is discussed. The reaction features of the packed bed of pellets are analyzed under different conditions by varying the key parameters.     

Macroscopic cylindrical governing equations: transport process in unsaturated porous media with a line heat source

The paper outlines a formulation of the cylindrical transport equations which describe simultaneous heat and mass transfer in unsaturated porous materials when a line heat resource is embedded in the medium. A macroscopic continuum mechanics approach is adopted to derive the coupled continuity, momentum and energy equations. Hydrodynamic law such as Darcy's law and Darcy–Buckingham theorem are utilized to simplify the continuity and momentum equations of fluid flow. Migration of liquid due to surface tension effects is modeled in the analysis. The effects of phase change on the heat transfer are also included in the energy equation. The constituent equations are expanded in the cylindrical co‐ordinate system. The resulting equations reported in this paper are found to agree with equations reported in this paper are found to agree with equations obtained by other researchers who used volume‐averaging techniques to study similar phenomena in unsaturated porous materials. Copyright © 1999 John Wiley & Sons, Ltd.     

Nonlinear Hyperbolic Rigid Heat Conductor of the Coleman Type

A one-dimensional nonlinear hyperbolic homogeneous isotropic rigid heat conductor proposed by Coleman is analyzed using the method of weakly nonlinear geometric optics. For such a model the law of conservation of energy, the dissipation inequality, the Cattaneo's equation, and a generalized energy-entropy relation with a parabolic variation of the energy and entropy along the heat-flux axis, are postulated. First, it is shown that the model can be described by a non-homogeneous quasi-linear hyperbolic matrix partial differential equation of the first order for an unknown vector u = (θ, Q) ^T , where θ and Q are the dimensionless absolute temperature and heat-flux fields, respectively. Next, the Cauchy problem for the matrix equation with a weakly perturbed initial condition is formulated, and an asymptotic solution to the problem in terms of the amplitudes σ_α (α = 1, 2) that satisfy a pair of nonlinear first order partial differential equations, is obtained. The Cauchy problem is then solved in a closed form when the initial data are suitably restricted. Numerical examples are included.     

Numerical and Optical Analysis of Solar Power Level Adaptable Solar Reactor

Solar energy is an abundant renewable energy resource that can be used to provide high process heat necessary to run thermochemical processes for production of various solar fuels and commodities. In a solar reactor, sunlight is concentrated into a receiver through a small opening called the aperture. However, obtaining and maintaining semiconstant high temperatures inside a solar reactor is a challenge. This is because the incident solar radiation can fluctuate depending on the position of the sun and the weather conditions. For fixed aperture size reactors, changes in incident solar flux directly affect the temperature inside the reactor. This paper presents a novel solar reactor with variable aperture mechanism that is designed and manufactured at our lab. Radiation heat transfer analysis of this reactor concept is studied via Monte Carlo (MC) ray tracing. MC ray tracing module is coupled to a steady-state one-dimensional energy equation solver. Energy equation is solved for the wall and gas, accounting for the absorption, emission, and convection. Incoming direct flux values for a typical day are obtained from National Renewable Energy Lab database. Results show that for a perfectly insulated reactor, the average temperature of the working fluid may be kept appreciably constant throughout the day if aperture diameter is varied between 3 cm and 1.5 cm for incoming fluxes starting with 400 W/m² at 05:12 a.m., reaching peak value of 981 W/m² at noon, and eventually receiving 400 W/m² at 6:58 p.m., which can make the solar reactor run about 13 hr continuously at 1500 K semiconstant temperature.