Showing papers in "Numerical Heat Transfer Part B-fundamentals in 2007"

The Atomistic Green's Function Method: An Efficient Simulation Approach for Nanoscale Phonon Transport

Abstract: This article presents a general formulation of an atomistic Green's function (AGF) method. The atomistic Green's function approach combines atomic-scale fidelity with asymptotic treatment of large-scale (bulk) features, such that the method is particularly well suited to address an emerging class of multiscale transport problems. A detailed mathematical derivation of the phonon transmission function is provided in terms of Green's functions and, using the transmission function, the heat flux integral is written in Landauer form. Within this theoretical framework, the required inputs to calculate heat flux are equilibrium atomic locations and an appropriate interatomic potential. Relevant algorithmic and implementation details are discussed. Several examples including a homogeneous atomic chain and two heterogeneous atomic chains are included to illustrate the applications of this methodology.

A Level Set Method for Analysis of Film Boiling on an Immersed Solid Surface

Abstract: A numerical method is presented for simulating film boiling on an immersed (or irregularly shaped) solid surface. The level set formulation for tracking the phase interfaces is modified to include the effect of phase change at the liquid–vapor interface and to treat the no-slip condition at the fluid–solid interface. The boundary or matching conditions at the phase interfaces are accurately imposed by incorporating the ghost fluid approach based on a sharp-interface representation. The numerical method is tested through computations of bubble rise in a stationary liquid, single-phase fluid flow past a circular cylinder, and film boiling on a horizontal cylinder.

Calculations of Steady and Pulsating Impinging Jets—An Assessment of 13 Widely used Turbulence Models

Abstract: Computational fluid dynamics (CFD) calculations of impinging jets have been performed with 13 widely spread Reynolds-averaged Navier-Stokes (RANS) turbulence models using commercial CFD software. It was the aim of these calculation to assess how heat transfer and flow structure can be predicted at different Reynolds numbers and different nozzle-to-plate distances. The only model able to predict correctly the laminar–turbulent transition occurring at small nozzle-to-plate distance was the SST k − ω model (“transitional flow option”). Most of the other models can only satisfactorily predict heat transfer in the turbulent wall jet region. When the SST k − ω model with transitional flow option is applied for calculation of pulsating jets, the tendencies are predicted correctly.

Modeling Transient Heat Transfer Using SPH and Implicit Time Integration

Abstract: In this article, a two-dimensional transient heat conduction problem is modeled using smoothed particle hydrodynamics (SPH) with a Crank-Nicolson implicit time integration technique. The main feature of this work is that it applies implicit time stepping, an unconditionally stable Crank-Nicolson approach, in the thermal conduction simulation of liquid-phase diffusion (LPD) semiconductor crystal growth. This SPH simulation is compared with the equivalent finite-volume results. As well, two transient thermal conduction test problems are simulated using both explicit and Crank-Nicolson schemes, and their results compared with the analytical solutions. One of the current drawbacks of SPH is that explicit time-stepping algorithms, such as predictor-corrector methods or leapfrog methods, require extremely small time steps for a stable simulation. Using implicit time integration opens SPH up to a much larger class of practical problems in applied mechanics.

Benchmark Solution for Time-Dependent Natural Convection Flows with an Accelerated Full-Multigrid Method

Abstract: A computational procedure is presented with an accelerated full-multigrid scheme for an efficient modeling of time-dependent buoyancy-driven flows. The smoother is the iterative red-and-black successive overrelaxation (RBSOR) scheme. In order to improve the convergence, an acceleration parameter, Γ, is implemented in the classical full-multigrid procedure. It is shown that an optimal value of Γ = 3.75 minimizes the number of iterations needed for convergence. Numerical results are presented and compared with available investigations for an 8:1 differentially heated enclosure and a square heated cavity. Solutions for Prandtl number Pr = 0.71, Rayleigh number Ra = 3.4 × 105 for the 8:1 heated enclosure, and 105 ≤ Ra ≤ 109 for the square cavity are presented and show excellent agreement.

Solution of Radiative Heat Transfer in 2-D Geometries by a Modified Finite-Volume Method Based on a Cell Vertex Scheme Using Unstructured Triangular Meshes

Abstract: A new finite-volume method (FVM) based on a cell vertex scheme and associated to a new modified exponential scheme in which temperature is approximated by linear interpolation using nodal values is proposed to solve the radiative transfer equation in gray absorbing media. The application of the FVM to unstructured triangular meshes is detailed and discussed. The developed code is validated with benchmark cases and applied to pure radiative problem. The approach shows very good performance for the wall heat transfer evaluation. The results indicate that good accuracy is obtained on coarse computational grids, and that solution errors diminish rapidly as the grid is refined.

Numerical Simulation of Subcooled Nucleate Boiling by Coupling Level-Set Method with Moving-Mesh Method

Abstract: A new numerical procedure coupling the level-set method with the moving-mesh method to simulate subcooled nucleate pool boiling is proposed. Numerical test problems have validated this new method. The simulation of bubble dynamics during nucleate boiling under liquid subcooling shows that this novel adaptive method is more accurate in determining interfacial heat transfer than a computational method based on uniform grids with the same number of mesh points.

Second-Order Radiative Transfer Equation and Its Properties of Numerical Solution Using the Finite-Element Method

Abstract: The original radiative transfer equation is a first-order integrodifferential equation, which can be taken as a convection-dominated equation. The presence of the convection term may cause nonphysical oscillation of solutions. This type of instability can occur in many numerical methods, including the finite-difference method and the finite-element method, if no special stability treatment is used. To overcome this problem, a second-order radiative transfer equation is derived, which is a diffusion-type equation similar to the heat conduction equation for an anisotropic medium. The consistency of the second-order radiative transfer equation with the original radiative transfer equation is demonstrated. The perturbation characteristics of error are analyzed and compared for both the first- and second-order equations. Good numerical properties are found for the second-order radiative transfer equation. To show the properties of the numerical solution, the standard Galerkin finite-element method is employed ...

A Coupled Incompressible Flow Solver on Structured Grids

Abstract: This article deals with the formulation, implementation, and testing of a fully coupled velocity–pressure algorithm for the solution of laminar incompressible flow problems. The tight velocity–pressure coupling is developed within the context of a collocated structured grid, and the systems of equations involving velocity and pressure are solved simultaneously. The pressure and momentum equations are derived in a way similar to a segregated SIMPLE algorithm [1], yielding an extended set of diagonally dominant equations. An algebraic multigrid solver is used to accelerate the solution of the extended system of equations. The performance of the newly developed coupled algorithm is evaluated by solving three test problems showing the effects of grid size, mesh skewness, large pressure gradients, and large source terms on the convergence behavior. Results are presented in the form of convergence history plots and tabulated values of the maximum number of required iterations, the total CPU time, and the CPU ti...

Transient Radiation and Conduction Heat Transfer in a Gray Absorbing-Emitting Medium Applied on Two-Dimensional Complex-Shaped Domains

Abstract: This article is devoted to transient radiation and conduction heat transfer in a gray absorbing-emitting medium in a two-dimensional complex-shaped domain using unstructured triangular meshes. The radiative transfer equation (RTE) is solved by using a new finite-volume method (FVM) based on a cell vertex scheme and associated to a modified exponential scheme. The PHAML (Parallel Hierarchical Adaptive MultiLevel) code is used to solve the energy balance equation using low- or high-order finite elements. Several benchmark cases including steady and transient states and applied to different geometries are used to validate the developed code. Results show very good agreement.

Discrete Ordinates Radiation Element Method for Radiative Heat Transfer in Three-Dimensional Participating Media

Abstract: A new algorithm, the discrete ordinates radiation element method (DOREM), for modeling radiative heat transfer in inhomogeneous three-dimensional participating media is described. The DOREM uses advantages of the both the radiation element method (REM) and the discrete ordinates method. Benchmark comparisons are conducted against several radiation models. The DOREM successfully implements radiative heat transfer simulations precisely, since false scattering never occurs. The DOREM has advantages of computational speed against Monte Carlo, and the CPU time and the memory size of the DOREM are 82 times faster and 767 times smaller at the maximum than that of the REM.

A Compact Momentum Interpolation Procedure for Unsteady Flows and Relaxation

Abstract: An improved version of the momentum interpolation approach for calculating velocities at the cell faces in nonstaggered grids is proposed The procedure is developed for unsteady flows and takes into account the inclusion of relaxation Regarding the integration in time, the first-order Euler and the second-order Euler and Adams-Moulton schemes are analyzed The proposed procedure results in a compact, easy-to-implement expression which is shown to provide the desirable performance: Spurious oscillations of pressure are avoided; converged, steady solutions are independent from relaxation coefficient and time-step size; and the accuracy of the discretization is not negatively affected

An Implicit Fixed-Grid Method for the Finite-Element Analysis of Heat Transfer Involving Phase Changes

Abstract: The finite-element method is widely used for thermal numerical simulation of heat treatment, casting, or welding processes. The modeling of phase changes requires the simulation of highly nonlinear problems due to latent heat effects. In this article, a finite-element procedure is developed for modeling latent heat effects using an implicit time discretization for transient heat conduction problems involving phase changes for which the enthalpy can be supposed to be a function of temperature only. It is also developed for stationary convection-diffusion problems. Examples are presented to show the efficiency of the method for isothermal and anisothermal transformations. Finally, the method is extended to take couplings with metallurgical transformation kinetics into account.

Model Reduction for Heat Conduction with Radiative Boundary Conditions using the Modal Identification Method

Abstract: A model reduction for a conductive material with radiative boundary condition is carried out using the modal identification method. For this type of system, we show that the classical application of the modal identification method may lead to nonacceptable reduced models as a result of some low sensitivities and parameter correlations. Hence, a new reduced model formulation is stated to improve the model reduction process. Some numerical test cases illustrate the study. It is found that the reduced models fit very well with the original system, reducing the order up to 99.5%.

A General Dynamic Linear Tensor-Diffusivity Subgrid-Scale Heat Flux Model for Large-Eddy Simulation of Turbulent Thermal Flows

Abstract: In this article, a general dynamic linear tensor diffusivity model is proposed for representing the subgrid-scale (SGS) heat flux (HF) The tensor diffusivity for the model is an inhomogeneous linear function of the resolved strain and rotation rate tensors, and includes three conventional dynamic SGS HF modeling approaches as special cases In contrast to the dynamic SGS eddy diffusivity modeling approach, the proposed model admits more degrees of freedom for representing the SGS thermal diffusivity, allows for nonalignment between the SGS HF and resolved temperature gradient, and consequently provides a more realistic geometric representation of the SGS heat flux To validate the proposed modeling approach, numerical simulations have been performed based on a combined forced- and natural-convention flow in a vertical channel with a Reynolds number and a Grashof number Gr = 96 × 105 In comparison with the reported direct numerical simulation data and the results obtained using the conventional dynamic

An Improved Numerical Scheme for the SIMPLER Method on NonOrthogonal Curvilinear Coordinates: SIMPLERM

Abstract: In this article, an improved numerical algorithm named SIMPLERM is proposed for incompressible fluid flow computations on the nonstaggered and nonorthogonal curvilinear grid system. In the proposed algorithm, the contravariant velocities are chosen as the cell face velocities and the Cartesian components as the primary variables. The velocity under-relaxation factor is incorporated into the momentum interpolation, and special treatment is adopted to avoid the underrelaxation factor dependence of the velocity solution. In addition, a 1 − δ pressure difference is introduced into the interfacial contravariant velocity determination. Compared with the existing implementation methods of the SIMPLE family on non-staggered and nonorthogonal grids, the SIMPLERM algorithm can guarantee the coupling between velocity and pressure, underrelaxation independence of the solution, and satisfaction of the conservation law, while still possessing sufficient robustness.

An Efficient and Robust Numerical Scheme for the SIMPLER Algorithm on Non-Orthogonal Curvilinear Coordinates: CLEARER

Abstract: In this article, an Improved SIMPLER (CLEARER) algorithm is formulated to solve the incompressible fluid flow and heat transfer on the nonstaggered, nonorthogonal curvilinear grid system. By virtue of a modified momentum interpolation method in calculating the interface contravariant velocity in both the predictor step and the corrector step, the coupling between pressure and velocity is fully guaranteed, and the conservation law is also satisfied. A second relaxation factor is introduced in the corrector step, of which the convergent solution is independent. By setting the second relaxation factor less than the underrelaxation factor for the velocity to some extent, both the convergence rate and robustness can be greatly enhanced. Meanwhile, the CLEARER algorithm can also overcome the severe grid nonorthogonality. With the simplified pressure-correction equation, the convergent solution can still be obtained even when the intersection angle among grid lines is as low as 1°, which may provide valuable gui...

Improvement of SIMPLER Algorithm for Incompressible Flow on Collocated Grid System

Abstract: In this article an Improved SIMPLER (CLEARER) algorithm is proposed to solve incompressible fluid flow and heat transfer problems. Numerical study shows with the CLEARER algorithm on a collocated grid, in the correction stage the velocities on the main nodes are overcorrected with the pressure correction, which lowers the convergence rate; hence a second relaxation factor is introduced to overcome this disadvantage. By setting this factor less than the underrelaxation factor for velocities, the convergence performance can be significantly enhanced; meanwhile, the robustness can also be increased. Four numerical examples with reliable solutions are computed to validate the CLEARER algorithm, and the results show that this algorithm can predict the numerical results accurately. Compared with the SIMPLER algorithm, CLEARER can enhance the convergence rate greatly, and in some cases it only needs as little as 17% of the iterations required by SIMPLER to reach the same convergence criterion.

Effect of Various Thermal Boundary Conditions on Natural Convection in a Trapezoidal Cavity with Linearly Heated Side Wall(s)

Abstract: A penalty finite-element-based study has been carried out for natural-convection flow in a trapezoidal cavity with uniformly heated bottom wall and linearly heated and/or cooled vertical wall(s) in the presence of an insulated top wall. For linearly heated side walls, symmetry in flow patterns is observed, whereas secondary circulation is observed for the linearly heated left wall and cooled right wall. The local Nusselt number indicates reversal of heat flow at the side walls or the left wall. The average Nusselt number versus Rayleigh number illustrates that the overall heat transfer rate at the bottom wall is larger for the linearly heated left wall and cooled right wall.

Numerical Investigation of Rarefied Diatomic Gas Flow and Heat Transfer in a Microchannel Using DSMC with Uniform Heat Flux Boundary Condition—Part I: Numerical Method and Validation

Abstract: For flows associated with microelectromechanical systems (MEMS), the heat flux specified (HFS) boundary condition exists broadly. However, problems with the HFS boundary condition have not been well realized in the simulations of microchannel flows using the direct-simulation Monte Carlo (DSMC) method. In the present work, inverse temperature sampling (ITS) is used to deal with diatomic gaseous flow and heat transfer in a microchannel. This technique provides an approach to calculate the molecular reflective characteristic temperature from the molecular incident energy and the heat flux at the wall boundary. Coupled with the DSMC method, this diatomic molecule ITS technique can be used to treat the HFS boundary conditions in the DSMC method. Verification indicates that the proposed diatomic molecule ITS method can accurately simulate the gaseous flow and heat transfer. In Part II of this work, the proposed method is applied to demonstrate general microchannel gaseous flow properties under uniform heat flu...

Numerical Investigation of Rarefied Diatomic Gas Flow and Heat Transfer in a Microchannel Using DSMC with Uniform Heat Flux Boundary Condition—Part II: Applications

Abstract: In the first part of this work (Part I), we presented and validated the DSMC-HFS method, which can be used to deal with heat flux specified boundary conditions in DSMC simulations. In this article, the method is applied to demonstrate the general properties of rarefied diatomic gaseous flow in a microchannel under uniform heat flux boundary conditions. The effects of wall heat flux on gaseous flow and heat transfer characteristics are investigated numerically and discussed in detail. It can be concluded from the present research that gaseous rarefication and compressibility increase with the increase of the wall heat flux. Gas acceleration at higher wall heat flux is more obvious than that at lower wall heat flux. The high wall heat flux reduces the mass flow rate and elevates the heat transfer ability except at the channel inlet.

Decoupled Reconstruction Method for Simultaneous Estimation of Temperatures and Radiative Properties in a One-Dimensional, Gray, Participating Medium

Abstract: In the decoupled reconstruction method, using an improved Tikhonov regularization method, the temperature distribution in a participating medium is reconstructed from the boundary temperature image, and the radiative properties (absorption and scattering coefficients) are updated from the measured radiative intensity image. These two steps are taken alternately until convergence is reached. The distributions of temperature and radiative properties for two one-dimensional cases are reconstructed by the method from the boundary temperature and intensity images disturbed by measuremental errors with standard deviations up to 0.05, and the method shows good accuracy and robustness.

A Hybrid Cartesian/Immersed-Boundary Finite-Element Method for Simulating Heat and Flow Patterns in a Two-Roll Mill

Abstract: This article describes numerical investigations of the flow and heat patterns in a two-roll mill using the immersed-boundary finite-element method over a fixed Cartesian grid. The second-order projection method is used to advance the solution in time, and a structured linear-triangle element is employed for the spatial discretization. An easily implemented interpolation scheme is adopted to allow accurate imposition of the boundary conditions on an arbitrary shape. Two numerical experiments are carried out, including two-roll-mill flow generated by the two inner cylinders rotating independently in fixed locations and moving around the center of cavity. The physical characteristics, streamline topologies, and temperature contours are discussed for a range of the rotating velocities and Reynolds numbers. The accuracy and robustness of the developed numerical model are validated by results obtained from the unstructured finite element method.

On Parallel Nonequilibrium Molecular Dynamics Simulations of Heat Conduction in Heterogeneous Materials with Three-Body Potentials: Si/Ge Superlattice

Abstract: Parallelization strategies for nonequilibrium molecular dynamics (NEMD) simulations of heat conduction in heterogeneous materials are presented. In particular, a previously published algorithm involving the pair decomposition of three-body potentials is extended for heterogeneous materials. In addition, a novel and linear scaling scheme, also based on pair decomposition of three-body terms, is introduced for the calculation of the heat flux. The distributed-computing-based implementation of this algorithm is outlined and its speed-up characteristics are demonstrated to be close to ideal. Example NEMD simulations using the new algorithm are performed for the Si/Ge superlattice based on the three-body Stillinger-Weber potential.

A Comparative Study of Modeling of Radiative Heat Transfer using MOL Solution of DOM with Gray Gas, Wide-Band Correlated-k, and Spectral Line-Based Weighted Sum of Gray Gases Models

Abstract: A radiation code based on the method of lines (MOL) solution of the discrete ordinates method (DOM) for the prediction of radiative heat transfer in nongray absorbing-emitting media was developed by incorporation of two different gas spectral radiative property models, wide-band correlated-k (WBCK) and spectral line-based weighted sum of gray gases (SLW) models. Predictive accuracy and computational efficiency of the code were assessed by applying it to one- and two-dimensional test problems and benchmarking its steady-state predictions against line-by-line (LBL) solutions and measurements available in the literature. In order to show the improvements accomplished by these two spectral models over and above the ones obtained by gray gas approximation, predictions obtained by spectral models were also compared with those of the gray gas (GG) model. Comparisons reveal that the MOL solution of the DOM with the SLW model produces the most accurate results for radiative heat fluxes and source terms, at the exp...

Estimation of surface conditions for nonlinear inverse heat conduction problems using the hybrid inverse scheme

Abstract: A hybrid numerical method involving the Laplace transform technique and finite-difference method in conjunction with the least-squares method and actual experimental temperature data inside the test material is proposed to estimate the unknown surface conditions of inverse heat conduction problems with the temperature-dependent thermal conductivity and heat capacity. The nonlinear terms in the differential equations are linearized using the Taylor series approximation. In this study, the functional form of the surface conditions is unknown a priori and is assumed to be a function of time before performing the inverse calculation. In addition, the whole time domain is divided into several analysis subtime intervals and then the unknown estimates on each subtime interval can be predicted. In order to show the accuracy and validity of the present inverse scheme, a comparison among the present estimates, direct solution, and actual experimental temperature data is made. The effects of the measurement errors, ...

Particle Transport in Microchannels

Abstract: Particle transport in microchannel is presented. This article focuses on situations in which the sizes of the particles are comparable to the sizes of the channels. These solid bodies are sufficiently large that momentum is exchanged between the bodies and the flowing fluid. As a result, the solid bodies affect the fluid flow significantly, and vice versa, resulting in a transient process in which the motions of the solid bodies and the flow field are strongly coupled. The flow field and the particulate flow must then be solved simultaneously. The solid bodies are modeled as a fluid constraint to move with rigid body motion. The solid–fluid interface is described using a distance function. For demonstration purposes, the finite-volume method is used to solve the resulting set of governing equations. The present approach is validated against (1) flow around stationary, (2) flow around forced rotating, (3) flow around freely rotating cylinders, and (4) sedimentation of a circular cylinder under gravity. Fin...

Solving the Temperature Distribution Field in Nonlinear Heat Conduction Problems Using the Hopfield Neural Network

Abstract: This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in nonlinear heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the nonlinear heat conduction problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is used to solve the heat equation for conduction problems with a power-law nonlinearity. The results confirm that the proposed CHNN scheme provides an accurate means of solving the transient temperature distributions of nonlinear heat conduction problems on a real-time basis.

Modeling Convection-Driven Diffusion-Controlled Wet Chemical Etching Using a Total-Concentration Fixed-Grid Method

Abstract: A total-concentration fixed-grid method is presented to model the convection-driven wet chemical etching process. The proposed method is analogous to the enthalpy method used in the modeling of melting and solidification problems. A total concentration which is the sum of the unreacted etchant concentration and the reacted etchant concentration is defined. The governing equation based on the newly defined total concentration includes the interface condition. Hence the etchfront position can be found implicitly using the proposed method. The reacted etchant concentration is used to predict the etch front position while etching progresses. Since the grid size is fixed, there is no grid velocity, unlike the case with existing moving-grid approaches. Cartesian grids can be used to capture the complicated etch front evolved during etching. In this article, a two-dimensional, incompressible, Newtonian fluid with an infinitely fast reaction at the interface is considered. For demonstration purposes, a finite-vol...

Investigation of glazing behavior in a fire environment using a spectral discrete ordinates method for radiative heat transfer

Abstract: Radiative heat transfer plays a major role in the analysis of glazing behavior in fires, but its rigorous modeling has received little attention. In the present study, a spectral radiative heat transfer model, based on the discrete ordinates method (DOM), is developed and employed to analyze heat transfer and the transient temperature distribution in a glazing structure subjected to fire heat flux. Comparisons are made between model predictions and literature experimental data; acceptable agreements are found. The study also investigates the influence of the glass properties and geometry on the temperature and time to breakage.