COMPARISON OF INTERFACE-FOLLOWING TECHNIQUES FOR NUMERICAL ANALYSIS OF PHASE-CHANGE PROBLEMS

The Semi-Implicit Method for Pressure-Linked Equations (SIMPLE)algorithm for heat transfer and fluid flow problems is extended to time-periodic situations. A vectorized line group method for solving the system of associated algebraic equations in a rectangular two-dimensional computational domain is developed to speed up the computations. A multiblock procedure with the line group method is used to solve a piston-driven oscillating heat transfer problem. The numerical results obtained show some interesting new phenomena and agree with analytical results where such comparisons are possible.     

ITERATIVE METHOD FOR THE NUMERICAL PREDICTION OF HEAT TRANSFER IN PROBLEMS INVOLVING LARGE DIFFERENCES IN THERMAL CONDUCTIVITIES

Heal exchange that occurs between materials with largely differing thermal conductivities is commonly encountered in engineering practice. Conventional iterative solution methods perform poorly for the numerical solution of such problems. A block correction procedure, designed for enhancing the convergence of iterative solution methods, is used in conjunction with the line-by-line iterative solution method. The overall solution algorithm is a multigrid strategy with two grid levels. Results of computations for test problems indicate that the proposed solution procedure enables efficient solution of heat transfer problems with large conductivity differences for which the conventional line-by-line method proves ineffective.     

Modeling of Heat Transfer Across the Interface in Two-Fluid Flows

A model is presented in this article to deal with heat transfer across the interface separating two immiscible fluids. It is suitable to be incorporated into interface-tracking methods, such as volume-of-fluid (VOF) methods, because a sharp interface is available in these approaches. The temperature at the interface and the heat flux through it are calculated in such a way that the continuity of the two properties at the contact surface is satisfied explicitly. With use of these values, the temperature either at the centroid or on a face of the interface cell can be estimated, which serves as Dirichlet boundary condition for the energy equation. The temperature field is then calculated by solving the energy equations for the two fluids simultaneously in an implicit way. This method is first assessed via testing on two heat conduction problems in which two solids are in contact. Good agreement between numerical solutions and theory is obtained. To demonstrate its capability, it is applied to two kinds of heat transfer problems, one being the collapse of a heated water column in a cavity, and the other the falling of a molten tin droplet in an oil tank. The effect of fluid flow on the heat transfer is clearly illustrated.     

EFFECT OF INCLINED VORTEX GENERATORS ON HEAT TRANSFER ENHANCEMENT IN A THREE-DIMENSIONAL CHANNEL

Three-dimensional unsteady flow and heat transfer in a channel with inclined block shape vortex generators mounted on one side of a channel flow are investigated for different Reynolds numbers, Re _H= 400 - 1500 , and Pr = 0.71. This study was informed to gain an understanding of the flow phenomena and calculate the heat transfer and pressure drop for different Reynolds numbers. The effect of computational domain, angle of incidence, size of the vortex generator, and the discretization schemes on the results are also investigated. Simulations use an incompressible finite volume code, based on a fractional step technique with a multigrid pressure Poisson solver and a nonstaggered grid arrangement.     

A multidomain multigrid pseudospectral method for incompressible flows

Pseudospectral method has the merit of high accuracy and the defects of simple geometry suitability and low computational efficiency. To remedy the two defects, a multidomain multigrid Chebyshev pseudospectral method is proposed and validated through the numerical solution of two-dimensional incompressible Navier-Stokes equations in the primitive variable formulation. To facilitate the implementation of the multidomain multigrid method, the IP_N-IP_N method is utilized to approximate the velocity and pressure with the same degree of Chebyshev polynomials within each subdomain, and an interface/boundary condensation method is developed to implement the pseudospectral operators of multigrid at the interface/boundary of subdomains. The accuracy and efficiency of the proposed method are first validated by numerical solutions of the lid-driven cavity problem. The numerical results are in good agreement with the benchmark solutions, and the speeding up of multigrid is 4–9 compared against the single grid. Then the capability of the proposed method for even more complex geometries with a close/open boundary is demonstrated by numerical solutions of several typical problems. The proposed method is quite generic and can be extended to the high accuracy and efficiency solution of three-dimensional incompressible/compressible, unsteady/steady fluid flows and heat transfer problems.     

Recommendations on Enhancing the Efficiency of Algebraic Multigrid Preconditioned GMRES in Solving Coupled Fluid Flow Equations

The algebraic multigrid (AMG) algorithm as a preconditioner to the Krylov subspace methods has drawn the attention of many researchers in solving fluid flow and heat transfer problems. However, the efficient employment of this solver needs experience, because users have to quantify several important parameters. In this work, we choose a hybrid finite-volume element method and quantify the optimum magnitudes for those parameters. To generalize our results, two sets of fluid flow governing equations, the thermobuoyant flow and confined diffusion flame, are studied and the optimum values are determined. The results indicate that the AMG can be very effective if a proper storage method is chosen.     

A HYBRID FINITE ELEMENT-FINITE DIFFERENCE METHOD FOR SOLVING THREE-DIMENSIONAL HEAT TRANSPORT EQUATIONS IN A DOUBLE-LAYERED THIN FILM WITH MICROSCALE THICKNESS

W e develop a finite element?finite difference method for solving three-dimensional heat transport equations in a double-layered thin film with microscale thickness. The implicit scheme is solved by using a preconditioned Richardson iteration, so that only two block tridiagonal linear systems with unknowns at the interface are solved for each iteration. W e then apply a parallel Gaussian elimination procedure to solve these two block tridiagonal linear systems and develop a domain decomposition algorithm for thermal analysis of the double-layered thin film. Numerical results for thermal analysis of a gold layer on a chromium padding layer are obtained.     

MULTIBLOCK IMPLEMENTATION STRATEGY FOR A 3-D PRESSURE-BASED FLOW AND HEAT TRANSFER SOLVER

This article reports on a multiblock implementation of a general three-dimensional single-block computational fluid dynamics code, which is developed in a nonorthogonal, structured, collocated finite-volume grid system, and incorporates a range of turbulence models. The multiblock implementation is essentially block-unstructured, each block having its own local coordinate system unrelated to those of its neighbors. Any of the blocks may interface with more than one neighbor along any block face. Interblock communication is handled by inner-boundary connection information (receive and send point index arrays) and effected via two-layer dummy cells along interblock boundaries. This communication procedure is easy to extend to parallel computation. The implementation of the algorithm, which takes the advantage of Fortran 90, employs a method to keep most of the single-block code unchanged. Two cases are presented to validate the implementation, and another case with a block number ranging from 1 to 160 blocks is presented for test of the influence of the multiblocking on the convergence rate and execution time.     

A Parallel Multigrid Finite-Volume Solver on a Collocated Grid for Incompressible Navier-Stokes Equations

Multigrid techniques are widely used to accelerate the convergence of iterative solvers. Serial multigrid solvers have been efficiently applied to a broad class of problems, including fluid flows governed by incompressible Navier-Stokes equations. With the recent advances in high-performance computing (HPC), there is an ever-increasing need for using multiple processors to solve computationally demanding problems. Thus, it is imperative that new algorithms be developed to run the multigrid solvers on parallel machines. In this work, we have developed a parallel finite-volume multigrid solver to simulate incompressible viscous flows in a collocated grid. The coarse-grid equations are derived from a pressure-based algorithm (SIMPLE). A domain decomposition technique is applied to parallelize the solver using a Message Passing Interface (MPI) library. The multigrid performance of the parallel solver has been tested on a lid-driven cavity flow. The scalability of the parallel code on both single- and multigrid solvers was tested and the characteristics were analyzed. A high-fidelity benchmark solution for lid-driven cavity flow problem in a 1,024 × 1,024 grid is presented for a range of Reynolds numbers. Parallel multigrid speedup as high as three orders of magnitude is achieved for low-Reynolds-number flows. The optimal multigrid efficiency is validated, i.e., the computational cost is shown to increase proportionally with the problem size.     

FLUID FLOW CALCULATIONS USING A MULTIGRID METHOD AND AN IMPROVED DISCRETIZATION SCHEME

In this study the use of a multigrid method with an improved discretization scheme has been investigated for the solution of the fluid flow equations. The flux-spline scheme has been employed for convection-diffusion in the transport equations. The solution algorithm is based on a coupled solution of the continuity and momentum equations in conjunction with a multigrid method. Two test problems have been considered to assess the performance of the solution procedure. The flux-spline scheme leads to accurate solutions without requiring excessively fine grids. The solution algorithm is shown to be robust and rapidly convergent.     

MULTIGRID COMPUTATION OF NATURAL CONVECTION IN ENCLOSURES WITH A CONDUCTIVE BAFFLE

Conjugate heat transfer has been investigated in a two-dimensional square enclosure with a conducting vertical baffle of finite thickness and varying height. The horizontal end walls are assumed to be adiabatic, and the vertical watts are at constant but different temperatures. Calculations are made by a staggered, finite volume multigrid procedure. The performance of the multigrid method in accelerating the convergence rate is remarkable by comparison with the usual iterative method. The influence of inserting a baffle into the buoyancy-driven square cavity on the heat transfer, as well as on the temperature distribution and velocity field, has been obtained for various Rayleigh numbers Ra, solid /fluid conductivity ratio k? dimensionless baffle height H, and baffle location L. Predicted flow patterns and isotherms indicate that the effect of inserting baffles of varying height upon the overall heat transfer and local temperature profiles in the cavity is limited, except when the height of baffle is large (H > 0.5). The effect of conductivity is also found to be marginal. However, both the height and the conductivity become very important when the baffle is located very near the hot wall or cold walls.     

MIXED CONVECTION FROM A MULTIBLOCK HEATER IN A CHANNEL WITH IMPOSED THERMAL MODULATION

Enhancement of mixed-convection heat transfer in a multiblock heater arrangement in a channel is studied. At the most upstream heated block, a time-periodic heat generation is present, while the heat generation is constant in other heater blocks. The explicit effect of using thermal modulation (time-periodic heat generation) at the upstream heater is examined by acquiring comprehensive numerical solutions. The heat transfer enhancement is pronounced at the heaters at far downstream, and the augmentation is maximized when resonance is realized. The resonance frequency is close to the natural frequency of the system, which scales with the time for the main stream to travel through the interblock region. Plots are illustrated to demonstrate the formations of a pair of circulations downstream of the most upstream heater, which leads to identifying the natural frequency. The increase in the overall pressure drop is calculated. The benefit of heat transfer augmentation, as opposed to the increased pressure drop, is assessed to justify the use of thermal modulation in the upstream heater.     

CALCULATION OF HEAT TRANSFER IN A RADIALLY ROTATING COOLANT PASSAGE

The three-dimensional flow field and heat transfer in a radially rotating coolant passage are studied numerically. The passage chosen has a square cross section with smooth isothermal walls of finite length. The axis of rotation is normal to the flow direction with the flow radially outward. The effects of Coriolis forces, centrifugal buoyancy, and fluid Reynolds number on the flow and heat transfer have all been considered. The analysis has been performed by using a fully elliptic, three-dimensional, body-fitted computational fluid dynamics code based on pressure correction techniques. The numerical technique employs a multigrid iterative solution procedure and the standard κ ? ε turbulence model for both the hydrodynamics and heat transfer. The effect of rotation is included by considering the governing equations of motion in a relative frame of reference that moves with the passage. The consequence of rotation is to bring higher velocity fluid from the core to the trailing surface, thereby increasing both the friction and heat transfer at this face. At the same time, the heat transfer is predicted to decrease along the leading surface. The effect of buoyancy is to increase the radial velocity of the fluid, thus generally increasing the heat transfer along both the leading and trailing surfaces. These effects and trends that have been predicted are in agreement with experimental heat transfer data available in the literature [1,2]. The quantitative agreement with the data was also found to be quite satisfactory.     

A lattice Boltzmann model for interphase conjugate heat transfer

A lattice Boltzmann model is proposed with a newly modified equilibrium distribution function for solving the conservation form of the energy equation to treat the interphase conjugate heat transfer problems under both steady state and unsteady state. The temperature and heat flux continuity conditions at the interface can be inherently satisfied without needing any additional treatments, such as iterative computation, correcting procedure for the incoming distribution function, and the complicated calculation procedure for the source term, to account for the interphase conjugate heat transfer. The implementation of the present LB model, therefore, is more straightforward and more efficient than those in most previous models, especially for problems with complex interfaces. The applicability and accuracy of the proposed LB model were evaluated by some benchmark problems including both simple flat interface and complex interface geometry. The results show excellent agreements with analytical solutions or finite volume results, demonstrating that the present model can serve as a promising numerical technique for dealing with fluid flow and heat transfer in complex heterogeneous systems.     

A Fast Numerical Simulation for Modeling Simultaneous Metal Flow and Solidification in Thin Cavities Using the Lubrication Approximation

A numerical algorithm for modelling steady flow of liquid metal accompanied by solidification in a thin cavity is presented. The problem is closely related to a die cast process and in particular to the metal flow phenomenon observed in thin ventilation channels. Using the fact that the rate of metal flow in the channel is much higher than the rate of solidification, a numerical algorithm is developed by treating the metal flow as steady in a given time-step while treating the heat transfer in the thickness direction as transient. The flow in the thin cavity is treated as two dimensional after integrating the momentum and continuity equations over the thickness of the channel, while the heat transfer is modelled as a one-dimensional phenomenon in the thickness direction. The presence of a moving solid-liquid interface introduces non-linearity in the resulting set of equations, and which are solved iteratively. The location and shape of the solid-liquid interface are found as a part of the solution. The staggered grid arrangement is used to discretize the flow governing equations and the resulting set of partial differential equations is solved using the SIMPLE algorithm. The thickness direction heat-transfer problem accompanied by phase change is solved using a control volume formulation. The results are compared with the predictions of the commercial software FLOW3D® which solves the full three-dimensional set of flow and heat transfer equations accompanied with solidification. The Reynolds's lubrication equations accompanied by the through-the-thickness heat loss and solidification model can be successfully implemented to analyze flow and solidification of liquid metals in thin channel during the die cast process. The results were obtained with significant savings in CPU time.     

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.     

A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer

A lattice Boltzmann algorithm for fluid–solid conjugate heat transfer is developed. A new generalized heat generation implement is presented and a “half lattice division” treatment for the fluid–solid interaction and energy transport is proposed, which insures the temperature and heat flux continuities at the interface. The new scheme agrees well with the classical CFD method for predictions of flow and heat transfer in a heated thick-wall microchannel with less mesh number and less computational costs.     

Fully implicit method for coupling multiblock meshes with nonmatching interface grids

A newly developed efficient and fully implicit method for multiblock mesh coupling that preserves the convergence characteristics of single-block meshing is presented. The technique is developed in the context of an unstructured pressure-based collocated finite-volume method, is applicable to both segregated and coupled flow solvers, and is ideal for code parallelization. The discretization at interfaces is performed in a separate step to stitch the regions sub-matrices into a global matrix. By solving the global matrix, the solution achieved to the multiregion problem is exactly the one that would result from a single-mesh discretization. The method is tested by solving three laminar flow problems. Solutions are obtained by meshing the domain as one block or by subdividing it into a number of blocks with non-matching grids at interfaces. Results show the very tight coupling at interfaces with a convergence rate that is independent of the number of blocks used.     

Effect of moving distance of a moving block on heat transfer in a channel flow

A numerical investigation of heat transfer rate of an insulated block moving on a heated surface in a channel was studied. The study mainly investigated the effect of block moving distance on the heat transfer rate of heated surface. This subject belongs to a kind of moving boundary problems, and the modified Arbitrary Lagrangian Eulerian method is suitable for solving this subject. The results show that the block moving distance affects the flow and thermal fields remarkably. The heat transfer rate of heated surface increases proportionally to the increment of the block moving distance, when the block moving distance is larger than a critical value.     

A COMPARATIVE ASSESSMENT WITHIN A MULTIGRID ENVIRONMENT OF SEGREGATED PRESSURE-BASED ALGORITHMS FOR FLUID FLOW AT ALL SPEEDS

This article deals with the evaluation of six segregated high-resolution pressure-based algorithms, which extend the SIMPLE, SIMPLEC, PISO, SIMPLEX, SIMPLEST, and PRIME algorithms, originally developed for incompressible flow, to compressible flow simulations. The algorithms are implemented within a single grid, a prolongation grid, and a full multigrid method and their performance assessed by solving problems in the subsonic, transonic, supersonic, and hypersonic regimes. This study clearly demonstrates that all algorithms are capable of predicting fluid flow at all speeds and qualify as efficient smoothers in multigrid calculations. In terms of CPU efficiency, there is no global and consistent superiority of any algorithm over the others, even though PRIME and SIMPLEST are generally the most expensive for inviscid flow problems. Moreover, these two algorithms are found to be very unstable in most of the cases tested, requiring considerable upwind bleeding (up to 50%) of the high-resolution scheme to promote convergence. The most stable algorithms are SIMPLEC and SIMPLEX. Moreover, the reduction in computational effort associated with the prolongation grid method reveals the importance of initial guess in segregated solvers. The most efficient method is found to be the full multigrid method, which resulted in a convergence acceleration ratio, in comparison with the single grid method, as high as 18.4.