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.     

A Multidomain Chebyshev Pseudo-Spectral Method for Fluid Flow and Heat Transfer from Square Cylinders

A simple multidomain Chebyshev pseudo-spectral method is developed for two-dimensional fluid flow and heat transfer over square cylinders. The incompressible Navier-Stokes equations with primitive variables are discretized in several subdomains of the computational domain. The velocities and pressure are discretized with the same order of Chebyshev polynomials, i.e., the P_N-P_N method. The Projection method is applied in coupling the pressure with the velocity. The present method is first validated by benchmark problems of natural convection in a square cavity. Then the method based on multidomains is applied to simulate fluid flow and heat transfer from square cylinders. The numerical results agree well with the existing results.     

A novel immersed boundary/Fourier pseudospectral method for flows with thermal effects

ABSTRACT

A novel immersed boundary method (IBM) for flows with thermal effects is proposed, combining high accuracy and low computational cost, provided by the Fourier pseudospectral method (FPSM), for the possibility of handling complex and nonperiodical geometries using the IBM. With focus on incompressible flow problems modeled by Navier-Stokes, mass, and energy equations, the method of manufactured solutions is used for the numerical verification of Dirichlet boundary conditions imposed via the IBM. Then, the proposed method is applied on two different 2-D cases: (1) energy transfer due to natural convection in a square cavity, and (2) an annulus between horizontal concentric cylinders nonuniformly heated. Good agreement with available data in the literature has been achieved.     

A MULTIGRID ALGORITHM FOR A MULTIBLOCK PRESSURE-BASED FLOW AND HEAT TRANSFER SOLVER

A multigrid algorithm is developed along with an implicit multiblock pressure-based solver for calculating flow and heat transfer problems on nonorthogonal grids. The implicit treatment of the block interface has proven to be important for the efficiency of a multiblock method. In this study, the implicitness is adopted for all grid levels during a multigrid solution process. As a result, the block interface becomes invisible in the final solutions and the convergence is not adversely affected. The proposed multiblock multigrid algorithm is demonstrated on several representative two-dimensional flow and heat transfer problems. The results show that order-of-magnitude saving in computational time can be achieved with the multigrid algorithm.     

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.     

A novel parallel two-step algorithm based on finite element discretization for the incompressible flow problem

Based on a two-step finite element method and fully overlapping domain decomposition technique, a parallel two-step algorithm for the incompressible flow problem is introduced and analyzed. In the new algorithm, all of the computations are performed in parallel on global composite meshes which are fine around the subdomain that we concentrate on but coarse elsewhere. Each processor first solves an original problem based on the P₁–P₁ stabilized finite element method and then solves a generalized Stokes problem based on the P₂–P₂ stabilized finite element method on a global composite mesh. The proposed algorithm is easy to implement. What’s more, the new algorithm can yield more accurate solutions compared with the numerical solution obtained by the common two-step method. Numerical tests are presented to verify the efficiency and validity of the proposed algorithm.     

MULTIGRID SOLUTION OF UNSTEADY NAVIER-STOKES EQUATIONS USING A PRESSURE METHOD

Abstract

A multigrid relaxation method is applied to a pressure-based implicit procedure to solve the unsteady, incompressible Navier-Stokes equations. The present multigrid method is a correction scheme according to Brandt. This method is used to solve the scalar matrices resulting from the finite-volume formulation and uses flux averaging as the restriction operator. The accuracy and computational efficiency are demonstrated with a steady-state driven cavity flow and an unsteady flow over a circular cylinder case. The results are compared with single-grid results using the OrthoMin conjugate gradient method and experimental data     

Parametric Study of an Improved Gamma Differencing Scheme Based on Normalized-Variable Formulation for Low-Speed Flow With Artificial Compressibility Technique

Based on the normalized-variable formulation (NVF), the modified GAMMA (MGAMMA) scheme previously devised for compressible flow calculations is now incorporated into an incompressible multigrid solver using the artificial compressibility (AC) technique on an unstructured grid. The MGAMMA scheme used in the present work is parameterized in order to further assess its accuracy and convergence compared to other well-known second-order high-resolution (HR) schemes, such as GAMMA and MINMOD. Testing is performed on three sets of problems: (1) advection of four scalar profiles; (2) flow in the Binnie-Green nozzle; and (3) flow past the NACA 0012 and NACA 4412 airfoils. It is shown that when lowering the diffusion-controlled parameter (β _m ), which serves as an attempt to reduce the numerical diffusion of the HR schemes tested, only the MGAMMA scheme is able to provide a converged solution while attaining solution accuracy.     

Development of a Compact and Accurate Discretization for Incompressible Navier-Stokes Equations Based on an Equation-Solving Solution Gradient,Part III: Fluid Flow Simulations on Staggered Polygonal Grids

A compact and accuracy discretization of incompressible Navier-Stokes equations on staggered polygonal grids is presented in this article. It is a sequel to our efforts in developing a feasible solution procedure to simulate incompressible flow problems in complicated domains. By taking advantage of the discretization procedure for the convection-diffusion equation described in our previous work, difference counterparts of the Navier-Stokes equations can be obtained on staggered polygonal grids. Additional ingredients of pressure–velocity coupling and boundary conditions for velocity gradients in the solution procedure are also described. Several test problems are solved to illustrate the feasibility of the present formulation. From the numerical results obtained, it is evident that the proposed scheme is a useful tools to simulate incompressible flow field in arbitrary domains.     

Second-Law Analysis for an Inclined Channel Containing Porous-Clear Fluid Layers by Using the Differential Transform Method

In this study, convection in a porous medium for a laminar, incompressible, non-Darcy model flow in an inclined channel has been investigated. The flow field considered is composed of porous and clear viscous layers. The solutions are carried out for both clear fluid and porous regions by using the differential transform method (DTM). For the solutions of governing equations, constant values for some parameters such as angle of inclination (φ), porous parameter (σ), and the ratio of the heights of two layers (h) are assigned. In order to verify the applied solution technique, the results obtained are compared to the already existing ones evaluated by perturbation method. It is noticed that the results by two methods are in agreement for small values of Brinkman number (Br). However, for higher values of Br, the solutions carried out by perturbation method lose accuracy but the results of the DTM are still valid. The entropy generation number (N _s ) is derived and plotted by using dimensionless velocity and temperature profiles. One of the advantages of this study to similar studies is to give an open form series solution, which gives a tractable and easily applicable recurative form of nonlinear field equations. In similar studies, it is said that the equations are solved; however, neither solution technique nor accuracy or applicability of given technique are clear. In this work, these are well documented.     

Using a Parallel Spatial/Angular Agglomeration Multigrid Scheme to Accelerate the FVM Radiative Heat Transfer Computation—Part II: Numerical Results

A parallel spatial/angular agglomeration multigrid methodology, employing the full approximation scheme (FAS), is developed to accelerate the finite-volume method for the prediction of radiative heat transfer in absorbing, emitting, and scattering gray media. The methodology for the spatial, angular, and combined spatial/angular agglomeration multigrid procedure was analyzed in Part I of this study. In this second part the proposed numerical scheme is validated against benchmark test cases, demonstrating its capability for considerably improved computational performance, especially in problems including scattering media and reflecting boundaries.     

Simultaneous determination of the heat source and the initial data by using an explicit Lie-group shooting method

Abstract

In this article, an explicit Lie-group shooting method (LGSM) is developed to solve the time-dependent heat source and the initial data for backward heat conduction problems. To recover both unknown data simultaneously, it is very difficult to obtain a stable solution by explicit or implicit schemes. To solve these problems by using conventional numerical schemes, numerical iterative regularization techniques and numerical integration techniques are necessary. To avoid these numerical techniques and to increase the computational efficiency, an explicit LGSM is developed. According to the solution of the quadratic equation of the LGSM, the initial condition can be directly obtained by using the final condition and boundary conditions at the initial time and final time. Using the reciprocal relationship of the solutions for the initial condition and the final condition, the proposed algorithm can avoid numerical integration and numerical iteration. Additionally, a closed-form formula from a two-point Lie-group equation can be directly used to calculate the heat source term. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and stable solution, even with noisy measurement data, by comparing the estimation results with the existing literature.     

NUMERICAL METHOD FOR HEAT TRANSFER,FLUID FLOW,AND STRESS ANALYSIS IN PHASE-CHANGE PROBLEMS

This work presents a finite-volume method for simultaneous prediction of physical phenomena occurring during a solid / liquid phase change, including buoyancy-driven flow in the liquid, deformation and stresses in the solid, and heat transfer in both the liquid and solid parts of the solution domain. The liquid is treated as a Newtonian incompressible fluid and it is assumed that the solid behaves as a thermoelastic body, although other constitutive equations for liquid and / or solid could easily be incorporated. The method solves integral equations of mass, momentum, and energy balance discretized on numerical meshes consisting of cells of arbitrary polyhedral shape. The method is validated by comparing numerical results with analytical solutions and available measurement data.     

Development of a mass-preserving level set redistancing algorithm for simulation of rising bubble

Abstract

This article is aimed to simulate the gas-liquid flow of rising bubbles with a mass-preserving level set method. To resolve the topological changes of gas-liquid interface where the classic finite difference scheme may yield oscillation solutions, the spatial terms in the level set advection equation will be approximated by an optimized compact reconstruction weighted essentially non-oscillatory (OCRWENO) scheme. This scheme achieves high-order accuracy in smooth regions, and meanwhile avoid numerical oscillation near discontinuities. Two benchmark problems including vortex flow and deforming field are chosen to compare the present simulation with previous numerical researches. Several rising bubble problems are validated by the proposed level set method.     

A New High-Precision Boundary-Type Meshless Method for Calculation of Multidomain Steady-State Heat Conduction Problems

This article presents the idea for calculating 2-D steady-state heat conduction problems with multidomain combination by employing the virtual boundary meshless least-square method. Being different from the conventional virtual boundary-element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Thus, the proposed method has the advantages of both the boundary-type meshless method and the virtual boundary element method. Since the configuration of the virtual boundary requires a certain preparation, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2-D problems) to reduce the complexity and difficulty of calculus without loss of accuracy, while the “vertex question” existing in the BEM can be avoided. Numerical examples show that the proposed method is more precise than several other numerical methods while selecting fewer degrees of freedom. In addition, its numerical stability is also verified by computing several cases.     

Assessment of different spatial/angular agglomeration multigrid schemes for the acceleration of FVM radiative heat transfer computations

ABSTRACT

The development and comparison of different parallel spatial/angular agglomeration multigrid schemes to accelerate the finite volume method, for the prediction of radiative heat transfer, are reported in this study. The proposed multigrid methodologies are based on the solution of radiative transfer equation with the full approximation scheme coupled with the full multigrid method, considering different types of sequentially coarser spatial and angular resolutions as well as different V-cycle types. The encountered numerical tests, involving highly scattering media and reflecting boundaries, reveal the superiority of the nested scheme along with the V(2,0)-cycle-type strategy, while they highlight the significant contribution of the angular extension of the multigrid technique.     

Dual solutions for flow and radiative heat transfer of a micropolar fluid over stretching/shrinking sheet

A boundary layer analysis is presented for the flow and radiative heat transfer of an incompressible micropolar fluid over stretching/shrinking sheet with power-law surface velocity and temperature distributions. Dual solutions are analytically obtained firstly by homotopy analysis method (HAM). It is found that dual solutions not only exist for the shrinking flow as reported in the previous literatures, but also exist for the stretching flow. The special case of the first branch (K = 0, classical Newtonian fluid) is compared with the existing numerical results of stretching flow in good agreement. Our results show that both solutions are physically meaningful (two solutions are closely related to each other), unlike the results previously reported that only one solution is acceptable. Moreover, the effects of the material parameter K, the radiative Prandtl number Pr_n, the velocity exponent parameter m and the temperature exponent parameter λ on the flow and heat transfer characteristics are analyzed in detail.     

Combined influence of cross-diffusion and variable fluid properties on free convection flow past a vertical cone

The free convective flow of an incompressible viscous fluid over an isothermal vertical cone with variable viscosity and variable thermal conductivity is examined in the presence of the Soret and Dufour effects. As thermal and solutal boundary conditions at the cone's surface, the constant temperature and concentration (WTC) and constant heat and mass flux (HMF) cases are taken into account. The successive linearization method is applied to linearize a system of nonlinear differential equations that describes the flow under investigation. The numerical solution for the resulting linear equations is attained by means of the Chebyshev spectral method. The obtained numerical results are compared and found to be in good agreement with previously published results. The impact of significant parameters on the heat and mass transfer rates is evaluated and presented graphically for the WTC and HMF situations. In both cases, the Soret number increases the skin friction coefficient and rate of heat transfer while decreasing the Sherwood number. With an increase in the Dufour parameter, the coefficient of skin friction and Sherwood numbers increase while the heat transmission rate decreases.     

Numerical modeling of interfacial heat and mass transport phenomena during a phase change using ANSYS-Fluent

ABSTRACT

Numerical modeling of phase change requires accurate estimations of heat and mass transport at the interface. The present work develops a model in ANSYS-Fluent with user-defined functions to address phase change with a planar interface. An interface boundary method determines the heat fluxes with the exact location of the interface without interpolation functions. Five cases are analyzed based on the classic Stefan problem for validating the model. The numerical model is validated against closed form theoretical solutions with agreement within 0.55%. This work can be extended to include curvature effects and the interaction between the interface and heterogeneous surfaces.     

Simulations of Droplet Spreading and Solidification Using an Improved SPH Model

Smoothed particle hydrodynamics (SPH) method as one of the meshless Lagrangian methods has been widely used to simulate problems with free surface. The traditional SPH method suffers from so-called tensile instability, which may eventually result in numerical instability or complete blowup during the simulation of bubble/droplet dynamics. A new pressure-correction equation is proposed to efficiently transport the local pressure to the neighboring area during the impact of incompressible/compressible fluid and reduce the disorder of particle distribution. Consequently, the accuracy and the efficiency of the SPH method can be dramatically improved. New treatments to the surface tension and solidification are also proposed to manipulate SPH particles near the free surface and the solidification interface. The improved SPH method has been used to simulate droplet impact, spreading, and solidification. It is evident that the new method can handle the droplet contraction problem without causing numerical instability. The numerically predicted flattening ratio of the splat due to droplet impact is in good agreement with the analytical prediction. The results demonstrate that the improved SPH model is a powerful tool to study droplet spreading and solidification.