Fourier Analysis of the Effect of Momentum Interpolation on the SIMPLE Series Formulated on a Collocated Grid

The error amplification matrices of three SIMPLE series formulated on a staggered grid and a collocated grid using three momentum interpolation methods are derived. The effects of grid layout and momentum interpolation methods on convergence properties of SIMPLE series are checked. The results indicate that the convergence conditions are available by damping out the highest and lowest errors. The SIMPLE series show almost identical convergence rates under a wide range of useful velocity relaxation factors, regardless of grid layout and interpolation method. However, the high-frequency reduction shows significant variation with grid layout and interpolation method, which maybe referred to in multigrid application.     

The Physical Influence Scheme applied to staggered unstructured grids for solving fluid flow problems

Abstract

In this article, a finite volume formulation for solving the Navier-Stokes equations using unstructured hybrid grids and a staggered arrangement of variable pressure and velocity is presented. In this manner, a tight spatial pressure-velocity coupling is ensured without compromising geometrical flexibility. A second contribution of this work lays upon proposing a suitable interpolation function for the convected velocities in the momentum equations. For this purpose, we employ the so-called “Physical Influence Scheme (PIS),” which fully incorporates all physical phenomena involved, namely convection, diffusion, and pressure effects. Each term corresponding to each physical phenomenon is carefully addressed. We present numerical experiments that show the suitability of the PIS for the staggered finite volume formulation and highlight the importance of including diffusion and pressure effects (in addition to convection) into the interpolation function.     

A New Stability-Guaranteed Second-Order Difference Scheme

Based on the stability-controllable second-order difference (SCSD) scheme, a new stability-guaranteed second-order difference (SGSD) scheme is proposed whose merits are absolutely stable and adaptive. Its numerical accuracy is at least no less than that of the central difference (CD) and second-order upwind difference (SUD) schemes and sometimes higher than that of the QUICK scheme. The SGSD scheme can automatically choose a different difference scheme according to the available local field information in difference space or time. It automatically approaches the central difference scheme where or when diffusion is dominant, and approaches the second-order upwind difference scheme where or when convection is dominant. Computations for two benchmark problems using the SGSD and the other three schemes show its feasibility in engineering computations.     

SOLUTION OF THE STEADY NAVIER-STOKES EQUATIONS BY THE PRESSURE CORRECTION METHOD IN THE ALE GRID

The arbitrary Lagrangian-Eulerian (ALE) grid system possesses some computational merits over the fully staggered and collocated grids in solving the fluid flow problem with primitive variables. However, severe pressure wiggle problems will occur when a standard centered difference scheme is adopted in this grid. In this article, a split velocity concept is introduced to eliminate the checkerboard pressure field in the ALE grid. In this scheme, the cell face velocities are corrected to account for the true pressure gradients by a simple algebraic arrangement of adjacent velocities. The consequence of cell face velocities is analogous to that from momentum interpolation in the collocated grid; however, the split velocities do not produce deterioration of the total mass flux in the original velocity field. Computed results are compared with other numerical predictions and available experimental data for developing channel flow, lip-driven cavity flow, sudden expansion flow, and flow over a backward-facing step. It is shown that this method successfully simulates complex flow situations with different flow characteristics. Finally, comparison of convergent rate with other grid systems is also discussed in this article.     

Conservation Properties of Unstructured Finite-Volume Mesh Schemes for the Navier-Stokes Equations

The Navier-Stokes equations describe fluid flow by conserving mass and momentum. There are two main mesh discretizations for the computation of these equations, the collocated and staggered schemes. Collocated schemes locate the velocity field at the same grid points as the pressure one, while staggered discretizations locate variables at different points within the mesh. One of the most important characteristic of the discretization schemes, aside from accuracy, is their capacity to discretely conserve kinetic energy, specially when solving turbulent flow. Hence, this work analyzes the accuracy and conservation properties of two particular collocated and staggered schemes by solving various problems.     

NUMERICAL SOLUTION OF NAVIER-STOKES EQUATIONS WITH NONSTAGGERED GRIDS USING FINITE ANALYTIC METHOD

A new method referred to here as the momentum weighted interpolation method (MWIM) is introduced to solve the incompressible Navier-Stokes equations with nonstaggered grids. This new method is compared to other methods that employ nonstaggered and staggered grids, in terms of computational effort, number of iterations, and accuracy. It is found that MWIM with nonstaggered grids is as accurate as the staggered grid methods, is easier to implement, and is computationally more efficient. The finite analytic method is used to discretize the governing equations of the fluid flow.     

IMPLEMENTATION OF CLEAR ALGORITHM ON COLLOCATED GRID SYSTEM AND APPLICATION EXAMPLES

ABSTRACT

Implementation of the CLEAR algorithm on a collocated grid system is conducted. Detailed discussion of the previous momentum interoperation method (MIM) is given to analyze the condition to get a unique converged solution that is independent of relaxation factor for steady flow. Six numerical examples on nonstaggered grids of forced-convective fluid flow and natural convection are provided to compare the convergence performance between CLEAR and SIMPLER. The domain extension method, widely used on staggered grids to deal with mildly irregular domains, is further refined to meet the requirement of collocated grids. It is shown that on a collocated grid the CLEAR algorithm can also greatly enhance the convergence rate, based on iteration number and CPU time consumed, compared with the SIMPLER algorithm with similar robustness.     

Stability-Controllable Second-Order Difference Scheme for Convection Term

INTanDUCTIONIntherealmofcomputati0na1fluiddynamicsandconvectiveheattransfer,false-diffusionandconvec-tiveinstabilityarethemajorobstaclestotheachieve-melltofaccurateandeconomicalsoluti0nt0thegov-erningequations.Theconvectiveinstabilityisassrvciatedwiththepresenceofalargeconvectiontermintheequationandischaracterizedbyanoscilla-tionofthesolutionfromnodet0n.de[1'2].Thisin-stabilitydiffersfromtheinstabilitycausedbylargetimestepintransientsystem(instabilityofinitialproblem),andmaybeencountereddu…     

热质渗透壁面饱和多孔介质通道流动与热质传递的数值模拟

利用数值模拟方法研究了多孔介质中存在温度梯度、浓度梯度并具有热质渗透壁面时的受迫对流对传热传质的影响。采用有限容积法在同位网格上离散控制多孔介质内流体流动与热质传递方程守恒方程(即N-S)，对流项采用二阶精度的QUICK格式，扩散项采用中心差分格式。利用SIMPLE算法求解压力和速度耦合问题。利用所发展的程序研究了在不同孔隙率，不同的温度、浓度边界条件下，流场、温度场和浓度场以及Nu和Sh的变化规律。     

基于可压缩SIMPLE算法的叶栅通道湍流流场的数值模拟

开发了二维任意曲线坐标系下、采用同位网格布局、能计算跨音速可压缩流动的SIMPLE方法计算程序。为提高精度,采用QUICK格式、CUI格式在不同的网格分布下计算。为了验证本文方法对低速到跨音速范围内的粘性流动的数值分析效果,对几个典型算例进行了数值试验,计算结果与文献和实验数据吻合较好。     

Inlet conditions effects on vertical wall jets in forced and mixed convection regimes

In this paper a numerical investigation of a laminar isothermal or nonisothermal two dimensional plane wall jet is carried out. Special attention has been paid to the effect of the inlet conditions at the nozzle exit on the hydrodynamic and thermal characteristics of the flow, in both convection regimes: forced and mixed. Two velocities profiles at the nozzle exit are used: uniform profile and parabolic profile. The Prandtl number effect on the jet flow characteristics is also analyzed in the case of forced convection regime.The system of governing equations is solved with an implicit finite difference scheme. For numerical stability we use a staggered non-uniform grid. The obtained results show, for the two convection modes, that the inlet conditions affect the flow in the immediate neighbourhood of the nozzle (core region) in which the flow is governed mainly by the inertia forces. In the established region the results become independent of the flow inlet conditions. Secondly, the effect of the Prandtl number is significant in the plume region in which the jet flow is governed by buoyant forces.     

Benchmark solutions for natural convection in a cubic cavity using the high-order time-space method

Benchmark numerical solutions for a three-dimensional natural convection heat transfer problem in a cubical cavity are presented in this paper. The 3-D cavity has two differentially heated and isothermal vertical walls and also four adiabatic walls. The computations are conducted for three Rayleigh numbers of 10⁴, 10⁵ and 10⁶. The filled fluid is with air and the Prandtl number is fixed at 0.71. The computed results are efficiently obtained by using the time-space method, which was proposed by Saitoh (1991) as a highly efficient and fast solver for general heat transfer and fluid flow problems. In our computations, the high-accuracy finite differences of a fourth-order were employed for the spatial discretization of governing equations and boundary conditions. In addition the third-order backward finite difference was used in timewise discretization. The resultant converged flow and temperature characteristics are also presented. The spatial grid dependency of the solutions was examined on a uniform grid. In addition, the grid-independent benchmark solutions were obtained by Richardson extrapolation for three cases. The present benchmark solutions will be useful for checking the performance and accuracy of any numerical methodologies.     

Numerical and experimental investigation of melting of ice involving natural convection

Two‐dimensional transient melting of ice in a rectangular enclosure was numerically and experimentally investigated. Natural convection in the liquid phase due to the temperature dependency of water density was considered in the numerical model. The implicit finite difference method with fixed staggered grid approach was utilized. The SIMPLER algorithm was followed for the solution of pressure and velocity fields in the liquid phase. The prediction of the model was found to be satisfactory through preliminary experimentation. Copyright © 2002 John Wiley & Sons, Ltd.     

A staggered grid arrangement for solving incompressible flows with hybrid unstructured meshes

This work presents an alternative to the discretization of the Navier–Stokes equations using a finite volume method for hybrid unstructured grids with a staggered grid arrangement of variables. It has developed a numerical scheme, analogous to the element-based finite volume method, for the solution of 2-D incompressible fluid flow problems using several coupling strategies. All velocity components are stored at each face of the elements (pressure control volumes), following the usual procedure of staggering velocity and pressure. With this staggered arrangement, the balance of mass and momentum is satisfied, simultaneously, for the same set of variables, rendering numerical stability when compared to the nonstaggered arrangement.     

Heat transfer in particulate flows with Direct Numerical Simulation (DNS)

A Direct Numerical Simulation (DNS) method has been developed to solve the heat transfer equations for the computation of thermal convection in particulate flows. This numerical method makes use of a finite difference method in combination with the Immersed Boundary (IB) method for treating the particulate phase. A regular Eulerian grid is used to solve the modified momentum and energy equations for the entire flow region simultaneously. In the region that is occupied by the solid particles, a second particle-based Lagrangian grid is used, which tracks particles, and a force density function or an energy density function is introduced to represent the momentum interaction or thermal interaction between particle and fluid. The numerical methods developed in this paper have been validated extensively by comparing the present simulation results with those obtained by others.     

Study on Momentum Interpolation Methods with Curvilinear Collocated Grids in Single-Phase and Two-Phase Flows

To avoid checkerboard pressure in numerical simulations by using collocated grids, the momentum interpolation method was initially proposed by Rhie and Chow and subsequently revised by various researchers. All the momentum interpolation methods are classified into two categories, which are called method A and method B in this article. The performance of the two kinds of methods is evaluated for both single- and two-phase flow in the present work. It is found that method B is more robust for the numerical simulation of two-phase flow.     

Unsteady conjugate forced convection heat/mass transfer in ensembles of spherical particles with cell models

Numerical methods are used to investigate the transient, conjugate, forced convection heat/mass transfer in multiparticle systems at low to moderate Reynolds numbers. The interparticle interactions have been accounted for by using the simple cell models. The momentum and heat/mass balance equations were solved numerically in spherical coordinates system by a finite difference method. The values considered for the sphere Reynolds number are Re < 100. The computations were focused on the influence of the voidage and physical properties ratios on the heat/mass transfer rate for sphere Peclet number, 10 ? Pe ? 1000.     

The Effect of a Blockage on Forced Convection Heat Transfer From a Heated Square Cylinder to Power-Law Fluids

Forced convection heat transfer characteristics of a long, heated square cylinder blocking the flow of a power-law fluid in a channel is numerically investigated in this study. In particular, the role of the power-law index n, Reynolds number Re, Prandtl number Pr, and blockage ratio β(=B/H) on the rate of heat transfer from a square cylinder in a channel has been studied over the following ranges of conditions: 0.5 ≤ n ≤ 1.8, 60 ≤ Re ≤ 160, β = 1/4, 1/2, and 0.7 ≤ Pr ≤ 50. A semi-explicit finite-volume method is used on a nonuniform collocated grid arrangement. The third-order QUICK and the second-order central difference schemes are used to discretize the convective and diffusive terms, respectively, in the momentum and energy equations. Irrespective of the type of behavior of fluid (different values of n), the average Nusselt number increases as the blockage ratio increases. Similar to the unconfined flow configuration, the average Nusselt number increases monotonically with Reynolds and Prandtl numbers for both values of the blockage ratio and for all values of power-law index considered here. Further insights into the heat transfer phenomenon are provided by presenting isotherm contours in the vicinity of the cylinder for a range of values of the Reynolds number, Prandtl number, and power-law index for the two values of β considered in this work.     

A NUMERICAL STUDY OF SOLIDIFICATION IN THE PRESENCE OF A FREE SURFACE UNDER MICROGRAVITY CONDITIONS

A numerical study of the relative importance of Marangoni effects under microgravity conditions is presented. The mathematical formulation adopted is based on the enthalpy porosity method. One of the advantages of the fixed grid method is that a unique set of equations and boundary conditions is used for the whole domain, including both solid and liquid phases. The governing equations written in a vorticity-velocity formulation are discretized using a finite volume technique on a staggered grid. A fully implicit method has been adopted for the mass and momentum equations, while the temperature field is solved separately in order to evaluate the variation in the local liquid mass fraction. The resulting algebraic system of equations is solved using a preconditioned BI-CGStab method. Numerical results modelling the free surface, including the effects on it of Marangoni convection, are presented. The influence of the presence of argon in the gap above the free surface is investigated. During the numerical simulations presented in this paper 161 2 41 and 641 2 161 uniform meshes on the whole computational domain for values of Marangoni number ( Ma ) up to 16,120 and Rayleigh number ( Ra ) of 5 have been used.     

A finite difference method for studying thermal deformation in a 3D thin film exposed to ultrashort pulsed lasers

Ultrashort pulsed lasers have been attracting worldwide interest in science and engineering communities. Studying the thermal deformation induced by ultrashort pulsed lasers is important for preventing thermal damage. This article presents a new numerical method for studying thermal deformation in a 3D thin film exposed to ultrashort pulsed lasers. The method is obtained based on the parabolic two-step model and implicit finite difference schemes on a staggered mesh. It accounts for the coupling effect between lattice temperature and strain rate, as well as for the hot electron-blast effect in momentum transfer. In particular, a fourth-order compact scheme is developed for evaluating those stress derivatives in the dynamic equations of motion. The method allows us to avoid non-physical oscillations in the solution. Its performance is demonstrated by a numerical example.