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.     

SPECIAL TREATMENT OF PRESSURE CORRECTION BASED ON CONTINUITY CONSERVATION IN A PRESSURE-BASED ALGORITHM

The special features of pressure-correction equations and their effects on the performance of the SIMPLE algorithm have been systematically investigated based on the concept of continuity conservation. Except for use of the same iterative method as for the momentum equations, iterative solution of the pressure-correction equation has special features in three respects: initial values, boundary conditions (BCs), and iterative procedure. First, the initial values in each outer loop are independent and should be reset as zeroes. Second, the BCs are fully reverse to that of velocity: Dirichlet velocity BCs correspond to Neumann BCs of pressure correction, and Neumann velocity BCs lead to pressure-correction Dirichlet BCs. Third, more inner iterations for the pressure-correction equation are required to better satisfy continuity conservation. Dealing properly with these features can greatly improve the efficiency of the SIMPLE algorithm. Computational results and comparisons have shown that global mass conservation BCs are favorable to convergence, but may be slowed down by the local conservation BCs. During the course of convergence, the BCs of the pressure-correction equation are vital: only correct BCs can boost convergence, incorrect BCs cannot. Increasing the inner iterations of the pressure-correction equation will significantly decrease the outer-loop iterations, and therefore effectively improve the performance of the SIMPLE algorithm.     

Avoiding under-relaxations in SIMPLE algorithm

Abstract

The SIMPLE algorithm is devised by interpolating the mass continuity and non-advective momentum equations, provoking apparent simplicity and clarity in the formulation. The SIMPLE variant entitled as the SIMPLE-AC scheme convokes an artificial compressibility (AC) parameter to augment the diagonal dominance of discretized pressure-correction equation. Both methods are characteristically pressure-based, employing a cell-centered finite-volume Δ-formulation on a non-orthogonal collocated grid. A dual-dissipation scheme accompanied by limiting factors, repressing local extrema into the cell-face velocity and pressure, is used to unravel the issue of pressure-velocity decoupling; both SIMPLE and SIMPLE-AC schemes maintain an equivalent scaling (e.g., primary and auxiliary pseudo-time steps remain the same) with the cell-face dissipation and nodal influence coefficients. The phenomenal progress embedded in both contrivances facilitate an avoidance of the pervasive velocity/pressure under-relaxation. However, the SIMPLE-AC algorithm is benefited with using a higher CFL number, enhanced robustness, and convergence compared with the SIMPLE method.     

Doubly fed induction generator‐based variable‐speed wind turbine: Proposal of a simplified model under a faulty grid with short‐duration faults

The aim of this study is to provide a simplified model of a variable‐speed wind turbine (VSWT) with the technology of a doubly fed induction generator (DFIG), which operates under faulty grid conditions. A simplified model is proposed, which consists of a set of electrical and mechanical equations that can be easily modeled as simplistic electrical circuits. It makes it an excellent tool to achieve fault ride‐through capability of grid‐connected VSWT with DFIGs. Both symmetrical and unsymmetrical grid faults, which cause symmetrical and unsymmetrical voltage sags, have been applied to the system in order to validate the model. The proposed simplified model has been compared with the traditional full‐order model under multiple sags (different durations and depths), and the results reveal that both models present similar accuracy. As the idea is to reduce the computational time required to simulate the machine behavior under faulty grid conditions, the proposed model becomes suitable for that purpose. The analytical study has been validated by simulations carried out with MATLAB .     

A DIRECT PARALLEL ALGORITHM FOR THE EFFICIENT SOLUTION OF THE PRESSURE-CORRECTION EQUATION OF INCOMPRESSIBLE FLOW PROBLEMS USING LOOSELY COUPLED COMPUTERS

The numerical simulation of time-accurated complex flows needs large computational resources. For the case of incompressible flows, the solution of the pressure-correction equation is typically the main bottleneck, especially on loosely coupled parallel computers such as PC clusters. An algorithm intended to solve this problem is presented. It is a variant of the Schur complement method that uses direct solvers for each subdomain and for the interface equation. The inverse of the interface matrix is evaluated and stored in parallel. Simulation of turbulent natural convection is used as a benchmark to show its potential and limitations.     

AN IMPROVED SIMPLE METHOD ON A COLLOCATED GRID

A pressure-based method characterized by the SIMPLE algorithm is developed on a nonorthogonal collocated grid for solving two-dimensional incompressible fluid flow problems, using a cell-centered finite-volume approximation. The concept of artificial density is combined with the pressure Poisson equation that provokes density perturbations, assisting the transformation between primitive and conservative variables. A nonlinear explicit flux correction is utilized at the cell face in discretizing the continuity equation, which functions effectively in suppressing pressure oscillations. The pressure-correction equation principally consolidates a triplicate-time approach when the Courant number CFL > 1. A rotational matrix, accounting for the flow directionality in the upwinding, is introduced to evaluate the convective flux. The numerical experiments in reference to a few familiar laminar flows demonstrate that the entire contrivance executes a residual smoothing enhancement, facilitating an avoidance of the pressure underrelaxation. Consequently, included benefits are the use of larger Courant numbers, enhanced robustness, and improved overall damping properties of the unfactored pseudo-time integration procedure.     

An accurate numerical solution study of three-dimensional natural convection in a box

This paper describes the application of the finite difference method to the simulation of three-dimensional natural convection in a box. The velocity–vorticity formulation is employed to represent the mass, momentum, and energy conservations of the fluid medium. We employ a fractional time marching technique for solving seven field variables involving three velocity, three vorticity and one temperature components. By using the fast Fourier transform (FFT) and a tridiagonal matrix algorithm (TDMA), the velocity Poisson equations are advanced in space along with the continuity equation, thus solving efficiently and easily the diagonally dominant tridiagonal matrix equations. Both vorticity and energy equations are discretized through an explicit method (Adams–Bashforth central difference scheme) as a simplified numerical scheme for solving 3D problems, which otherwise requires enormous computational effort. A natural convection in a box for the Rayleigh number equal to 10⁴, 10⁵, 10⁶ and 10⁷ as well as As = L_x / L_z aspect ratios varying from 0.25 to 4 is investigated. It is shown that the benchmark results for temperature and flow fields could be obtained using the present algorithm.     

Natural convective flow in an inclined lid-driven enclosure with a heated thin plate in the middle

Natural convection heat transfer from a heated thin plate located in the middle of a lid-driven inclined square enclosure has been analyzed numerically. Left and right of the cavity are adiabatic, the two horizontal walls have constant temperature lower than the plate’s temperature. The study is formulated in terms of the vorticity-stream function procedure and numerical solution was performed using a fully higher-order compact (FHOC) finite difference scheme on the 9-point 2D stencil. Air was chosen as a working fluid (Pr = 0.71). Two cases are considered depending on the position of heated thin plate (Case I, horizontal position; Case II, vertical position). Governing parameters, which are effective on flow field and temperature distribution, are Rayleigh number values (Ra) ranging from 10³ to 10⁵ and inclination angles γ (0° ? γ < 360°). The fluid flow, heat transfer and heat transport characteristics were illustrated by streamlines, isotherms and Nusselt number (Nu). It is found that fluid flow and temperature fields strongly depend on Rayleigh numbers and inclination angles. Further, for the vertical located position of thin plate heat transfer becomes more enhanced with lower γ at various Rayleigh numbers.     

Fourier Analysis of the Effect of Grid Non-Orthogonality on the SIMPLE Algorithm

In this article, the error amplification matrix is developed for the SIMPLE algorithm formulated on the nonorthogonal grid by Fourier decomposition. The effect of grid skewness on the convergence properties of the SIMPLE algorithm is investigated in terms of robustness and convergence rate. As the grid nonorthogonality increases, the robustness of the SIMPLE algorithm, characterized by the convergence range of the pressure relaxation factor for a given velocity relaxation factor, weakens, and an effective remedy is to take the cross pressure correction terms into account rather than omit them, which is in agreement with the conclusions reached by Peric. In the case of convergence, the convergence rate seems to be strongly dependent on flow field, and the optimal rate can be reached when the grid direction coincides well with the local flow direction. The analysis also suggests that a collocated layout should be adopted in preference to a staggered layout, due to its better robustness on a nonorthogonal grid.     

A Modified Pressure-Correction Scheme for the SIMPLER Method,MSIMPLER

A new method is proposed to accelerate the convergence rate for the SIMPLER algorithm by artificially changing the underrelaxation term to match the dependent variable to be solved. Based on this idea, a new pressure-correction equation is derived, and the modified algorithm is named MSIMPLER. Five numerical experiments show that the MSIMPLER algorithm can appreciably enhance the convergence rate for cases of low and moderate underrelaxation factors with good robustness.     

NUMERICAL AERODYNAMIC SIMULATION OF STEADY AND TRANSIENT FLOWS AROUND TWO-DIMENSIONAL BLUFF BODIES USING THE NONSTAGGERED GRID SYSTEM

The time-averaged Navier-Stokes equations are solved numerically by a finite-volume method and applied to study flow around two-dimensional bluff bodies. The finite-volume equations are formulated in strong conservative form on a general, nonorthogonal grid system. The resulting equations are then solved by an implicit, time marching, pressure-correction based algorithm. If the flow problem has a steady state solution, then it is obtained by taking sufficient time steps until Ike flow field remains unchanged with time. As test cases for the developed methodology, two problems are selected; one has a steady state solution and the other has only a transient solution. Numerical predictions are obtained with the standard k-? turbulence model for the steady state, turbulent flow problem. The k-? model was able to predict the major, experimentally observed flow characteristics including the small separation bubble near the rear end of the body selected for the steady state test case. For the transient test case, the algorithm correctly captured the transient nature of the problem. However, agreement with the experimental results was only moderate because of the lower order differencing scheme employed in the method.     

Estimation of computational grid for thermal convection simulation with lattice Boltzmann method

Accurate, quantitative but not empirical estimation of computational grids helps quickly formulate appropriate computational schemes and shorten the preprocessing of simulations. In this paper, some formulas are proposed to limit a certain range of computational grid N_L for the thermal convection simulations with double distribution function lattice Boltzmann method (DDF-LBM). These formulas are induced from the analysis of relationships among DDF-LBM mathematical limits, mesoscopic physic limits, and flow boundary layer limits, with certain nondimensional parameters Pr, Ra, and Ma. After discussing the essence of the common way in which Ma value is increased to enhance the simulating stability of DDF-LBM, it is confirmed that above formulas also benefit the equivalence between the grid number and increased Ma value. To verify the above formulas, the simulations of Rayleigh–Bénard convection in a square enclosure filled with air at Ra?=?10⁴–10⁸ have been performed. The results coincide well with those in other published references, which suggests the validity of the present study.     

A Coupled Pressure-Based Co-Located Finite-Volume Solution Method for Natural-Convection Flows

A pressure-based coupled solution method based on a finite-volume discretization is presented. The method uses a cell-centered co-located variable arrangement on a nonorthogonal two-dimensional structured grid. The coupled algebraic analogs of the mass, momentum, and energy conservation equations for incompressible flow are solved. In addition to coupling the mass and momentum equations, the energy equation is coupled to the velocities via a Newton-Raphson linearization of the energy advection terms. The momentum equations are coupled to the energy equation via an implicit temperature in the Boussinesq approximation. The convergence behavior of the new method is demonstrated on the solution of steady, laminar natural convection in an annulus for Prandtl numbers of 0.707 and 13,050 at a Rayleigh number of 1 × 10⁶. A significant reduction in the number of iterations to convergence is obtained with the new method compared to a method with only velocity-to-temperature coupling and a method with energy and momentum decoupled. An improvement to the new method was obtained by using an approach that uses a delayed time-step increase and a modified face temperature value estimation.     

Computational Analysis of Reacting Flows in Afterburner

Abstract

Gas turbine afterburner is used during take-off, combat, maneuvers, and emergencies when the aircraft engine needs more thrust than normal. A 60° sector full-scaled afterburner, with extended domain of three times the nozzle diameter in the axial direction and two times the nozzle diameter in the radial direction, is modeled. The numerical calculations are performed using SIMPLE algorithm and k–ε model has been used for turbulence. Kerosene (C₁₂H₂₃) is taken as fuel and virtual injectors are specified for fuel injection. Energy equation and species transport with the Discrete Phase model is selected for computations. Maximum density of 1.25?kg/m³ is observed and the density of the fluid reduced to 0.2?kg/m³ at the exit of nozzle after combustion. The desired Mach number of 1.1 could be observed at the exit of the nozzle. The CO₂ mass fraction increased from 0 to 0.075 whereas the O₂ mass fraction decreased from 0.23 to 0.145 from the inlet to the exit of afterburner. The maximum temperature of 2500?K is observed radially at 0.2?m, from the center of the afterburner and axially at a distance of 0.9?m of afterburner. The obtained results are validated with published experimental and computational fluid dynamics results.     

喷油策略对汽油压燃燃烧及碳烟生成过程影响的数值模拟研究

基于三维计算流体动力学(CFD)软件CONVERGE,耦合甲苯掺比燃料(toluene reference fuel,TRF)简化动力学机理及多步现象学碳烟模型,建立汽油压燃(GCI)的数值模拟模型。通过改变气道喷射比例、主喷时刻和预主喷间隔研究了高负荷条件下气道喷射结合缸内直喷的喷油策略对GCI燃烧及碳烟生成过程的影响。研究结果表明,增加气道喷射比例、提前主喷时刻和增大预主喷间隔都能够缩短燃烧持续期,使放热更为集中,从而降低碳烟排放;改变气道喷射比例对碳烟成核及表面生长有较大的影响,主喷时刻提前能够提高氧化速率。当气道喷射比例为40%,主喷时刻为-8°,预主喷间隔为15°时,碳烟排放为0.015 1g/(kW·h),相比试验基准工况降低了33.8%,而最大压升率也控制在可接受的范围内。     

APPLICATION OF A FRACTIONAL-STEP SCHEME AND FINITE-VOLUME METHOD FOR SIMULATING FLOW PAST A SURFACE-MOUNTED MIXING TAB

The fractional-step scheme and finite-volume method are applied on a structured body-fitted grid to simulate the flow passing over a trapezoidal tab mounted on a flat plate. The implementation of boundary conditions on tab surfaces is greatly simplified with this grid system. Due to grid nonorthogonality, however, discretization of Navier-Stokes equations leads to linear systems with complicated coefficient matrices. For the problem size in this work, performance data indicate that parallel operations occupy about 98.38 % of the simulation, giving rise to a maximum parallel speedup of S p, max , 61.73. The flow passing over the trapezoidal tab is simulated at a Reynolds number Re = 600 based on the inlet free-stream velocity and the tab height, and the results are compared with a particle image velocimetry (PIV) measurement with the same parameters. The simulation successfully captures the vortex structures in the tab wake as observed in the experiments. Comparisons of the instantaneous flow patterns, the mean velocity, and second-order moments also show good agreement. The simulation and PIV experiment also produce a similar shear-stress distribution along the streamwise direction at the flat plate.     

Laminar and turbulent natural convection from a stack of thin hollow horizontal cylinders: A numerical study

Three dimensional natural convection from a stack of thin hollow horizontal cylinders has been investigated numerically in both laminar and turbulent regimes of Rayleigh number (Ra) spanning in the range 10⁴ to 10⁸ and 10¹⁰ to 10¹³, respectively. In the present study, the length to diameter ratio (L/D) of the cylinders is considered in the range 0.5–20. Thin horizontal hollow cylinders of three, six, and ten in numbers are arranged in a triangular manner to form three different types of stacks of same length scale. The full Navier-Stokes equation along with the energy equation are solved to predict the flow pattern and heat loss from the cylinders. The present computational study is able to capture very interesting buoyancy plume structures around the stack of short and long hollow cylinders. The average Nusselt number (Nu) shows a positive dependence on Ra for all L/D. In addition, Nu for a stack of three cylinders is found to be marginally higher than the stack of six cylinders followed by ten cylinders. Empirical correlations are developed for Nu as a function of Ra and L/D, which would be beneficial to academic researchers as well as practicing engineers.     

Simplified edge isolation of buried contact solar cells

The aim of this work is to implement simple edge isolation techniques in buried contact solar cell (BCSC) process by preserving the active cell area. Here we present results of two simplified edge isolation techniques for BCSC and they are compared with the standard process incorporating mechanical edge isolation using a dicing saw. The first technique is chemical wet etching of the solar cell's rear side in an inline system recently developed by University of Konstanz and Rena. The second technique is edge removal carried out in a fluoride/oxide radicals environment of a Asyntis plasma etcher. While the shunt resistance R_sh obtained with wet etching is between 1500 and 7000 Ω cm², the standard process shows R_sh values ranging from 2100–6300 Ω cm². The R_sh after plasma processing is between 1000 and 3600 Ω cm². These cell results show that both wet and plasma etching achieve results close to mechanical edge isolation.However, a slight reduction of short circuit current is observed for the cells undergone standard as well as plasma processing. This is due to the presence of floating volume shunts formed at the rear n–p⁺ junction, which are not removed by either the standard or plasma process. These shunts do not influence the IV-curve of the solar cells and are nearly invisible with conventional thermography, as they are not connected to the front side emitter grid. Hence, light-modulated lock-in thermography measurements were carried out to analyse these shunts.     

Natural convection in tilted rectangular enclosures with a vertically situated hot plate inside

A numerical study of laminar natural convection in tilted rectangular enclosures that contain a vertically situated hot plate is performed. The plate is very thin and isothermal on both lateral ends, and it acts as a heat source within the medium. Three surfaces of the rectangular enclosure are insulated while one lateral surface is cold. Navier–Stokes equations, continuity equation and the energy equation, along with the Boussinesq approximation, are expressed in the form of vorticity-transport equations. All the pertinent equations are solved using the finite volume method with SIMPLE algorithm. The Rayleigh numbers and the tilt angle of the enclosure are ranged from 10⁵ to 10⁷ and from 0° to 90°, respectively. The aspect ratios of the rectangular enclosures that are considered in this study are A = 1 and A = 2. The isotherms and streamlines are produced for various Rayleigh numbers and geometrical conditions, and steady-state Nusselt numbers are computed. The steady-state plate-surface-averaged Nusselt numbers are computed for each case as a function of Rayleigh number and other non-dimensional geometrical parameters and a correlation useful for practical problems was derived.     

A Simplified Model with a Hybrid Analytical-Numerical Solution for Predicting the Unsteady Conjugate Heat Transfer Process in Pipelines

This work presents a new, simplified and yet efficient model for studying the transient conjugate heat transfer process in pipelines. The simplified model consists of assuming the thermal transport by fluid flow in bulk form and the pipe–wall heat conduction as two-dimensional. The scale analysis of the dimensionless equations resulting from the simplified model allows for the identification of two predicting parameters useful in determining when axial diffusion in the pipe wall is important, ψ = Pe Γ/ξ < 1 and λ = 4 Nu E ²/ξ < 1, and when radial diffusion is important, ψ > 1 and λ > 1. When these criteria are satisfied, in which case the existing analytical solution becomes invalid, the simplified model proposed here becomes an efficient alternative to the full-scale numerical simulations required for solving the problem. The scale analysis predictions and the hybrid analytical-numerical solutions of the simplified model, involving the Laplace transform and the finite-volume method, are validated first by comparison against the existing analytical solution, and then by comparison against experimental results of turbulent flow, showing excellent agreement in both cases.