Cell Peclet Number Research Articles

Monotonic unbounded schemes transformer (MUST): an approach to remove undershoots and overshoots in family of unbounded schemes using finite-volume method

PurposeThis paper presents a Monotonic Unbounded Schemes Transformer (MUST) approach to bound/monotonize (remove undershoots and overshoots) unbounded spatial differencing schemes automatically, and naturally. Automatically means the approach (1) captures the critical cell Peclet number when an unbounded scheme starts to produce physically unrealistic solution automatically, and (2) removes the undershoots and overshoots as part of the formulation without requiring human interventions. Naturally implies, all the terms in the discretization equation of the unbounded spatial differencing scheme are retained.Design/methodology/approachThe authors do not formulate new higher-order scheme. MUST transforms an unbounded higher-order scheme into a bounded higher-order scheme.FindingsThe solutions obtained with MUST are identical to those without MUST when the cell Peclet number is smaller than the critical cell Peclet number. For cell Peclet numbers larger than the critical cell Peclet numbers, MUST sets the nodal values to the limiter value which can be derived for the problem at-hand. The authors propose a way to derive the limiter value. The authors tested MUST on the central differencing scheme, the second-order upwind differencing scheme and the QUICK differencing scheme. In all cases tested, MUST is able to (1) capture the critical cell Peclet numbers; the exact locations when overshoots and undershoots occur, and (2) limit the nodal value to the value of the limiter values. These are achieved by retaining all the discretization terms of the respective differencing schemes naturally and accomplished automatically as part of the discretization process. The authors demonstrated MUST using one-dimensional problems. Results for a two-dimensional convection–diffusion problem are shown in Appendix to show generality of MUST.Originality/valueThe authors present an original approach to convert any unbounded scheme to bounded scheme while retaining all the terms in the original discretization equation.

Computational Investigation of Micro-channel Heat Sink with Rectangular Shape Obstacles

Nowadays a lot of interest is given to the geometrical modification of heat sink systems to cool down the electronic components. To improve the performance index of the heat sinks, the use of geometrical features with different shapes and at different locations on the surface can be a valuable approach. In this paper, the effect of rectangular shape obstacles on the micro channel heat sink (MCHS) performance is studied. Due to surface features, vortex is developed which helps to increase the heat transfer rate. Numerical modeling software Comsol Multiphysics with heat transfer in fluid physics is used to investigate the characteristics of a micro-channel heat sink. The numerical result shows that the heat transfer rate can be improved through an appropriate arrangement of rectangular shape obstacles, on the heat sink. Numerical analysis and the comparison is carried out for micro-channel heat sink with and without obstacles. In this paper, various parameters like temperature rise, cell Peclet number and Mean effective thermal conductivity are studied.

Computational Evaluation of Mixing Performance in 3-D Swirl-Generating Passive Micromixers

Computational Fluid Dynamics (CFD) tools are used to investigate fluid flow and scalar mixing in micromixers where low molecular diffusivities yield advection dominant transport. In these applications, achieving a numerical solution is challenging. Numerical procedures used to overcome these difficulties may cause misevaluation of the mixing process. Evaluation of the mixing performance of these devices without appropriate analysis of the contribution of numerical diffusion yields over estimation of mixing performance. In this study, two- and four-inlet swirl-generating micromixers are examined for different mesh density, flow and molecular diffusivity scenarios. It is shown that mesh densities need to be high enough to reveal numerical diffusion errors in scalar transport simulations. Two-inlet micromixer design was found to produce higher numerical diffusion. In both micromixer configurations, when cell Peclet numbers were around 50 and 100 for Reynolds numbers 240 and 120, the numerical diffusion effects were tolerable. However, when large cell Peclet number scenarios were tested, it was found that the molecular diffusivity of the fluid is completely masked by false diffusion errors.

EFDC 모형을 이용한 댐 붕괴류 수치모의 및 매개변수 민감도 분석

In this study, a series of numerical simulation of dam break flow was conducted using EFDC model, and input conditions including cell size, time step, and turbulent eddy viscosity were considered to analyze parameter sensitivity. In case of coarse mesh layout, the propagated length of the shock wave front was <TEX>${\Delta}_x$</TEX> longer than that of other mesh layouts, and the velocity results showed jagged edge, which can be cured by applying fine grid mesh. Turbulent eddy viscosity influenced magnitude of the maximum velocity passing through gate up to 20% and the cell Peclet number less than 2.0 ensured no numerical oscillations.

On the Issues of Asymmetry Observed in Heat Transfer &amp; Material Flow in Friction Stir Welding

In this paper, a standard CFD code, COMSOL is used to describe the 3-dimensional modeling of friction stir welding using threaded tool and based on fully sticking friction. A standard threaded tool profile is used for the full flow analysis including heat generation and heat flow. Temperature distribution is analyzed for full sticking and sliding condition. The effect of different operating parameters on the temperature distribution was performed.The importance of cell Peclet number is also investigated in the temperature distribution on advancing and retreading side. The results are in reasonable agreement with the predictions.

A Conservative Pressure-Correction Method on Collocated Grid for Low Mach Number Flows

A novel extension to SMAC scheme is proposed for variable density flows under low Mach number approximation. The algorithm is based on a predictor—corrector time integration scheme that employs a projection method for the momentum equation. A constant-coefficient Poisson equation is solved for the pressure following both the predictor and corrector steps to satisfy the continuity equation at each time step. The proposed algorithm has second order centrally differenced convective fluxes with upwinding based on Cell Peclet number while diffusive flux are viscous fourth order accurate. Spatial discretization is performed on a collocated grid system that offers computational simplicity and straight forward extension to curvilinear coordinate systems. The algorithm is kinetic energy preserving. Further in this paper robustness and accuracy are demonstrated by performing test on channel flow with non-Boussinesq condition on different temperature ratios.

Anti-diffusion in central difference scheme at steady-state transport equation

This study shows that the central difference scheme causes anti-diffusion when applied to a steady-state one-dimensional scalar transport equation. This behaviour is opposite to the upwind scheme which adds numerical diffusion. An iterative anti-diffusion correction algorithm is developed, which eliminates anti-diffusion and reduces cell-Peclet number. This correction, in turn, eliminates or minimises numerical oscillation. This correction scheme is formulated for the inhomogenous equation and for two-dimensional problem as well.

A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity

We propose and analyse a symmetric weighted interior penalty method to approximate in a discontinuous Galerkin framework advection―diffusion equations with anisotropic and discontinuous diffusivity. The originality of the method consists in the use of diffusivity-dependent weighted averages to better cope with locally small diffusivity (or equivalently with locally high Peclet numbers) on fitted meshes. The analysis yields convergence results for the natural energy norm that are optimal with respect to mesh size and robust with respect to diffusivity. The convergence results for the advective derivative are optimal with respect to mesh size and robust for isotropic diffusivity, as well as for anisotropic diffusivity if the cell Peclet numbers evaluated with the largest eigenvalue of the diffusivity tensor are large enough. Numerical results are presented to illustrate the performance of the proposed scheme.

Analysis of stability and accuracy of finite-difference schemes on a skewed mesh

Numerical stability and accuracy of finite-difference schemes on a skewed non-uniform mesh are investigated to provide guidelines for mesh design and for devising appropriate solution methods when mesh skewness is unavoidable. In the current analysis, a linear advection–diffusion equation in a Cartesian coordinate system is transformed into a curvilinear one corresponding to a skewed mesh. A finite-difference approximation of the transformed equation leads to the leading-order error terms which are responsible for time-step restrictions and numerical instability. A truncation error analysis of central-difference approximation reveals the effects of mesh non-uniformity and skewness on the solution quality. In addition, a modified wavenumber analysis is performed for the central- and upwind-difference schemes coupled with time-integration methods, to examine the effects of mesh skewness angle and flow direction relative to the mesh angle on the numerical stability. In general, severe mesh skewness leads to restrictions on the allowable cell Peclet number and time-step size and also increases phase and amplitude errors.

Stabilized Galerkin approximation of convection-diffusion-reaction equations: discrete maximum principle and convergence

We analyze a nonlinear shock-capturing scheme for H1-conforming, piecewise-affine finite element approximations of linear elliptic problems. The meshes are assumed to satisfy two standard conditions: a local quasi-uniformity property and the Xu-Zikatanov condition ensuring that the stiffness matrix associated with the Poisson equation is an M-matrix. A discrete maximum principle is rigorously established in any space dimension for convection-diffusion-reaction problems. We prove that the shock-capturing finite element solution converges to that without shock-capturing if the cell Peclet numbers are sufficiently small. Moreover, in the diffusion-dominated regime, the difference between the two finite element solutions super-converges with respect to the actual approximation error. Numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

FOUR-POINT INTERPOLATION SCHEMES FOR CONVECTIVE FLUXES APPROXIMATION

Two four-point interpolation schemes, FPTOI and FPFOI, have been proposed for approximating convective fluxes. In these new schemes, three points in the upstream direction and one in the downstream direction are used for interpolation. The FPTOI scheme has third-order accuracy and smaller odd truncation error terms (seventh derivative), whereas the FPFOI scheme has fourth-order accuracy and a larger critical cell Peclet number (16/5) compared with the QUICK scheme. In order to assess their capabilities, the two schemes are applied to three test problems, one linear and two nonlinear cases, all involving the simultaneous presence of convective dominance and flow-to-grid skewness. The two new schemes have high-order accuracy and further have demonstrated, in computations of flow cases, the ability to preserve better numerical stability behaviors in comparison with the QUICK scheme.

PRECONDITIONING TECHNIQUES FOR THE BiCGSTAB ALGORITHM USED IN CONVECTION-DIFFUSION PROBLEMS

In this work, we investigate the techniques for computing the preconditioning matrices used in the conjugate gradient method. The physical problem we consider is the convection-diffusion problem with high cell Peclet number, which is differenced by the third-order upwind-biased scheme. The preconditioning matrix is computed by the traditional Incomplete Lower-Upper (ILU) decomposition of the coefficient matrix or the coefficient matrix that is produced by the first-order upwind difference. Since the upwind schemes are one-sided-biased, the factorization error is greatly influenced by the arrangement of the variable in the linear system. At high cell Peclet number, we can reduce the factorization error significantly by ordering the variable according to the local flow direction. This technique, when used with the ILU decomposition of the coefficient matrix produced by the first-order difference, is found to have steady convergence rate with no serious wiggles at high cell Peclet number.

SOLUTION OF THE ADVECTION-DIFFUSION EQUATION USING A COMBINATION OF DISCONTINUOUS AND MIXED FINITE ELEMENTS

When transport is advection-dominated, classical numerical methods introduce excessive artificial diffusion and spurious oscillations. Special methods are required to overcome these phenomena. To solve the advection‒diffusion equation, a numerical method is developed using a discontinuous finite element method for the discretization of the advective terms. At the discontinuities of the approximate solution, numerical advective fluxes are calculated using one-dimensional approximate Riemann solvers. The method is stabilized with a multidimensional slope limiter which introduces small amounts of numerical diffusion when sharp concentration fronts occur. In addition, the diffusive term is discretized using a mixed hybrid finite element method. With this approach, numerical oscillations are completely avoided for a full range of cell Peclet numbers. The combination of discontinuous and mixed finite elements can be easily applied to 2D and 3D models using various types of elements in regular and irregular meshes. Numerical tests show good agreement with 1D and 2D analytical solutions. This approach is compared at the same time with two different numerical methods, a standard mixed finite method and a finite volume approach with high-resolution upwind terms. Regular and irregular meshes are used for the numerical tests to study the mesh effects on the numerical results. Our data show that in all cases this approach performs well. © 1997 by John Wiley & Sons, Ltd.

Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms

Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long-time behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid. We introduce a modification of higher order Godunov methods that possesses the correct asymptotic behavior, allowing the use of coarse grids (large cell Peclet numbers). The idea is to build into the numerical scheme the asymptotic balances that lead to this behavior. Numerical experiments on 2 × 2 systems verify our analysis.

A Combined Laplace Transform and Streamline Upwind Approach for Nonideal Transport of Solutes in Porous Media

A new numerical method for nonideal transport of solutes in porous media is developed by first using the Laplace transform to eliminate the time dependency and then the streamline upwind scheme to solve the spatial equations. The transient solution is ultimately recovered by an efficient quotient difference inversion algorithm. By introducing complex‐valued artificial dispersion in the weighting functions, characteristics of the transient solutions have been successfully addressed. The optimum of the complex‐valued artificial dispersion has been derived for one‐dimensional problems. In multidimensional cases the streamline upwind scheme is modified by adding complex‐valued artificial dispersion along the streamline. Within this numerical scheme the grid orientation problems have been successfully treated. The limitations on the cell Peclet number and on the Courant number were greatly relaxed. Both one‐dimensional and two‐dimensional numerical examples are used to illustrate applications of this technique.

On the computational efficiency of certain upwinding schemes

The computational efficiency of certain upwinding schemes for the discretization of the one-dimensional advection dominated transport equation are examined. In particular, the first order upwind, the second order upwind, the weighted scheme, and the QUICK scheme are considered. The total computational cost for a prescribed accuracy of these schemes was compared using a general purpose algorithm for the integration of time dependent partial differential equations using the method of lines. The method of lines approach enables us to utilize the well developed methodology for the assessment of computational efficiency of ODE integrators. Moreover, in the time integration, the resulting system of ODEs is integrated by an efficient variable order/stepsize numerical integration scheme. The results indicate that for small cell Peclet number, central differences are superior. For moderate values of the cell Peclet number the QUICK scheme is the most attractive. Under high advective flow conditions, however, the second order upwinding is the most reliable.

A Triangular Finite Volume Approach With High‐Resolution Upwind Terms for the Solution of Groundwater Transport Equations

A finite volume approach is developed for the solution of the contaminant transport equation in groundwater. By defining a triangular control volume over which the dependent variable of the governing equation is averaged, the scheme combines the flexibility in handling complex geometries intrinsic to finite element methods with the simplicity of finite difference techniques. High‐resolution upwind schemes are employed for the discretization of the advective terms. The technique is based on the concept of “monotone interpolation” to ensure the monotonicity preserving property of the scheme, and on the exact solution of local Riemann problems at the interface between neighboring control volumes. In this way, numerical oscillations are completely avoided for a full range of cell Peclet numbers. Together with the discretization of the dispersive fluxes, an approximation is obtained that is locally first order, but globally of second‐order accuracy. As compared to usual upwind schemes, much smaller amounts of numerical viscosity are introduced when sharp concentration fronts occur. A number of numerical tests show good agreement with analytical solutions. A hypothetical problem involving nonequilibrium reaction terms is solved to illustrate the applicability and robustness of the proposed formulation for solving the groundwater transport equations.

Application of Higher-Order Methods in Compositional Simulation

Summary A third-order differencing scheme is described and compared with several alternative methods commonly used to discretize the convection term of the component conservation equations of compositional simulators. This new method can be used at higher cell Peclet numbers than those used with other common convective differencing schemes, resulting in a substantial saving in computation time for the same accuracy. Comparisons are made with analytical solutions and with 2D and 3D simulation results for single-point upstream weighting, two-point upstream weighting, and Chaudhari's method.

An upwind Galerkin finite element analysis of linear induction motors

An upwind Galerkin finite-element method (GFEM) for electromagnetic problems in moving media is proposed. In the conventional GFEM, when the cell Peclet number becomes greater than the critical value of 2, many undesirable flux loops due to oscillatory solutions appear in the flux plots. It is shown that the upwind GFEM can stabilize such oscillations successfully. Although the extra-fine subdivision of elements makes it possible to suppress such oscillations without upwinding, the 13 mm length of a mesh in the present model must be subdivided into 2.43 mu m in order that the Peclet number in the back iron shall not exceed the critical value of 2 or there will be a need for enormous computation time and memories. For this reason, the present GFEM is the most promising numerical method for convective diffusion equation solvers with larger cell Peclet number.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Numerical Analysis and Experiments of Convective Heat Flow in Fire Compartment

Theoretical and experimental numerical analysis shows the capabilities of finite difference calculations of fine induced natural convective heat flow in a fire compartment. The viscous, heat conductive, compressible fluid is represented using a K-e model. Because a two-point upwind difference scheme gives numerical viscosity, the computational results are suspect at large velocities. The practical stability limits and truncation errors for finite difference equations approximating fire flows have been analyzed. The sensitivities of numerical solutions have been evaluated by the theoretical and experimental numerical analysis. Based on results of numerical experiments we propose a reasonable time interval and space mesh size criteria which also considers CPU time. Furthermore, we have introduced the effective maximum cell Reynolds number (Re∗) for the equation of motion, or the effective maximum cell Peclet number (Pe∗) for the equation of energy. We propose that the values of Re∗ and Pe∗ indicate the trust which may be placed on the approximate solutions.