High-order Discretization Schemes Research Articles

THE ISSUES OF SOLUTION APPROXIMATION IN HIGHER-ORDER SCHEMES ON DISTORTED GRIDS

The impact of grid cell geometry on the accuracy of a high order discretization is studied. The issues of solution approximation are investigated on unstructured grids where grid cells are present that are almost degenerate. It will be demonstrated that high-order discretization schemes which employ compact discretization stencil are less sensitive to the geometry of a distorted grid in comparison with those over expanded stencils.

CFD modelling of stirred tanks: Numerical considerations

The aim of this paper is to demonstrate that the Multiple Reference Frames (MRF) impeller rotation model and the standard k– ε turbulence model, as commonly used in engineering CFD simulations of stirred tanks, can accurately model turbulent fluid flow provided very fine grids coupled with higher-order discretization schemes are used. The MRF model has been found to give adequate results for the steady-state simulation of stirred tanks but the k– ε turbulence model generally under or over-predicts turbulence. In this study the CFD software Fluent 6 is used to simulate flow in a small baffled tank of standard geometry agitated by a Rushton turbine impeller. Simulations are conducted on four grids of significantly different resolution using the upwind, central and QUICK discretization schemes. CFD model results are evaluated in terms of the predicted flow field, power number, mean velocity components and turbulent kinetic energy using published experimental data. The general flow field and mean fluid velocity predictions are not strongly influenced by either the grid resolution or discretization scheme. However, turbulent kinetic energy predictions are strongly influenced by both the grid resolution and discretization scheme. In this study a grid consisting of nearly 2 million control volumes in one half of a 15 cm diameter stirred tank, combined with a high-order discretization scheme, was required to accurately predict the turbulent kinetic energy. These represent conditions which are considerably more numerically intensive than used in most similar studies and suggests that the poor predictions of turbulence obtained using the k– ε turbulence model, often noted in the literature, may be due to numerical errors rather than inadequacies in the turbulence model.

A charge‐scaling implementation of the variational electrostatic projection method

Two new charge-scaling methods for efficient modeling of the solvated macromolecular environment in hybrid QM/MM calculations of biological reactions are presented. The methods are extensions of the variational electrostatic projection (VEP) method, and allows a subset of atomic charges in the external environment to be adjusted to mimic, in the active dynamical region, the electrostatic potential and field due to the large surrounding macromolecule and solvent. The method has the advantages that it offers improved accuracy, does not require the use of a three-dimensional grid or auxiliary set of fitting points, and requires only minor molecular simulation code modifications. The VEP-cs and VEP-RVM+cs methods are able to attain very high accuracy (relative force errors of 10(-7) or better with appropriate choice of control parameters), and take advantage of a recently introduced set of high-order discretization schemes and Gaussian exponents for boundary element solvation and VEP methods. The methods developed here serve as potentially powerful tools in the arsenal of computational techniques used in multiscale computational modeling problems.

Space-Time Mapping Analysis of Airfoil Nonlinear Interaction With Unsteady Inviscid Flow

A new computational method of space-time mapping analysis is applied to nonlinear, inviscid computation of unsteady airfoil response to an upstream flow with a finite amplitude of a time-harmonic vortical perturbation. The unique feature of the method is that it solves the unsteady problem as a steady-state one, by treating the time coordinate identically to space directions. The periodic nature of the flow is used to design the mesh covering one period of the vortical gust in the time direction, with a periodic boundary condition applied at the time-inflow and time-outflow boundaries. The computed solution is, thus, driven directly to the final long-time periodic solution using a pseudotime marching with a high-order discretization scheme. Computed unsteady aerodynamic and aeroacoustic airfoil responses are compared against available numerical solutions. Results obtained for a high-amplitude impinging gust identify zones in the computational domain where nonlinear response effects appear most significant.

High-order discretization schemes for biochemical applications of boundary element solvation and variational electrostatic projection methods

A series of high-order surface element discretization schemes for variational boundary element methods are introduced. The surface elements are chosen in accord with angular quadrature rules for integration of spherical harmonics. Surface element interactions are modeled by Coulomb integrals between spherical Gaussian functions with exponents chosen to reproduce the exact variational energy and Gauss's law for a point charge in a spherical cavity. The present work allows high-order surface element expansions to be made for variational methods such as the conductorlike screening model for solvation and the variational electrostatic projection method for generalized solvent boundary potentials in molecular simulations.

Computational Electromagnetism

Computational Electromagnetism

Steady-State Rate-Based Simulation of Packed Reactive Distillation: Spatial Discretization

This paper investigates issues related to the numerical simulation of rate-based models for catalytic distillation processes. Special emphasis is placed on spatial discretization in the solution of steady-state models describing packed reactive distillation columns. A general rate-based model for packed reactive distillation is briefly presented, along with its underlying assumptions. This mathematical model is then discretized along its spatial dimensions using different finite-difference schemes. In this paper, steady-state simulations are considered to assess various properties of the different discretization methods. Using a tert-amyl methyl ether (TAME) packed reactive distillation as a case study, we show that a cell-based approach, similarly to a first-order finite-difference approximation, is inefficient in converging to the solution of the mathematical model. To address this problem, a higher-order discretization scheme is used, and its advantages are illustrated. This latter method is of interes...

A modified version of the discrete ordinates method for radiative heat transfer modelling

The standard discrete ordinates method (SDOM) suffers from two major sources of inaccuracy, the ray effects and false scattering. False scattering may be significantly reduced using high order discretization schemes, while ray effects originated from abrupt changes of wall temperatures are mitigated by the modified discrete ordinates method (MDOM). However, the MDOM is insensitive to ray effects caused by sharp gradients of the emissive power of the medium. Therefore, a new version (NMDOM) is proposed to overcome this shortcoming. The accuracy of the SDOM, MDOM and NMDOM are investigated, along with the interaction between ray effects and false scattering. It is concluded that the NMDOM yields accurate solutions even for coarse spatial and angular discretizations.

Comparison of method of lines and finite difference solutions of 2‐D Navier–Stokes equations for transient laminar pipe flow

AbstractPerformances of method of lines (MOL) and finite difference method (FDM) were tested from the viewpoints of solution accuracy and central processing unit (CPU) time by applying them to the solution of time‐dependent 2‐D Navier–Stokes equations for transient laminar flow without/with sudden expansion and comparing their results with steady‐state numerical predictions and measurements previously reported in the literature. Predictions of both methods were obtained on the same computer by using the same order of spatial discretization and the same uniform grid distribution. Axial velocity and pressure distribution in pipe flow and steady‐state reattachment lengths in sudden expansion flow on uniform grid distribution predicted by both methods were found to be in excellent agreement. Transient solutions of both methods for pipe flow problem show favourable comparison and are in accordance with expected trends. However, non‐physical oscillations were produced by both methods in the transient solution of sudden expansion pipe flow. MOL was demonstrated to yield non‐oscillatory solutions for recirculating flows when intelligent higher‐order discretization scheme is utilized for convective terms. MOL was found to be superior to FDM with respect to CPU and set‐up times and its flexibility for incorporation of other conservation equations. Copyright © 2001 John Wiley & Sons, Ltd.

The effect of numerical diffusion on simulation of isolated bubbles in a gas–solid fluidized bed

A two-fluid hydrodynamic model of a two-dimensional gas–solids fluidized bed is used to calculate the shape of an isolated bubble. The calculations are done by applying four different discretization schemes to the convection term. The bubble shape calculated by using the first-order upwind (FOW) scheme is unphysical and pointed, where as the bubble shape calculated by the second-order schemes SMART, Superbee, and MINMOD is physically realistic and rounded. Grid independence of the solutions is established. Superbee and SMART are found to give the most accurate results for a given grid resolution. Higher order discretization schemes are implemented with a downwind factor (DWF) method and with a deferred correction (DC) method. The DC method is significantly faster than the DWF method.

A multigrid method with higher-order discretization schemes

SUMMARY The implementation of the multigrid method into the SIMPLE algorithm presents interesting aspects concerning the mass fluxes conservation on coarser grids, the k ‐o turbulence model and the higher-order discretization schemes. Higher-order discretization schemes for the convection terms are increasingly used in order to guarantee accuracy in demanding engineering applications. However, when used in single-grid algorithms, their convergence is considerably slower compared with the first-order schemes. Unbounded higher-order schemes offer maximum accuracy, but quite often they do not converge due to their oscillatory behaviour. This paper demonstrates the dual function of the multigrid method: reduction of CPU time and stabilization of the iterating procedure, making it possible to perform computations with the third-order accurate QUICK scheme in all cases. The method is applied to the calculation of two- and three-dimensional flows with or without turbulence modelling. The results show that the convergence rate of the present algorithm does not deteriorate when QUICK is used and that, if applied on complex engineering cases, large gains in computational time can be achieved. Copyright © 2001 John Wiley & Sons, Ltd.

Direct Computation of Jet Noise Produced by Large-Scale Axisymmetric Structures

A methodology is presented for directly calculating the noise emission associated with large-scale structures in a supersonic jet. The nonlinear governing equations are solved in a computational domain that encompasses both the jet e ow and the acoustic near e eld. A high-order discretization scheme is used along with careful boundary treatment to capture the disturbance e eld accurately. Nonlinear interactions among the various frequency modes of the e ow structure were found to alter the development of each mode and, hence, ine uence its radiation pattern. Comparingthecalculatedradiationpatterntoexperimentalobservationsindicatesthattheaxisymmetricstructure contributes preferentially in the forward direction, whereas the azimuthal structure is associated with a higher emission angle and with a stronger effect on jet spreading. Sensitivity of the radiated sound e eld to the type of incoming disturbances is studied.

Separation-induced transition to turbulence : Second-moment closure modelling

The paper presents some results of application of a low-Re-number second-moment closure (SMC) to modelling the laminar-to-turbulent transition induced by a separation bubble. The same model, tested earlier in a number of low and high-Re-number flows, was found also to reproduce reasonably well several cases of bypass transition, as well as cyclic sequence of laminarization and turbulence revival in oscillating flows at transitional Re numbers, without any artificial transition triggering. The focus of the paper is on separation-induced transition in flow over a flat plate with a circular leading edge, and on a plane surface on which a laminar separation bubble was generated by imposed suction on the wall-opposite boundary. The results show acceptable agreement with available experimental data, large-eddy and direct numerical simulations (LES, DNS). The importance of applying higher-order discretization schemes for reproducing both the bubble and the transition is also discussed.

The computation of turbulent flow through stationary and rotating U-bends with rib-roughened surfaces

This study focuses on the numerical computation of turbulent flow through stationary and rotating U-ducts of square cross-section, with rib-roughened inner and outer walls along the straight sections. Square-sectioned ribs, normal to the flow direction are employed in a staggered arrangement. The rib height-to-diameter ratio is 0.1 and the rib pitch-to-rib height ratio is 10. The bend curvature ratio (R c /D) is 0.65 and the flow Reynolds number is 100000. The duct rotates about an axis parallel to that of curvature, at a rotation number (R 0 ≡ ± 0.2). The preliminary computations presented suggest that grid requirements can be significantly reduced, through a combination of body fitted grids and high-order convective discretisation schemes in all transport equations. The presence of regions of flow separation, necessitates the use of low Re second-moment closures

A Systematic Computational Methodology Applied to a Three-Dimensional Film-Cooling Flowfield

Numerical results are presented for a three-dimensional discrete-jet in crossflow problem typical of a realistic film-cooling application in gas turbines. Key aspects of the study include: (1) application of a systematic computational methodology that stresses accurate computational model of the physical problem, including simultaneous, fully elliptic solution of the crossflow, film-hole, and plenum regions; high-quality three-dimensional unstructured grid generation techniques, which have yet to be documented for this class of problems; the use of a high-order discretization scheme to reduce numerical errors significantly; and effective turbulence modeling; (2) a three-way comparison of results to both code validation quality experimental data and a previously documented structured grid simulation; and (3) identification of sources of discrepancy between predicted and measured results, as well as recommendations to alleviate these discrepancies. Solutions were obtained with a multiblock, unstructured/adaptive grid, fully explicit, time-marching, Reynolds-averaged Navier–Stokes code with multigrid, local time stepping, and residual smoothing type acceleration techniques. The computational methodology was applied to the validation test case of a row of discrete jets on a flat plate with a streamwise injection angle of 35 deg, and two film-hole length-to-diameter ratios of 3.5 and 1.75. The density ratio for all cases was 2.0, blowing ratio was varied from 0.5 to 2.0, and free-stream turbulence intensity was 2 percent. The results demonstrate that the prescribed computational methodology yields consistently more accurate solutions for this class of problems than previous attempts published in the open literature. Sources of disagreement between measured and computed results have been identified, and recommendations made for future prediction of film-cooling problems.

A Convergence Analysis of an H-Version Finite-Element Method with High-Order Elements for Two-Dimensional Elasto-Plasticity Problems

In this paper, we will give an h-version finite-element method for a two-dimensional nonlinear elasto-plasticity problem. A family of admissible constitutive laws based on the so-called gauge-function method is introduced first, and then a high-order h-version semidiscretization scheme is presented. The existence and uniqueness of the solution for the semidiscrete problem are guaranteed by using some special properties of the constitutive law, and finally we will show that as the maximum element size ${h}\longrightarrow 0$, the solution of the semidiscrete problem will converge to the solution of the continuous problem. The high-order h-version discretization scheme introduced here is unusual. If the partition of the spatial space only has rectangles or parallelograms involved, then there would not be any limit on the element degree. However, if the partition of the spatial space has some triangular elements, then only certain combinations of finite-element spaces for displacement and stress functions can be used. The discretization scheme also provides a useful idea for applications of hp-version or high-order h-version finite-element methods for two-dimensional problems where the elasto-plastic body is not a polygon, such as a disk or an annulus.