MacCormack Scheme Research Articles

The MacCormack difference scheme and radiation transport

The MacCormack scheme and its variants proposed by Kassar and Ali (Comput. Methods Appl. Mech. Engrg. 106 (1993) 395) are applied to the radiative transfer problem studied by Stone and Mihalas (J. Comput. Phys. 100 (1992) 402). It is shown that the usual form of MacCormack scheme gives phase advanced results in the case of an optically thick medium and that the wave speed deteriorates further as the time step is decreased. On the other hand, the proposed variants, although requiring shorter time steps, give increased wave speed accuracy. Three hybrid variants are proposed of which the first two give more accurate wave speed for the time step size used by Stone and Mihalas. The results presented for the error prone optically thick medium, are excellent when flux corrected transport is applied to enhance the resolution of the radiation front.

A multiphase formulation for two phase flows

This paper investigates the multi‐phase behaviour of droplets injected into a nozzle at two separate wall locations. The physical features of the droplets (rate of mass, density and radius) at each injector location are identical. This system can be described by a two‐phase Eulerian—Eulerian approach that yields classical systems of equations: three for the gaseous phase and three for the dispersed droplet phase. An underlying assumption in the two phase model is that no interaction occurs between droplets. The numerical solution of the model (using the MacCormack scheme) indicates however that the opposite jets do interact to form one jet. This inconsistency is overcome in the current paper by associating the droplets from a given injection location with a separate phase and subsequently solving equations describing a multiphase system (here, three‐phase system). Comparison of numerical predications between the two‐phase and the multiphase model shows significantly different results. In particular the multiphase model shows no jet interaction.

NONLINEAR DYNAMICS OF HEAT TRANSFER ENHANCEMENT USING EDDY PROMOTERS

ABSTRACT This investigation concerns the nonlinear dynamics of heat transfer from a surface using an upstream eddy promoter, A numerical model is developed for the coupled fluid flow and heat transfer process based on a modified MacCormack scheme. Numerical simulations are carried out to determine the response and heat transfer enhancement due to the promoter. The average heal transfer from a cavity floor is seen to be increased by a factor of approximately five over the “unpromoted” flow. Another interesting feature of the study is the nonlinear viscous flow dynamics from the cylinder-wall interaction which differ significantly from the familiar cylinder-free stream patterns.

A comparison of numerical methods applied to non‐linear adsorption columns

AbstractEight numerical schemes (first‐order upstream finite difference, MacCormack, explicit Taylor–Galerkin, random choice, flux‐corrected transport, ENO, TVD, and Euler–Lagrange methods) are compared on the basis of their computational efficiency for one‐dimensional non‐linear convection–diffusion problems. For the ideal chromatographic equation for which an exact solution exists, errors plotted against computational times show that the best methods are the random choice, Euler–Lagrange and flux‐corrected MacCormack methods. Even when significant diffusion is added to the model, steep gradients are possible because of non‐linearities. In such an instance, the random choice and flux‐corrected transport methods give the best performance. One can now tackle more complicated problems and refer to this comparative study in order to choose an adequate numerical method which will provide sufficiently accurate results at a reasonable cost.

One-Dimensional Modeling and Measurement of Pulsating Gas-Solid Flow in Tubes

Abstract Pulsating turbulent gas-particle flow in a circular tube is investigated by both experimental and numerical methods. In experiments, the ensemble averaged centerline gas and particle velocities ate measured by using laser Doppler anemometry. The amplitude and the frequency of pulsations are controlled via the diameter and the RPM of a rotating butterfly valve. It was found that significant variations could be obtained along the axial position in the amplitude and the phase of the fluid velocity deviation from its mean. Both the amplitude and the phase shift behavior was a function of the imposed pulsating frequency and amplitude. Particle velocity measurements showed that the slip velocities between the fluid and particles are dependent on frequency and position along the axial direction. The experiments are simulated using a one-dimensional transient model which consists of one-dimensional compressible flow equations in an Eulerian, and particle momentum equation in a Lagrangian frame of reference. The explicit MacCormack method is used for the gas phase and Gear's method is used for the particle phase equations. An assessment of the one-dimensional, transient model is presented. It is shown that the experimentally observed dependence of both the fluid and the particle velocities on the frequency of forced oscillations as well as the significant variations with the axial distance can be simulated well by this simple model. The model can predict the instantaneous slip velocity between the fluid and solid particles in good agreement with measurements. This information is essential for combustion analysis of pulsed combustors utilizing pulverized coal.

Computation of wind flow around a tall building and the large-scale vortex structure

A numerical study of three-dimensional wind flow around a tall building is presented in this paper. The solution is obtained by solving weakly compressible flow equations, along with Smagorinsky's subgrid-scale turbulent model. The numerical scheme is based on MacCormack's predictor-corrector explicit finite volume method. First, the numerical model was verified by testing a shear flow around surface-mounted cube, then a detailed study was carried out for a shear flow around a taller building model (width:length:height = 1:0.889:4.667). The main task of this paper is to explore the large-scale vortex structure and the unsteady behavior of flow around a tall building, which are still not well understood.

Computation of wind flow around a tall building and the large-scale vortex structure

A numerical study of three-dimensional wind flow around a tall building is presented in this paper. The solution is obtained by solving weakly compressible flow equations, along with Smagorinsky's subgrid-scale turbulent model. The numerical scheme is based on MacCormack's predictor-corrector explicit finite volume method. First, the numerical model was verified by testing a shear flow around surface-mounted cube, then a detailed study was carried out for a shear flow around a taller building model (width:length:height = 1:0.889:4.667). The main task of this paper is to explore the large-scale vortex structure and the unsteady behavior of flow around a tall building, which are still not well understood.

The stiff conservation laws and the MacCormack difference scheme

In the modelling of nonequilibrium gas dynamics, such as detonation and radiation hydrodynamics, the system of conservation laws governing the inviscid flow contains source terms representing chemistry. Numerical computations of such equations with dominant source terms, using the usual form of MacCormack predictor corrector method, exhibit errors in wave speed. Leveque and Yee (J. Comput. Phys. 86 (1990) 187) have used a model scalar equation to illustrate the severe error in the wave speed in computations using the MacCormack scheme. Here, modifications in the MacCormack scheme are proposed that remove these errors. Fourier analysis, for a linear model equation, and numerically computed results of the model problem of Leveque and Yee are presented to show that the modified schemes gives the correct wave speed. Comparison with the MacCormack scheme is made regarding stability and phase error. Our proposed schemes give correct wave speed irrespective of the source strength; however, the time step required for stable results decreases inversely with the strength of the source term.

Comparison of FCT with other numerical schemes for the burgers equation

The flux corrected transport (FCT) method is used to solve the linear and nonlinear Burgers Equations and to compare with other commonly used explicit methods such as the central difference (CD), the upwind (UW), and the predictor corrector or MacCormack (MC) schemes. The results show that the FCT scheme exhibits the best behavior for the linear as well as nonlinear cases. The amount of numerical diffusion is the least for the FCT scheme compared to the other schemes and more importantly there are no non-physical oscillations.

DEVELOPMENT OF HIGH-ACCURACY CONVECTION SCHEMES FOR SEQUENTIAL SOLVERS

The applicability of high-resolution schemes, notably TVD schemes, to resolve sharp flow gradients using a sequential solution approach, commonly found in pressure-based algorithms, is explored. These high-resolution schemes have been widely applied to systems of equations such as Euler equations and Navier-Stokes equations solved in a simultaneous manner, by assigning the eigenvalues of the Jacobian matrix of the system as the characteristic speeds of signal propagation. It is demonstrated that by extending these high resolution shock capturing schemes to a sequential solver, which treats the system of equations as a collection of scalar conservation equations, the speed of signal propagation in the solution has to be coordinated by assigning the local convection speed as the characteristic speed for the whole system. Consequently, a higher amount of dissipation (compared to the case when the system is solved in a simultaneous manner) is required to eliminate the oscillations near the discontinuities. Special treatment of the pressure terms, which are now treated as source terms in the sequential solver, such as an extension of MacCormack's predictor-corrector method and an operator splitting technique such as Strong's time-splitting method, leads to an improvement of accuracy in the solution pro files.

Computations and experiments for a multiple normal shock/boundary-layer interaction

Results from a numerical investigation of a Mach 1.61 multiple normal shock wave/turbulent boundary-layer interaction are compared to wall static pressure and laser Doppler velocimeter measurements. The computations used the explicit, time-dependent, second-order accurate MacCormack scheme to solve the mass-averaged Navier-Stokes equations. Turbulence was modeled by means of the Baldwin-Lomax algebraic model and the Wilcox-Rubesin two-equation model. The computation with the Wilcox-Rubesin model was able to capture the major features of the normal shock train and accurately predicted the flow reacceleration mechanisms which occur between shocks. However, this computation failed to accurately predict the level of flow separation under the first shock. The Baldwin-Lomax computation displayed a more limited ability to capture the features of this shock train flow.

Comparison of Leapfrog, Smolarkiewicz, and MacCormack Schemes Applied to Nonlinear Equations

The MacCormack scheme is a finite-difference scheme widely used in the aerospace simulations. It is a two-step algorithm, and contains a small amount of implicit numerical diffusion that makes it numerically stable without having to use any explicit filtering. It uses a nonstaggered grid. A detailed comparison with the leapfrog and Smolarkiewicz schemes is presented using the nonlinear advection equation and the Euler equations for a variety of conditions at different Courant numbers. Of the schemes tested, the unfiltered leapfrog is the least acceptable for the solution of nonlinear equations. Although it is numerically stable for linear problems, when used to solve nonlinear equations (without using any explicit filtering) it becomes numerically unstable or nonlinearly unstable. Furthermore, it introduces large phase errors, and produces better results with small Courant numbers. The MacCormack scheme is nonlinearly stable, produces modest amounts of numerical dispersion and diffusion, has no phase speed error, and works best at Courant numbers close to 1. When applied to nonlinear equations, the Smolarkiewicz scheme exhibits the least amount of numerical diffusion but more numerical dispersion than the MacCormack scheme. For stability it requires Courant numbers equal to or smaller than 0.5. In practical applications, we recommend the MacCormack scheme for the solution of the nonlinear equations, and either the Smolarkiewicz or the MacCormack scheme for equations involving conservation of a passive scalar.

A MacCormack scheme for incompressible flow

MacCormack's explicit predictor-corrector scheme is extended for incompressible flow on marker-and-cell grids. Cell-centered velocities are first constructed by taking a normal velocity component from one of two possible cell faces for each coordinate direction. The predictor phase then proceeds in the usual manner employed for cell-centered grids. Central differencing is used for diffusion terms, and one-sided differencing (in the same direction as the face-to-center velocity transfer) is used for advective terms. At the end of the predictor phase, cell-centered velocity increments are transferred back to the cell faces from which the normal velocity components were taken, and the pressure is adjusted to enforce conservation of mass. The corrector phase follows the same pattern, but the directions for the face-to-center velocity transfer and the one-sided differencing are the reverse of those in the predictor. The extended MacCormack method requires only one control volume to be defined for each grid cell. This affords the convenience of a cell-centered scheme with the tight mass conservation of a staggered grid. Computed results are presented for a straight channel, a driven cavity, a backstep, and a circular cylinder.

Navier-Stokes study of supersonic cavity flowfield with passive control

A computational investigation of the supersonic turbulent flow past a two-dimensiona l rectangular cavity with passive venting is described. The effect of passive venting was included through the use of a porous surface over a vent chamber in the floor of the cavity. The passive venting was numerically simulated by the use of a linear form of the Darcy pressure-velocity law. The time-accurate solutions of the two-dimensional, Reynoldsaveraged, Navier-Stokes equations were generated using the explicit MacCormack scheme. The capability Of the numerical scheme is first demonstrated by the computations of an open and closed cavity without passive venting. The results of these computations also provide a reference case for the passive venting computations. The effect of passive venting on the closed cavity is then demonstrated and analyzed. These results show that the passive venting dramatically changes the closed cavity flow to nearly an open cavity flow. The free shear layer formed between the high-speed outer flow and slower inner flow is seen to bridge the cavity completely j resulting in an open cavity flow. The passive venting velocities are determined to be less than 5% of the freestream velocities, and largely confined to the upstream and downstream portions of the cavity floor. The computational results show good agreement with available experimental data.

Elastic wave propagation using fully vectorized high order finite‐difference algorithms

Two finite‐difference schemes for solving the elastic wave equation in heterogeneous two‐dimensional media are implemented on a vector computer. A modified Lax‐Wendroff scheme that is second‐order accurate both in time and space and is a version of the MacCormack scheme that is second‐order accurate in time and fourth‐order in space. The algorithms are based on the matrix times vector by diagonals technique that is fully vectorized and is described using a novel notation for vector supercomputer operations. The technique described can be implemented on a vector processor of modest dimensions and increase the applicability of finite differences. The two difference operators are compared and the programs are tested for a simple case of standing sinusoidal waves for which the exact solution is known and also for a two‐layer model with a line source. A comparison of the results for an actual well‐to‐well experiment verifies the usefulness of the two‐dimensional approach in modeling the results.

Navier-Stokes Calculations of Transonic Flows Past Cavities

Presented in this paper is a computational investigation of subsonic and transonic flows past three-dimensional deep and transitional cavities. Simulations of these self-induced oscillatory flows have been generated through time-accurate solutions of the Reynolds averaged, full Navier-Stokes equations, using the explicit MacCormack scheme. The Reynolds stresses have been included through the Baldwin-Lomax algebraic turbulence model with certain modifications. The computational results include instantaneous and time averaged flow properties. The results of an experimental investigation have been used not only to validate the time-averaged results, but also to investigate the effects of varying the Mach number and the incoming boundary-layer thickness. Time series analyses have been performed for the instantaneous pressure values on the cavity floor and compared with the results obtained by a predictive formula. While most of the comparisons have been favorable, some discrepancies have been observed, particularly on the rear face. The present results help understanding the three-dimensional and unsteady features of the separations, vortices, the shear layer, as well as some of the aeroacoustic phenomena of compressible cavity flows.

Reflection of shock wave from a compression corner in a particle-laden gas region

We investigated in this paper the progression of a shock-wave reflected from a compression corner in a particle-laden gas medium using a TVD class numerical technique and a MacCormack scheme. For a gas-only flow, the numerical results agreed well with the existing experimental data, suggesting that the gas phase is correctively solved. The effect of particle size and mass fraction ratio is investigated for a dilute gas-particle flow. It has been shown that the shock-wave diffraction and the flow configuration after the shock can become remarkably different from the gas-only flow depending on the particle parameters. Relaxation phenomenon due to the momentum drag and the heat exchange between the gas and the particle phases is explained.

The stability of MacCormack's method for the scalar advection equation

We show that MacCormack's method for the scalar advection equation u t = au x + bu y is stable if (aΔt/Δx) 2 + (bΔt/Δy) 2 ≤ 2 1+ 1 8 −2≈ 1 8 . This bound on the mesh ratios is not optimal, since numerical sampling shows that .3266 will do when ( aΔ t/Δ x) 2 = ( bΔ t/Δ y) 2.

Hyperbolic heat conduction with convection boundary conditions and pulse heating effects

This paper describes the numerical simulation of hyperbolic heat conduction with convection boundary conditions. The effect of a step heat loading, a sudden pulse heat loading, and a pulse internal heat source are considered in conjunction with convection boundary conditions. Two methods of solution are presented for predicting the transient behavior of the propagating thermal disturbances. In the first method, MacCormack's predictor-corrector method is employed for integrating the hyperbolic system of equations. Next the transfinite element method, which employs specially tailored elements, is used for accurately representing the transient response of the propagating thermal wavefronts. The agreement between the results of various numerical test cases not only validates the representative behavior of the thermal wave fronts but also provides an understanding of the representative behavior due to convection boundary conditions and varied heating effects.

HYPERBOLIC STEFAN PROBLEM WITH APPLIED SURFACE HEAT FLUX AND TEMPERATURE-DEPENDENT THERMAL CONDUCTIVITY

Abstract The hyperbolic Stefan problem with an applied surface heat flux and temperature-dependent thermal conductivity is solved numerically for a semi-infinite slab using Mac-Cormack's predictor-corrector method. Solutions are presented for cases where the melt temperature is both below and above the instantaneous jump in surface temperature at time t = O+. The interface condition, surface temperature, and internal temperatures are presented for different Stefan numbers and melt temperatures, as well as thermal conductivity both increasing and decreasing with temperature. The results obtained from the hyperbolic solution are compared with those obtained from the parabolic solution.