MacCormack Scheme Research Articles

Absorbing boundary technique for open channel flows

SUMMARY An absorbing boundary condition is formulated and applied to the one-dimensional open channel flow equations in conjunction with an explicit MacCormack scheme. The physical flow domain has been truncated by introducing an artificial pseudo-boundary. By using an appropriate boundary condition on a truncated domain, it is shown that, for flow containing shocks, the solution can be accelerated to its stationary profile with no loss of accuracy. Copyright © 2000 John Wiley & Sons, Ltd.

Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state

Unsteady flow in collapsible tubes has been widely studied for a number of different physiological applications; the principal motivation for the work of this paper is the study of blood flow in the jugular vein of an upright, long-necked subject (a giraffe). The one-dimensional equations governing gravity- or pressure-driven flow in collapsible tubes have been solved in the past using finite-difference (MacCormack) methods. Such schemes, however, produce numerical artifacts near discontinuities such as elastic jumps. This paper describes a numerical scheme developed to solve the one-dimensional equations using a more accurate upwind finite volume (Godunov) scheme that has been used successfully in gas dynamics and shallow water wave problems. The adapatation of the Godunov method to the present application is non-trivial due to the highly nonlinear nature of the pressure–area relation for collapsible tubes.The code is tested by comparing both unsteady and converged solutions with analytical solutions where available. Further tests include comparison with solutions obtained from MacCormack methods which illustrate the accuracy of the present method.Finally the possibility of roll waves occurring in collapsible tubes is also considered, both as a test case for the scheme and as an interesting phenomenon in its own right, arising out of the similarity of the collapsible tube equations to those governing shallow water flow.

Prediction of cutoff noise in centrifugal fans

The most important noise in centrifugal fans is aerodynamic noise, in which cutoff noise is the most main component. Cutoff noise is mainly caused by the fluctuating pressure on the cutoff in the volute impacted by the nonuniform flow from the exit of the rotating impeller. A numerical method is presented for prediction of cutoff noise in centrifugal fans. The contents of the method are mainly the following. (1) The fluctuating pressure serving as the source of cutoff noise is given out by using the existing computation software of 3D viscous flow in centrifugal fans and the time-frozen hypothesis. (2) The sound field in the volute of a centrifugal fan and the sound power levels of cutoff noise are computed by using the forth-order MacCormack scheme and making some important numerical measures. The sound field in the volute and the sound power levels of cutoff noise for three practical volutes with different cutoff clearance have been computed and the acoustic pressure distribution in space and change with time are discussed. Finally, the sound power levels of cutoff noise have been measured. The errors between the predicted and the tested sound power levels of cutoff noise are less than 3 dB. [Work supported by NSF.]

CHARACTERISICS OF SURGING DISCHARGE CHANNEL WITH BASIN AND A NUMERICAL SIMULATION METHOD

In an intake and discharge channel of an electric power plant or a LNG plant, surging phenomena of water may cause a siphon break or an overflow at pits, and then may give a damage to the facility. A new numerical simulation method to predict surging was developed. In the method, the “slot model” was applied to the pressurized flow, and the MacCormack's scheme is used to simulate the mixed flow condition of sub-critical and supercritical flow. The water level in a discharge basin which is usually set at the end of the discharge channel can be calculated simultaneously. The laboratory tests were carried out to verify the simulation model for three types of channels. Three types of channels were a pressurized channel, an open channel and a channel which had a vertical bend at the end. All the channels have a discharge basin at the end. It is shown that the model can simulate an unsteady surging phenomenon for almost all types of channels.

Numerical Simulation of GM Refrigerator Operating at about 4 K.

This paper describes the numerical simulation of a GM refrigerator operating at about 4 K. The simulation model is composed of a regenerator, an expansion space, and a cooling stage. The volumetric change of the expansion space and the actual thermophysical properties of the working fluid, helium are considered. The basic equations are mode up of one-dimensional fluid equations and an energy equation of the regenerator material and the cooling stage. The fluid equations are expressed in the general coordinate system of which a coordinate axis moves with time to take account of the volumetric change of the expansion space. These basic equations are differentiated by using the TVD MacCormack method. In order to verify the simulation model, the theoretical validity of the calculation results was checked and calculation results were compared with results of experiments. As a result it was confirmed that the simulation model is appropriate.

A Computational Fluid Dynamics and Heat Transfer Model for Gaseous Core and Very High Temperature Gas-Cooled Reactors

A computational approach to the solution of Navier-Stokes equations for the thermal and flow fields of very high temperature gas-cooled and gaseous core reactors is presented. An implicit-explicit, finite volume, MacCormack method, in conjunction with the Gauss-Seidel line iteration procedure, is utilized to solve axisymmetric, thin-layer Navier-Stokes equations. An enthalpy rebalancing scheme is implemented to allow the convergence solutions to be obtained with the application of a wall heat flux. The subsonic and supersonic flows of helium in a very high temperature gas-cooled reactor and uranium tetrafluoride (UF4) in a gaseous core reactor under variable boundary conditions (such as adiabatic, isothermal, and constant heat flux) are calculated. The numerical results are compared with other published results and experimental-based correlations. The good agreement with empirical correlations indicates the usefulness of the presented model for the prediction of the flow and temperature distribution under the convective and radiative heat transfer environment of very high temperature gas-cooled and gaseous core reactors.

Evaluation of a High-Accuracy MacCormack-Type Scheme Using Benchmark Problems

Due to their inherent dissipation and stability, the MacCormack scheme and its variants have been widely used in the computation of unsteady flow and acoustic problems. However, these schemes require many points per wavelength in order to propagate waves with a reasonable amount of accuracy. Recently, much progress has been made in improving the wave propagation qualities of MacCormack-type schemes. In this work, the linear wave propagation characteristics of MacCormack-type schemes are compared using several of the CAA Benchmark Problems.

FINITE-DIFFERENCE SOLUTION OF HYPERBOLIC HEAT CONDUCTION WITH TEMPERATURE-DEPENDENT PROPERTIES

A numerical evaluation of the temperature field in an infinite solid medium that surrounds a cylindrical surface is presented. An unsteady and uniform heat flux density is prescribed at the cylindrical surface, and Cattaneo-Vernotte's constitutive equation for the heat flux density is supposed to hold. The hyperbolic differential problem is solved by MacCormack's predictor-corrector method by assuming that both the thermal conductivity and the specific heat are temperature-dependent. Then, the results of the numerical evaluation are compared with the analytical solution that is available in the literature for the special case of constant thermophysical properties.

Flow around a Cylinder: Shallow-Water Modeling with Diffusion-Dispersion

The equations for two-dimensional (2D), horizontal flow are obtained by vertical depth-integration of the continuity and momentum equations. Diffusion-dispersion terms appear, containing turbulent and dispersion stresses. Turbulent stresses are expressed with the eddy-viscosity concept; dispersion stresses are evaluated using velocity distributions along and across the curved streamlines. The differential equations are solved using the MacCormack scheme. The numerical simulation was done for three runs; the expected alteration of the flow field around the cylinder is evident, notably the wake behind the cylinder. For comparison, two similar runs were performed in a laboratory channel. The velocity vectors upstream from the cylinder and along its sides are in reasonably good agreement. However, very close to the cylinder, the simulation sometimes underpredicts the velocities. Downstream from the cylinder, agreement is satisfactory both inside and outside the wake. The flow depths at the centerline both upstream and downstream from the cylinder are also in good agreement; however, along the cylinder circumference, the simulated flow depths are lower than the observed ones.

Simulation of border irrigation system using explicit MacCormack finite difference method

The hydrodynamic, zero-inertia and kinematic wave models for simulating all phases of the border irrigation event are presented. The explicit second-order accurate MacCormack method is used for solving the governing equations. A technique for implementing the boundary conditions that are consistent with the numerical scheme is discussed. All the three models do not require a special treatment such as moving grid, deformable grid, subgrid technique, etc., to accurately simulate the advancing and receding fronts. The results of all the three models namely hydrodynamic finite difference model (HDFD), zero-inertia finite difference model (ZIFD) and kinematic wave finite difference model (KWFD) are compared with observed advance and recession times for four border irrigation events available in literature. A very good comparison of results is observed. The relative merits and demerits of the models are discussed. The HDFD model is found to be more suitable for simulation of all the four phases of border irrigation events.

A NUMERICAL SIMULATION OF HYDRAULIC JUMP SHAPES ON WIDENING STEEP SLOPE CHANNELS

The MacConnack scheme has some advantages in calculations of flow in steep slope channels withhydraulic jumps. However, in point of calculation stability and so foirth, the MacCormack scheme has not become a method which is fully generalized.We made experiments on hydraulic jump shapes on widening steep slope channels and simulated calculations on experimental results. The comparison between the experimental results and calculated ones indicates the validity of simulation method.

DEM Based Overland Flow Routing Model

A physically based, distributed rainfall-runoff event model is developed to route overland flows from flat agricultural watersheds. The model works on a cell basis and routes overland flows from one cell to the next following the maximum downslope directions. The model is able to consider spatially-varied data of soils, crops, land slopes, and aspects, which can be extracted from geographic information systems (GIS) and from digital elevation models (DEMs). Because of this feature, the model can be used for evaluating the impacts of agricultural practices on surface runoff. To describe overland flows on flat watersheds, the model uses the diffusion wave approximation of the St. Venant equations for computing the hydrograph. The computation is accomplished using the MacCormack scheme, a second order accurate numerical method. The model was tested against analytic solutions of the kinematic wave equations and was applied to route the overland flows across Goodwater Creek, a USDA research watershed. The model was calibrated using 26 events and verified using 11 events. The results show that the model works well.

APPLICABILITY OF 2-D NUMERICAL MODEL BASED ON MACCORMACK SCHEME TO SHALLOW WATER FLOW

A 2-D numerical model for unsteady free surface flows based on MacCormack Scheme, that incorporats 4-step algorism and Jameson's artificial viscosity, is developed in the orthogonal as well as generalized curvilinear coordinate system. The model is applied to the flows around groin (s) in which super-as well as sub-critical flows coexist and to the flow in a meandering channel. It is found from comparisons between experimental and numerical results that the model performs well when three dimensionality of a flow is moderate.

PRACTICAL HYDRAULIC SIMULATION MODEL FOR UNDERGROUND DRAINAGE SYSTEM

A practical numerical simulation model is presented for the underground drainage network system with a large scale drainage pumping station. In this model, the hypothetical slot at the crown of the pipe is installed to analyze the pressurized flow, and the MacCormack scheme for numerical solution technique is applied to simulate the mixed flow condition of sub-critical and super-critical flow. The model is verified by comparison with monitoring water levels of laboratory tests and exiting drainage network system with pump station. From these calculation results, it is shown that the simulation model is very useful to simulate unsteady free surface-pressurized flows and surge phenomenon of the pipe network system including the steep slope pipe

Numerical Simulation of GM Refrigerator Operating at about 4K.

This paper describes the numerical simulation of a GM refrigerator operating at about 4K. The simulation model is composed of a regenerator, an expansion space, and a cooling stage. The volumetric change of the expansion space and the actual thermophysical properties of the working fluid, helium are considered. The basic equations are made up of one-dimensional fluid equations and an energy equation of the regenerator material and the cooling stage. The fluid equations are expressed in the general coordinate system of which a coordinate axis moves with time to take account of the volumetric change of the expansion space. These basic equations are differentiated by using the TVD MacCormack method. In order to verify the simulation model, the theoretical validity of the calculation results was checked and calculation results were compared with results of experiments. As a result it was confirmed that the simulation model is appropriate.

Analytical and numerical results on the attenuation and dispersion of an acoustic wave through a two‐phase flow

Presents validation of the Eulerian approach for unsteady two‐phase flows, whose behaviour depends on the coupling between the two phases, on the basis of the study of attentuation and dispersion of an acoustic wave propagating into a one dimensional two‐phase flow. This approach and the corresponding numerical aspects are accurate enough for later applications in more complex geometries, where “vortex shedding” phenomena take place. Attenuation and dispersion of a pressure wave in a two‐phase medium of rest was previously studied by Temkin and Dobbins. Present work is an extension of this theory to the case of a two‐phase flow. This theoretical approach leads to a numerical solution of the problem. Compares the derived results with those obtained from a direct numerical simulation based on MacCormack scheme in a finite volume formulation. Verifies that analytical and numerical approaches are in good agreement.

Analytic evaluation of finite difference methods for compressible direct and large eddy simulations

We investigate the performance of several different numerical methods using the von Neumann method for the compressible Euler Equations. The results are used to analytically estimate the error due to the numerical schemes with an approximate spectrum of the turbulence obtained from isotropic homogeneous turbulence theory. The final spectrum is calculated from the linearized von Neumann analysis and a characteristic time scale of turbulence. In this study, we compare the MacCormack scheme (second and fourth order), the original Flux-Corrected-Transport (FCT) scheme, a second-order FCT scheme, a fourth-order Runge-Kutta, centered-space scheme, a second order implicit upwind scheme, a high order implicit upwind method and a recent sixth-order Padé method. Most methods have been used in prior studies involving large eddy or direct numerical simulations. The results show significant differences in the stability and resolution characteristics of the various methods. In conjunction with conventional tests for compressible flow methods (such as performance of the Riemann problem), the present approach may provide improved a priori evaluation of proposed methods for LES and DNS investigations.

The simulation of gas dynamics in engine manifolds using non-linear symmetric difference schemes

Symmetric difference schemes, such as the two-step Lax—Wendroff method or MacCormack's method, are used extensively in engine simulation codes in order to achieve second-order accuracy. These schemes, however, produce spurious oscillations when shock waves and contact surfaces are encountered and require the addition of non-linear terms (flux limiters) to cope with such phenomena. The paper reviews briefly the general theory behind the formulation of such nonlinear algorithms and examines the performance of two schemes in particular. The flux corrected transport (FCT) technique has been adopted by several research groups for this purpose and appears to operate satisfactorily for many flows in engine manifolds. It is shown in the present work that, under some circumstances, this method can distort the solution and produce large errors in mass conservation. The symmetric flux limiter due to Davis, based on the total variation diminishing (TVD) criterion, is shown here to avoid such problems and requires similar computational effort to the FCT approach.

Simulation of ultrasonic pulse propagation through the abdominal wall

Ultrasonic pulse propagation through the abdominal wall has been simulated using a model for two-dimensional propagation through anatomically realistic tissue cross sections. The time-domain equations for wave propagation in a medium of variable sound speed and density were discretized to obtain a set of coupled finite-difference equations. These difference equations were solved numerically using a two-step MacCormack scheme that is fourth-order accurate in space and second-order accurate in time. The inhomogeneous tissue of the abdominal wall was represented by two-dimensional matrices of sound speed and density values. These values were determined by processing scanned images of abdominal wall cross sections stained to identify connective tissue, muscle, and fat, each of which was assumed to have a constant sound speed and density. The computational configuration was chosen to simulate that of wavefront distortion measurements performed on the same specimens. Qualitative agreement was found between those measurements and the results of the present computations, indicating that the computational model correctly depicts the salient characteristics of ultrasonic wavefront distortion in vivo. However, quantitative agreement was limited by the two-dimensionality of the computation and the absence of detailed tissue microstructure. Calculations performed using an asymptotic straight-ray approximation showed good agreement with time-shift aberrations predicted by the full-wave method, but did not explain the amplitude fluctuations and waveform distortion found in the experiments and the full-wave calculations. Visualization of computed wave propagation within tissue cross sections suggests that amplitude fluctuations and waveform distortion observed in ultrasonic propagation through the abdominal wall are associated with scattering from internal inhomogeneities such as septa within the subcutaneous fat. These observations, as well as statistical analysis of computed and observed amplitude fluctuations, suggest that weak fluctuation models do not fully describe ultrasonic wavefront distortion caused by the abdominal wall.

Numerical simulation of vortex-shedding phenomenon in a channel with flow induced through porous wall

Abstract Segmented solid propellant rocket motors tend to develop unpredicted pressure and thrust oscillations that could be attributed to a periodic phenomenon. An experimental and numerical assessment of the stability of a solid propellant motor was conducted. This numerical computation was based on an experimental study carried out on a cold-flow, reduced-scale, bidimensional duct with a complex internal geometry (obstacles, outflowing cavities, submerged nozzle). The objectives of our study are to characterize the internal flow (mean, fluctuating induced by injection through a porous wall and the shear layer created at the top of the second obstacle. The code employed solves the unsteady, compressible two-dimensional (2-D) Navier-Stokes equations in a laminar regime by a predictor-corrector MacCormack scheme. Three grids are developed for the evaluation while respecting the complexity of the internal geometry. The results obtained allow us to separate the flow into two zones: one, laminar, upstream of the obstacle; the other, disturbed, downstrea, in which vortex structures develop. The phenomena of instabilities (vortex-shedding, pairing) are in accordance with the first two acoustic longitudinal modes of the chamber. Last, the numerical computations are discussed and systematically compared with the experimental results.