MacCormack Scheme Research Articles

Analysis of metal extrusion by the Finite Volume Method

Abstract Present work examines and validates the novel numerical scheme to calculate the velocity, stress, strain, pressure and strain rate fields of metal plastic flow in direct extrusion processes by employing the finite volume method, FVM. Traditionally, the classical methods such as the upper-bound, slab, slip-line and more recently the Finite Element Method have been largely applied in metal extrusion analysis. However, recently the FVM has been applied and published by the authors for analysis of metal plastic flow, concluding that direct extrusion of metals could be mathematically modelled by the plastic flow formulation similar to an incompressible non-linear viscous fluid. Tannehill et al. suggested that viscous fluid flow can be numerically simulated by FVM, obeying the mass, momentum and energy conservation equations and boundary conditions. Hence, the governing equations of metal plastic flow in Euler approach were discretized by FVM, using the Explicit MacCormack Method in structured, fixed and collocated mesh. SIMPLE method was applied to attain the necessary pressure-velocity coupling. These new numerical scheme was applied to the analysis of direct hot extrusion process of Al 6351 and Al 6060 aluminium alloys. The velocity and other variables fields achieved fast convergence and a good agreement with experimental results from visioplasticity tests by the grid stripe pattern technique and Forge 2008 software. The MacCormack Method applied to metal extrusion revealed consistent results without the need of artificial viscosity as required by the compressible fluid flow simulation approaches. Therefore, present numerical results confirm that FVM with MacCormack method together with Euler formulation approach and SIMPLE method can be applied satisfactory in the solution of metal forming processes.

One Dimensional Numerical Simulation of Aggradation-Degradation in a Channel Using Finite Difference Method Case Study Chashma Right Bank Canal (CRBC)

A one-dimensional MacCormack explicit finite difference model is developed for simulating hydraulics and bed changes in irrigation channels. The Saint-Venant equations describing unsteady flow in open channels and the continuity equation for the conservation of sediment mass are numerically solved. These equations are highly nonlinear and therefore do not have analytical solutions. For this purpose the MacCormack scheme is used. The scheme is second order accurate; it is a coupled solution as it is a two-step predictor corrector method. Model gives results in terms of bed level changes, flow depth and discharge provided physical boundaries of the system are valid for simulation time. Model execution and accuracy is very sensitive to time step and stability. The simulated results show a good agreement with previous studies in the downstream section, and can predict an average value of measured profile in the upstream section. The application of this model to Chashma Right Bank Canal in Pakistan and results of the model are compared with the published results gave very convincing result.

Numerical investigation of hydrogen detonation suppression with inert particle in pipelines

To numerically study the propagation process of detonation of H2O2N2 mixture and explosion suppression process with inert particle explosion, this paper developed theoretical models for detonation of one-dimensional combustible system and explosion suppression with inert particle based on two-fluid model by considering elementary chemical reactions, shock wave as well as momentum and energy transport effects between gas and particle phases. Meanwhile, Riemann invariant was derived on the basis of the one-dimensional problem. Gas and particle phase equations were solved using TVD and MacCormack schemes, respectively, Stiffness of equations was addressed by time-splitting scheme. Runge-Kutta method was employed to solve chemical reaction. Based on these equations, numerical method was constructed to analyze detonation of H2O2N2 mixture and explosion suppression process. The results showed that under certain initial conditions, high-temperature fireball induced shock wave formed and subsequently developed into detonation. After loading suitable concentration of inert particle into a certain explosion suppression area, combustion flame decelerated due to the transfer of momentum and energy between gas and particle phases, and distance between shock wave and flame increased gradually to lose energy support. Meanwhile, inert particles also absorbed shock wave energy. The two effects proceeded simultaneously and mutually fed back to rapidly attenuate shock waves, thereby successfully suppressing explosion. The inert particle density, concentration and diameter were critical important to explosion suppressant. Besides the product, reactant and some mainly radical components affected the detonation velocity. Especially, OH radicals were present almost only in the combustion reaction zone, indicating their significant impact on the chemical reaction rates.

High-order compact MacCormack scheme for two-dimensional compressible and non-hydrostatic equations of the atmosphere

This study is devoted to application of the fourth-order compact MacCormack scheme to spatial differencing of the conservative form of two-dimensional and non-hydrostatic equation of a dry atmosphere. To advance the solution in time a four-stage Runge–Kutta method is used. To perform the simulations, two test cases including evolution of a warm bubble and a cold bubble in a neutral atmosphere with open and rigid boundaries are employed. In addition, the second-order MacCormack and the standard fourth-order compact MacCormack schemes are used to perform the simulations. Qualitative and quantitative assessment of the numerical results for different test cases exhibit the superiority of the fourth-order compact MacCormack scheme on the second-order method.

Efficient two-phase mass-conserving level set method for simulation of incompressible turbulent free surface flows with large density ratio

An efficient three-dimensional two-phase flow model for simulation of free surface flows is developed. The method is based in the solution of Navier-Stokes equation for quasi-incompressible flows. The equations in generalized coordinates are solved by extension of the fully explicit MacCormack scheme, second order in time and fourth order in space. The free surface is implicitly captured by the zero Level Set of a smooth function and by the Ghost Fluid Method to capture accurately shape discontinuities for properties in the vicinity of the interface. Finally, the Volume of Fluid method is used to ensure mass conservation. Turbulence is described by large eddy simulation, where only the large-scale eddies are solved, while the small scales are modeled by using the selective structure function subgrid-scale model. Boundary shapes are represented through the immersed boundary method on the Cartesian grid. The numerical model is validated by some free surface problems. Numerical predictions by all cases are in good agreement with experimental data.

Viriato: A Fourier–Hermite spectral code for strongly magnetized fluid–kinetic plasma dynamics

We report on the algorithms and numerical methods used in Viriato, a novel fluid–kinetic code that solves two distinct sets of equations: (i) the Kinetic Reduced Electron Heating Model (KREHM) equations (Zocco and Schekochihin, 2011) (which reduce to the standard Reduced-MHD equations in the appropriate limit) and (ii) the kinetic reduced MHD (KRMHD) equations (Schekochihin et al., 2009). Two main applications of these equations are magnetized (Alfvénic) plasma turbulence and magnetic reconnection. Viriato uses operator splitting (Strang or Godunov) to separate the dynamics parallel and perpendicular to the ambient magnetic field (assumed strong). Along the magnetic field, Viriato allows for either a second-order accurate MacCormack method or, for higher accuracy, a spectral-like scheme composed of the combination of a total variation diminishing (TVD) third order Runge–Kutta method for the time derivative with a 7th order upwind scheme for the fluxes. Perpendicular to the field Viriato is pseudo-spectral, and the time integration is performed by means of an iterative predictor–corrector scheme. In addition, a distinctive feature of Viriato is its spectral representation of the parallel velocity-space dependence, achieved by means of a Hermite representation of the perturbed distribution function. A series of linear and nonlinear benchmarks and tests are presented, including a detailed analysis of 2D and 3D Orszag–Tang-type decaying turbulence, both in fluid and kinetic regimes.

Propagation and damping of slow MHD waves in a flowing viscous coronal plasma

We investigate the propagation of slow MHD waves in a flowing viscous solar coronal plasma. The compressive viscosity and steady flow along and opposite to the wave propagation are taken into account to study the damping of slow waves. We numerically solve the MHD equations by MacCormack method to examine the effect of steady flow on the damping of slow MHD waves in viscous solar coronal plasma. Amplitude of velocity perturbation and damping time of slow waves decrease with the increase in the value of Mach number. Flow causes a phase shift in the perturbed velocity amplitude and an increase in wave period. The damping of slow waves in flowing viscous plasma is stronger than the damping of waves in viscous plasma. Slow wave in backward flow damps earlier than the wave in forward flow.

Numerical Simulation of Compressible Two-Phase Condensing Flows

In the present paper, the hybrid AUSM-van Leer scheme is extended to solve the governing equations of twophase condensing flows. The method of moments with the classical homogeneous nucleation theory is used to model the non-equilibrium condensation phenomenon. Firstly, the hybrid method is validated using two test cases (i.e. Laval nozzle and rotor-tip cascade) and the results are compared with the MacCormack method. Then the hybrid method is used to solve two other problems (i.e. wavy channel and VKI stage). Based on the numerical results of the paper, the hybrid AUSM-van Leer scheme is an accurate method to simulate twophase transonic flows with nucleation. If the super cooling degree reaches to its maximum value, the nonequilibrium condensation begins and wetness fraction increases suddenly. Also across a shock the wetness fraction decreases due to evaporation of the droplets.

Analytical and numerical solutions of the Local Inertial Equations

Neglecting the convective terms in the Saint-Venant Equations (SVE) in flood hydrodynamic modelling can be done without a loss in accuracy of the simulation results. In this case the Local Inertial Equations (LInE) are obtained. Herein we present two analytical solutions for the Local Inertial Equations. The first is the classical instantaneous Dam-Break Problem and the second a steady state solution over a bump. These solutions are compared with two numerical schemes, namely the first order Roe scheme and the second order MacCormack scheme. Comparison between analytical and numerical results shows that the numerical schemes and the analytical solution converge to a unique solution. Furthermore, by neglecting the convective terms the original numerical schemes remain stable without the need for adding entropy correction, artificial viscosity or special initial conditions, as in the case of the full SVE.

Comparison of Numerical Schemes for LWR Model under Heterogeneous Traffic Conditions

First order macroscopic model like Lighthill-Whitham-Richards (LWR) has been extensively studied and applied for various homogeneous traffic problems. Applicability and adaptability of LWR models to various heterogeneous traffic conditions is still under exploration. Finding solutions for the macroscopic models using analytical methods is a complicated task, numerical approximations are used. The present study attempts to understand the suitability of different numerical schemes for a traffic conditions in detail. Various first order and second order numerical schemes were chosen for numerical integration. Derivation of the numerical scheme, several important issues like accuracy, stability and convergence of each scheme were discussed. Simulated variables like flow, density and speeds were compared with the original data collected from the two different urban arterials with and without bottlenecks in Delhi, India. The comparison of the results of various numerical schemes shows that Lax-Friedrichs and MacCormack schemes produced better results and more stable than the other schemes.

On numerical modeling of the contaminant transport equations of the wetland hydrology and water quality model WETSAND

The reliability of the MacCormack finite difference method for solving the contaminant transport equations of wetland model WETSAND is investigated. WETSAND solves the coupled advection-dispersion-reaction equations for the nitrogen cycle, total nitrogen and total phosphorus concentrations by using the implicit finite difference method. In addition to the implicit scheme, the MacCormack algorithm is implemented within WETSAND. Then, the results obtained by using the MacCormack algorithm are compared with the results obtained by using the implicit finite difference method for both synthetic examples and real data which is collected at the restored wetland site of Duke University at Sandy Creek watershed. Results show that the numerical methods are in good agreement. While the MacCormack scheme may be computationally more efficient for small velocities and dispersion coefficients (as is commonly the case for wetlands and lakes), much longer computational times are needed for the cases with high velocity and dispersion coefficient values (e.g., streams) since the magnitude of the time step has to be selected according to the CFL stability condition.

Research on the Muzzle Blast Flow with Gas-Particle Mixtures Based on Eulerian-Eulerian Approach

ABSTRACTThe issue on the muzzle blast flow with gas-particle mixtures was numerically investigated in this paper. The propellant gas in the cannon was assumed to be gas-particle mixtures consisting of a variety of gaseous species and particles. The model made use of the Eulerian-Eulerican approach, where the particle were modeled as a second fluid with parameters like bulk density, velocity and temperature, interacting with the gas flow. A high-resolution upwind scheme(AUSMPW+) and detailed reaction kinetics model were employed to solve the chemical non-equilibrium Euler equations for gas phase. The Euler equations for particle phase were solved by MacCormack scheme. The particle diameter and the mass fraction of particle were tested to show their effects on the development process of muzzle blast flow with gas-particle mixtures. The distribution of the main flow parameters of both gas and particle were obtained at different time intervals. The results show the evolution of the muzzle blast flow with gas-particle mixtures and demonstrate the effects of key parameter on the flow field of the gas-particle flow. This paper is a significant investigation for understanding the muzzle blast flow with gas-particle mixtures, which can provide valuable reference for the research on the muzzle blast flow.

A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling.

Haemodynamical simulations using one-dimensional (1D) computational models exhibit many of the features of the systemic circulation under normal and diseased conditions. Recent interest in verifying 1D numerical schemes has led to the development of alternative experimental setups and the use of three-dimensional numerical models to acquire data not easily measured in vivo. In most studies to date, only one particular 1D scheme is tested. In this paper, we present a systematic comparison of six commonly used numerical schemes for 1D blood flow modelling: discontinuous Galerkin, locally conservative Galerkin, Galerkin least-squares finite element method, finite volume method, finite difference MacCormack method and a simplified trapezium rule method. Comparisons are made in a series of six benchmark test cases with an increasing degree of complexity. The accuracy of the numerical schemes is assessed by comparison with theoretical results, three-dimensional numerical data in compatible domains with distensible walls or experimental data in a network of silicone tubes. Results show a good agreement among all numerical schemes and their ability to capture the main features of pressure, flow and area waveforms in large arteries. All the information used in this study, including the input data for all benchmark cases, experimental data where available and numerical solutions for each scheme, is made publicly available online, providing a comprehensive reference data set to support the development of 1D models and numerical schemes.

Comprehensive two-dimensional river ice model based on boundary-fitted coordinate transformation method

River ice is a natural phenomenon in cold regions, influenced by meteorology, geomorphology, and hydraulic conditions. River ice processes involve complex interactions between hydrodynamic, mechanical, and thermal processes, and they are also influenced by weather and hydrologic conditions. Because natural rivers are serpentine, with bends, narrows, and straight reaches, the commonly-used one-dimensional river ice models and two-dimensional models based on the rectangular Cartesian coordinates are incapable of simulating the physical phenomena accurately. In order to accurately simulate the complicated river geometry and overcome the difficulties of numerical simulation resulting from both complex boundaries and differences between length and width scales, a two-dimensional river ice numerical model based on a boundary-fitted coordinate transformation method was developed. The presented model considers the influence of the frazil ice accumulation under ice cover and the shape of the leading edge of ice cover during the freezing process. The model is capable of determining the velocity field, the distribution of water temperature, the concentration distribution of frazil ice, the transport of floating ice, the progression, stability, and thawing of ice cover, and the transport, accumulation, and erosion of ice under ice cover. A MacCormack scheme was used to solve the equations numerically. The model was validated with field observations from the Hequ Reach of the Yellow River. Comparison of simulation results with field data indicates that the model is capable of simulating the river ice process with high accuracy.

Numerical simulation of particle motion using a combined MacCormack and immersed boundary method

A numerical approach is presented for the direct numerical simulation of particle motion that combines the MacCormack scheme and the immersed boundary method. It exhibits the advantageous features of the explicit MacCormack scheme which is second-order accurate in time and space with simplicity in programing. The approach solves the compressible Navier–Stokes equations and uses the immersed boundary method to tackle the interactions between the fluid and the suspended particles. The force due to the interaction of two phases is computed via an elastic forcing method. The numerical approach is validated using uniform flow past a stationary circular cylinder, sedimentation of circular discs, and particle motion (orientation and translation) in unidirectional flows. Results are also compared to simulation obtained from a mixture model for solid particles for the same flow conditions.

A numerical model for the dynamics of one-velocity gas-vapor-droplet fluid

In this paper, the results of a numerical simulation of the dynamics of a gas-vapor-droplet system are presented. A mathematical model used is based on the equations of gas dynamics, which are numerically solved using the MacCormack scheme. The stationary distributions of pressure, velocity, temperature and partial densities of the mixture components are found by studying the flow around a cylinder.

Analysis of Aluminium Hot Extrusion by Finite Volume Method

A novel numerical approach to calculate strain rate, stress and velocity fields of axisymmetric hot extrusion process is presented and discussed. Traditionally, metal forming analysis has been performed by upper-bound, slab, slip-line and Finite Element (FEM) methods. However, Finite Volume Method (FVM) has been recently developed for metal plastic flow analysis. Hence, metal extrusion was modelled by flow formulation. Governing equations were discretized by FVM, using Explicit MacCormack Method in structured and collocated mesh. SIMPLE method was applied to attain necessary pressure-velocity coupling. This scheme was applied to hot extrusion of aluminium alloy 6351. Numerical results of velocity field from present FVM produced consistent and good agreement with FEM results, employing FORGE 2008. Present results also suggest that MacCormack and SIMPLE methods can be applied in the solution of metal forming processes.

Investigation of the Effects of Non-Linear and Non-Homogeneous Non-Fourier Heat Conduction Equations on Temperature Distribution in a Semi-Infinite Body

In this paper, the non-Fourier heat conduction in a semi-infinite body was examined. The heat wave non-Fourier heat conduction model was used for thermal analysis. Thermal conductivity was assumed temperature-dependent which resulted in a non-linear equation. The heat source was also considered temperature-dependent which resulted in a non-homogeneous equation. The Mac-Cormack predictor-corrector numerical method was employed to solve the equations. It was concluded that, the non-linear analysis of the non-Fourier heat transfer problems is of great importance. Also, the case which assumed a temperature-dependent heat source had a considerable difference with the case where a constant heat source was assumed.

Dynamics of the Working Medium of a Diffusion-Cooled Electric-Discharge CO2 Laser with Periodic Heat Release in the Axial Region of its Cylindrical Resonator

The dynamics of the gas mixture in the cylindrical resonator of a CO2 laser with periodic heat release in the region of its axis was defined on the basis of the numerical solution of the system of Navier–Stokes equations by the explicit MacCormack method with splitting of the initial operator in space directions. The space fields of the gas-dynamic functions of the gas mixture and the ranges of their change were determined, and the character of the averaged flow of this mixture was investigated.

Modeling of gas release following pipeline rupture: Proposing non-dimensional correlation

Conveying natural gas through a pipeline is widely accompanied by accidents in industries. It is important to perform hazard analysis by estimating the amount of released gas at the first step immediately after the happening. The present study proposes a new approach to evaluate the gas release rate and released mass by introducing non-dimensional equations for simulating quasi-one-dimensional transient compressible flow in a ruptured gas pipeline. These equations are derived from Euler equations. The hyperbolic governing equations are solved numerically by the explicit MacCormack's method. The influence of different dimensionless physical parameters on release rate are then investigated; among which relative pressure, relative hole diameter and friction term are the most important parameters. Results show that the release rate has the similar behavior for a wide range of operating pressures from 60 psig (0.4 MPa for distribution pipeline) to 1000 psig (6.9 MPa for transmission pipeline). Since choking condition happens in rupture area, release rate is not a function of pipeline pressure. Therefore, relative hole diameter and friction term are the only effective parameters in this case. By introducing an indicator and its criterion for long pipelines it could be seen that not only is this indicator dependent on geometry (i.e. ratio of length to diameter of pipe), but it also belongs to flow regime (i.e. friction coefficient). Finally, general relations for natural gas release rate in any physical condition are proposed by exploring effective dimensionless parameters on released mass.