MacCormack Scheme Research Articles

Performance study of a magnetohydrodynamic accelerator using air-plasma as working gas

The fundamental performance of a linear Faraday type magnetohydrodynamic (MHD) accelerator under chemical and thermal equilibrium conditions is studied in this paper. Quasi-one dimensional numerical simulation is performed to characterize acceleration along the specified channel. In order to solve the set of differential equations of magnetohydrodynamics, the MacCormack scheme is employed. In this present calculation, a working gas of air-plasma, consisting of diatomic molecules of nitrogen, oxygen and hydrogen, is considered in order to actualize application of the MHD accelerator propulsion system. Numerical results show the characteristics of flow velocity, gas temperature and electrical conductivity of the MHD accelerator with the working gas of air-plasma. The computations are compared with numerical simulation results using the inert gas of argon evaluated in past studies.

NUMERICAL SIMULATION OF LOCAL SCOURING BY COMPLEX FLOW OVER BACKWARD-FACING STEP

When a flow over backward-facing step or consolidation work includes both subcritical and supercritical flows, a very complicated local scouring is generated by interaction of a wave jump and a submerged jet at the downstream of these structures. Since the local scouring decreases the stability of these structures, it is a very important problem on disaster prevention. In this paper, we present a 2-dimensional numerical model which can reproduce the local scouring by the complex flow over backward-facing step. The numerical model is based on the MacCormack scheme, and the FAVOR method is introduced into the basic equations. A bed load and a suspended load are taken into consideration by the bed evolution model. The numerical results reproduce the local scouring by repetition of each flow well.

Numerical Simulation for the Performance of a Magnetohydrodynamic Accelerator

Recent results of our quasi-one-dimensional numerical simulation for the performance of an MHD accelerator under chemical and thermal equilibrium conditions are presented. For the present study, the working gas was changed to air-plasma from inert gas of argon plasma, which was studied previously to actualize the application of an MHD accelerator propulsion system. The numerical simulation confirmed the performance of MHD accelerators with segmented Faraday, Hall, and diagonal electrode connections. In order to solve the set of differential equations of magnetohydrodynamics, the MacCormack scheme is employed. The results show several parameters of MHD accelerator performance with air-plasma working gas. The calculation results are compared with previous results to clarify the effect of the difference in working gases.

A Numerical Study of Unsteady, Thermal, Glass Fiber Drawing Processes

An e-cient second-order stable numerical method is presented to solve the model partial difierential equations of thermal glass flber processing. The physical process and structure of the model equations are described flrst. The numerical issues are then clarifled. The heart of our method is a MacCormack scheme with ∞ux limiting. The numerical method is validated on a linearized isothermal model and by comparison with known exact stationary solutions. The numerical method is then generalized to solve the equations of motion of thermal glass flber drawing, exhibiting order of convergence. Further, the nonlinear PDE scheme is benchmarked against an independent linearized stability analysis of boundary value solutions near the onset of instability, which demonstrates the e-ciency of the method.

Experimental investigation and numerical validation of explosion suppression by inert particles in large-scale duct

A large-scale duct with an explosion suppressor was designed to investigate experimentally the explosion suppression by inert particles for a CH 4/O 2/N 2 mixture. The duct is 25 m long and has an internal diameter of 700 mm. Pressure and flame signals were recorded some distance away from ignitor in the duct. Pressure tracking lines of the shock front for the different inert particle cloud densities and the inert particle diameters were made. The measured results indicate that the shock front is decoupled from the flame front in the inert particle cloud, which leads to a suppression of explosion. Also, the experiments suggest that increasing the inert particle cloud density or decreasing the inert particle diameter can enhance the ability to suppress explosion. For the purpose of validation, a two-dimensional numerical model coupled with the element chemical reaction mechanism for the simulations of the CH 4/O 2/N 2 mixture explosion suppression by the inert particles has been developed. This model makes use of the second-order TVD scheme and the MacCormack scheme to calculate gas-phase and particle-phase equations, respectively. The Strang splitting technique is used to treat the stiffness due to the coupling of the governing equations, while the implicit Gear algorithm is used to treat the stiffness due to the chemical reactions. The effect of inert particle cloud density on explosion suppression was investigated using the model. The calculated results indicate that the accumulation of inert particles slows the propagation of the gas-phase shock front and results in explosion suppression. With increased inert particle cloud density, the explosion suppression is more prominent. The calculated results show a qualitative agreement with the measured results in the large-scale duct experiment.

Mathematical modelling of compressible viscous fluid flow in a male rotor‐housing gap of screw machines

AbstractThis contribution is devoted to the mathematical modelling of a compressible viscous fluid flow through a 2‐D model of the male rotor‐housing gap in screw machines. Numerical solution of the nonlinear conservative system of the compressible Navier‐Stokes equations is obtained by means of the cell‐centred finite volume formulation of the explicit two‐step TVD MacCormack scheme proposed by Causon on a structured quadrilateral grid using the own developed numerical code. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On numerical modeling of overland flow

The reliability of several finite difference numerical formulations for solving one-dimensional kinematic and diffusion wave equations that describe overland flow is investigated. Particular emphasis is placed on two numerical schemes which are second order accurate in space and time, one each from the explicit and implicit families, respectively: the MacCormack and the 4-point implicit schemes. The end numerical solutions are compared to analytical solutions used as benchmark data, and to implicit and explicit schemes widely used to solve the overland flow governing equations. The magnitude of the time step was selected from the CFL stability condition. The algorithms are used to solve two examples, and the results are compared in terms of accuracy and required computational time. In both cases, the results indicate that the MacCormack and 4-point implicit methods are superior in terms of accuracy to the other widely used schemes. The MacCormack and 4-point implicit methods are comparable in accuracy, but the MacCormack algorithm is computationally more efficient.

Adaptive mesh refinement and coarsening for aerodynamic flow simulations

AbstractA new mesh refinement technique for unstructured grids is discussed. The new technique presents the important advantage of maintaining the original grid skewness, thanks to the capability of handling hanging nodes. The paper also presents an interpretation of MacCormack's method in an unstructured grid context. Results for a transonic convergent–divergent nozzle, for a convergent nozzle with a supersonic entrance and for transonic flow over a NACA 0012 airfoil are presented and discussed. Copyright © 2004 John Wiley & Sons, Ltd.

One-dimensional hydrodynamic/sediment transport model applicable to steep mountain streams

A new one-dimensional (1-D) numerical model for calculating flow and sediment transport in steep mountain streams is developed. 3ST1D, which stands for Steep Stream Sediment Transport 1-D model, is applicable to unsteady flowconditions that occur over transcritical flowstream reaches such as flows over step-pool sequences. 3ST1D consists of two coupled components, the hydrodynamic and the sediment transport. The flow component is addressed here by solving the unsteady form of the Saint-Venant equations. The TotalVariation Diminishing Dissipation (TVD)-MacCormack scheme, which is a shock-capturing scheme capable of rendering the solution oscillation free, is employed here to approximate the hydrodynamic solution over transcritical flow stream reaches. The sediment component of the model accounts for multifractional sediment transport and incorporates a series of various incipient motion criteria and frictional formulas applicable to mountain streams. In addition, sediment entrainability is estimated based ...

One-dimensional hydrodynamic/sediment transport model applicable to steep mountain streams

A new one-dimensional (1-D) numerical model for calculating flow and sediment transport in steep mountain streams is developed. 3 ST1D, which stands for Steep Stream Sediment Transport 1-D model, is applicable to unsteady flow conditions that occur over transcritical flow stream reaches such as flows over step-pool sequences. 3ST1D consists of two coupled components, the hydrodynamic and the sediment transport. The flow component is addressed here by solving the unsteady form of the Saint-Venant equations. The Total Variation Diminishing Dissipation (TVD)-MacCormack scheme, which is a shock-capturing scheme capable of rendering the solution oscillation free, is employed here to approximate the hydrodynamic solution over transcritical flow stream reaches. The sediment component of the model accounts for multifractional sediment transport and incorporates a series of various incipient motion criteria and frictional formulas applicable to mountain streams. In addition, sediment entrainability is estimated based on a state-of-the art formula that accounts for the bed porosity, turbulent bursting frequency, probability of occurrence of strong episodic turbulent events, and sediment availability in the unit bed area. The model at the end of each time step predicts the flow depth, velocity and shear stress distribution within a cell and calculates changes in bed evolution and grain size distribution. The overall performance of the model is evaluated by comparing its predictions with observations from two flume studies, two field investigations and against the predictions of the quasi-steady model of Lopez and Falcon developed for mountain streams. A sensitivity analysis is performed to assess the effects of cell size and Manning’s roughness coefficient in the predictive ability of the model.

ラバールノズル内の気泡流の数値解析(S25-1 気泡力学に関する基礎解析とその応用(1),S25 気泡力学に関する基礎解析とその応用)

The two-phase bubbly flow in a Laval nozzle is numerically studied with a two-fluid and three-pressure model of averaged equations for two-phase bubbly flow. The governing equations are extended to quasi-one-dimensional flows, and solved with MacCormack scheme. As a result of numerical simulation of subsonic bubbly flow in a Laval nozzle, the distributions of pressure, flow velocity, density and volume fraction rate in each phase are clarified.

EVALUATION OF GAS-KINETIC SCHEMES FOR SOLVING 1D INVISCID COMPRESSIBLE FLOW PROBLEMS

In this paper, two classes of gas-kinetic schemes are investigated for one-dimensional compressible inviscid flow, namely, the Kinetic Flux Vector Splitting (KFVS) scheme and the Bhatnagar-Gross-Krook (BGK) scheme. Second-order high-resolution scheme for both methods are also developed for calculating flows containing discontinuities. This is achieved by means of reconstructing the initial data via Monotone Upstream-Centered Schemes for Conservation Laws (MUSCL) approach. The Total Variation Diminishing (TVD) shock capturing properties of these second-order schemes are achieved through the use of non-linear limiters. In addition, a multistage TVD Runge-Kutta method is employed for the time integration of the finite volume gas-kinetic scheme. Exclusive consideration is focused on the BGK scheme, which yields a better numerical result in comparison with the KFVS scheme. Three typical one-dimensional inviscid flow problems containing shocks, namely, steady flow in a divergent nozzle, the unsteady shock tube problem, and two interacting blast waves problem are analyzed numerically with the gas-kinetic schemes. Numerical results from the first-order and second-order gas-kinetic schemes are presented and compared with the exact solutions. Other computed results such as those from the Steger-Warming's Flux Vector Splitting scheme, Roe's Flux Difference Splitting scheme, MacCormack's scheme, and high-order compact Van Leer's Flux Splitting Scheme are also presented as comparisons to the gas-kinetic schemes. In addition, the effects of grid sizes on the numerical results of the gas-kinetic schemes are also investigated.

Engineering Model for Simulation of Debris Cloud Propagation Inside Gas-Filled Pressure Vessels

Space debris causes a severe hazard to the International Space Station. Because of the possibility for the pressure vessels onboard space station to fail catastrophically being impacted these stored energy devices play a central role in determining the survivability of space station when subject to space debris impact. Recent studies of pressure vessel reaction to hypervelocity impact showed that an interaction between traveling debris cloud and gas medium inside the pressure vessel has a dramatic influence on a vessel rear wall deformation and failure. The purpose of this work is to determine the parameters of fragments that travel across the vessel diameter and act on the vessel rear wall, as well as parameters of a shock wave generated in the process of interaction between the debris cloud and gas. For evaluation of debris and gas parameters near the leading edge of the cloud the approach suggested by Nigmatulin is used. According to this approach the adiabatic two-phase flow is considered. The cloud-gas interaction is described by the set of nonlinear hyperbolic partial differential equations. This system is solved numerically by MacCormack method. The model proposed enables us to simulate both the fragments velocities and pressure amplitude of the shock wave. This engineering model and the code developed can be used for both the pressure vessel designing and analysis of its survivability.

Finite-volume component-wise TVD schemes for 2D shallow water equations

Four finite-volume component-wise total variation diminishing (TVD) schemes are proposed for solving the two-dimensional shallow water equations. In the framework of the finite volume method, a proposed algorithm using the flux-splitting technique is established by modifying the MacCormack scheme to preserve second-order accuracy in both space and time. Based on this algorithm, four component-wise TVD schemes, including the Liou–Steffen splitting (LSS), van Leer splitting, Steger–Warming splitting and local Lax–Friedrichs splitting schemes, are developed. These schemes are verified through the simulations of the 1D dam-break, the oblique hydraulic jump, the partial dam-break and circular dam-break problems. It is demonstrated that the proposed schemes are accurate, efficient and robust to capture the discontinuous shock waves without any spurious oscillations in the complex flow domains with dry-bed situation, bottom slope or friction. The simulated results also show that the LSS scheme has the best numerical accuracy among the schemes tested.

Numerical boundary conditions for sound scattering simulation

A hierarchy of boundary algorithms based both on a decomposition of the flow field and on a characteristic formulation of the conservation equations is introduced for the computation of sound wave scattering by vortical flows. The robustness of the proposed algorithms is analysed. Sound and vorticity wave reflexion at an open boundary are estimated for various mean flows and different sound wave fields. A direct assessment of this algorithm coupled with an interior 2–4 Mac Cormack scheme for sound scattering computations based on the 2D inviscid gas dynamics equations is performed using comparison with theoretical scattering results.

A parallel hydrodynamic model for shallow water equations

A parallel implementation of a finite difference model for solving two-dimensional, time-dependent, open channel flows is presented. The algebraic equations resulting from the finite difference discretization of the two dimensional shallow water flow equations are solved by using explicit MacCormack scheme. The parallel code has been implemented on distributed–shared memory system, by using domain decomposition techniques. The message passing interface (MPI) protocols are incorporated for inter processor data communication. The effect of using two different geometry partitions is investigated. A comparison of the wallclock time of the code between these two partitions is made, and code performances with respect to different number of processors are presented.

1446 3 圧力 2 流体モデルによる気泡流中を伝わる衝撃波の数値シミュレーション

The shock wave propagation in a bubbly liquid is numerically studied with a two-fluid and three-pressure model of averaged equations for two-phase bubbly flow. A surface average pressure at the bubble wall is introduced in addition to the volume average pressures for gas and liquid phases so as to deal with the local high pressure caused by the cavitation bubble collapse in high-speed bubbly flows. The governing equations are solved with MacCormack scheme. The influence of initial void fraction on the propagation of shock wave is clarified. Furthermore, it is shown that the difference of the propagation characteristics between the compression and expansion waves appears to be conspicuous more and more as the initial void fraction increase.

Origination of In‐Phase Oscillations of Thin Plates with Aeroelastic Interaction

Synchronization of oscillations of thin elastic plates that are walls of a gas‐filled channel is considered. The gas motion is described by a system of Navier–Stokes equations, which is solved using the second‐order MacCormack method with time splitting. The motion of the channel walls is described by a system of geometrically nonlinear dynamic equations of the theory of this plates, which is solved by the finite‐difference method. Kinematic and dynamic contact conditions are imposed at the interface between the media. A numerical experiment is performed to determine typical dynamic regimes and study the transition of the aeroelastic system to in‐phase oscillations.

RANDOM OSCILLATIONS OF AN ANTIPHASE-EXCITED AEROELASTIC SYSTEM WITH SYNCHRONIZATION

We examine synchronization of the oscillatory motion of thin elastic cylindrical plates forming the walls of a channel filled by a gas. The gas motion in the channel is described by a system of Navier–Stokes equations solved by the MacCormack method of second-order accuracy. The motion of the channel walls is described by a system of dynamic, geometrically nonlinear equations of the thin-shell theory; this system is solved by the finite difference method. Kinematic and dynamic contact conditions are set at the interface between the media. By means of a numerical experiment, possible scenarios of the transition of the aeroelastic system to in-phase oscillations were identified, and the regime of random oscillations in the system with synchronization under antiphase external excitation was found.

Mathematical modelling of compressible inviscid fluid flow through a sealing gap in the screw compressor

In this article, the essential steps of the numerical simulation of compressible inviscid fluid flow in a sealing gap of the screw compressor starting from the description of a mathematical model to its final numerical solution are presented. The mathematical model of compressible inviscid flow is described by the conservative system of the Euler equations. For the numerical solution of this system, the cell-centred finite volume formulation of the explicit two-step MacCormack scheme with Jameson’s artificial dissipation was used.