MacCormack Scheme Research Articles

ESTIMATION AND LOCALIZATION OF MAXIMUM TEMPERATURE IN A TUBULAR CHEMICAL REACTOR BY LUENBERGER STATE OBSERVER

This study deals with an observer built for distributed parameter systems described by nonlinear representations. This observer was applied to a tubular chemical reactor in order to estimate the maximum temperature of the reactant mixture and its position in stationary and transient regimes. This estimation uses a Luenberger observer, while MacCormack's numerical method was used for resolution of the partial differential equations. Furthermore, to take into account the nonlinearity of the system, the gain of this estimator was proposed as a function of the position along the chemical reactor. This observer requires the values of both concentration and temperature at the inlet but only of temperature at the outlet. The convergence and robustness of this estimator were experimentally tested with initialization and modeling errors.

High resolution numerical schemes for solving kinematic wave equation

Summary This paper compares the stability, accuracy, and computational cost of several numerical methods for solving the kinematic wave equation. The numerical methods include the second-order MacCormack finite difference scheme, the MacCormack scheme with a dissipative interface, the second-order MUSCL finite volume scheme, and the fifth-order WENO finite volume scheme. These numerical schemes are tested against several synthetic cases and an overland flow experiment, which include shock wave, rarefaction wave, wave steepening, uniform/non-uniform rainfall generated overland flows, and flow over a channel of varying bed slope. The results show that the MacCormack scheme is not a Total Variation Diminishing (TVD) scheme because oscillatory solutions occurred at the presence of shock wave, rarefaction wave, and overland flow over rapidly varying bed slopes. The MacCormack scheme with a dissipative interface is free of oscillation but with considerable diffusions. The Godunov-type schemes are accurate and stable when dealing with discontinuous waves. Furthermore the Godunov-type schemes, like MUSCL and WENO scheme, are needed for simulating surface flow from spatially non-uniformly distributed rainfalls over irregular terrains using moderate computing resources on current personal computers.

Study of Numerical Simulation of RF Wave in Magnetized Plasma

Wave heating is an important way to implement the controlled fusion. The propagation and absorbability of wave in the plasma are described by the two-fluid equations and Maxwell’s equations. A numerical algorithm based on the FDTD method and the MacCormack scheme is designed. The numerical simulation based on parallel computing is carried out to verify the validity of the algorithm and simulation.

Resonance of Free-Surface Waves Provoked by Floodgate Maneuvers

AbstractPrediction of damage caused by floodgate maneuvers is treated by the determination of factors amplifying the magnitude of generated damage. The identification of vulnerable areas is obtained by analogy with the water-hammer phenomenon in pressurized pipes. The numerical model is based on the Boussinesq assumption for transient flow in open channels and solved by the Mac-Cormack scheme. Transient flow concerns a rectangular channel where the initial steady state is uniform. Transient regime is provoked by the movement of a floodgate at the upstream extremity of the channel producing sinusoidal fluctuations of depth. A condition of floodgate closure is imposed at the downstream extremity. The applied excitation has permitted analysis of the fluid structure interaction, examination of pulsations provoking the resonance of the free-surface waves, and study of the evolution of depths according to time along the channel. According to the frequency of the movement of the floodgate at the upstream end, di...

Unsteady compressible flows in channel with varying walls

This study deals with numerical solution of a 2D and 3D unsteady flows of a compressible viscous fluid in 2D and 3D channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall, nearly closing the channel during oscillations. The channels shape is a simplified geometry of the glottal space in the human vocal tract. Goal is numerical simulation of flow in the channels which involves attributes of real flow causing acoustic perturbations. The system of Navier-Stokes equations closed with static pressure expression for ideal gas describes the unsteady laminar flow of compressible viscous fluid. The numerical solution is implemented using the finite volume method and the predictor-corrector MacCormack scheme with artificial viscosity using a grid of quadrilateral cells. The unsteady grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian-Eulerian method. The application of developed method for numerical simulations of flow fields in the 2D and 3D channels, acquired from a developed program, are presented for inlet velocity u=4.12 m/s, inlet Reynolds number Re=4481 and the wall motion frequency 100 Hz.

Numerical solution of compressible and incompressible unsteady flows in channel inspired by vocal tract

This study deals with the numerical solution of a 2D unsteady flow of a viscous fluid in a channel for low inlet airflow velocity. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The channel is a simplified model of the glottal space in the human vocal tract. Four governing systems are considered to describe the unsteady laminar flow of a viscous fluid in the channel. The numerical solution is implemented using the finite volume method (FVM) and the predictor–corrector MacCormack scheme with artificial viscosity using a grid of quadrilateral cells. The unsteady grid of quadrilateral cells is considered in the form of conservation laws using the Arbitrary Lagrangian–Eulerian method. The numerical simulations of flow fields in the channel, acquired from a developed program, are presented for inlet velocity uˆ∞=4.12ms−1 and Reynolds number Re∞=4481 and the wall motion frequency 100 Hz.

Numerical Treatment of a Modified MacCormack Scheme in a Nondimensional Form of the Water Quality Models in a Nonuniform Flow Stream

Two mathematical models are used to simulate water quality in a nonuniform flow stream. The first model is the hydrodynamic model that provides the velocity field and the elevation of water. The second model is the dispersion model that provides the pollutant concentration field. Both models are formulated in one-dimensional equations. The traditional Crank-Nicolson method is also used in the hydrodynamic model. At each step, the flow velocity fields calculated from the first model are the input into the second model as the field data. A modified MacCormack method is subsequently employed in the second model. This paper proposes a simply remarkable alteration to the MacCormack method so as to make it more accurate without any significant loss of computational efficiency. The results obtained indicate that the proposed modified MacCormack scheme does improve the prediction accuracy compared to that of the traditional MacCormack method.

Three-dimensional Steady State Interplanetary Solar Wind Simulation in Spherical Coordinates with a Six-component Grid

采用二阶MacCormack差分格式, 利用稳态的磁流体(MHD)方程组在球坐标系六片网格下模拟研究了行星际太阳风. 六片网格系统能有效避免极区奇性和网格收敛性. 迭代按径向方向推进求解, 很大程度上减少了计算量, 节约了计算时间. 内边界条件根据太阳与行星际观测确定, 比较测试了5种内边界条件, 模拟给出了1922卡林顿周的背景太阳风结构. 几种内边界条件所得模拟结果与行星际观测基本吻合. 太阳风速度采用McGregor 等的经验公式给出, 磁场由水平电流片(HCCS)模型得到, 密度和温度分别根据动量守恒和气压守恒得到, 研究表明采用这样的边界条件模拟结果最佳.

Numerical experiments modelling turbulent flows

The work aims at investigation of the possibilities of modelling transonic flows mainly in external aerodynamics. New results are presented and compared with reference data and previously achieved results. For the turbulent flow simulations two modifications of the basic k − ω model are employed: SST and TNT. The numerical solution was achieved by using the MacCormack scheme on structured non-ortogonal grids. Artificial dissipation was added to improve the numerical stability.

Accurate Simulation of Contaminant Transport Using High-Order Compact Finite Difference Schemes

Numerical simulation of advective-dispersive contaminant transport is carried out by using high-order compact finite difference schemes combined with second-order MacCormack and fourth-order Runge-Kutta schemes. Both of the two schemes have accuracy of sixth-order in space. A sixth-order MacCormack scheme is proposed for the first time within this study. For the aim of demonstrating efficiency and high-order accuracy of the current methods, some numerical experiments have been done. The schemes are implemented to solve two test problems with known exact solutions. It has been exhibited that the methods are capable of succeeding high accuracy and efficiency with minimal computational effort, by comparisons of the computed results with exact solutions.

Dependence on the sectional shape of tube for propagation behavior of three-dimensional detonation

This paper is concerned with numerical simulation on propagation behavior of three-dimensional detonations in square tubes. Influence on motion trajectories of high pressure region (shock triple points) due to the change of aspect ratio of tube is solved by a finite difference method which consists of the semi-implicit MacCormack method and Non-MUSCLE type TVD scheme in order to prevent numerical stiffness and Gibb's phenomena. Reaction mechanism utilized in this study is the Nagoya model constructed for the stoichiometric hydrogen and oxygen gas mixture. It was found from numerical results that distinction in propagation mode could be explained from the relation between the geometric form of the tube section and the behavior of triple points. Moreover, it was shown that the transition of the configuration and the mode of propagation might not be settled causing the aspect ratio of the section.

Numerical and experimental study on parameters distribution inside a two-phase ejector

The two-phase flow process in an ejector was numerically and experimentally studied using R141b as a working fluid. A modified one-dimensional gas–liquid ejector model was proposed to remedy the defect in the traditional one. Gas–liquid boundary layer regions were discussed and used to close the model. Mac Cormack method is used to discrete controlling equations of gas–liquid two-phase flow in the ejector. The radial distribution of velocity and temperature, the variation of void fraction, the axial velocity variation and the influence of primary steam pressure on the mixing process were predicted with the numerical model. An experimental rig was set up to validate the model by comparing the experimental pressure distribution in the ejector with the calculating one.

Numerical simulation of two-phase reactive flow with moving boundary

Purpose – In computational fluid dynamics for two-phase reactive flow of interior ballistic, the conventional schemes (MacCormack method, etc.) are known to introduce unphysical oscillations in the region where the gradient is high. This paper aims to improve the ability to capture the complex shock wave during the interior ballistic cycle. Design/methodology/approach – A two-phase flow model is established to describe the complex physical process based on a modified two-fluid theory. The solution of model is obtained including the following key methods: an approximate Riemann solver to construct upwind fluxes, the MUSCL extension to achieve high-order accuracy, a splitting approach to solve source terms, a self-adapting method to expand the computational domain for projectile motion and a control volume conservation method for the moving boundary. Findings – The paper is devoted to applying a high-resolution numerical method to simulate a transient two-phase reactive flow with moving boundary in guns. Several verification tests demonstrate the accuracy and reliability of this approach. Simulation of two-phase reaction flow with a projectile motion in a large-caliber gun shows an excellent agreement between numerical simulation and experimental measurements. Practical implications – This paper has implications for improving the ability to capture the complex physics phenomena of two-phase flow during interior ballistic cycle and predict the combustion details, such as the flame spreading, the formation of pressure waves and so on. Originality/value – This approach is reliable as a prediction tool for the understanding of the physical phenomenon and can therefore be used as an assessment tool for future interior ballistics studies.

Modelling and simulation of cross flow grain dryers

A mathematical model, algorithm, and computer program were developed to simulate the performance of cross flow grain dryers and cross flow dryers with energy saving. The mass and heat transfer processes were described by a system of four non-linear partial differential equations. This system of equations was solved by the MacCormack method with time splitting. The Neumann method was used to determine convergence. The source-terms in these equations were computed by auxiliary semi-empirical equations obtained by experimental data from thin layer drying. Equipment developed to obtain these data permitted variation of the initial air humidity, temperature, and velocity. Fixed bed drying experiments were conducted to validate the model. Simulations using various control regimens were made to determine the impact on energy consumption and cross flow dryer performance due to recycling air exhausted from various stages of the dryer. An iterative process was used to determine the initial conditions at the entrance to each section of the dryer. The computer simulations were used to evaluate the non-uniformity of temperature and grain moisture content distributions in dryers, the duration of the drying process and the energy efficiency for each geometry and control regimen.

Comparison Results between MacCormack Scheme and Steger Warming Scheme for the Case of Supersonic Flow Pass through Divergent Nozzle

The present work focuses on the development of computer code which allow the engine designer investigate the flow behavior through a divergent nozzle. The flow is assumed as quasi one dimensional and steady inviscid compressible flow. In other word the flow is governed by compressible Euler equation depending on the local Mach number. Solving such a mixed type of equation is difficult. The boundary between hyperbolic and elliptic region is not clear. To avoid such difficulty can be done through treating a steady flow problem as unsteady flow problems with the boundary condition is fixed with respect to time. As unsteady flow problem, the Euler equation becomes hyperbolic type of partial differential equation with respect to time. This approach allows one to solve the unsteady flow problem goes to a steady state solution. There are various methods can used for solving hyperbolic type of equation, such as MacCormacks scheme, Steger Warmings scheme, Beam Warmings Scheme and TVD Scheme. The present work used MacCormacks scheme and Steger Warming scheme will be used to investigate the flow behavior through a divergent nozzle. Comparison with analytic solution shows that Steger Warmings scheme gives a better result than MacCormacks scheme.

Numerical study of shock wave dynamics in a gas suspension

We describe a process of pressure discontinuity breakdown in a gas suspension, dynamics of which is simulated by a system of equations of motion for a two-temperature two-speed monodisperse medium without phase transition and coagulation. The carrying medium is described by the Navier-Stokes equation system. A disperse phase is simulated by equations of mass, momentum and internal energy conservation. The system is reduced to a dimensionless form, it is written in the generalized coordinates and is solved by the explicit MacCormack method with a conservative correction scheme applied to each phase to obtain a monotonous solution. We describe an effect of decreasing the time intensity of a shock wave when its attitude is retained. The time of the pressure differential value relaxation at the direct shock wave front depends on the volumetric content and dispersion degree of the gas suspension.

Parallel tsunami simulation and visualization on tiled display wall using OpenGL Shading Language

Tsunami simulation consists of fluid dynamics, numerical computations, and visualization techniques. Nonlinear shallow water equations are often used to model the tsunami propagation. Tsunami inundation is modeled by adding the friction slope to the conservation of momentum. The two-step second-order finite difference MacCormack numerical method can solve these equations. It is well suited for nonlinear equations and simpler for related application development. In addition, the finite difference method can be computed in parallel. The programmable graphics hardware allows general-purpose computing on graphics processing units (GPUs) to solve the MacCormack method in parallel to speed up the simulation. Tsunami simulation data can be loaded as textures data in graphics memory, the computation processes can be written as shader programs using OpenGL Shading Language so that the operations can be computed by graphics processors in parallel. We developed different versions of the tsunami simulation and visualization programs: (i) CPU-based, and (ii) CPU–GPU collaboration to implement the MacCormack numerical method. The performance results showed that graphics hardware accelerated simulation had a significant improvement in the execution time of each computation step. Real-time simulation and visualization are made possible by the programmable shaders. Furthermore, we achieved high-performance parallel visualization on a tiled display wall with a cluster of computers.

Deterministic unsteady and aeroelastic flow simulations with high-order FVM schemes

During the last few years, unsteady and aeroelastic simulations have become a standard part of CFD computations in aeronautical applications. They are usually used with various types of turbulent models the most of which are derived from system of the Reynolds-Averaged Navier–Stokes equations (RANS). However, the RANS system itself is derived using time-averaging and hence it neglects certain type of unsteadiness in its own nature. Therefore it is necessary to investigate what is the ability of the contemporary numerical methods (i.e. combinations of numerical schemes and turbulence models) to simulate various types of unsteady flows. The authors carried out a series of numerical tests of two types of unsteady flows in external aerodynamics. Namely the transonic compressible flow over harmonically oscillating NACA 0012 aerofoil and subsonic compressible flow over the same aerofoil with oscillations induced by the flow itself (i.e. considering the aeroelastic interaction). Three different high-order finite volume schemes with various contemporary turbulence models were chosen for the testing. In the first researched case the computations were carried out using: the Modified Causon’s scheme “as reported by Furmanek and Kozel (Kybernetika 45(4):567–579, 2009)” (derived from TVD form of the classical MacCormack scheme) and the solver edge developed by the Swedish defence and research agency FOI (n-stage Runge–Kutta scheme, central differences with Jameson’s artificial dissipation, geometric multigrid, implicit residual smoothing). The tested models of turbulence were: the one-equation Spalart–Allmaras model, the two-equation Kok’s TNT model and the EARSM model. Unfortunately, none of these models produced results able to serve as a reliable numerical simulation of the first researched case. The main cause is probably the interaction between the turbulent boundary layer and shock-wave formed on the top side of the aerofoil. For the purpose of further investigation the Modified Causon’s scheme was adapted for modelling aeroelastic interaction between aerofoil and flow with one degree of freedom. Results of this simulation correspond better to the real flow, but it is necessary to carry out another tests to find out what is the exact cause of the observed problems.

Flows in convergent channel: comparison of numerical results of different mathematical models

This study deals with numerical solution of 2D unsteady flow of compressible and incompressible viscous fluid in convergent channel for low inlet airflow velocity. Three mathematical models based on system of Navier-Stokes equations for laminar flow are presented and numerical results of the models in the channel are compared. The unsteadiness of the flow is caused by a prescribed periodic motion of a part of the channel wall with large amplitudes, nearly closing the channel during oscillations. The numerical solution is implemented using the finite volume method and the predictor-corrector MacCormack scheme with Jameson artificial viscosity using a grid of quadrilateral cells. The unsteady grid of quadrilateral cells is considered in the form of conservation laws using Arbitrary Lagrangian-Eulerian method. The numerical results, acquired from a developed program, are presented for inlet velocity \(\hat{u}_{\infty }=4.12\) ms\(^{-1}\) and Reynolds number Re\(_{\infty } = 4 \times 10^3\) and the wall motion frequency 100 Hz.

Gas flow control method of recycling gas drilling technology

Recycling gas drilling is a new drilling technology. The paper based on the theory of gas transient flow, a set of theoretic models of gas flow control at the export of annulus are set up in combinations of the principles of the seepage flow mechanics and the characteristics of recycling gas drilling technique. According to the basic laws which the flowing gas should comply with, the properties of various gas flows and the dynamic pressure are portrayed when the gas zone is penetrated. Two control schemes are proposed, namely constant pressure method and constant flow rate method. The Mac Cormack method and the characteristic line method are used to solve the theoretic models. The process of drilling gas zone was analysed using numerical simulation. The results of study can play a positive role in guiding the field operation.