MacCormack Scheme Research Articles

Granular roll waves along a long chute: From formation to collapse

Most simulation findings regarding granular roll waves are based on the analysis of steady uniform flows with an initial artificial perturbation. The difference in the scenario used in our study is that roll waves are generated by applying a zero-mean fluctuant flux inflow condition over the upstream boundary. Only the upstream section of the inclined chute is covered with a sheet of granules with no supplementary application of a boundary condition. Starting with the depth-averaged governing framework, the μ(Fr,h)-dynamic basal friction law, and the μ(I)-rheology-based second-order viscous terms, and adopting a second-order shock-capturing total variation diminishing–MacCormack scheme, we captured self-destabilized roll waves generated in an unsteady granular chute flow. In the simulated unsteady flow, three stages—wave spreading, wave forming and wave-collapse—are identified and discussed. The simulation results are confirmed by observations in laboratory granular flow experiment and natural avalanches from other literatures.

Interval uncertain optimization for interior ballistics based on Chebyshev surrogate model and affine arithmetic

This article proposes an interior ballistic interval optimization method with the consideration of parameter uncertainty. Interior ballistic parameters such as charge mass, web thickness, powder aperture and chamber volume are considered as design variables and described by interval number. A one-dimensional two-phase interior ballistic model is constructed, and the MacCormack scheme is used to calculate the interior ballistic parameters. A Chebyshev surrogate model is constructed to replace the one-dimensional two-phase interior ballistic model and applied to the optimization process. For each set of design variables, affine arithmetic is introduced to calculate the interval bounds of the performance index of the interior ballistic, avoiding nested double-loop optimization and improving computational efficiency. The Pareto-optimal solution is searched for by an improved non-dominant sorting genetic algorithm. An example is given to demonstrate the effectiveness of the proposed method.

A unified depth-averaged approach for integrated modeling of surface and subsurface flow systems

A two-dimensional depth-averaged continuum framework is developed for simulating the interaction of flows outside and within porous media. A unique set of generalized equations applicable to both surface and subsurface systems is derived originating from conservation laws of microscopic variables with first volume-averaging and then integrating vertically. The model is free of ad-hoc coupling as interface conditions are satisfied implicitly owing to the generalized nature of the governing equations. The set of depth-integrated equations (unconditionally hyperbolic) is resolved by employing a Total-Variation-Diminishing MacCormack scheme. The developed model is validated with four one-dimensional experimental test-cases (containing both Darcy and non-Darcy regimes) and applied to a demonstrative two-dimensional scenario having various interface conditions. Results obtained from the proposed framework highlights the potential applicability of the generalized model for diverse time-scales and vertical/ sloping interface conditions.

Comparative Study on Various Numerical Schemes for Solving Nozzle Flow Problems

The inviscid compressible flows represent the flow model of high-speed flow with the viscous effects can be ignored. Such flow phenomena appear when a high Reynolds number flow past trough a streamlined body at relative a low angle attack. The most commercial passenger airplane is operated at such flow condition, as results the aircraft industries have already put their efforts to develop the capability for solving the governing equation of inviscid compressible flow. The governing equation of inviscid compressible flow which is called as the Euler Equation can solved numerically. There are various numerical schemes had been developed and the most of them are dedicated for capturing the presence of shock wave phenomena which is normally appearing at high speed flow. The present work presents a comparative study over various numerical schemes which they are already been developed for solving the Euler equations. Those numerical schemes are the MacCormack Scheme, the Fourth Order Runge Kutta Scheme, The First and Second Order Flux Splitting Steger Warming Scheme, Flux Splitting Bram Van Leer Scheme, the Harten Yee TVD scheme, Roe Sweby TVD scheme, Davies – Yee TVD scheme and the Modified Four Order Runge Kutta and Harten Yee TVD scheme. These nine numerical schemes are applied to the case of flow past through nozzle. The flow phenomena may appear along the nozzle may be the form completely as an isentropic flow or the flow with shock wave. These two flow phenomena are depended the boundary condition of the flow problem under considered. The implementation to case of isentropic flow along the nozzle, those nine numerical schemes works well, and the number of iterations required nearly the same between one with the other except for the Fourth Order Runge Kutta Scheme. In the case of flow with shock wave, the Fourth Order Runge Kutta Scheme, Second Order Steger Warming and the Bram Van Leer Scheme are failing. The presence of shock wave, each numerical scheme requires more iteration number to achieve a convergent solution. The Modified Runge Kutta - Harten Yee scheme may represent an appropriate a numerical scheme for solving the flow problem with the presence of shock wave.

Assessment of streamline-based pressure reconstruction for PIV data in supersonic flows with shock waves

In this paper, a streamline-based (SLB) pressure reconstruction method for particle image velocimetry (PIV) data, which has great potential to reduce the error propagation across relatively strong shock wave (compared with the conventional spatial integration method) and decrease computation cost and operation difficulty [compared with the MacCormack method (Liu et al. 2017, 2019)] in supersonic flows, is comprehensively analyzed and evaluated by numerical and experimental velocity fields. First, the theoretical error model caused by particle relaxation is built by an oblique shock wave. Then, the feasibility of this method in five typical kinds of shock flows is verified based on a simulated Mach interaction velocity field generated by two wedges of 17° and 22°. Besides, the effects of random velocity error, streamline resolution, velocity spatial resolution and out-of-plane component on the pressure reconstruction are also analyzed in detail. Finally, the PIV measurements of oblique and bow shock waves with a free stream Mach number of 2.91 are conducted for experimental validation. These results confirm the feasibility, robustness and good accuracy of the SLB method in supersonic flows with relatively strong shock waves.

Hardware Acceleration of Tsunami Wave Propagation Modeling in the Southern Part of Japan

In order to speed up the calculation of tsunami wave propagation, the field-programmable gate array (FPGA) microchip is used. This makes it possible to achieve valuable performance gain with a modern regular personal computer. The two half-step MacCormack scheme was used herein for numerical approximation of the shallow water system. We studied the distribution of tsunami wave maximal heights along the coast of the southern part of Japan. In particular, the dependence of wave maximal heights on the particular tsunami source location was investigated. Synthetic 100 × 200 km sources have realistic parameters corresponding to this region. As observed numerically, only selected parts of the entire coast line are subject to dangerous tsunami wave amplitudes. The particular locations of such areas strongly depend on the location of the tsunami source. However, the extreme tsunami heights in some of those areas can be attributed to local bathymetry. The proposed hardware acceleration to compute tsunami wave propagation can be used for rapid (say, in a few minutes) tsunami wave danger evaluation for a particular village or industrial unit on the coast.

Numerical simulation for salinity intrusion measurement models using the MacCormack finite difference method with lagrange interpolation

This study aims to develop numerical simulation of one-dimensional advection- diffusion equation. We propose two different methods for salinity intrusion measurement in a stream. In the first method, the forward time centered space (FTCS) is used. In the second method, the MacCormack scheme is applied. The results from these methods can be used as salinity intrusion measurement compared with the first set of exact numbers. Then, we interpolate function at left boundary by using the MacCormack scheme compared with the second set of exact numbers. The results prove that the interpolate function can actually be used. Moreover, the parameters are used and tested with technique of the MacCormack Scheme in order to simulate salinity intrusion measurement methods. After comparing results of both methods with the first set of exact numbers, we found that the MacCormack scheme is the most suitable method.

A three‐level time‐split MacCormack method for two‐dimensional nonlinear reaction‐diffusion equations

SummaryA three‐level explicit time‐split MacCormack method is proposed for solving the two‐dimensional nonlinear reaction‐diffusion equations. The computational cost is reduced thank to the splitting and the explicit MacCormack scheme. Under the well‐known condition of Courant‐Friedrich‐Lewy (CFL) for stability of explicit numerical schemes applied to linear parabolic partial differential equations, we prove the stability and convergence of the method in L∞(0,T;L2)‐norm. A wide set of numerical evidences which provide the convergence rate of the new algorithm are presented and critically discussed.

Feedback control of a spacecraft electro-dynamic thruster

In this paper, the possibility of feedback servo control of a spacecraft thruster's specific impulse by a boundary feedback system is theoretically considered. The motivation to introduce feedback control is two-fold. The first is to stabilize any inherent plasma instabilities and the second is regulate the output specific impulse of the thruster. Two cases are considered: Electro-thermal thrusters and Electro-dynamic thrusters. The chamber inlet temperature in the case of the electro-thermal thruster or the boundary electric field potential responsible for generation of the velocity of the plasma ions in the case of an electrodynamic thruster, are controlled by feedback so as to regulate the thruster's specific impulse. By introducing typical disturbances in the plasma ionization voltage, it is shown using a two-dimensional fluid model and a suitable boundary feedback law, where the chamber inlet temperature or applied boundary potential is proportional to the error in the specific impulse and the desired specific impulse, that the specific impulse of the thruster may be regulated and held constant. The robustness of the control system is numerically tested, by a two dimensional simulation model using McCormacks's method. The Navier-Stokes equations, including the magneto-hydrodynamic variables were discretized and simulated, using the explicit MacCormack method for a typical nozzle domain. The numerical results for the open and closed-loop velocity fields were obtained and the specific impulse was computed from these fields. It was thus shown that not only the stability of the plasma is realized but also that the specific impulse is regulated as desired.

Evaluation of vertical groundwater flux from borehole temperatures: a case study in the Leizhou Peninsula, South China

A least-square solver is presented for groundwater flux estimation using the one-dimensional conduction–advection equation, which was numerically solved using the MacCormack scheme. This approach was utilized to estimate groundwater fluxes from observed borehole temperatures under conditions of surface warming in both rural and urban areas in Leizhou Peninsula, China. The estimated groundwater recharge rate is 0.423 m/year, while the discharge rate is 0.269 m/year. The groundwater recharge rates were obtained in the north part of the study area, and the discharge locations were on Nansan, Donghai and Luzhou Islands, which fit favorably with the regional groundwater flow pattern. Relatively high correlations and low root mean square errors were detected between observed and computed borehole temperature–depth profiles, indicating the estimation is reasonable. The misfit between the observed and computed data might be associated with the uneven distribution of ground surface temperature (GST) and vertical soil heterogeneity. The urban heat island effect contributes an increment of 0.006 °C per year to the GST in the study area, which induces a 0.96 °C higher subsurface temperature in both recharge and discharge areas. The utilization of surface air temperature (SAT) as the upper boundary might underestimate groundwater fluxes, which might lead to inaccurate information for regional groundwater resource management.

Mode conversion and damping of MHD waves in stratified viscous solar atmosphere

We study the mode conversion and damping of MHD waves in a gravitationally stratified low-$\beta $ to high-$\beta $ isothermal atmosphere of the Sun permeated by a uniform vertical magnetic field. We solve the MHD equations by MacCormack method to examine the effect of viscosity on amplitudes of physical quantities such as velocity, mass density, gas pressure and magnetic field before and after the $\beta \approx 1$ layer. It is found that mode conversion occurs at the point $z \approx -1.8$ in $\beta \approx 1$ layer and viscosity does not influence the mode conversion phenomenon. In the presence of viscosity, the amplitudes of vertical and horizontal velocity components, horizontal and vertical magnetic field components, gas pressure and mass density decrease and decay of amplitude is stronger for higher values of viscosity. In the low $\beta $ plasma, damping length decreases with the increase in viscosity but remains almost unchanged in the high $\beta $ plasma. When the driving frequency increases, conversion from slow to fast mode decreases.

A novel three-level time-split MacCormack scheme for two-dimensional evolutionary linear convection-diffusion-reaction equation with source term

ABSTRACT This study presents a three-level explicit time-split MacCormack method to compute approximate solutions of two-dimensional time-dependent linear convection-diffusion-reaction equations with source term. The difference operators split the two-dimensional problem into two pieces so that each subproblem is easily solvable using the original MacCormack approach. Second order accuracy in time and fourth-order convergence in space are achieved by the application of the Taylor series expansion. The proposed algorithm minimizes the computational time, computer memory requirement and is easy to implement. Under a suitable time-step restriction, both stability and error estimates of the numerical scheme are deeply analysed in -norm. Numerical evidences which confirm the theoretical analysis are considered and discussed.

An efficient three‐level explicit time‐split scheme for solving two‐dimensional unsteady nonlinear coupled Burgers' equations

SummaryThis work deals with the numerical solutions of two‐dimensional viscous coupled Burgers' equations with appropriate initial and boundary conditions using a three‐level explicit time‐split MacCormack approach. In this technique, the differential operators split the two‐dimensional problem into two pieces so that the two‐step explicit MacCormack scheme can be easily applied to each subproblem. This reduces the computational cost of the algorithm. For low Reynolds numbers, the proposed method is second‐order accurate in time and fourth‐order convergent in space, whereas it is second‐order convergent in both time and space for high Reynolds numbers problems. This observation shows the utility and efficiency of the considered method compared with a broad range of numerical schemes widely studied in the literature for solving the two‐dimensional time‐dependent nonlinear coupled Burgers' equations. A large set of numerical examples that confirm the theoretical results are presented and critically discussed.

Simulation of Salinity Intrusion in Chao Phraya River by a Developed Mathematical Model and MacCormack Scheme with Cubic Spline Interpolation

The purpose of this research was to develop a mathematical model for controlling salinity intrusion in Chao Phraya River, Thailand, using one-dimensional advection-diffusion equation. There have been many research studies that applied a mathematical model called dispersion model to estimating salinity concentration. Our proposed dispersion model is simple, using very few parameters, but can simulate salinity intrusion adequately. We used MacCormack scheme to approximate salinity intrusion and cubic spline interpolation technique to approximate field data including initial salinity concentrations and the concentrations at the left boundary. The MacCormack scheme and Cubic Spline interpolation technique were quite suitable for approximating real data, and the proposed mathematical model was able to predict salinity intrusion adequately.

Application of the Implicit MacCormack Scheme for Computation of Supersonic Turbulent Jets Using the Parabolized Navier—Stokes Equations

An effective unconditionally stable numerical method for solving equations describing the flow of supersonic turbulent jets is developed. The method does not require the inversion of block tridiagonal systems of algebraic equations. The results obtained according to this method are in fairly close agreement with the calculations results on the basis of other methods and experimental data.

A model for simulation of debris flow sedimentation in slit detention-dam reservoirs

Debris flow is a natural phenomenon that requires appropriate structural measures such as detention dams to reduce hazards resulting from it. The objective of this paper is to present a two-dimensional numerical model to investigate debris flow deposition in the reservoir of the detention slit dams. The governing equations include shallow water equations considering the bed level variation, sediment continuity, and debris-flow transport equations. Shallow water equations were discretized using the MacCormack scheme and then Banihabib Scheme was applied for solving sediment continuity equation. The governing equations were solved in an iterative procedure and as a result, flow properties and bed level variations were obtained in silt dam reservoir. The model performance was evaluated against experimental and numerical models in two aspects including the outflow hydrograph and reservoir bed level variations or trapping efficiency. The model performed quite well in outflow hydrograph simulation, and the comparison of the bed level variations or trapping efficiency illustrated that the simulated values were acceptable. A sensitivity analysis of the parameters was carried out to analyze the influence of parameters variation on outflow and sedimentation volume. The results of this sensitivity analysis revealed sedimentation is more sensitive to the parameter variation than outflow hydrograph. Generally, the proposed numerical method can provide insights on the debris flow deposition in slit dam reservoir, which could be useful for slit dam design.

Experimental comparison of PIV-based pressure measurements in supersonic flows with shock waves

This paper evaluates the performance of the novel MacCormack method for particle image velocimetry (PIV)-based pressure reconstruction in real supersonic PIV experiments that include oblique shock waves generated by three different wedges of 10°, 15°, and 20° with a free stream Mach number of 2.91. Besides, the precision of the widely used spatial integration methods and Poisson method for compressible flows is compared with that of the MacCormack method in the supersonic flows. The results show the good performance of the MacCormack method and its significant advantage in complex shock flows. The results are also consistent with the conclusions based on the synthetic velocity fields by (Liu et al 2017 Exp. Fluids 6 1–22), and further confirm the reliability of this method. The non-conservative spatial integration method usually achieves larger pressure distributions behind shock waves than theoretical values while the conservative spatial integration method performs much better but still fails to obtain accurate results in the strong shock flow. The Poisson method does not have obvious optimization compared to the spatial integration method. However, the pressure field reconstructed by the time-marching MacCormack method is constantly in good agreement with theory, which demonstrates the feasibility and advantage of this method in complex supersonic flows with strong shock waves.

Seismic wavefield modelling in two‐phase media including undulating topography with the modified Biot/squirt model by a curvilinear‐grid finite difference method

ABSTRACTIn this paper, we deduced the corresponding first‐order velocity–stress equation for curvilinear coordinates from the first‐order velocity–stress equation based on the modified Biot/squirt model for a two‐dimensional two‐phase medium. The equations are then numerically solved by an optimized high‐order non‐staggered finite difference scheme, that is, the dispersion relation preserving/optimization MacCormack scheme. To implement undulating free‐surface topography, we derive an analytical relationship between the derivatives of the particle velocity components and use the compact finite‐difference scheme plus a traction‐image method. In the undulating free surface and the undulating subsurface interface of two‐phase medium, the complex reflected wave and transmitted wave can be clearly recognized in the numerical simulation results. The simulation results show that the curvilinear‐grid finite‐difference method, which uses a body‐conforming grid to describe the undulating surface, can accurately reduce the numerical scattering effect of seismic wave propagation caused by the use of ladder‐shaped grid to fit the surfaces when undulating topography is present in a two‐phase isotropic medium.

Assessment of some numerical methods for estimating the parameters of the one-dimensional advection\u2013dispersion model

This study appraised optimisations of numerical solutions of the one-dimensional advection–dispersion model (AD-Model) to synthetic data generated using an analytical solution. The motivation for the work was to identify reliable methods for estimating stream solute transport parameters from observed events in small rivers. Numerical solutions of the AD-Model must contend with several effects that might disturb the solution, with the introduction of numerical diffusion and numerical dispersion being particularly important issues. This poses a problem if physical dispersion is being identified by optimising model coefficients using observations of solute transport from field experiments. The discretisation schemes used were the Backward-Time/Centred-Space, Crank–Nicolson, Implicit QUICK, MacCormack and QUICKEST methods. Optimisations were obtained for several grid resolutions by keeping the time step constant whilst varying the space step: the range of Peclet number, Pe, was 1.5–12.0. Generally, increasing the space step led to poorer estimated coefficients and poorer fits to the synthetic concentration profiles. For Pe < 5 only Crank–Nicolson, MacCormack and QUICKEST gave reliable optimised dispersion coefficients: those from Backward-Time/Centred-Space and Implicit QUICK being significantly underestimated. For Pe > 5 Crank–Nicolson and MacCormack gave slightly overestimated dispersion coefficients whilst the other methods gave significantly underestimated dispersion coefficients. These findings were generally consistent with the known presence of numerical diffusion and numerical dispersion in the methods.

NUMERICAL SIMULATIONS FOR NON CONSERVATIVE HYPERBOLIC SYSTEM. APPLICATION TO TRANSIENT TWO-PHASE FLOW WITH CAVITATION PHENOMENON

A numerical method for simulating transient flows of gas-liquid mixtures is proposed. The mathematical model, established for a suspension of gas bubbles in liquid, includes an equation taking into account the relative velocity between the gas and liquid. A numerical technique based on the Mac Cormack scheme combined with the method of characteristics is presented. Theoretical results for transients initiated by a rapid closing valves are compared with measurements. A good agreement is found particularly for large values of initial dissolved gas concentration.