MacCormack Scheme Research Articles

Modelling Time-Varying Subsurface Seepage in Unconfined Sloping Aquifers: A Comprehensive Analysis of Groundwater Flow with Nonlinear Advection-Diffusion Equations

A stream-aquifer subsurface seepage model is developed to estimate fluctuations in the water table in an unconfined sloping aquifer which receives subsurface seepage from adjacent water bodies and time-varying vertical recharge. The groundwater flow is approximated by a nonlinear advection-diffusion equation subjected to time-dependent boundary conditions. Two numerical schemes namely Mac Cormack scheme and Du Fort Frankel scheme are used for solving the nonlinear flow equations. A Model is developed by solving the nonlinear Boussinesq equation using the Laplace transform technique. The combined effects of bed slope, stream rise rate, and vertical recharge are demonstrated using an illustrative example. To assess the validity of linearization, numerical solutions are compared with analytical solutions using an extensive value of aquifer parameters. Simulation is done using hypothetical data and results are compared with the previous literature. The results indicate that the approximate analytical solution agrees with the numerical solution for bed sloping angle within the range – 5 deg to 5 deg

Assessment of Multi-Physical Fields Reconstruction Approaches in Three-Dimensional Supersonic Flow with Shocks

The paper evaluates the application of several multi-physical fields reconstruction approaches based on the Tomographic Particle Image Velocimetry (PIV) in three-dimensional supersonic flows with shock waves, including the pressure, temperature, and density. The MacCormack method and the Flux Vector Splitting method with outstanding performance in two-dimensional supersonic flow are extended into three-dimensional forms. The conventional Poisson method is also considered because of its widespread application and high accuracy in subsonic and transonic flow. All of these approaches are established by the conservative Navier-Stokes equations and solved by the time-marching iteration process, combined with the perfect gas law and adiabatic flow assumption. The performances are evaluated by numerical velocimetry data. To simulate the typical three-dimensional features, the cases of uniform freestream at Mach 0.78 and 3 past a cone with a 20° half-angle are selected to obtain a sufficient spanwise velocity component. The results confirm the feasibility of the PIV-based reconstruction methods in the conical flow field. The conventional Poisson method performs well only in the subsonic case while the Flux Vector Splitting method has a better performance in supersonic flow, including higher accuracy, stability, and efficiency, with a low-level root-mean-square error of 0.449% and local maximum relative error of 1%.

Numerical modeling and simulation of water hammer phenomena using the MacCormack method

ABSTRACT Water hammer refers to pressure fluctuations caused by changes in fluid speed in hydraulic systems. This research proposes a model using the MacCormack method to study and control water hammer, specifically focusing on managing valve closure to reduce shock wave pressure during transient flow. Numerical simulations compare linear and quadratic closure laws for both rapid and gradual valve closure, with the goal of identifying the optimal laws. The findings show that with faster closure times, the overpressure at the valve section decreases linearly in the second quarter of the return period (t4). The application of the quadratic law results in reduced pressures at the valve compared to the linear law, with a maximum pressure difference of 0.05 bar between the highest and lowest values. When searching for the optimal valve closure law to mitigate overpressure, it is found that the exponent ‘m’ falls within the range of 1.117–1.599. As the slow closing time increases gradually, the range of variation for ‘m’ decreases. Furthermore, in the case of underpressure prevention, the exponent ‘m’ ranges from 0.95 to 1.419, and the range of ‘m’ remains relatively constant with the increase in closing time.

A dynamic one-dimensional model for simulating unsteady air-water stratified flow in sewer pipes.

Ventilation is paramount in sanitary and stormwater sewer systems to mitigate odor problems and avert pressure surges. Existing numerical models have constraints in practical applications in actual sewer systems due to insufficient airflow modeling or suitability only for steady-state conditions. This research endeavors to formulate a mathematical model capable of accurately simulating various operational conditions of sewer systems under the natural ventilation condition. The dynamic water flow is modeled using a shock-capturing MacCormack scheme. The dynamic airflow model amalgamates energy and momentum equations, circumventing laborious pressure iteration computations. This model utilizes friction coefficients at interfaces to enhance the description of the momentum exchange in the airflow and provide a logical explanation for air pressure. A systematic analysis indicates that this model can be easily adapted to include complex boundary conditions, facilitating its use for modeling airflow in real sewer networks. Furthermore, this research uncovers a direct correlation between the air-to-water flow rate ratio and the filling ratio under natural ventilation conditions, and an empirical formula encapsulating this relationship is derived. This finding offers insights for practical engineering applications.

Comparison of Computational Methods and Approaches Applied in Formulation of Boundary Conditions in Lagrange’s Ballistic Problem

This paper compares algorithms for constructing a differential solution to the gas dynamics equations in the surrounding of a moving boundary. In order to compare the algorithms, the Lagrange problem, also known as the piston problem, was chosen as a test problem. A comparison was made between the author’s algorithm based on the method of characteristics and the classical approach with a fictitious cell using the solution of the Riemann problem. Calculations were carried out for the case of subsonic and supersonic gas motion. Comparisons were made on a fixed grid with a dynamically expanding cell (Euler grid) and on a uniformly expanding grid (Arbitrary Lagrangian-Eulerian – ALE grid). The problem was solved using numerical schemes of second order in time and space Lax - Wendroff type: the Richtmyer scheme and the MacCormack scheme. The results of calculations using the characteristics method were used as a benchmark solution. The comparative analysis carried out allows conclusions to be drawn regarding the accuracy of the individual approaches, as well as the scope of their applicability.

Building Treatments in Numerical Modeling of Dam Break Induced Flow in Urban Area

Dam break is one of the most dangerous disasters, which cause a large mass of water from the reservoir is suddenly released to the downstream. Although dam break events are still rare, the occurrences can cause severe damage to the downstream area, especially if the downstream is an urban area. Urban flood due to dam break has a different characteristic compared to the flood in nature. Modeling the flood is good way to learn the urban flood characteristic. Therefore, further studies are needed to develop an urban dam break model. Numerical modeling is one of the most effective methods to simulate the flood. In numerical modeling, buildings can be modeled with 3 approaches, as a wall boundary, as an area with high manning value, and as a contour. This paper discusses about two-dimensional flow of a dam break which flows through a sloping channel and single building, particularly comparing 3 approaches to modeled the building. The case is based on laboratory experiment of previous study. The model was developed with 2D Shallow Water Equations (SWE) which are based on the conservation of mass and momentum derived from Navier Stokes equation. The equations then discretized using Mac-Cormack numerical scheme. Numerical filter is used to reduce the numerical instability or oscillation in the model. The numerical model results are then compared to the experimental result from previous study. The accuracy of the numerical model was checked using Root Mean Square Error (RMSE) to determine the best methods to modeled the building. The results indicate that modeling the building as a wall boundary have the best results with lowest RMSE value and also shown the closest comparison of water level compared to the experimental results. By conducting this study, it is hoped to assist the urban dam break modeling in the future.

Computational Study of Supersonic Flow Over a Flat Plate With Protrusion

Motivated by a developmental approach of using micro-actuated surface protrusions to control/maneuver slender bodies in supersonic flight, this paper presents a detailed study of a two dimensional laminar supersonic flow over a flat plate with a surface protrusion. The flow field is computed by solving the Navier-Stokes equations using the finite difference method with the particle velocity upwinding scheme (PVUS) for spatial discretization, and the explicit Mac Cormack scheme for temporal integration. A range of free stream Mach numbers (2.0 - 4.5), Reynolds numbers (1000 - 100000) and protrusion heights (0.866% to 8.66% of the characteristic length) has been considered for a thorough parametric study. The parametric study indicates that the oblique shock structure and strength are influenced by all the variables to varying extents. However, the shock location is substantially altered by the protrusion height, Reynolds number, and protrusion shape, while the influence of Mach number is only marginal. An increase in the protrusion height results in an increased wall pressure as well as increased separation lengths on both sides of the protrusion. In contrast, an increase in Mach number increases the wall pressure, but moves the separation point marginally towards the protrusion. Increase in Reynolds number and protrusion bluffness (triangular → trapezoidal → rectangular) increases separation lengths on both sides of the protrusion. In addition to taking a close look at the flow physics, particularly in the protrusion vicinity, considerable attention has also been devoted to important design parameters such as wall pressure, skin friction and the overall normal and tangential force coefficients.

Numerical Study of Benchmark Experiments of Supersonic Compressor Cascades

Benchmark cases of turbulent supersonic flow through compressor cascades were computed using a modern upwind scheme (Roe with MUSCL pre-processing), with the k-ω turbulence model, and the classical MacCormack method. The test cases derive from experiments through ARL SL 19 blade cascades at Detroit Diesel Allison (DDA) at the design Mach number, and at an off-design condition from ONERA. A verification of the codes was obtained by computing an inviscid flow through a wedge cascade at similar conditions. Predictions are in very good agreement for the DDA test case for the blade surface pressure distributions. Cleaner solutions were obtained using the Roe scheme. The MacCormack solution exhibits oscillations corresponding to oscillations of the shock-induced separating boundary layer on the pressure surface near the trailing edge. Wake flow Mach number distributions were very well predicted, but not flow angle distributions. Predictions are less accurate for the ONERA test case which was at a slightly higher Mach number and which moves shock locations sufficiently to cause a more sensitive response from the suction surface boundary layer.

A Predictor–Corrector Compact Difference Scheme for a Nonlinear Fractional Differential Equation

In this work, a predictor–corrector compact difference scheme for a nonlinear fractional differential equation is presented. The MacCormack method is provided to deal with nonlinear terms, the Riemann–Liouville (R-L) fractional integral term is treated by means of the second-order convolution quadrature formula, and the Caputo derivative term is discretized by the L1 discrete formula. Through the first and second derivatives of the matrix under the compact difference, we improve the precision of this scheme. Then, the existence and uniqueness are proved, and the numerical experiments are presented.

Rusanov’s Third-Order Accurate Scheme for Modeling Plasma Oscillations

A modification of the well-known Rusanov third-order accurate scheme is proposed for modeling nonrelativistic oscillations of a cold plasma. Only first- and second-order accurate schemes were used earlier for similar computations in Eulerian variables. In the case of a test problem with a smooth solution, the errors of the constructed scheme are investigated and compared with the errors of the MacCormack scheme. For the problem of free plasma oscillations induced by a short intense laser pulse, numerical results are presented concerning the conservation of energy and an additional function for both schemes and the accuracy of the electron density in the center of the domain. It is concluded that the Rusanov scheme is superior theoretically, although the MacCormack scheme is more suitable for applications, primarily, for computations of long-lived processes and cold plasma oscillations similar to actual ones. A theoretical analysis of approximation and stability, together with experimental observations of quantitative characteristics of the error for the most sensitive quantities, significantly improves the reliability of the computations.

Pulsating pressurization of two-phase fluid in a pipe filled with water and a little gas

Although two-phase flows containing gas and water have received extensive attention, the pulsating pressurization effect of a two-phase fluid in a pipe is unclear and the influence of the gas-phase content has not been revealed. This paper discusses the pulsating pressurization of such a two-phase fluid. First, the two-phase Navier–Stokes equations are derived and an algorithm is developed based on MacCormack's method. The reliability of the algorithm is examined and validated using Poiseuille's theory and existing experimental two-phase flow data. Finally, the influence of several key factors is discussed, including the gas-phase fraction and pipe slenderness. Our results show that a significant pulsating supercharging phenomenon occurs when the gas-phase fraction is less than 10−3. When the gas-phase fraction is greater than this critical value, the pulsating supercharging effect decreases significantly with the increasing gas-phase fraction. The equivalent elastic modulus of the two-phase fluid rapidly decreases as the gas-phase fraction increases, and the pressure disturbance is absorbed by the gas bubbles, causing an apparent weakening of the pulsating supercharging effect. Thus, decreasing the gas-phase content can enhance the pulsating supercharging effect. The pipe slenderness has a very limited influence on the pulsating pressurization process, and the maximum reduction is only 7.3% for slenderness ratios of up to 2000. Moreover, we derive and propose a new mathematical expression for the inlet boundary that is applicable to gas–liquid two-phase flows. To our knowledge, this paper extends the pulsating pressurization range from the single-phase to two-phase fluid for the first time and reports different physical phenomena and regularity. The present research clarifies the pulsating pressurization phenomenon in two-phase flows, providing a valuable reference for pulsating pressurization design.

Numerical modeling of dam break induced flow through multiple buildings in an idealized city

This study presents two-dimensional dam break numerical modeling through an urban area. There are three approaches to defining a building in numerical modeling, as a wall boundary, contour, and Manning, which have been widely used. However, there is no research that compares and evaluates the accuracy of each approach. Thus, this study aims to determine the best approach, in term of accuracy, and evaluate its reliability by comparing the numerical modeling approach's results to experiment results. The model was developed based on the Saint-Venant equations, then solved using the MacCormack method with a numerical filter. The authors first utilized a single-building experiment to determine the best approach. The results indicated that the wall boundary approach had the highest accuracy based on MDAPE, with the lowest value of 13.47%, followed by Manning with 16.48%, and Contour with 21.46%. A multiple-building case is then used to evaluate the best approach's reliability. The results show high similarity to the experiment results. However, the wall boundary approach cannot be used when the building is submerged or has a height variation, since these approaches cannot define the building's height. Moreover, Manning approach cannot be considered as a building approach, since the approach does not reflect the behaviour of the building. Thus, the authors recommend the wall boundary approach when the building does not have a height variation and is not submerged, and the contour approach otherwise.

Free-surface elastic wave injection and wavefield reconstruction with applications to elastic reverse time migration

Accurately injecting and extrapolating multicomponent seismic data is a major step in elastic reverse time migration (RTM) and full-waveform inversion. The theoretical basis of this process is the seismic representation theorem. We propose a numerical approach based on the collocated-grid MacCormack finite-difference (FD) scheme to implement the injection and extrapolation. The MacCormack FD scheme defines wavefield parameters on the same grid. When dealing with the boundary condition in the representation theorem, it is more advantageous than the commonly used staggered-grid FD scheme which defines the stress and velocity on offset grids; thus, interpolations are required to link the boundary values to nearby nodes. In contrast, with the MacCormack scheme, the traction and particle velocity data can be directly injected into the boundary nodes as equivalent concentrated forces, force dipoles, and double couples. The accuracy of this approach is validated by comparing the regenerated wavefield with the original wavefield. Our primary purpose is to develop an injection method for elastic RTM, but this approach also can be used to numerically investigate other interesting issues in the elastic wave injection, including errors and artifacts caused by injection from unclosed boundaries, incomplete boundary values, and uncertain velocity models. In particular, seismic data are usually acquired on the ground surface where the traction-free condition is naturally maintained. We simulate a land acquisition scenario by generating a synthetic multicomponent data set on a traction-free surface. The reconstructed wavefields from velocity data are compared with the original forward-propagated waves to evaluate the accuracy of the boundary treatments. Two multicomponent elastic RTM examples are presented to demonstrate the applications of the proposed injection method. By combining the regenerated receiver-side wavefield and the vectorized imaging condition, the PP, PS, SP, and SS images are calculated.

Dry and wet boundary treatment and improvement of a TVD‐MacCormack scheme in shallow water flow

AbstractTo solve shallow water equation, this paper proposes a simple and easy‐to‐operate dry‐wet boundary treatment method based on the total variation diminishing (TVD)‐ MacCormack scheme. The method requires to judge dry and wet nodes before the calculation of prediction step and correction step, respectively. Then, the dry nodes are fictitiously celled and the topographic variables are reset. Moreover, previous researches show that when the same differential scheme was used, the left and right traveling waves showed over predicted computational fluxes during the downstream dry bed flow evolution, which led to distorted values and non‐real physical phenomena. To solve the problem, the difference scheme for prediction step and correction step is modified, and a new finite difference scheme improvement method is proposed. Finally, the numerical solutions are compared with the analytical solution results by five classical cases to verify the rationality of the proposed method in this paper.

MacCormack method for solving one-dimensional bed-load sediment transport model

In this work, we investigate the numerical solution of one-dimensional bed-load sediment transport model using two steps finite difference method which so-called MacCormack method. Bed-load sediment transport model is composed by the shallow water equation and Exner equation. The Meyer-Peter and Muller (MPM) formula and Wu formula will be used to determine the Grass factor of the bed-load sediment transport. These governing equations will be discretized into predictor and corrector steps of the MacCormack method. The numerical results of the MacCormack method will be validated with an analytical solution of the bed-load sediment transport model. In addition, the MacCormack solution will also be compared with experimental solutions and another numerical method solutions that have existed previously. The numerical results based on MacCormack method give excellent results in which the numerical and the analytical results are hardly differentiated with RMSE of around 00042 or 4,2 .

Approach to the Numerical Study of Wave Processes in a Layered and Fractured Porous Media in a Two-Dimensional Formulation

A new approach to the numerical study of arbitrary waveform impulses in a layered porous and fractured-porous medium in a two-dimensional formulation has been developed. Layers can have different characteristics and contain fractures. A computer implementation of the mathematical model based on the finite-difference MacCormack method has been completed. A number of test calculations have been carried out confirming the reliability of the numerical solutions obtained. The possibility of using the proposed approach to solve problems of wave dynamics is shown.

The Optimal Sine Pulse Frequency of Pulse Hydraulic Fracturing for Reservoir Stimulation

Pulse hydraulic fracturing (PHF) is a key technique for reservoir stimulation. PHF can well accelerate the rupture of rock. However, the supercharging mechanism of PHF is not fully understood. The main reason is that the pressure distribution and its variation, especially the peak pressure characteristics, are unclear inside the pipe and fissure. The present research focuses on the sine pulse applied at the inlet of a pipe or fracture to reveal the variation regularity of peak pressure with the pulse frequency, amplitude, pipe length, diameter and wave speed. First, the weakly compressible Navier–Stokes equations were developed to simulate the variation of fluid pressure. The computation codes were developed using the MacCormack method validated by the existing experimental data. Then, the sine pulse effect was studied inside the pipe and fissure. Last, a new frequency model was built to describe the relationship between the optimal pulse frequency, wave speed and pipe length. The results show that there is a family of frequencies at which the peak pressure of the endpoint can be significantly enhanced and that these frequencies are the optimal pulse frequency. It is found that the optimal pulse frequency depends on the pipe or fissure length and wave speed. At the optimal pulse frequency, the peak pressure at the endpoint can be increased by 100% or more, and the cavitation phenomenon occurs. However, the peak pressure decreases when with the decrease in the pipe diameter and fissure departure due to the friction drag effect of the wall. These new landmark findings are very important for the PHF technique. In addition, a new universal frequency model is built to predict the optimal sine pulse frequency. The present research shows the variation regularity of the fluid pressure inside the pipe and develops a sine frequency-controlled method, providing a potential guide for reservoir stimulation.

Pulse supercharging phenomena in a water-filled pipe and a universal prediction model of optimal pulse frequency

A water hammer is an important natural phenomenon that can be used to fracture rock with enhanced local water pressure. The oscillatory injection of a column of water can be used to make a pipe water hammer. However, the optimal injection frequency to create a water hammer has not yet been found. The main reason for this is that the distribution of fluid pressure and its variation are unclear inside a pipe. In this study, we demonstrate for the first time that there can be significant supercharging phenomena and a law governing their appearance in a water-filled pipe. We first find the optimal pulse frequency to reproduce the supercharging process. We also clarify the supercharging mechanism at an optimal frequency. First, a simplified pipe model is adopted, and weakly compressible Navier–Stokes equations are developed to simulate the flow of water in pulse hydraulic fracturing (PHF). The computation code is developed using the MacCormack method, which has second-order accuracy in time and space. The computation codes and program are validated using experimental data of weakly compressible flows. Then, the square pulse effects are studied inside a pipe, including the effects of pulse frequency, amplitude, pipe length, diameter, and wave speed. Finally, a new universal frequency model is built to describe the relationship among optimal pulse frequency, wave speed, and pipe length. The results show that in square PHF, there is a family of frequencies for which the fluid peak pressure can be significantly enhanced, and these frequencies include the optimal pulse frequency. The optimal frequency of a square pulse depends on the pipe length and wave speed. At the optimal pulse frequency, the maximum peak pressure of the fluid can be increased by 100% or more, and cavitation occurs. These new landmark findings are very valuable for understanding pulse supercharging in an internal water wave. In addition, a new universal frequency model is built to predict optimal pulse frequency. This study identifies an evolution law of peak pressure inside a pipe and proposes a practical frequency-control model for the first time, which can provide a theoretical guide for PHF design.

Effect of sediment transport, flow depth and infiltration on soil moisture profiles in irrigation furrows

A numerical model was used to analyze the effect of sediment transport and flow depth on soil moisture profiles from a simulation of the hypothetical irrigation furrows. The Saint Venant and sediment continuity equations were solved by using MacCormack scheme based on finite difference method to analyze the overland flow and sediment transport, and Richards equation was used to analyze the subsurface flow, which was solved by using a mass conservative fully implicit finite difference method. Flow depth significantly affected the moisture profiles in irrigation channels while the effect of sediment-laden water on the moisture profiles was observed insignificant. The model was used to study the effect of infiltration on moisture profiles and soil water retention curve (SWRCs) in irrigation furrows. It was observed that infiltration significantly affected and altered the moisture profiles and soil water retention curve (SWRC). Among the different soil parameters (av and nv ) and saturated hydraulic conductivity () significantly affected the moisture profiles, while and influenced the soil water retention curve (SWRC) as compared to other retention parameters.

THE SIMULATION OF ONE-DIMENSIONAL SHALLOW WATER WAVE EQUATION WITH MACCORMACK SCHEMES

Many practical problems can be modeled using the one-dimensional shallow water wave equation. Therefore, the solution to the one-dimensional shallow water wave equation will be discussed to solve this problem. The research method used was the study of literature related to the shallow water wave equation and its solution method. The one-dimensional shallow water wave equation can be derived from the law of conservation of mass and the law of conservation of momentum. In this study, one of the finite difference methods will be discussed, namely the MacCormack method. The MacCormack method consists of two steps, namely the predictor and corrector steps. The MacCormack method was used to perform numerical simulations of the pond and tsunami models for one-dimensional (1D) shallow water wave equations with flat and non-flat topography. The simulation results showed that the channel's topography could affect the water surface's height and velocity. At the same time, a channel with a non-flat topography had a slower water velocity than the water velocity of a channel with a flat topography.