McCormack Scheme Research Articles

Numerical investigation of multiphase flow in a channel

Due to the occurrence of failures are divided into design, production and operational. Structural failures occur due to imperfection or violation of the established rules and (or) norms for the design of an object. Production - arise as a result of improper assignment of technological processes for the manufacture or restoration of parts and assembly of a car or are the result of a violation of the accepted technology, unsatisfactory quality of the material of parts or coatings applied to them, the use of insufficiently accurate measuring instruments, failure to meet technical requirements for the manufacture and assembly of elements, as well as the manufacture and assembly of elements and the object as a whole. In the article, basic information about the principles of calculations for solving a system of multiphase Navier–Stokes equations using the control volume method is provided. It is shown that the relationship between velocities and pressure is found using the procedure of SIMPLE. For the numerical solution of this problem, the McCormack scheme was applied. A comparison has been made between each other and with the experimental data. For turbulence, the Spalart–Allmares model was used. And also in the work the movement of sedimentary fluid in different diameters and different Reynolds numbers was studied.

Several different ways to increase the accuracy of the numerical solution of the first order wave equation

The article presents several different ways to increase the accuracy of the numerical solution of differential equations. The comparison of schemes with different accuracy for the first order wave equation problem is presented. The schemes of the first order of upwind scheme, the second order of accuracy of McCormack, the third order of accuracy of Warming–Cutler–Lomax, and the scheme of the fourth order of accuracy of Abarbanel–Gotlieb–Turkel were applied. The condensed computational grids for the McCormack scheme are used, and the results using an adaptive grid for the McCormack scheme were compared.

An investigation of induced free-surface wave oscillations in prismatic open-channel

ABSTRACT This paper reported on the free-surface wave behavior in a rectangular open-channel, induced by sluice-gate maneuvers. Numerical computations used the McCormack scheme to solve the one-dimensional extended St-Venant model embedding the Prandtl power-law momentum correction coefficient. The fundamental pulsation of the hydraulic system was inspected using the analogy with the water-hammer theory, known in pressurized-pipe flows. Different flow scenarios were reported; including the superposition between (i) downstream water-hammer maneuver and sine excitation of upstream flow depth and (ii) sinusoidal fluctuations of upstream lateral inflow and downstream flow-depth. Results highlighted that such sluice-gates maneuvers involved severe scenarios leading to significant amplifications of the depth peak and crest values above the initial value.

Numerical investigation towards the improvement of hydraulic-jump prediction in rectangular open-channels

ABSTRACT This study probed into the numerical characterization of a rapidly varied free-surface flow behavior including a hydraulic jump onset. The extended One-Dimensional St.-Venant Equations, embedding the Boussinesq add-on term, were employed to describe a free-surface wave behavior. The numerical computations were based on a second-order precision in time and fourth-order in space explicit finite-difference scheme ((2/4)-dissipative numerical scheme). The computed results were then compared with those issued from the McCormack – based alternative solver and experimental data quoted in the literature. The findings revealed that the developed solver allowed an accurate prediction of the amplitude and location characteristics of the hydraulic jump. In addition, they suggested that the (2-4)-dissipative scheme-based algorithm was more practical than the alternative shock capturing methods in terms of prediction accuracy, implementation simplicity, and calculation time consumption. Unlike the McCormack scheme – based solver, the proposed algorithm provided numerical signals free of numerical oscillations in the vicinity of the steep gradient involved by hydraulic jumps. Yet, it required more computational time than the McCormack scheme – based alternative.

Improved Numerical Method for Aw-Rascle Type Continuum Traffic Flow Models

Continuum traffic flow models are essentially nonlinear hyperbolic systems of partial differential equations. Except for limited specific cases, these systems must be solved numerically. Mathematical structure of continuum models can be different for each class of models. As a result, suitable numerical schemes for some classes may not be efficient for others. In this study, an improved numerical method is proposed for a class of second-order traffic flow models. The method is based on McCormack scheme, which is a widely-used method for non-homogeneous second-order traffic flow models, but fails to produce reasonable results for homogeneous models including Aw-Rascle type models which are the focus of this paper. It is shown that this is mainly due to spurious numerical oscillations. Smoothing methods to overcome these issues are studied and applied. Central dispersion and artificial viscosity (AV) methods are incorporated into the standard McCormack scheme and tested. To reduce numerical diffusion, a total variation diminishing Runge-Kutta time stepping scheme is applied. Results show the capability of the proposed methods, and especially the AV method, to eliminate the oscillations of the standard McCormack scheme as well as controlling numerical diffusion.

PECULIARITIES OF MATHEMATICAL MODELING OF THE FLOW WHEN SELECTING THE GENERAL SIZES OF A BRIDGE CROSSING WITH GROUP HOLES

The problems of mathematical modeling of hydrodynamics of open flows are analyzed in the article when general sizes of bridge crossings with group holes are assigned. Most of the existing methods for calculating branched flows are based on the use of one-dimensional models, it does not allow to describe sufficiently the hydrodynamic processes occurring in zones with separated flows. Therefore, in the calculation of bridge crossings with group holes, two-dimensional models that take into account the features of the morphology of natural watercourses, the uneven distribution of the average velocities along the vertical, the phenomenon of flow separation, as well as the change in rate along the flow due to the separation of the flow is suggested. The existing experimental and theoretical studies are analyzed, the flow structure and the physical model of the flow branching process in the zone of influence of bridge crossings with group holes are considered. Based on the fundamental equations of flow transfer, a mathematical two-dimensional model of the flow of open flows is presented, which takes into account the influence of secondary currents. To substantiate the zone of influence of bridge crossings with group holes, an equation for vorticity is proposed that reflects the distribution of vortex structures. For the closure of the equations of motion and continuity, algebraic relations for the Reynolds stresses in conjunction with the two-parameter k-ε model for depth-averaged values are used, which is modified to take into account the effect of the curvature of the flow. The method of numerical realization of the proposed mathematical model is based on the predictor-corrector method using the McCormack scheme, modified by splitting the model equations with respect to spatial coordinates and time. The method of successive upper relaxation based on the Gauss-Seidel iterative method is used to realize the algebraic relations of turbulent stress transfer. The conclusion is made about the expediency of practical application of this model

Experimental Verification of Storm Sewer Transient Flow Simulation

Abstract The paper focuses mainly on laboratory investigations of transient and transcritical flow in a single pipe of a sewer system. The aim of this paper is to present a comparison between pressure values calculated by an improved McCormack scheme and those measured at the hydraulic laboratory of the Gdańsk University of Technology, which were observed inside a pipe in an experiment for water flow with pressurization. The analysis proves that the numerical approach applied to flow simulation is a sufficiently accurate and reliable technique for predicting the basic parameters of storm sewer flow

Lax-Wendroff and McCormack Schemes for Numerical Simulation of Unsteady Gradually and Rapidly Varied Open Channel Flow

Abstract Two explicit schemes of the finite difference method are presented and analyzed in the paper. The applicability of the Lax-Wendroff and McCormack schemes for modeling unsteady rapidly and gradually varied open channel flow is investigated. For simulation of the transcritical flow the original and improved McCormack scheme is used. The schemes are used for numerical solution of one dimensional Saint-Venant equations describing free surface water flow. Two numerical simulations of flow with different hydraulic characteristics were performed - the first one for the extreme flow of the dam-break type and the second one for the simplified flood wave propagation problem. The computational results are compared to each other and to arbitrary solutions.

A Numerical Model for Two-Dimensional Flood Routing in Complex Terrains

In this paper, a new powerful numerical hydrodynamic in-house model is presented and tested. The model simulates flood routing in two dimensions. It is based on the solution of Shallow Water Equations using the Finite Difference Method according to the explicit McCormack numerical scheme which has shock capturing capability. The innovation of the proposed model lies in the modification of McCormack scheme by incorporating artificial viscosity through a diffusion factor in order to avoid oscillations as proposed by various researchers. Additionally, a threshold of water depth is introduced in order to distinguish the wet and dry cells of the computational domain. The model is capable of producing maps for the inundation extent, water depths and depth-averaged water velocities. Finally, the paper presents extensive testing of the model by comparison with analytical solution, experimental results and with the output of another software package in real world flood simulation studies.

An efficient and flexible solver for the simulation of the morphodynamics of fast evolving flows on coarse sediment beaches

SUMMARYA well‐balanced total variation diminishing–McCormack scheme is used to simulate the fast evolving flow on a mobile coarse sediments bed. The scheme is chosen because of its shock capturing capabilities and its relative simplicity, which allow different sediment transport formulae to be slotted in easily. A typical example of the kind of flows treated here is bore‐driven wave run‐up. The analogy with a dam‐break on a mobile bed is used here to analyze the performance of the model. The model solves the nonlinear shallow water equations coupled with the Exner sediment balance equation for the mobile bed. Quasi‐analytical solutions to this problem for different expressions for instantaneous sediment discharge formulae are used to test the performance of the scheme. Together with the existing solution for the Grass formula, a further solution is obtained for a different formula. Numerical tests were also carried out for a further formula that is an industry standard. The agreement of the results with the solutions is very good and consistent results were obtained for all the formulae tested. Copyright © 2011 John Wiley & Sons, Ltd.

상류이송형 McCormack 기법의 개발

본 연구에서는 McCormack 기법의 2차 정확도 및 단순성의 장점과 불연속 흐름을 해석할 수 있는 상류이송기법의 장점을 결합하여 상류이송형 McCormack 기법을 새로이 개발하였다. 이 기법은 생성항을 효과적으로 처리할 수 있는 장점도 지니고 있다. 본 연구에서 개발된 기법을 해석해를 가진 가상적인 하도에 적용한 결과 기존 McCormack 기법에서 발생하던 수치진동없이 해석해를 잘 재현할 수 있는 것으로 나타났다. 또한 실제 하천에 대한 적용을 위해 하상과 하폭의 변화가 매우 심한 한강 하류구간에 적용한 결과 기존 McCormack 기법이 해석할 수 없는 정상류나 부정류를 모두 잘 해석할 수 있는 것으로 나타났다. 본 연구에서 개발한 상류이송형 McCormack 기법은 복잡한 자연하도의 흐름해석을 위해 효과적으로 사용될 수 있을 것으로 판단된다. In this study, the upwind McCormack scheme is introduced to combine the advantage of McCormack scheme, the second order accuracy and simplicity, and the advantage of the upwind scheme, to be applied to the discontinuous flows. This scheme also has another advantage of treating the source terms effectively. This model is approved through applying to the discontinuous flow case with the analytical solution, and the natural river with very strong source terms. Applications of the upwind McCormack scheme developed in this paper show good agreements with the analytical solution without numerical oscillation in existing McCormack scheme. Futhermore, applications to the natural river, the lower Han river with strong variation of bed and width, also show good results in case of both steady flow and unsteady flow. The upwind McCormack scheme in this study will be used for the analysis of flow in natural rivers effectively.

Numerical Modeling in Conjugate Formulation of the Temperature State of the Leading Edge of Hypersonic Flying Vehicles under Different Methods of Organization of the Cooling System

Heat transfer in the region of the leading edge of a hypersonic flying vehicle has been calculated by finite‐difference integration of the Reynolds equations closed by the low‐Reynolds version of the k–e model of turbulence. Integration has been performed using the explicit–implicit McCormack scheme. It is shown that blowing of a cooling agent through a porous insert can lead to a substantial decrease of heat flux. The Darcy equation was used in solution of the conjugate problem.

COMPUTATION OF RUNOFF IN HYPOTHETICAL MICROCATCHMENT USING DYNAMIC ROUTING

ABSTRACT In the present work, the runoff in an independent hypothetical micro-catchment due to rainfall is computed using a mathematical model. One-dimensional Saint Venant equations are used as the governing equations. These equations are solved numerically by an explicit finite-difference method—McCormack Scheme. The analysis is performed for different intensities of rainfall, perviousness coefficients and cloud movement. The results are also compared with the hydrograph derived using discrete kernel approach for all rainfall intensities.

Radial Flow Modeling for Estimating Level-Basin Irrigation Parameters

Previous research in the estimation of surface irrigation parameters from advance time or surface profile observations has focused on the development of solutions for parallel surface flow conditions. This study was conducted to assess the usefulness of a polar model simulating radial flow in the estimation of infiltration and roughness parameters of a point-inflow level-basin irrigation event. The two-step explicit McCormack scheme was applied to the solution of the governing equations. The method of characteristics was used at boundary nodes. Point-wise semiimplicit discretization of the friction and geometric source terms was used. Results of this model compared well with those obtained with a two-dimensional Cartesian model simulating radial flow. Gradient and nongradient minimization techniques were applied to determine pairs of infiltration and roughness parameters. The estimation of the parameters is as feasible for radial flow as it is for parallel flow.

Numerical Modeling of Flood Propagation through System of Reservoirs

An explicit numerical flow‐routing model is developed based on the second order McCormack scheme. This paper describes the use of the finite difference scheme for the solution of the St. Venant equations of the one‐dimensional unsteady open channel flow, and the treatment of the boundary problem by the method of characteristics. A case study is presented using the system of reservoirs and the fluvial network of tributaries of the left‐hand side of the Ebro River valley in Spain. To get an estimate of the regulating action provided by those reservoirs during the strong rainfall of the first days of November 1982, the registered inflow and outflow hydrographs are used for the simulation of the propagation of the waves along the watercourses. Results of flow computations are presented and compared with field measurements to show that this model can be considered a suitable technique for hydraulic studies involving rapidly varying flows.