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A numerical model and results of calculations of convective-film cooling of a plate with a double-sided flow around it by high-temperature gas and cooling air are presented. The mathematical model is based on the hybrid 2-D RANS method in a nonstationary formulation for a viscous compressible heat-conducting gas using velocity and thermal wall functions with conjugation of fluid dynamics and heat transfer problems. [ABSTRACT FROM AUTHOR]

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented.

In this paper we propose an explicit predictor-corrector finite difference scheme to numerically solve one-dimensional conservation laws with discontinuous flux function appearing in various physical model problems, such as traffic flow and two-phase flow in porous media. The proposed method is based on the second-order MacCormack finite difference scheme and the solution is obtained by correcting first-order schemes. It is shown that the order of convergence is quadratic in the grid spacing for uniform grids when applied to problems with discontinuity. To illustrate some properties of the proposed scheme, numerical results applied to linear as well as non-linear problems are presented. [ABSTRACT FROM AUTHOR]

In this study, the effect of infiltration on overland flow is explored by incorporating infiltration into the flood routing equations. Particularly, different types of infiltration methods, i.e. the Horton Method, the Integrated Horton Method and the Infiltration Index Method are integrated into the kinematic wave equation. Infiltration capacity curve and infiltration rate are calculated along with the hydrographs. Moreover, the same parameters are determined by incorporating the Integrated Horton method into different forms of Saint-Venant Equations to evaluate the effect of different routing methods on overland flow hydrograph. MacCormack explicit finite difference scheme is employed in solving the coupled infiltration-overland flow differential equations. For each wave routing method, hydrographs of overland flow with infiltration and with no infiltration are compared and discussed under constant and variable rainfall intensities and for different soil types. Results show that infiltration methods affect the overland flow hydrographs significantly. [ABSTRACT FROM AUTHOR]

The purpose of this research was to develop a numerical model of one-dimensional advection-diffusion equation for estimating salinity level in Lower Chao Phraya River, Thailand. The main numerical technique was MacCormack finite-difference scheme with initial and boundary conditions approximated by cubic spline interpolation. The estimates matched satisfactorily with actual measurement data from several salinity monitoring stations along the river. The final model was first used by Thailand's Metropolitan Waterworks Authority for planning salinity control in 2018. [ABSTRACT FROM AUTHOR]

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 MacCormack 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.

A numerical method for simulating transient ows 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 MacCormack scheme combined with the method of characteristics for the boundary conditions is presented. Theoretical results for transients initiated by a rapid closing valves are compared with measure- ments. A good agreement is found particularly for large values of initial dissolved gas concentration. [ABSTRACT FROM AUTHOR]

Although process-based distributed hydrological models (PDHMs) are evolving rapidly over the last few decades, their extensive applications are still challenged by the computational expenses. This study attempted, for the first time, to apply the numerically efficient MacCormack algorithm to overland flow routing in a representative high-spatial resolution PDHM, i.e., the distributed hydrology-soil-vegetation model (DHSVM), in order to improve its computational efficiency. The analytical verification indicates that both the semi and full versions of the MacCormack schemes exhibit robust numerical stability and are more computationally efficient than the conventional explicit linear scheme. The full-version outperforms the semi-version in terms of simulation accuracy when a same time step is adopted. The semi-MacCormack scheme was implemented into DHSVM (version 3.1.2) to solve the kinematic wave equations for overland flow routing. The performance and practicality of the enhanced DHSVM-MacCormack model was assessed by performing two groups of modeling experiments in the Mercer Creek watershed, a small urban catchment near Bellevue, Washington. The experiments show that DHSVM-MacCormack can considerably improve the computational efficiency without compromising the simulation accuracy of the original DHSVM model. More specifically, with the same computational environment and model settings, the computational time required by DHSVM-MacCormack can be reduced to several dozen minutes for a simulation period of three months (in contrast with one day and a half by the original DHSVM model) without noticeable sacrifice of the accuracy. The MacCormack scheme proves to be applicable to overland flow routing in DHSVM, which implies that it can be coupled into other PHDMs for watershed routing to either significantly improve their computational efficiency or to make the kinematic wave routing for high resolution modeling computational feasible. [ABSTRACT FROM AUTHOR]

River ice is a natural phenomenon in cold regions, influenced by meteorology, geomorphology, and hydraulic conditions. River ice processes involve complex interactions between hydrodynamic, mechanical, and thermal processes, and they are also influenced by weather and hydrologic conditions. Because natural rivers are serpentine, with bends, narrows, and straight reaches, the commonly-used one-dimensional river ice models and two-dimensional models based on the rectangular Cartesian coordinates are incapable of simulating the physical phenomena accurately. In order to accurately simulate the complicated river geometry and overcome the difficulties of numerical simulation resulting from both complex boundaries and differences between length and width scales, a two-dimensional river ice numerical model based on a boundary-fitted coordinate transformation method was developed. The presented model considers the influence of the frazil ice accumulation under ice cover and the shape of the leading edge of ice cover during the freezing process. The model is capable of determining the velocity field, the distribution of water temperature, the concentration distribution of frazil ice, the transport of floating ice, the progression, stability, and thawing of ice cover, and the transport, accumulation, and erosion of ice under ice cover. A MacCormack scheme was used to solve the equations numerically. The model was validated with field observations from the Hequ Reach of the Yellow River. Comparison of simulation results with field data indicates that the model is capable of simulating the river ice process with high accuracy.

Flood risk assessment is generally studied using flood simulation models; however, flood risk managers often simplify the computational process; this is called a 'simplification strategy'. This study investigates the appropriateness of the 'simplification strategy' when used as a flood risk assessment tool for areas prone to flash flooding. The 2004 Boscastle, UK, flash flood was selected as a case study. Three different model structures were considered in this study, including: (1) a shock-capturing model, (2) a regular ADI-type flood model and (3) a diffusion wave model, i.e. a zero-inertia approach. The key findings from this paper strongly suggest that applying the 'simplification strategy' is only appropriate for flood simulations with a mild slope and over relatively smooth terrains, whereas in areas susceptible to flash flooding (i.e. steep catchments), following this strategy can lead to significantly erroneous predictions of the main parameters-particularly the peak water levels and the inundation extent. For flood risk assessment of urban areas, where the emergence of flash flooding is possible, it is shown to be necessary to incorporate shock-capturing algorithms in the solution procedure, since these algorithms prevent the formation of spurious oscillations and provide a more realistic simulation of the flood levels. [ABSTRACT FROM AUTHOR]

The basal entrainment of debris flows play a significant role in its amplification and the final volume of deposition could be several to hundred times of its initial volume. In this paper, the shallow water equations coupled with basal material entrainment applied in debris flows have been given to demonstrate the amplification. A new basal entrainment model taking advantage of both Coulomb and Voellmy frictional laws are proposed, providing a unified formula to simulate the stop-and-go process of debris flows. Both time and space second-order MacCormack-TVD finite difference method is suggested to solve the coupled equations. Numerical comparisons with USGS flume experiment and Hongchun gully debris flow in Wenchuan earthquake-induced area are carried out to prove its effectiveness. It is established that the momentum exchange term between the flows and the basal materials has a significant influence on the dynamic characteristics and the entrainment effects are essential to model the dynamic process of debris flows in an earthquake-induced area. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We have investigated the oscillations and separation of the gas suspension moving in an acoustical resonator representing an open plane channel, in which the direction of medium oscillations is perpendicular to the flow direction. Computational modeling of the processes in the above resonator has been carried out. The drift conditions for the solid phase in a dispersed flow under the action of nonlinear acoustic fields have been determined. The results of calculations of the gas suspension dynamics and the distribution of the solid fraction density depending on the carrier phase velocity for various instants of time are presented. [ABSTRACT FROM AUTHOR]

Abstract: A two-dimensional mountainous mass flow dynamic procedure solver (Massflow-2D) using the MacCormack-TVD finite difference scheme is proposed. The solver is implemented in Matlab on structured meshes with variable computational domain. To verify the model, a variety of numerical test scenarios, namely, the classical one-dimensional and two-dimensional dam break, the landslide in Hong Kong in 1993 and the Nora debris flow in the Italian Alps in 2000, are executed, and the model outputs are compared with published results. It is established that the model predictions agree well with both the analytical solution as well as the field observations. [Copyright &y& Elsevier]

Abstract: A generic three-dimensional process-based model is presented, aimed at simulation of the construction of turbidite fan stratigraphy by low-density turbidity current events. It combines theoretical formulations on density flow and sediment transport of multiple grain sizes to simulate 2-DH turbidity current flow and sedimentation over arbitrary topography. The model is solved on a rectangular grid by means of a robust and efficient second-order finite-difference approximation. A high-resolution shock-capturing technique is employed to accurately model the speed and shape of the discontinuous flow front. In this paper, the implementation of such a high-resolution scheme is explained in detail. Efficiency and robustness of the numerical solution are tested by comparing modeled flow behavior and sedimentation to measurements from flume-tank experiments in which turbidity currents interacted with obstacles representative of a tectonically deformed basin floor. Results illustrate that the interaction of the flow with the obstacles is realistically simulated, and that the experimental deposit geometries are reasonably well reproduced. Stability of the model depends on the length of the computational time step and the properties of the adopted flux limiter function. Run time of the simulations is somewhat shorter than the real-time duration of the experiments, which is deemed acceptable considering the small computational time step which must be adopted to keep the model stable at this small scale. At the much larger scale of real-world turbidity currents a much larger computational time step can be adopted, which will speed up simulation considerably. [Copyright &y& Elsevier]

Summary: In this paper, vertically averaged free-surface and subsurface flows are linked and solved simultaneously in a two-dimensional numerical model for predicting flood flows. A TVD-MacCormack scheme is used to solve the shallow water equations for free surface flows, while the standard MacCormack scheme is employed to solve the transient Boussinesq equations for unconfined groundwater flows. The dynamic linking of the surface/subsurface models enables the interactions between the surface water flow and the neighbouring groundwater flow in the horizontal plane to be studied. The developed model is firstly verified against the analytical solutions and experimental measurements. The model is then used to investigate the influence of buildings on flood flows, where the buildings are modelled as porous media. This approach of modelling buildings is compared with two other commonly used methods: the first being to represent the buildings by solid blocks, and the second is to include the buildings by increasing the local roughness. It has been found that the combined surface/subsurface model provides a high degree of flexibility in representing the buildings in a flood flow simulation. [Copyright &y& Elsevier]

Abstract: Stream water quality modeling often involves numerical methods to solve the dynamic one-dimensional advection–dispersion–reaction equations (ADRE). There are numerous explicit and implicit finite difference schemes for solving these problems, and two commonly used schemes are the MacCormack and Saulyev schemes. This paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency. Using advection dominated (high Peclet number) problems as test cases, performances of the revised schemes are compared to performances of five classic schemes: forward-time/centered-space (FTCS); backward-time/centered-space (BTCS); Crank–Nicolson; and the traditional MacCormack and Saulyev schemes. All seven of the above numerical schemes were tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel, and performances were considered with respect to stability, accuracy, and computational efficiency. Results indicated that both the modified Saulyev and the MacCormack schemes, which are named the Saulyev c and MacCormack c schemes respectively, greatly improved the prediction accuracy over the original ones. The computation efficiency in terms of CPU time was not impacted for the Saulyev c scheme. The MacCormack c scheme demonstrated increased time consumption but was still much faster than implicit schemes. [Copyright &y& Elsevier]

Summary: The shallow water equations (SWEs) are solved numerically in general curvilinear coordinates for predicting flood flows. Second-order accuracy is achieved both in time and space through the use of the MacCormack scheme, in conjunction with a total variation diminishing (TVD) modification. The method is highly efficient as no local characteristic transformation is needed. The possible numerical imbalance induced by the inconsistent evaluation of the flux gradient and source terms is removed by employing the non-conservative form of the SWEs. All of the flow regimes and their transitions can be modelled. Specific considerations have been given to simulating flows in practical environments. The friction–slope term is discretized semi-implicitly when the water depth is smaller than a prescribed value. The check for flooding/drying (wet/dry status of the cell) is performed at each time step. Validation tests include steady flows along a deflected wall and a converging channel, unsteady wave diffraction around a circular cylinder, and the Malpasset dam-break flood in 1959. The results indicate that the present model is able to predict accurately the shock fronts, even over initially dry beds with friction and abrupt slope changes. [Copyright &y& Elsevier]

Abstract: A total variation diminishing (TVD) modification of the MacCormack scheme is developed for simulating shallow water dynamics on a uniform Cartesian grid. Results obtained using conventional and deviatoric forms of the conservative non-linear shallow water equations (SWEs) are compared for cases where the bed has a varying topography. The comparisons demonstrate that the deviatoric form of the SWEs gives more accurate results than the conventional form, in the absence of numerical balancing of the flux-gradient and source terms. A further comparison is undertaken between the TVD-MacCormack model and an alternating direction implicit (ADI) model for cases involving steep-fronted shallow flows. It is demonstrated that the ADI model is unable to predict trans-critical flows correctly, and artificial viscosity has to be introduced to remove spurious oscillations. The TVD-MacCormack model reproduces all flow regimes accurately. Finally, the TVD-MacCormack model is used to predict a laboratory-scale dyke break undertaken at Delft University of Technology. The predictions agree closely with the experimental data, and are in excellent agreement with results from an alternative Godunov-type model. [Copyright &y& Elsevier]

In this study, we propose a new numerical method, named as Traction Image method, to accurately and efficiently implement the traction-free boundary conditions in finite difference simulation in the presence of surface topography. In this algorithm, the computational domain is discretized by boundary-conforming grids, in which the irregular surface is transformed into a ‘flat’ surface in computational space. Thus, the artefact of staircase approximation to arbitrarily irregular surface can be avoided. Such boundary-conforming gridding is equivalent to a curvilinear coordinate system, in which the first-order partial differential velocity-stress equations are numerically updated by an optimized high-order non-staggered finite difference scheme, that is, DRP/opt MacCormack scheme. To satisfy the free surface boundary conditions, we extend the Stress Image method for planar surface to Traction Image method for arbitrarily irregular surface by antisymmetrically setting the values of normal traction on the grid points above the free surface. This Traction Image method can be efficiently implemented. To validate this new method, we perform numerical tests to several complex models by comparing our results with those computed by other independent accurate methods. Although some of the testing examples have extremely sloped topography, all tested results show an excellent agreement between our results and those from the reference solutions, confirming the validity of our method for modelling seismic waves in the heterogeneous media with arbitrary shape topography. Numerical tests also demonstrate the efficiency of this method. We find about 10 grid points per shortest wavelength is enough to maintain the global accuracy of the simulation. Although the current study is for 2-D P-SV problem, it can be easily extended to 3-D problem. [ABSTRACT FROM AUTHOR]

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

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