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The construction of the limiter is a critical factor in the traditional Total Variation Diminishing (TVD) scheme. Among the classical limiters, superbee has the lowest numerical dissipation, but it can lead to over-compression in smooth regions and excessive artificial steepening at discontinuities and critical points if the computation time is prolonged. Classical limiters like Minmod, van Leer, van Albada, and MC fail to distinguish between different wave types, and they can even cause numerical oscillations for multi-critical value problems with prolonged computation times. A class of adaptive limiters has been created by combining classical limiters with superbee. This adaptive limiter can achieve second-order accuracy in smoothed regions and effectively reduce over-compression and excessive artificial steepening for long computation times. Analytical and numerical results show that the MUSCL scheme with an adaptive limiter is efficient.

The efficiency of calculating the outgoing shock waves arising in liquid (water) as a result of collapse of a spherical cavitation bubble by the Godunov method and its UNO and TVD modifications with and without the use of a discontinuous reconstruction of the numerical solution in the vicinity of the shock front is investigated. The dynamics of the vapor in the bubble and the surrounding liquid is governed by the equations of gas dynamics closed by wide-range equations of state. The thermal conductivity of both fluids, the heat and mass transfer across the bubble surface are taken into account. Moving computational grids refined to the explicitly traced bubble surface are applied. It is found that all the three methods considered show quite good resolution of the generation and the initial stage of the propagation of the outgoing shock wave even on relatively coarse grids. A rather fine grid with 128000 cells in liquid allows one to obtain satisfactory description of the further propagation and transformation of the shock wave until its front radius becomes approximately equal to (i) the initial bubble radius when using the Godunov method and (ii) ten initial bubble radii when using the UNO and TVD modifications. It is revealed that the discontinuous reconstruction of the numerical solution can significantly increase the numerical efficiency. In particular, with applying the reconstruction, the results of the Godunov method on the indicated fine grid satisfactorily reproduce the profile of the shock wave in the vicinity of its maximum until its front radius becomes equal to ten initial bubble radii. [ABSTRACT FROM AUTHOR]

The results of numerical modeling of propagation of an outgoing shock wave produced by the collapse of a cavitation bubble in water are compared with the experimental data from [O. Lindau and W. Lauterborn, AIP Conf. Proc. 524, 385 (2000)]. Those data are the shock wave pressure maxima at a number of points in the vicinity of the bubble at different distances from the bubble center. They were derived in the case the initial bubble radius is 1.92 mm, the liquid pressure is 1 bar, and the liquid temperature is C. In the numerical modeling, the liquid flow and the dynamics of the vapor inside the bubble are governed by the equations of gas dynamics closed with wide-range equations of state. The model takes into account the heat conductivity, the evaporation and condensation on the bubble surface. The governing equations are solved by the Godunov method and its UNO- and TVD-modifications of increased order of accuracy with moving grids refined to the bubble surface. Discontinuous reconstruction of the numerical solutions in the vicinity of the shock wave front is applied. It is shown that the results of the numerical modeling by all the three methods with using the discontinuous reconstructions are very close to one another and well agree with the experimental data. [ABSTRACT FROM AUTHOR]

The present work proposes a new morphodynamic model for flow simulation in open channels. The governing one dimensional (1D) shallow water equations along with the Exner sediment continuity equation are solved here using coupled approach. A TVD based MacCormack predictor corrector method is used here for solution of the coupled governing equations. The developed numerical model is used to replicate several test cases of known experimental and approximate solutions. These cases include bed aggradation due to sediment overloading, bed degradation due to sediment shut-off, dam break flow over erodible bed and movable bump due to subcritical and transcritical flows. The results obtained are found to be excellent and free from oscillations near sharp gradient. The numerical model is also found to be capable of capturing the wave-fronts of both water surface and bed sediment. Quantitative analysis of the results in terms of two statistical parameters establishes the potential of the present model in more complicated practical problems. [ABSTRACT FROM AUTHOR]

To study wave propagation in continuous media, the classical Godunov method, which is stable and monotonic, is widely used. However, due to the first order of accuracy, it can lead to strong smearing of jumps, contact discontinuities, and other features of the solution in regions with large gradients. The possibility of increasing the computational efficiency of the elastic-plastic waves in a body in comparison with the Godunov method by applying its TVD- and UNO-modifications has been investigated in this paper. The UNO-modification is strictly second-order accurate, whereas the TVD-modification loses that accuracy at the solution extrema due to exactly satisfying the condition of total variation diminishing, while the UNO-modification meets it approximately. The plastic state of the body has been described by the Mises condition. Estimation of the efficiency of the considered modifications has been carried out by comparing the results of their application to a number of problems with the results obtained by the Godunov method. Problems of the propagation of 1D plane elastic-plastic waves have been considered. Those waves result in a body from the action of a pressure pulse on its surface or the given impulsive displacement of the surface. It has been shown that on the same computational grids the results of the considered modifications are much better than those obtained by the Godunov method. In particular, the width of smearing the jump-like fronts of both the elastic and plastic waves is significantly less. At that, the UNO-modification is more preferable because the TVD-modification tends to "cut'' the solution extrema.

An improved r -factor algorithm for implementing total variation diminishing (TVD) scheme has been proposed for the gradient smoothing method (GSM) using unstructured meshes. Different from the methods using structured meshes, for the methods using unstructured meshes, generally the upwind point cannot be clearly defined. In the present algorithm, the value of upwind point has been successfully approximated for unstructured meshes by using the GSM with different gradient smoothing schemes, including node GSM (nGSM) midpoint GSM (mGSM) and centroid GSM (cGSM). The present method has been used to solve hyperbolic partial differential equation discontinuous problems, where three classical flux limiters (Superbee, Van leer and Minmod) were used. Numerical results indicate that the proposed algorithm based on mGSM and cGSM schemes can avoid the numerical oscillation and reduce the numerical diffusion effectively. Generally the scheme based on cGSM leads to the best performance among the three proposed schemes in terms of accuracy and monotonicity. [ABSTRACT FROM AUTHOR]

This paper proposes a new high-resolution finite volume method for solving the two-dimensional (2D) solute transport equation using an unstructured mesh. A new simple r-factor algorithm is introduced into the Total Variation Diminishing flux limiter to achieve a more efficient yet accurate high-resolution scheme for solving the advection term. To avoid the physically-meaningless negative solutions resulted from using the Green-Gauss theorem, a nonlinear two-point flux approximation scheme is adopted to deal with the anisotropic diffusion term. The developed method can be readily coupled with a two-dimensional finite-volume-based flow models under unstructured triangular mesh. By integrating with the ELCIRC flow model, the proposed method was verified using three idealized benchmark cases (i.e., advection of a circle-shaped solute field, advection in a cyclogenesis flow field and transport of a initially square-shaped solute plume), and further applied to simulate the non-reactive solute transport process driven by irregular tides in the Deep Bay, eastern Pearl River Estuary of China. These cases are also simulated by models using other existing methods, including different r-factor for advection term and the Green-Gauss theorem for diffusion term. The comparison between the results from the new method and those from other existing methods demonstrated the new method could describe advection induced concentration shock and discontinuities, and anisotropic diffusion at high resolution without providing spurious oscillations and negative values. [ABSTRACT FROM AUTHOR]

Total Variation Diminishing (TVD) schemes are low dissipative and high resolution schemes but bounded by stability criterion CFL

Abstract: The development of the software package FlowModellium designed for simulating high-speed flows of continuum medium taking into account nonequilibrium chemical reactions is described. The numerical method and the two-level parallel algorithm used in the package are presented. Examples of computations are discussed. [ABSTRACT FROM AUTHOR]

Explicit formulation of one-dimensional Euler equation with TVD scheme is bounded by the stability criterion CFL<1. This limitation does not allow large time steps and considerably increases simulation time. Large time step schemes have been developed for explicit formulations which are stable for larger CFL. TVD schemes are nonoscillatory but suffer with smearing of discontinuities. Artificial compression method (ACM) derived by Harten is appropriate to deal with smearing problem. Present work focuses this deficiency and present LTS TVD scheme results with ACM to investigate its behavior and consequences. One-dimensional shock tube problem for SOD, LAX, and inverse shock boundary conditions is solved to reveal the effect of ACM on large time step scheme at complex flow regions, namely expansion fan, contact, and shock wave. Results for all the three test cases with proposed methodology are satisfactory with some minor issues. Results depict that proper descriptions of parameters in ACM are very important, especially in the case of inverse shock boundary condition. It is concluded that large time step scheme with ACM effectively reduces numerical dissipation along with shorter simulation time. Such an approach for handling smearing problem in LTS TVD schemes proves its vital role in the design and analysis phase and provides more opportunities to examine in shorter time period. However, only one-dimensional problem is solved in present studies. Further studies should also be carried out with two-dimensional and three-dimensional test cases in order to reinforce currently computed results. [ABSTRACT FROM AUTHOR]

Simulation of plasmas in electromagnetic fields requires numerical solution of a kinetic equation that describes the time evolution of the particle distribution function. In this paper we propose a finite volume scheme based on integral relation for Poisson brackets to solve the Liouville equation, the most fundamental kinetic equation. The proposed scheme conserves the number of particles, maintains the total-variation-diminishing (TVD) property, and provides high-quality numerical results. Other types of kinetic equations may be also formulated in terms of Poisson brackets and solved with the proposed method including the transport equations describing the acceleration and propagation of Solar Energetic Particles (SEPs), which is of practical importance, since the high energy SEPs produce radiation hazards. The proposed scheme is demonstrated to be accurate and efficient, which makes it applicable to global simulation systems analyzing space weather. • The control volume approach to a kinetic equation conserves the number of particles. • Being an equation for advection along the Hamiltonian trajectory the Liouville equation has TVD property. • High-resolution conservative scheme is constructed for wide class of kinetic equation. • The scheme is applicable to energetic particle transport through the heliosphere. [ABSTRACT FROM AUTHOR]

A finite-difference TVD scheme is presented for problems in nonequilibrium wave dynamics of heterogeneous media with different velocities and temperatures but with identical pressures of the phases. A nonlinear form of artificial viscosity depending on the phase relaxation time is proposed. The computed solutions are compared with exact self-similar ones for an equilibrium heterogeneous medium. The performance of the scheme is demonstrated by numerical simulation with varying particle diameters, grid sizes, and particle concentrations. It is shown that the scheme is efficient in terms of Fletcher's criterion as applied to stiff problems. [ABSTRACT FROM AUTHOR]

In this article, a characteristic-based dual-phase-lag numerical model based on finite difference method has been developed to predict the microscopic heating response in time as well as consideration of the micro-structured effect. High-order TVD (Total Variation Diminishing) schemes being oscillation-free can yield high-order accurate solutions without introducing wiggles and therefore are utilised in this work. A multi-domain approach integrated within the dual-phase-lag numerical model allows the computation of microscopic conjugate heat transfer problems. Effects of different phase-lag values on the behaviour of heat transfer are investigated. The model is capable of predicting temperature patterns transiting from the wave nature of heat propagation to additional diffusion being experienced within different solid regions via phonon–electron interaction or phonon scattering. [ABSTRACT FROM AUTHOR]

Variation Diminishing (TVD) schemes are low dissipative and high resolution schemes but bounded by stability criterion CFL<1 for explicit formulation. Stability criteria for explicit formulation limits time stepping and thus increase computational cost (computational time, machine cost). Research in the field of large time step (LTS) scheme is an active field for last three decades. In present work, Zhan Sen Qian's modified form of Harten LTS TVD scheme is studied and used to solve one dimensional benchmark test cases. SOD and LAX cases of shock tube problem are solved to understand the behavior of modified large time step scheme in regions of discontinuities and strong shock waves. The numerical results are found to be in good agreement with analytical results, except slight oscillations near contact discontinuity for larger values of K. Results also reveal that the discrepancy between numerical and analytical results near expansion fan, contact discontinuity and shock grows for larger values of K. Increase in discrepancy is due to the increase in truncation error. Truncation error strongly depends on step size and step size increases as CFL (or K) increases. In present work, the correction into the numerical formulation of characteristic transformation is discussed and the inverse characteristic transformations are performed using local right eigen vector in each cell interface location. This idea of extending Harten's large time step method for hyperbolic conservation laws proved to be very useful as the results shows that the modified scheme is a high resolution low dissipative and efficient scheme for 1D test cases. [ABSTRACT FROM AUTHOR]

In most TVD schemes, the r-factors were proposed according to the cell-centered (CC) finite volume method (FVM) framework for the numerical approximation to the convective term. However, it is questionable whether those r-factors would be appropriate and effective for the vertex-centered (VC) FVM. In the paper, we collected five kinds of r-factor formulae and found out that only three of those, respectively by Bruner (1996), Darwish and Moukalled (2003) and Cassuli and Zanolli (2005) can be formally extended to a context of the VC FVM. Numerical tests indicate that the TVD schemes and r-factors, after being extended and introduced to a context of the VC FVM, maintained their similar characteristics as in a context of the CC FVM. However, when the gradient-based r-factors and the SUPERBEE scheme were applied simultaneously, non-physical oscillations near the sharp step would appear. In the transient case, the oscillations were weaker in a context of the VC FVM than those in a context of the CC FVM, while the effect was reversed in the steady case. To eliminate disadvantages in the gradient-based r-factor formula, a new modification method by limiting values on the virtual node, namely ϕ in the paper, was validated by the tests to effectively dissipate spurious oscillations. [ABSTRACT FROM AUTHOR]

This paper focuses on the behavior of shock waves formed in an underexpanded impinging jet. The jet becomes underexpanded when the pressure ratio exceeds the critical value across the convergent nozzle discharging it. When a flat plate is placed perpendicularly to the jet, the strong shock wave called ‘plate shock' appears in the flow field near the plate. The underexpanded jet discharged from a nozzle into the atmosphere is often used for industrial applications, e.g., an assist gas of laser cutting and a cooling jet in glass tempering process. The jet impinges on the work piece and spreads out on its surface. So this study concentrates on the relation between the jet oscillation and the location of the work piece and especially discusses behavior of shock formed in jet. The impinging jet was formed under conditions of different nozzle pressure ratios and nozzle-plate spacings and was visualized using the shadowgraph and schlieren methods. The instantaneous shape of jet was measured through analyzing the pictures taken at random under each condition and the oscillation pattern was examined. In addition, under the same conditions, a numerical study was carried out using TVD scheme. As a result, it was found that the oscillation pattern of plate shock depends on its location and the nozzle pressure ratio.

Probability density evolution method is extended to analysis of fatigue reliability. The joint probability density evolution equation of random parameters and fatigue damage is derived based on the principle of preservation of probability, and a finite difference algorithm in terms of TVD theory is presented. For a given damage threshold, cycle-dependent fatigue reliability can be calculated by the proposed method without assuming the probabilistic distribution of fatigue damage in advance. Two validation examples indicate that the proposed method is able to give reasonable results for constant-amplitude loading and variable-amplitude loading. The predicted fatigue reliability under constant-amplitude loading shows a considerable accuracy. In addition, reliability isolines of fatigue damage can be used to predict the fatigue life with a specified failure probability. [ABSTRACT FROM AUTHOR]

In the present paper, the effects of non-equilibrium condensation on the lift divergence Mach number with the angle of attack in a transonic 2-D moist air flow of NACA0012 are investigated using a numerical analysis with a total variation diminishing scheme. In the case of T = 298 K and α = 3°, the lift divergence Mach number M decreases with increasing Ф up to 40%, whereas beyond Ф = 40%, the lift divergence Mach number is nearly constant at M = 0.75. On the other hand, for the case of α = 6°, M increases with increasing Ф up to 47%, and then M begins to decrease with an increase in Ф. Because of the attenuating effect of non-equilibrium condensation on the shock stall, for the case of α = 3°, C generally decreases with an increase in Ф, whereas for M∞ = 0.85, which is appreciably larger than the lift divergence Mach number, C increases with an increase in Ф. In the case of M∞ = 0.85, regardless of Ф, it is shown that the location of the maximum Mach number at the upper wall side is nearly constant at x/c = 0.4. It is found that the Mach number for the minimum C decreases with an increase in Ф. [ABSTRACT FROM AUTHOR]

SUMMARY An extension of a total variation diminishing (TVD) scheme to solve one-dimensional (1D) blood flow for human circulation is proposed. This method is simple as it involves only a few modifications to existing shock-capturing TVD schemes. We have applied the method to a wide range of test cases including a complete simulation of the human vascular network. Excellent solutions have been demonstrated for problems involving varying and discontinuous mechanical properties of blood vessels. For 1D network simulations, the method has been shown to agree well with the reported computational results. Finally, the method has been demonstrated to compare favorably with in vivo experiments set up to study the impact of circle of Willis anomalies on flow patterns in the cerebral arterial system. Copyright © 2015 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

• Differential, discrete and numerical stability analysis of 1D vertical annular flow • Closures analyzed: Surface tension; dynamic pressure; momentum flux parameter (CL) • Stronger impact of CL than other closures in a broadband of wave frequencies • CL greater than 1 leads to more regular waves and more uniform flow field • Low pass filter provided by CL is similar in linear and non-linear wave regimes For several industrial applications involving two-phase gas-liquid flows, the one-dimensional Two–Fluid Model equations are typically solved for flow simulation. To ensure reliable predictions, it is very important to assess the stability properties and grid dependency of one-dimensional formulations, which are known to be largely affected by the closure relations and numerical schemes employed. In this work, a stability analysis of the transient one-dimensional Two–Fluid Model was performed for vertical annular flows. A viscous approach of differential and discretized formulations was analysed. The influence of the momentum flux parameter, interfacial pressure jump due to surface tension and a dynamic pressure model were investigated. The analytical results were compared to those obtained by numerical solution of the model equations with the Finite Volume Method, for various experimental configurations taken from the literature. Results showed that closure models largely affect the wave frequencies and the growth rates captured by the model. The surface tension term introduced a cut-off frequency in the differential formulation rendering the model well-posed and effectively stabilizing short waves. The introduction of the dynamic pressure term considered here did not affect the cut-off values. However, the wave growth rates decreased compared to the case without this term. The liquid momentum flux parameter greater than 1 presented a stronger influence on the evolution of interfacial waves in a broadband of frequencies than the other closures and leads to more regular waves and a more uniform flow field but much higher values may excessively damp the solution, which blocks the natural formation of larger waves in vertical annular flows. The numerical solution of the model equations showed very good agreement with the discrete stability analysis performed and was able to capture the wave frequencies and associated (linear) growth rates in the wave formation region. Concerning the momentum flux parameter, the bandwidth of the most damped disturbances in the linear and nonlinear region was similar. This suggests that the frequency response in this case was not very different from linear to nonlinear wave regimes. The methodology proved to be an important tool for further development of closure models and numerical schemes applied to vertical annular flows. Graphical Abstract [Display omitted] [ABSTRACT FROM AUTHOR]

The two-fluid model with kinetic theory of granular flow is implemented into the open source CFD package OpenFOAM®. The effect of total variation diminishing (TVD) convection schemes is investigated by simulating two bubbling fluidized beds. Five TVD schemes are employed to discretize the convection terms of phase velocity and solid volume fraction. Simulated results of the two test cases give reasonable agreement with the experimental data in the literature. For the discretization of the phase velocity convection terms, the five schemes give quite similar time-averaged radial profiles of particle axial velocity. The predicted bubbles in the bed with a central jet are not influenced by the different schemes. For the discretization of the solid volume fraction convection terms, the limitedLinear01, Sweby01 and vanLeer01 schemes give the converged and reasonable solutions, whereas the SuperBee01 and MUSCL01 schemes diverge the solutions. When using the faceLimited gradient scheme the convection scheme becomes more diffusive. [ABSTRACT FROM AUTHOR]

The hydrodynamics of geophysical flows in oceanic shelves, estuaries, and rivers are often studied by solving shallow water equations under either hydrostatic or non-hydrostatic assumptions. Although the hydrostatic models are quite accurate and cost-efficient for many practical applications, there are situations when the fully hydrodynamic models are preferred despite a larger cost for computations. The present numerical model is implemented by the finite volume method (FVM) based on unstructured grids. The model can be efficiently switched between hydrostatic and non-hydrostatic modules. The case study shows that for waves pro-pagating along the bar a criterion with respect to the shallowness alone, the ratio between the depth and the wave length, is insufficient to warrant the performance of shallow flow equations with a hydrostatic approach and the nonlinearity in wave dynamics can be better accounted with a hydrodynamic approach. Besides the prediction of the flows over complex bathymetries, for instance, over asymmetrical dunes, by a hydrodynamic approach is shown to be superior in accuracy to the hydrostatic simulation. [ABSTRACT FROM AUTHOR]

Within shallow water equations, the numerical simulation problem of the generation and propagation of shallow waves caused by a running bottom rupture is considered. The rupture is a trench with a width increasing with time. As a result of fluid drainage into the trench, waves are formed. The basic problem parameters influencing the characteristics of the formed waves are found. [ABSTRACT FROM AUTHOR]

The problem of steady outflow of rarefied gas from a reservoir into vacuum through a long cylindrical tube of circular cross-section at a constant temperature is considered on the basis of the kinetic S-model. The kinetic equation is solved numerically using a conservative second-order method. The basic calculated quantity is the gas flow rate through the channel; the flowfields are also studied. The solutions obtained are compared with the known results. [ABSTRACT FROM AUTHOR]

The kinetic S-model is used to study the steady rarefied gas flow through a long pipe of variable cross section joining two tanks with arbitrary differences in pressure and temperature. The kinetic equation is solved numerically by applying a second-order accurate conservative method on an unstructured mesh. The basic quantity to be computed is the gas flow rate through the pipe. The possibility of finding a solution based on the assumption of the plane cross sectional flow is also explored. The resulting solutions are compared with previously known results. [ABSTRACT FROM AUTHOR]

A transonic flow with a non-equilibrium condensation past NACA0012 profile whose aspect ratio AR is 1.0 with the angle of attack was analyzed by numerical analysis using a TVD scheme, and investigated using an intermittent indraft type supersonic wind tunnel. Transonic flows of 0.70-0.90 in free stream Mach number with variations of Φ and α were tested. For the same M and α, the increase in stagnation relative humidity Φ caused a decrease in the drag coefficient of profile (that is total) which is composed of the components of form, viscous, wave and condensation; however, the lift coefficient up to Φ = 50% increased in the opposite direction. As an example, in the case of M = 0.83, Φ = 50%, α = 3○ and T = 298 K, the decreasing rate of the coefficient of profile drag and the increasing rate of the lift coefficient compared to the case of Φ = 0% caused by non-equilibrium condensation amounted to 65% and 52%, respectively. In addition, for the same Φ and α, as the free stream Mach number M increased, at first, the lift coefficient increased slightly, and then suddenly severely dropped, and finally remained nearly constant. The suddenly dropped free stream Mach number in C became larger with an increase of Φ. It turned out that the drag coefficients of form and viscous were almost independent of Φ. The contribution of wave drag to the coefficient of profile drag for M = 0.83, Φ = 30%, α = 3° and T = 298 K amounted to approximately 79%, and in the case of Φ = 60%, the contribution of the non-equilibrium condensation to the reduction in the coefficient of the profile drag compared to the case without condensation amounted to 75%. Especially, for the case of Φ = 0% and α = 0°, there was an oscillatory flow region around M = 0.87. [ABSTRACT FROM AUTHOR]

We present a numerical model based on the hydro-morphodynamical coupling to study coastal sandbar migration. In order to improve both nonlinear and dispersive wave processes in relatively shallow water, we developed a finite element model based on the Legendre polynomials and on the Extended Boussinesq model. This model reproduces the propagation of wave trains with a high degree of accuracy on a greater range of depths than the standard Boussinesq models. We also implemented the Total Variation Diminishing schemes to improve the quality of the computed hydrodynamic fields, especially in areas where sharp flow gradients occurred. The coupled morpho-hydrodynamical model is then used to simulate the migration of real sandbars observed at Rousty beach (Mediterranean French coast). For verification the model results are compared with field measurements obtained from a small-scale field campaign carried out over two years at Rousty beach, and the results of this comparison are thoroughly discussed and analyzed. [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: This paper presents a high-resolution numerical method for solving mass transport problems involving advection and anisotropic diffusion in shallow water based on unstructured mesh. An alternating operator-splitting technique is adopted to advance the numerical solution with advection and diffusion terms solved separately in two steps. By introducing a new r-factor into the Total Variation Diminishing (TVD) limiter, an improved finite-volume method is developed to solve the advection term with significant reduction of numerical diffusion and oscillation errors. In addition, a coordinate transformation is introduced to simplify the diffusion term with the Green-Gauss theorem used to deal with the anisotropic effect based on unstructured mesh. The new scheme is validated against three benchmark cases with separated and combined advection and diffusion transport processes involved. Results show that the scheme performs better than existing methods in predicting the advective transport, particularly when a sharp concentration front is in presence. The model also provides a sound solution for the anisotropic diffusion phenomenon. Anisotropic diffusion has been largely neglected by existing flow models based on unstructured mesh, which usually treat the diffusion process as being isotropic for simplicity. Based on the flow field provided by the ELCIRC model, the developed transport model was successfully applied to simulate the transport of a hypothetical conservative tracer in a bay under the influence of tides. [Copyright &y& Elsevier]

Abstract: A large time step (LTS) TVD scheme originally proposed by Harten is modified and further developed in the present paper and applied to Euler equations in multidimensional problems. By firstly revealing the drawbacks of Harten’s original LTS TVD scheme, and reasoning the occurrence of the spurious oscillations, a modified formulation of its characteristic transformation is proposed and a high resolution, strongly robust LTS TVD scheme is formulated. The modified scheme is proven to be capable of taking larger number of time steps than the original one. Following the modified strategy, the LTS TVD schemes for Yee’s upwind TVD scheme and Yee–Roe–Davis’s symmetric TVD scheme are constructed. The family of the LTS schemes is then extended to multidimensional by time splitting procedure, and the associated boundary condition treatment suitable for the LTS scheme is also imposed. The numerical experiments on Sod’s shock tube problem, inviscid flows over NACA0012 airfoil and ONERA M6 wing are performed to validate the developed schemes. Computational efficiencies for the respective schemes under different CFL numbers are also evaluated and compared. The results reveal that the improvement is sizable as compared to the respective single time step schemes, especially for the CFL number ranging from 1.0 to 4.0. [Copyright &y& Elsevier]

Abstract: Numerical solution of the advection–dispersion equation, used to evaluate transport of solutes in porous media, requires discretization schemes for space and time stepping. We examine use of quadratic upstream interpolation schemes QUICK, QUICKEST, and the total variation diminution scheme ULTIMATE, and compare these with UPSTREAM and CENTRAL schemes in the HYDRUS-1D model. Results for purely convective transport show that quadratic schemes can reduce the oscillations compared to the CENTRAL scheme and numerical dispersion compared to the UPSTREAM scheme. When dispersion is introduced all schemes give similar results for Peclet number Pe < 2. All schemes show similar behavior for non-uniform grids that become finer in the direction of flow. When grids become coarser in the direction of flow, some schemes produce considerable oscillations, with all schemes showing significant clipping of the peak, but quadratic schemes extending the range of stability tenfold to Pe < 20. Similar results were also obtained for transport of a non-linear retarded solute transport (except the QUICK scheme) and for reactive transport (except the UPSTREAM scheme). Analysis of transient solute transport show that all schemes produce similar results for the position of the infiltration front for Pe = 2. When Pe = 10, the CENTRAL scheme produced significant oscillations near the infiltration front, compared to only minor oscillations for QUICKEST and no oscillations for the ULTIMATE scheme. These comparisons show that quadratic schemes have promise for extending the range of stability in numerical solutions of solute transport in porous media and allowing coarser grids. [Copyright &y& Elsevier]

The problem of finding an interpolating image between two given images in an image sequence is considered. The problem is formulated as an optimal control problem governed by a transport equation, i.e. we aim at finding a flow field which transports the first image as close as possible to the second image. This approach bears similarities with the Horn and Schunck method for optical flow calculation but in fact the model is quite different. The images are modeled in the space of functions of bounded variation and an analysis of solutions of transport equations in this space is included. Moreover, the existence of optimal controls is proven and necessary conditions are derived. Finally, two algorithms are given and numerical results are compared with existing methods. The new method is competitive with state-of-the-art methods and even outperforms several existing methods. [ABSTRACT FROM AUTHOR]

The breakup of an axisymmetric viscous jet is considered in the lubrication approximation. The discretised equations are solved on a fixed equidistant one-dimensional Eulerian grid. The governing equations are implemented in a conservative second order accurate total variation diminishing (TVD) scheme, preventing the numerical diffusivity. Singularities that occur at pinchoff and coalescence are regularised by a small modification on the surface tension. The modification is of the order of the spatial step Δx. This regularisation ensures that the solution of the presented numerical model converges to the exact solution of the breakup of a jet in the lubrication approximation. The results of the presented numerical model agree quantitatively with the analytical solution of the Rayleigh-Plateau instability, and with experimental results on the final stage of the Rayleigh-Plateau instability. [ABSTRACT FROM AUTHOR]

It is important to understand how water waves propagate through water channels in order to reduce inundation damages induced by surges and/or tsunami run-up. For this purpose, a two-dimensional numerical model is developed based on Nonlinear Shallow Water Equations. In order to appropriately describe the strongly nonlinear hydrodynamics, a high-order TVD scheme is implemented. Model capabilities are confirmed through comparison with existing analytical studies including the dam-break problems, run-up and back-wash on a sloping bathymetry. The model is then applied to the study on the inundation of bore propagating in a channel of parabolic cross-section from a side breach. [ABSTRACT FROM AUTHOR]

Abstract: A numerical scheme is presented for accurate simulation of fluid flow using the lattice Boltzmann equation (LBE) on unstructured mesh. A finite volume approach is adopted to discretize the LBE on a cell-centered, arbitrary shaped, triangular tessellation. The formulation includes a formal, second order discretization using a Total Variation Diminishing (TVD) scheme for the terms representing advection of the distribution function in physical space, due to microscopic particle motion. The advantage of the LBE approach is exploited by implementing the scheme in a new computer code to run on a parallel computing system. Performance of the new formulation is systematically investigated by simulating four benchmark flows of increasing complexity, namely (1) flow in a plane channel, (2) unsteady Couette flow, (3) flow caused by a moving lid over a 2D square cavity and (4) flow over a circular cylinder. For each of these flows, the present scheme is validated with the results from Navier–Stokes computations as well as lattice Boltzmann simulations on regular mesh. It is shown that the scheme is robust and accurate for the different test problems studied. [Copyright &y& Elsevier]

A high-order accurate method for analyzing two-dimensional rarefied gas flows is proposed on the basis of a nonstationary kinetic equation in arbitrarily shaped regions. The basic idea behind the method is the use of hybrid unstructured meshes in physical space. Special attention is given to the performance of the method in a wide range of Knudsen numbers and to accurate approximations of boundary conditions. Examples calculations are provided. [ABSTRACT FROM AUTHOR]

As recognized previously, a minimum-length nozzle has the smallest possible throat-to-exit length that is still capable of maintaining uniform supersonic flow at the nozzle exit. In the present study, for the flow of moist air through a nearly minimum-length nozzle designed by the method of characteristics, the effects of nonequilibrium condensation on the uniformity of flow properties, the momentum efflux, and the flow distortion at the nozzle exit plane are discussed by experiment and numerical analysis of a third-order Total Variation Diminishing (TVD) finite difference scheme. The onset and zone of nonequilibrium condensation in a minimum-length nozzle are quite different from those of a general convergent-divergent supersonic nozzle. We know that the uniformity of flow properties at the nozzle exit with regard to the flow with nonequilibrium condensation in a minimum-length nozzle cannot be guaranteed. On the other hand, owing to the positions of the onset of condensation at the incident region of expansion waves from the sharp corner just downstream of the nozzle throat, the deceleration gradient and magnitude of heat released from the process of nonequilibrium condensation to the surrounding of ϕ0=60% are greater than those of ϕ0=70% in the case of T0=290K. Furthermore, it has been determined that the decrease in efflux of momentum from the nozzle exit for the stagnation relative humidity of ϕ0=70%(T0=290K), which corresponds to the case with nonequilibrium condensation shock, is 6.8% smaller than that of isentropic expansion. [ABSTRACT FROM AUTHOR]

For the shallow water equations in the first approximation (Saint-Venant equations), a TVD scheme is developed for shock-capturing computations of open channel flows with discontinuous waves. The scheme is based on a special nondivergence approximation of the total momentum equation that does not involve integrals related to the cross-section pressure force and the channel wall reaction. In standard divergence difference schemes, most of the CPU time is spent on the computation of these integrals. Test computations demonstrate that the discontinuity relations reproduced by the scheme are accurate enough for actual discontinuous wave propagation to be numerically simulated. All the qualitatively distinct solutions for a dam collapsing in a trapezoidal channel with a contraction in the tailwater area are constructed as an example. [ABSTRACT FROM AUTHOR]

The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408-463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241-282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82-104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199-224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255-283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks. [ABSTRACT FROM AUTHOR]

Abstract: Sensitivity analysis is an important tool which can be used to investigate the stability of a process perturbed by parameter changes and uncertainty impacts. In this work the unsteady sensitivity equations for complex looped pipe networks are solved. Special attention is focused on the coupled version of these equations, with the direct problem. For this purpose a splitting method using a Total Variation Diminishing (TVD) scheme with very good quality of stability is set up and validated on a benchmark pipe network. [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]

Abstract: A parametric study of combustor inlet configuration for supersonic combustion ramjet (Scramjet) engine has been conducted by solving two-dimensional full Navier–Stokes equations. The main stream is air of Mach 5 entering through the configured inlet of the combustor and gaseous hydrogen is injected from the configured jet on the side wall. The parameters included are air stream angle and injection angle. On the effect of air stream angle, strong interaction between main and injecting flows can be observed for smaller angle causing sharp increase in mixing efficiency on the top of injector. Also high momentum of air stream towards the side wall causes no recirculation at the upstream of injector and the system becomes unable for flame holding. For the variation of injection angle, results show that in upstream of injector the mixing is dominated by recirculation and in downstream the mixing is dominated by mass concentration of hydrogen. Upstream recirculation is dominant for injecting angle 60° and 90°. Incorporating the various effects, perpendicular injection shows the maximum mixing efficiency and its large upstream recirculation region has a good flame holding capability. [Copyright &y& Elsevier]