Impact of a discretization scheme on an autoignition time in LES of a reacting droplet-laden mixing layer

Discretization schemes based on the normalized variable diagram (NVD) and total variation diminishing (TVD) schemes are applied to the solution of the radiative transfer equation (RTE) in unstructured grids. The success of the application of these schemes to Cartesian grids has been previously demonstrated. However, their extension to unstructured grids is not straightforward. A few methods to overcome this difficulty have been proposed, and successfully applied to the solution of a scalar transport equation and to the Navier-Stokes equations. These methods are applied here to the solution of the RTE, along with a new one, recently proposed, for several test cases in which the analytical solution or other reliable numerical solutions are available for comparison. The results demonstrate that although the NVD and TVD schemes are much more accurate than the step and mean flux interpolation schemes, their order of accuracy in the case of unstructured grids is lower than in Cartesian grids. Moreover, in contrast to Cartesian grids, the NVD and TVD schemes are not strictly bounded in unstructured grids, and unphysical solutions may occur. The alternative method proposed to implement the NVD and TVD schemes is generally more accurate than previous ones, but also computationally more demanding.

A comparative study of the performance of TVB (total variation bounded), TVD (total variation diminishing), and ENO (essentially non-oscillatory) schemes for the Euler equations was conducted. All the schemes are constructed by applying the characteristic flux difference splitting method to a modified flux which has either TVB, TVD, or ENO property. A third-order ENO scheme using reconstruction via primitive function approach is also described. Numerical results for one-dimensional and two-dimensional gasdynamic problems indicate that ENO scheme performs better than the other two while the TVB has the least satisfactory results. A modified eigenvalue approach is proposed to improve the TVB scheme. Results using this approach for TVD, TVB, and ENO schemes are also included.

This study presents the implementation of a scalar transport algorithm in the recently developed General Ocean Model (GOM), a three-dimensional, unstructured grid, finite volume/finite difference model. Solving the advection–diffusion transport equation is an essential part of any ocean circulation model since the baroclinic density gradient distinguishes saline water from freshwater. To achieve both high accuracy and computational efficiency, we adopted a second-order semi-implicit Total Variation Diminishing (TVD) scheme. The TVD approach, known for its ability to suppress non-physical oscillations near steep gradients, provides a higher-fidelity representation of salinity fronts without introducing significant numerical artifacts. The TVD algorithm is constructed with the first-order Upwind scheme, which is known for suffering from excessive numerical diffusion, and the higher-order anti-diffusive flux term. The implemented transport algorithm is evaluated using two standard test cases, an ideal lock exchange problem and a U-shaped channel problem, and it is further applied to simulate salinity dynamics in Mobile Bay, Alabama. The model results from both the first-order Upwind and second-order TVD schemes are compared. The results indicate that the TVD scheme marginally improves the resolution of salinity fronts while maintaining computational stability and efficiency. The implementation enables a flexible and straightforward transition between the first-order scheme, which is faster than the second-order scheme, and the second-order scheme, which is less diffusive than the first-order scheme, enhancing the GOM’s capability for realistic and efficient salinity simulations in a tidally driven estuarine system.

In recent studies, the type of shock-capturing scheme called total variation-diminishing (TVD) scheme, was proposed by Harten, and was applied widely in gas dynamics. The TVD scheme has second-order accuracy, is oscillation-free across discontinuities, and does not require additional artificial viscosity. Therefore it was introduced to solve hydrodynamics for free-surface flows, in particular, for recent complex dam-break flows. Delis and Skeels made a comparison with several different TVD schemes (i.e., symmetric, upwind, TVD-MacCormack, and MUSCL scheme) to predict one-dimensional (1D) dam-break flows. Based on the above research results, a high-resolution algorithm model is proposed for the solution of the 2D shallow water equations by adopting the numerical flux for the TVD scheme, the Strang type operator splitting method and the finite volume method (FVM).And this model was used to predict the flood evolution process caused by the instantaneous partial dam-break, and the simulating results indicate that this model is fairly effective for simulating dam-break flood waves.

High-resolution total variation diminishing (TVD) schemes in the framework of the finite volume method are presented and evaluated for hydraulic shock wave modeling. Three approximate Riemann solvers, namely the FVS, Roe and Osher schemes, are extended to high-resolution TVD schemes based on the direct MUSCL-Hancock (DMH) slope limiter approach. The TVD schemes are then used to develop numerical models to compute water depth and velocity. The numerical models developed are then verified through simulations of the dam-break flows, the oblique hydraulic jump, and the shock-on-shock interaction. The numerical models with TVD schemes are capable of capturing discontinuous shock waves without any spurious oscillation. A comparison of numerical efficiency shows that the Osher-DMH scheme coupled with van Leer limiter performs the best among the proposed TVD schemes. Applications of the Osher-DMH scheme to flows of partial dam-break experiments have shown that the simulated water depths agree well with the measured data.

This article presents some advantages using a shape-preserving total variation diminishing (TVD) advection scheme in an ecosystem model. The superbee flux-limiter has been used to the second-order Lax–Wendroff advection scheme to make it TVD. We performed simulations for three shelf sea regions with different characteristic time scales, namely, the North Sea, the Barents Sea, and the Baltic Sea. To explore the advantages, we also performed reference runs with the much simpler and computationally cheaper upwind advection scheme. Frontal structures are much better resolved with the TVD scheme. The bottom salinity in the Baltic Sea stays at realistic values throughout the 10-year simulation with the TVD scheme, while with the upwind scheme, it drifts towards lower values and the permanent haline stratification in the Baltic is almost completely eroded within one seasonal cycle. Only when applying TVD for both the vertical and horizontal advections the model succeeded to preserve haline stratification in the decadal simulation. Lower trophic level patterns are far better reproduced with the TVD scheme, and for the estimated cod larval survival, the advantages seem to be even stronger. Simulations using the TVD-derived prey fields identified distinct regions such as Dogger Bank to favor potential larvae survival (PLS), which did not appear as particularly favorable in the upstream simulations. The TVD scheme needs about 25 % more time on the central processing unit (CPU) in case of a pure hydrodynamic setup with only two scalar state variables (Barents Sea application). The additional CPU time cost increases for a coupled physical–biological model application with a large number of state variables. However, this is more than compensated by all the advantages found, and, hence, we conclude that it is worthwhile to use the TVD scheme in our ecosystem model.

A new family of high accuracy Total Variation Diminishing (TVD) schemes has been developed. Members of the family include the conventional second-order TVD upwind scheme, various other second-order accurate TVD schemes with lower truncation error, and even a third-order accurate TVD approximation. All the schemes are defined with a five-point grid bandwidth. In this paper, the new algorithms are described for scalar equations, systems, and arbitrary coordinates. Selected numerical results are provided to illustrate the new algorithms and their properties.

Numerical simulations of acoustic waves in a shear layer and in an idealized combustion chamber using high resolution Total Variation Diminishing (TVD) schemes has been carried out to study the effects of inherent scheme dissipation and dispersion errors of this class of problems. The numerical results are compared against available exact solutions to quantify these errors. Several popular TVD limiters widely used in the Computational Fluid Dynamics (CFD) community have been assessed. Osher-Chakravarthy limiters are modified so that they can be used in explicit schemes. Among all the limiters investigated, Osher-Chakravarthy third-order limiter is identified as having performed the best. It is also found that all TVD schemes have exceptionally small dispersive errors.

In this study, the following total variation diminishing (TVD) schemes for solving the Navier-Stokes equations have been tested: the Chakravarthy and Szema (1985) upwind biased TVD scheme, the Harten's upwind TVD scheme described by Yee et al. (1983), and the Yee's (1985) symmetric TVD scheme. The schemes have been compared using three test cases. The first case was the one-dimensional shock tube problem which tested the shock-capturing abilities of the schemes. Chakravarthy's and Harten's schemes gave similar results which were found to be more accurate than the results from Yee's scheme. The second case was a compressible boundary layer which tested the schemes's abilities to solve fiscous flows. In this case, the three schemes yielded almost identical results. Finally, the shock/boundary-layer interaction case studied experimentally by Hakkinen et al. (1959) was computed. Here, Chakravarthy's and Yee's schemes compared most favorably with the published data, with Yee's scheme giving slightly better results.

A Total Variation Diminishing (TVD) scheme for multi-species transport with first-order reaction network in multidimensional space is discussed in this article. The partial differential equations which describe this multi-species transport with chain reactions are in the form of a coupled system. This system is then solved by the TVD scheme with various flux limiters. The numerical diffusion controlled by the flux limiters is explained in detail. The stability and consistency conditions of the TVD scheme is also derived. The relation between the flux limiters and mesh parameters is obtained through stability conditions. A necessary condition for a scheme to be TVD is also derived.

Total variation diminishing (TVD) schemes have been recently introduced for the calculation of the one-dimensional unsteady flow in the inlet and outlet pipes of internal combustion engines. This paper describes the flux difference splitting technique (with first- or second-order upwind fluxes) for the classic TVD schemes. To avoid problems at nodes with a section change, a new TVD scheme is developed. This paper further describes a method to impose the boundary condition at the pipe end, independent of the numerical scheme used. This is shown for a reservoir inlet of the pipe and a subsonic outlet flow. For two test cases (the shock-tube and the tapered-pipe calculation), the new TVD algorithm is compared with the classic TVD schemes. The evaluation shows that the new cell—vertex TVD scheme with superbee limiter in two stage form combines a high accuracy with an exact representation of the mass flow in each of the nodes.

An upwind total variation diminishing (TVD) scheme and a predictor-corrector symmetric TVD scheme were used to numerically simulate the blast wave diffraction on a stationary object. The objective is to help design an optimum configuration so that lateral motion is minimized and at the same time vortex shedding and flow separation are reduced during a blast wave encounter. Results are presented for a generic configuration for both a coarse grid and a fine grid to illustrate the global and local diffraction flow fields. Numerical experiments for the shock wave reflection on a wedge are also included to validate the current approach. Numerical study indicated that these TVD schemes are more stable and produced higher shock resolution than classical shock capturing methods such as the explicit MacCormack scheme.

Three approaches to building one-dimensional shape-preserving advection schemes, based on TVD (total variation diminishing) schemes, on positive schemes, and on the universal limiter, are shown to lead to the same constraints on the fluxes between grid boxes. Thus, although they have slightly different conceptual bases, the three approaches lead to mathematically equivalent schemes.

A comparison of different numerical algorithms used in commercial codes for the calculation of the one-dimensional unsteady flow in the pipes of the inlet and exhaust systems of internal combustion engines is presented in this work. The comparison is made between the Method Of Characteristics (MOC), different Lax-Wendroff schemes, first order upwind schemes and the newest TVD (Total Variation Diminishing) schemes. These algorithms are representative for the complete evolution noticed in the computer codes from the beginning of their use to the present state of the art. Two models of realistic problems in engine simulation tasks are considered: the shock tube calculation (so called Sod’s problem) and the calculation in a tapered pipe. The first test case simulates the exhaust valve opening and releasing a pressure (shock)wave in the exhaust manifold while the other test case covers any gradual variation in the cross section of the manifold pipes. For both test cases computed results are compared with an exact solution and computer time and accuracy are evaluated. None of the examined schemes is completely satisfactory. They either show too much overshoots (for the first test case), or they have local discretization errors (at the section changes of the second test case). A new TVD scheme is proposed that does not introduce any of the foregoing inaccuracies. With this scheme overshoots and dips are eliminated and mass balances are fulfilled, while maintaining high accuracy. [S0742-4795(00)00304-5]

A TVD (Total Variation Diminishing) scheme based on a compressible Navier-Stokes equation was applied to axisymmetric flow in a low pressure impactor with a circular nozzle. This TVD scheme was a type combined to a two-step MacCormack scheme. The shock was clearly observed at the half distance between the outlet of jet and the impaction plate. The critical stagnation pressure ratio of the impactor designed by Hering et al. was found to be about 0.40. The separation efficiency curve of the impactor was not dependent on the stagnation pressure ratio, nor the distance between the jet and the plate when the stagnation pressure ratio was less than 0.70.