A comparative study of the efficiency of jet schemes

Journal of Hydroinformatics, 2006

Using the interpolation polynomial method, major upwind explicit advection–diffusion schemes of up to fifth-order accuracy are rederived and their properties are explored. The trend emerges that the higher the order of accuracy of an advection scheme, the easier is the task of scheme stabilization and wiggling suppression. Thus, for a certain range of the turbulent diffusion coefficient, the stability interval of third- and fifth-order up-upwind explicit schemes can be extended up to three units of the Courant number (0≤c≤3). Having good phase behavior, advection odd-order schemes are stable within a single computational cell (0≤c≤1). By contrast, even-order schemes are stable within two consecutive grid-cells (0≤c≤2), but exhibit poor dispersive properties. Stemming from the finding that considered higher-order upwind schemes (even, in particular) can be expressed as a linear combination of two lower-order ones (odd in this case), the best qualities of odd- and even-order algorithm...

Arxiv preprint astro-ph/9807241, 1998

Simple modifications for higher-order Godunovtype difference schemes are presented which allow for accurate advection of multi-fluid flows in hydrodynamic simulations. The constraint that the sum of all mass fractions has to be equal to one in every computational zone throughout the simulation is fulfilled by renormalizing the mass fractions during the advection step. The proposed modification is appropriate for any difference scheme written in conservation form. Unlike other commonly used methods it does not violate the conservative character of the advection method. A new steepening mechanism, which is based on modification of interpolation profiles, is used to reduce numerical diffusion across composition discontinuities. Additional procedures are described, which are necessary to enforce monotonicity. Several numerical experiments are presented which demonstrate the capability of our Consistent Multi-fluid Advection (CMA) method in case of smooth and discontinuous distributions of fluid phases and under different hydrodynamic conditions. It is shown that due to the reduced diffusivity of the proposed scheme the abundance of some heavy elements obtained from hydrodynamic simulations of type II supernova explosions can change by a factor of a few in the most extreme cases.

20th AIAA Computational Fluid Dynamics Conference, 2011

Higher-order discretizations have the potential to reduce the computational cost required to achieve a desired error level. In this study, we consider higher-order discretizations of the conservation equations suitable for unstructured, triangular grids. In particular, the methods studied include continuous (SUPG/GLS) and classical discontinuous Galerkin (DG) finite element methods, the correction procedure via reconstruction (CPR) formulations of the DG and spectral volume methods, and cell and vertex-centered finite volume (FV) algorithms. This paper presents subsonic and supersonic, inviscid results for a canonical set of aerodynamic applications. Error convergence and computational performance of these discretizations are compared, and preliminary results indicate that the methods perform relatively similarly. When singularities are present in the flow solutions and uniformly refined meshes are used, all methods fail to achieve optimal convergence rates, and the performance benefits of the higher-order discretizations are reduced; adaptive meshing improves the efficiency of the higher-order method and recovers optimal convergence rates.

Computer Methods in Applied Mechanics and Engineering, 1987

In this note we present results of an accuracy analysis of a recent characteristic-based Galerkin method suited for advection-dominated problems. The analysis shows that the numerical propagation characteristics of the explicit time-stepping scheme which uses linear basis functions for spatial discretization are superior to those of the related classical Lax-Wendroff method and the implicit Crank-Nicolson scheme. The model is subjected to three analytical test problems which embrace many essential realistic features of environmental and coastal hydrodynamic applications: pure advection of a steep Gaussian profile, dispersion of a continuous source in an oscillating flow, and long-wave propagation with bottom frictional dissipation in a rectangular channel. The numerical results demonstrate that the accuracy achieved with the present scheme is excellent and comparable to that of a characteristic-based finite difference scheme which uses Hermitian cubic interpolating polynomials. The results reported herein suggest strongly further use and testing of this robust model in engineering practice.

Applied numerical mathematics, 2003

We consider the classical problem of a two-dimensional laminar jet of incompressible fluid flowing into a stationary medium of the same fluid [H. Schlichting, Boundary-Layer Theory, McGraw-Hill, 1979]. The equations of motion are the same as the boundary layer equations for flow over an infinite flat plate, but with different boundary conditions. It has been shown [A.R. Ansari et al., Parameter robust numerical solutions for the laminar free jet, submitted] that using an appropriate piecewise uniform mesh, numerical solutions together with their scaled discrete derivatives are obtained which are parameter (i.e., viscosity ν) robust with respect to both the number of mesh nodes and the number of iterations required for convergence. We prove that there do not exist fitted operator schemes which converge ν-uniformly if the fitting coefficients are independent of the problem data.

paper describes studies to quantify the numerical errors caused by 'false diffusion', and to compare the performance of alternative numerical schemes for describing elliptic convective flow and heat transfer, within supersonic jets mixing into supersonic or subsonic streams. Results obtained are presented and discussed. Eleven schemes were considered in this study, but converged solutions were obtained with only five of them. Results obtained with the successful schemes are presented and discussed. It is concluded that for the high-shearing, high-velocity flows considered, the 'upwind' differencing scheme is probably the best choice, despite its dissipative nature and that the numerical errors associated with its use are no more significant than those introduced by uncertainties in the turbulence models. Pressure WaYe /.-Jet axls . . 7 0 Crown Copyright, 1987. FIG. 1. Flowfield around the nozzle exit.

International Journal of Computational Fluid Dynamics, 2003

2014

The purpose of this work is to compare two numerical formulations for unstructured grids that achieve high-order spatial discretization for compressible aerodynamic flows. High-order methods are necessary on the analysis of complex flows to reduce the number of mesh elements one would otherwise need if using traditional second-order schemes. In the present work, the 2-D Euler equations are solved numerically in a finite volume, cell centered context. The third-order Weighted Essentially Non-Oscillatory (WENO) and Spectral Finite Volume (SFV) methods are considered in this study for the spatial discretization of the governing equations. Time integration uses explicit, Runge-Kutta type schemes. Two literature test cases, one steady and one unsteady, are considered to assess the resolution capabilities and performance of the two spatial discretization methods. Both methods are suitable for the aerospace applications of interest. However, each method has characteristics that excel over the other scheme. The present results are valuable as a form of providing guidelines for future developments regarding high-order methods for unstructured meshes.

This Philosophiae Doctor thesis presents the motivation, objectives and reasoning behind the undertaken project. This research, study the capability of compressible Implicit Large Eddy Simulation (ILES) in predicting free shear layer flows, under different free stream regimes (Static and Co-flow jets). Unsteady flows or jet flows are non-uniform in structure, temperature, pressure and velocity. Turbulent mixing is of particular importance for the developing of this class of flows. As a shear layer is formed immediately downstream of the jet exhaust, an early linear instability involving exponential growth of small perturbations is introduced at the jet discharge. Beyond this development stage, in the non-linear Kelvin-Helmholtz instability region large scale vortex rings roll up, and their dynamics of formation and merging become the defining feature of the transitional shear flow into fully developed regime. This class of flows is particularly relevant to numerical predictions, as the extreme nature of the flow in question is considered as a benchmark; however, experimental data should be selected carefully as some results are controversial. To qualify the behaviour of unsteady flows, some important criteria have been selected for the analysis of the flow quantities at different regions of the flow field (average velocities, Reynolds stresses and dissipation rates). A good estimation of high-order statistics (Standard Deviation, Skewness and Kurtosis) correspond to mathematical steadiness and convergence of results. From the physical point of view, similarity analysis between jet's wake sections reveals physical steadiness in results. Spectral analysis of the different regions of the flow field could be used as a sign that the energy cascade is correctly predicted or efficiently enough since this is where the smallest scales are usually present and which in effect require to be modelled by the different numerical schemes. The flow solver has been reviewed and improved. The former, a revised version of i Abstract ii the reconstruction numerical schemes (WENO 5 th and WENO 9 th orders) has been performed and tested, the correspondent results have been compared against analytical data; the latter, correction of the method to compute the Jacobian of the transformation (singularity correction), by changing from the standard algebraic to geometric method, and augmented with transparent boundary condition, giving mathematical and physical meaning to the obtained results. The flow solver improvements and review have been verified and validated through simulations of a compressible Convergent-Divergent Nozzle (CDN), and the standard and a modified version of the Shock tube test cases, where the results are gained with minimal modelling effort. The study of numerical errors associated with the simulations of turbulent flows, for unsteady explicit time step predictions, have been performed and a new formula proposed. Ten different computational methods have been employed in the framework of ILES and computations have been performed for a jet flow configuration for which experimental data and DNS are available. It can be seen that a numerical error bar can be defined that takes into account the errors arising from the different numerical building blocks of the simulation method. The effects of different grids, Riemann solvers and numerical reconstruction schemes have been considered, however, the approach can be extended to take into account the effects of the initial and boundary conditions as well as subgrid scale modelling, if applicable. From the physical analysis several observations were established, revealing that differences in terms of jet's core size are not an important parameter in terms of quantification and qualification of predictions, in other words, data should be reduced to the jet's inertial reference system. Moreover, the comparative study has been performed to identify the differences between Riemann solvers (CBS and HLLC), Low Mach number Limiting/Corrections (LMC), numerical reconstruction schemes (MUSCL and WENO) and spatial order of accuracy (2 nd-order LMC, 5 th-order LMC and 9 th-order schemes) in combination with the most efficient cost/resolution discretization level (Medium mesh). The comparisons between results reveals for the Static and Co-Flow jets that the CBS MUSCL 5 th-order LMC and the HLLC MUSCL 5 th-order LMC as the most accurate schemes in predicting this class of flows, accordingly. Furthermore, the selected numerical methods show to be in accordance with the empirical (Static) and experimental (Co-flow) results in terms of resonance frequency and/or Strouhal number; also, the expected behaviour in terms of spectral energy decay rate throughout the jet's central line is observed. To conclude the study of the Static jet case, a possible explanation for the jet's Abstract iii buoyancy effect is presented. First, I am grateful and thankful to my parents for their economical support throughout the duration of the Ph.D. Words are not enough to show my appreciation and I thank them for the sacrifices they have done towards me. Also my deepest gratitude goes to my relatives for their understanding and always being there for me when in need, in particular to my Pearl. I sincerely thank my supervisor Professor Dimitris Drikakis for his effort, guidance and patience he has invested as well as invited me as his Ph.D. student. His keen advice and expertise guided my Ph.D. from it's beginnings all the way to the very end. It goes without saying, his open mind in terms of being receptive to different approaches in order to achieve a realistic solution, even when we got a five minutes meeting.

1998

Conventional explicit nite di erence schemes for the advection equations are subject to the time step restrictions dictated by the CFL condition. In many situations, time step sizes are not chosen to satisfy accuracy requirements but rather to satisfy the CFL condition. In this paper we present explicit algorithms which are stable far beyond the CFL restriction. Similar or even better accuracy can be achieved with a much larger time step size. The idea is matching the stencil and the real domain of dependence by characteristic analysis. Numerical tests are done for linear advection equations as well as the Burgers equation.

Atmospheric Environment, 2001

A new numerical algorithm using quintic splines is developed and analyzed: quintic spline Taylor-series expansion (QSTSE). QSTSE is an Eulerian #ux-based scheme that uses quintic splines to compute space derivatives and Taylor series expansion to march in time. The new scheme is strictly mass conservative and positive de"nite while maintaining high peak retention. The new algorithm is compared against accurate space derivatives (ASD), Galerkin "nite element techniques, and the Bott scheme. The cases presented include classical rotational "elds, deformative "elds, as well as a full-scale aerosol model. Research shows that QSTSE presents signi"cant improvements in speed and oscillation suppression against ASD. Furthermore, QSTSE predicts some of the most accurate results among the schemes tested.

2007

This article was originally published in a journal published by Elsevier, and the attached copy is provided by Elsevier for the author's benefit and for the benefit of the author's institution, for non-commercial research and educational use including without limitation use in instruction at your institution, sending it to specific colleagues that you know, and providing a copy to your institution's administrator.

Journal of Applied Mathematics, 2013

Three numerical methods have been used to solve the one-dimensional advection-diffusion equation with constant coefficients. This partial differential equation is dissipative but not dispersive. We consider the Lax-Wendroff scheme which is explicit, the Crank-Nicolson scheme which is implicit, and a nonstandard finite difference scheme (Mickens 1991). We solve a 1D numerical experiment with specified initial and boundary conditions, for which the exact solution is known using all these three schemes using some different values for the space and time step sizes denoted byhandk, respectively, for which the Reynolds number is 2 or 4. Some errors are computed, namely, the error rate with respect to theL1norm, dispersion, and dissipation errors. We have both dissipative and dispersive errors, and this indicates that the methods generate artificial dispersion, though the partial differential considered is not dispersive. It is seen that the Lax-Wendroff and NSFD are quite good methods to ...

ArXiv, 2021

In this paper, we present a fully local second-order upwind scheme, applicable on generic meshes. This is done by hybridisation, which is achieved by introducing unknowns on each edge of the mesh. By doing so, fluxes only depend on values associated to a single cell, and thus, this scheme can easily be applied even on cells near the boundary of the domain. Another advantage of hybridised schemes is that static condensation can be employed, leading to a very efficient implementation. A convergence analysis, which also covers a flux-limited TVD variant of the scheme, is then presented. Numerical results are also given in order to compare this with a hybridised first-order upwind scheme and a classical cell-centered second-order upwind type scheme.

Journal of Computational Physics, 2006

A new, conservative semi-Lagrangian formulation is proposed for the discretization of the scalar advection equation in flux form. The approach combines the accuracy and conservation properties of the Discontinuous Galerkin (DG) method with the computational efficiency and robustness of Semi-Lagrangian (SL) techniques. Unconditional stability in the von Neumann sense is proved for the proposed discretization in the one-dimensional case. A monotonization technique is then introduced, based on the Flux Corrected Transport approach. This yields a multi-dimensional monotonic scheme for the piecewise constant component of the computed solution that is characterized by a smaller amount of numerical diffusion than standard DG methods. The accuracy and stability of the method are further demonstrated by two-dimensional tracer advection tests in the case of incompressible flows. The comparison with results obtained by standard SL and DG methods highlights several advantages of the new technique.