Issures in Discontinuous High-Order Methods: Broadband Wave Computation and Viscous Boundary Layer Resolution

A new discontinuous formulation named Correction Procedure via Reconstruction (CPR) was developed for conservation laws. CPR is an efficient nodal differential formulation unifying the discontinuous Galerkin (DG), spectral volume (SV) and spectral difference (SD) methods, is easy to implement. In this thesis, we deal with two issues: the efficient computation of broadband waves, and the proper resolution of a viscous boundary layer with the high-order CPR method. A hybrid discontinuous space including polynomial and Fourier bases is employed in the CPR formulation in order to compute broad-band waves. The polynomial bases are used to achieve a certain order of accuracy, while the Fourier bases are able to exactly resolve waves at a certain frequency. Free-parameters introduced in the Fourier bases are optimized in order to minimize both dispersion and dissipation errors by mimicking the dispersion-relation-preserving (DRP) method for a one-dimensional wave problem. For the one-dimensional wave problem, the dispersion and dissipation properties and the optimization procedure are investigated through a wave propagation analysis. The optimization procedure is verified with a wave propagation analysis and several numerical tests. The two-dimensional wave behavior is investigated through a wave propagation analysis and the wave propagation properties are verified with a numerical test of the two-dimensional acoustic wave equation. In order to understand the mesh size requirement to resolve a viscous boundary layer using CPR method, grid resolution studies are performed. . It is well known that the mesh size, which is defined from non-dimensional wall distance y^+=1, gives accepted results to simulate viscous boundary layer problem for 2nd order finite volume method. For high-order CPR formulation, what grid size y^+ is required to match the results from the 2nd order finite volume method with y^+=1. 1D and 2D burger\u27s equation are used as the viscous boundary layer model problem. Skin friction is used as the indicator of accuracy for the resolution of a boundary layer. Keywords: (Correction Procedure via Reconstruction), A Hybrid Discontinuous Space, Wave Propagation Analysis, Grid Resolution Study, Method of Manufactured Solution

Structural schemes for one dimension stationary equations

In this paper, we propose a new paradigm for finite differences numerical methods, based on compact schemes to provide high order accurate approximations of a smooth solution. The method involves its derivatives approximations at the grid points and the construction of structural equations deriving from the kernels of a matrix that gathers the variables belonging to a small stencil. Numerical schemes involve combinations of physical equations and the structural relations. We have analysed the spectral resolution of the most common structural equations and performed numerical tests to address both the stability and accuracy issues for popular linear and non-linear problems. Several benchmarks are presented that ensure that the developed technology can cope with several problems that may involve non-linearity.S. Clain acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/00324/2020. R. M. S. Pereira acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/04650/2020. P. A. Pereira acknowledges the financial support by Portuguese Funds through Foundation for Science and Technology (FCT) in the framework of the Strategic Funding UIDB/00013/2020. Diogo Lopes acknowledges the financial support by national funds (PIDDAC), through the FCT – Fundação para a Ciência e a Tecnologia and FCT/MCTES under the scope of the projects UIDB/05549/2020 and UIDP/05549/2020. S. Clain and R. M.Pereira acknowledge the financial support by FEDER – Fundo Europeu de Desenvolvimento Regional, through COMPETE 2020 – Programa Operacional Fatores de Competitividade, and the National Funds through FCT, project N◦. POCI-01-0145-FEDER-028118

CABARET in the ocean gyres

Author Posting. © The Author(s), 2009. This is the author's version of the work. It is posted here by permission of Elsevier B.V. for personal use, not for redistribution. The definitive version was published in Ocean Modelling 30 (2009): 155-168, doi:10.1016/j.ocemod.2009.06.009.A new high-resolution Eulerian numerical method is proposed for modelling quasigeostrophic ocean dynamics in eddying regimes. The method is based on a novel, second-order non-dissipative and lowdispersive conservative advection scheme called CABARET. The properties of the new method are compared with those of several high-resolution Eulerian methods for linear advection and gas dynamics. Then, the CABARET method is applied to the classical model of the double-gyre ocean circulation and its performance is contrasted against that of the common vorticity-preserving Arakawa method. In turbulent regimes, the new method permits credible numerical simulations on much coarser computational grids.Supports from the Royal Society of London and from the Mary Sears Visitor Grant are acknowledged by SK with gratitude. The work of VG was supported by the Russian Foundation for Basic Research (RFBR), grant 06-01-00819a. Funding for PB was provided by the NSF grant 0725796

ULTRA-SHARP nonoscillatory convection schemes for high-speed steady multidimensional flow

For convection-dominated flows, classical second-order methods are notoriously oscillatory and often unstable. For this reason, many computational fluid dynamicists have adopted various forms of (inherently stable) first-order upwinding over the past few decades. Although it is now well known that first-order convection schemes suffer from serious inaccuracies attributable to artificial viscosity or numerical diffusion under high convection conditions, these methods continue to enjoy widespread popularity for numerical heat transfer calculations, apparently due to a perceived lack of viable high accuracy alternatives. But alternatives are available. For example, nonoscillatory methods used in gasdynamics, including currently popular TVD schemes, can be easily adapted to multidimensional incompressible flow and convective transport. This, in itself, would be a major advance for numerical convective heat transfer, for example. But, as is shown, second-order TVD schemes form only a small, overly restrictive, subclass of a much more universal, and extremely simple, nonoscillatory flux-limiting strategy which can be applied to convection schemes of arbitrarily high order accuracy, while requiring only a simple tridiagonal ADI line-solver, as used in the majority of general purpose iterative codes for incompressible flow and numerical heat transfer. The new universal limiter and associated solution procedures form the so-called ULTRA-SHARP alternative for high resolution nonoscillatory multidimensional steady state high speed convective modelling

Large eddy simulations of turbulent flows and aeroacoustics noise predictions using high-order methods

Large eddy simulation (LES) has received increased attention in industrial applications over the past few decades for challenging vortex-dominated turbulent flows. Direct numerical simulations (DNS) have also been used to study interesting flow physics at low to moderate Reynolds numbers. This is due to the advancements in computational algorithms and computing power of modern computers which paved the way for simulating more practical flow problems. In its 2030 vision, NASA has predicted that scale resolving simulations will be increasingly used for vortex-dominated turbulent flow simulations such as rotorcraft flows and turbomachinery flows in aircraft engines. Multiple international workshops on high-order CFD methods have conclusively demonstrated the advantage of high-order methods over 1st and 2nd order ones in accuracy/efficiency for such scale-resolving simulations due to their lower dispersion and dissipation errors. In this dissertation, we analyze the performance of high-order CFD methods for LES using Fourier analysis techniques. We also propose new ideas and approaches for studying the dispersion/dissipation of high-order multi-degree of freedom methods. In addition, we study aspects of mesh resolution requirements for DNS and LES of turbomachinery flows using the high-order flux reconstruction/correction procedure via reconstruction (FR/CPR) method. Finally, we offer an efficient implementation of the Ffowcs-Williams & Hawkings (FWH) acoustic analogy formulation in a hybrid framework with the FR/CPR method for jet noise predictions of supersonic jets

ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA)

The proceedings of the Benchmark Problems in Computational Aeroacoustics Workshop held at NASA Langley Research Center are the subject of this report. The purpose of the Workshop was to assess the utility of a number of numerical schemes in the context of the unusual requirements of aeroacoustical calculations. The schemes were assessed from the viewpoint of dispersion and dissipation -- issues important to long time integration and long distance propagation in aeroacoustics. Also investigated were the effect of implementation of different boundary conditions. The Workshop included a forum in which practical engineering problems related to computational aeroacoustics were discussed. This discussion took the form of a dialogue between an industrial panel and the workshop participants and was an effort to suggest the direction of evolution of this field in the context of current engineering needs