Analysis of a new high resolution upwind compact scheme

This paper describes the development of a second generation of low dispersion numerical schemes in the context of a nite volume discretization of the governing uid dynamic equations. Until recently, nite volume methods had not been endowed with techniques designed to mitigate numerical dispersion errors. The purpose of this study is to present a new family of low dispersion nite volume numerical schemes, the Type 2 formulation. The Type 1 family of schemes was presented in an earlier work. As in any nite volume method, the surface integrals are computed by using ow eld properties at the cell faces, interpolated from values stored at the cell nodes. The new family of schemes uses an interpolation procedure designed to optimize the di erence in sinusoidal wave properties occurring across a given cell interface. This procedure renders a more accurate representation of the wave v ariable di erences at higher wavenumbers. This new family of schemes is better suited for application in the ux di erence splitting algorithms commonly found in computer programs designed for computational uid dynamics. It is of particular interest that the new schemes mitigate numerical dispersion errors for both interface variables and for the di erences in wave v ariables across a cell interface. A n umber of classical acoustics problems are solved, and where possible, comparisons with other numerical and exact solutions are shown. It is especially signi cant that this new method has low n umerical dispersion and may be easily retro tted into existing nite volume codes.

20th INTERNATIONAL CONGRESS ON SOUND AND VIBRATION (ICSV20), BANGKOK, THAILAND, 2013

A class of higher-order compact upwind biased finite difference (FD) schemes based on the Taylor Series Expansion is developed in this work. The numerical accuracy of this family of compact upwind biased FD schemes is demonstrated by means of Fourier analysis wherein it is shown that these FD schemes have very low dispersion errors over a relatively large bandwidth of wave numbers as compared to explicit or non-compact schemes. The advantage of its use over compact central FD schemes is seen by the presence of inbuilt dissipation which damps only the unresolved high-frequency waves, thereby avoiding the need to use explicit artificial selective damping (ASD) terms. Compact downwind FD schemes are also formulated to obtain boundary closure schemes for expressing the spatial derivatives adjacent to and at the boundary nodes, thereby obtaining an overall FD scheme for the entire computational domain in terms of simple matrix operation. The stability of overall FD scheme in context with its use in the pseudo-characteristic formulation (PCF) of 1-D linearized Euler equations (with super-imposed mean flow) and anechoic boundary conditions is established by means of an eigenvalue analysis.

Aerospace Science and Technology, 2011

Based on the curvilinear grids, grid-optimized upwind dispersion-relation-preserving (GOUPDRP) finite difference schemes are studied in the paper. GOUPDRP finite difference schemes can automatically resolve waves with much shorter wavelengths and need not consider the filter or the explicit dissipation terms, and have much smaller dispersive and dissipative errors. Recently, GOUPDRP schemes on the non-uniform Cartesian meshes have been developed and tested by authors of this paper, others optimized upwind finite schemes just only on the uniform Cartesian grids also have been investigated; nevertheless, practical problems existing in aeroacoustics are unusually restricted to the uniform or non-uniform Cartesian grids, the corresponding computational grids could be curvilinear ones owing to the different complex geometries. In contrast to other optimized upwind schemes on the uniform Cartesian grids, GOUPDRP schemes can be easily extended to the curvilinear meshes. In order to determine the optimization coefficients of GOUPDRP schemes on the curvilinear grids, a general curvilinear transformation is introduced from physical domain to computational domain. Mathematic formulations related to GOUPDRP schemes on the curvilinear meshes are also derived in detail in the paper. To demonstrate the effectiveness of GOUPDRP schemes, benchmark problems are solved using the curvilinear grids; calculated solutions are compared with their analytical counterparts. Numerical results show that GOUPDRP schemes on the curvilinear grids are highly efficient and relatively potential to solve practical problems in aeroacoustics.

Computers & Fluids, 2012

Residual-Based Compact (RBC) schemes are reviewed and applied to realistic compressible flow problems. The principle and advantages of high-order RBC schemes are first presented. Then, an implementation in the elsA code of the 3rd-order RBC scheme is used for RANS calculations of the transonic flow over the ONERA M-6 wing and of the flow in a CME2 compressor stage. Finally, the 7th-order RBC scheme is applied to the complete Euler equations for computing a planar aeroacoustic problem and the propagation of a spinning acoustic mode in an axisymetric aeroengine inlet. Numerical results demonstrate the accuracy and efficiency of the RBC approach.

Journal of Scientific …, 2004

A variety of aeroacoustic problems involve small-amplitude linear wave propagation. Highorder schemes have the accuracy and the low dispersion and dissipation wave propagation properties that are required to model linear acoustic waves with minimal spatial resolution. This review presents a selection of high-order finite difference time-explicit schemes for aeroacoustic applications. A scheme selection method based on the computational cost for a given accuracy level is proposed.

Applied Mathematics and Computation, 2010

This article presents a time-accurate numerical method using high-order accurate compact finite difference scheme for the incompressible Navier-Stokes equations. The method relies on the artificial compressibility formulation, which endows the governing equations a hyperbolic-parabolic nature. The convective terms are discretized with a third-order upwind compact scheme based on flux-difference splitting, and the viscous terms are approximated with a fourth-order central compact scheme. Dual-time stepping is implemented for time-accurate calculation in conjunction with Beam-Warming approximate factorization scheme. The present compact scheme is compared with an established non-compact scheme via analysis in a model equation and numerical tests in four benchmark flow problems. Comparisons demonstrate that the present third-order upwind compact scheme is more accurate than the non-compact scheme while having the same computational cost as the latter.

This volume contains the proceedings of the ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA). CAA is a relatively new area of research addressing issues relevant to the acoustic propagation of sound generated by fluid flow. Advances in computer technology make addressing these issues in a more detailed manner a possibility. Such advances allow the treatment of the fully nonlinear propagation problem as well as the direct computation of the acoustic sources, the sound generation problem. These possibilities are expected to benefit both the validation of models that have been developed as well as help develop better models for more complex flows. In situations where calculations are not prohibitively expensive a direct computation of the acoustic and source fields becomes possible.

13th AIAA/CEAS Aeroacoustics Conference (28th AIAA Aeroacoustics Conference), 2007

The large disparity between the time and length scales of an acoustically active flow field, and the ones of the resulting generated acoustic field, is an issue in computational aeroacoustics (CAA). Numerical schemes used to calculate the time and space derivatives in CAA should exhibit a low dispersion and dissipation error. This paper presents the implementation of a high-order finite difference scheme to model CAA problems. The numerical scheme consists of a sixth-order prefactored compact scheme coupled with the 4-6 Low Dispersion and Dissipation Runge-Kutta (LDDRK) time marching scheme. Onesided explicit boundary stencils are implemented and a buffer zone is used to eliminate spurious numerical waves. Results are in good agreement with a 1-D advection equation benchmark problem. This constitutes a preliminary validation for the scheme to explore and investigate the acoustic propagation of sound generated aerodynamically.

A modification of the Roe scheme aimed at low Mach number flows is discussed. It improves the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. No conflict is observed between the reduced dissipation and the accuracy or stability of the scheme in any of the investigated test cases ranging from low Mach number potential flow to hypersonic viscous flow around a cylinder.