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Two dimensional analysis of hybrid spectral/finite difference schemes for linearized compressible Navier-Stokes equations

Two dimensional analysis of hybrid spectral/finite difference schemes for linearized compressible Navier-Stokes equations
Two dimensional analysis of hybrid spectral/finite difference schemes for linearized compressible Navier-Stokes equations

This study seeks to compare different combinations of spatial dicretization methods under a coupled spatial temporal framework in two dimensional wavenumber space. The aim is to understand the effect of dispersion and dissipation on both the convection and diffusion terms found in the two dimensional linearized compressible Navier–Stokes Equations (LCNSE) when a hybrid finite difference/Fourier spectral scheme is used in the x and y directions. In two dimensional wavespace, the spectral resolution becomes a function of both the wavenumber and the wave propagation angle, the orientation of the wave front with respect to the grid. At sufficiently low CFL number where temporal discretization effects can be neglected, we show that a hybrid finite difference/Fourier spectral schemes is more accurate than a full finite difference method for the two dimensional advection equation, but that this is not so in the case of the LCNSE. Group velocities, phase velocities as well as numerical amplification factor were used to quantify the numerical anisotropy of the dispersion and dissipation properties. Unlike the advection equation, the dispersion relation representing the acoustic modes of the LCNSE contains an acoustic terms in addition to its advection and viscous terms. This makes the group velocity in each spatial direction a function of the wavenumber in both spatial directions. This can lead to conditions for which a hybrid Fourier spectral/finite difference method can become less or more accurate than a full finite difference method. To better understand the comparison of the dispersion properties between a hybrid and full FD scheme, the integrated sum of the error between the numerical group velocity Vgrp,full∗ and the exact solution across all wavenumbers for a range of wave propagation angle is examined. In the comparison between a hybrid and full FD discretization schemes, the fourth order central (CDS4), fourth order dispersion relation preserving (DRP4) and sixth order central compact (CCOM6) schemes share the same characteristics. At low wave propagation angle, the integrated errors of the full FD and hybrid discretization schemes remain the same. At intermediate wave propagation angle, the integrated error of the full FD schemes become smaller than that of the hybrid scheme. At large wave propagation angle, the integrated error of the full FD schemes diverges while the integrated error of the hybrid discretization schemes converge to zero. At high reduced wavenumber and sufficiently low CFL number where temporal discretization error can be neglected, it was found that the numerical dissipation of the viscous term based on the CDS4, DRP4, CCOM6 and isotropy optimized CDS4 schemes (CDS4 opt) schemes was lower than the actual physical dissipation, which is only a function of the cell Reynolds number. The wave propagation angle at which the numerical dissipation of the viscous term approaches its maximum occurs at π/ 4 for the CDS4, DRP4, CCOM6 and CDS4 opt schemes.

Computational Aero-acoustics (CAA), Explicit Runge–Kutta (RK) schemes, Finite difference, Fourier analysis, Fourier spectral
1573-7691
Tan, Raynold
c4070330-043a-420a-97fe-3d18142b6087
Ooi, Andrew
666f76c1-0f81-4233-baaf-ac51d58c6931
Sandberg, Richard D.
3085eb4d-a164-46c1-9998-82c3aac5cf7c
Tan, Raynold
c4070330-043a-420a-97fe-3d18142b6087
Ooi, Andrew
666f76c1-0f81-4233-baaf-ac51d58c6931
Sandberg, Richard D.
3085eb4d-a164-46c1-9998-82c3aac5cf7c

Tan, Raynold, Ooi, Andrew and Sandberg, Richard D. (2021) Two dimensional analysis of hybrid spectral/finite difference schemes for linearized compressible Navier-Stokes equations. Journal of Scientific Computing, 87 (2), [42]. (doi:10.1007/s10915-021-01442-x).

Record type: Article

Abstract

This study seeks to compare different combinations of spatial dicretization methods under a coupled spatial temporal framework in two dimensional wavenumber space. The aim is to understand the effect of dispersion and dissipation on both the convection and diffusion terms found in the two dimensional linearized compressible Navier–Stokes Equations (LCNSE) when a hybrid finite difference/Fourier spectral scheme is used in the x and y directions. In two dimensional wavespace, the spectral resolution becomes a function of both the wavenumber and the wave propagation angle, the orientation of the wave front with respect to the grid. At sufficiently low CFL number where temporal discretization effects can be neglected, we show that a hybrid finite difference/Fourier spectral schemes is more accurate than a full finite difference method for the two dimensional advection equation, but that this is not so in the case of the LCNSE. Group velocities, phase velocities as well as numerical amplification factor were used to quantify the numerical anisotropy of the dispersion and dissipation properties. Unlike the advection equation, the dispersion relation representing the acoustic modes of the LCNSE contains an acoustic terms in addition to its advection and viscous terms. This makes the group velocity in each spatial direction a function of the wavenumber in both spatial directions. This can lead to conditions for which a hybrid Fourier spectral/finite difference method can become less or more accurate than a full finite difference method. To better understand the comparison of the dispersion properties between a hybrid and full FD scheme, the integrated sum of the error between the numerical group velocity Vgrp,full∗ and the exact solution across all wavenumbers for a range of wave propagation angle is examined. In the comparison between a hybrid and full FD discretization schemes, the fourth order central (CDS4), fourth order dispersion relation preserving (DRP4) and sixth order central compact (CCOM6) schemes share the same characteristics. At low wave propagation angle, the integrated errors of the full FD and hybrid discretization schemes remain the same. At intermediate wave propagation angle, the integrated error of the full FD schemes become smaller than that of the hybrid scheme. At large wave propagation angle, the integrated error of the full FD schemes diverges while the integrated error of the hybrid discretization schemes converge to zero. At high reduced wavenumber and sufficiently low CFL number where temporal discretization error can be neglected, it was found that the numerical dissipation of the viscous term based on the CDS4, DRP4, CCOM6 and isotropy optimized CDS4 schemes (CDS4 opt) schemes was lower than the actual physical dissipation, which is only a function of the cell Reynolds number. The wave propagation angle at which the numerical dissipation of the viscous term approaches its maximum occurs at π/ 4 for the CDS4, DRP4, CCOM6 and CDS4 opt schemes.

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Accepted/In Press date: 17 February 2021
Published date: 18 March 2021
Additional Information: Funding Information: The authors would like to thank the University of Melbourne for funding this research. Publisher Copyright: © 2021, The Author(s).
Keywords: Computational Aero-acoustics (CAA), Explicit Runge–Kutta (RK) schemes, Finite difference, Fourier analysis, Fourier spectral

Identifiers

Local EPrints ID: 447546
URI: http://eprints.soton.ac.uk/id/eprint/447546
ISSN: 1573-7691
PURE UUID: 73be98f6-deed-407e-804d-470957d09419

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Date deposited: 15 Mar 2021 17:40
Last modified: 17 Mar 2024 06:23

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Contributors

Author: Raynold Tan
Author: Andrew Ooi
Author: Richard D. Sandberg

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