High resolution conjugate filters for the simulation of flows
Y. C. Zhou, G. W. Wei
TL;DR
The paper tackles the challenge of achieving high-resolution, low-dissipation simulations of flows with shocks and high-frequency content. It introduces a CFOR scheme built on a Hermite kernel realized via the discrete singular convolution (DSC) framework, coupling high-pass derivative filters with a conjugate low-pass filter to suppress spurious oscillations. The authors provide Fourier analyses and extensive 1D/2D tests, showing that the CFOR-Hermite method attains near-optimal resolution and excellent shock-capturing with very low point-per-wavelength requirements, outperforming prior CFOR realizations in certain shock-dominated scenarios. The work suggests that CFOR-Hermite can enable accurate large-scale CFD with reduced mesh requirements, offering a flexible, high-resolution alternative to traditional shock-capturing schemes.
Abstract
This paper proposes a Hermite-kernel realization of the conjugate filter oscillation reduction (CFOR) scheme for the simulation of fluid flows. The Hermite kernel is constructed by using the discrete singular convolution (DSC) algorithm, which provides a systematic generation of low-pass filter and its conjugate high-pass filters. The high-pass filters are utilized for approximating spatial derivatives in solving flow equations, while the conjugate low-pass filter is activated to eliminate spurious oscillations accumulated during the time evolution of a flow. As both low-pass and high-pass filters are derived from the Hermite kernel, they have similar regularity, time-frequency localization, effective frequency band and compact support. Fourier analysis indicates that the CFOR-Hermite scheme yields a nearly optimal resolution and has a better approximation to the ideal low-pass filter than previously CFOR schemes. Thus, it has better potential for resolving natural high frequency oscillations from a shock. Extensive one- and two-dimensional numerical examples, including both incompressible and compressible flows, with or without shocks, are employed to explore the utility, test the resolution, and examine the stability of the present CFOR-Hermite scheme. Extremely small ratio of point-per-wavelength (PPW) is achieved in solving the Taylor problem, advancing a wavepacket and resolving a shock/entropy wave interaction. The present results for the advection of an isentropic vortex compare very favorably to those in the literature.