A Class of High Order Compact Schemes with Good Spectral Resolution for Aeroacoustics
In this paper, we design a class of linear compact schemes based on the cell-centered compact scheme of ((Lele, J Comput Phys 103:16–42, 1992). These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the func
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Abstract In this paper, we design a class of linear compact schemes based on the cell-centered compact scheme of ((Lele, J Comput Phys 103:16–42, 1992). These schemes equate a weighted sum of the nodal derivatives of a smooth function to a weighted sum of the function on both the grid points and the cell-centers. Through systematic Fourier analysis and numerical tests, we observe that the schemes have good properties of high order, high resolution, and low dissipation. It is an ideal class of schemes for the simulation of multiscale problems such as aeroacoustics and turbulence. Keywords Compact scheme aeroacoustics
High
order
High
resolution
Computational
1 Introduction Direct numerical simulation (DNS) and large eddy simulation (LES) are two important methods to reveal the mechanism of multiscale problems such as turbulence and aeroacoustics. DNS for multiscale problems requires that the numerical grid should be fine enough to resolve the structure of smallest scales. However, due to the limitation of computational resources, most DNS studies have been carried out with marginal grid resolution. Besides the common problems in DNS of turbulence, there are computational issues that are unique to aeroacoustics (Tam 1995). First, the aerodynamic noise is broadband and the spectrum is fairly wide. Second, the amplitudes of the physical variables of the aerodynamic noise are far smaller than those of the mean flow. Third, the distance from the noise X. Liu (&) S. Zhang State Key Laboratory of Aerodynamics, China Aerodynamics Research and Development Center, Mianyang 621000 Sichuan, China e-mail: [email protected]
Y. Zhou et al. (eds.), Fluid-Structure-Sound Interactions and Control, Lecture Notes in Mechanical Engineering, DOI: 10.1007/978-3-642-40371-2_35, Ó Springer-Verlag Berlin Heidelberg 2014
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source to the location of interest in aeroacoustic problems is quite long. To ensure that the computed solution is uniformly accurate over such a long propagation distance, the numerical scheme should have minimal numerical dispersion, dissipation, and anisotropy. The most influential compact schemes for derivatives, interpolation, and filtering were proposed by Lele (1992). Through systematic Fourier analysis, it is shown that these compact schemes have spectral-like resolution for short waves. In this paper, we propose a new idea to design the compact scheme based on the cell-centered compact scheme of Lele (1992). Instead of using only the values on cell centers, both the values of cell centers and grid nodes are used on the right hand side of compact schemes. Both the accuracy order and the wave resolution property are improved significantly. Numerical tests show that this is an ideal scheme for the DNS for multiscale problems.
2 Central Compact Schemes In this section, we present the methodology to design central compact schemes (CCS). We start our work from the cell-centered compact scheme (CCCS) proposed by Lele (1992). Then we extend this scheme to a class of higher orde
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