Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-26T05:59:33.555Z Has data issue: false hasContentIssue false

A Memory-Saving Algorithm for Spectral Method of Three-Dimensional Homogeneous Isotropic Turbulence

Published online by Cambridge University Press:  20 August 2015

Qing-Dong Cai*
Affiliation:
LTCS and CAPT, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China
Shiyi Chen*
Affiliation:
LTCS and CAPT, Department of Mechanics and Aerospace Engineering, College of Engineering, Peking University, Beijing 100871, China Department of Mechanical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218, USA
*
Corresponding author.Email:[email protected]
Get access

Abstract

Homogeneous isotropic turbulence has been playing a key role in the research of turbulence theory. And the pseudo-spectral method is the most popular numerical method to simulate this type of flow fields in a periodic box, where fast Fourier transform (FFT) is mostly effective. However, the bottle-neck in this method is the memory of computer, which motivates us to construct a memory-saving algorithm for spectral method in present paper. Inevitably, more times of FFT are needed as compensation. In the most memory-saving situation, only 6 three-dimension arrays are employed in the code. The cost of computation is increased by a factor of 4, and that 38 FFTs are needed per time step instead of the previous 9 FFTs. A simulation of isotropic turbulence on 20483 grid can be implemented on a 256G distributed memory clusters through this method.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Taylor, G. I., Statistical theory of turbulence, Proc. Roy. Soc. London. A., 151 (1935), 421–444.Google Scholar
[2]Kolmogorov, A. N., The local structure of turbulence in incompressible viscous fluid for very large Reynolds number, Dokl. Acad. Nauk. SSSR., 30 (1941), 301–305. also: Turbulence and stochastic process: Kolmogorov’s ideas 50 years on, Pro. Math. Phys. Sci., 434 (1991), 9–13.Google Scholar
[3]Kolmogorov, A. N., On degeneration of isotropic turbulence in an incompressible viscous liquid, Dokl. Acad. Nauk. SSSR., 31 (1941), 538–540.Google Scholar
[4]Kolmogorov, A. N., Dissipation of energy in locally isotropic turbulence, Dokl. Acad. Nauk. SSSR., 32 (1941), 16–19. also: Turbulence and stochastic process: Kolmogorov’s ideas 50 years on, Pro. Math. Phys. Sci., 434 (1991), 15–17.Google Scholar
[5]Orszag, S. A., Numerical simulation of incompressible flows within simple boundaries: I. Galerkin (spetral) representations, Stu. Appl. Math., 50 (1971), 293–327.Google Scholar
[6]Canuto, C., Hussaini, M. Y., Quarteron, A., and Zang, T. A., Spectral Methods in Fluid Dynamics, Springer-Verlag, 1987.Google Scholar
[7]Siggia, E. D., Numerical study of small-scale intermittency in three-dimensional turbulence, J. Fluid. Mech., 107 (1981), 375–406.Google Scholar
[8]Kerr, R. M., Higher-order derivative correlations and the alignment of small-scale structures in isotropic numerical turbulence, J. Fluid. Mech., 153 (1985), 31–58.CrossRefGoogle Scholar
[9]Kerr, R. M., Velocity, scalar and transfer spectra in numerical turbulence, J. Fluid. Mech., 211 (1990), 309–332.Google Scholar
[10]She, Z.-S., Jackson, E., and Orszag, S. A., Intermittent vortex structures in homogeneous isotropic turbulence, Nature., 344 (1990), 226–228.CrossRefGoogle Scholar
[11]Chen, S., and Shan, X., High-resolution turbulent simulations using the connection machine-2, Comput. Phys., 6 (1992), 643–646.Google Scholar
[12]Chen, S., Doolen, G. D., Kraichnan, R. H., and She, Z. S., On statistical correlations between velocity increments and locally averaged dissipation in homogeneous turbulence, Phys. Fluids. A., 5 (1993), 458–463.CrossRefGoogle Scholar
[13]Wang, L. P., Chen, S., Brasseur, J. G., and Wyngaard, J. C., Examination of hypotheses in the Kolomogorov refined turbulence theory through high-resolution simulations, part 1. velocity field, J. Fluid. Mech., 309 (1996), 113–156.Google Scholar
[14]Yeung, P. K., and Zhou, Y., Universality of the Kolmogorov constant in numerical simulations of turbulence, Phys. Rev. E., 56 (1997), 1746–1752.Google Scholar
[15]Yeung, P. K., and Zhou, Y., Numerical study of rotating turbulence with external forcing, Phys. Fluids., 10 (1998), 2895–2909.CrossRefGoogle Scholar
[16]Yokokawa, M., Itakura, K., Uno, A., Ishihara, T., and Kaneda, Y., 16.4-Tflops direct numerical simulation of turbulence by aFourier spectral method onthe Earth Simulator, in: Conference on High Performance Networking and Computing, Proceedings of the 2002 ACM/IEEE conference on Supercomputing, Baltimore, Maryland, 2002, 1–17.Google Scholar
[17]Ishihara, T., Yoshida, K., and Kaneda, Y., Anisotropic velocity correlation spectrum at small scales in a homogeneous turbulent shear flow, Phys. Rev. Lett., 88 (2002), 154501.CrossRefGoogle Scholar
[18]Kaneda, Y., Ishihara, T., Yokokawa, M., Itakura, K., and Uno, A., Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box, Phys. Fluids., 15 (2003), L21–L24.Google Scholar
[19]Gotoh, T., Fukayama, D., and Nakano, T., Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation, Phys. Fluids., 14 (2002), 1065–1081.CrossRefGoogle Scholar
[20]Kaneda, Y., and Ishihara, T., High-resolution direct numerical simulation of turbulence, J. Turbul., 7 (2006), N20.CrossRefGoogle Scholar