湍流数值模拟的准谱方法分析
Analysis of pseudo-spectral methods used for numerical simulation of turbulence
摘要：
本文利用全局谱分析（GSA）作为检验数值方法精度的工具，对对流和对流-扩散方程的正则问题进行了研究，得到了这些问题的精确解。类似地，通过直接数值模拟（DNS）计算的湍流事件通常使用Kolmogorov的均匀各向同性湍流理论结果进行校准，如Turbulence -U中所示。Frisch，Cambridge Univ. Press，UK（1995）.然而，用于模拟该问题的数值方法没有校准，如通过使用对流和/或对流-扩散方程的GSA。这是一个例外，在“过渡和湍流模拟的关键评估-Sengupta，T.K.，在IUTAM Symp.转捩和湍流的计算、建模和控制进展，第491-532页，世界科学。Pte.有限公司、新加坡（2016年）"，在对流方程的帮助下提倡这种校准。对于湍流，极端事件的特征在于存在小于柯尔莫哥洛夫长度尺度的长度尺度，柯尔莫哥洛夫长度尺度是不被转换为热的最大波数的启发式限制。随着计算机能力的提高，最近已经报道了许多使用准谱方法的模拟，其中空间离散化在傅立叶谱空间中进行，时间离散化采用两阶段的龙格-库塔（RK 2）方法。但没有分析报告，以确保这种模拟的高精度。本文在傅立叶谱框架下分析了对流和对流-扩散方程的几种多级龙格-库塔法。我们确定的RK 2-傅立叶谱方法使用GSA的主要误差来源，并展示了如何避免这种错误，并指定数值参数，以达到最高的精度，捕捉极端事件的湍流。
英文摘要：
Global spectral analysis (GSA) is used as a tool to test the accuracy of numerical methods with the help of canonical problems of convection and convection-diffusion equation which admit exact solutions. Similarly, events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with theoretical results of homogeneous isotropic turbulence due to Kolmogorov, as given in Turbulence -U. Frisch, Cambridge Univ. Press, UK (1995). However, numerical methods for the simulation of this problem are not calibrated, as by using GSA of convection and/or convection-diffusion equation. This is with the exception in "A critical assessment of simulations for transitional and turbulence flows-Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, pp 491-532, World Sci. Publ. Co. Pte. Ltd., Singapore (2016)", where such a calibration has been advocated with the help of convection equation. For turbulent flows, an extreme event is characterized by the presence of length scales smaller than the Kolmogorov length scale, a heuristic limit for the largest wavenumber present without being converted to heat. With growing computer power, recently many simulations have been reported using a pseudo-spectral method, with spatial discretization performed in Fourier spectral space and a two-stage, Runge-Kutta (RK2) method for time discretization. But no analyses are reported to ensure high accuracy of such simulations. Here, an analysis is reported for few multi-stage Runge-Kutta methods in the Fourier spectral framework for convection and convection-diffusion equations. We identify the major source of error for the RK2-Fourier spectral method using GSA and also show how to avoid this error and specify numerical parameters for achieving highest accuracy possible to capture extreme events in turbulent flows.
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