Modal analysis of turbulent flows simulated with spectral element method

Leveraging the high-order nature of spectral element methods, this work introduces a methodology that enables instantaneous, local analysis of turbulent flows through a transformation to a modal space. It also introduces a dynamic explicit modal filter (DEMF) for removing the excess energy in large-eddy simulation of turbulent flows. The transformation is achieved through the application of the discrete Chebyshev transform to nodal solution values, and the resulting modes are employed to characterize and distinguish different turbulent flow properties. Implementing direct numerical simulation (DNS), a qualitative explanation of how modes can assess flow directionality and turbulence properties is provided by considering the modal representation at different locations for different turbulent flows. Additionally, reducing the resolution from DNS, it is shown how modes can qualitatively assess local flow resolvedness. A quantitative assessment of local flow resolvedness is conducted by calculating sequential energy levels from the modes and then comparing them between DNS and under-resolved cases. These energy levels correspond to different scales of motion and their behavior changes as turbulence intensifies and decays. It is observed that flow under-resolvedness manifests itself in the increased magnitudes of higher energy levels corresponding to small scales of motion. Finally, the new DEMF is discussed, which is triggered locally by comparing the local Kolmogorov length scale and the average grid spacing. The filter removes the excess energy in the detected under-resolved regions by selectively removing the energy of the modes that contribute to higher energy levels.