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Spacetime-spectral analysis of flowfields

Vilas J. Shinde

Abstract

The classical Fourier analysis of a time signal, in the discrete sense, provides the frequency content of signal under the assumption of periodicity. Although the original signal can be exactly recovered using an inverse transform, the time dependence of the spectrum remains inaccessible. There exist various time-frequency analysis techniques, such as the short time fast Fourier transform and wavelets, but those are fundamentally limited in achieving high resolution in both the time and frequency domains concurrently. For spatiotemporal flowfields, the frequency based modal decompositions generally provide spatial modes with a temporal counterpart that evolves at a constant frequency. However, an accurate time-local spectral contribution and its variation over time are highly desired to better understand the intermittent/extreme events, for instance, in turbulent flowfields. To this end, this paper presents a spectral mode decomposition that yields spectral-space and spectral-time modes, where the latter along with the associated spectral energies provide a spectrogram that is at the resolution of flowfields. The spectral modes get both the frequency and energy ranks. The numerical examples demonstrate the use of technique not only in spacetime-frequency analysis but also in reduced-order modeling and denoising applications.

Spacetime-spectral analysis of flowfields

Abstract

The classical Fourier analysis of a time signal, in the discrete sense, provides the frequency content of signal under the assumption of periodicity. Although the original signal can be exactly recovered using an inverse transform, the time dependence of the spectrum remains inaccessible. There exist various time-frequency analysis techniques, such as the short time fast Fourier transform and wavelets, but those are fundamentally limited in achieving high resolution in both the time and frequency domains concurrently. For spatiotemporal flowfields, the frequency based modal decompositions generally provide spatial modes with a temporal counterpart that evolves at a constant frequency. However, an accurate time-local spectral contribution and its variation over time are highly desired to better understand the intermittent/extreme events, for instance, in turbulent flowfields. To this end, this paper presents a spectral mode decomposition that yields spectral-space and spectral-time modes, where the latter along with the associated spectral energies provide a spectrogram that is at the resolution of flowfields. The spectral modes get both the frequency and energy ranks. The numerical examples demonstrate the use of technique not only in spacetime-frequency analysis but also in reduced-order modeling and denoising applications.
Paper Structure (8 sections, 20 equations, 12 figures)

This paper contains 8 sections, 20 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Spacetime flowfields and Fourier transform
  4. Spectral mode decomposition
  5. Lid-driven cavity flow
  6. Turbulent shockwave boundary layer interaction flow
  7. Full/reduced rank reconstruction of flowfields
  8. Concluding remarks

Figures (12)

  • Figure 1: Spectral mode decomposition of lid-driven cavity at $Re=12000$. (a) An instantaneous DNS flowfield of the horizontal velocity component. (b) Spectral energies of the SMD modes. (c) Phase portraits of the SMD time-modes. (d) Real part of the spatial spectral mode at $f=0.4$. (e) Imaginary part of the spatial spectral mode at $f=0.4$. (d) Real part of the spatial spectral mode at $f=0.8$. (e) Imaginary part of the spatial spectral mode at $f=0.8$.
  • Figure 2: Spectrogram, $|\varphi_f|^2$, of lid-driven cavity flow at $Re=12000$.
  • Figure 3: Spectral mode decomposition of lid-driven cavity at $Re=15000$. (a) Spectral energies of the SMD modes. (b) Phase portraits of the SMD time-modes at $f=0.24$ versus $f=0.38$. (c) Phase portraits of the SMD time-modes at $f=0.24$ versus $f=0.62$. (d) Real parts of the spatial spectral mode at $f=0.24$. (e) Real part of the spatial spectral mode at $f=0.38$. (d) Real part of the spatial spectral mode at $f=0.62$. (e) Real part of the spatial spectral mode at $f=0.76$.
  • Figure 4: Spectrogram, $|\varphi_f|^2$, of lid-driven cavity flow at $Re=15000$.
  • Figure 5: Turbulent shockwave boundary layer interaction at Mach $M=2.7$ and Reynolds number of $Re_\delta=54600$, shown in terms of instantaneous flowfields of (a) the density gradient magnitude $|\nabla\rho|$, and (b) the streamwise velocity $u_1$.
  • ...and 7 more figures