Statistical properties of Fourier modes in isotropic turbulence

Fourier representations of turbulent velocity fields present a natural way of studying turbulent flows, as they allow for the understanding of its dynamics in terms of different scales, represented by different Fourier modes. Numerical methods like Direct Numerical Simulations often use spectral methods, which rely on this Fourier representation to advance the solution in the wavenumber space. Because all the information in the velocity field is contained in its Fourier representation, an understanding of the statistical properties and evolution of these Fourier modes can not only provide deep insight into the dynamics of turbulent flows, but also inform modeling methods that operate in Fourier space, such as the recently introduced Selected Eddy Simulations (SES). SES solves only a subset of Fourier modes at a time and evolves the rest using simple dynamics.

In this work, we study the statistical properties of the Fourier modes of velocity and acceleration for moderate to high Reynolds numbers. Time-averaged and shell-averaged statistics are calculated for different wavenumbers and Reynolds numbers, and their scaling and universality are studied. We also analyze the time series of individual modes to determine their autocorrelation, cross-correlation, and governing time-scales. The time scale is indicative of the dominant physical process at each scale. Finally, these results are used to make stochastic models for the evolution of these modes.