Symmetry-reduced Dynamic Mode Decomposition of Near-wall Turbulence

Abstract

Data-driven dimensionality reduction methods such as proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have proven to be useful for exploring complex phenomena within fluid dynamics and beyond. A well-known challenge for these techniques is posed by the continuous symmetries, e.g. translations and rotations, of the system under consideration as drifts in the data dominate the modal expansions without providing an insight into the dynamics of the problem. In the present study, we address this issue for the pressure-driven flow in a rectangular channel by formulating a continuous symmetry reduction method that eliminates the translations simultaneously in the streamwise and spanwise directions. As an application, we consider turbulence in a minimal flow unit at a Reynolds number (based on the centerline velocity and half-channel height) Re = 2000 and compute the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding data windows of varying durations. SRDMD of channel flow reveals episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.

Symmetry-Aware Autoencoders: s-PCA and s-nlPCA

“…Furthermore, a modulation in the inhomogeneous direction, which does not change the alignment of the samples, can be used in cases such as turbulent channel flow, see e.g. Marensi et al [16], to avoid infinite shifts due to a vanishing Fourier amplitude for the first complex coefficient [4]. In a conventional application of the method of slicing this modulation has to be user-supplied and carefully tailored, whereas in our network it is learned as part of the network training.…”

“…This means that any solution (𝑢, 𝑣, 𝑝) (𝑥 + 𝑠, 𝑦) is a solution to the Navier-Stokes equations if (𝑢, 𝑣, 𝑝) (𝑥, 𝑦) is a solution, see e.g. Marensi et al [16].…”

“…However, the weights of our network only converge to a pure sine and cosine function in the translated, homogeneous direction. No constraint is placed on the modulation of the weights in the non-translated directions, but such a modulation may be desirable under certain circumstances, see Marensi et al [16].…”

“…We anticipate the same behavior here, even though a prediction on the shape of the weights in the homogeneous direction 𝑦 cannot be made. According to Marensi et al [16], a dependence of the transformation parameter on the non-translated, inhomogeneous directions may be required, if discontinuities of the predicted shift are to be avoided. For Kolmogorov flow, this dependence is not necessary, since the amplitude of the first complex Fourier coefficient never reaches zero.…”

“…The method is only valid when the amplitude of the first Fourier mode is non-zero, since the shift of a sample with zero amplitude would become infinite. This problem is known to arise in turbulent channel flow [16] and can be circumvented by introducing a handcrafted modulation of the Fourier basis in the inhomogeneous direction. A different way to approach advection-dominated problems is to split the problem into advection modes, determined through optimal mass transfer.…”