Symmetry-reduced dynamic mode decomposition of near-wall turbulence

Abstract

Data-driven dimensionality reduction methods such as proper orthogonal decomposition and dynamic mode decomposition 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 fluid flows in rectangular channels by formulating a continuous symmetry reduction method that eliminates the translations in the streamwise and spanwise directions simultaneously. We demonstrate our method by computing the symmetry-reduced dynamic mode decomposition (SRDMD) of sliding windows of data obtained from the transitional plane-Couette and turbulent plane-Poiseuille flow simulations. In the former setting, SRDMD captures the dynamics in the vicinity of the invariant solutions with translation symmetries, i.e. travelling waves and relative periodic orbits, whereas in the latter, our calculations reveal episodes of turbulent time evolution that can be approximated by a low-dimensional linear expansion.

Transition to chaos and modal structure of magnetized Taylor–Couette flow

“…This is not a concerning issue here, since we used DMD not for reduced-order modelling of the system, but as a diagnostic tool for its dynamical behaviour. The future work will include relating the temporal evolution of the modes into a nonlinear model, together with improving robustness of the presented DMD method by taking into account flow symmetries [ 20 , 21 ], or harnessing statistical properties of the flow [ 22 ]. Such a model could provide a quantitative description of nonlinear interactions accompanying the transition to MRI turbulence.…”

Symmetry-Reduced Low-Dimensional Representation of Large-Scale Dynamics in the Asymptotic Suction Boundary Layer

“…Having in mind its application to the ASBL, we introduce the slicing method in channel geometry following Ref. [23]. That is, we consider flow fields periodically extended in streamwise direction, x, and spanwise direction, z, and bounded in the wall-normal direction, y.…”

“…Symmetry reduction using the method of slices has already carried out successfully for high-dimensional dynamical systems, such as numerical simulations of pipe flow [? 22] and recently in combination with DMD to plane Couette and plane Poiseuille flow simulations up to friction Reynolds number Re τ = 205 with translation symmetries in streamwise and spanwise direction [23].…”

“…In this paper, we apply symmetry-reduced DMD (SRDMD [23]) to data obtained from DNS of the ASBL at Re τ ≈ 330, in order to construct lowdimensional representations of the dynamics of a large-scale persistent coherent structure in the flow. This large-scale low-momentum region extends through the simulation domain in streamwise direction and drifts transversely through the computational domain.…”

Stochastic Model of Organizational State Transitions in a Turbulent Pipe Flow

“…The existence of spatial transitions among the OS modes suggests, within the perspective of dynamical systems, that the turbulent pipe flow could be described as a chaotic attractor and its unstable periodic orbits in a phase space of much reduced dimensionality [7][8][9][10][11]. In connection with this circle of ideas, we are motivated to study the OS transitions in the framework of stochastic processes, focusing particular attention on their recurrent dynamics.…”