Dynamics of a Data-Driven Low-Dimensional Model of Turbulent Minimal Pipe Flow

TL;DR

The paper demonstrates that turbulent minimal-pipe-flow dynamics at $\,\mathrm{Re}=2500\$ can be faithfully represented by a data-driven low-dimensional manifold learned through POD, nonlinear autoencoders, and stabilized neural ODEs (DManD). This framework reduces the effective dimensionality from $\mathcal{O}(10^5)$ to $\mathcal{O}(10)$ without sacrificing key dynamics, achieving accurate short-time trajectory tracking and long-time statistics, including Reynolds stresses and energy balance. A damping regularization stabilizes latent dynamics, and the model enables discovery of exact coherent states (ECS) by providing effective initial conditions for full DNS ECS solvers, yielding 17 new ECS and several long-period relative periodic orbits. The work highlights the potential of data-driven manifold dynamics for turbulence in complex shear flows and points toward parameter-transferability and control-oriented applications with partial observations.

Abstract

The simulation of turbulent flow requires many degrees of freedom to resolve all the relevant times and length scales. However, due to the dissipative nature of the Navier-Stokes equations, the long-term dynamics are expected to lie on a finite-dimensional invariant manifold with fewer degrees of freedom. In this study, we build low-dimensional data-driven models of pressure-driven flow through a circular pipe. We impose the `shift-and-reflect' symmetry to study the system in a minimal computational cell (e.g., smallest domain size that sustains turbulence) at a Reynolds number of 2500. We build these models by using autoencoders to parametrize the manifold coordinates and neural ODEs to describe their time evolution. Direct numerical simulations (DNS) typically require on the order of O(105) degrees of freedom, while our data-driven framework enables the construction of models with fewer than 20 degrees of freedom. Remarkably, these reduced order models effectively capture crucial features of the flow, including the streak breakdown. In short-time tracking, these models accurately track the true trajectory for one Lyapunov time, while at long-times, they successfully capture key aspects of the dynamics such as Reynolds stresses and energy balance. Additionally, we report a library of exact coherent states (ECS) found in the DNS with the aid of these low-dimensional models. This approach leads to the discovery of seventeen previously unknown solutions within the turbulent pipe flow system, notably featuring relative periodic orbits characterized by the longest reported periods for such flow conditions.

Figures (15)

• Figure 1: Schematic representation of the three-dimensional pipe flow system. Panel (a) shows a snapshot of the magnitude of the velocity field. For visualisation purposes, the entire pipe is shown, although calculations in this work is restricted in the shift-and-reflect symmetry subspace with $m_p=4$ ( whose boundaries are highlighted with solid red lines). Panel (b) represents the energy in the axially dependent modes only (k non-zero). This quantity decays rapidly after relaminarisations.
• Figure 2: (a) Eigenvalues of the POD modes sorted in descending order. (b) Reconstruction of four components of the Reynolds stresses from the DNS and the data projected onto 512 POD modes. The curves correspond to $\left \langle u_z^2 \right \rangle, \left \langle u_\theta^2 \right \rangle, \left \langle u_r u_z \right \rangle$ and $\left \langle u_\theta^2 \right \rangle$, from top to bottom, respectively.
• Figure 3: Nonlinear reduction with autoencoders: (a) Relative error on test data for POD coefficients, standard and hybrid autoencoders as a function of the latent dimension $d_h$. For each dimension, results for two standard and two hybrid autoencoders are reported. (b) Reconstruction of $\left \langle \left \| a_n \right \|^2 \right \rangle$ (mean-squared POD coefficient amplitudes) for the test data from 512 POD modes and the standard and hybrid autoencoders at $d_h = 20$. (c) Components of the Reynolds stresses from the DNS and using autoencoders with $d_h=20$. (d) Two-dimensional representation of the flow field in a $z-\theta$ plane ($r = 0.496$) with $u_z$ for the DNS and reconstructed using the hybrid autoencoder at $d_h = 20$.
• Figure 4: Normalised kinetic energy of the system for the DNS and DManD model with $d_h = 20$ up to $t=200$shown for two random initial conditions, corresponding to panels (a) and (b), respectively). Panels (c) and (d) represent two-dimensional representation of the dynamics in the $z-\theta$ plane ($r = 0.496$) with $u_z$ for the DNS and DManD model for IC corresponding to panels (a) and (b), respectively. The vertical dashed line marks one Lyapunov time. We refer the reader to the supplemental materials to view a video of the trajectory corresponding to panel (a).
• Figure 5: Self-sustaining process in DManD model corresponding to the initial condition shown in Figure 4a. Isosurfaces of the streamwise velocity fluctuations are displayed for $u_z' = 0.05$ (blue, representing fast streaks) and $u_z' = -0.05$ (red, representing low-speed streaks). Additionally, each snapshot includes isosurfaces of the $\lambda_2$ criterion with a threshold of $\lambda_2 = 0.1$, shown in green, to highlight vortical structures. Only a quarter of the domain is shown.