Energy transfer and budget analysis for transient process with phase-averaged reduced-order model

TL;DR

The paper introduces a phase-averaged reduced-order modeling framework for transient flows, built on time-varying eigenmodes extracted by a phase-controlled DMD (tDMDpc) and a phase-averaged energy-budget derivation. The method enables instantaneous, frequency-based decomposition of transient dynamics and reveals how energy transfers from a base flow to dominant oscillations, cascading to higher harmonics via triadic interactions. Through a canonical cylinder wake transition, the work demonstrates that phase control stabilizes the transient modal evolution, and the phase-averaged ROM reproduces both linear growth and nonlinear energy transfer patterns, linking recirculation-region dynamics to modal growth. The approach offers a data-driven, operator-based lens on transient energy transfer that can extend to more complex flows and other modal decompositions, with potential implications for flow control and prediction.

Abstract

We derive a phase-averaged representation of transient flows based on the eigenmodes of a data-driven linear operator that approximates the Navier-Stokes dynamics. In performing phase averaging, it is assumed that, at each instant during the transient evolution, the eigenmode amplitude remains invariant, while only the complex phase angle differs among distinct realizations of the transient process. From this modal-phase perspective, the linear operator is defined as the best-fit operator that represents phase-different transient evolutions. By introducing a time-varying dynamic mode decomposition with a phase-control strategy formulated from this modal-phase perspective, time-varying eigenmodes are extracted from numerical simulations. In this formulation, the transient process is decomposed into time-varying eigenmodes, phase-shift angles, and amplitude coefficients. Furthermore, by averaging the Navier-Stokes equations over the phase-shift angle, a frequency-domain form of the equations can be derived at any given instant, assuming that the phase-shift angle is time-independent. This frequency-domain representation reveals the instantaneous energy budget and the presence of energy transfer through triadic interactions. The proposed analysis is demonstrated using a canonical example of two-dimensional flow around a circular cylinder transitioning from a steady to an unsteady state. The time-varying dynamic mode decomposition with phase control is shown to capture the transient evolution of the frequency components accurately. In addition, the temporal evolution of the energy budget and transfer distribution reveals that transient growth processes exhibit different time-dependent characteristics of energy transfer, even in cylinder flows at Reynolds numbers that eventually lead to a periodic state.

Figures (40)

• Figure 1: Computational grids around a circular cylinder.
• Figure 2: Schematic of tDMDpc. By changing the temporal phase of the mode in the initial flow and time progressing independently by CFD, a transient flow with different phases can be obtained.
• Figure 3: $x$-direction velocity fields of the base flow at $\Rey=100$ obtained from the regular grid. The base flow is obtained by imposing symmetry on $y=0$.
• Figure 4: Base flow for various $\Rey$ from the regular grid. (a) $0$-line of the $x$-direction velocity component of the base flow (steady flow) at various $\Rey$. (b) The $x$-direction length of the recirculation region, $L_\text{recirc}$. The $x$-direction length of the $0$ line expands linearly with increasing $Re$.
• Figure 5: Spatial distribution of $x$-direction component of eigenmode $(\boldsymbol{\varphi}_{f_1})_x$ and its absolute value: (a) and (b) $\Rey=40$, (c) and (d) $60$, (e) and (f) $100$, and (g) and (h) $150$. The black line indicates the zero line of $x$-direction velocity from the base flow. All modes were obtained from the regular grid.