Space-time nonlinear reduced-order modelling for unsteady flows

TL;DR

The paper addresses the challenge of modeling statistically stationary turbulent flows with reduced-order models by formulating a space-time nonlinear ROM that uses a Spectral Proper Orthogonal Decomposition (SPOD) basis and Galerkin projection to obtain a nonlinear algebraic system for amplitude coefficients. It solves this system through a robust gradient-based optimization that minimizes a residual-based objective to enforce momentum conservation in the reduced subspace, incorporating the fundamental frequency as an optimization variable. Demonstrated on a high-$Re$ 2D lid-driven cavity, the method yields multiple ROM solutions that capture dominant flow structures and long-time statistics without closure corrections, although energy distribution near truncation boundaries is overpredicted. The work highlights the potential of space-time ROMs for efficient, statistics-based turbulence modeling and discusses initialization, cost, and future closure strategies necessary to extend to 3D flows and broader parameter studies.

Abstract

This work investigates projection-based Reduced-Order Models (ROMs) formulated in the frequency domain, employing a space-time basis constructed with Spectral Proper Orthogonal Decomposition to efficiently represent dominant spatio-temporal coherent structures. Although frequency domain formulations are well suited to capturing time-periodic solutions, such as unstable periodic orbits, this study focusses on modelling statistically stationary flows by computing long-time solutions that approximate their underlying statistics. In contrast to traditional ROMs based solely on spatial modes, a space-time formulation achieves simultaneous reduction in both space and time. This is accomplished by Galerkin projection of the Navier-Stokes equations onto the basis using a space-time inner product, yielding a quadratic algebraic system of equations in the unknown amplitude coefficients. Solutions of the ROM are obtained by identifying amplitude coefficients that minimise an objective function corresponding to the sum of the squares of the residuals of the algebraic system across all frequencies and modes, quantifying the aggregate violation of momentum conservation within the reduced subspace. A robust gradient-based optimisation algorithm is employed to identify the minima of this objective function. The method is demonstrated for chaotic flow in a two-dimensional lid-driven cavity at $Re=20{,}000$, where solutions with extended temporal periods approximately fifteen times the dominant shear layer time scale are sought. Even without employing closure models to represent the truncated spatio-temporal triadic interactions, multiple ROM solutions are found that successfully reproduce the dominant dynamical flow features and predict the statistical distribution of turbulent quantities with good fidelity, although they tend to overpredict energy at spatio-temporal scales near the truncation boundary.

Figures (15)

• Figure 1: Four snapshots of the vorticity field of the fully-developed state separated by a time interval $\Delta t = 0.4$, panel (a). Time history of kinetic energy ($\mathcal{K}$) and turbulent kinetic energy ($\mathcal{K}'$) with the corresponding Probability Density Function (PDF), panels (b-c).
• Figure 2: Spectrum of SPOD eigenvalues, up to mode 18, coloured by the mode index, panel (a). The vertical lines identify energy peaks at $f^k= 0.586, 1.211, 1.797$. The decay of eigenvalues for the mean component and these three energy-peak frequencies and their cumulative energy are shown in panels (b) and (c).
• Figure 3: Real part of the $x$-component of the first three SPOD modes for the mean component and the three peak frequencies marked by vertical lines in \ref{['fig:SPOD_spectra_truncation']}(a).
• Figure 4: Potential strategies for selecting SPOD modes for ROM R95, panel (a), and R90, panel (b), overlaid on the heat map of the SPOD eigenvalues. The solid green boundary corresponds to the nominal truncation boundary, where all eigenvalues are globally ranked prior to truncation. In practice, modes falling within the rectangular region enclosed by the green dash-dotted line are retained.
• Figure 5: The squared magnitude of the 300 sets of amplitude coefficients (light grey circles), obtained from protocol A (top panels) and protocol B (bottom panels). These are compared to the SPOD eigenvalues (solid lines) for the first three SPOD modes. The red triangles denote coefficients from one of the initial guesses and the green dots denote the average over all guesses.