A space–time projection framework for approximate periodic orbit computation

This work investigates projection-based reduced space–time models formulated in the frequency domain, employing space–time basis functions constructed with spectral proper orthogonal decomposition (SPOD) to represent dominant spatio-temporal structures. Although frequency-domain formulations are well suited to capturing time-periodic solutions, such as unstable periodic orbits, this study focuses on computing long-time approximate periodic solutions of statistically stationary flows whose statistics resemble those of the underlying chaotic attractor. In contrast to reduced-order models based solely on spatial modes, a space–time formulation achieves simultaneous reduction in both space and time via Galerkin projection of the Navier–Stokes equations onto the SPOD basis using a space–time inner product, yielding a quadratic algebraic system for the amplitude coefficients. Approximate solutions of the reduced system are obtained by identifying coefficients that minimise an objective function corresponding to the sum of squared residuals across all frequencies and modes, quantifying violation of momentum conservation within the reduced subspace. A robust gradient-based optimisation algorithm is used to identify minima. The method is demonstrated for chaotic flow in a two-dimensional lid-driven cavity at a Reynolds number equal to 20 000, where solutions with extended temporal periods approximately fifteen times the dominant shear-layer time scale are sought. Even without closure models to represent truncated spatio-temporal triadic interactions, multiple reduced-order solutions are found that identify approximate periodic orbits which reproduce dominant dynamical flow features and recover with good fidelity to the statistical distribution of the training data used to construct the basis, although these solutions tend to overpredict energy near the truncation boundary.