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We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional approaches relying on identifying all short, fundamental Unstable Periodic Orbits to compute ergodic averages via cycle expansion are computationally prohibitive for high-dimensional fluid systems. Our framework stems from the observation in Lasagna, Phys. Rev. E (2020), that a single unstable periodic orbit with a period sufficiently long to span a large fraction of the attractor captures the statistical properties of chaotic trajectories. Given the difficulty of identifying unstable periodic orbits for high-dimensional fluid systems, approximate trajectories residing in a low-dimensional subspace are instead constructed using resolvent modes, which inherently capture the temporal periodicity of unstable periodic orbits. The amplitude coefficients of these modes are adjusted iteratively with gradient-based optimisation to minimise the violation of the projected governing equations, producing trajectories that approximate, rather than exactly solve, the system dynamics. A first attempt at utilising this framework on a chaotic system is made here on the Lorenz 1963 equations, where resolvent analysis enables an exact dimensionality reduction from three to two dimensions. Key observables averaged over these trajectories produced by the approach as well as probability distributions and spectra rapidly converge to values obtained from long chaotic simulations, even with a limited number of iterations. This indicates that exact solutions may not necessary to approximate the system's statistical behaviour, as the trajectories obtained from partial optimisation provide a sufficient ``sketch'' of the attractor in state space.

Secondary flows are generated when a lateral variation of the topography, such as streamwise aligned ridges, is imposed to a turbulent wall-bounded flow. In this case, the flow field is characterized by vortical structures developing along the streamwise direction known as Prandt's vortices of the second kind (Prandtl 1965). As demonstrated in previous experimental and numerical works, the strength and flow organization of these turbulent structures depend on the ridge shape. In this paper, the effect of the ridge geometry on the generation of secondary currents is investigated using the linearised RANS-based model proposed by Zampino (2022). Here, symmetric channels with rectangular, triangular and elliptical ridges are studied and the secondary flows are compared in order to highlight the main differences and similarities. The analogies of the flow organisation between the three geometries suggest that the secondary currents do not depend on the ridge shape when the ridges are small and isolated, and the strength of secondary flows collapses when properly scaled with the mean ridge height. Finally, the generation of secondary flows and the effect of the ridge shape on the flow organisation is studied in detail for the complex geometries. In particular, for trapezoidal ridges, we observed that tertiary flows emerge for the ridges where the scaling behaviour does not hold.

A rapid predictive tool based on the linearised Reynolds-averaged Navier-Stokes equations is proposed in this work to investigate secondary currents generated by streamwise-independent surface topography modulations in turbulent channel flow. The tool is derived by coupling the Reynolds-averaged momentum equation to the Spalart-Allmaras transport equation for the turbulent eddy viscosity, using a nonlinear constitutive relation for the Reynolds stresses to capture correctly secondary motions. Linearised equations, describing the steady flow response to arbitrary surface modulations, are derived by assuming that surface modulations are shallow. Since the equations are linear, the superposition principle holds and the flow response induced by an arbitrary modulation can be obtained by combining appropriately the elementary responses obtained over sinusoidal modulations at multiple spanwise length scales. The tool permits a rapid exploration of large parameter spaces characterising structured surface topographies previously examined in the literature. Here, channels with sinusoidal walls and with longitudinal rectangular ridges are considered. For sinusoidal walls, a large response is observed at two spanwise wavelengths scaling in inner and outer units respectively, mirroring the amplification mechanisms in turbulent shear flows observed from transient growth analysis. For longitudinal rectangular ridges, the model suggests that the analysis of the response and the interpretation of the topology of secondary structures is facilitated when the ridge width and the gap between ridges are used instead of other combinations proposed in the literature.