Ati S. Sharma

We propose a 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 [D. Lasagna, Phys. Rev. E 102, 052220 (2020)2470-004510.1103/PhysRevE.102.052220] 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 optimization to minimize the violation of the projected governing equations, producing trajectories that approximate, rather than exactly solve, the system dynamics. An attempt at utilizing 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 be necessary to approximate the system's statistical behavior, as the trajectories obtained from partial optimization provide a sufficient "sketch" of the attractor in state space.

Shock waves interacting with turbulent boundary layers on wings can result first in self-sustained flow unsteadiness and eventually in structural vibration. Due to its importance to modern wing design and aircraft certification, the transonic flow physics continue to be investigated intensively. Herein we focus the discussion on three main aspects. First, we assess a practical implementation of an iterative resolvent algorithm in the linear harmonic incarnation of an industrial computational fluid dynamics code for computing optimal forcing and response modes. This heavily relies on the efficient solution of large sparse linear systems of equations. Second, we showcase its application as a predictive tool to detect transonic buffet flow unsteadiness, well before a global stability analysis can first identify its dynamics through weakly damped eigenmodes, using the NASA common research model at wind-tunnel conditions. Third, we discuss its ability to uncover modal physics, not identifiable through global stability analysis, revealing higher-frequency wake and wingtip vortex modes while shedding some light on the elusive finite wing equivalent of the aerofoil buffet mode. We demonstrate that earlier computational limitations of resolvent analysis, when solving the truncated singular value decomposition using matrix-forming methods with direct matrix factorisation, have been overcome ready for industrial use.

One of the outstanding problems in the dynamical systems approach to turbulence is to find a sufficient number of invariant solutions to characterize the underlying dynamics of turbulence [ Annu. Rev. Fluid Mech. 44, 203 (2012)]. As a practical matter, the solutions can be difficult to find. To improve this situation, we show how to find periodic orbits and equilibria in plane Couette flow by projecting pseudorecurrent segments of turbulent trajectories onto the left-singular vectors of the Navier-Stokes equations linearized about the relevant mean flow (resolvent modes). The projections are, subsequently, used to initiate Newton-Krylov-hookstep searches, and new (relative) periodic orbits and equilibria are discovered. We call the process project-then-search and validate the process by first applying it to previously known fixed point and periodic solutions. Along the way, we find new branches of equilibria, which include bifurcations from previously known branches, and new periodic orbits that closely shadow turbulent trajectories in state space.

New equilibrium solution branches for plane Couette flow are reported which add to the inventory of known solution branches. The exact solutions are found by projecting known equilibria onto the resolvent modes of McKeon & Sharma (J. Fl. Mech., vol. 658, 2010, pp. 336-382) to generate approximate solutions that are subsequently used as seeds in a Newton-Krylov-hookstep search. Searches initialised with these projections have a convergence rate of 96%, leading to the discovery of 16 new equilibria. The low-dimensional nature of the resulting equilibria is attributed to the projection generated by the resolvent model; resolvent modes for a given equilibrium solution span a low-dimensional space which the solution approximately inhabits. This property is exploited to generate new branches of equilibria in plane Couette flow, and to jump to known branches. The new branches include bifurcations from previously known bifurcation curves and a disconnected bifurcation curve which displays interesting behaviour when continued in Reynolds number. The unstable manifolds of the new exact coherent states are also computed to expand the known state space structure of plane Couette flow.

Previous work has established the usefulness of the resolvent operator that maps the terms nonlinear in the turbulent fluctuations to the fluctuations themselves. Further work has described the self-similarity of the resolvent arising from that of the mean velocity profile. The orthogonal modes provided by the resolvent analysis describe the wall-normal coherence of the motions and inherit that self-similarity. In this contribution, we present the implications of this similarity for the nonlinear interaction between modes with different scales and wall-normal locations. By considering the nonlinear interactions between modes, it is shown that much of the turbulence scaling behaviour in the logarithmic region can be determined from a single arbitrarily chosen reference plane. Thus, the geometric scaling of the modes is impressed upon the nonlinear interaction between modes. Implications of these observations on the self-sustaining mechanisms of wall turbulence, modelling and simulation are outlined. This article is part of the themed issue ‘Toward the development of high-fidelity models of wall turbulence at large Reynolds number’.

Previous work has established the usefulness of the resolvent operator that maps the terms nonlinear in the turbulent fluctuations to the fluctuations themselves. Further work has described the self-similarity of the resolvent arising from that of the mean velocity profile. The orthogonal modes provided by the resolvent analysis describe the wall-normal coherence of the motions and inherit that self-similarity. In this contribution, we present the implications of this similarity for the nonlinear interaction between modes with different scales and wall-normal locations. By considering the nonlinear interactions between modes, it is shown that much of the turbulence scaling behaviour in the logarithmic region can be determined from a single arbitrarily chosen reference plane. Thus, the geometric scaling of the modes is impressed upon the nonlinear interaction between modes. Implications of these observations on the self-sustaining mechanisms of wall turbulence, modelling and simulation are outlined.