The first part of this paper reviews the application of the sum-of-squares-of-polynomials technique to the problem of global stability of fluid flows. It describes the known approaches and the latest results, in particular, obtaining for a version of the rotating Couette flow a better stability range than the range given by the classic energy stability method. The second part of this paper describes new results and ideas, including a new method of obtaining bounds for time-averaged flow parameters illustrated with a model problem and a method of obtaining approximate bounds that are insensitive to unstable steady states and periodic orbits. It is proposed to use the bound on the energy dissipation rate as the cost functional in the design of flow control aimed at reducing turbulent drag.

International audienceA model for the instantaneous wall-shear-stress distribution is presented for zero-pressure-gradient turbulent boundary layers. The model, based on empirical and theoretical considerations, is able to reconstruct a statistically representative fluctuating wall-shear-stress time-series, , using only the low-frequency content of the streamwise velocity measured in the logarithmic region, away from the wall. Results, including spectra and second-order moments, show that the model is capable of successfully capturing Reynolds number trends as observed in other studies

This paper introduces a new method for proving global stability of fluid flows through the construction of Lyapunov functionals. For finite dimensional approximations of fluid systems, we show how one can exploit recently developed optimization methods based on sum-of-squares decomposition to construct a polynomial Lyapunov function. We then show how these methods can be extended to infinite dimensional Navier-Stokes systems using robust optimization techniques. Crucially, this extension requires only the solution of infinite-dimensional linear eigenvalue problems and finite-dimensional sum-of-squares optimization problems.We further show that subject to minor technical constraints, a general polynomial Lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower-bound on the maximum Reynolds number for which a flow is globally stable, if the flow does remain globally stable for Reynolds numbers at least slightly beyond the energy stability limit. Such polynomial functions can be searched for efficiently using the SOS technique we propose.

Abstract. We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.

By introducing a notion of an ideal large-scale filter, a formal statement is given of the hypothesis of the quasi-steady quasi-homogeneous nature of the interaction between the large and small scales in the near-wall part of turbulent flows. This made the derivations easier and more rigorous.A method is proposed to find the optimal large-scale filter by multi-objective optimization, with the first objective being a large correlation between large-scale fluctuations near the wall and in the layer at a certain finite distance from the wall, and the second objective being a small correlation between the small scales in the same layers. The filter was demonstrated to give good results. Within the quasi-steady quasi-homogeneous theory expansions for various quantities were found with respect to the amplitude of the large-scale fluctuations. Including the higher-order terms improved the agreement with numerical data. Interestingly, it turns out that the quasisteady quasi-homogeneous theory implies a dependence of the mean profile log-law constants on the Reynolds number. The main overall result of the present work is the demonstration of the relevance of the quasi-steady quasi-homogeneous theory for near-wall turbulent flows.

Two conceptual frameworks for the origin of the streaky pattern in near-wall developed turbulent flows are compared. According to the framework that dominated the research over several decades, the pattern of streaks is dictated by the pattern of wall-normal motions via the lift-up mechanism. Various concepts within this framework describe the wall-normal motions as induced by longitudinal vortices, hairpin vortices, vortex packets, etc. According to the newly emerging conceptual framework, the combined action of lift-up of the mean profile, mean shear, and viscous diffusion has its own pattern-forming properties. The pattern of streaks is dictated by these linear effects to a much greater extent than by the pattern of the wall-normal motions. Numerical results supporting the new conceptual framework are presented. An approximate approach for calculating the streak spacing within the new framework is proposed. It is shown to have a significant predictive ability.

In this paper we consider the rôle that numerical computations -in particular Galerkin approximations -can play in problems modelled by the 3d Navier-Stokes equations, for which no rigorous proof of the existence of unique solutions is currently available. We prove a robustness theorem for strong solutions, from which we derive an a posteriori check that can be applied to a numerical solution to guarantee the existence of a strong solution of the corresponding exact problem.We then consider Galerkin approximations, and show that if a strong solution exists the Galerkin approximations will converge to it; thus if one is prepared to assume that the Navier-Stokes equations are regular one can justify this particular numerical method rigorously.Combining these two results we show that if a strong solution of the exact problem exists then this can be verified numerically using an algorithm that can be guaranteed to terminate in a finite time.We thus introduce the possibility of rigorous computations of the solutions of the 3d Navier-Stokes equations (despite the lack of rigorous existence and uniqueness results), and demonstrate that numerical investigation can be used to rule out the occurrence of possible singularities in particular examples.2

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