High-Reynolds-number weakly stratified flow past an obstacle

Published versio

Quasi-steady quasi-homogeneous description of the scale interactions in near-wall turbulence

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 quasi-steady 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

High-Reynolds-number Batchelor-model asymptotics of a flow past an aerofoil with a vortex trapped in a cavity

Published versio

Inviscid Batchelor-model flow past an airfoil with a vortex trapped in a cavity

Published versio

Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application

With the goal of providing the first example of application of a recently proposed method, thus demonstrating its ability to give results in principle, global stability of a version of the rotating Couette flow is examined. The flow depends on the Reynolds number and a parameter characterising the magnitude of the Coriolis force. By converting the original Navier-Stokes equations to a finite-dimensional uncertain dynamical system using a partial Galerkin expansion, high-degree polynomial Lyapunov functionals were found by sum-of-squares-of-polynomials optimization. It is demonstrated that the proposed method allows obtaining the exact global stability limit for this flow in a range of values of the parameter characterising the Coriolis force. Outside this range a lower bound for the global stability limit was obtained, which is still better than the energy stability limit. In the course of the study several results meaningful in the context of the method used were also obtained. Overall, the results obtained demonstrate the applicability of the recently proposed approach to global stability of the fluid flows. To the best of our knowledge, it is the first case in which global stability of a fluid flow has been proved by a generic method for the value of a Reynolds number greater than that which could be achieved with the energy stability approach

Finding unstable periodic orbits: A hybrid approach with polynomial optimization

We present a novel method to compute unstable periodic orbits (UPOs) that optimize the infinite-time average of a given quantity for polynomial ODE systems. The UPO search procedure relies on polynomial optimization to construct nonnegative polynomials whose sublevel sets approximately localize parts of the optimal UPO, and that can be used to implement a simple yet effective control strategy to reduce the UPO's instability. Precisely, we construct a family of controlled ODE systems, parameterized by a scalar k, such that the original ODE system is recovered for k=0 and such that the optimal orbit is less unstable, or even stabilized, for k>0. Periodic orbits for the controlled system can often be more easily converged with traditional methods, and numerical continuation in k allows one to recover optimal UPOs for the original system. The effectiveness of this approach is illustrated on three low-dimensional ODE systems with chaotic dynamics

Expensive control of long-time averages using sum of squares and Its application to a laminar wake flow

The paper presents a nonlinear state-feedback con- trol design approach for long-time average cost control, where the control effort is assumed to be expensive. The approach is based on sum-of-squares and semi-definite programming techniques. It is applicable to dynamical systems whose right-hand side is a polynomial function in the state variables and the controls. The key idea, first described but not implemented in (Chernyshenko et al. Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problem of optimizing a cost function involving long-time averages is replaced by an optimization of the upper bound of the same average. As such, controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. The present paper introduces a method resolving the well-known inherent non-convexity of this kind of optimization. The method is based on the formal assumption that the control is expensive, from which it follows that the optimal control is small. The resulting asymptotic optimization problems are convex. The derivation of all the polynomial coefficients in the controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The proposed approach is applied to the problem of designing a full-information feedback controller that mitigates vortex shedding in the wake of a circular cylinder in the laminar regime via rotary oscillations. Control results on a reduced-order model of the actuated wake and in direct numerical simulation are reported

Turbulent skin-friction reduction by wavy surfaces

Direct numerical simulations of fully-developed turbulent channel flows with wavy walls are undertaken. The wavy walls, skewed with respect to the mean flow direction, are introduced as a means of emulating a Spatial Stokes Layer (SSL) induced by in-plane wall motion. The transverse shear strain above the wavy wall is shown to be similar to that of a SSL, thereby affecting the turbulent flow, and leading to a reduction in the turbulent skin-friction drag. The pressure- and friction-drag levels are carefully quantified for various flow configurations, exhibiting a combined maximum overall-drag reduction of about 0.5%. The friction-drag reduction is shown to behave approximately quadratically for small wave slopes and then linearly for higher slopes, whilst the pressure-drag penalty increases quadratically. Unlike in the SSL case, there is a region of increased turbulence production over a portion of the wall, above the leeward side of the wave, thus giving rise to a local increase in dissipation. The transverse shear-strain layer is shown to be approximately Reynolds-number independent when the wave geometry is scaled in wall units

Polynomial sum of squares in fluid dynamics: a review with a look ahead.

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