Sergei Chernyshenko

The mechanism of streak formation in near-wall turbulence

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.

A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations

In this paper we consider the role that numerical computations—in particular Galerkin approximations—can play in problems modeled by the three-dimensional (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 com...

Flow Models for a Vortex Cell

The flow inside a vortex trapping cavity is simulated by a suite of models: point vortex, Prandtl-Batchelor flow, and Reynolds-averaged Navier-Stokes equations. The scope is to ascertain to what extent an inviscid model can be used to design vortex cells. It turns out that the Prandtl-Batchelor flow, with an appropriate jump in the Bernoulli constant across the dividing streamline, gives an acceptable representation of the solution found by the Reynolds-averaged Navier-Stokes equations, which in turn compares well with experimental results when an appropriate turbulence model is selected.

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.

Trapped vortices and a favourable pressure gradient

It is shown that there exist bodies such that in two-dimensional steady inviscid incompressible flow the pressure gradient is favourable over the entire surface of the body, and the lift is non-zero, if the body is immersed in a uniform stream and there are also two trapped point vortices.

Global Stability Analysis of Fluid Flows using Sum-of-Squares

In this paper we present a new method for assessing the stability of finite-dimensional approximations to the Navier-Stokes equation for fluid flows. Approximations to the Navier-Stokes equation typically take the form of a set of linear ODEs with an additional bilinear term that conserves the total energy of the system state. We suggest a structured method for generating Lyapunov functions using sum-of-squares optimization that exploits this energy conservation property. We provide a numerical example demonstrating the use of this technique to assess the stability of a model of a shear flow between infinite parallel plates.

Streaks and vortices in near-wall turbulence

This paper presents evidence that organization of wall-normal motions plays almost no role in the creation of streaks. This evidence consists of the theory of streak generation not requiring the existence of organized vortices, extensive quantitative comparisons between the theory and direct numerical simulations, including examples of large variation in average spacing of the streaks of different scalars simultaneously present in the flow, and an example of the scalar streaks in an artificially created purely random flow.

Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization

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 c...

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 characterizing 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 characterizing 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.

Turbulent flow and heat transfer in eccentric annulus

An eccentric annular duct is a prototype element in a number of engineering applications. Numerous modeling and experimental efforts have been made to investigate the details of the flow field and heat transfer characteristics in such kind of ducts [1, 2, 3]. As for the turbulent flow in eccentric annular duct, it is interesting from a fundamental point of view since it presents an ideal model for investigating inhomogeneous turbulent flows, where the conditions of turbulence production vary significantly within the cross-section.