arXiv subjects

Sergei Chernyshenko

Publications and source records attributed to Sergei Chernyshenko.

12 recordsLinked to original sources

arXiv2025

Representation and Stability Analysis of 1D PDEs with Periodic Boundary Conditions

PDEs with periodic boundary conditions are frequently used to model processes in large spatial environments, assuming solutions to extend periodically beyond some bounded interval. However, solutions to these PDEs often do not converge to a unique equilibrium, but instead converge to non-stationary trajectories existing in the nullspace of the spatial differential operator (e.g. $\frac{\partial^2}{\partial x^2}$). To analyse this convergence behaviour, in this paper, it is shown how such trajectories can be modeled for a broad class of linear, 2nd order, 1D PDEs with periodic as well as more general boundary conditions, using the Partial Integral Equation (PIE) representation. In particular, it is first shown how any PDE state satisfying these boundary conditions can be uniquely expressed in terms of two components, existing in the image and the nullspace of the differential operator $\frac{\partial^2}{\partial x^2}$, respectively. An equivalent representation of linear PDEs is then derived as a PIE, explicitly defining the dynamics of both state components. Finally, a notion of exponential stability is defined that requires only one of the state components to converge to zero, and it is shown how this stability notion can be tested by solving a linear operator inequality. The proposed methodology is applied to examples of heat and wave equations, demonstrating that exponential stability can be verified with tight bounds on the rate of decay.

math.AP↗

arXiv2022

Global stability of fluid flows despite transient growth of energy

Verifying nonlinear stability of a laminar fluid flow against all perturbations is a central challenge in fluid dynamics. Past results rely on monotonic decrease of a perturbation energy or a similar quadratic generalized energy. None show stability for the many flows that seem to be stable despite these energies growing transiently. Here a broadly applicable method to verify global stability of such flows is presented. It uses polynomial optimization computations to construct non-quadratic Lyapunov functions that decrease monotonically. The method is used to verify global stability of 2D plane Couette flow at Reynolds numbers above the energy stability threshold found by Orr in 1907. This is the first global stability result for any flow that surpasses the energy method.

physics.flu-dyn↗

arXiv2021

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 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.

math.DS↗

arXiv2020

Extension of QSQH theory of scale interaction in near-wall turbulence to all velocity components

The QSQH theory is extended to all three velocity components taking into account the fluctuations of the direction of the large-scale component of the wall friction. This effect is found to be significant. It explains the large sensitivity of the fluctuations of longitudinal and spanwise velocities to variations in the Reynolds number in comparison with the sensitivity of the mean velocity, the Reynolds stress, and the wall-normal velocity fluctuations. The analysis shows that the variation of the longitudinal velocity fluctuations with the Reynolds number is dominated by the variation of the amplitude and wall-normal-scale modulation of the universal mean velocity profile by the outer, large-scale, Reynolds-number-dependent motions. The variation of spanwise velocity fluctuations is dominated by the fluctuations of the direction of the large-scale component of the wall friction. The Reynolds number dependence of the other second moments is not dominated by these mechanisms because the mean wall-normal velocity and the mean spanwise velocity are zero. Explicit relationships between the differences in the second moments of velocity in any two high-Reynolds-number near-wall flows were derived. The comparisons gave a satisfactory agreement for the root-mean square of the wall-parallel velocity components in the range of the distances from the wall where modulation by large-scale motions dominates. Relationships between the differences of the constants of the logarithmic law, the shape of the mean velocity profile, and the differences of the second moments of velocity caused by the differences in large-scale motions were derived and estimated quantitatively.

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arXiv2020

Finding extremal periodic orbits with polynomial optimisation, with application to a nine-mode model of shear flow

Tobasco et al. [Physics Letters A, 382:382-386, 2018; see https://doi.org/10.1016/j.physleta.2017.12.023] recently suggested that trajectories of ODE systems that optimize the infinite-time average of a certain observable can be localized using sublevel sets of a function that arise when bounding such averages using so-called auxiliary functions. In this paper we demonstrate that this idea is viable and allows for the computation of extremal unstable periodic orbits (UPOs) for polynomial ODE systems. First, we prove that polynomial optimization is guaranteed to produce auxiliary functions that yield near-sharp bounds on time averages, which is required in order to localize the extremal orbit accurately. Second, we show that points inside the relevant sublevel sets can be computed efficiently through direct nonlinear optimization. Such points provide good initial conditions for UPO computations. As a proof of concept, we then combine these methods with a single-shooting Newton-Raphson algorithm to study extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow. We discover three previously unknown families of UPOs, one of which simultaneously minimizes the mean energy dissipation rate and maximizes the mean perturbation energy relative to the laminar state for Reynolds numbers approximately between 81.24 and 125.

math.DS↗

arXiv2017

Relationship between the methods of bounding time averages

The problem of finding bounds of time-averaged characteristics of dynamical systems, such as the bound on the mean energy dissipation rate in a turbulent flow governed by incompressible Navier-Stokes equations, is considered. It is shown that the direct method described by Seis (J. Fluid Mech., 777:591-603, 2015) and the auxiliary functional method by Chernyshenko et al. (Phil. Trans. R. Soc. B., 372:20130350, 2014) are related and can lead to the same bound. The well-known background flow method of Doering and Constantin is equivalent to the auxiliary functional method with a quadratic auxiliary functional. The known implementations of the direct method apparently also correspond to quadratic auxiliary functionals. The findings are illustrated by the analysis of the plane Couette flow. Three routes of further progress using non-quadratic auxiliary functionals and at the same time allowing to utilise the experience accumulated with the background flow method are proposed: making the balance parameter dependent on the energy, making the background flow time-dependent in a specific way, and adding helicity to the auxiliary functional.

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arXiv2016

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed.

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arXiv2016

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 control 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.

math.OC↗

arXiv2015

Long-time Average Cost Control of Polynomial Systems: A Sum-of-squares-based Small-feedback Approach

The two main contributions of this paper are a proof of concept of the recent novel idea in the area of long-time average cost control, and a new method of overcoming the well-known difficulty of non-convexity of simultaneous optimization of a control law and an additional tunable function. A recently-proposed method of obtaining rigorous bounds of long-time average cost is first outlined for the uncontrolled system with polynomials of system state on the right-hand side. In this method the polynomial constraints are relaxed to be sum-of-squares and formulated as semi-definite programs. It was proposed to use the upper bound of long-time average cost as the objective function instead of the time-average cost itself in controller design. In the present paper this suggestion is implemented for a particular system and is shown to give good results. Designing the optimal controller by this method requires optimising simultaneously both the control law and a tunable function similar to the Lyapunov function. The new approach proposed and implemented in this paper for overcoming the inherent non-convexity of this optimisation is based on a formal assumption that the amplitude of control is small. By expanding the tunable function and the bound in the small parameter, the long-time average cost is reduced by minimizing the respective bound in each term of the series. The derivation of all the polynomial coefficients in controller is given in terms of the solvability conditions of state-dependent linear and bilinear inequalities. The resultant sum-of-squares problems are solved in sequence, thus avoiding the non-convexity in optimization. The proposed approach is implemented for a simple model of oscillatory vortex shedding behind a cylinder.

math.OC↗

arXiv2013

Drag reduction by a solid wall emulating spanwise oscillations. Part 1

A new idea for turbulent skin-friction reduction is proposed, wherein the shape of the solid wall is designed to create the spanwise pressure gradient acting similarly to the well-known method of drag reduction by in-plane spanwise wall motion. Estimates based on the assumption of similarity with drag reduction effect of in-plane wall motion suggest that drag reduction of about 2.4% can be achieved in the flow past a wavy wall, with the crests forming an angle of about 38 degrees with the main flow direction, and the wave period in the main flow direction equal to about 1500 wall units. The required height of the wall waves have to be adjusted to achieve the same intensity of the spanwise motion as that created by an in-plane moving wall of the same wavelength and with peak wall velocity equal to 2 wall units. Further research is being conducted in order to determine this height. Suggestions are made for further research on confirming the feasibility of the proposed method and on optimising the wall shape.

physics.flu-dyn↗

arXiv2011

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

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.

math.OC↗

arXiv2008

Master-modes in 3D turbulent channel flow

Turbulent flow fields can be expanded into a series in a set of basic functions. The terms of such series are often called modes. A master- (or determining) mode set is a subset of these modes, the time history of which uniquely determines the time history of the entire turbulent flow provided that this flow is developed. In the present work the existence of the master-mode-set is demonstrated numerically for turbulent channel flow. The minimal size of a master-mode set and the rate of the process of the recovery of the entire flow from the master-mode set history are estimated. The velocity field corresponding to the minimal master-mode set is found to be a good approximation for mean velocity in the entire flow field. Mean characteristics involving velocity derivatives deviate in a very close vicinity to the wall, while master-mode two-point correlations exhibit unrealistic oscillations. This can be improved by using a larger than minimal master-mode set. The near-wall streaks are found to be contained in the velocity field corresponding to the minimal master-mode set, and the same is true at least for the large-scale part of the longitudinal vorticity structure. A database containing the time history of a master-mode set is demonstrated to be an efficient tool for investigating rare events in turbulent flows. In particular, a travelling-wave-like object was identified on the basis of the analysis of the database. Two master-mode-set databases of the time history of a turbulent channel flow are made available online at http://www.dnsdata.afm.ses.soton.ac.uk/. The services provided include the facility for the code uploaded by a user to be run on the server with an access to the data.

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