## Publications

99 results found

Chernyshenko S, 2023, Background flow hidden in a bound for Nusselt number, *Physica D: Nonlinear Phenomena*, Vol: 445, ISSN: 0167-2789

The well-known background flow method for finding bounds for time-averaged characteristics of dynamical systems, proposed by Doering and Constantin (1994, 1995) is a special case of the auxiliary functional method of Chernyshenko et al. (2014). Chernyshenko (2022) proved that bounds obtained by the direct method described by Seis (2015) can be obtained also by the auxiliary functional method and, therefore, by the background flow method when the auxiliary functional is quadratic. This brief note outlines the technique by which the background flow and more generally the auxiliary functional can be obtained when a proof of a bound for infinite time average by the direct method is known, by applying this technique to the case of the bound on the Nusselt number for infinite-Prandtl-number Rayleigh–Bénard convection obtained by Otto and Seis (2011).

Jiao Y, Chernyshenko S, Hwang Y, 2022, A driving mechanism of near-wall turbulence subject to adverse pressure gradient in a plane Couette flow, *Journal of Fluid Mechanics*, Vol: 941, ISSN: 0022-1120

The effect of adverse pressure gradient (APG) on near-wall turbulence is studied, with a particular attention to the turbulence production mechanism. A plane turbulent Couette flow is considered for several values of constant APG in the lower wall region. A direct numerical simulation (DNS) in a large computational domain shows that turbulence near the lower wall continues to exist even at sufficiently large APGs. On increasing the APG, the cross-streamwise turbulence intensities increase, and the near-wall streaks gradually disappear. A linear analysis using the optimal transient growth indicates that the APG inhibits the generation of near-wall streaks due to the significant reduction of the mean shear in the region near the lower wall. The turbulent fluctuation dynamics beyond the linear regime is studied with a DNS in a minimal flow unit. The near-wall self-sustaining process involving streaks is significantly weakened or destroyed as APG increases, while the turbulent fluctuations become more isotropic and localised. Using a conditional averaging analysis, a new mechanism of near-wall turbulence production under strong APG is uncovered. This mechanism is initiated by the wall-normal nonlinear transport of an outer wall-normal velocity fluctuation to the near-wall region. The transported wall-normal velocity fluctuation is subsequently amplified via the Orr mechanism, resulting in the non-zero turbulence production involving spatially localised vortical structures. This mechanism is also confirmed by DNS of the flow in a large computational domain, where strong correlation between the wall-normal nonlinear transport and turbulence production is observed.

Fuentes F, Goluskin D, Chernyshenko S, 2022, Global Stability of Fluid Flows Despite Transient Growth of Energy, *PHYSICAL REVIEW LETTERS*, Vol: 128, ISSN: 0031-9007

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Lakshmi MV, Fantuzzi G, Chernyshenko SI,
et al., 2021, Finding unstable periodic orbits: A hybrid approach with polynomial optimization, *Physica D: Nonlinear Phenomena*, Vol: 427, ISSN: 0167-2789

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.

Chernyshenko S, 2021, Extension of QSQH theory of scale interaction in near-wall turbulence to all velocity components, *JOURNAL OF FLUID MECHANICS*, Vol: 916, ISSN: 0022-1120

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- Citations: 4

Jiao Y, Hwang Y, Chernyshenko S, 2021, The Orr mechanism in transition of parallel shear flow, *Physical Review Fluids*, Vol: 6, ISSN: 2469-990X

The Orr mechanism is revisited to understand its precise role in the transition of plane Couette flow. By considering homogeneous shear flow and plane Couette flow, it is identified that the Orr mechanism induces a lift-up effect which significantly amplifies spanwise velocity. An optimal perturbation analysis for an individual velocity component reveals that the amplification of spanwise velocity is most active at the streamwise length comparable to the given spanwise length of the perturbation. The relevance of this mechanism to transition is subsequently examined in plane Couette flow. To this end, a set of initial conditions, which combines the optimal perturbation for spanwise velocity with the one for all the velocity components, is considered by varying their amplitudes. Two representative transition scenarios are found: oblique and streak transitions. In the former, the spanwise velocity perturbation amplified with the Orr mechanism initiates both streak amplification and breakdown, whereas in the latter, its role is limited only to the streak breakdown at the late stage of transition. As such, the oblique transition offers a route to turbulence energetically more efficient than the streak transition, at least for the cases examined in the present paper. Finally, the oblique transition is found to share many physical similarities with the transition by the minimal seed.

Lakshmi MV, Fantuzzi G, Fernández-Caballero JD,
et al., 2020, Finding extremal periodic orbits with polynomial optimization, with application to a nine-mode model of shear ﬂow, *SIAM Journal on Applied Dynamical Systems*, Vol: 19, Pages: 763-787, ISSN: 1536-0040

Tobasco et al. [Phys. Lett. A, 382:382–386, 2018] 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 Netwon–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.

Iyer A, Witherden F, Chernyshenko S,
et al., 2019, Identifying eigenmodes of averaged small-amplitude perturbations to turbulent channel flow, *Journal of Fluid Mechanics*, Vol: 875, Pages: 758-780, ISSN: 0022-1120

Eigenmodes of averaged small-amplitude perturbations to a turbulent channel flow — which is one of the most fundamental canonical flows — are identified for the first time via an extensive set of high-fidelity GPU-accelerated direct numerical simulations. While the system governing averaged small-amplitude perturbations to turbulent channel flow remains unknown, the fact such eigenmodes can be identified constitutes direct evidence that it is linear. Moreover, while the eigenvalue associated with the slowest-decaying anti-symmetric eigenmode mode is found to be real, the eigenvalue associated with the slowest-decaying symmetric eigenmode mode is found to be complex. This indicates that the unknown linear system governing the evolution of averaged small-amplitude perturbations cannot be self-adjoint, even for the case of a uni-directional flow. In addition to elucidating aspects of the flow physics, the findings provide guidance for development of new unsteady Reynolds-averaged Navier-Stokes turbulence models, and constitute a new and accessible benchmark problem for assessing the performance of existing models,which are used widely throughout industry.

Chernyshenko SI, Zhang C, Butt H,
et al., 2019, A large-scale filter for applications of QSQH theory of scale interactions in near-wall turbulence, *Fluid Dynamics Research*, Vol: 51, ISSN: 0169-5983

An outlook on the recently proposed quasi-steady quasi-homogeneous (QSQH) theory of the effect of large-scale structures on the near-wall turbulence is provided. The paper focuses on the selection of the filter, which defines the large-scale structures. It gives a brief overview of the QSQH theory, discusses the filter needed to distinguish between large and small scales, and the related issues of the accuracy of the QSQH theory, describes the probe needed for using the QSQH theory, and outlines the procedure of extrapolating the characteristics of near-wall turbulence from medium to high Reynolds numbers.

Ghebali S, Chernyshenko SI, Leschziner MA, 2019, Turbulent-drag reduction by oblique wavy wall undulations, ERCOFTAC Series, Pages: 545-551

Reducing the turbulent skin-friction drag over civilian aircraft is a potentially high-reward target, as this drag component accounts for about half of the total drag in cruise conditions. Thus, even modest reductions convert into material savings, resulting in significant cuts in costs. Active-control techniques can be remarkably effective at suppressing turbulence and drag, but pose major engineering challenges in terms of actuation, efficient operation, reliability and maintainability. In contrast, passive techniques based on riblets are easier to implement, but face important durability and maintenance limitations related to the extremely small spacing of the grooves. The alternative passive-control method that is the subject of the present paper was first proposed in Chernyshenko (Drag reduction by a solid wavy wall emulating spanwise oscillations. Part 1. [physics.flu-dyn](arXiv:1304.4638 ), (2013), [1]). The key characteristic of the method is that it involves wavy surface undulations directed obliquely to the mean flow and having wave lengths two orders of magnitude larger than riblets, and would thus be much more practical to manufacture and maintain.

Ghebali S, Chernyshenko SI, Leschziner M, 2017, Can large-scale oblique undulations on a solid wall reduce the turbulent drag?, *Physics of Fluids*, Vol: 29, Pages: 105102-1-105102-15, ISSN: 1070-6631

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. However, some important differences with respect to the SSL case are brought to light too. In particular, the phase variations of the turbulent properties are accentuated and, 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 pressure- and friction-drag levels are carefully quantified for various flow configurations, exhibiting a combined maximum overall-drag reduction of about 0.6%. 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. The transverse shear-strain layer is shown to be approximately Reynolds-number independent when the wave geometry is scaled in wall units.

Ghebali S, Chernyshenko SI, Leschziner MA, 2017, Can large-scale oblique undulations on a solid wall reduce the turbulent drag?, *Physics of Fluids*, Vol: 29, Pages: 105102-105102

Direct numerical simulations of fully-developed turbulent channel flows withwavy walls are undertaken. The wavy walls, skewed with respect to the mean flowdirection, are introduced as a means of emulating a Spatial Stokes Layer (SSL)induced by in-plane wall motion. The transverse shear strain above the wavywall is shown to be similar to that of a SSL, thereby affecting the turbulentflow, and leading to a reduction in the turbulent skin-friction drag. Thepressure- and friction-drag levels are carefully quantified for various flowconfigurations, exhibiting a combined maximum overall-drag reduction of about0.5%. The friction-drag reduction is shown to behave approximatelyquadratically for small wave slopes and then linearly for higher slopes, whilstthe pressure-drag penalty increases quadratically. Unlike in the SSL case,there is a region of increased turbulence production over a portion of thewall, above the leeward side of the wave, thus giving rise to a local increasein dissipation. The transverse shear-strain layer is shown to be approximatelyReynolds-number independent when the wave geometry is scaled in wall units.

Ghebali S, Chernyshenko S, Leschziner M, 2017, Turbulent drag reduction by wavy wall, 10th International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2017

Fully-developed turbulent flow in channels with oblique wavy walls is analysed, from a drag-reduction perspective, by means of Direct Numerical Simulations (DNS). The wavy geometry is chosen to emulate the shear strain produced by a Spatial Stokes Layer (SSL) generated by oscillatory wall motion. As the cost of performing a parametric optimisation is prohibitive, an alternate solution is presented, based on a linear model of a perturbed plane-channel flow, using a turbulent viscosity. Flow properties and levels of drag reduction or increase are reported for various configurations.

Chernyshenko S, 2017, Relationship between the methods of bounding time averages

The problem of finding bounds of time-averaged characteristics of dynamicalsystems, such as for example the bound on the mean energy dissipation rate in aturbulent flow governed by incompressible Navier-Stokes equations, isconsidered. It is shown that both the well-known background flow method ofDoering and Constantin and the direct method proposed by Seis in 2015correspond to the same quadratic storage functional in the framework of theindefinite storage functional method. In particular, a background flow can befound corresponding to the linear functional used in the direct method and viceversa. It is shown that any bound obtained with the background flow method canalso be obtained by the direct method. The reverse is true subject to anadditional constraint. The relative advantages of the three methods arediscussed.

Huang D, Jin B, Lasagna D,
et al., 2017, Expensive control of long-time averages using sum of squares and Its application to a laminar wake flow, *IEEE Transactions on Control Systems Technology*, Vol: 25, Pages: 2073-2086, ISSN: 1558-0865

The paper presents a nonlinear state-feedback con-trol design approach for long-time average cost control, where thecontrol effort is assumed to be expensive. The approach is basedon sum-of-squares and semi-definite programming techniques. Itis applicable to dynamical systems whose right-hand side is apolynomial function in the state variables and the controls. Thekey idea, first described but not implemented in (Chernyshenkoetal.Phil. Trans. R. Soc. A, 372, 2014), is that the difficult problemof optimizing a cost function involving long-time averages isreplaced by an optimization of the upper bound of the sameaverage. As such, controller design requires the simultaneousoptimization of both the control law and a tunable function,similar to a Lyapunov function. The present paper introducesa method resolving the well-known inherent non-convexity ofthis kind of optimization. The method is based on the formalassumption that the control is expensive, from which it followsthat the optimal control is small. The resulting asymptoticoptimization problems are convex. The derivation of all thepolynomial coefficients in the controller is given in terms ofthe solvability conditions of state-dependent linear and bilinearinequalities. The proposed approach is applied to the problemof designing a full-information feedback controller that mitigatesvortex shedding in the wake of a circular cylinder in the laminarregime via rotary oscillations. Control results on a reduced-ordermodel of the actuated wake and in direct numerical simulationare reported.

Chernyshenko SI, Zhang C, Butt H, et al., 2017, Extrapolating statistics of turbulent flows to higher Re using quasi-steady theory of scale interaction in near-wall turbulence, 10th International Symposium on Turbulence and Shear Flow Phenomena, TSFP 2017

A new technique for extrapolating statistical characteristics of near-wall turbulence from medium to higher Re is outlined. Results for extrapolating the velocity two-point correlation from Reτ= 2003 to Reτ= 4179 and for the parameters of an optimized comb probe for detecting the large-scale velocity component required for applying the technique in practice are presented.

Lasagna D, Huang D, Tutty OR,
et al., 2016, Sum-of-Squares approach to feedback control of laminar wake flows, *Journal of Fluid Mechanics*, Vol: 809, Pages: 628-663, ISSN: 0022-1120

In this paper 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. (Phil. Trans. R. Soc. Lond. A, vol. 372, 2014, 20130350), 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 and efficient 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 first 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 resolved kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, total energy efficiency and the physical control mechanisms identified are analysed in detail. Key elements of the methodology, implications and future work ar

Fantuzzi G, Goluskin D, Huang D,
et al., 2016, Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization, *SIAM Journal on Applied Dynamical Systems*, Vol: 15, Pages: 1962-1988, ISSN: 1536-0040

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.

Fantuzzi G, Goluskin D, Huang D,
et al., 2016, Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization, *SIAM Journal on Applied Dynamical Systems*, Vol: 15, Pages: 1962-1988, ISSN: 1536-0040

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.

Zhang C, Chernyshenko SI, 2016, Quasi-steady quasi-homogeneous description of the scale interactions in near-wall turbulence, *Physical Review Fluids*, Vol: 1, ISSN: 2469-990X

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.

Lasagna D, Tutty OR, Chernyshenko S, 2016, Flow regimes in a simplified Taylor-Couette-type flow model, *European Journal of Mechanics B - Fluids*, Vol: 57, Pages: 176-191, ISSN: 0997-7546

In this paper we introduce a simplified variant of the well-known Taylor–Couette flow. The aim is to develop and investigate a model problem which is as simple as possible while admitting a wide range of behaviour, and which can be used for further study into stability, transition and ultimately control of flow. As opposed to models based on ordinary differential equations, this model is fully specified by a set of partial differential equations that describe the evolution of the three velocity components over two spatial dimensions, in one meridian plane between the two counter-rotating coaxial cylinders. We assume axisymmetric perturbations of the flow in a narrow gap limit of the governing equations and, considering the evolution of the flow in a narrow strip of fluid between the two cylinders, we assume periodic boundary conditions along the radial and axial directions, with special additional symmetry constraints. In the paper, we present linear stability analysis of the first bifurcation, leading to the well known Taylor vortices, and of the secondary bifurcation, which, depending on the type of symmetries imposed on the solution, can lead to wave-like solutions travelling along the axial direction. In addition, we show results of numerical simulations to highlight the wide range of flow structures that emerge, from simple uni-directional flow to chaotic motion, even with the restriction placed on the flow.

Lasagna D, Huang D, Tutty OR, et al., 2016, Controlling Fluid Flows with Positive Polynomials, 35th Chinese Control Conference (CCC), Publisher: IEEE, Pages: 1301-1306, ISSN: 2161-2927

Huang D, Chernyshenko S, 2015, Long-Time Average Cost Control of Polynomial Systems: A Sum-of-Squares-Based Small-Feedback Approach, 8th ASME Annual Dynamic Systems and Control Conference (DSCC 2015), Publisher: AMER SOC MECHANICAL ENGINEERS

This paper provides a proof of concept of the recent novel idea in the area of long-time average cost control. Meanwhile, a new method of overcoming the well-known difficulty of nonconvexity of simultaneous optimization of a control law and an additional tunable function is given. First, a recently-proposed method of obtaining rigorous bounds of long-time average cost is 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.

Huang D, Chernyshenko S, Lasagna D, et al., 2015, Long-time average cost control of polynomial systems: a sum of squares approach, 2015 European Control Conference (ECC), Publisher: IEEE, Pages: 3244-3249

This paper provides a numerically tractable approach for long-time average cost control of nonlinear dynamical systems with polynomials of system state on the right-hand side. First, a recently-proposed method of obtaining rigorous bounds of long-time average cost is outlined for the uncontrolled system, where the polynomial constraints are strengthened to be sum-of-squares and formulated as semi-definite programs. As such, it allows to use any general (polynomial) functions to optimize the bound. Then, a polynomial type state feedback controller design scheme is presented to further suppress the long-time average cost. The derivation of state feedback controller is given in terms of the solvability conditions of state-dependent bilinear matrix inequalities. Finally, the mitigation of oscillatory vortex shedding behind a cylinder is addressed to illustrate the validity of the proposed approach.

Huang D, Chernyshenko SI, Goulart P,
et al., 2015, Sum-of-squares of polynomials approach to nonlinear stability of fluid flows: an example of application, *Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences*, Vol: 471, Pages: 1-18, ISSN: 0080-4630

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.

Huang D, Chernyshenko S, 2015, Low-order State-feedback Controller Design for Long-time Average Cost Control of Fluid Flow Systems: A Sum-of-squares Approach, 34th Chinese Control Conference (CCC), Publisher: IEEE, Pages: 2479-2484, ISSN: 2161-2927

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Huang D, Chernyshenko S, 2015, Long-time Average Cost Control of Stochastic Systems Using Sum of Squares of Polynomials, 34th Chinese Control Conference (CCC), Publisher: IEEE, Pages: 2344-2349, ISSN: 2161-2927

Chernyshenko SI, Goulart P, Huang D,
et al., 2014, Polynomial sum of squares in fluid dynamics: a review with a look ahead, *Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 372, Pages: 1-18, ISSN: 1364-503X

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.

Chernyshenko S, Huang D, Goulart P, et al., 2013, Nonlinear Stability Analysis of Fluid Flow using Sum of Squares of Polynomials, 11th International Conference of Numerical Analysis and Applied Mathematics (ICNAAM), Publisher: AMER INST PHYSICS, Pages: 265-268, ISSN: 0094-243X

Vodop'yanov IS, Nikitin NV, Chernyshenko SI, 2013, Turbulent drag reduction by spanwise oscillations of a ribbed surface, *FLUID DYNAMICS*, Vol: 48, Pages: 461-470, ISSN: 0015-4628

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