Nonlinear stability analysis of fluid flow using sum of squares of polynomials

Chernyshenko, S. I.; Huang, Deqing; Goulart, Paul J.; Lasagna, Davide; Tutty, O.R.

doi:10.1063/1.4825472

Cited by 3 publications

(6 citation statements)

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“…[35,38,39,10,3]), and the software packages YALMIP [25] or SOSTOOLS [29] can assist in formulating SoS constraints as SDPs. More details on convex optimization and the link between SoS polynomials and SDPs can be found in [5,32,26], while examples using SoS programming to study dynamical systems can be found in [32,30,31,37,14,7,16,40,18].…”

Section: B)mentioning

confidence: 99%

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Fantuzzi¹,

Goluskin²,

Huang³

et al. 2016

SIAM J. Appl. Dyn. Syst.

Self Cite

106

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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.

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Section: B)mentioning

confidence: 99%

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Fantuzzi¹,

Goluskin²,

Huang³

et al. 2016

SIAM J. Appl. Dyn. Syst.

Self Cite

106

View full text Add to dashboard Cite

show abstract

“…However, an attempt to do this was not successful [36]. This might imply that the quality of the uncertainty bounds derived and used in [32,33,36] deteriorates as the dimension of the uncertain system increases.…”

Section: (Iii) Application To a Version Of Rotating Couette Flowmentioning

confidence: 99%

“…A bound in the form |Θ| 2 ≤ p(a, q 2 ) = c 1 |a| 2 q 2 + c 2 q 4 , where | · | represents the standard Euclidean norm in R n , is also available [32], and an important improvement described in [33] gives a simple method of calculating the constants c 1 and c 2 . The next step is to abandon u s entirely, keeping only the bounds in terms of a and q 2 on the functions of u s .…”

Section: Applying Sum Of Squares For Stability Analysis Of Fluid Flowsmentioning

confidence: 99%

“…The first application of this approach to a fluid flow is described in [33,36]. The rotating Couette flow is a well-known benchmark for flow stability studies [37].…”

Section: (Iii) Application To a Version Of Rotating Couette Flowmentioning

confidence: 99%

“…Assuming that the gap between the cylinders is much smaller than the cylinder radius, a Cartesian coordinate system x := (x, y, z) can be introduced in such a way that the axis of rotation is parallel to the z-axis, whereas the circumferential direction corresponds to the x-axis. Only flows independent of x were considered in [33,36]. The flow velocity was represented as (y + u , v , w ), so that u = (u , v , w ) was the velocity perturbation.…”

Section: (Iii) Application To a Version Of Rotating Couette Flowmentioning

confidence: 99%

See 2 more Smart Citations

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

Chernyshenko

Goulart

Huang

et al. 2014

Phil. Trans. R. Soc. A.

Self Cite

181

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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.

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Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number

Fantuzzi

Wynn

2017

J. Fluid Mech.

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Using the energy method we investigate the stability of pure conduction in Pearson’s model for Bénard–Marangoni convection in a layer of fluid at infinite Prandtl number. Upon extending the space of admissible perturbations to the conductive state, we find an exact solution to the energy stability variational problem for a range of thermal boundary conditions describing perfectly conducting, imperfectly conducting, and insulating boundaries. Our analysis extends and improves previous results, and shows that with the energy method global stability can be proven up to the linear instability threshold only when the top and bottom boundaries of the fluid layer are insulating. Contrary to the well-known Rayleigh–Bénard convection set-up, therefore, energy stability theory does not exclude the possibility of subcritical instabilities against finite-amplitude perturbations

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Nonlinear stability analysis of fluid flow using sum of squares of polynomials

Cited by 3 publications

References 0 publications

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

Bounds for Deterministic and Stochastic Dynamical Systems using Sum-of-Squares Optimization

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

Exact energy stability of Bénard–Marangoni convection at infinite Prandtl number

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