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

Abstract

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.

Harbingers and latecomers – the order of appearance of exact coherent structures in plane Poiseuille flow

“…All of these states, as well as exact coherent structures with different flow fields contribute to the temporal evolution, and the network that forms with increasing Reynolds numbers and that can then carry the turbulent dynamics. Very little is known about the lowest possible Reynolds number for the appearance of coherent structures, and there are hardly any methods for determining them reliably [39][40][41] However, the state T W E has the potential to be the lowest possible state in PPF flow by analogy to the lowest lying state in plane Couette flow. Identifcation of lower lying exact coherent structures in either flow should therefore also have implications for the other flow.…”

On long-term boundedness of Galerkin models

“…The finding of a 'background flow' representing the coordinate shift (3.4) might remedy the restrictiveness of the energy function but is not necessarily enough for the proof (Doering & Constantin 1994). A promising approach of the construction of less restrictive Lyapunov functions is pursued by Goulart & Chernyshenko (2012) and Chernyshenko et al (2014) based on the polynomial sum of squares approach of Parrilo (2003).…”

Long-time average cost control of polynomial systems: A sum of squares approach

“…While optimization of an upper bound with control [7] does not involve a Lyapunov function, it does involve a similar function, and it shares the same difficulty of nonconvexity. In the present work we present an iterative polynomial type state feedback controller design scheme for the long-time average upper-bound control.…”

“…To overcome this complexity, it was proposed [7] to use an upper bound for the long-time average cost instead of the long-time average cost itself in cases when such an upper bound is easier to calculate. The idea is based on the hope that the control reducing an upper bound for a quantity will also reduce the quantity itself.…”

“…Without going into technical details, it should be noted that these methods are based on optimizing quadratic functionals, which is a significant constraint resulting in the bounds often being far from tight. In [7] a new method of finding bounds for long-time averages was proposed, which allows a trade-off between the quality of bound and the complexity of its calculation. The method utilizes the use of the sum of squares (SOS) decomposition of polynomials and semidefinite programming (SDP).…”