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 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

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 con- trol 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

Flow along a long thin cylinder

Two different approaches have been used to calculate turblent flow along a long thin culinder where the flow is aligned with the cylinder. A boundary-layer code is used to predict the mean flow for very long cylinders (length to ratio of up to O(106)), with the effects of the turbulence estimated through a turbulence model. Detailed comparison with experimental results shows that the mean properties of the flow are predicted within experimental accuracy. The boundary-layer model predicts that, sufficiently far downstream, the surface shear stress will be (almost) constant. This is consistent with experimental results from long cylinders in the form of sonar arrays. A periodic Navier-Stokes problem is formulated, and solutions generated for the boundary-layer model and experiments. Strongly turbulent flow occurs only near the surface of the cylinder, with relatively weak turbulence over most of the boundary layer. For a thick boundary layer with the boundary-layer thickness much larger than the cylinder radius, the mean flow is effectively constant near the surface, in both temporal and spatial frameworks, while the outer flow continues to develop in time or space. Calculations of the circumferentially averaged surface pressure spectrum sho that, in physical terms, as the radius of the cylinder decreases, the surface noise from the turbulence increases, with the maximum noise at a Reynolds number of O(103). An increase in noise with a decrease in radius (Reynolds number) is consistent with experimental results

High Reynolds number flow in collapsible pipes and channels

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Flow regimes in a simplified Taylor-Couette-type flow model

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