Davide Lasagna

Multi-time delay, multi-point Linear Stochastic Estimation of a cavity shear layer velocity from wall-pressure measurements

In this paper, the recently introduced multi-time-delay linear stochastic estimation technique is thoroughly described, focusing on its fundamental properties and potentialities. In the multi-time-delay approach, the estimate of the temporal evolution of the velocity at a given location in the flow field is obtained from multiple past samples of the unconditional sources. The technique is applied to estimate the velocity in a cavity shear layer flow, based on wall-pressure measurements from multiple sensors. The cavity flow was investigated by performing simultaneous measurements of a single hot-wire probe, traversed on a fine grid in the shear layer, and of multiple wall-mounted condenser microphones in the cavity region. The paper compares classical high-order single-time-delay estimation approaches with the multi-time-delay technique, which is significantly more accurate as it produces a much lower mean-square estimation error, thus providing a faithful reconstruction of the time-evolution of the velocity field. This improved accuracy is strongly dependent on the number n of past wall-pressure samples used in the estimate. In this paper, we also demonstrate that the estimated velocity field only contains the signature of the relevant flow mechanisms which correlate well with the wall-pressure, while incoherent components are filtered out. The multi-time-delay approach successfully captures the spatio-temporal dynamics of the velocity fluctuation distribution in the shear layer, as it clearly resolves the dynamics of the relevant flow structures. The effect of the number of sensors used in the estimate was also considered. In general, it was evident that use of more sensors leads to better accuracy, but as the number n of past values increases, the gain becomes marginal.

Nonlinear Multi-Time-Delay Stochastic Estimation: Application to Cavity Flow and Turbulent Channel Flow

A nonlinear extension of the multi-time-delay stochastic estimation technique is presented. The proposed approach consists of the design of nonlinear prediction filters based on artificial neural networks or, for smaller problems, on Volterra expansions of the measured wall variable. The application to two different flows is discussed. The first case is the estimation of the temporal dynamics of the velocity fluctuations in a cavity shear layer in low subsonic conditions from wall-pressure measurements. The second case is the estimation of the streamwise velocity fluctuations in the buffer layer of a fully developed turbulent channel flow from wall shear stress measurements. It is shown that the accuracy of the nonlinear technique is application dependent as it is significantly affected by the underlying nonlinear nature of the flow investigated. In particular, we show that, for the cavity shear layer case, the improvement is marginal and it does not appear to be worth the additional computational complex...

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 characterizing 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 characterizing 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.

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

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 , 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 are finally discussed.

Streamwise vortices originating from synthetic jet?turbulent boundary layer interaction

The interaction between a flat plate turbulent boundary layer and a synthetic jet issuing from a rectangular slot slanted with respect to the free stream was studied experimentally using digital particle image velocimetry. Instantaneous flow fields were sampled in a cross-plane downstream of the slot. Results concerning the effects of varying the synthetic jet velocity ratio at fixed stroke length L0 and yaw angle, and the effects of varying the orifice yaw angle β at a fixed frequency are presented. The formation of a pair of counter-rotating vortical structures, completely embedded in the boundary layer, was observed in the mean flow field when the slot was aligned with the cross-flow. As the slot yaw angle was increased, the leeward vortex intensified while the other became weaker. These vortical structures are the traces of streamwise vortices forming upstream, at the slot exit, during the blowing phases. As the jet velocity ratio and the slot yaw angle were increased the vortices grew in size and intensity. The vortex identification technique showed that these vortical structures are intermittently present in the instantaneous flow fields with a percentage growing with the frequency but not influenced by the yaw angle. Conditional averages showed that while the rotational core of the identified vortices is nearly unaffected, their outer region is greatly modified and grows in size and intensity as the jet velocity ratio and the yaw angle increases.

Virtual Shaping on NACA 0015 by Means of a High Momentum Coefficient Synthetic Jet

Results concerning flow control on a NACA 0015 airfoil using high power synthetic jets are presented for low incidences and for Reynolds numbers ranging from 132000 to 425000. The forcing was operated through a spanwise slit positioned near the leading edge at x/c = 1.25% or at x/c = 10% on the upper surface. Static pressure distribution measurements around the airfoil, wake surveys and smoke flow visualizations were performed. Pressure distributions were significantly modified around the injection location, showing an area of intense suction which increased the lift and strongly affected the drag. Flow visualizations highlighted that the intense suction was due to a virtual shaping effect caused by the formation of a recirculation bubble capable of displacing the streamlines. Low momentum deficits in the wake velocity distributions and, in certain conditions, jet-like flow were observed for the forced cases. Finally, a scaling law relating the bubble size to the forcing intensity is proposed.

Expensive Control of Long-Time Averages Using Sum of Squares and Its Application to A Laminar Wake Flow

This 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 semidefinite 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 by Chernyshenko et al., 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, a controller design requires the simultaneous optimization of both the control law and a tunable function, similar to a Lyapunov function. This paper introduces a method resolving the well-known inherent nonconvexity 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.

Noise Filtering for Wall-Pressure Fluctuations in Measurements Around a Cylinder With Laminar and Turbulent Flow Separation

This paper proposes two different noise cancellation techniques for cleaning wall-pressure fluctuations signals. These fluctuations are measured around a circular cylinder with laminar and turbulent flow separation. The noise cancellation techniques are based on Wiener and adaptive filters and use the signals of pressure transducers mounted in a cross section of the cylinder and the signal of a free-field sensor opportunely located upstream. First, synthetic signals are used in order to validate the procedure. Then, both techniques are applied to the experimental data. Specific attention is paid to the filter order, optimized by a method introduced in this paper. Both filter types showed a selective behavior preserving the essence of the fluid dynamic phenomena characterizing the flow fields at each Reynolds number tested, especially when laminar separation occurs.

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

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.

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

A recently proposed approach to investigating the stability of fluid flows with respect to perturbation of arbitrary large amplitude is applied to a version of the stability problem for the rotating Couette flow. An analytic method of obtaining one of the bounds required in the approach is proposed, and calculations are performed. The obtained results constitute a proof of concept of the approach and, to the best of our knowledge, it is also the first case when global stability of fluid flow was proved by a generic method for the value of a Reynolds number greater than what could be achieved with the energy stability approach.