Laurent Cordier

Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced-order model

In this paper we investigate the optimal control approach for the active control and drag optimization of incompressible viscous flow past circular cylinders. The control function is the time angular velocity of the rotating cylinder. The wake flow is solved in the laminar regime Re=200 with a finite-element method. Due to the CPU and memory costs related to the optimal control theory, a proper orthogonal decomposition (POD) reduced-order model (ROM) is used as the state equation. The key enablers to an accurate and robust POD ROM are the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem and the use of a snapshot ensemble for POD based on chirp-forced transients. Since the POD basis represents only velocities, we minimize a drag-related cost functional characteristic of the wake unsteadiness. The optimization problem is solved using Lagrange multipliers to enforce the constraints. 25% of relative drag reduction is found when the Navier-Stokes equations are controlled using a harmonic control function deduced from the optimal solution determined with the POD ROM. Earlier numerical studies concerning mean drag reduction are confirmed: it is shown, in particular, that without a sufficient penalization of the control input, our approach is energetically inefficient. The main result is that cost-reduction factors of 100 and 760 are obtained for the CPU time and the memory, respectively. Finally, limits of the performance of our approach are discussed.

Optimal control of the cylinder wake in the laminar regime by trust-region methods and POD reduced-order models

In this paper we investigate the optimal control approach for the active control of the circular cylinder wake flow considered in the laminar regime (Re = 200). The objective is the mean drag minimization of the wake where the control function is the time harmonic angular velocity of the rotating cylinder. When the Navier-Stokes equations are used as state equation, the discretization of the optimality system leads to large scale discretized optimization problems that represent a tremendous computational task. In order to reduce the number of state variables during the optimization process, a Proper Orthogonal Decomposition (POD) Reduced Order Model (ROM) is then derived to be used as state equation. Since the range of validity of the POD ROM is generally limited to the vicinity of the design parameters in the control parameter space, we propose to use the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reduced order models during the optimization process. Benefiting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with a high a fidelity model. A lot of computational work is indeed saved because the optimization process is now based only on low-fidelity models. When the TRPOD is applied to the wake flow configuration, this approach leads to a relative mean drag reduction of 30% for reduced numerical costs.

Examination of large-scale structures in a turbulent plane mixing layer. Part 1. Proper orthogonal decomposition

Large-scale structures in a plane turbulent mixing layer are studied through the use of the proper orthogonal decomposition (POD). Extensive experimental measurements are obtained in a turbulent plane mixing layer by means of two cross-wire rakes aligned normal to the direction of the mean shear and perpendicular to the mean flow direction. The measurements are acquired well into the asymptotic region. From the measured velocities the two-point spectral tensor is calculated as a function of separation in the cross-stream direction and spanwise and streamwise wavenumbers. The continuity equation is then used for the calculation of the non-measured components of the tensor. The POD is applied using the cross-spectral tensor as its kernel. This decomposition yields an optimal basis set in the mean square sense. The energy contained in the POD modes converges rapidly with the first mode being dominant (49% of the turbulent kinetic energy). Examination of these modes shows that the first mode contains evidence of both known flow organizations in the mixing layer, i.e. quasi-two-dimensional spanwise structures and streamwise aligned vortices. Using the shot-noise theory the dominant mode of the POD is transformed back into physical space. This structure is also indicative of the known flow organizations.

Examination of large-scale structures in a turbulent plane mixing layer. Part 2. Dynamical systems model

The temporal dynamics of large-scale structures in a plane turbulent mixing layer are studied through the development of a low-order dynamical system of ordinary differential equations (ODEs). This model is derived by projecting Navier–Stokes equations onto an empirical basis set from the proper orthogonal decomposition (POD) using a Galerkin method. To obtain this low-dimensional set of equations, a truncation is performed that only includes the first POD mode for selected streamwise/spanwise ( k 1 / k 3 ) modes. The initial truncations are for k 3 = 0; however, once these truncations are evaluated, non-zero spanwise wavenumbers are added. These truncated systems of equations are then examined in the pseudo-Fourier space in which they are solved and by reconstructing the velocity field. Two different methods for closing the mean streamwise velocity are evaluated that show the importance of introducing, into the low-order dynamical system, a term allowing feedback between the turbulent and mean flows. The results of the numerical simulations show a strongly periodic flow indicative of the spanwise vorticity. The simulated flow had the correct energy distributions in the cross-stream direction. These models also indicated that the events associated with the centre of the mixing layer lead the temporal dynamics. For truncations involving both spanwise and streamwise wavenumbers, the reconstructed velocity field exhibits the main spanwise and streamwise vortical structures known to exist in this flow. The streamwise aligned vorticity is shown to connect spanwise vortex tubes.

Cluster-based reduced-order modelling of a mixing layer

We propose a novel cluster-based reduced-order modelling (CROM) strategy of unsteady flows. CROM combines the cluster analysis pioneered in Gunzburgers group (Burkardt et al. 2006) and and transition matrix models introduced in fluid dynamics in Eckhardts group (Schneider et al. 2007). CROM constitutes a potential alternative to POD models and generalises the Ulam-Galerkin method classically used in dynamical systems to determine a finite-rank approximation of the Perron-Frobenius operator. The proposed strategy processes a time-resolved sequence of flow snapshots in two steps. First, the snapshot data are clustered into a small number of representative states, called centroids, in the state space. These centroids partition the state space in complementary non-overlapping regions (centroidal Voronoi cells). Departing from the standard algorithm, the probabilities of the clusters are determined, and the states are sorted by analysis of the transition matrix. Secondly, the transitions between the states are dynamically modelled using a Markov process. Physical mechanisms are then distilled by a refined analysis of the Markov process, e.g. using finite-time Lyapunov exponent and entropic methods. This CROM framework is applied to the Lorenz attractor (as illustrative example), to velocity fields of the spatially evolving incompressible mixing layer and the three-dimensional turbulent wake of a bluff body. For these examples, CROM is shown to identify non-trivial quasi-attractors and transition processes in an unsupervised manner. CROM has numerous potential applications for the systematic identification of physical mechanisms of complex dynamics, for comparison of flow evolution models, for the identification of precursors to desirable and undesirable events, and for flow control applications exploiting nonlinear actuation dynamics.

Large-Scale-Structure Identification and Control in Turbulent Shear Flows

The control of turbulent shear flows can be achieved through the control of the large-scale structures. Indeed, even in fully turbulent shear flows, these large-scale structures are known to be of primary importance for most of the flow characteristics. Due to the turbulent character of flows under practical and industrial interests, the detection, analysis, prediction and control of the large-scale structures are quite complex. We present, in this chapter, several tools available for accessing these large-scale structures. Non conditional, stochastic methods, based on correlations, are preferentially detailed. In particular, the Proper Orthogonal Decomposition and the Linear Stochastic Estimation are described. As an illustration of the potentialities of these objective stochastic approaches, the particular case of the turbulent plane mixing layer is derived as a guide-line all along this chapter. In its last part this chapter will focus onto the way by which, using POD, low dimensional models of the structures can be derived, in the framework of a closed loop control of these structures. This approach, deeply related to the notion of deterministic chaos, is then emphasized. The important problem of the closure (mean flow/turbulent flow) appearing in this context is recalled. The last part of this chapter is devoted to a point that is specific and of importance for these stochastic approaches. The limits for the representation of the flow by these models when the flow configuration evolves have to be given. The methods able to take into account this evolution are of crucial importance and under rapid development.

Reduced-order models for closed-loop wake control

We review a strategy for low- and least-order Galerkin models suitable for the design of closed-loop stabilization of wakes. These low-order models are based on a fixed set of dominant coherent structures and tend to be incurably fragile owing to two challenges. Firstly, they miss the important stabilizing effects of interactions with the base flow and stochastic fluctuations. Secondly, their range of validity is restricted by ignoring mode deformations during natural and actuated transients. We address the first challenge by including shift mode(s) and nonlinear turbulence models. The resulting robust least-order model lives on an inertial manifold, which links slow variations in the base flow and coherent and stochastic fluctuation amplitudes. The second challenge, the deformation of coherent structures, is addressed by parameter-dependent modes, allowing smooth transitions between operating conditions. Now, the Galerkin model lives on a refined manifold incorporating mode deformations. Control design is a simple corollary of the distilled model structure. We illustrate the modelling path for actuated wake flows.

Development of a nonlinear eddy-viscosity closure for the triple-decomposition stability analysis of a turbulent channel

The analysis of the instabilities in an unsteady turbulent flow is undertaken using a triple decomposition to distinguish between the time-averaged field, a coherent wave and the remaining turbulent scales of motion. The stability properties of the coherent scale are of interest. Previous studies have relied on prescribed constants to close the equations governing the evolution of the coherent wave. Here we propose an approach where the model constants are determined only from the statistical measures of the unperturbed velocity field. Specifically, a nonlinear eddy-viscosity model is used to close the equations, and is a generalisation of earlier linear eddy-viscosity closures. Unlike previous models the proposed approach does not assume the same dissipation rate for the time- and phase-averaged fields. The proposed approach is applied to a previously published turbulent channel flow, which was harmonically perturbed by two vibrating ribbons located near the channel walls. The response of the flow was recorded at several downstream stations by phase averaging the probe measurements at the same frequency as the forcing. The experimentally measured growth rates and velocity profiles, are compared to the eigenvalues and eigenvectors resulting from the stability analysis undertaken herein. The modes recovered from the solution of the eigenvalue problem, using the nonlinear eddy-viscosity model, are shown to capture the experimentally measured spatial decay rates and mode shapes of the coherent scale.

On the generation of a reverse von Kármán street for the controlled cylinder wake in the laminar regime

In this Brief Communication we are interested in the maximum mean drag reduction that can be achieved under rotary sinusoidal control for the circular cylinder wake in the laminar regime. For a Reynolds number equal to 200, we give numerical evidence that partial control restricted to an upstream part of the cylinder surface may considerably increase the effectiveness of the control. Indeed, a maximum value of relative mean drag reduction equal to 30% is obtained when applying a specific sinusoidal control to the whole cylinder, where up to 75% of reduction can be obtained when the same control law is applied only to a well-selected upstream part of the cylinder. This result suggests that a mean flow correction field with negative drag is observable for this controlled flow configuration. The significant thrust force that is locally generated in the near wake corresponds to a reverse von Karman vortex street as commonly observed in fish-like locomotion or flapping wing flight. Finally, the energetic efficiency of the control is quantified by examining the power saving ratio: it is shown that our approach is energetically inefficient. However, it is also demonstrated that for this control scheme the improvement of the effectiveness generally occurs along with an improvement of the efficiency.

Drag Minimization of the Cylinder Wake by Trust-Region Proper Orthogonal Decomposition

In this paper we investigate the optimal control approach for the active control of the circular cylinder wake flow considered in the laminar regime (Re = 200). The objective is the minimization of the mean total drag where the control function is the time harmonic angular velocity of the rotating cylinder. When the Navier-Stokes equations are used as state equations, the discretization of the optimality system leads to large scale discretized optimization problems that represent a tremendous computational task. In order to reduce the number of state variables during the optimization process, a Proper Orthogonal Decomposition (POD) Reduced-Order Model (ROM) is then derived to be used as state equation. Since the range of validity of the POD ROM is generally limited to the vicinity of the design parameters in the control parameter space, we propose to use the Trust-Region Proper Orthogonal Decomposition (TRPOD) approach, originally introduced by Fahl (2000), to update the reduced-order models during the optimization process. Benefiting from the trust-region philosophy, rigorous convergence results guarantee that the iterates produced by the TRPOD algorithm will converge to the solution of the original optimization problem defined with the Navier-Stokes equations. A lot of computational work is indeed saved because the optimization process is now based only on low-fidelity models. The key enablers to an accurate and robust POD ROM for the pressure and velocity fields are the extension of the POD basis functions to the pressure data, the introduction of a time-dependent eddy-viscosity estimated for each POD mode as the solution of an auxiliary optimization problem, and the inclusion in the POD ROM of different non-equilibrium modes. When the TRPOD algorithm is applied to the wake flow configuration, this approach converges to the minimum predicted by an open-loop control approach and leads to a relative mean drag reduction of 30% for reduced numerical costs (a cost reduction factor of 1600 is obtained for the memory and the optimization problem is solved approximately 4 times more quickly).