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

Abstract

In this paper, optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime ($\mathit{Re}=200$). The control parameters are the amplitude and the frequency of the time-harmonic cylinder rotation. In order to reduce the size of the discretized optimality system, a proper orthogonal decomposition (POD) reduced-order model (ROM) is derived to be used as state equation. We then propose to employ the trust-region proper orthogonal decomposition (TRPOD) approach, originally introduced by Fahl [M. Fahl, Trust-region methods for flow control based on reduced order modeling, Ph.D. Thesis, Trier University, 2000], to update the reduced-order models during the optimization process. A lot of computational work is saved because the optimization process is now based only on low-fidelity models. A particular care was taken to derive a POD ROM for the pressure and velocity fields with an appropriate balance between model accuracy and robustness. The key enablers are the extension of the POD basis functions to the pressure data, the use of calibration methods for the POD ROM and the addition in the POD expansion of several non-equilibrium modes to describe various operating conditions. 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% at reduced cost.

Introduction

During the last decade, the optimal control theory 1 has emerged as a new approach to solve active flow control and aerodynamic shape design problems. Indeed, these problems can be reduced [2] to the minimization or maximization of an objective functional $\mathcal{J}$ according to n control or design parameters $\mathit{c}=(c_1\text{,}{c}_2\text{,}\dots \text{,}{c}_n)$ under some constraints. However, whatever the specific class of numerical methods generally considered (methods of descent type, stochastic methods), the computational costs (CPU and memory) related to the resolution of optimization problems are so important that they become unsuited to the applications of flow control for three-dimensional turbulent flows. The application in an immediate future of active control to complex flows is thus conditioned by the development of approximate models of the system [3]. The objective of these surrogate models [4] is to capture the essence of the physics of the controlled system while reducing the costs associated to the solution of the non-linear state equations. As a result, there have been many studies devoted to the development of Reduced-Order Models (ROM) that serve as low-dimensional approximation models to the large-scale discretized Navier–Stokes equations [5], for a review of the different reduced-order modelling techniques. The model reduction method discussed in this paper fall in the category of reduced basis approaches. For the reduced bases, a number of choices exist [6], for a presentation: Lagrange basis, Hermite basis, Taylor basis, proper orthogonal decomposition (POD) basis [7], [8], Krylov basis [9], Centroidal Voronoi Tessellations (CVT) basis [10], balanced POD basis [11], etc. Today, the most popular reduced-order modelling approach for complex systems in fluid mechanics is based on POD. This study is restricted to this case: we consider that the unsteady non-linear dynamics of the flow is modelled via a reduced-order model based on POD (POD ROM).

The POD (and other similar techniques of ROM) can be viewed as a method of information compression. Essentially, the POD algorithm try to remove “redundant” information (if any) from the data base. As a consequence, the ability of POD modes to approximate any state of a complex system is totally dependent of the information originally contained in the snapshot set used to generate the POD functions. Thus, a POD basis cannot contain more information than that contained in the snapshot set. The generation of “good” snapshot set is then crucial to the success of use of POD ROM approach in a bifurcation analysis [12], [13], [14] or more generally in an optimization setting. Since the POD basis is intrinsic to a particular flow, we need to give special attention to adapt the POD ROM (and the POD basis naturally) to changes in physics when the flow is altered by control. This central question is discussed in more details in [15] where two strategies are evidenced for use of POD ROM in an optimization setting. A first approach consists in distributing uniformly in the control parameter space the snapshot ensemble to be used for POD. However, in this case, a lot of runs of the high-dimensional code would be necessary to generate the snapshots and that more especially as the number of the control parameters is important. Therefore, developing systematic and rational methodologies for generating good snapshots set is a critical enabler for effective reduced-order modelling, since a POD basis is only as good as the snapshot set used to generate it. Very recently, it was demonstrated in [10] that Centroidal Voronoi Tessellations could be one method of intelligent sampling in parameter space. Failing this, a simpler method to implement is to generate generalized POD functions by forcing the flow with an ad-hoc time-dependent excitation that is rich in transients [16]. The second approach consists of an adaptive method in which new snapshots are regularly determined during the optimization process when the effectiveness of the existing POD ROM to represent accurately the controlled flow is considered to be insufficient [17], [18], [19].

At this point, two key questions still remain:

• How to decide automatically whether or not a POD ROM has to be adapted to a new flow configuration?

• Can we demonstrate under certain conditions (which should ideally be most general as possible) that the optimal solution based on the POD ROM corresponds to a local optimizer for the original problem?

The main drawback of this second approach is that for adaptively updating a reduced basis during an optimization process, new solves of the high-dimensional approximations of the Navier–Stokes equations need to be done. Since these simulations are costly, this approach is not appropriate for real-time control flow.

Due to its simple geometry and its representative behavior of separated flows [20], the cylinder wake flow has been broadly studied this past decade to experiment some control methods that could be used later in more complex engineering configurations. The majority of these studies were motivated by the experiments of [21] where 80% of relative¹ mean drag reduction was empirically found at $\mathit{Re}=15\text{,}000$ by unsteady rotary oscillation of the cylinder. This experimental work was followed by a series of numerical [22], [23], [24], [25], [26], [27], [28], [29], [30] and experimental investigations [31], [32], [33], [34]. Recently, due to the maturity of control theory, optimization methods and computational fluid dynamics, optimal and suboptimal approaches attracted increased attention in flow control setting [35], [36], [37]. For example, in [38], [39], [40] the optimal control theory was used with the two-dimensional Navier–Stokes equations as the state equation to control by rotary oscillation the unsteady wake of the cylinder (see Table 1 for the characteristics of these approaches). An attractive element of the optimal control approach is that the control design is explicitly based on the cost functional. However, the very large computational costs (CPU and memory), involved in the resolution of the optimality system commonly used in the optimal control theory [2], prevent to solve routinely optimization problems based on the three-dimensional Navier–Stokes equations.² For cutting down these numerical costs different approaches are possible [3], for a review. One promising approach is to first develop POD ROM to approximate the fluid flow and then to optimize exactly the reduced-order models as it was already discussed in Section 1.1. A general discussion of the use of approximation models in optimization can be found in [44]. In this study, we want to develop a low-cost optimal control approach for the drag minimization of the cylinder wake with rotary motion for control law (see Fig. 1). In addition, as opposed to what was made in [15], where the cost functional to be minimized was not the drag but a drag-related cost function (the turbulent kinetic energy contained in the wake), we will directly take here for cost functional the mean drag coefficient (viscous and pressure contributions). Then, to reduce as much as possible the computational costs associated to the present study, the flow is considered two-dimensional and in the laminar regime. However, the methodology presented here that consists of combining the optimal control approach and a POD ROM should easily be expanded to three-dimensional and turbulent flows.

In their numerical investigation of the controlled wake flow by rotary oscillation of the cylinder, Protas and Wesfreid [30] argued that in the supercritical regime, the effectiveness of the control in terms of drag reduction increases with the Reynolds number. This important result was recently confirmed by a study of our group [45] which showed analytically that the power necessary to control the wake by unsteady rotation varied, for fixed values of the control parameters A and ${\mathit{St}}_{\text{f}}$, like the inverse of the square root of the Reynolds number. Therefore, since the wake flow remains two-dimensional up to a value of the Reynolds number approximately equal to 190 where a spanwise supercritical Hopf bifurcation occurs and where the three-dimensional effects appear [46], [47], the “optimal” value of the Reynolds number for our two-dimensional study is slightly lower than 200. However for facilitating the comparisons with the results of the literature, a Reynolds number of 200 is considered. According to the observations of [38], the control minimizing the drag generates vortices that are less energetic than those produced by the stationary cylinder. An energetic criterion seems to be well adapted to the investigation of drag reduction. Therefore, due to the energetic optimality of convergence of the POD basis [7], [8], [48], the choice of POD to develop a reduced-order model of the controlled unsteady flow seems to be well adapted. A similar approach was already considered in [16], [49] to control the wake flow at a supercritical Reynolds number of 100.

Finally, we need to choose between the two opposite strategies discussed at the end of Section 1.1. If we want to develop active flow control method that can be used for real-time, on-line feedback control, our interest is to include in the snapshot set all the information needed during the optimization process or at least as much information as we can, and then to generate the reduced-order basis. Following this approach the POD functions are determined once for all at the beginning of the optimization process and no refresh is realized. This method was successfully applied to control the cylinder wake flow in [15]. It was demonstrated that an accurate and robust POD ROM can be derived using a snapshot ensemble for POD-based on chirp-forced transients of the flow. Moreover, 25% of relative drag reduction was found when the Navier–Stokes equations were controlled using an harmonic control law deduced from the optimal solution determined with the POD ROM. However, the excitation used to determine the generalized POD functions lacks of justifications and, with this particular approach, there is no mathematical assurance that the optimal solution based on the POD ROM corresponds to a local optimizer for the high-fidelity model. The same remark can be made concerning the approaches presented in [17], [18] and [19]. Indeed, in these articles, a new POD ROM is determined when the control law does not evolve sufficiently with the previous model. With this strategy, there is not any proof that the control which is finally obtained is solution of the initial problem of optimization. Therefore, in this paper, we propose to use a specific adaptive method called trust-region proper orthogonal decomposition (TRPOD) to update the reduced-order models during the optimization process. This approach, originally introduced by [50], benefits from the trust-region philosophy [51], for an introduction. Then, 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, the so-called global convergence³ of the trust-region methods. Moreover, in [15], the POD basis used to derive the reduced-order models represented only velocities. Therefore, a drag-related cost functional characteristic of the wake unsteadiness was minimized. Since the pressure term contributes to approximately 80% of the total drag coefficient for a Reynolds number equal to 200, here, a pressure POD basis was determined (Section 4.1), allowing us to consider the total drag as objective functional in our optimal control approach (Section 4.3).

The main objective of this paper is to demonstrate in a simple flow control configuration that the use of the TRPOD algorithm can be successful to determine at least a local minimizer of the original problem. To supplement this main result, we will also give an estimate of the computational savings that can be obtained by a POD ROM based optimal control approach compared with the more “classical” approach where the Navier–Stokes equations are used for constraints [38], [39], [40]. Consequently in this study our main concern is not to determine the control law with the maximum energetic efficiency as it can be characterized for example by the power saving ratio (PSR) [40], for a definition or hereafter in Section 5.2.3. As far as we know (see Table 1), the work presented in [40] is the only one which considers for cost functional the sum of the drag power and the control power thus making it possible to determine an optimal solution that is by construction energetically efficient. In the other works, the cost of the control is not considered or at best as a regularization parameter. This discussion will be developed in Section 5.2.3 where we compare the energetic efficiency of the different approaches.

This manuscript is organized as follows. Section 2 begins with the introduction of the generic controlled flow configuration. In the next two subsections, a mathematical expression of the mean drag coefficient is first introduced (Section 2.2), then an open-loop control study of the cylinder wake is carried out (Section 2.3). The optimization by trust-region methods and POD reduced-order models is presented in Section 3 where the trust-region POD (TRPOD) algorithm is formally introduced. Following the philosophy of trust-region methods, a robust surrogate function for the mean drag coefficient is then constructed in Section 4. The key enablers are the extension of the POD basis to the pressure field (Section 4.1) and the introduction of non-equilibrium modes in the POD expansion to represent different operating conditions (Section 4.2). Finally, we formulate an optimal control problem for the POD ROM (Section 5.1) and present the numerical results of the mean drag minimization of the cylinder wake flow obtained by a suboptimal and an optimal (TRPOD) POD-based adaptive controllers (Section 5.2).