Thomas R. Bewley

Flow control: new challenges for a new Renaissance

Abstract As traditional scientific disciplines individually grow towards their maturity, many new opportunities for significant advances lie at their intersection. For example, remarkable developments in control theory in the last few decades have considerably expanded the selection of available tools which may be applied to regulate physical and electrical systems. These techniques hold great promise for several applications in fluid mechanics, including the delay of transition and the regulation of turbulence. Such applications of control theory require a very balanced perspective, in which one considers the relevant flow physics when designing the control algorithms and, conversely, takes into account the requirements and limitations of control algorithms when designing both reduced-order flow models and the fluid–mechanical systems to be controlled themselves. Such a balanced perspective is elusive, however, as both the research establishment in general and universities in particular are accustomed only to the dissemination and teaching of component technologies in isolated fields. To advance, we must not toss substantial new interdisciplinary questions over the fence for fear of them being “outside our area”; rather, we must break down these very fences that limit us, and attack these challenging new questions with a Renaissance approach. In this spirit, this paper surveys a few recent attempts at bridging the gaps between the several scientific disciplines comprising the field of flow control, in an attempt to clarify the authors perspective on how recent advances in these constituent disciplines fit together in a manner that opens up significant new research opportunities.

DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms

Direct numerical simulations (DNS) and optimal control theory are used in a predictive control setting to determine controls that effectively reduce the turbulent kinetic energy and drag of a turbulent flow in a plane channel at Re τ = 100 and Re τ = 180. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control. The algorithm used for the control optimization is based solely on the control objective and the nonlinear partial differential equation governing the flow, with no ad hoc assumptions other than the finite prediction horizon, T , over which the control is optimized. Flow relaminarization, accompanied by a drag reduction of over 50%, is obtained in some of the control cases with the predictive control approach in direct numerical simulations of subcritical turbulent channel flows. Such performance far exceeds what has been obtained to date in similar flows (using this type of actuation) via adaptive strategies such as neural networks, intuition-based strategies such as opposition control, and the so-called ‘suboptimal’ strategies, which involve optimizations over a vanishingly small prediction horizon T + → 0. To achieve flow relaminarization in the predictive control approach, it is shown that it is necessary to optimize the controls over a sufficiently long prediction horizon T + [gsim ] 25. Implications of this result are discussed. The predictive control algorithm requires full flow field information and is computationally expensive, involving iterative direct numerical simulations. It is, therefore, impossible to implement this algorithm directly in a practical setting. However, these calculations allow us to quantify the best possible system performance given a certain class of flow actuation and to qualify how optimized controls correlate with the near-wall coherent structures believed to dominate the process of turbulence production in wall-bounded flows. Further, various approaches have been proposed to distil practical feedback schemes from the predictive control approach without the suboptimal approximation, which is shown in the present work to restrict severely the effectiveness of the resulting control algorithm. The present work thus represents a further step towards the determination of optimally effective yet implementable control strategies for the mitigation or enhancement of the consequential effects of turbulence.

Optimal and robust control and estimation of linear paths to transition

Optimal and robust control theories are used to determine effective, estimator-based feedback control rules for laminar plane channel flows that effectively stabilize linearly unstable flow perturbations at Re = 10000 and linearly stable flow perturbations, characterized by mechanisms for very large disturbance amplification, at Re = 5000. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control, and the flow measurement is derived from the wall skin friction. The control objective, beyond simply stabilizing any unstable eigenvalues (which is relatively easy to accomplish), is to minimize the energy of the flow perturbations created by external disturbance forcing. This is important because, when mechanisms for large disturbance amplification are present, small-amplitude external disturbance forcing may excite flow perturbations with sufficiently large amplitude to induce nonlinear flow instability. The control algorithms used in the present work account for system disturbances and measurement noise in a rigorous fashion by application of modern linear control techniques to the discretized linear stability problem

Feedback Control of Turbulence

A brief review of current approaches to active feedback control of the fluctuations arising in turbulent flows is presented, emphasizing the mathematical techniques involved. Active feedback control schemes are categorized and compared by examining the extent to which they are based on the governing flow equations. These schemes are broken down into the following categories: adaptive schemes, schemes based on heuristic physical arguments, schemes based on a dynamical systems approach, and schemes based on optimal control theory applied directly to the NavierStokes equations. Recent advances in methods of implementing small scale flow control ideas are also reviewed.

A general framework for robust control in fluid mechanics

Abstract The application of optimal control theory to complex problems in fluid mechanics has proven to be quite effective when complete state information from high-resolution numerical simulations is available [P. Moin, T.R. Bewley, Appl. Mech. Rev., Part 2 47 (6) (1994) S3–S13; T.R. Bewley, P. Moin, R. Temam, J. Fluid Mech. (1999), submitted for publication]. In this approach, an iterative optimization algorithm based on the repeated computation of an adjoint field is used to optimize the controls for finite-horizon nonlinear flow problems [F. Abergel, R. Temam, Theoret. Comput. Fluid Dyn. 1 (1990) 303–325]. In order to extend this infinite-dimensional optimization approach to control externally disturbed flows in which the controls must be determined based on limited noisy flow measurements alone, it is necessary that the controls computed be insensitive to both state disturbances and measurement noise. For this reason, robust control theory, a generalization of optimal control theory, has been examined as a technique by which effective control algorithms which are insensitive to a broad class of external disturbances may be developed for a wide variety of infinite-dimensional linear and nonlinear problems in fluid mechanics. An aim of the present paper is to put such algorithms into a rigorous mathematical framework, for it cannot be assumed at the outset that a solution to the infinite-dimensional robust control problem even exists. In this paper, conditions on the initial data, the parameters in the cost functional, and the regularity of the problem are established such that existence and uniqueness of the solution to the robust control problem can be proven. Both linear and nonlinear problems are treated, and the 2D and 3D nonlinear cases are treated separately in order to get the best possible estimates. Several generalizations are discussed and an appropriate numerical method is proposed.

Observed mechanisms for turbulence attenuation and enhancement in opposition-controlled wall-bounded flows

Opposition control is a simple method used to attenuate near-wall turbulence and reduce drag in wall-bounded turbulent flows [H. Choi, P. Moin, and J. Kim, J. Fluid Mech. 262, 75 (1994)]. This method employs blowing and suction at the wall in opposition to the wall-normal fluid velocity a small distance from the wall. Results from direct numerical simulations of turbulent channel flow indicate that, when the control at the wall is based on detection of the wall-normal velocity in a plane sufficiently close to the wall, the opposition control strategy establishes a “virtual wall,” i.e., a plane that has approximately no through flow, halfway between the detection plane and the wall. As a consequence, drag is reduced about 25%. When the detection plane is at a greater distance from the wall, a virtual wall is not established, and the blowing and suction increase the drag significantly.

Relaminarization of Reτ=100 turbulence using gain scheduling and linear state-feedback control

The first successful application of linear full-state feedback optimal control theory to consistently relaminarize turbulent channel flow at Re τ =100 with full state information and gain scheduling is reported. The actuation is zero-net mass-flux blowing and suction on the channel walls. Two key issues central to the success of this strategy are: (a) the choice of the mean-flow profile about which the equations are linearized for the computation of the linear feedback gains, and (b) the choice of an objective function which targets the control effort on the flow perturbations of interest. A range of mean-flow profiles between the laminar and fully turbulent profiles and a weighted energy measure which targets flow perturbations in the near-wall region were found to provide effective feedback gains. A gain-scheduling strategy to tune the feedback gains to the nonstationary mean-flow profile is introduced, resulting in consistent relaminarization of the turbulent flow in all realizations tested.

State estimation in wall-bounded flow systems. Part 1. Perturbed laminar flows

In applications involving the model-based control of transitional wall-bounded flow systems, it is often desired to estimate the interior flow state based on a history of noisy measurements from ...

A fundamental limit on the balance of power in a transpiration-controlled channel flow

This paper is a direct sequel to Bewley & Aamo ( J. Fluid Mech ., vol. 499, 2004, pp. 183–196). It was conjectured in that paper, based on the numerical evidence available at that time, that the minimum drag of a constant mass flux channel flow might in fact be that of the laminar flow. This conjecture turned out to be false; Min et al . ( J. Fluid Mech ., vol. 558, 2006, 309318) discovered a curious control strategy which in fact reduces the time-averaged drag to sub-laminar levels. The present paper establishes rigorously that the power of the control input applied at the walls is always larger than the power saved (due to drag reduction below the laminar level) for any possible control distribution, including that proposed by Min et al . (2006), thus establishing that, energetically (that is accounting for the power saved due to drag reduction and the power exerted by application of the control), the optimal control solution is necessarily to relaminarize the flow.

Performance of a linear robust control strategy on a nonlinear model of spatially developing flows

This paper investigates the control of self-excited oscillations in spatially developing flow systems such as jets and wakes using