Energy efficiency and performance limitations of linear adaptive control for transition delay

Abstract

A reactive control technique with localised actuators and sensors is used to delay the transition to turbulence in a flat-plate boundary-layer flow. Through extensive direct numerical simulations, it is shown that an adaptive technique, which computes the control law on-line, is able to significantly reduce skin-friction drag in the presence of random three-dimensional perturbation fields with linear and weakly nonlinear behaviour. An energy budget analysis is performed in order to assess the net energy saving capabilities of the linear control approach. When considering a model of the dielectric-barrier-discharge (DBD) plasma actuator, the energy spent to create appropriate actuation force inside the boundary layer is of the same order as the energy gained from reducing skin-friction drag. With a model of an ideal actuator a net energy gain of three orders of magnitude can be achieved by efficiently damping small-amplitude disturbances upstream. The energy analysis in this study thus provides an upper limit for what we can expect in terms of drag-reduction efficiency for linear control of transition as a means for drag reduction.

On the wave-cancelling nature of boundary layer flow control

“…The control action is analyzed via direct numerical simulations (DNS) with the pseudo-spectral code SIMSON [12], which gives a high numerical accuracy. The same code has been used in several works by this group, both for simulations of laminar/transitional boundary layers [16] and turbulent flows [15].…”

“…For applications in boundary layer transition control, the reader is referred to the works of Fabbiane et al [16,18], and the work of Bagheri et al [2] which presents a nice introduction to optimal control for applications in fluid mechanics. The success of LQG controllers is related to their optimality, the design being made in two steps, from the solution of two Riccati equations for the estimation and control problems, which guarantee stability, as long as the system is observable and controllable.…”

“…Extension to the 3D problem is made via a multiple-input-multiple-output (MIMO) strategy [16]. Given that the control law is linear and there is an equal number of sensors and actuators, (M + 1), the total number of transfer functions to be considered, in the 3D case, would be (M + 1) 2 .…”

“…The control action is then decided by means of measuring the input and acting to minimize a given quantity at the objective position. This can be accomplished using static compensators, such as the linear-quadratic-Gaussian (LQG) regulator [16,25]. Differently from adaptive schemes, which have been more recently applied due to their increased robustness to plant uncertainties [17,45], LQG is computed off-line, using the reduced-order model, and remains unchanged during its operation on the nonlinear system.…”

“…With the currently available actuators, the efficiency of closed-loop control is close to the break-even point [16]. Furthermore, the design of the controller based on the ERA model, however efficient, does not provide an interpretation of the states of the system, and therefore, the assessment of the position of sensors and actuators cannot be done without performing simulations using the full system, which, in general, could be computationally demanding.…”