TURBULENT FRICTION-DRAG REDUCTION CONTROL-SCHEME REQUIRING ONLY STREAMWISE WALL SHEAR SIGNAL

Abstract

Proc. 3rd Int. Symp. on Turbulence and Shear Flow Phenomena Sendai, Japan, June 25-27, 2003 We propose a new algebraic control scheme for drag reduction in wall-turbulence, which requires the streamwise wall-shear signal only. By assuming continuously distributed sensors and actuators, the controller is designed to reduce the near-wall Reynolds shear stress that is directly responsible for the turbulent skin friction drag. Intuitive and suboptimal control schemes are considered. The derived control laws are assessed by means of direct numerical simulation of turbulent pipe flow at Reτ 180. A clear drag reduction symptom associated with a negative near-wall Reynolds stress is observed when the control scheme derived by the suboptimal control theory is applied. INTRODUCTION For successful development of an active feedback control system for drag reduction in wall-bounded turbulent flow, the effectiveness of the control algorithm used as well as the performance of the hardware components such as sensors and actuators is of great importance. Control schemes may be classified into two types, i.e., explicit and implicit schemes. The explicit scheme is one in which the control input of the actuator i, φi , is given explicitly, e.g., φi( x, t) = F[s j( x, t′ | t′ ≤ t)], where s j is the sensor information and F is a mapping function. On the other hand, the implicit scheme, such as the optimal control (e.g., Bewley et al., 2001) only describes a relation to be satisfied (i.e. the control input minimizing the cost functional) and requires iterative procedures to determine the control input. While such implicit schemes are useful to explore the possibility of drag reduction control, the explicit schemes are easier to be implemented in the real applications. In the last decade, various explicit control algorithms were developed and assessed by using direct numerical simulation (DNS) of controlled turbulent flow. Choi et al. (1994) proposed so-called the opposition control, in which blowing/suction velocity is given at the wall so as to oppose the velocity components at a virtual detection plane located abovethe wall. They attained about 25 % drag reduction in their DNS of turbulent channelflow at low Reynoldsnumbers. Subsequently, several attempts were made to develop control algorithms using the information measurable at the wall. Lee et al. (1997) used a neural network and obtained an algorithm in which the control input is given as a weighted sum of the spanwise wall-shear stresses, ∂w/∂y|w , measured around the actuator. Lee et al. (1998) derived series of analytical solutions of the control input to minimize the cost function in the framework of the suboptimal control. Their DNS of channel flow at Reτ 110 showed 16-22% drag reduction when ∂w/∂y|w (in this case, the control law is quite similar to that obtained by using the neural network mentioned above) or the wall pressure, pw, was used as the sensor signal. From a practical point of view, it is desirable to use the streamwise wall-shear stress, τw = ∂u/∂y|w, or pw (or both) as a sensor signal because a streamwise wall-shear stress sensor (Yoshino et al., 2003) and a wall pressure sensor (Lofdahl et al., 1996) of sufficiently small size and high frequency response are becoming available. For the use of pw, in addition to the work by Lee et al. (1998), Koumoutsakos(1999) presented an algorithm to suppress the vorticity flux, and succeeded to reduce the friction drag in his DNS. For the use of τw, however, development of effective algorithm seems more difficult. Lee et al. (1998) also presented a suboptimal solution aiming at reduction of τw. This algorithm uses τw as the sensor signal only, but the friction drag (i.e., τw) was not reduced by that algorithm. Very recently, Lee et al. (2001) applied a two-dimensional linear-quadratic-Gaussian (LQG) controller to a linearized NavierStokes equation. About 10 % drag reduction was attained in their DNS of a channel flow at Reτ 100. They also attained 17 % drag reduction by making an ad hoc extension. Morimoto et al. (2002) employed a weighted sum of τw as the control input and optimized the weights by using the genetic algorithm (GA). The excellent gene (i.e., the pattern of weights) led to 12 % drag reduction in a channel flow at Reτ 100. The previous suboptimal control using the streamwise wall-shear signal only targeted at direct suppression of the streamwise wallshear. Namely, the cost functional may be expressed as