# Near-Field Aeroacoustics of Spanwise Forcing on a Transonic Wing: A DNS Study
## 1 Introduction  
Flow control is the branch of fluid mechanics where techniques to manipulate a flow in view of a technological objective are studied. Among the most pursued goals of flow control, aerodynamic drag reduction and noise reduction stand out as crucial technological challenges. The former is particularly relevant in the turbulent regime, where the comparatively large wall shear stresses lead to a large consumption of energy required to drive the flow, for instance, as energy required to pump a fluid through a duct or to propel a body through a still fluid. A large number of drag-reducing techniques have been developed over the last decades; they are typically classified as 'passive' or 'active', depending on whether or not energy is introduced in the flow [1]. Losses due to skin-friction are particularly difficult to abate, as they are intrinsic of near-wall turbulence and descend directly from the interaction between the structure of turbulence and the solid wall. A classic example of a passive skin-friction drag reduction technique is represented by riblets, which are probably the most promising passive approach to date. Riblets are tiny streamwise corrugations of the wall surface which yield skin-friction reduction up to 9% in a turbulent channel flow at low speed [2]. This figure is bound to reduce significantly down to 1-2% of the aerodynamicdrag once a typical aircraft in cruise flight is considered [3]. Hence, to achieve larger drag reduction, active approaches are considered. Among them, a class known as'spanwise forcing' stands out for its robustness, performance, and simplicity. A recent review of spanwise forcing for drag reduction can be found in Ref. [4]. In this type of forcing, the boundary condition of the spanwise component of the flow velocity at the wall is imposed as an unsteady, spatially periodic distribution. The most efficient technique of this class is the use of streamwise traveling waves of spanwise velocity (StTw), introduced by [PERSON] et al. [5]. Within this framework, the spanwise velocity component of the velocity at the wall varies in time and space according to  
\[w_{w}(x,t)=A\sin(x_{x}x-\omega t) \tag{1}\]  
where \(A\) is the amplitude of the forcing, \(x_{x}\) and \(\omega\) are the spatial and temporal frequencies of the waves, and \(x\) and \(t\) denote the streamwise and temporal coordinates, respectively. StTw modify the near-wall velocity profile and are capable of reducing skin-friction drag with rates up to 48% for intermediate values of \(A\); furthermore, this kind of control yields the largest net energy savings among the active techniques [5]. Experimental realizations of such forcing were provided in a pipe flow [6]; the actuator was later adapted to planar geometry [7; 8]. [PERSON] and [PERSON] [9] characterized the spatially discrete waveform that is typical of these experiments. The effects of the Reynolds number (\(Re\)) on skin-friction reduction induced by StTw was assessed in the thorough study by [PERSON] et al. [10]. The compressibility effects in a plane channel flow were assessed by [PERSON] et al. [11], who showed that no beneficial or detrimental effects derive from compressibility, once the comparison with the unmanipulated flow is carried out properly. The performance of StTw in non-planar geometries was analyzed by [PERSON] et al. [12]. The latter study carried out a preliminary analysis on the effects of StTw applied on a transonic wing slab: they employed direct numerical simulations (DNSs) to simulate the turbulent flow around the wing at a cruise Mach number and at a relatively small value of the Reynolds number. Their analysis was recently extended by [PERSON] et al. [13], who carried out a parametric study on the same setup by varying the control input of the forcing and the angle of attack of the airflow. It was shown that the drag of the airfoil alone can be reduced up to 4%; extrapolating the savings to a full aircraft, a potential reduction of 12% in the total drag force of the aircraft at a negligible power cost was estimated.