## 1 Introduction  
Efficient energy use is a key aspect in our efforts to reduce the greenhouse gas emissions and increase the powertrain efficiencies around the world. A substantial portion of the global energy consumption is dedicated to transportation. Here, aerodynamic drag plays a critical role, significantly impacting the energy consumption in this sector, which is still largely dominated by fossil fuels. Reducing drag forces involves overcoming two main types of drag: pressure drag and skin friction drag. For streamlined objects, skin friction forces typically outweigh pressure forces, making the reduction in skin friction drag the most appealing. However, skin friction, inherently tied to the boundary layer flow of a fluid over a surface, is particularly challenging to reduce. A closer examination reveals that laminar boundary layers have significantly lower skin friction than turbulent boundary layers. As the Reynolds number (Re) increases, there is a rapid rise in the skin friction coefficient (\(C_{f}\)), further complicating the effort to minimize drag. Delaying the transition from laminar to turbulent flow results in a net lower skin friction drag, but the mechanisms governing this transition are complex and depend on the type of flow and ambient conditions, as is often the case in fluid control problems [1]. These transition mechanisms, first described by [PERSON] at the end of the 19 th century, have since been thoroughly studied. In the case of boundary layer flow, two main classes of transition are described: one describing a natural transition and the other one describing a bypass approach [2; 3]. When there is no modal growth observed during the transition, it can be categorized as a \"bypass\" transition. This type is most often due to the growth of an initial perturbation, e.g., free stream disturbances or surface roughness. Its nonmodal method bypasses the modal growth, as observed in the former category, the natural transition. This transition is found when there are no or only very small environmental disturbances. This modal approach is distinguished by the exponential amplification (or decay) of the disturbances within a certain range of forcing frequencies and Re numbers. The initial unstable eigenmode is characterized by exponentially growing traveling waves referred to as Tollmien-Schlichting (TS) waves [4; 5]. This research primarily concentrates on natural transition as this boundary layer instability transition type is understood much better and is much more suitable for flow control. It is also more fundamentally and practically significant for addressing the challenges related to real-life applications, such as moving vehicles in cruise conditions. A significant body of research is devoted to active methods of attenuating these TS waves, such as suction systems, wall motion, or surface heating [6]. However, a passive method proves to be more attractive as this mitigates the need for additional energy-consuming parts such as sensors or actuators, making it much easier to practically implement. In this field, pioneering work by [PERSON] et al. [7] showed that creating streamwise elongated streaks can effectively dampen the growth of these TS waves in such a way that postponing the transition is possible. This streak generation, however, is an intricate mechanism that, if not carefully controlled, could create the opposite effect, i.e., advancing the transition location. The fundamental process behind transition delay is the \"lift-up\" mechanism, as described in [PERSON] [8]. Streaks are created by a relatively weak pair of streamwise vortices that are counter-rotating. The resulting low- and high-speed spanwise velocity variations are created by the alternating effect of lifting up the low-speed fluid from the near-wall boundary region and the down-pulling of the higher-speed velocity region of the upper boundary layer. This interchange in momentum leads to the formation of streaks.