*2.1.1. Natural transition*

Given there was an inflection point in the velocity profile, it was proposed that all boundary layer profiles were unstable according to inviscid stability theory. Prandtl [9] later explained a physical prediction of transition and then Tollmien [10] proved it mathematically that viscous instability waves (often identified as Tollmien-Schlichting (hereinafter TS) waves) could cause a laminar boundary layer to destabilize. But these findings were not accepted by aerodynamic researchers until the study performed by Schubauer and Skramstad [11], since the early experimental investigation for transition prediction had large free-stream turbulence level.

Now, the free-stream turbulence level is accepted as low when it is less than 1% (<1%) [12]. In that flow case, a laminar boundary layer becomes linearly unstable when the critical Reynolds number is increased, showing TS waves that have started to grow.

The sketch of the natural transition process is indicated in **Figure 1**. The flow in the area indicated by number 1 is laminar. In the area denoted by number 2, TS waves start to grow. After that point, the transition to turbulence may not be concluded every time due to the slow growth of TS waves. In aerodynamic literature, the regular transition starts to occur after the nonlinear waves have taken place. It can be said that the transition to turbulence is inevitable

**Figure 1.** The sketch after Kurelek [13], Bertolotti [14] and Schlichting and Gersten [15] with regard to top view of simplified flat plate boundary layer transition. (1) Laminar flow, (2) TS waves, (3) three-dimensional waves and Λ-structure formation, (4) vortex breakdown, (5) turbulent spots formation, (6) turbulent flow.

after Λ-structures and three-dimensional disturbance defined as nonlinear waves in the area of number 3 and 4. Once those structures form, spots spread in all directions, resulting in the existing turbulent boundary layer as shown in numbers of 5 and 6, respectively.
