e^N method

The e^N method works by superimposing small disturbances on the flow, considering it to be laminar. The assumption is made that both the original and the newly disturbed flow satisfy the Navier-Stokes equations. This disturbed flow can be linearised and described with a perturbation equation. This equation may have unstable solutions. Any such case where a disturbance is caused where the perturbation equation has unstable solutions can be considered unstable, and hence could lead to a transition point. This method assumes a flow parallel to the boundary layer with a constant shape, which will not always be the case in analysis. The method can be used to determine the local (in)stability at the span-wise position, however. If a local transition occurs, it must also occur under the same circumstances on the global frame. This analysis can be repeated for multiple span-wise stations. As the transition point is determined by the first point where this happens, only the point closest to the leading edge where this happens is sought for.

A two-dimensional disturbance stream function can be defined as ${\displaystyle \Psi (x,y,t)=\phi (y)e^{-\alpha _{i}x}e^{i(\alpha _{r}x-\omega t)}}$, from which the disturbance velocity components in the x- and y directions follow from ${\displaystyle u'={\frac {\partial \Psi }{\partial y}};v'=-{\frac {\partial \Psi }{\partial x}}}$. Here the circular frequency ω is taken to be the real in the disturbance stream, and the wave number α complex. Hence, in the case of an instability, the complex part of the wave number needs to be positive for there to be a growing disturbance. Any prior disturbance passing through will be amplified by ${\displaystyle \sigma _{a}=\ln {\frac {a}{a_{0}}}=\int _{x0}^{x}-\alpha _{i}dx}$, where x0 is the value of x where the disturbance with frequency ω first becomes unstable (known as the Orr-Sommerfeld equation). Experiments by Smith and Gamberoni, and later by Van Ingen have shown that transition occurs when the amplification factor (being the critical amplification factor) equals 9. For clean wind tunnels and for atmospheric turbulence, the critical amplification factors equals 12 and 4 in that order.

Experiments have shown that the largest factors affecting the position where this happens are the shape of the velocity profile over the lift-generating surface, the Reynolds number, and the frequency or wavelength of the disturbances itself.^[1]

Behind the transition point in a boundary layer the mean speed and friction drag increases.