Engineering approach to prediction of the laminar-turbulent transition in the boundary layer (e^N method)

The most popular and effective engineering methods of transition predictions are based on the idea of the e^N method.

Since the amplification is of the exponential form, the corresponding amplification factor with distribution of x will look like as given on the Figure. Note that starting from the maximum point, this Tollmien-Schlichting wave is not amplified anymore and it starts to be diminishing. But this is just one frequency. If I choose any other frequency, I can have such lines that’ll rise later. We have infinite number of Tollmien-Schlichting waves with different frequencies and we have individual curve of each of them.

The idea of e^N method for transition prediction was published by Van Ingen in 1956.

The above formula describes the cumulative amplitude growth that is determined for a bunch of modes with different frequencies. One can obtain as a result a family of lines, showing how the amplification factor of each mode grows downstream the boundary layer (BL). In principle, if I have a boundary layer, then in BL we have an excitation of all possible frequencies. Then, one can define the envelope for this family, which is a line effectively parametrized by the frequency. (see Figure) 

We assume that the actual amplification of disturbances in the BL will be inside this envelope. So if I choose a line, which sort of has all this infinitely many lines underneath, then I will be on the same side. So if I can calculate this envelope line, which goes on the top of all individual lines, I’ll have sort of a border of a maximum possible local amplification that may in principle happen in any cross section on the TBL. The hypothesis behind it also says that the actual transition point will be when this envelope will reach a certain value of this logarithm. For example, when the flow is in a very calm atmosphere, where the lever of turbulence outside the BL is very-very small, then the BL will survive without being stabilized such amplification and a local amplification up to say 9 orders of magnitude of initial disturbance. So this logarithm is 9.

But if the external disturbance field will be much higher, then that means that from the very beginning the disturbances are large, so it’s enough e.g. to amplify them by 4^th order of magnitude, or maybe 5^th order of magnitude, before they actually cause a transition to the TBL.

Physically the idea is primitive but surprisingly it works very well. Over the years, people gained a huge experience in manipulating this amplification factor and e.g. those Aerodynamists who use for aerodynamic design software like Xfoil, they know very well which amplification factor with this e^N criterion to choose. Dependently on the different conditions for which this or another airfoil is actually designed. If it’s designed for a glider, which supposed to fly in a very calm atmosphere, that’s the situation it supposed to get its best of its performance, then the one should choose this very-very small. The level of this is not about the level of external atmosphere but it’s also about the quality of the surface of the wing. If it’s very well polished, like a “glass”, which is most typical for modern gliders then this number can reach ever 11, for the measure of transition. But if we have something like a small wind turbine, small like, someone puts it on a farm, there is send around, there are insects on it, probably bird was hit by the blade, there is feather here and there on it, so the quality is very-very bad, it could be even lowered to around 4, especially if this contamination of the surface is close to the leading edge.

There is another important issue. If you have an airfoil and you’re worrying about making it dirty, then if this dirtiness is in front, near the leading edge, it’s really dangerous. If it’s far away, near the trailing edge, it’s not dangerous. This is pretty obvious. Initially the thickness of the BL is very-very small, meaning that even a small obstacle from the point of view of the laminar BL is huge. The same obstacle, when flown to the airfoil somewhere near the trailing edge, for the BL which is now 10 times thicker, is simply less sensitive.

Reference:

• Aerodynamics lecture, by Jacek Szumbarski, Warsaw University of Technology