Transition

The boundary layer changes character from smooth, steady motion (laminar flow) to turbulent motion with unsteadiness and eddies on a very small scale.

Several things may be responsible for the transition between these two states. These include: surface roughness, adverse pressure gradients, and wing sweep.

NASA TP 2256 contains flight data on the allowable surface roughness and waviness for laminar flow. NACA TN 4363, A Simplified Method for Determination of Critical Height of Distributed Roughness Particles for Boundary Layer Transition, contains another discussion. A rule of thumb from Hoerner is that a roughness element should be at least 25% of the boundary layer displacement thickness to cause transition, although this depends on the form of the roughness elements. Two-dimensional trips, such as tape strips must be much larger than 3-D trips. Wing sweep also encourages transition through the cross-flow instability. Some discussion appears in the above NASA TP, but methods for predicting this are, in general, very complex. People often run large computer codes that take quite some time to master and their results are often questioned.

The usual reason for transition on airfoils is the presence of an adverse pressure gradient. (Increasing Cp with x.) When this becomes severe enough, or long enough, transition is likely to occur.

As a rule of thumb, boundary layers at high Reynolds numbers can withstand very little adverse gradient and one may assume that once the gradient is adverse transition will occur.

On a more quantitative basis, several transition criteria have been proposed and are in wide use. These are empirically-based and work over a restricted Reynolds number range. Two of the most popular are Michel's and Granville's criteria. Michel's criteria states that transition will occur when the Reynolds number based on momentum thickness exceeds a certain value which depends on the local Reynolds number:

The relationship holds for Re_x values between 10⁵ and 40x10⁶.
Unfortunately, in practice, the left side increases at about the same rate as the right hand side, making the determination of the transition location inaccurate. But when an adverse gradient exists, the left side grows much more rapidly and transition is reasonably predicted. (Note that when gradients are not large transition can be expected at values of Re_x near 3 million. This is very much a ballpark estimate, of course.)

A somewhat more refined approach to transition prediction involves an analysis of the stability of the boundary layer equations to disturbances.

The basic ideas for this approach were described by Osborne Reynolds (of Reynolds number fame) and Lord Rayleigh (of Rayleigh number fame). Their hypothesis was that, above a critical Reynolds number, small disturbances to the laminar boundary layer, are not damped by viscosity but rather amplified until the laminar character of the flow disappears. One starts with the unsteady Navier-Stokes equations and assumes that the unsteady terms are very small and are written as a Fourier series with unknown coefficients. Upon substitution of these equations into the unsteady part of the NS equations, one obtains a 4th order, linear ODE, known as the Orr-Sommerfeld equation. If the mean velocity profile and Reynolds number are specified, this leads to an eigenvalue problem for each assumed value of a disturbance wavelength. In 1957, A.M.O. Smith did several experiments to correlate the location of transition with the value of the real part of this eigenvalue. He found that transition generally occurred when the disturbances had grown by about 8000 times (e⁹) their value at the point of neutral stability.

This sort of linear stability theory (eⁿ method) is now routinely used to estimate transition location with n between 9 and 11.