Juggler sequence

In fluid dynamics, a Tollmien–Schlichting wave (often abbreviated T-S wave) is a streamwise instability which arises in a viscous boundary layer. It is one of the more common methods by which a laminar boundary layer transitions to turbulence. The waves are initiated when some disturbance (sound, for example) interacts with leading edge roughness in a process known as receptivity. These waves are slowly amplified as they move downstream until they may eventually grow large enough that nonlinearities take over and the flow transitions to turbulence.

These waves, originally discovered by Ludwig Prandtl, were further studied by two of his former students, Walter Tollmien and Hermann Schlichting for whom the phenomenon is named.

Physical Mechanism

In order for a boundary layer to be absolutely unstable (have an inviscid instability), it must satisfy Rayleigh's criterion; namely $D^{2}U=0$ Where $D$ represents the y-derivative and $U$ is the free stream velocity profile. In other words, the velocity profile must have an inflection point to be unstable.

It is clear that in a typical boundary layer with a zero pressure gradient, the flow will be unconditionally stable; however, we know from experience this is not the case and the flow does transition. It is clear, then, that viscosity must be an important factor in the instability. It can be shown using energy methods that

$\frac{DE}{Dt}=-\underset{V}{\int }{u}^{\prime }{v}^{\prime }\left(\frac{dU}{dy}\right)-\frac{1}{R}\underset{V}{\int }{\left(\mathrm{\nabla }{\overrightarrow{v}}^{\prime }\right)}^2$

The rightmost term is a viscous dissipation term and is stabilizing. The left term, however, is the Reynolds stress term and is the primary production method for instability growth. In an inviscid flow, the $u^{\prime }$ and $v^{\prime }$ terms are orthogonal, so the term is zero, as one would expect. However, with the addition of viscosity, the two components are no longer orthogonal and the term becomes nonzero. In this regard, viscosity is destabilizing and is the reason for the formation of T-S waves.

Transition Phenomena

Initial Disturbance

In a laminar boundary layer, if the initial disturbance spectrum is nearly infinitesimal and random (with no discrete frequency peaks), the initial instability will occur as two-dimensional Tollmien–Schlichting waves, travelling in the mean flow direction if compressibility is not important. However, three-dimensionality soon appears as the Tollmien–Schlichting waves rather quickly begin to show variations. There are known to be many paths from Tollmien–Schlichting waves to turbulence, and many of them are explained by the non-linear theories of flow instability.

Final Transition

A shear layer develops viscous instability and forms Tollmien–Schlichting waves which grow, while still laminar, into finite amplitude (1 to 2 percent of the freestream velocity) three-dimensional fluctuations in velocity and pressure to develop three-dimensional unstable waves and hairpin eddies. From then on, the process is more a breakdown than a growth. The longitudinally stretched vortices begin a cascading breakdown into smaller units, until the relevant frequencies and wave numbers are approaching randomness. Then in this diffusively fluctuating state, intense local changes occur at random times and locations in the shear layer near the wall. At the locally intense fluctuations, turbulent 'spots' are formed that burst forth in the form of growing and spreading spots — result of which is a fully turbulent state downstream.