Laminar-turbulent transition

The plume from an ordinary candle transitions from laminar to turbulent flow in this Schlieren photograph.

The process of a laminar flow becoming turbulent is known as laminar-turbulent transition. This is an extraordinarily complicated process which at present is not fully understood. However, as the result of many decades of intensive research, certain features have become gradually clear, and it is known that the process proceeds through a series of stages. "Transitional flow" can refer to transition in either direction, that is laminar-turbulent transitional or turbulent-laminar transitional flow.

While the process applies to any fluid flow, it is most often used in the context of boundary layers due to their ubiquity in real fluid flows and their importance in many fluid-dynamic processes.

History

Reynolds’ experiment on fluid dynamics in pipes

Reynolds’ observations of the nature of the flow in his experiments

Osborne Reynolds demonstrated the transition to turbulent flow in a classic experiment in which he examined an outlet from a large water tank through a small tube. At the end of the tank there was a stopcock used to vary the water speed inside the tube. The junction of the tube with the tank was nicely rounded. A filament of colored fluid was introduced at the mouth. When the water was slow, the filament remained distinct through the entire length of the tube. When the speed was increased, the filament broke up at a given point and diffused throughout the cross-section. Reynolds identified the governing parameter, the dimensionless Reynolds number. The point at which the color diffuses throughout the tube is the transition point from laminar to turbulent.

Reynolds found that the transition occurred between Re = 2000 and 13000, depending on the smoothness of the entry conditions. When extreme care is taken, the transition can even happen with Re as high as 40000. On the other hand, Re = 2000 appears to be about the lowest value obtained at a rough entrance.^[1]

Reynolds' publications in fluid dynamics began in the early 1870s. His final theoretical model published in the mid-1890s is still the standard mathematical framework used today. Examples of titles from his more groundbreaking reports:

Improvements in Apparatus for Obtaining Motive Power from Fluids and also for Raising or Forcing Fluids. (1875)

An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels. (1883)

On the dynamical theory of incompressible viscous fluids and the determination of the criterion. (1895)

Transition stages in a boundary layer

The path from receptivity to laminar-turbulent transition as illustrated by Morkovin, 1994.^[2]

A boundary layer can transition to turbulence through a number of paths. Which path is realized physically depends on the initial conditions such as initial disturbance amplitude and surface roughness. The level of understanding of each phase varies greatly, from near complete understanding of primary mode growth to a near-complete lack of understanding of bypass mechanisms.

Receptivity

The initial stage of the natural transition process is known as the Receptivity phase and consists of the transformation of environmental disturbances – both acoustic (sound) and vortical (turbulence) – into small perturbations within the boundary layer. The mechanisms by which these disturbances arise are varied and include freestream sound and/or turbulence interacting with surface curvature, shape discontinuities and surface roughness. These initial conditions are small, often unmeasurable perturbations to the basic state flow. From here, the growth (or decay) of these disturbances depends on the nature of the disturbance and the nature of the basic state. Acoustic disturbances tend to excite two-dimensional instabilities such as Tollmien–Schlichting waves (T-S waves), while vortical disturbances tend to lead to the growth of three-dimensional phenomena such as the crossflow instability.^[3]

Numerous experiments in recent decades have revealed that the extent of the amplification region, and hence the location of the transition point on the body surface, is strongly dependent not only upon the amplitude and/or the spectrum of external disturbances but also on their physical nature. Some of the disturbances easily penetrate into the boundary layer whilst others do not. Consequently, the concept of boundary layer transition is a complex one and still lacks a complete theoretical exposition.

Primary mode growth

If the initial, environmentally-generated disturbance is small enough, the next stage of the transition process is that of primary mode growth. In this stage, the initial disturbances grow (or decay) in a manner described by linear stability theory.^[4] The specific instabilities that are exhibited in reality depend on the geometry of the problem and the nature and amplitude of initial disturbances. Across a range of Reynolds numbers in a given flow configuration, the most amplified modes can and often do vary.

There are several major types of instability which commonly occur in boundary layers. In subsonic and early supersonic flows, the dominant two-dimensional instabilities are T-S waves. For flows in which a three-dimensional boundary layer develops such as a swept wing, the crossflow instability becomes important. For flows navigating concave surface curvature, Görtler vortices may become the dominant instability. Each instability has its own physical origins and its own set of control strategies - some of which are contraindicated by other instabilities – adding to the difficulty in controlling laminar-turbulent transition.

Secondary instabilities

The primary modes themselves don't actually lead directly to breakdown, but instead lead to the formation of secondary instability mechanisms. As the primary modes grow and distort the mean flow, they begin to exhibit nonlinearities and linear theory no longer applies. Complicating the matter is the growing distortion of the mean flow, which can lead to inflection points in the velocity profile a situation shown by Lord Rayleigh to indicate absolute instability in a boundary layer. These secondary instabilities lead rapidly to breakdown. These secondary instabilities are often much higher in frequency than their linear precursors

See also

References

Wikimedia Commons has media related to Laminar to turbulent flow transition.