Notes on Computational Fluid Dynamics: General Principles

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6.6 Scales of turbulence

In Sec. 6.2 , turbulence was pictured as a mass of intertwined eddies which are rotating and distorting. The rotation, i.e. vorticity, mostly originates at solid boundaries and propagates locally (Sec. 6.3 ) through boundary layers (Sec. 6.4 ), which can separate and shed into the ﬂow ﬁeld (Sec. 6.5 ).

Flow disturbances trigger instabilities which cause vortices to stretch, compress and/or ‘break up’. At a suﬃciently high $\text{[math]}$ , coherent ﬂow structures rapidly disintegrate into a mass of turbulent eddies. This process is described in Richardson’s poem:⁸

Big whorls have little whorls that feed oﬀ their velocity, and little whorls have lesser whorls and so on to viscosity — in the molecular sense.

In other words, large eddies created by instabilities become progressively smaller until they reach a size where the dissipation of their kinetic energy due to (molecular) viscosity is signiﬁcant. The loss of kinetic energy causes these eddies to “die out” before they can become any smaller.

Kolmogorov microscales

The Kolmogorov microscales⁹ describe the smallest scales that can exist in turbulent ﬂow. They can be derived from heuristic arguments as follows. Energy dissipates as heat at a rate of $\text{[math]}$ per unit mass, obtained from Eq. (2.58 ) and switching from ‘per unit volume’ by replacing $\text{[math]}$ by $\text{[math]}$ .

Using scale similarity arguments (Sec. 2.21 ) the average rate of dissipation of energy $\text{[math]}$ can then be estimated for the smallest scales, characterised by length $\text{[math]}$ and speed $\text{[math]}$ , to be

(6.7)

where “ $\text{[math]}$ ” means ”of the order of magnitude”.

Dissipation occurs when the Reynolds number, based on the Kolmogorov scales, is of the order unity, i.e. $\text{[math]}$ . Combined with Eq. (6.7 ) this yields the Kolmogorov microscales below.

$pict\relax \special {t4ht=$

At the largest scales, eddies generated from the mean ﬂow can be characterised by a length $\text{[math]}$ and ﬂow speed $\text{[math]}$ (of the scale of turbulent ﬂuctuations). The kinetic energy per unit mass of these eddies $\text{[math]}$ .

Experiments show the lifetime of an eddy is of the order of the time for one revolution, the turn-over time $\text{[math]}$ . Therefore the rate of transfer of kinetic energy from larger to smaller eddies is $\text{[math]}$ .

This rate of transfer of energy between scales must match the rate of dissipation, Eq. (6.7). Equating the two energy rates yields the correspondence between the Kolmogorov scales and the largest scales in terms of a large scale $\text{[math]}$ :

3=4 1=4 1=2 lRe ; v u Re ; =v (l=u)Re : \relax \special {t4ht=

(6.9)

⁸Lewis Fry Richardson, Weather prediction by numerical process, 1922.

⁹Andrey Nikolaevich Kolmogorov, The local structure of turbulence in incompressible viscous ﬂuid for very large Reynolds numbers, ﬁrst published in Russian in Dokl. Akad. Nauk SSSR 30, 1941.

Notes on CFD: General Principles - 6.6 Scales of turbulence