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Read about Kolmogorov’s theory of turbulence of flows




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Richardson's notion of turbulence was that a turbulent flow is composed by "eddies" of different sizes. The sizes define a characteristic length scale for the eddies, which are also characterized by flow velocity scales and time scales (turnover time) dependent on the length scale. The large eddies are unstable and eventually break up originating smaller eddies, and the kinetic energy of the initial large eddy is divided into the smaller eddies that stemmed from it. These smaller eddies undergo the same process, giving rise to even smaller eddies which inherit the energy of their predecessor eddy, and so on. In this way, the energy is passed down from the large scales of the motion to smaller scales until reaching a sufficiently small length scale such that the viscosity of the fluid can effectively dissipate the kinetic energy into internal energy.

In his original theory of 1941, Kolmogorov postulated that for very high Reynolds numbers, the small scale turbulent motions are statistically isotropic (i.e. no preferential spatial direction could be discerned). In general, the large scales of a flow are not isotropic, since they are determined by the particular geometrical features of the boundaries (the size characterizing the large scales will be denoted as L). Kolmogorov's idea was that in the Richardson's energy cascade this geometrical and directional information is lost, while the scale is reduced, so that the statistics of the small scales has a universal character: they are the same for all turbulent flows when the Reynolds number is sufficiently high.

Thus, Kolmogorov introduced a second hypothesis: for very high Reynolds numbers the statistics of small scales are universally and uniquely determined by the viscosity ( ) and the rate of energy dissipation ( ). With only these two parameters, the unique length that can be formed by dimensional analysis is

.

This is today known as the Kolmogorov length scale (see Kolmogorov microscales).

A turbulent flow is characterized by a hierarchy of scales through which the energy cascade takes place. Dissipation of kinetic energy takes place at scales of the order of Kolmogorov length , while the input of energy into the cascade comes from the decay of the large scales, of order L. These two scales at the extremes of the cascade can differ by several orders of magnitude at high Reynolds numbers. In between there is a range of scales (each one with its own characteristic length r) that has formed at the expense of the energy of the large ones. These scales are very large compared with the Kolmogorov length, but still very small compared with the large scale of the flow (i.e. ). Since eddies in this range are much larger than the dissipative eddies that exist at Kolmogorov scales, kinetic energy is essentially not dissipated in this range, and it is merely transferred to smaller scales until viscous effects become important as the order of the Kolmogorov scale is approached. Within this range inertial effects are still much larger than viscous effects, and it is possible to assume that viscosity does not play a role in their internal dynamics (for this reason this range is called "inertial range").



Hence, a third hypothesis of Kolmogorov was that at very high Reynolds number the statistics of scales in the range are universally and uniquely determined by the scale r and the rate of energy dissipation .

The way in which the kinetic energy is distributed over the multiplicity of scales is a fundamental characterization of a turbulent flow. For homogeneous turbulence (i.e., statistically invariant under translations of the reference frame) this is usually done by means of the energy spectrum function , where k is the modulus of the wavevector corresponding to some harmonics in a Fourier representation of the flow velocity field u(x):

,

where û(k) is the Fourier transform of the flow velocity field. Thus, E(k)dk represents the contribution to the kinetic energy from all the Fourier modes with k < |k| < k+ dk, and therefore,

,

where is the mean turbulent kinetic energy of the flow. The wavenumber k corresponding to length scale r is . Therefore, by dimensional analysis, the only possible form for the energy spectrum function according with the third Kolmogorov's hypothesis is

,

where C would be a universal constant. This is one of the most famous results of Kolmogorov 1941 theory, and considerable experimental evidence has accumulated that supports it.




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