Kolmogorovs theory of turbulence

This appears to be in agreement with experimental observations. Since the viscosity term has the highest derivative in the Navier-Stokes equation the limit v — 0 is singular and corresponds to divergent velocity gradients. This is an example of a dissipative anomaly in which the time-reversal symmetry, that is broken for the viscous flow, is not restored in the limit of vanishing viscosity. 

Using the dissipation rate (e) as the only relevant flow parameter in the v — 0 limit, from dimensional considerations the typical fluctuations of the velocity field over a distance l and the corresponding characteristic timescales r(Z), usually interpreted as eddy turnover time , can be estimated as 

in the Re — oo limit the turbulent velocity field is a rough non-differentiable function Sv(l) l1/3 for l -h. 0. This expression identifies the so-called Holder exponent as a = 1/3. It is easy to see that the energy transfer rate 8v(1)2/t(1) is independent of the length scale l and applying the first of the above formulas to the integral scale (L) one can relate the energy dissipation rate to the large scale properties of the flow as u3/l. 

The results of this dimensional analysis are supported by an exact result obtained by Kolmogorov for the third-order longitudinal structure function. The structure functions are moments of velocity differences measured at a given distance. More specifically the structure function of order n, Sn(l), is defined as 

Brackets denote spatial or ensemble averages. Starting from the Navier-Stokes equation and assuming a statistically isotropic and homogeneous velocity field one can derive the Karman-Howarth equation for the longitudinal structure functions as 

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The dependence of the Reynolds number on the local velocity can be established in terms of the Kolmogorov theory of turbulence. 

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