Kolmogorov microscale

The Kolmogorow microscale is the smallest scale when considering the energy cascade of a turbulent flow .

According to Richardson , the spectrum of the turbulent flow is broken down into three wavelength ranges :

• the injection area takes energy to
• the inertial area transports them and
• In the dissipation range of the smallest wavelengths, the energy is converted into heat by friction .

In 1941 Kolmogorow found not only a universal formula for the spectral power density in the inertial range, the so-called 5/3 law: ${\displaystyle P(k)}$

${\displaystyle P(k)\sim k^{-5/3}}$

where k is the circular wave number , but also described the dissipation range referred to as the microscale by Kolmogorow , which only depends on the mean value of the dissipation rate per unit of mass and on the kinematic viscosity of the fluid : ${\displaystyle \epsilon }$ ${\displaystyle \nu }$

Kolmogorov length scale	${\displaystyle \eta =\left({\frac {\nu ^{3}}{\epsilon }}\right)^{1/4}}$
Kolmogorov time scale	${\displaystyle \tau _{\eta }=\left({\frac {\nu }{\epsilon }}\right)^{1/2}}$
Kolmogorov speed scale	${\displaystyle u_{\eta }=\left(\nu \epsilon \right)^{1/4}}$

In his theory, Kolmogorow assumes that the length scale is the same for every turbulent flow, i.e. only depends on and . The definition of the scale can be obtained with the help of this premise and a dimensional analysis . Since the dimension of the kinematic viscosity is length ² / time and the dimension of the rate of dissipation per unit mass is length ² / time ³ , the relationship is obtained as a combination to obtain the dimension of time . ${\displaystyle \epsilon }$ ${\displaystyle \nu }$ ${\displaystyle \tau _{\eta }=(\nu /\epsilon )^{1/2}}$

Because of the assumption of a constant mean dissipation rate, his approach is a molecular field approximation .

swell

• MT Landahl, E. Mollo-Christensen: Turbulence and Random Processes in Fluid Mechanics , Cambridge, 2nd edition, 1992.

Individual evidence

1. ↑ Uwe Schimpf: Fourier analysis of microscale temperature fluctuations on the water surface. Diploma thesis at the University of Heidelberg. (No longer available online.) May 1996, archived from the original on February 13, 2012 ; Retrieved December 5, 2010 . Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice.