Kolmogorov microscales

Visualization of the output of a 2D computational fluid dynamics simulation solving the Navier Stokes equations with a scalar transport term. The flow is initialized with turbulence generated using Kolmogorov scaling and the velocity field then moves from left to right at 1 grid cell per second. A point source is situated upwind to simulate the dispersion of a scalar plume in this velocity field

In fluid dynamics, Kolmogorov microscales are the smallest scales in turbulent flow. At the Kolmogorov scale, viscosity dominates and the turbulence kinetic energy is dissipated into thermal energy. They are defined^[1] by

Kolmogorov length scale	$\eta =\left({\frac {\nu ^{3}}{\varepsilon }}\right)^{1/4}$
Kolmogorov time scale	$\tau _{\eta }={\sqrt {\frac {\nu }{\varepsilon }}}$
Kolmogorov velocity scale	$u_{\eta }=\left(\nu \varepsilon \right)^{1/4}$

where

$ε$ is the average rate of dissipation of turbulence kinetic energy per unit mass, and
$ν$ is the kinematic viscosity of the fluid.

Typical values of the Kolmogorov length scale, for atmospheric motion in which the large eddies have length scales on the order of kilometers, range from 0.1 to 10 millimeters; for smaller flows such as in laboratory systems, $η$ may be much smaller.^[2]

In 1941, Andrey Kolmogorov introduced the hypothesis that the smallest scales of turbulence are universal (similar for every turbulent flow) and that they depend only on $ε$ and $ν$ .^[3] The definitions of the Kolmogorov microscales can be obtained using this idea and dimensional analysis. Since the dimension of kinematic viscosity is length²/time, and the dimension of the energy dissipation rate per unit mass is length²/time³, the only combination that has the dimension of time is $\tau _{\eta }={\sqrt {\tfrac {\nu }{\varepsilon }}}$ which is the Kolmogorov time scale. Similarly, the Kolmogorov length scale is the only combination of $ε$ and $ν$ that has dimension of length.

Alternatively, the definition of the Kolmogorov time scale can be obtained from the inverse of the mean square strain rate tensor, $\tau _{\eta }={\tfrac {1}{\sqrt {2\langle E_{ij}E_{ij}\rangle }}},$ which also gives $\tau _{\eta }={\sqrt {\tfrac {\nu }{\varepsilon }}}$ using the definition of the energy dissipation rate per unit mass $\varepsilon =2\nu \langle E_{ij}E_{ij}\rangle .$ Then the Kolmogorov length scale can be obtained as the scale at which the Reynolds number ( $Re$ ) is equal to 1,

\mathrm {Re} ={\frac {UL}{\nu }}={\frac {(\eta /\tau _{\eta })\eta }{\nu }}=1.

Kolmogorov's 1941 theory is a mean field theory since it assumes that the relevant dynamical parameter is the mean energy dissipation rate. In fluid turbulence, the energy dissipation rate fluctuates in space and time, so it is possible to think of the microscales as quantities that also vary in space and time. However, standard practice is to use mean field values since they represent the typical values of the smallest scales in a given flow. In 1961, Kolomogorov published a refined version of the similarity hypotheses that accounts for the log-normal distribution of the dissipation rate.^[4]

References

^ M. T. Landahl; E. Mollo-Christensen (1992). Turbulence and Random Processes in Fluid Mechanics (2nd ed.). Cambridge University Press. p. 10. ISBN 978-0521422130.
^ George, William K. "Lectures in Turbulence for the 21st Century." Department of Thermo and Fluid Engineering, Chalmers University of Technology, Göteborg, Sweden (2005).p 64 [online] http://www.turbulence-online.com/Publications/Lecture_Notes/Turbulence_Lille/TB_16January2013.pdf
^ Kolmogorov, A. N. (1941). "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers". Dokl. Akad. Nauk SSSR. 31: 99–101.
^ Kolmogorov, A. N. (1961). "A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number". Journal of Fluid Mechanics. 13 (1): 82–85. doi:10.1017/S0022112062000518.

[1] M. T. Landahl; E. Mollo-Christensen (1992). Turbulence and Random Processes in Fluid Mechanics (2nd ed.). Cambridge University Press. p. 10. ISBN 978-0521422130.

[2] George, William K. "Lectures in Turbulence for the 21st Century." Department of Thermo and Fluid Engineering, Chalmers University of Technology, Göteborg, Sweden (2005).p 64 [online] http://www.turbulence-online.com/Publications/Lecture_Notes/Turbulence_Lille/TB_16January2013.pdf

[3] Kolmogorov, A. N. (1941). "Local structure of turbulence in an incompressible fluid at very high Reynolds numbers". Dokl. Akad. Nauk SSSR. 31: 99–101.

[4] Kolmogorov, A. N. (1961). "A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number". Journal of Fluid Mechanics. 13 (1): 82–85. doi:10.1017/S0022112062000518.

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See also

References