Kolmogorov microscales

{{short description|Smallest length scales in turbulent fluid flow}}

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{{cite book|title=Turbulence and Random Processes in Fluid Mechanics|year=1992|publisher=Cambridge University Press|isbn=978-0521422130|author=M. T. Landahl|edition=2nd|author2=E. Mollo-Christensen |page=10}} by

class="wikitable" | Kolmogorov length scale | $\backslash eta\; =\; \backslash left(\; \backslash frac\{\backslash nu^3\}\{\backslash varepsilon\}\; \backslash right)^\{1/4\}$

Kolmogorov time scale | $\backslash tau\_\backslash eta\; =\; \backslash sqrt\{\; \backslash frac\{\backslash nu\}\{\backslash varepsilon\}\; \}$

Kolmogorov velocity scale | $u\_\backslash eta\; =\; \backslash left(\; \backslash nu\; \backslash varepsilon\; \backslash right)^\{1/4\}$

where

• {{mvar|ε}} is the average rate of dissipation of turbulence kinetic energy per unit mass, and
• {{mvar|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, {{mvar|η}} may be much smaller.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

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 {{mvar|ε}} and {{mvar|ν}}.{{cite journal |last1=Kolmogorov |first1=A. N. |title=Local structure of turbulence in an incompressible fluid at very high Reynolds numbers |journal=Dokl. Akad. Nauk SSSR |date=1941 |volume=31 |pages=99-101}} 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 $\backslash tau\_\backslash eta=\backslash sqrt\{\; \backslash tfrac\; \backslash nu\; \backslash varepsilon\; \}$ which is the Kolmogorov time scale. Similarly, the Kolmogorov length scale is the only combination of {{mvar|ε}} and {{mvar|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, $\backslash tau\_\backslash eta\; =\; \backslash tfrac\{1\}\{\backslash sqrt\{2\; \backslash langle\; E\_\{ij\}\; E\_\{ij\}\; \backslash rangle\}\},$ which also gives $\backslash tau\_\backslash eta\; =\; \backslash sqrt\{\; \backslash tfrac\; \backslash nu\; \backslash varepsilon\; \}$ using the definition of the energy dissipation rate per unit mass $\backslash varepsilon\; =\; 2\; \backslash nu\; \backslash langle\; E\_\{ij\}\; E\_\{ij\}\; \backslash rangle\; .$ Then the Kolmogorov length scale can be obtained as the scale at which the Reynolds number ({{math|Re}}) is equal to 1,

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

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.{{cite journal |last1=Kolmogorov |first1=A. N. |title=A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number |journal=Journal of Fluid Mechanics |date=1961 |volume=13 |issue=1 |pages=82-85 |doi=10.1017/S0022112062000518}}