Stiff equation: Difference between revisions
en>Wolfrock →A-stability: Fix bug in equation. Now the equation appears. Perhaps this is what the IP editor was trying to do. |
en>Anita5192 Undid revision 580760434 by 83.233.184.109 (talk)Reverted good faith edits. This was grammatically and mathematically more correct before. See talk page. |
||
| Line 1: | Line 1: | ||
'''Kolmogorov microscales''' are the smallest [[scale (ratio)|scale]]s in [[Turbulence|turbulent flow]]. At the Kolmogorov scale, viscosity dominates and the turbulent kinetic energy is dissipated into heat. They are defined<ref>{{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|coauthors=E. Mollo-Christensen|page=10}}</ref> by | |||
{| class="wikitable" | |||
| Kolmogorov length scale | |||
| <math>\eta = \left( \frac{\nu^3}{\epsilon} \right)^{1/4}</math> | |||
|- | |||
| Kolmogorov time scale | |||
| <math>\tau_\eta = \left( \frac{\nu}{\epsilon} \right)^{1/2}</math> | |||
|- | |||
| Kolmogorov velocity scale | |||
| <math>u_\eta = \left( \nu \epsilon \right)^{1/4}</math> | |||
|} | |||
where <math>\epsilon</math> is the average rate of dissipation of [[turbulence kinetic energy]] per unit mass, and <math>\nu</math> is the [[kinematic viscosity]] of the fluid. | |||
In his 1941 theory, [[Andrey Kolmogorov]] introduced the idea that the smallest scales of [[turbulence]] are universal (similar for every [[turbulent flow]]) and that they depend only on <math>\epsilon</math> and <math>\nu</math>. The definitions of the Kolmogorov microscales can be obtained using this idea and [[dimensional analysis]]. It therefore is not a theory derived from first principles. Since the dimension of kinematic viscosity is length<sup>2</sup>/time, and the dimension of the [[energy dissipation]] rate per unit mass is length<sup>2</sup>/time<sup>3</sup>, the only combination that has the dimension of time is <math> \tau_\eta=(\nu / \epsilon)^{1/2}</math> which is the Kolmorogov time scale. Similarly, the Kolmogorov length scale is the only combination of <math>\epsilon</math> and <math>\nu</math> that has dimension of length. | |||
The Kolmogorov 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. | |||
==See also== | |||
*[[Taylor microscale]] | |||
*[[Integral length scale]] | |||
*[[Batchelor scale]] | |||
== References == | |||
{{reflist}} | |||
{{DEFAULTSORT:Kolmogorov Microscales}} | |||
[[Category:Turbulence]] | |||
Revision as of 17:02, 8 November 2013
Kolmogorov microscales are the smallest scales in turbulent flow. At the Kolmogorov scale, viscosity dominates and the turbulent kinetic energy is dissipated into heat. They are defined[1] by
| Kolmogorov length scale | |
| Kolmogorov time scale | |
| Kolmogorov velocity scale |
where is the average rate of dissipation of turbulence kinetic energy per unit mass, and is the kinematic viscosity of the fluid.
In his 1941 theory, Andrey Kolmogorov introduced the idea that the smallest scales of turbulence are universal (similar for every turbulent flow) and that they depend only on and . The definitions of the Kolmogorov microscales can be obtained using this idea and dimensional analysis. It therefore is not a theory derived from first principles. 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 which is the Kolmorogov time scale. Similarly, the Kolmogorov length scale is the only combination of and that has dimension of length.
The Kolmogorov 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.
See also
References
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534