Stiff equation - Revision history

en>December Rush: /* See also */

2014-12-17T19:24:54Z

152.77.240.76: /* Motivating example */ correction of math print (not sure of it)

2014-02-07T11:23:45Z

Motivating example: correction of math print (not sure of it)

← Older revision		Revision as of 13:23, 7 February 2014
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	'~~''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		Emilia Shryock is my name but you can contact me something you like. Minnesota is where he's been living for years. Body building is what my family members and I appreciate. Bookkeeping is my occupation.<br><br>Here is my web-site - diet meal delivery ([http://Flark.it/healthymealsdelivered68340 look at this web-site])
	~~{\| 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]]~~

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.

2013-11-08T16:02:20Z

Undid revision 580760434 by 83.233.184.109 (talk)Reverted good faith edits. This was grammatically and mathematically more correct before. See talk page.

← Older revision		Revision as of 18:02, 8 November 2013
Line 1:		Line 1:
	~~I am Dusty from Epiquerez~~. ~~I am learning to play~~ the ~~Trumpet~~. ~~Other hobbies~~ are ~~Bird watching~~.<br><br>~~Here~~ is ~~my site~~ ... ~~Fifa 15 coin generator~~ (~~[http:~~//~~colts-Hockey~~.~~com~~/~~home~~/~~userinfo~~.~~php?uid~~=~~476232 please click the following article~~])		'''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]]

en>Wolfrock: /* A-stability */ Fix bug in equation. Now the equation appears. Perhaps this is what the IP editor was trying to do.

2012-07-28T02:07:47Z

A-stability: Fix bug in equation. Now the equation appears. Perhaps this is what the IP editor was trying to do.

New page

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← Older revision		Revision as of 21:24, 17 December 2014
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	~~Emilia Shryock is my name but you can contact me something you like~~. ~~Minnesota is where he's been living for years~~. ~~Body building is what my family members and~~ I ~~appreciate~~. ~~Bookkeeping is my occupation~~.<br><br>~~Here is~~ my ~~web-site - diet meal delivery (~~[http://~~Flark~~.it/~~healthymealsdelivered68340 look at this web-site~~])		Hello! I am Regena. I smile that I could unify to the whole globe. I live in Italy, in the south region. I dream to go to the different countries, to get familiarized with interesting people.<br><br>my homepage; [http://www.Weka-Computer.de/?document_srl=1092782 fifa 15 coin hack]