Reynolds stress - Revision history

Admin at 10:39, 26 January 2023

2023-01-26T10:39:18Z

← Older revision		Revision as of 12:39, 26 January 2023
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	Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates (think of ''x'' as radial, ''y'' as poloidal, and ''z'' as toroidal):		Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates (think of ''x'' as radial, ''y'' as poloidal, and ''z'' as toroidal):
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, {{ISBN\|9780750310307}}</ref>

	:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\frac{1}{\rho}\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math>		:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\frac{1}{\rho}\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math>

Arturo at 12:58, 5 August 2011

2011-08-05T12:58:52Z

← Older revision		Revision as of 14:58, 5 August 2011
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	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>

	:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math>		:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\frac{1}{\rho}\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math>

	Averaging over a [[Flux surface\|magnetic surface]] (i.e., over ''y''), the right-hand side cancels:		Averaging over a [[Flux surface\|magnetic surface]] (i.e., over ''y''), the right-hand side cancels:

Admin: Reverted edits by 178.20.71.246 (Talk) to last revision by Admin

2011-07-16T09:54:15Z

Reverted edits by 178.20.71.246 (Talk) to last revision by Admin

← Older revision		Revision as of 11:54, 16 July 2011
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	It ~~was dark when I woke~~. This is a ~~ray~~ of ~~sunishne~~.		In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence.
			<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>

			Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates (think of ''x'' as radial, ''y'' as poloidal, and ''z'' as toroidal):
			<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>

			:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math>

			Averaging over a [[Flux surface\|magnetic surface]] (i.e., over ''y''), the right-hand side cancels:

			:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math>

			It may seem as if one has lost all information concerning the background field.
			However, this is not true, as the choice of the ''x,y,z'' coordinate system depends, precisely, on the background magnetic field (and, in particular, on the cited flux surfaces).
			The corresponding anisotropy is in fact essential to the effectiveness of the Reynolds Stress mechanism.

			Now, writing the flow as the sum of a mean and a fluctuating part

			:<math>u = \bar{u} + \tilde{u}</math>

			one obtains

			:<math>\frac{\partial \bar{u}_y}{\partial t} + \nabla_x \left \langle \tilde{u}_x \tilde{u}_y \right \rangle = 0</math>

			Here, the Reynolds stress tensor appears:

			:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>

			Thus, a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, <math>\tilde{u}_x</math> and <math>\tilde{u}_y</math> must be ''correlated'' for this to work.
			This correlation occurs naturally in the presence of a background (mean) gradient driving turbulent transport.

			Once the laminar flow shear develops, it may suppress small-scale turbulence, leading to a reduction of transport.
			<ref>[http://link.aps.org/doi/10.1103/RevModPhys.72.109 P. W. Terry, ''Suppression of turbulence and transport by sheared flow'', Rev. Mod. Phys. '''72''' (2000) 109–165]</ref>

			== See also ==

			* [[H-mode]]
			* [[Internal Transport Barrier]]

			== References ==
			<references />

178.20.71.246: gJJifhclsZeKmuMmxR

2011-07-16T08:28:10Z

gJJifhclsZeKmuMmxR

← Older revision		Revision as of 10:28, 16 July 2011
Line 1:		Line 1:
	~~In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence.~~		It was dark when I woke. This is a ray of sunishne.
	~~<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>~~

	~~Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates (think of ''x'' as radial, ''y'' as poloidal, and ''z'' as toroidal):~~
	~~<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>~~

	~~:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = -\nabla_y P + \frac{1}{\rho} \left ( \vec{j} \times \vec{B} \right )_y</math>~~

	~~Averaging over a [[Flux surface\|magnetic surface]] (i.e., over ''y''), the right-hand side cancels:~~

	~~:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math>~~

	It ~~may seem as if one has lost all information concerning the background field~~.
	~~However, this is not true, as the choice of the ''x,y,z'' coordinate system depends, precisely, on the background magnetic field (and, in particular, on the cited flux surfaces).~~
	~~The corresponding anisotropy~~ is ~~in fact essential to the effectiveness of the Reynolds Stress mechanism.~~

	~~Now, writing the flow as the sum of a mean and a fluctuating part~~

	~~:<math>u = \bar{u} + \tilde{u}</math>~~

	~~one obtains~~

	~~:<math>\frac{\partial \bar{u}_y}{\partial t} + \nabla_x \left \langle \tilde{u}_x \tilde{u}_y \right \rangle = 0</math>~~

	~~Here, the Reynolds stress tensor appears:~~

	~~:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>~~

	~~Thus,~~ a ~~non-zero value~~ of ~~the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, <math>\tilde{u}_x</math> and <math>\tilde{u}_y</math> must be ''correlated'' for this to work.~~
	~~This correlation occurs naturally in the presence of a background (mean) gradient driving turbulent transport.~~

	~~Once the laminar flow shear develops, it may suppress small-scale turbulence, leading to a reduction of transport.~~
	~~<ref>[http://link.aps.org/doi/10.1103/RevModPhys.72.109 P. W. Terry, ''Suppression of turbulence and transport by sheared flow'', Rev. Mod. Phys~~. ~~'''72''' (2000) 109–165]</ref>~~

	~~== See also ==~~

	* [[H-mode]]
	* [[Internal Transport Barrier]]

	~~== References ==~~
	~~<references />~~

Admin at 10:05, 13 July 2011

2011-07-13T10:05:26Z

← Older revision		Revision as of 12:05, 13 July 2011
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	:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>		:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>

	Thus, a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, <math>\tilde{u}_x</math> and <math>\tilde{u}_y</math> must be ''correlated'' for this to work~~, which will depend on~~ the ~~details~~ of ~~the~~ (~~equations describing~~ the) turbulence.		Thus, a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, <math>\tilde{u}_x</math> and <math>\tilde{u}_y</math> must be ''correlated'' for this to work.
			This correlation occurs naturally in the presence of a background (mean) gradient driving turbulent transport.

			Once the laminar flow shear develops, it may suppress small-scale turbulence, leading to a reduction of transport.
			<ref>[http://link.aps.org/doi/10.1103/RevModPhys.72.109 P. W. Terry, ''Suppression of turbulence and transport by sheared flow'', Rev. Mod. Phys. '''72''' (2000) 109–165]</ref>

	== See also ==		== See also ==

Admin at 19:09, 12 July 2011

2011-07-12T19:09:52Z

← Older revision		Revision as of 21:09, 12 July 2011
Line 2:		Line 2:
	<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>		<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>

	Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates:		Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates (think of ''x'' as radial, ''y'' as poloidal, and ''z'' as toroidal):
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>

Admin at 19:08, 12 July 2011

2011-07-12T19:08:32Z

← Older revision		Revision as of 21:08, 12 July 2011
Line 2:		Line 2:
	<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>		<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>

	Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor:		Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor, in slab coordinates:
	<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>		<ref>R. Balescu, ''Aspects of Anomalous Transport in Plasmas'', Institute of Physics Pub., Bristol and Philadelphia, 2005, ISBN 9780750310307</ref>

Line 10:		Line 10:

	:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math>		:<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math>

			It may seem as if one has lost all information concerning the background field.
			However, this is not true, as the choice of the ''x,y,z'' coordinate system depends, precisely, on the background magnetic field (and, in particular, on the cited flux surfaces).
			The corresponding anisotropy is in fact essential to the effectiveness of the Reynolds Stress mechanism.

	Now, writing the flow as the sum of a mean and a fluctuating part		Now, writing the flow as the sum of a mean and a fluctuating part

Admin at 14:37, 12 July 2011

2011-07-12T14:37:14Z

← Older revision		Revision as of 16:37, 12 July 2011
Line 23:		Line 23:
	:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>		:<math>R_{xy} = \left \langle \tilde{u}_x \tilde{u}_y \right \rangle</math>

	~~and it is clear that~~ a non-zero value of the ''gradient'' of the Reynolds stress (of fluctuating flow components) can drive a laminar flow.		Thus, a non-zero value of the gradient of the Reynolds stress (of fluctuating flow components) can drive a laminar flow. Obviously, <math>\tilde{u}_x</math> and <math>\tilde{u}_y</math> must be ''correlated'' for this to work, which will depend on the details of the (equations describing the) turbulence.

	== See also ==		== See also ==

Admin at 14:13, 12 July 2011

2011-07-12T14:13:34Z

← Older revision		Revision as of 16:13, 12 July 2011
Line 1:		Line 1:
	In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence.		In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence.
			<ref>[http://dx.doi.org/10.1088/0741-3335/43/10/308 S.B. Korsholm et al, ''Reynolds stress and shear flow generation'', Plasma Phys. Control. Fusion '''43''' (2001) 1377]</ref>

	Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor:		Starting from the incompressible momentum balance equation, neglecting the dissipative pressure tensor:

Admin at 16:47, 30 August 2009

2009-08-30T16:47:06Z

← Older revision		Revision as of 18:47, 30 August 2009
Line 23:		Line 23:

	and it is clear that a non-zero value of the ''gradient'' of the Reynolds stress (of fluctuating flow components) can drive a laminar flow.		and it is clear that a non-zero value of the ''gradient'' of the Reynolds stress (of fluctuating flow components) can drive a laminar flow.

			== See also ==

			* [[H-mode]]
			* [[Internal Transport Barrier]]

	== References ==		== References ==
	<references />		<references />