Reynolds stress: Difference between revisions

(4 intermediate revisions by 3 users not shown)

Line 3:

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 ) = -\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:

Line 27:

:<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 ==

@@ Line 3: / Line 3: @@
 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 ) = -\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:
@@ Line 27: / Line 27: @@
 :<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 ==