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Reynolds stress: Difference between revisions
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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 | :<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]] ( | Averaging over a [[Flux surface|magnetic surface]] (i.e., over ''y''), the right-hand side cancels: | ||
:<math>\frac{\partial | :<math>\frac{\partial u_y}{\partial t} + \nabla_x \left ( u_x u_y \right ) = 0</math> | ||
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 | ||
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one obtains | one obtains | ||
:<math>\frac{\partial \bar{u} | :<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: | Here, the Reynolds stress tensor appears: | ||
:<math>R_{xy} = \left \langle \tilde{u} | :<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. | 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. | ||