Introduction: In the context of fusion plasmas, the Reynolds stress is a mechanism for generation of sheared flow from turbulence
Definition: For a homogeneous fluid and an incompressible flow, the flow velocities are split into a mean part and a fluctuating part using Reynolds decomposition.
With u(x,t) being the flow velocity vector having components ui in the xicoordinate direction (with xi denoting the components of the coordinate vector X) The mean velocities ui are determined by either time averaging, spatial averaging or ensemble averaging, depending on the flow under study. Further ui’notes the fluctuating (turbulence) part of the velocity.
The components τ'ij of the Reynolds stress tensor are defined as:
With ρ the fluid density, taken to be non-fluctuating for this homogeneous fluid. Another – often used – definition, for constant density, of the Reynolds stress components is:
Which has the dimensions of velocity squared, instead of stress.
Averaging and the Reynolds stress: To illustrate, Cartesian vector index notation is used. For simplicity, consider an incompressible fluid: Given the fluid velocity as a function of position and time, write the average fluid velocity as , and the velocity fluctuation is . Then the conventional ensemble rules of averaging are that
One splits the Euler equations or the Navier-Stokes equations into an average and a fluctuating part. One finds that upon averaging the fluid equations, a stress on the right hand side appears of the form this is the Reynolds stress, conventionally written
The divergence of this stress is the force density on the fluid due to the turbulent fluctuations.
Discussion: The question then is what is the value of the Reynolds stress? This has been the subject of intense modeling and interest, for roughly the past century. The problem is recognized as a closure problem, akin to the problem of closure in the BBGKY hierarchy. A transport equation for the Reynolds stress may be found by taking the outer product of the fluid equations for the fluctuating velocity, with itself.
Typically, the average is formally defined as an ensemble average as in statistical ensemble theory. However, as a practical matter, the average may also be thought of as a spatial average over some length scale, or a temporal average. Note that, while formally the connection between such averages is justified in equilibrium statistical mechanics by the ergodic theorem, the statistical mechanics of hydrodynamic turbulence is currently far from understood. In fact, the Reynolds stress at any given point in a turbulent fluid is somewhat subject to interpretation, depending upon how one defines the average.