Reynolds averaging stress

For a random velocity field $u(x,t,\omega)$ with finite second moments, let $v=\mathbb E u$ . The Reynolds averaging stress in the covariance convention is

C=\mathbb E[(u-v)\otimes(u-v)] =\mathbb E[u\otimes u]-v\otimes v.

It is the covariance matrix of velocity, using expectation and the outer product. If averaging commutes with the derivatives and each realization solves Navier–Stokes, then

\partial_t v+\operatorname{div}(v\otimes v)+\nabla\mathbb E p =\nu\Delta v+\mathbb E f-\operatorname{div}C.

Sign convention

For every vector $a$ , $a^TCa=\mathbb E|a\cdot(u-v)|^2\geq0$ . If a defect tensor is instead placed as $+\operatorname{div}R$ on the right, then $R=-C$ . A general defect tensor need not be an actual covariance.

References

De Lellis and Székelyhidi, The h-principle and the equations of fluid dynamics (2012).