 Solution

Let the mean streamwise velocity be $U(y)$ and set $S=dU/dy>0$. The relevant covariance is $\overline{u^{\prime }{v}^{\prime }}<0$ on this half of the channel. The work of the Reynolds stress against the mean shear rate givesThe eddy viscosity relation is $-\overline{u^{\prime }{v}^{\prime }}={\nu }_{T}S$, soThese are the kinematic-stress conventions, with the covariance measured per unit mass; inserting a stress measured in force per area would require division by mass density.