As far as I recon, what Reynolds did was actually velocity splitting to its mean and relative value (as it is in this case), with respect to spatial coordinates, not time. Hence I see rather mistakenly assumed, that Reynolds did time-averaging. It is more likely spatial, and it even brings more sense to Reynolds stress tensor as occurring from differences in velocities in two adjacent volumes, not in two different time instances. As obvious as it may seem, that the solution obtained with time averaging for N-S equations brings about same results as in Reynolds paper, the difference lies within averaging respect (temporal vs. spatial in his original paper).
I would gladly hear anyone else's opinion in this matter.
- see, Osborne Reynolds, On the Dynamical Theory of Incompressible Viscous Fluid and the Determination of the Criterion, Phil. Trans., vol.186, p.123, 1895