The first Rcr as a possible measure of the entrainment length in a 2D steady wake

At a fixed distance from the body which creates the wake, entrainment is only seen to increase with the Reynolds number (R) up to a distance of almost 20 body scales. This increase levels up to a Reynolds number close to the critical value for the onset of the first instability. The entrainment is observed to be almost extinguished at a distance which is nearly the same for all the steady wakes within the R range here considered, i.e. [20-100], which indicates that supercritical steady wakes have the same entrainment length as the subcritiacal ones. It is observed that this distance is equal to a number of body lengths that is equal to the value of the critical Reynolds number (∼47), as indicated by a large compilation of experimental results. A fortiori of these findings, we propose to interpret the unsteady bifurcation as a process that allows a smooth increase-redistribution of the entrainment along the wake according to the weight of the convection over the diffusion. The entrainment variation along the steady wake has been determined using a matched asymptotic expansion of the Navier-Stokes velocity field [D. Tordella, M. Belan, Phys. Fluids 15 (7) (2003) 1897] built on criteria that include the matching of the transversal velocity produced by the entrainment process.