Streamwise evolution of the entrainment in a steady two-dimensional bluff-body wake

An asymptotic representation for entrainment in a two-dimensional steady wake behind a circular cylinder is presented. The Reynolds number R ranges from 20 to 100. The representation is obtained from an asymptotic Navier-Stokes solution that includes the region of the wake where the non-parallelism of the streamlines is not yet negligible. The entrainment is determined as the flow rate variation in the streamwise direction. It is maximum at the beginning of the intermediate region just downstream of the symmetric counter rotating attached eddies. Moving downstream, it decreases continuously to zero, which is the asymptotic value in the far field. The entrainment increases with the Reynolds number. In terms of the normalized volumetric flow rate, the spatial evolution of the entrainment depends on the Reynolds number up to a distance of almost 20 body scales. Afterwards, the Reynolds dependence becomes weak. In the R range here considered, the entrainment can be considered negligible at a normalized distance from the body in between 50 − 60, that is, a distance value of the same order of magnitude of R. This result is in agreement with the scaling often adopted in wake stability studies to represent the slow spatial variation in a multiscaling framework.