The RNS/Prandtl equations and their link with other asymptotic descriptions: Application to the wall shear stress scaling in a constricted pipe

In this paper, a steady laminar axisymmetrical flow in a straight constricted pipe is considered. The RNS/Prandtl equations are presented as an asymptotic limit of the Navier-Stokes equations. This set of equations is shown to include at first order several asymptotic descriptions of the full Navier-Stokes equations: the Blasius regime, interacting boundary layer theory, triple deck theory, the Poiseuille regime and double deck theory. These theories are all characterised by a constant pressure in each cross section. Thus, these equations are able to describe the transitions between flow regions that correspond to different classical asymptotic descriptions or regimes that are usually done with the full Navier-Stokes equations. One potential application is to predict the order of magnitude of the wall shear stress in a constricted pipe. This prediction will be compared with Navier-Stokes computations for a case of a severe constriction.