A two-dimensional vortex sheet model for a temporal free shear layer: integrating beyond the Moore singularity by the technique of viscosity switch smoothing

Abstract

The Navier-Stokes equations are the fundamental equations governing the dynamics of Newtonian

uids, if one neglects theirs derivation from the Boltzmann equation of molecular dynamics. The inviscid Navier-Stokes equations are known as the Euler equations. The Reynolds number of the ow, which is the ratio of inertial to frictional forces in the uid, is in nite for an inviscid

uid. For very low Reynolds number ows it is possible to obtain some exact and approximate solutions of the Navier-Stokes equations. Modelling turbulent ows is a di cult problem in uid dynamics. The Navier-Stokes equations are not amenable to mathematical analysis, and they cannot be used to predict detailed consequences or the emergence of randomness at high Reynolds number. The theory of Richardson and Kolmogorov views turbulence as a cascade of eddies or coherent structures. However, most of these studies are statistical and are not likely to answer the lack of universality. The theory of chaos in uid ows, originated by Lorenz, views turbulence as a sensitive dependence on initial conditions. Simple nonlinear equations with analytical solutions and prescribed initial conditions were found to exhibit chaotic and apparently random behaviour. Turbulent scales have a fractal like distribution. However, the randomness generated by the Navier-Stokes equations may be due to (hidden) intrinsic reasons. This framework cannot be used to predict global variables in a turbulent ow, for example the Reynolds number dependence of the resistance coe cient, in a pipe of given radius and pressure gradient.