Does the stationary viscous flow around a circular cylinder exist for large Reynolds numbers? A numerical solution via variational imbedding

We propose an approach to identifying the solutions of the steady incompressible Navier–Stokes equations for large Reynolds numbers. These cannot be obtained as initial-value problems for the unsteady system because of the instability of the latter. Our approach consists of replacing the original steady-state problem for the Navier–Stokes equations by a boundary-value problem for the Euler–Lagrange equations for minimization of the quadratic functional of the original equations. This technique is called Method of Variational Imbedding (MVI) and in this case it leads to a system of higher-order partial differential equations, which is solved by means of an operator-splitting method. As a featuring example we consider the classical flow around a circular cylinder which is known to lose stability as early as for Re=40. We find a stationary solution with recirculation zone for Reynolds numbers as large as Re=200. Thus, new information about the possible hybrid flow regimes is obtained.