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In this paper we review some classes of exact solutions of the Navier-Stokes equations under a time-independent external straining flow, centering around the celebrated Burgers vortex. The objectives are (i) to clarify the relationship between them and (ii) to examine them as models of turbulence. Particularly we study the Lundgren spiral model for turbulence in the presence of azimuthal vorticity (that is, with axial velocity). The implication on linear stability of the Burgers vortex is briefly discussed.

A survey on a class of exact solutions of the Navier-Stokes equations and a model for turbulence