The theory and application of Navier-Stokeslets (NSlets)

Abstract

Consider a closed body moving in an unbounded fluid that decays to rest in the far-field and governed by the incompressible Navier-Stokes equations. By considering a translating reference frame, this is equivalent to a uniform flow past the body. A velocity representation is given as an integral distribution of Green’s functions of the Navier-Stokes equations which we shall call NSlets. The strength of the NSlets is the same as the force distribution over the body boundary. An expansion for the NSlet is given with the leading-order term being the Oseenlet. To test the theory, the following three two-dimensional steady flow benchmark applications are considered. First, consider uniform flow past a circular cylinder for three cases: low Reynolds number, high Reynolds number, and also intermediate Reynolds numbers at values 26 and 36. These values are chosen because the flow is still steady and has not yet become unsteady. For the low Reynolds number, approximate the NSlet by the leading order Oseenlet term. For the high Reynolds number, approximate the NSlet by the Eulerlet which is the leading order Oseenlet in the high Reynolds number limit. For the intermediate Reynolds numbers, approximate the NSlet by an Eulerlet close to its origin and an Oseenlet further away. Second, consider uniform flow past a slender body with elliptical cross section with Reynolds number Re ∼ 106 and approximate the NSlet by the Eulerlet. Finally, consider the Blasius problem of uniform flow past a semi-infinite flat plate and consider the first three terms in the NSlet approximation.