Asymptotic solutions for two-dimensional low Reynolds number flow around an impulsively started circular cylinder

Abstract

The unsteady flow field of an incompressible viscous fluid around an impulsively started cylinder with slow motion is studied in detail. Integral expressions are derived from the nonlinear vorticity equation, and are solved by the method of matched asymptotic expansions. To complete the matching process five regions are necessary and their regions are essentially governed by the following relations: (i) the initial flow is unsteady Stokes flow (I), (ii) the early transient flow near the cylinder is steady Stokes flow (II), but the far-field flow is unsteady Stokes flow (III), so that Stokes&–Oseen-like matching is necessary, and (iii) as time increases the inertia terms become significant far downstream; thus the far flow is unsteady Oseen flow (IV), but the flow near the cylinder is steady Stokes flow (V), so that the matching of the Stokes–Oseen equations is necessary. The asymptotic analytical solutions are given for five flow fields around a circular cylinder. Also presented are the drag coefficient, the vorticity, and the streamline. The drag coefficient is verified quantitatively by comparing with earlier theories of the initial flow and the steady flow. The streamline patterns calculated show the generation of a circulating zone close to the circular cylinder just as for the transient flow around a sphere, and the difference between two-dimensional and three-dimensional flows is discussed.

Generalized fundamental solutions for unsteady viscous flows

“…. The results obtained by Price & Tan [5], who only considered a special case in which the direction of   t Ω is fixed and parallel to that of 1 e all the time, can be recovered from the generalized fundamental solution (7) by simply putting…”

“…Sano [6] obtained a long-time representation for the force on a sphere in response to an impulsively Jian-Jun Shu 2 started flow at a small but finite Reynolds number using the method of matched asymptotic expansions. Nakanishi et al [7] treated a two-dimensional version of the problem. Lovalenti & Brady [8,9] extended Sano's result to a step change in the free-stream velocity using a reciprocal theorem.…”

On the low-Reynolds-number flow about two rotating circular cylinders

“…(ii) As obtained by Nakanishi et al (1997) for a simple circular cylinder, we expect (by virtue of (2.13)) that ζ (iii) Any non-integral term arising on the right-hand side of (3.20) or (3.21) is seen to obey ∇ 2 o F =0o r∇ 2 o F =c 2 F respectively if one sets ∇ 2 o = ∂ 2 /∂X 2 + ∂ 2 /∂Y 2 . Accordingly, the outer aproximations…”

“…As explained in Nakanishi et al (1997) and Ueda et al (2001), one thus needs to distinguish five significant local domains, in the sense of Kida (1991) and Kida & Take (1995), in the (r, p)-plane, where r = |x|. These domains, depicted in figure 3 and labelled (I) to (V), are such that:…”

“…( 3.3)3.2. Governing integral representations and assumptions for the long-time behaviour It should be possible, as achieved in Nakanishi et al (1997) for one circular cylinder, to derive asymptotic estimates of (ψ,ζ )i nd o m a i ns (I)-(V) as ǫ vanishes. However, in this work we restrict the analysis to the long-time behaviour of the flow and thus consider (ψ,ζ )i ndomains (IV) and (V) only.…”