The unsteady laminar boundary layer on an axisymmetric body subject to small-amplitude fluctuations in the free-stream velocity

Abstract

The effect of small amplitude, time-periodic, freestream disturbances on an otherwise steady axisymmetric boundary layer on a circular cylinder is considered.Numerical solutions of the problem are presented, and an asymptotic solution, valid far downstream along the axis of the cylinder is detailed. Particular emphasis is placed on the unsteady eigensolutions that occur far downstream, which turn out to be very different from the analogous planar eigensolutions. These axisymmetric eigensolutions are computed numerically and also are described by asymptotic analyses valid for low and high frequencies of oscillation. 0L)10

On a class of unsteady, non-parallel, three-dimensional disturbances to boundary-layer flows

“…Simply stated, we investigate theξ → ∞ form of eigensolutions to this system. The analysis closely follows that of Lam & Rott (1960), Ackerberg & Phillips (1972) and Goldstein (1983) for two-dimensional unsteady disturbances (of infinite wavenumber) to Blasius flow, and of Duck (1991) for the corresponding axisymmetric configuration.…”

“…On the other hand as ξ (or ξ) → ∞, unsteadiness effects undoubtedly play a leading role -the ξ 2 multiplying the unsteady terms in (5.2), (5.4) ensures this. In the Appendix it is shown that sufficiently far downstream, solutions to (5.2)-(5.4) take a form similar to decaying, unsteady eigensolutions found in the context of two-dimensional boundary layers by Lam & Rott (1960), Ackerberg & Phillips (1972) and Goldstein (1983) (these also have some similarity with those found in corresponding axisymmetric boundary layers by Duck 1991). The key result is that the unsteady eigen-behaviour is dominated by a term of the form e −Λξ 3 , where Λ is dependent on ω, and is such that Re{Λ} > 0, irrespective of ω.…”

“…The analysis may be continued, with, in general g(ξ) ∼ξ r (see Appendix A of Goldstein 1983, andDuck 1991 for an analogous calculation), but the key result here is the super-exponential decay with ξ; the primary difference with the two-dimensional context, however, is the slow (algebraic) decay ofψ 1 (and hence alsoû 1 ) with σ.…”