The lee-wave régime for a slender body in a rotating flow

Abstract

The uniform motion of a closed, axisymmetric body along the axis of an unbounded, rotating, inviscid, incompressible fluid is considered on Long's hypotheses that: the flow is steady; the flow is uniform far upstream of the body; the inertial waves excited by the body cannot propagate upstream. The appropriate similarity parameters arek, an inverse Rossby number based on the body length, and δ, the slenderness ratio of the body. It is conjectured that an upper bound to the parametric régime in which the solution implied by Long's hypotheses remains valid, saykδ≡k<k_c, is determined by the first occurrence, with increasingk, of a local reversal of the flow.A general solution for the stream function is established in terms of an assumed distribution of dipoles along the axis of the body. The disturbance upstream of the body is found to be proportional to the product ofk²and the dipole moment (total dipole strength) and to fall off only as the inverse distance, as compared with the inverse cube of the distance for a potential flow. The corresponding wave drag is found to depend on the power spectrum of the dipole distribution in the axial wave-number interval (0,k) and to be a monotonically decreasing function of the axial velocity for fixed angular velocity. Asymptotic solutions for prescribed bodies are established in the following limits: (i)K→ 0 with δ fixed; (ii) δ → 0 withkfixed; (iii)k→ ∞ withkδfixed. Both the upstream disturbance and the wave drag in the limit (i) depend essentially on the dipole moment of the body with respect to a uniform, potential flow. The limit (ii) is analogous to conventional slender-body theory and yields a dipole density that is proportional to the cross-sectional area of the body. The limit (iii) leads to a singular integral equation that is solved to determinek_cand the dipole moment for a slender body.The results are applied to a sphere and a slender ellipsoid. The upstream axial velocity and the drag coefficient based on Stewartson's results for a sphere are found to differ significantly from Maxworthy's (1969) measurements, presumably in consequence of viscous separation effects. Maxworthy's measured values of upstream axial velocity are found to agree with the theoretical values for an equivalent ellipsoid, based on the sphere plus its upstream wake, fork[lsim ]k_c.