G121 渦輪の速度に対する一般公式

概要

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A large-Reynolds-number asymptotic formula is established for translation speed of a thin axisymmetric vortex ring, with circulation Γ, propagating in a viscous incompressible fluid of kinematic viscosity ν. Saffman's firstorder formula is extended to О((ν/Γ)^<3/2>). This formula also applies to a steady inviscid vortex ring of a general distribution of vorticity and is an extension to О(ε^3) of Fraenkel-Saffman's first-order formula. Here ε is the ratio of the core radius σ_0 to the ring radius R_0. We use the method of matched asymptotic expansions, but an appeal to the integral identity of total kinetic energy and the conservation law of the hydrodynamic impulse enables us to skip higher-order asymptotics of the Navier-Stokes equations. It sufficies to calculate the dipole field at О(ε) and a correction of vorticity О(ε^2). Comparison of the present formula with direct numerical simulations for a viscous vortex ring starting from a Gaussian vorticity distribution demonstrates a significant improvement in approximation.
2001-07-31