Turbulence Structure Around a Columnar Vortex - RDT and Vortex Wave Excitation

Abstract

In this paper, the structure of initially isotropic homogeneous turbulence around solid and fluid columnar vortices (with circulation Г and radius σ) is investigated using rapid distortion theory (RDT). We assume that the turbulence velocity u ₀ is small compared with Г/σ, that the length scale L is such that (L/u ₀ ) ≫ σ/Г and that the Reynolds number is very large. The initial condition is a columnar vortex embedded in a field of initially homogeneous isotropic turbulence (Fig la). In order to investigate the different elements of the phenomena, namely vortex stretching by differential rotation, blocking of external fluctuations by the vortex and wave generation on the vortex, we consider three idealisations of the columnar vortex. For the first “solid cylinder model: (SC)”, the core of a columnar vortex is replaced by a solid cylinder rotating with a constant angular velocity Ω = Г/2πσ ². The fluid motion outside the core is irrotational with the circulation F. The turbulent eddies are stretched by the differential rotation induced around the columnar vortex and the azimuthal component of turbulent vorticity grows algebraically, in proportion to time t, while the other components do not grow. At the same time the eddies are blocked by the presence of the vortex. Thus, RDT also enables us to simulate the change in the random velocity fields around the vortex. It is found that the turbulent eddies are wrapped around the columnar vortex to form fine-scale structures like vortex rings, with similar properties to those computed by [1], who conducted direct numerical simulations at moderately high Reynolds number using a spectral method (128³) (Fig 2). What they observed are 1) azimuthal alignment of the external vorticity, 2) excitation of bending waves on the columnar vortex and 3) the tendency of the external vorticity to form vortex rings which become axisymmetric and interact with each other. Although the third of these manifestations is the result of nonlinear interactions between eddies of the external turbulence under the action of the straining motions, the first two are consistent with the RDT results.