MULTI MODES FOR THE VORTEX SHEET-TUBE TRANSFORMATION PROCESS AND VISCOELASTIC EFFECT

Abstract

Primary elements which constitute the turbulent flow field are the vortex sheets and the vortex tubes [1, 2]. The vortex tubes are often formed by rolling-up of the vortex sheets. It is generally considered that this rolling-up is attributable to the Kelvin-Helmholtz instability, e.g. [3]. The aim of the present study is to reveal the details of the process for formation of the vortex tube along the vortex sheet, and subsequent formation of hierarchical cluster structure of the tubes. Then, we explore the implication of the occurrence of this transformation process for turbulence energy cascade [2]. In the present study, the eigenvalues were ordered so that the eigenvalue, the eigenvector of which is maximally aligned with the vorticity vector is chosen as the z component, the largest remaining eigenvalue as the + component [1]. The eigenvalues of the strain-rate tensor are denoted as σz and σ±. We have utilized the DNS data for the incompressible decaying/forced homogeneous isotropic turbulence. In an early stage in the decaying case, several flat sheets, the lateral extents of which were several times of the integral length, emerged. With lapse of time, the sheet-tube transformation abundantly occurred along the sheet and the clusters of tubes were formed. The direction of the axis of the vortex tube, however, was often perpendicular to the vorticity vectors along the vortex sheet. This vorticity configuration is similar to that in Kawahara et al. [4]. Appearance of this vorticity configuration is inconsistent with the stability analysis for the stagnation-point flow [3], in which it was shown that the vorticity component perpendicular to the direction of the diverging flow decays, and that the parallel component can grow. Detailed analysis revealed that the tube was not formed by the rolling-up of a single sheet, but formed through the interaction of the two different sheets. Initially, these vortex sheets were placed perpendicular to each other. Depending on the alignment of vorticity vectors along the two sheets, three modes of configurations were found. When the vorticity vector on the two sheets were perpendicular (Mode T), the region with σz 0 and the vortex-stretching term in the transverse (+) direction is positive, the low-pressure region gradually accumulated to form the tube, the axis of which was in the direction transverse (+) to the sheet. In the end, the spiral vortex sheet emanating from the tube with transverse vorticity was formed. In another vorticity configuration, the vorticity vectors along the two sheets were parallel (Mode P), and the axis of the tube formed in mode P was parallel to the vorticity vectors on the sheets. The circulation around the tube in Mode P was generally large, and the tube in Mode P persisted for a rather long period of time. This