Stretched vortices – the sinews of turbulence; large-Reynolds-number asymptotics

Abstract

A large-Reynolds-number asymptotic theory is presented for the problem of a vortex tube of finite circulation [Gcy ] subjected to uniform non-axisymmetric irrotational strain, and aligned along an axis of positive rate of strain. It is shown that at leading order the vorticity field is determined by a solvability condition at first-order in ε = 1/R[Gcy ] where R[gcy ] = [gcy ]/ν. The first-order problem is solved completely, and contours of constant rate of energy dissipation are obtained and compared with the family of contour maps obtained in a previous numerical study of the problem. It is found that the region of large dissipation does not overlap the region of large enstrophy; in fact, the dissipation rate is maximal at a distance from the vortex axis at which the enstrophy has fallen to only 2.8% of its maximum value. The correlation between enstrophy and dissipation fields is found to be 0.19 + O(ε2). The solution reveals that the stretched vortex can survive for a long time even when two of the principal rates of strain are positive, provided R[gcy ] is large enough. The manner in which the theory may be extended to higher orders in ε is indicated. The results are discussed in relation to the high-vorticity regions (here described as ‘sinews’) observed in many direct numerical simulations of turbulence.

Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

“…In Section 2.1 it is shown that an unusual decoupling takes place between ω 3 and W with γ satisfying a nonlinear equation which is related to the second z-derivative of the pressure p zz which is constrained to be spatially uniform. The second stage is to solve these equations for ω 3 and W . To construct stretched vortex solutions of Burgers type it is necessary to be more specific with the velocity field in Eq.…”

“…(42) to which we turn in Section 3.2. Before this, however, we discuss axisymmetric solutions for 3 and W.…”

“…We take a similar approach to that of the previous section for the Navier-Stokes equations where axisymmetric solutions for ω 3 …”

“…Stretched vortices of Burgers type, which are exact solutions of the Navier-Stokes equations, are often used as typical solutions to illustrate the tube-sheet paradigm of modern turbulence theory [1][2][3][4][5][6][7][8][9]. In reality they are pseudo-3D in nature as they are composed of 2D flows superimposed on a 3D vorticity free strain field, a fact exploited by Lundgren in his transformation [2].…”

“…Depending on whether a tube or shear layer symmetry has been chosen, corresponding to uni-axial or bi-axial strain [3], they have the properties that the vorticity ω ω ω firstly lies respectively either along the axis of the tube or in the plane of the layer and secondly that it aligns with an eigenvector of the strain matrix S. While numerical simulations [10,11] and experiments [12] have shown that, in a spatially averaged sense, alignment of the vorticity vector ω ω ω with the intermediate eigenvector of S is favoured (for a list of references see [13]), it is clear that local vorticity accumulation and alignment processes are more complicated than this [14,15]. For instance, the solution for the Burgers vortex has the drawback that it is stretched by a strain field that is decoupled from the flow around it and that its vorticity is unidirectional [1][2][3]. Complicated vortical structures caused by both vortex stretching and compression have a dynamic complexity that requires a more subtle theoretical explanation (see the recent review by Pullin and Saffman [16]).…”