Small-scale structure of the Taylor–Green vortex

Abstract

& Frisch (1980), by a shift of origin. It offers the advantage of the symmetric Fourier representation (1 2 ) . Early-time behaviour of the inviscid flowThe symmetries of the TG vortex are listed in appendix A. Here we emphasize only those that help to visualize the qualitative features of the flow and those that may be important in making this flow atypical of general three-dimensional flow. First, for all times, no fluid crosses any of the boundaries x , y or z = nn, where n is an integer.Therefore the flow can be visualized as flow in the box 0 < x , y , z < n with impermeable stress-free faces. I n the following discussion, the region 0 < x, y , z < n is termed the impermeable box, as it confines the flow, while the region 0 < x , y , z < 2n is termed the periodicity box, as it reflects the periodicity of the Fourier series (1.2). Also, because of the symmetries listed in appendix A, the flow at any point in space can be inferred from its values in the fundamental box 0 < x, y, z < in.Secondly, if near each face we write the velocity field in terms of components parallel or perpendicular to the face, i.e. u = u,, + vl, then ul and au,,/an vanish on the face. This implies that the vorticity on each face is normal to that face so it may be written w = @, where fi is the unit normal. Note that g must vanish on all edges of the box where faces meet. I n contrast, a general incompressible flow will have only isolated points of vanishing vorticity. (Both velocity and vorticity also vanish for all time a t the centre x = y = z = in.)Thirdly, the vanishing of ul and au,,/an on each face also implies that the tensor V v is partly diagonal. One principal axis of the strain rate or symmetric part of this tensor is then perpendicular to the face. Furthermore, the magnitude of the strain rate along this axis determines the fractional growth rate of the normal vorticity on the face, i.e. d % = --~r * v --lnlcl. an ' Idt (2.1) 14-2

Evolution of vortex-surface fields in viscous Taylor–Green and Kida–Pelz flows

“…Parameters of the runs in the present study are listed in table 1. As shown in figure 1, the dissipation rate ε = ν ∇u : ∇u in both flows increases at early time and approaches a peak after a finite time, which can indicate the transition from initial smooth flow to turbulent flow at high Reynolds numbers (Brachet et al 1983;Boratav & Pelz 1994).…”

“…Here, the iso-surfaces of φ v at negative contour levels are not shown for clarity except in figure 5(a). It is noted that all the surfaces at particular contour levelsφ v = ±C have the same dynamics but they move in different directions under the TG symmetries (Brachet et al 1983). In figures 5(b) and 5(c), the vortex surfaces are stretched and gradually approach to each other, but the topology of both the vortex surfaces and vortex lines on the surfaces is unchanged.…”

“…They developed VSFs for both initial Taylor-Green (TG) (see Taylor & Green 1937;Brachet et al 1983) and Kida-Pelz (KP) (see Kida 1985;Boratav & Pelz 1994) velocity fields which display simple topology and geometry.…”

“…In addition, the TG and KP symmetries (Brachet et al 1983;Kida 1985) were utilized to reduce the computational time and scale. Parameters of the runs in the present study are listed in table 1.…”

3D Euler about a 2D symmetry plane

“…It is anticipated that in addition to these much higher resolution anti-parallel calculations, there will soon be new high-resolution calculations of the Kida-Pelz flow (anticipated in these proceedings by the contribution of Grafke et al [16]) and the Taylor-Green calculations initiated by Brachet et al (1983) [17] will soon be continued by Brachet. If these prove to be singular, and anti-parallel, it is possible that they will reproduce the scalings hinted at here.…”