S. Goto

S. Kida

National Institute for Fusion Science (NIFS-Repository)

between them.Fig.! Cluster of small-scale coherent vortical structuresin turbulence. The vorticity vector inside each vortex tubepoints approximately from marked end to unmarked one.(a)(b)Fig.2 Cross-section of an anti-parallel vortex pair in-dicated by solid circles. (a) Arrows: relative velocity tothe center. Hyperbolic stagnation points are denoted bysolid squares. Contours: normal components of vortic-ity. Positive regions are shaded. (b) Lines: projections ofmaterial line elements near the cross-section. Contours:magnitude of their stretching rates.Reference1) Bathelor, G. K., Proc. Roy. Soc. London A 213(1952) 349.2) Kida, S. and Goto, 2. Phys. Fluids 14 (2002) 352.3) Goto, S. and Kida, S. Fluid Dyn. Res. (2003) in press.A line (surface) which always consists of a same set offluid particles is called a material line (surface). It hasbeen intuitively predicted1) and numerically confirmed2)that the total length (area) of a material line (surface)in turbulence grows exponentially in time. Since thisstrong stretching is closely related with the strong tur-bulent mixing, which is an important problem not onlytheoretically but also in applications, the stretching ofmaterial objects has been intensively studied by manyauthors. However, the physical mechanism of neitherstrong mixing nor strong stretching of material objectsin turbulence has been understood well.It is well-known, on the other hand, that there exitsmall-scale coherent vortical structures universally inturbulence. The coherent vortical structures have tubu-lar shapes with radius of 0(5"7) and lengths ranging be-tween "7 and'c. Here, "7 is the Kolmogorov length, theminimum length scale in isotropic turbulence, and ,c isthe integral length, the maximum length scale. A typ-ical cluster of the coherent vortical structures is shownin Fig.I. It may be observed clearly that the tubularvortices tend to align to each other in an anti-parallelmanner. This strong tendency of anti-parallel alignmentof coherent vortices is confirmed also quantitatively (seeRef.3).An important inherent feature of an anti-parallel vor-tex pair is that there are two hyperbolic stagnationpoints on the cross-section which moves with the pair.Around the stagnation points, magnitudes of eigenval-ues of strain tensor are large, and the streamlines on thecross-section are nearly parallel to the first eigenvector ofthe tensor. It is these features of an anti-parallel vortexpair that serve a physical explanation of strong stretch-ing of material objects in turbulence, which is summarizeas follows. [1] Material lines - since a material surfacemay be regarded as a set of material lines, the physicalmechanism of their strong stretching is the same as thatof lines - are likely to align to quasi-stationary stream-lines, i.e., streamlines on the frame moving with a vortexpair. [2] Since the streamlines are nearly parallel to thestrain-tensor eigenvectors around the hyperbolic stagna-tion points, and since the eigenvalues are large there,material lines are strongly stretched around the stagna-tion points. We plot a typical turbulent velocity field andthe appearance of material lines stretching on the cross-section of a vortex pair in Fig.2, which strongly supportsthe above argument.In conclusion, universally existing coherent tubularvortical structures in turbulence tend to align to eachother in an anti-parallel manner, and such an anti-parallel pair of vortices sustain the strong stretching ofmaterial objects around the hyperbolic stagnation pointsGoto, S., Kida, S.§ 25. Material Object Stretching InTurbulence346

Material Object Stretching in Turbulence

Abstract

Similar works

Full text

Available Versions

National Institute for Fusion Science (NIFS-Repository)