4D visualization of isotropic turbulence and dynamics ofhigh-enstrophy structures
Yuji Hattori1 and Takashi Ishihara2
121

Institute of Fluid Science, Tohoku University, Japan
Graduate School of Engineering, Nagoya University, Japan

Introduction

It is well known that in the high Reynolds number turbulence there exist thin tube-like vortical structures whoseradii are order of the Kolmogorov length scale. Direct numerical simulations have been successful in revealing theirvarious features (see e.g. Ishihara et al.[1]). Althoughstatistics of turbulence is extensively investigated, thereare less works on the geometry and dynamics of the vortical structures. Moisy and Jiménez[2] characterized thegeometry and spatial distribution of high-vorticity regions
by means of box-counting methods. Bermejo-Moreno etal.[3] extracted structures of high enstrophy and characterized them by geometrical analysis; distribution of bloblike, tube-like and sheet-like structures was studied in somedetail. The results so far, however, are mostly based oneither instantaneous or averaged ﬁelds. The dynamics or
temporal evolution of the ﬁne-scale vortical structures remains unexplored.
In this paper we study the dynamics of high-enstrophy
structures in isotropic turbulence by 4D visualization andgeometric analysis. Our aim is to understand the temporal
evolution of vortical structures in turbulence and thereby
elucidate the interaction of vortical structures, which includes deformation, reconnection, merging and cancellation; these should play an important role in the cascade
process of turbulence. We are particularly interested in the
dynamical equilibrium of the ﬁne-scale vortical structures,
which is closely related with the statistical properties turbulence and would be helpful in improving the accuracy ofturbulence models.
24D visualization

The numerical method is essentially the same with Ishihara et al.[4]. The three-dimensional Navier-Stokes equations are solved by Fourier spectral method. The total
number of modes is N 3 = 10243 . The microscale Reynolds
number is Reλ ≈ 358, while the Kolmogorov
length is√
kmax η ≈ 1.59 where kmax = 483 ≈ 2N/3 is the maximum wavenumber.
The DNS data are visualized by Realization WorkSpace(RWS) of the Advanced Fluid Information Research Center (AFIRC), Institute of Fluid Science, Tohoku University. RWS is a virtual reality system for three-dimensionalvisualization (Fig. 1); the screen extends both in front andon the ﬂoor with total size of 4.5[m]×4.0[m]. The large
disk space of AFIRC allows us to store a suﬃcient numberof data for animation of the three-dimensional ﬁeld (4Dvisualization). In the present study we choose an isosurface of the magnitude of vorticity for the three-dimensional

ﬁeld. The 4D visualization has shown various ways of interaction of the tube-like vortical structures: deformation,
reconnection, merging, cancellation etc. This led us to investigate the temporal evolution of the tube-like vorticalstructures, which are called high-enstrophy structures in
the following to make it clear that vortical structures withsmall magnitude of vorticity are excluded.
3

Analysis of high-enstrophy structures

In order to study the temporal evolution of vorticalstructures in turbulence, we introduce a simple deﬁnitionof high-enstrophy structures. First we choose a thresholdωc for the magnitude of vorticity. Two grid points are said
to be neighbors if the distance between them is smaller
than 2∆x, where ∆x = 2π/N is the grid spacing. Twogrid points are regarded as connected if there is a sequenceof neighbors between them. Then a set of connected gridx) > ωcpoints S is called a high-enstrophy structure if ω(xfor all x ∈ S. There are a few disadvantages due to thesimplicity of the deﬁnition. For instance, there are many
tiny structures consisting of a few points; one third of thestructures have points less than 10. The above simple definition, however, is preferred for the present purpose since
it is not time-consuming for temporal analysis.
As a ﬁrst step toward studying the temporal evolution
we show some basic properties of high-enstrophy structuresin an instantaneous ﬁeld of isotropic turbulence. Figure 2
shows the volume fraction of the largest high-enstrophy
structure, or in other words, the maximum volume of thestructures divided by the total volume of the structuresfor ωc . Here the volume is deﬁned by the number n ofpoints in the structure multiplied by the volume of unit
cell (∆x)3 . For small ωc the largest structure occupies almost the whole region of ω > ωc ; in other words the wholeregion ω > ωc is almost connected. A transition is observed between 1 < ωc /ω < 3. For large ωc the volume
fraction is close to zero, which implies that we can identify disconnected high-enstrophy structures. Also shownin Fig. 2 is the number of high-enstrophy structures. It
takes maximum around ωc /ω = 1.8, for which the volume
fraction is 0.78, a rather large value. In the following weset ωc /ω = 3.
Figure 3 shows examples of high-enstrophy structures
obtained by the present method. There are a number of
tube-like structures like the one shown in left. However,some are combination of such tube-like structures which
are interacting as shown in right.
Relation between the volume and area of the highenstrophy structures is shown by scatter plot in Figure
4. The area is approximately proportional to the numberFig. 3. Examples of high-enstrophy structure.

Fig. 1. 4D visualization in Realization WorkSpace.

1.0

0.8

8e+04

0.6

6e+04

0.4

4e+04

0.2

2e+04

0.0

ns/n1e+05
number of structures

fraction

0.7
1.0

0e+00
01

23
ωc/ω

45

0.5
0.4
0.3
100

101

102

103
n104

105

106

Fig. 4. Scatter plot of n and ns /n.

6

Acknowledgment
Fig. 2. Volume fraction of maximum high-enstrophy structure (+, left axis) and number of high-enstrophy structures(×, right axis).

ns of surface points of the structure, where a point in S iscalled a surface point of S if at least one of the adjacentpoints is out of S. Although the data points are widely
spread, we observe a crude scaling law ns /n ∝ n−1/6 . If
we apply this scaling to cylindrical tubes, the radius andthe length scale as n1/6 and n2/3 , respectively. The valueof the radius is estimated as r ∼ 2(n/ns )1/2 ∆x, the maximum of which is ∼ 3∆x. The length is also estimated asl ∼ ns ∆x/(4π) ∼ 102 ∆x for ns ∼ 103 . These values are
consistent with thin tube-like vortical structures although
not all of the high-enstrophy structures are tube-like.

4Conclusion

The high-enstrophy structures in isotropic turbulence
are investigated by 4D visualization using a virtual reality system. As a step toward studying in detail the dynamics of the high-enstrophy structures geometrical analysis of an instantaneous ﬁeld is performed. Our simple
deﬁnitions are successfully applied to identify the highenstrophy structures and their surfaces. Temporal evolution and interaction of the structures are currently investigated and the results will be reported in the conference.

Numerical simulations are performed on the scalar parallel computing system of the Advanced Fluid Information
Research Center (AFIRC), Institute of Fluid Science, Tohoku University. Part of the work was carried out under
the Collaborative Research Project of the Institute of Fluid
Science, Tohoku University.
References[1] T. Ishihara, T. Gotoh and Y. Kaneda, “Study of HighReynolds Number Isotropic Turbulence by Direct Numerical Simulation,” Annu. Rev. Fluid Mech. 41, pp. 165–180(2009).
[2] F. Moisy and J. Jiménez, “Geometry and clustering ofintense structures in isotropic turbulence,” J. Fluid Mech.
513, pp. 111–133 (2004).
[3] I. Bermejo-Moreno, D. I. Pullin and K. Horiuti, “Geometry of enstrophy and dissipation, grid resolution eﬀects
and proximity issues in turbulence,” J. Fluid Mech. 620,
pp. 121–166 (2009).
[4] T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura and A.
Uno, “Small-scale statistics in high-resolution direct numerical simulation of turbulence: Reynolds number dependence of one-point velocity gradient statistics,” J. Fluid
Mech. 592, pp. 335–366 (2007).