S. Goto

S. Kida

National Institute for Fusion Science (NIFS-Repository)

Reﬂective ﬂakes (mica or aluminum ﬂakes, commer-cial products such as Kalliroscope, etc) are widely usedto visualize ﬂuid motions in the laboratory. An exampleis shown in Fig. 1(a), which is a ﬂake visualization of asteady ﬂow in a precessing sphere (Fig. 2). In this exper-iment, a small amount (about 13 ppm) of TiO2-coatedmica, whose size is about (10 µm)2 × 0.1 µm, is seeded tothe water conﬁned in the sphere. The incident rays area thin laser sheet which runs through the centre of thesphere perpendicularly to the spin axis (see Fig. 2). Re-ﬂected light by the ﬂakes is recorded by a digital camerafrom the perspective parallel to the spin axis. Since thevisualized pattern is stationary, we may suppose that theﬂow is steady. However, can we extract the informationof ﬂow structures from the visualized pattern?This fundamental problem has been investigated byquite a few authors (see the references in Ref. 1), butthere is no clear answer. In order to investigate themechanism of ﬂake visualizations, we derive1), under theassumption that ﬂakes are inﬁnitely thin elliptic diskswithout inertia, the governing equations of ﬂakes:dXdt= u�X(t), t�(1)for the translational motion of the position vector X(t),anddNdt= −�N ×∇��N · u�×N (2)for the rotational motion of the normal vector N(t).Here, u(x, t) denotes the ﬂuid velocity at position xand time t. Note that (2) indicates that the temporalevolution of ﬂake orientations is identical to that of theinﬁnitesimal material surface elements.Then, we may numerically reproduce the labora-tory visualization (Fig. 1a) by tracking ﬂakes accordingto (1) and (2) in the steady ﬂow which is simulated bya spectral method in the precisely same ﬂow conditionsas in the experiment. Since (1) indicates that the spa-tial distribution of ﬂakes in an incompressible ﬂuid isuniform if it is initially so, the pattern observed in ﬂakevisualizations stems from their non-uniform orientations.Therefore, ﬂake orientations at each point on the lasersheet are numerically determined to simulate the inten-sity of the observed light based on the probability forﬂakes to reﬂect the incident rays to the observer. Ob-tained result is shown in Fig. 1(b), which is in excellentagreement with the laboratory result in Fig. 1(a). Wemay, therefore, conclude that the motion of ﬂakes is welldescribed by (1) and (2).By further detailed numerical investigations, it isshown that ﬂake orientations are isotropic on the threeconcentric bright circles in Fig. 1, whereas those oﬀ thecircles are strongly anisotropic. Hence, the three circlesare always bright irrespective of the angles of incidentrays and perspective, but the observed light intensity inthe latter regions depends on the angles. It is conﬁrmedboth by laboratory experiments and numerical simula-tions that observed patterns are indeed diﬀerent for dif-ferent angles of the incident rays. This result implies thata visualized image by a single pair of incident rays andperspective cannot distinguish these essentially diﬀerentregions, and we need to use multiple light sources or per-spectives to determine ﬂake orientations. The possibilityof the identiﬁcation of ﬂow structures, especially coher-ent structures in turbulence, based on thus determinedﬂake orientations is under investigation.(a)(b)Fig. 1: Flake visualization of a steady ﬂow in the pre-cessing sphere (Fig. 2). (a) Laboratory experiment. (b)Numerical simulation based on (1) and (2).ΩsSpin axisΩp Precession axisFig. 2: Precessing sphere.1) Goto, S., Kida, S., and Fujiwara, S.: J. Fluid Mech.(2011) in press.374§2. Flow Visualizations by Reflective FlakesGoto, S. (Okayama Univ.), Kida, S. (Doshisya Univ.)

Flow Visualizations by Reflective Flakes

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