Susumu Goto

Turbulence driven by precession in spherical and slightly elongated spheroidal cavities

Motivated by the fascinating fact that strong turbulence can be sustained in a weakly precessing container, we conducted a series of laboratory experiments on the flow in a precessing spherical cavity, and in a slightly elongated prolate spheroidal cavity with a minor-to-major axis ratio of 0.9. In order to determine the conditions required to sustain turbulence in these cavities, and to investigate the statistics of the sustained turbulence, we developed an experimental technique to conduct high-quality flow visualizations as well as measurements via particle image velocimetry on a turntable and by using an intense laser. In general, flows in a precessing cavity are controlled by two non-dimensional parameters: the Reynolds number Re (or its reciprocal, the Ekman number) which is defined by the cavity size, spin angular velocity, and the kinematic viscosity of the confined fluid, and the Poincare number Po, which is defined by the ratio of the magnitude of the precession angular velocity to that of the s...

Analytical calculation of four-point correlations for a simple model of cages involving numerous particles.

Dynamics of a one-dimensional system of Brownian particles with short-range repulsive interaction (diameter σ) is studied with a liquid-theoretical approach. The mean square displacement, the two-particle displacement correlation, and the overlap-density-based generalized susceptibility are calculated analytically by way of the Lagrangian correlation of the interparticulate space, instead of the Eulerian correlation of density that is commonly used in the standard mode-coupling theory. In regard to the mean square displacement, the linear analysis reproduces the established result on the asymptotic subdiffusive behavior of the system. A finite-time correction is given by incorporating the effect of entropic nonlinearity with a Lagrangian version of mode-coupling theory. The notorious difficulty in derivation of the mode-coupling theory concerning violation of the fluctuation-dissipation theorem is found to disappear by virtue of the Lagrangian description. The Lagrangian description also facilitates analytical calculation of four-point correlations in the space-time, such as the two-particle displacement correlation. The two-particle displacement correlation, which is asymptotically self-similar in the space-time, illustrates how the cage effect confines each particle within a short radius on one hand and creates collective motion of numerous particles on the other hand. As the time elapses, the correlation length grows unlimitedly, and the generalized susceptibility based on the overlap density converges to a finite value which is an increasing function of the density. The distribution function behind these dynamical four-point correlations and its extension to three-dimensional cases, respecting the tensorial character of the two-particle displacement correlation, are also discussed.

Turbulent mixing in a precessing sphere

By numerically simulating turbulent flows at high Reynolds numbers in a precessing sphere, we propose a method to enhance the mixing of a fluid confined within a smooth cavity by its rotational motion alone. To precisely evaluate the mixing efficiency, we extend the quantification method proposed by Danckwerts [“The definition and measurement of some characteristics of mixtures,” Appl. Sci. Res. A 3, 279–296 (1952)] to the case in which only a finite number of fluid particle trajectories can be known. Our accurate numerical tracking of fluid particles in the flow, which is controlled by the Reynolds number (an indicator of the spin rate) and the Poincare number (the precession rate), shows the following results. First, the mixing process on the time scale normalized by the spin period is independent of the Reynolds number as long as it is high enough for the flow to be developed turbulence. Second, fastest mixing is achieved under weak precession (Poincare number ≈0.1); in such cases, perfect mixing requi...

Calculation of displacement correlation tensor indicating vortical cooperative motion in two-dimensional colloidal liquids

As an indicator of cooperative motion in a system of Brownian particles that models two-dimensional colloidal liquids, a displacement correlation tensor is calculated analytically and compared with numerical results. The key idea for the analytical calculation is to relate the displacement correlation tensor, which is a kind of four-point space-time correlation, to the Lagrangian two-time correlation of the deformation gradient tensor. Tensorial treatment of the statistical quantities, including the displacement correlation itself, allows capturing the vortical structure of the cooperative motion. The calculated displacement correlation also implies a negative long-time tail in the velocity autocorrelation, which is a manifestation of the cage effect. Both the longitudinal and transverse components of the displacement correlation are found to be expressible in terms of a similarity variable, suggesting that the cages are nested to form a self-similar structure in the space-time.

Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the two-dimensional case, the linear theory gives description of vortical cooperative motion.

Local equilibrium hypothesis and Taylor's dissipation law

To qualitatively investigate the validity of Kolmogorov local equilibrium hypothesis and the Taylor dissipation law, we conduct direct numerical simulations of the three-dimensional turbulent Kolmogorov flow. Since strong scale-by-scale (i.e. Richardson-type) energy cascade events occur quasi-periodically, the kinetic energy of the turbulence and its dissipation rate evolve quasi-periodically too. In this unsteady turbulence driven by a steady force, instantaneous values of the dissipation rate obey the scaling recently discovered in wind tunnel experiments (Vassilicos 2015 Ann. Rev. Fluid Mech. 47 95–114) instead of the Taylor dissipation law. The Taylor dissipation law does not hold because the local equilibrium hypothesis does not hold in a relatively low wave-number range. The breakdown of this hypothesis is caused by the finite time needed for the energy at such large scales to reach the dissipative scale by the scale-by-scale energy cascade.

Displacement correlation as an indicator of collective motion in one-dimensional and quasi-one-dimensional systems of repulsive Brownian particles

While the slow dynamics in glassy liquids are known to be accompanied by collective motions undetectable with static structure factor and requiring four-point space-time correlations for their detection, it is usually difficult to calculate such correlations analytically. In the present study, a system of Brownian particles in a (quasi-)one-dimensional passageway is taken as an example to demonstrate the usefulness of displacement correlation. In the purely one-dimensional case (known as the single-file diffusion) with overtaking forbidden, the diffusion slows down and collective motion is captured by displacement correlation both calculated here numerically and analytically. On the other hand, displacement correlation vanishes if overtaking is allowed, which leads to normal diffusion.

Sub critical transition to turbulence in three-dimensional Kolmogorov flow

We study Kolmogorov flow on a three dimensional, periodic domain with aspect ratios fixed to unity. Using an energy method, we give a concise proof of the linear stability of the laminar flow profile. Since turbulent motion is observed for high enough Reynolds numbers, we expect the domain of attraction of the laminar flow to be bounded by the stable manifolds of simple invariant solutions. We show one such edge state to be an equilibrium with a spatial structure reminiscent of that found in plane Couette flow, with stream wise rolls on the largest spatial scales. When tracking the edge state, we find an upper and a lower branch solution that join in a saddle node bifurcation at finite Reynolds number.

Response function of turbulence computed via fluctuation-response relation of a Langevin system with vanishing noise

For a shell model of the fully developed turbulence and the incompressible Navier-Stokes equations in the Fourier space, when a Gaussian white noise is artificially added to the equation of each mode, an expression of the mean linear response function in terms of the velocity correlation functions is derived by applying the method developed for nonequilibrium Langevin systems [Harada and Sasa, Phys. Rev. Lett. 95, 130602 (2005)]. We verify numerically for the shell-model case that the derived expression of the response function, as the noise tends to zero, converges to the response function of the noiseless shell model.

Collective Motion of Repulsive Brownian Particles in Single-File Diffusion with and without Overtaking

Subdiffusion is commonly observed in liquids with high density or in restricted geometries, as the particles are constantly pushed back by their neighbors. Since this “cage effect” emerges from many-body dynamics involving spatiotemporally correlated motions, the slow diffusion should be understood not simply as a one-body problem but as a part of collective dynamics, described in terms of space–time correlations. Such collective dynamics are illustrated here by calculations of the two-particle displacement correlation in a system of repulsive Brownian particles confined in a (quasi-)one-dimensional channel, whose subdiffusive behavior is known as the single-file diffusion (SFD). The analytical calculation is formulated in terms of the Lagrangian correlation of density fluctuations. In addition, numerical solutions to the Langevin equation with large but finite interaction potential are studied to clarify the effect of overtaking. In the limiting case of the ideal SFD without overtaking, correlated motion with a diffusively growing length scale is observed. By allowing the particles to overtake each other, the short-range correlation is destroyed, but the long-range weak correlation remains almost intact. These results describe nested space–time structure of cages, whereby smaller cages are enclosed in larger cages with longer lifetimes.