Takeshi Matsumoto

Numerical simulation of two-dimensional Faraday waves with phase-field modelling

A fully nonlinear numerical simulation of two-dimensional Faraday waves between two incompressible and immiscible fluids is performed by adopting the phase-field method with the Cahn–Hilliard equation due to Jacqmin ( J. Comput. Phys. , vol. 155, 1999, pp. 96–127). Its validation is checked against the linear theory. In the nonlinear regime, qualitative comparison is made with an earlier vortex-sheet simulation of two-dimensional Faraday waves by Wright, Yon & Pozrikidis ( J. Fluid Mech. , vol. 400, 2000, pp. 1–32). The vorticity outside the interface region is studied in this comparison. The period tripling state, which is observed in the quasi-two-dimensional experiment by Jiang, Perlin & Schultz ( J. Fluid Mech. , vol. 369, 1998, pp. 273–299), is successfully simulated with the present phase-field method.

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.

Lagrangian singularities of steady two-dimensional flow

The Lagrangian complex-space singularities of the steady Eulerian flow with stream function sin x 1 cos x 2 are studied by numerical and analytical methods. The Lagrangian singular manifold is analytic. Its minimum distance from the real domain decreases logarithmically at short times and exponentially at large times.

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.

One-dimensional hydrodynamic model generating a turbulent cascade

As a minimal mathematical model generating cascade analogous to that of the Navier-Stokes turbulence in the inertial range, we propose a one-dimensional partial-differential-equation model that conserves the integral of the squared vorticity analog (enstrophy) in the inviscid case. With a large-scale random forcing and small viscosity, we find numerically that the model exhibits the enstrophy cascade, the broad energy spectrum with a sizable correction to the dimensional-analysis prediction, peculiar intermittency, and self-similarity in the dynamical system structure.

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.

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.

Large-scale lognormality in turbulence modeled by the Ornstein-Uhlenbeck process.

Lognormality was found experimentally for coarse-grained squared turbulence velocity and velocity increment when the coarsening scale is comparable to the correlation scale of the velocity [Mouri et al., Phys. Fluids 21, 065107 (2009)]. We investigate this large-scale lognormality by using a simple stochastic process with correlation, the Ornstein-Uhlenbeck (OU) process. It is shown that the OU process has a similar large-scale lognormality, which is studied numerically and analytically.

Numerical simulation of Faraday waves oscillated by two-frequency forcing

We perform a numerical simulation of Faraday waves forced with two-frequency oscillations using a level-set method with Lagrangian-particle corrections (particle level-set method). After validating the simulation with the linear stability analysis, we show that square, hexagonal, and rhomboidal patterns are reproduced in agreement with the laboratory experiments [Arbell and Fineberg, “Two-mode rhomboidal states in driven surface waves,” Phys. Rev. Lett. 84, 654–657 (2000) and “Temporally harmonic oscillons in Newtonian fluids,” Phys. Rev. Lett. 85, 756–759 (2000)]. We also show that the particle level-set’s high degree of conservation of volume is necessary in the simulations. The numerical results of the rhomboidal states are compared with weakly nonlinear analysis. Difficulty in simulating other patterns of the two-frequency forced Faraday waves is discussed.