Yukio Kaneda

Energy dissipation rate and energy spectrum in high resolution direct numerical simulations of turbulence in a periodic box

High-resolution direct numerical simulations (DNSs) of incompressible homogeneous turbulence in a periodic box with up to 40963 grid points were performed on the Earth Simulator computing system. DNS databases, including the present results, suggest that the normalized mean energy dissipation rate per unit mass tends to a constant, independent of the fluid kinematic viscosity ν as ν→0. The DNS results also suggest that the energy spectrum in the inertial subrange almost follows the Kolmogorov k−5/3 scaling law, where k is the wavenumber, but the exponent is steeper than −5/3 by about 0.1.

Small-scale statistics in high-resolution direct numerical simulation of turbulence : Reynolds number dependence of one-point velocity gradient statistics

One-point statistics of velocity gradients and Eulerian and Lagrangian accelerations are studied by analysing the data from high-resolution direct numerical simulations (DNS) of turbulence in a periodic box, with up to 4096 3 grid points. The DNS consist of two series of runs; one is with k max η∼ 1 (Series 1) and the other is with k max η∼2 (Series 2), where k max is the maximum wavenumber and η the Kolmogorov length scale. The maximum Taylor-microscale Reynolds number R λ in Series 1 is about 1130, and it is about 675 in Series 2. Particular attention is paid to the possible Reynolds number (Re) dependence of the statistics. The visualization of the intense vorticity regions shows that the turbulence field at high Re consists of clusters of small intense vorticity regions, and their structure is to be distinguished from those of small eddies. The possible dependence on Re of the probability distribution functions of velocity gradients is analysed through the dependence on R λ of the skewness and flatness factors (S and F). The DNS data suggest that the R λ dependence of S and F of the longitudinal velocity gradients fit well with a simple power law: S∼-0.32R λ 0.11 and F∼1.14R λ 0.34 , in fairly good agreement with previous experimental data. They also suggest that all the fourth-order moments of velocity gradients scale with R λ similarly to each other at R λ >00, in contrast to R λ < 100. Regarding the statistics of time derivatives, the second-order time derivatives of turbulent velocities are more intermittent than the first-order ones for both the Eulerian and Lagrangian velocities, and the Lagrangian time derivatives of turbulent velocities are more intermittent than the Eulerian time derivatives, as would be expected. The flatness factor of the Lagrangian acceleration is as large as 90 at R λ ≈430. The flatness factors of the Eulerian and Lagrangian accelerations increase with R λ approximately proportional to R λ αE and R λ αL , respectively, where α E ≈0.5 and α L ≈1.0, while those of the second-order time derivatives of the Eulerian and Lagrangian velocities increases approximately proportional to R λ βE and R λ βL , respectively, where β E ≈1.5 and β L ≈3.0.

16.4-Tflops Direct Numerical Simulation of Turbulence by a Fourier Spectral Method on the Earth Simulator

The high-resolution direct numerical simulations (DNSs) of incompressible turbulence with numbers of grid points up to 40963 have been executed on the Earth Simulator (ES). The DNSs are based on the Fourier spectral method, so that the equation for mass conservation is accurately solved. In DNS based on the spectral method, most of the computation time is consumed in calculating the three-dimensional (3D) Fast Fourier Transform (FFT), which requires huge-scale global data transfer and has been the major stumbling block that has prevented truly high-performance computing. By implementing new methods to efficiently perform the 3D-FFT on the ES, we have achieved DNS at 16.4 Tflops on 20483 grid points. The DNS yields an energy spectrum exhibiting a wide inertial subrange, in contrast to previous DNSs with lower resolutions, and therefore provides valuable data for the study of the universal features of turbulence at large Reynolds number.

Numerical simulation of semi-dilute suspensions of rodlike particles in shear flow

Abstract Computer simulation is carried out for the structural evolution of semidilute suspensions of non-Brownian, rodlike particles under shear flow. The short-range part of the hydrodynamic interaction is taken into account by the lubrication approximation, but the long-range part is neglected. For this model, the effect of the hydrodynamic interaction is found to be small. The major conclusions are (i) the excess viscosity is proportional to the number density n even in the region nL3 ≈ 40, where L is the length of the rod; (ii) the Folgar-Tucker constant C, which represents the deviation of each rod from the Jeffery orbit due to the hydrodynamic interaction, is of the order lO−7 ~ lO−4. Implications of these results in conjunction with existing theories and experiments are discussed.

High-resolution direct numerical simulation of turbulence

We performed high-resolution direct numerical simulations (DNS) of incompressible turbulence in a periodic box by using a Fourier spectral method with the number of grid points up to 40963. The simulations consist of two series: one with k maxη∼ 1 (series 1), and the other with k maxη∼ 2 (series 2), where k max is the highest wavenumber in each simulation, and η is the Kolmogorov length scale. In the 40963 DNS, the Taylor scale Reynolds number R λ ∼ 1200 and the ratio of L/η of the integral length scale L to η is approximately 2200, in series 1. The DNS data analysis reveals simple scaling of various spectra, and also sheds some light on (i) the energy spectrum at high R λ, (ii) the asymptotic R λ-dependence of the normalized energy dissipation rate, (iii) the anomalous scaling of the spectra of energy dissipation and enstrophy, and so on. After some preliminary remarks on the methods and limitations of the DNS, this paper presents a review of the DNS data analysis. Discussions are also made on some questions invoked by the DNS.

Lagrangian and Eulerian time correlations in turbulence

A dynamical analysis is made of the Lagrangian and Eulerian two‐time velocity correlations QL and QE for small time difference in turbulence at very high Reynolds number. The short‐time analysis yields a refinement of the so‐called random sweeping model approximation for the Eulerian correlation QE. An estimate of the Lagrangian frequency spectrum of the Lagrangian correlation QL in the inertial subrange is derived on the basis of a deductive Lagrangian renormalized approximation (LRA), which is consistent with the short‐time analysis. Estimates of some nondimensional constants associated with the Eulerian and Lagrangian time microscales are also obtained by making use of the energy spectrum predicted by the LRA.

Coherent vortices in high resolution direct numerical simulation of homogeneous isotropic turbulence: A wavelet viewpoint

Coherent vortices are extracted from data obtained by direct numerical simulation (DNS) of three-dimensional homogeneous isotropic turbulence performed for different Taylor microscale Reynolds numbers, ranging from Reλ=167 to 732, in order to study their role with respect to the flow intermittency. The wavelet-based extraction method assumes that coherent vortices are what remains after denoising, without requiring any template of their shape. Hypotheses are only made on the noise that, as the simplest guess, is considered to be additive, Gaussian, and white. The vorticity vector field is projected onto an orthogonal wavelet basis, and the coefficients whose moduli are larger than a given threshold are reconstructed in physical space, the threshold value depending on the enstrophy and the resolution of the field, which are both known a priori. The DNS dataset, computed with a dealiased pseudospectral method at resolutions N=2563,5123,10243, and 20483, is analyzed. It shows that, as the Reynolds number inc...

Suppression of vertical diffusion in strongly stratified turbulence

A spectral approximation for diffusion of passive scalar in stably and strongly stratified turbulence is presented. The approximation is based on a linearized approximation for the Eulerian two-time correlation and Corrsins conjecture for the Lagrangian two-time correlation. For strongly stratified turbulence, the vertical component of the turbulent velocity field is well approximated by a collection of Fourier modes (waves) each of which oscillates with a frequency depending on the direction of the wavevector. The proposed approximation suggests that the phase mixing among the Fourier modes having different frequencies causes the decay of the Lagrangian two-time vertical velocity autocorrelation, and the highly oscillatory nature of these modes results in the suppression of single-particle dispersion in the vertical direction. The approximation is free from any ad hoc adjusting parameter and shows that the suppression depends on the spectra of the velocity and fluctuating density fields. It is in good agreement with direct numerical simulations for strongly stratified turbulence.

Relative diffusion of a pair of fluid particles in the inertial subrange of turbulence

Turbulent diffusion of a pair of fluid particles in 3-dimensional homogeneous and isotropic turbulence was studied using a high-resolution direct numerical simulation (DNS) with 10243 grid points. The DNS showed that the mean square of the distance δx between the two fluid particles grows with time t as 〈|δx|2〉∼Cet3 in the inertial subrange, which is in agreement with Richardson (1926) and Obukhov (1941), where C≈0.7 and e is the mean dissipation rate per unit mass. A simple Lagrangian closure approximation for 〈|δx|2〉 is shown to be in good agreement with the DNS.

Inertial range structure of turbulent velocity and scalar fields in a Lagrangian renormalized approximation

The inertial range structure of turbulence is studied on the basis of an approximation derived by a systematic method of Lagrangian renormalized expansions. This method is also applied to the problem of a passive scalar field convected by turbulence, and some of its consequences are examined. Numerical values are obtained for various dimensionless constants in the inertial range including those in the k−5/3 spectrum law for the turbulent energy and the scalar field.