Mitsuo Yokokawa

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.

A 26.58 Tflops Global Atmospheric Simulation with the Spectral Transform Method on the Earth Simulator

A spectral atmospheric general circulation model called AFES (AGCM for Earth Simulator) was developed and optimized for the architecture of the Earth Simulator (ES). The ES is a massively parallel vector supercomputer that consists of 640 processor nodes interconnected by a single stage crossbar network with its total peak performance of 40.96 Tflops was achieved for a high resolution simulation (T1279L96) with AFES by utilizing the full 640-node configuration of the ES. The resulting computing efficiency is 64.9% of the peak performance, well surpassing that of conventional weather/climate applications having just 25-50% efficiency even on vector parallel computers. This remarkable performance proves the effectiveness of the ES as a viable means for practical applications.

Fragment molecular orbital study of the electronic excitations in the photosynthetic reaction center of Blastochloris viridis

All electron calculations were performed on the photosynthetic reaction center of Blastochloris viridis, using the fragment molecular orbital (FMO) method. The protein complex of 20,581 atoms and 77,754 electrons was divided into 1398 fragments, and the two‐body expansion of FMO/6‐31G* was applied to calculate the ground state. The excited electronic states of the embedded electron transfer system were separately calculated by the configuration interaction singles approach with the multilayer FMO method. Despite the structural symmetry of the system, asymmetric excitation energies were observed, especially on the bacteriopheophytin molecules. The asymmetry was attributed to electrostatic interaction with the surrounding proteins, in which the cytoplasmic side plays a major role.

Full Electron Calculation Beyond 20,000 Atoms: Ground Electronic State of Photosynthetic Proteins

A full electron calculation for the photosynthetic reaction center of Rhodopseudomonas viridis was performed by using the fragment molecular orbital (FMO) method on a massive cluster computer. The target system contains 20,581 atoms and 77,754 electrons, which was divided into 1,398 fragments. According to the FMO prescription, the calculations of the fragments and pairs of the fragments were conducted to obtain the electronic state of the system. The calculation at RHF/6-31G* level of theory took 72.5 hours with 600 CPUs. The CPUs were grouped into several workers, to which the calculations of the fragments were dispatched. An uneven CPU grouping, where two types of workers are generated, was shown to be efficient.

GridFMO — Quantum chemistry of proteins on the grid

A GridFMO application was developed by recoining the fragment molecular orbital (FMO) method of GAMESS with grid technology. With the GridFMO, quantum calculations of macro molecules become possible by using large amount of computational resources collected from many moderate-sized cluster computers. A new middleware suite was developed based on Ninf-G, whose fault tolerance and flexible resource management were found to be indispensable for long-term calculations. The GridFMO was used to draw ab initio potential energy curves of a protein motor system with 16,664 atoms. For the calculations, 10 cluster computers over the pacific rim were used, sharing the resources with other users via butch queue systems on each machine. A series of 14 GridFMO calculations were conducted for 70 days, coping with more than 100 problems cropping up. The FMO curves were compared against the molecular mechanics (MM), and it was confirmed that (1) the FMO method is capable of drawing smooth curves despite several cut-off approximations, and that (2) the MM method is reliable enough for molecular modeling.

Chapter 3 – DNS of Canonical Turbulence with up to 40963 Grid Points

Publisher Summary This chapter discusses direct numerical simulation (DNS) study of canonical turbulence. Incompressible turbulence obeying the Navier–Stokes (N–S) equation under periodic boundary conditions (BC) is widely regarded as one of the most canonical types of turbulences. It keeps the essence of tile turbulence dynamics: (1) the nonlinear convection effect associated with the fluid motion, (2) dissipativity, and (3) mass conservation, which is equivalent to the incompressibility or the so-called solenoidal condition in incompressible fluid. Underlying the study of turbulence in such a simple geometry is the idea of the Kolmogorov hypotheses, according to which the small scale statistics in fully developed turbulence at sufficiently high Reynolds number Re is universal and insensitive to the details of large scale conditions. The DNS of incompressible homogeneous turbulence was performed under periodic boundary conditions with the number of grid points up to 1024 3 on the VPP5000 system at the Information Technology Center, Nagoya University, and DNS up to 4096 3 grid points on the earth simulator (ES). The DNS is based on a spectral method free from alias error. Sustained performance of 16.4 Tflops was achieved in the DNS with 2048 3 grid points and double precision arithmetic on the ES.

High-Resolution Direct Numerical Simulation of Turbulence — Spectra of Fourth-Order Velocity Moments —

High-resolution direct numerical simulations (DNSs) of incompressible turbulence based on an alias-free spectral method were performed on the Earth Simulator. Statistics of turbulence are studied by a DNS on 10243 grid points with a special emphasis on the spectra of moments fourth order in velocity. A brief review is given on some results of the preliminary analysis of the data of DNSs with up to 20483 grid points.