Atsuya Uno

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.

The K computer : Japanese next-generation supercomputer development project

The K computer is a distributed memory supercomputer system consisting of more than 80,000 compute nodes which is being developed by RIKEN as a Japanese national project. Its performance is aimed at achieving 10 peta-flops sustained in the LINPACK benchmark. The system is under installation and adjustment. The whole system will be operational in 2012.

First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on the K computer

Real space DFT (RSDFT) is a simulation technique most suitable for massively-parallel architectures to perform first-principles electronic-structure calculations based on density functional theory. We here report unprecedented simulations on the electron states of silicon nanowires with up to 107,292 atoms carried out during the initial performance evaluation phase of the K computer being developed at RIKEN. The RSDFT code has been parallelized and optimized so as to make effective use of the various capabilities of the K computer. Simulation results for the self-consistent electron states of a silicon nanowire with 10,000 atoms were obtained in a run lasting about 24 hours and using 6,144 cores of the K computer. A 3.08 peta-flops sustained performance was measured for one iteration of the SCF calculation in a 107,292-atom Si nanowire calculation using 442,368 cores, which is 43.63% of the peak performance of 7.07 peta-flops.

Statistics of Energy Transfer in High-Resolution Direct Numerical Simulation of Turbulence in a Periodic Box

The statistics of energy transfer is studied by using the data of a series of high-resolution direct numerical simulations of incompressible homogeneous turbulence in a periodic box with the Taylor micro-scale Reynolds number R λ and grid points up to approximately 1130 and 4096 3 , respectively. The data show that the energy transfer T across the wave number k is highly intermittent and the skewness S and flatness F of T increase with k approximately as S ∝( k L ) α S , F ∝( k L ) α F in the inertial subrange, where α S ∼2/3, α F ∼1 and L the characteristic length scale of energy containing eddies. The comparison between the statistics of T , the energy dissipation rate e and its average e r over a domain of scale r shows that T is less intermittent than e, while there is a certain similarity between the probability distribution functions of T and e r .

Spectra of Energy Dissipation, Enstrophy and Pressure by High-Resolution Direct Numerical Simulations of Turbulence in a Periodic Box

The spectra of the squares of velocity quadratics including the energy dissipation rate e per unit mass, the enstrophy ω 2 and the pressure p were measured using the data obtained from direct numerical simulations (DNSs) of incompressible turbulence in a periodic box with number of grid points up to 2048 3 . These simulations were performed using the Earth Simulator computing system. The spectra for e, ω 2 and p exhibited a wave number range in which the spectra scaled with the wave number k as ∝ k - a . Exponent a for p was about 1.81, which is in good agreement with the value obtained by assuming the joint probability distribution of the velocity field to be Gaussian, while a values for e and ω 2 were about 2/3, and very different from the Gaussian approximation values.

Energy Spectrum in the Near Dissipation Range of High Resolution Direct Numerical Simulation of Turbulence

The energy spectrum in the near dissipation range of turbulence is studied by analyzing the data of a series of high-resolution direct numerical simulations of incompressible homogeneous turbulence...

K computer: 8.162 PetaFLOPS massively parallel scalar supercomputer built with over 548k cores

Many high-performance CPUs employ a multicore architecture with a moderate clock frequency and wide instruction issue, including SIMD extensions, to achieve high performance while retaining a practical power consumption. As demand for supercomputer performance grows faster than the rate that improvements are made to CPU performance, the total number of cores of high-end supercomputers has increased tremendously. Efficient handling of large numbers of cores is a key aspect in the design of supercomputers. Building a supercomputer with lower power consumption and significant reliability is also important from the viewpoints of cost and availability.

The design of ultra scalable MPI collective communication on the K computer

This paper proposes the design of ultra scalable MPI collective communication for the K computer, which consists of 82,944 computing nodes and is the world’s first system over 10 PFLOPS. The nodes are connected by a Tofu interconnect that introduces six dimensional mesh/torus topology. Existing MPI libraries, however, perform poorly on such a direct network system since they assume typical cluster environments. Thus, we design collective algorithms optimized for the K computer.On the design of the algorithms, we place importance on collision-freeness for long messages and low latency for short messages. The long-message algorithms use multiple RDMA network interfaces and consist of neighbor communication in order to gain high bandwidth and avoid message collisions. On the other hand, the short-message algorithms are designed to reduce software overhead, which comes from the number of relaying nodes. The evaluation results on up to 55,296 nodes of the K computer show the new implementation outperforms the existing one for long messages by a factor of 4 to 11 times. It also shows the short-message algorithms complement the long-message ones.