P. K. Yeung

Reynolds number dependence of Lagrangian statistics in large numerical simulations of isotropic turbulence

Lagrangian statistics are reported from a direct numerical simulation database with grid resolution up to 20483 and Taylor-scale Reynolds number approximately 650. The approach to Lagrangian Kolmogorov similarity at high Reynolds number is studied using both the velocity structure function and frequency spectrum. A significant scaling range is observed for the latter which is consistent with recent estimates of 6–7 for the scaling constant C 0. In contrast to some previous results at low Reynolds number, the current results suggest that at high Reynolds number the dissipation autocorrelation is a two-scale process influenced by both the Lagrangian velocity integral time scale and Kolmogorov time scale. Results on the logarithm of the pseudo-dissipation are in support of its modeling as a diffusion process with one-time Gaussian statistics. As the Reynolds number increases, the statistics of dissipation and enstrophy become more similar while their logarithms have significantly longer time scales.

On the role of vortical structures for turbulent mixing using direct numerical simulation and wavelet-based coherent vorticity extraction

The influence of vortical structures on the transport and mixing of passive scalars is investigated. Initial conditions are taken from a direct numerical simulation database of forced homogeneous isotropic turbulence, with passive scalar fluctuations, driven by a uniform mean gradient, are performed for Taylor microscale Reynolds numbers (R λ) of 140 and 240, and Schmidt numbers 1/8 and 1. For each R λ, after reaching a fully developed turbulent regime, which is statistically steady, the Coherent Vorticity Extraction is applied to the flow. It is shown that the coherent part is able to preserve the vortical structures with only less than 4% of wavelet coefficients while retaining 99.9% of energy. In contrast, the incoherent part is structureless and contains negligible energy. By taking the total, coherent and incoherent velocity fields in turn as initial conditions, new simulations were performed without forcing while the uniform mean scalar gradient is maintained. It is found that the results from simulations with total and coherent velocity fields as initial conditions are very similar. In contrast, the time integration of the incoherent flow exhibits its primarily dissipative nature. The evolutions of passive scalars at Schmidt numbers 1/8 and 1 advected by the total, coherent or incoherent velocity suggest that the vortical structures retained in the coherent part play a dominant role in turbulent transport and mixing. Indeed, the total and coherent flows give almost the same time evolution of the scalar variance, scalar flux and mean scalar dissipation, while the incoherent flow only gives rise to weak scalar diffusion.

On the communication complexity of 3D FFTs and its implications for Exascale

This paper revisits the communication complexity of large-scale 3D fast Fourier transforms (FFTs) and asks what impact trends in current architectures will have on FFT performance at exascale. We analyze both memory hierarchy traffic and network communication to derive suitable analytical models, which we calibrate against current software implementations; we then evaluate models to make predictions about potential scaling outcomes at exascale, based on extrapolating current technology trends. Of particular interest is the performance impact of choosing high-density processors, typified today by graphics co-processors (GPUs), as the base processor for an exascale system. Among various observations, a key prediction is that although inter-node all-to-all communication is expected to be the bottleneck of distributed FFTs, intra-node communication---expressed precisely in terms of the relative balance among compute capacity, memory bandwidth, and network bandwidth---will play a critical role.

Extreme events in computational turbulence.

Significance In the last decade or so, massive computations of turbulence performed by solving the exact equations of hydrodynamic turbulence have provided new quantitative data and enhanced our understanding. This paper presents results from the largest such computations, to date, devoted to the study of small scales. We focus on “extreme events” in energy dissipation and squared vorticity (enstrophy). For the Reynolds numbers of these simulations, events as large as 105 times the mean value are observed, albeit rarely. The structure of these extreme events is quite different from that believed previously. The major theme of this and related work is that huge fluctuations exist in hydrodynamic turbulence and that they have basic consequences to turbulence (as is the case in other nonlinear fields). We have performed direct numerical simulations of homogeneous and isotropic turbulence in a periodic box with 8,1923 grid points. These are the largest simulations performed, to date, aimed at improving our understanding of turbulence small-scale structure. We present some basic statistical results and focus on “extreme” events (whose magnitudes are several tens of thousands the mean value). The structure of these extreme events is quite different from that of moderately large events (of the order of 10 times the mean value). In particular, intense vorticity occurs primarily in the form of tubes for moderately large events whereas it is much more “chunky” for extreme events (though probably overlaid on the traditional vortex tubes). We track the temporal evolution of extreme events and find that they are generally short-lived. Extreme magnitudes of energy dissipation rate and enstrophy occur simultaneously in space and remain nearly colocated during their evolution.

Reynolds number scaling of velocity increments in isotropic turbulence

Using the largest database of isotropic turbulence available to date, generated by the direct numerical simulation (DNS) of the Navier-Stokes equations on an 8192^{3} periodic box, we show that the longitudinal and transverse velocity increments scale identically in the inertial range. By examining the DNS data at several Reynolds numbers, we infer that the contradictory results of the past on the inertial-range universality are artifacts of low Reynolds number and residual anisotropy. We further show that both longitudinal and transverse velocity increments scale on locally averaged dissipation rate, just as postulated by Kolmogorovs refined similarity hypothesis, and that, in isotropic turbulence, a single independent scaling adequately describes fluid turbulence in the inertial range.

Characteristics of backward and forward two-particle relative dispersion in turbulence at different Reynolds numbers

A new algorithm based on post-processing of saved trajectories has been developed and applied to obtain well-sampled backward and forward relative dispersion statistics in stationary isotropic turbulence, over a range of initial separations ranging from Kolmogorov to energy-containing scales. Detailed results are obtained over a range of Taylor-scale Reynolds numbers, up to 1000, which is higher than in recent work in the literature. Backward dispersion is faster, especially at intermediate times after the ballistic range and before long-time diffusive behavior is reached. Richardson scaling has been demonstrated for the mean-squared separation, and forward and backward Richardson constants estimated to be gf = 0.55 and gb = 1.5, which are close to or comparable to other estimates. However, because of persistent dissipation sub-range effects no corresponding scaling was observed for higher order moments of the separation. Analysis of the separation probability density function showed only transitory agree...

Direct numerical simulation of turbulent mixing at very low Schmidt number with a uniform mean gradient

In a recent direct numerical simulation (DNS) study [P. K. Yeung and K. R. Sreenivasan, “Spectrum of passive scalars of high molecular diffusivity in turbulent mixing,” J. Fluid Mech. 716, R14 (2013)] with Schmidt number as low as 1/2048, we verified the essential physical content of the theory of Batchelor, Howells, and Townsend [“Small-scale variation of convected quantities like temperature in turbulent fluid. 2. The case of large conductivity,” J. Fluid Mech. 5, 134 (1959)] for turbulent passive scalar fields with very strong diffusivity, decaying in the absence of any production mechanism. In particular, we confirmed the existence of the −17/3 power of the scalar spectral density in the so-called inertial-diffusive range. In the present paper, we consider the DNS of the same problem, but in the presence of a uniform mean gradient, which leads to the production of scalar fluctuations at (primarily) the large scales. For the parameters of the simulations, the presence of the mean gradient alters the ph...

GPU acceleration of a petascale application for turbulent mixing at high Schmidt number using OpenMP 4.5

Abstract This paper reports on the successful implementation of a massively parallel GPU-accelerated algorithm for the direct numerical simulation of turbulent mixing at high Schmidt number. The work stems from a recent development (Comput. Phys. Commun., vol. 219, 2017, 313–328), in which a low-communication algorithm was shown to attain high degrees of scalability on the Cray XE6 architecture when overlapping communication and computation via dedicated communication threads. An even higher level of performance has now been achieved using OpenMP 4.5 on the Cray XK7 architecture, where on each node the 16 integer cores of an AMD Interlagos processor share a single Nvidia K20X GPU accelerator. In the new algorithm, data movements are minimized by performing virtually all of the intensive scalar field computations in the form of combined compact finite difference (CCD) operations on the GPUs. A memory layout in departure from usual practices is found to provide much better performance for a specific kernel required to apply the CCD scheme. Asynchronous execution enabled by adding the OpenMP 4.5 NOWAIT clause to TARGET constructs improves scalability when used to overlap computation on the GPUs with computation and communication on the CPUs. On the 27-petaflops supercomputer Titan at Oak Ridge National Laboratory, USA, a GPU-to-CPU speedup factor of approximately 5 is consistently observed at the largest problem size of 819 2 3 grid points for the scalar field computed with 8192 XK7 nodes.

A dual communicator and dual grid-resolution algorithm for petascale simulations of turbulent mixing at high Schmidt number

Abstract A new dual-communicator algorithm with very favorable performance characteristics has been developed for direct numerical simulation (DNS) of turbulent mixing of a passive scalar governed by an advection–diffusion equation. We focus on the regime of high Schmidt number ( S c ), where because of low molecular diffusivity the grid-resolution requirements for the scalar field are stricter than those for the velocity field by a factor S c . Computational throughput is improved by simulating the velocity field on a coarse grid of N v 3 points with a Fourier pseudo-spectral (FPS) method, while the passive scalar is simulated on a fine grid of N θ 3 points with a combined compact finite difference (CCD) scheme which computes first and second derivatives at eighth-order accuracy. A static three-dimensional domain decomposition and a parallel solution algorithm for the CCD scheme are used to avoid the heavy communication cost of memory transposes. A kernel is used to evaluate several approaches to optimize the performance of the CCD routines, which account for 60% of the overall simulation cost. On the petascale supercomputer Blue Waters at the University of Illinois, Urbana–Champaign, scalability is improved substantially with a hybrid MPI-OpenMP approach in which a dedicated thread per NUMA domain overlaps communication calls with computational tasks performed by a separate team of threads spawned using OpenMP nested parallelism. At a target production problem size of 81923 (0.5 trillion) grid points on 262,144 cores, CCD timings are reduced by 34% compared to a pure-MPI implementation. Timings for 163843 (4 trillion) grid points on 524,288 cores encouragingly maintain scalability greater than 90%, although the wall clock time is too high for production runs at this size. Performance monitoring with CrayPat for problem sizes up to 40963 shows that the CCD routines can achieve nearly 6% of the peak flop rate. The new DNS code is built upon two existing FPS and CCD codes. With the grid ratio N θ ∕ N v = 8 , the disparity in the computational requirements for the velocity and scalar problems is addressed by splitting the global communicator MPI_COMM_WORLD into disjoint communicators for the velocity and scalar fields, respectively. Inter-communicator transfer of the velocity field from the velocity communicator to the scalar communicator is handled with discrete send and non-blocking receive calls, which are overlapped with other operations on the scalar communicator. For production simulations at N θ = 8192 and N v = 1024 on 262,144 cores for the scalar field, the DNS code achieves 94% strong scaling relative to 65,536 cores and 92% weak scaling relative to N θ = 1024 and N v = 128 on 512 cores.

Structure functions and applicability of Yaglom's relation in passive-scalar turbulent mixing at low Schmidt numbers with uniform mean gradient

An extensive direct numerical simulation database over a wide range of Reynolds and Schmidt numbers is used to examine the Schmidt number dependence of the structure function of passive scalars and the applicability of the so-called Yagloms relation in isotropic turbulence with a uniform mean scalar gradient. For the moderate Reynolds numbers available, the limited range of scales in scalar fields of very low Schmidt numbers (as low as 1/2048) is seen to lead to weaker intermittency, and weaker alignment between velocity gradients and principal strain rates. Strong departures from both Obukhov-Corrsin scaling for second-order structure functions and Yagloms relation for the mixed velocity-scalar third-order structure function are observed. Evaluation of different terms in the scalar structure function budget equation assuming statistical stationarity in time shows that, if the Schmidt number is very low, at intermediate scales production and diffusion terms (instead of advection) are major contributors in the balance against dissipation.