## Featured Story Corner## Turbulent Mixing at High Schmidt Number## —Dr. P.K. Yeung, Georgia Institute of TechnologyThe word "turbulence" is often used to describe certain situations of political unrest, economic uncertainty, labor strife, bumpy airplane rides, and, of course, many fluid motions occurring in nature and engineering. A common feature in all of these is the presence of disorder, complexity, and lack of predictability in detail even though some definite social or physical constraints do apply. In particular, fluctuations in fluid turbulence appear to be random despite being governed by partial differential equations that express basic and deterministic laws of conservation of mass, momentum, and energy. Although these characteristics of turbulence may seem undesirable, life without turbulence would be strange and difficult, if not impossible; for then coffee and cream will not mix, automobile engines may not run, pollutants will not disperse, and so forth, at least not in reasonable time. These examples show that the mechanisms by which turbulence causes efficient mixing and reduces spatial nonuniformities of momentum, heat, and chemical substances are central to our ability to understand natural phenomena and design improved engineering devices. Fluid turbulence is a Grand Challenge computational problem for which use of state-of-the-art cyberinfrastructure has always been of great importance. Even in its simplest form (without couplings with combustion chemistry, atmospheric phenomena, etc.) the inherent characteristic of disorder in turbulence is such that the flow is always unsteady in time and is always three-dimensional in space. Therefore, attempts to capture the full flow physics by numerical simulation require the flow variables to be calculated for a period of many time steps at a large set of grid points. More precisely, computational requirements are set by the range of scales over which disorderly fluctuations occur and interact in a nonlinear manner. The range of scales for the velocity fluctuations (which drive the mixing) is determined by a parameter called the Reynolds number, which depends on flow speed, body dimension or size of flow domain, and viscosity, and is high in most applications. In the case of turbulent mixing, we usually consider fluctuations of a "passive scalar"—which may, for example, be a substance concentration in a dilute mixture or small temperature difference—which does not affect the flow. The range of scales involved then depends further on the molecular diffusivity, which for heat transfer would be highest in liquid metals that conduct heat efficiently and lowest in organic liquids where molecular diffusion is slow and large temperature or concentration differences can develop. We usually refer to molecular diffusivity by the Schmidt number (Sc), which is the ratio of fluid viscosity to molecular diffusivity of the diffusing substance or property. The case of high Schmidt number (low diffusivity) is more difficult and less understood even in traditional experiments. Fortunately, advances in supercomputing have now made it feasible to simulate high-Sc mixing directly, as well as to provide detailed data for turbulent mixing over a wide range of molecular diffusivities.
Thanks to the availability of the 15.6-Teraflop DataStar at San Diego Supercomputer Center (SDSC), we have performed the world's largest simulation of turbulent mixing at high Schmidt number (low diffusivity), using as many as 2048 processors for 2048
The numerical method we use to solve the governing equations (Navier-Stokes for momentum and advection-diffusion equation for the scalars) is Fourier pseudospectral in space and second-order in time. The solution domain is a periodic cube which is divided among the parallel processors into "slabs" of equal size that each contain an integral number of planes. We use the IBM ESSL library for Fast Fourier Transforms and the standard MPI library for inter-processor communication calls. The code performs at more than 80% scalability with increasing number of processors both at fixed problem size and with problem size adjusted to match memory per processor used. The same code has been ported to the IBM Blue Gene at SDSC with literally no changes required, an encouraging sign for the future, because it scales even better there than on the DataStar. Furthermore, SDSC consultants, D. Pekurovsky and G. Chukkapalli, have worked with Georgia Tech PhD student, D.A. Donzis, to produce a new code that uses an alternative data decomposition that will allow us to use even more processors than presently. When the number of processors increases in the future, the new code will scale even better than our current production version, based on benchmarking studies.
With continuing generous resource allocations and strategic staff assistance at SDSC, and with new NSF investments in Cyberinfrasture on the horizon, the future is looking very bright. Currently our 2048 This research is part of a collaborative project with Professor K.R. Sreenivasan (Director of International Centre for Theoretical Physics, on leave from the College of Engineering at University of Maryland) supported by the National Science Foundation's Fluid Dynamics and Hydraulics Program. PK Yeung's research group has received approximately 2 million hours of CPU allocations at SDSC in the past two years; large computations focused on different aspects of turbulence are also carried out at two other national centers. He can be reached via email at pk.yeung@ae.gatech.edu |