Turbulent Mixing up Close

ook closely at almost any flowing air or water and you'll see that the flow is disorderly and filled with fluctuating eddies. A central characteristic of such turbulent flows is greatly enhanced mixing, which is clear to anyone who stirs cream into their coffee or watches a plume of pollution rising over a factory smokestack. Though such mixing is crucial in situations ranging from pollutant dispersion to blending fuel and oxidants for efficient combustion, a complete mathematical treatment of turbulent flows has remained an elusive goal. Now, numerical simulations on Blue Horizon by P.K. Yeung of the Georgia Institute of Technology are helping unravel these mysteries.

Turbulent fluid flow is often described as the most important physical problem that still awaits a full solution. It is everywhere, and yet characterized by challenging uncertainties that must be described in statistical terms. Who would place a bet on where that balloon a child releases will come to earth?

Despite their everyday familiarity, tur bulent flows have resisted the many research approaches seeking the scientific understanding necessary for accurate predictions. Although the motion of fluids like air and water is well described by the classical Navier-Stokes equations, which express basic principles of conservation of mass and momentum, mathematical solutions of these equations for turbulent flow have not yet been possible.

In turbulent flows, the fluid is "free" to move in many ways, as seen in the finely-wrought uniqueness of cloud formations. Indeed, disorderly fluctuations in turbulence span an enormous and continuous range of scales in both length and time, with largest scales near the overall physical size of the flow, down to tiny scales set by frictional (viscous) forces between adjacent elements of fluid. And the difficulty of the subject is due in large part to the complex and nonlinear interactions between these scales of motion, which moreover are inherently 3-D.

One approach that is making important progress is to perform detailed computer simulations of turbulent flows. Yeung, an associate professor in the School of Aerospace Engineering at Georgia Tech, has been conducting such simulations on Blue Horizon, NPACI's teraflops IBM RS/6000 SP at SDSC, with the dual objectives of increasing fundamental understanding of turbulence and helping develop statistical models of turbulent mixing and dispersion.

"My work involves taking a fundamental point of view, to numerically integrate the conservation equations for mass, momentum, and chemical species concentration," Yeung says. "We call this approach Direct Numerical Simulation, or DNS, because we're solving numerically the exact equations for the instantaneous fluctuations in the flow. The advantage of DNS is the ability to provide detail that would be difficult to get using other approaches." Following the simulations, DNS data for multiple runs can be analyzed to derive important statistical characteristics of the flow, as well as information that can be used to model complex real-world flows.

TURBULENCE BASICS

HELP ON THE BLUE HORIZON

GOING WITH THE FLOW

REFERENCES

P.K. Yeung. Lagrangian characteristics of turbulence and scalar transport in direct numerical simulations. 2000.J. Fluid Mechanics, (accepted).

TURBULENCE BASICS

Yeung's research is important basic science. Because it focuses on obtaining general results that apply to as wide a range of physical situations as possible, it does not reflect specific geometries like the wings of aircraft or engine combustion chambers. Yet because of the degree of detail available, especially for quantities difficult to measure in experiments, it provides valuable physical insights as well as an underlying knowledge base on which a lot of other science depends. His results are particularly useful for statistical models of mixing and dispersion applied to engineering and environmental problems such as dispersion of pollutants from point sources (Figure 1), and Yeung is involved in several active collaborations with U.S. and international researchers.

"Turbulence modelers require quantitative data," says Yeung. "Their models rely on certain assumptions about turbulent mixing and dispersion, and they need to know whether the approximations they are using are good or not. So one has to be very quantitative. And that's the strength of our direct numerical simulations--they provide both qualitative understanding and quantitative results."

Yeung's numerical simulations are also yielding new discoveries about the nature of turbulence itself. One such result relates to the question of how adjacent fluid elements disperse through the mixing action of turbulence. "If we have two infinitesimal pieces of the fluid, let's call them 'fluid particles,' that are initially close together, they're going to move apart due to the dispersion of the turbulence," explains Yeung. "And in our simulations, we can determine how quickly they do so. What we've found is that at an intermediate time in this process, when most particle pairs are still close together, there are some pairs that have drifted surprisingly far apart." Mathematically, this statistical property is expressed by large positive values of the skewness factor (Figure 2).

The implication of this observation is that, especially for a dangerous pollutant, it is not enough to know the average concentration or even the standard deviation--researchers need the full shape of the probability distribution. And simple assumptions about how far the pollutant will travel do not suffice. Some of it may be found much farther from the source than expected.

HELP ON THE BLUE HORIZON

Yeung and others have previously run simulations on turbulent flow in a periodic cube of fluid divided into 512 points on a side--about 130 million points in total--typically using 64 processors on previous IBM SP machines. "It's really a big step forward that the increased computing power of the Blue Horizon machine will allow us to use a larger number of grid points," Yeung says. "Now we'll be able to get data at a resolution of 1,024 grid points in each direction, which we couldn't do with the previous SP."

Yeung will make runs using 256 to 512 of Blue Horizon's 1,152 processors to compute this finer grid of over 1 billion points, a factor of eight larger than before. This is important because it can resolve a wider range of scales, and thus better simulate the conditions of real-world flows, which have a large range of interacting velocity and length scales. The wider this range of scales--corresponding to faster velocities or larger-scale flows--the higher the Reynolds number. In his new simulations, Yeung expects the Taylor-scale Reynolds number to be up to about 400, whereas previous simulations were limited to about 240.

While this number may not seem particularly high, the Reynolds number for the same flow can vary depending on the choice of length and time scales, and for his simulations, this Taylor-scale Reynolds number is actually higher than in many laboratory experiments. Indeed, it would correspond to a conventional Reynolds number of several hundred thousand in, say, a circular turbulent jet. Consequently, Yeung's simulations on Blue Horizon are yielding results that are of great use in modeling real-world turbulent flows.

Researchers are always limited in the Reynolds number and thus the size or speed of the flows they can simulate. As they build a database of results for increasingly higher cases, this allows more confident extrapolation to higher Reynolds number flows, which have great practical importance.

This research is an early use of Blue Horizon, now in full production at SDSC. "In getting this new machine running, we wanted knowledgeable and experienced users to work with us and the vendor to get everything operating correctly," says Giri Chukkapalli, a member of SDSC's Scientific Computing Department, who works with users to port and optimize their code for Blue Horizon.

"With Yeung's code, like a lot of people beginning to use this larger machine, there are issues of scaling their code to work efficiently with the larger number of processors," Chukkapalli says. "Many existing codes use 1-D decomposition schemes, and these don't always scale well, so we've developed a 2-D decomposition algorithm that makes more efficient use of the larger number of processors in this mixed-mode program model machine." And since the newer large machines seem to be settling on this program model, this code should remain portable in the future.

Figure 2. Unexpected Paths

The evolution in normalized time of the skewness factor (normalized third-order statistical moment) of the separation between fluid particle pairs, with varying initial separations that increase as they progress from yellow to purple (top to bottom). For fluid particles that are initially closest together (yellow curve), the large skewness value when most particle pairs are still close together means that a few pairs will be found quite far apart, suggesting that some of the pollutants in Figure 1 will spread unexpectedly far from the main path of the plume.

GOING WITH THE FLOW

Imagine riding along on that balloon released into the sky as it swirls and mixes on the wind. A strength of Yeung's flow simulations is that they naturally allow the use of this "ride-along" or Lagrangianof reference, which is extremely difficult to do in the laboratory. "Unlike in experiments, in DNS we know the full velocity field in 3-D space, and we have no difficulty keeping track of the identity of the fluid elements," says Yeung.

This is important because in studying the details of the mixing of polluted air with the surroundings, for example, being able to follow the history of flow properties along fluid element trajectories is highly useful. In fact, due to the growing speed and precision of these computer simulations, this kind of study is increasingly taking on some of the traditional role of wind tunnel experiments in providing basic data in the study of turbulence.

Another important benefit of this research is that the accumulating simulation runs are feeding a growing database, which is stored in SDSC's High Performance Storage System archival database. "This is useful because after the fact you can go back and find new things," Yeung says. Sometimes a collaborator will suggest calculating a new quantity for the flow from the simulations in the DNS database, and that can be done even long after the runs have been completed.

With the help of SDSC researchers to take full advantage of the Blue Horizon, this research continues to advance our understanding of the elusive basic science of turbulent flows at the same time that it provides data to improve the models for such practical uses as managing environmental pollution, improving combustion design, and innumerable other situations where turbulent fluid flows play a central role. --PT