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TURBULENCE CHALLENGES
SHOCKLET SCIENCE
COMPUTING COMPRESSIBLE TURBULENCE
 hockwaves like the loud "boom" following an explosion or the sonic boom of a supersonic aircraft are thin regions in a fluid across which pressure and density rise rapidly. When a shockwave abruptly accelerates an interface separating two different gases, this interface becomes unstable, so that tiny disturbances grow over time, ultimately leading to compressible turbulent mixing at the interface. Caltech researcher Ravi Samtaney and colleagues are using powerful supercomputers to model this kind of compressible turbulence more realistically than ever before, providing tools to study the details of phenomena from explosions to supernovae.
This research is part of the Department of Energy (DOE) sponsored Academic Strategic Alliances Program in the Accelerated Strategic Computing Initiative (ASCI), begun to conduct increasingly realistic testing in the virtual world that cannot be done in the real world. The basic studies that Samtaney, a senior research associate at Caltech, and colleagues Professor Dale Pullin and researcher Branko Kosovic of Caltech are conducting can be applied to modeling a range of applications. For example, in order for astronomers to better understand nebulae and supernovae (Figure 1), they need greater insight into shockwave dynamics. Samtaney's studies can also be of use in the fusion research known as Inertial Confinement Fusion (ICF), one possible approach to meeting the world's growing energy needs, as well as in supersonic combustion and other areas.
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TURBULENCE CHALLENGES
In studying turbulent flows, all important, real-world cases are beyond the reach of exact solutions, even with today's most powerful computers. This is due to the vast range of scales in turbulence-from near the overall size of the flow down to the very short scales where viscous forces dominate. Moreover, there are vigorous and nonlinear interactions across this wide range of scales. Thus, although researchers are principally interested in predicting the overall flow dynamics, they cannot ignore the influence of the smaller scales.
To overcome the difficulty of resolving the wide range of scales in turbulent motions, especially in real-world applications, techniques such as Large-Eddy Simulations (LES) are employed that include the larger-scale eddies but omit the smaller scales, which "fall through the cracks" of the simulation grids. "The challenge we face is to develop valid subgrid models that will capture the dynamics of the flow that are smaller than the resolution of the grid we can compute with, yet still reliably predict how these small-scale dynamics will affect the larger scale flow," said Samtaney.
The researchers have developed a model of the subgrid scale dynamics that they would like to apply to LES studies of compressible turbulence, including the rapidly growing Richtmyer-Meshkov instabilities on shockwave-accelerated interfaces that play an important role in explosive flows. However, they had to confirm that their model would work in decaying isotropic turbulence, a difficult case for LES.
To test their subgrid model, Samtaney and colleagues used "experimental data" obtained from highly realistic Direct Numerical Simulations (DNS) run on supercomputers, including NPACI's Blue Horizon. Because these simulations solve the full Navier-Stokes equations exactly, without approximations, and because of the wealth of information they provide, such simulations are increasingly used to gain insight into turbulent flows, and to test models such as the researchers' subgrid turbulence model.
As a first step, Samtaney compiled a database describing decaying isotropic turbulent flow within a cubic region using high-resolution DNS runs with a mesh of 256 points on a side, dividing the region into about 16 million cells and giving a detailed picture of the turbulence. Such high-resolution simulations are very computationally intensive, requiring long runs, even on large supercomputers like Blue Horizon.
Once the turbulence database had been compiled, the next step was to test their subgrid model to see whether it could reliably simulate turbulence in more complex cases. "To do this, we put our subgrid turbulence model into the equations for the turbulent flow, while running simulations at lower resolution, a mesh of only 32 on a side," Samtaney said. This divides the flow region into only about 32,000 cells, about 500 times less than the full-size simulations. Because this coarse resolution cannot directly resolve the smaller eddies, they must be accounted for by the subgrid model, hence testing whether it can predict the results from the exact, higher-resolution simulations.
Samtaney found that the subgrid model satisfactorily reproduced the overall, first-order results of the high-resolution DNS simulations in the database. This has given the researchers confidence that their subgrid model can be used in the larger LES codes for simulating real-world flows, including Richtmyer-Meshkov instabilities, in the ASCI Virtual Test Facility or astrophysics simulations.
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SHOCKLET SCIENCE
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Figure 2. Shocklets in Turbulent Compressible Flow
Volume rendering of velocity divergence in Direct Numerical Simulations of isotropic decaying turbulence. The long, stringy structures indicate shocklets or regions of large negative divergence, marking zones of strong compression. Note the similarity to nebula images (Figure 1). Such shockwaves are near-discontinuities in the flow where the flow decelerates from supersonic to subsonic across the shock front in aof reference attached to the shock. The presence of shocklets is challenging for computational models of compressible turbulent flows.
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"Our initial motivation was to develop a subgrid model that would enable us to conduct larger simulations, but when we did some simulations we saw that the velocity divergence field had the broadband spectra characteristic of shock discontinuities, that there were weak shocks in the flow-what we call shocklets," said Samtaney. The researchers also noticed that in astrophysical cases (Figure 1) there are structures that looked strikingly similar to the shocklet structures present in their simulations (Figure 2).
Having found shocklets in his exact DNS simulations, Samtaney needed to be sure that his subgrid model would still work in the presence of these shocklets. So the researchers started looking into the dynamics of the shocklets, distilling the research into the following question: "Given a box of turbulence at a given Taylor Reynolds number and turbulent Mach number, can we estimate the shocklet distribution?"
In their large DNS simulations of isotropic turbulence, the researchers were able to compute the statistical characteristics of the flow, including the shocklets. And while shocklets are present in the flow, it turns out that these structures are not very dynamically significant. Samtaney explains that they can be thought of as passive shocks that are being sloshed around by the flow-these shockwaves can compress the gas, but because they are relatively weak they have little effect on the overall turbulent flow.
"However, from a computational point of view, their existence is a challenge because they are near-discontinuities in the flow, and because we are using a nearly spectrally accurate method, you start to get ringing," said Samtaney. Because this numerical error can contaminate the results, the researchers had to ensure that this did not happen.
An interesting result of the DNS simulations is that "we were able to achieve quite a lot in terms of quantifying shocklets with a probability density function (PDF). In addition to being able to demonstrate that our subgrid model is valid even with shocklets, the essential result of this research is that we were able to get a model for the PDF of the shocklet strength, the statistics of shocklet occurrence, as a function of the Mach number or the pressure ratio across the shock, and neither of these has been done before," said Samtaney. The researchers also developed an algorithm to extract shocklets from the DNS simulations, finding stringy, "ribbon-like" surfaces (Figure 3).
"A question people often ask is what is the mechanism of generation of the shocklets," said Samtaney. One plausible mechanism involves two vortices that are counter-rotating so that the fluid flowing between them is accelerated, and when accelerated supersonically it can decelerate only through a shock, so the shocklets might be thought of as sitting in the throat of two vortices," said Samtaney. While this is one possible mechanism consistent with the elongated structure of shocklets, this area has yet to be explored in detail.
"Our overall goal was to see if our subgrid turbulence model would work in the presence of shocks, at least weak shocks, and we're pleased that it performs reasonably well," Samtaney said. Now the team is shifting its attention to using the subgrid model in large-eddy simulations of larger, real-world cases.
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COMPUTING COMPRESSIBLE TURBULENCE
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Figure 3. Shocklet Extraction Algorithm Results
Results of the algorithm developed to extract the presence of shocklets or weak shockwaves from the high-resolution Direct Numerical Simulations of turbulence. Note the similarity in shape to the long, stringy structures in Figure 2.
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In building their database of isotropic turbulence, the researchers carried out DNS simulations in a cubic domain. This approach resolves the full Navier-Stokes equations with no models or approximations using a 10th-order accurate finite-difference code. Samtaney explains that it is a compact finite difference code that can be thought of as being spatially implicit, that is, each step forward in time involves the velocities and other quantities over the whole space.
The researchers used a high resolution 256 cubed grid, dividing the region into some 32 million cells. Running on 128 processors of Blue Horizon, the runs typically took seven to 10 days, some 50 to 80 hours. "We had to run lots of diagnostics and that took a lot of time. We computed the DNS and the statistics on the fly, and stored the entire solution every 1 to 1.5 large eddy turnover time, saving the results in the HPSS here at caltech and at SDSC as well," Samtaney said.
A further benefit is that the DNS database of isotropic turbulence is a resource that can be used by other researchers to investigate various aspects of turbulence including verifying LES models. Because of the usefulness of the database and the large amount of work and computer time required to compile it, the researchers are making this database of turbulence solutions available so that others will not need to repeat this work. —PT 
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www.galcit.caltech.edu/~ravi/compturb.html
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Project Leader
Dan Meiron
Caltech
Participants
Ravi Samtaney,
Dale Pullin,
Branko Kosovic
Caltech
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