Reduced Dynamical Models of Turbulent Flow

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Turbulence affects us on a daily basis: through weather patterns, on airplanes and ships, and in pipelines. However, the theoretical understanding of turbulence remains an elusive but important scientific challenge. While the Navier-Stokes equations provide an accurate description of ordinary fluids, our ability to solve these equations for fluids that behave chaotically is very limited. The problem of predicting energy, transport, and drag in highly turbulent flows defies even the incredible speed and huge memory of today's massively parallel computers.A recent advance, called implicit dealiasing, significantly reduces memory usage and computation time compared with conventional simulation techniques. We propose to implement implicit dealiasing on massively parallel computers so that computation can be performed simultaneously with communication. Our adaptive algorithm is optimized for distributed networks of multicore processors. To benchmark our algorithms, Professor Wendell Horton at The University of Texas has generously given our group access to the sixth fastest supercomputer in the world, the Dell Zeus C8220 cluster (Stampede) at the Texas Advanced Computer Center. This project will give Canadian students opportunities to work with state-of-the-art supercomputers that are far more powerful than anything currently available in our country.Even with implicit dealiasing, however, the numerical simulation of fully developed turbulence is still beyond reach, because of the huge range of space and time scales. For this reason, a subgrid model is often adopted to approximate the effects of unresolved small scales on the larger retained scales. A reliable method for determining the effective eddy viscosity due to deleted subgrid modes constitutes a major unsolved problem for numerical simulations of turbulence, where one only evolves large scale modes. Reduced models allow a simulation on today's computers to mimic computations that will likely not be directly possible even on the computers of the next century.The recent development of a method for predicting the statistical properties of two-dimensional homogeneous fluid turbulence, called pseudospectral reduction, provides us with a new tool for achieving this goal. The statistics predicted by this reduced model agree remarkably well with large-scale numerical simulation data, even in flows containing long-lived vortices.The Russian mathematician Kolmogorov conjectured that turbulent energy transfer is independent of scale, due to a presumed underlying self-similarity in the energy transfer. We propose to compute this flux directly, building on the recently discovered fast partial Fourier transform, to verify the degree to which this self-similarity actually holds. The ultimate goal is to use this information to join pseudospectral reduction with a large-scale simulation, yielding a dynamic subgrid model that removes exactly the right amount of energy from each of the scales near the subgrid scale cutoff. This would avoid the bottleneck and overdamping effects that commonly plague subgrid models used in modern large eddy simulations.By greatly increasing the efficiency of numerical simulations of turbulence, this research will lead to an improved ability to understand, predict, and possibly control, turbulence from a mathematical perspective. In addition, the computational tools developed in this work open up new interdisciplinary collaboration opportunities for Canadian researchers and industry. For example, implicit dealiasing would be beneficial to many disciplines where convolutions are used, including data mining, image processing, and signal processing.