Numerical turbulence algorithms and code development




		Three-dimensional rendering of regions of strong vorticity in a 2,048³ hydrodynamic simulation. From left to right and top to bottom, successive zooms into the structures are shown. Note the small-scale vortex filaments and the development of clusters of vortex tubes at intermediate scales. As a reference in the last zoom (right), velocity field lines are shown in red. These figures show the complex flow structures that develop at very high Reynolds number (R_λ ∼1,300), and they provide an excellent test case for the study of turbulence as it appears in nature, the intricate dynamics of which lead to complex behavior and modified transport properties.

The Turbulence Numerics Team (TNT) staff includes Aimé Fournier (Project Scientist), Jonathan Graham (Graduate Student in Applied Mathematics, CU, supported by an NSF-CMG grant), Ed Lee (Graduate Student in Applied Mathematics, Columbia University), Pablo Mininni (Senior Postdoc, supported by an NSF-CMG grant), Annick Pouquet (Senior Scientist and Section Head), Duane Rosenberg (Software Engineer). Details about the general report below appear in the CISL Research Catalog organized by the last names of the research team members.

Turbulence science

We have pursued investigations of homogeneous and isotropic turbulence and turbulent structures at high Reynolds numbers, incorporating a broad variety of phenomena. For neutral fluids, we have run high Reynolds number (R_λ ∼800) simulations in an effort to study energy transfer between scales. We find that 20% of the energy flux in the small scales is due to interactions with the large-scale flow. A recent high-resolution (2,048³) hydrodynamic allowed for a more refined analysis at still higher Reynolds number (R_λ ∼1,300) showing that convergence to the asymptotic turbulence regime appears to be slow; that even though the nonlocal interactions diminish with Reynolds number, they are measureable with the highest resolution data we can currently afford.

This year has seen significant progress on applying and understanding regularizations as subgrid scale (SGS) models in the study of the Navier-Stokes equations. We find that certain regularizations may be useful as a model of unresolved scales, as they are able to reproduce the Navier-Stokes energy spectrum, and may offer a significant, though suboptimal (independent of Reynolds number) computational savings. Other regularizations are found to reproduce intermittency properties of these flows.

Using our recently developed spectral method for studying neutral and conducting incompressible fluids inside a rigid, "hard" spherical boundary, we have added the ability to study a richer set of physical effects that can more faithfully treat aspects of atmospheric flows and planetary and solar dynamos.

For conducting magnetohydrodynamic (MHD) flows, we have extended our previous results to show that the nonlocal behavior of the energy transfer in MHD is the result of a correlation between the velocity and magnetic fields. These MHD studies raise a number of interesting questions concerning the role of magnetic helicity, possible implications for dynamos, and fundamental aspects of universality in MHD.

In an important new development, we have carried out late-time analysis of a breakthrough MHD simulation at the unprecendented resolution of 1536³, and find that the energy spectrum is composed of two contributions, each moderately resolved. The appearance of this spectrum with two small-scale inertial ranges has never been observed before, and it corresponds to the interplay between Alfvén waves and turbulent eddies. It may be interpreted as a partial breakdown of universality in MHD.

Finally, we continue to investigate the ability of adaptive mesh refinement (AMR; see also below) to accurately model turbulent phenomena, partly to determine how well AMR can accelerate long time integration simulations, and also to establish a formal direct link between the AMR and the multi-resolution analysis of the flow and its structures.

Numerical turbulence algorithms and code development




		Contour plot of current density in an adaptive simulation of the Orszag-Tang vortex configuration. Red is high current, blue is low current, and the equivalent resolution is 512². Note the strong, thin current sheet at the center. This simluation demonstrates the ability of an explicit adaptive MHD spectral element solver to accurately capture small-scale structure in a flow that exhibits complex nonlinear behavior including intermittent magnetic reconnection.

The IMAGe Turbulence Numerics Team (TNT) develops both tools and models that enhance our capability to investigate geophysical turbulence, and it applies these capabilities to fundamental scientific objectives. This program complements the Geophysical Turbulence Program and focuses on the accurate simulation and understanding of fluid turbulence, as found in the atmosphere and for charged flows in the presence of magnetic fields. TNT research emphasizes simplified physical systems that still reproduce the complexity and multi-scale properties associated with turbulent flows but that allow for the highest possible Reynolds number. The code development and applications pertain directly to the NCAR strategic priorities of "Conducting computer science, computational science, applied mathematics, statistics, and numerical methods R&D" and "Developing and providing advanced services and tools."

TNT members have broad experience in developing a variety of algorithms for studying turbulence. Our highly scalable codes include a 2D and 3D pseudo-spectral hydro- and magnetohydrodynamics (MHD) code that may include a Hall current. These codes are proven to scale up to several thousand processors. These codes have also been modified to include a Lagrangian-averaged ("alpha") model that smooths the velocity locally and has proved useful for studies using very high Reynolds numbers. We also have a new (strictly) spectral method to solve the equations of hydrodynamics and MHD in a sphere, with which studies can be made of fluid turbulence (with rotation) at moderate Reynolds numbers with a variety of boundary conditions.

We have significantly modified the pseudo-spectral codes to make the addition of new physics easier, and to simplify the maintenance of the code as the number of users increases. Support for FFTW 3.x and new options for POSIX and MPI I/O were added. Now the pseudospectral code has a dynamical core, and the different solvers are picked at compile time by the user. A solver for rotating flows has already been written and tested, and simulations up to 5,123 grid points and Rossby numbers down to 0.05 were done. We are currently analyzing these results. Options for a solver for stratified flows are already in place, and our next task is to implement and test such a solver.

TNT continues work its high-order adaptive mesh refinement code, GASpAR. This year's activities have focused on developing an explicit incompressible MHD solver to extend the existing Burgers and Navier-Stokes solver capabilities. This new solver has been applied to a challenging turbulent MHD problem in the literature, and it was found to produce excellent agreement with established results. We also continue work on preconditioning in collaboration with IMAGe/CMG and have obtained preliminary results from an optimized additive Schwarz preconditioner that, in its first stage, will be applied to our iterative Krylov methods used with conforming elements. We have made a strategic decision to postpone adding modeling (e.g., so-called 'alpha' models) this year, and have instead made strides toward a full three-dimensional solver capability that we will pursue as well in FY2008.

We continue to develop new aspects of exploration and applicability for spectral elements, and to apply our extpertise in the area of spectral elements and numerical methods to areas of investigation that are of traditional interest to NCAR.

Finally, in collaboration with our visitor Marc Brachet, we have begun to develop and test a newly developed parallel version of a symmetric FFT algorithm for use in pseudo-spectral studies of problems that have the same symmetries as the so-called Taylor-Green vortices. We will continue our work on this project in FY2008 because it may enable us to achieve resolutions up to four times that of a standard pseudo-spectral code, and hence, to probe significantly higher Reynolds number flows.

TNT research is supported by NSF Core funding and partially by NSF grant CMG-0327888.