Mathematical Methods for Turbulent Flow

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The fundamental equations underlying the behaviour of ordinary fluids, due to Navier and Stokes, play an important role in science and engineering. However, when fluids behave chaotically, our ability to solve these equations is limited: the extreme range of interacting space and time scales makes mathematical analysis and direct numerical simulation difficult, even on massively parallel computers. We have recently discovered an exponential version of the classical four-stage Runge-Kutta integrator with stiff order 4, a feat that was previously thought to be impossible! We will generalize this technique to find exponential versions of arbitrary Runge-Kutta integrators and explore whether adaptive exponential integrators pairs, essential tools for turbulent shell models, could be useful in simulations of real fluids. Exponential integrators could also handle the stiff linearity that arises when implementing the pseudospectral method for complex geometries using a penalty method. Convolutions are at the heart of the pseudospectral method for simulating turbulent flow: they can be computed efficiently using the discrete fast Fourier transform (FFT). The cyclic nature of the FFT requires that non-fundamental harmonics, called aliases, be removed. Implicit dealiasing speeds up these computations by a factor of two and in 2D uses 2/3 (in 3D: 4/9) of the memory required by conventional zero padding. A recent parallelized version of implicit dealiasing handles an arbitrary number of input and output vectors, a crucial advance that is opening up many new applications, such as a clever formulation, due to Basdevant, of 2D convection that requires only 4 FFTs per Runge-Kutta stage, instead of the usual 5. We plan to apply implicit dealiasing to dissipationless regularizations of the Euler equations, to signal and image denoising, sparse FFTs, and more general boundary conditions. We also propose to apply it to compute partial FFTs, which arise in seismic imaging and turbulent flux profiles. The computed flux profiles could be used to calculate the damping rates needed to train turbulence subgrid models to remove the correct amount of energy from each retained scale. In view of recent interest in mapping out the attractor for 2D forced-dissipative turbulence under different forcing scenarios, we are attempting to saturate known function analytic constraints on the dynamics. The goal is to learn about invariant measures for turbulent flows. We would also like to use optimal transport theory to develop a simulation technique for 2D turbulence that respects rearrangement (Casimir) invariants. Turbulence can be visualized with the state-of-the-art 2D and 3D vector graphics language Asymptote. A proposed WebGL output format would bring the power of Asymptote to tablets and smart phones. We also propose a portable compressed binary format (v3d) for 3D vector graphics that supports vertex shading and a technique for implementing order-independent transparency.