Development of a hierarchical scale-adaptive large-eddy simulation method for the study of turbulence

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Studies indicate that atmospheric turbulence is responsible for substantial financial losses in our society. For instance, US aviation pays over $10 million per year to compensate for turbulence related rerouting of flights. In 2018, severe weather caused losses of over $1.9 billion in Canada. On the other hand, wind is a major source of renewable energy. Canada's wind energy capacity surpassed 12,000 megawatts in 2018. Canada has been a world leader in mathematics, fluid mechanics, and numerical weather prediction. Mathematical discoveries reveal how turbulence creates the drag on cars, airplanes, wind turbines, bridges, and buildings; and how it dictates the weather through its influence on large- scale atmosphere and oceans. My research program is dedicated to describing, mathematically, how the chaotic nature of turbulence gets organized into vortices across all scales. In particular, I will use the mathematical techniques that I develop to build a fully deterministic turbulence model. This innovative turbulence model will be applied to investigate common open problems in the physics of atmosphere, wind engineering, and aerospace. The research program will impact both computational sciences (adaptive numerical methods for nonlinear partial differential equations) and physics of fluid mechanics (turbulence modeling). In principle, existing turbulence models are stochastic and not designed to detect rapidly intensified vortices. This means that computational resources are being used inefficiently. The first objective extends our wavelet method, where the vortex stretching mechanism replaces the stochastic turbulence model. We can then derive effective equations from first principles, which focus only on the locally significant influence of turbulence, without needing to describe every little detail of the complex surroundings. The second and the third objective teach the computer model what aspects of the physics is absent in the current data which could intensify a vortex.    The proposal has two main strategies: 1) solve the effective equations of turbulence by combining the scale-adaptivity of wavelets with the vortex dynamics representation of turbulence; and 2) generate a database of simulated turbulent flows, which directly capture only the influence of atmospheric turbulence, and then use such data to learn how to design better buildings, wind farms, cars, airplanes, ships, etc.  This research program will train 2 PhD, 3 MSc, and 5 BSc students at the intersection between mathematical theory and engineering applications of the physics of fluids. They will learn mathematical innovations for creating new techniques and practical applications. The database and computer models we create and the knowledge gained in this research project will keep Canada at the forefront to deal with emerging challenges. This proposal contributes to an international effort to develop the next generation of artificially intelligent computational models.