A Spectrally Reduced Dynamic Turbulence Subgrid Model

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The understanding of fully developed turbulence remains an elusive challenge to researchers in science and engineering. Although the Navier-Stokes equation provides an accurate description of typical fluids, when it becomes chaotic at high speeds, our understanding is rudimentary. Indeed, substantial progress towards understanding this equation and its solutions constitutes one of the seven millennium challenges proposed by the Clay Mathematics Institute. Even approximate solutions of the problem defy modern computers, which lack the storage and speed necessary to accurately simulate turbulence in the Earth's atmosphere or around a moving body. Subgrid models are often used to allow a simulation on today's computer to mimic computations that will very likely not be directly possible even on the computers of the next century. However, most of the subgrid models used in practice are ad hoc devices that remove energy from the system at the smallest scales retained in the simulation. However, for a subgrid model to be faithful to the underlying physics, it must also inject energy at other scales, to describe the beating of small-scale waves. This important effect is neglected by many subgrid models in use today. The goal of this project is to develop reliable subgrid models for the next generation of turbulence simulations, based on a technique known as spectral reduction, which allows one to accurately estimate velocities at scales that are not directly retained in the simulation. Recently, a practical computational technique for applying spectral reduction to a one-dimensional analogue of the Navier-Stokes equation (with which it shares many properties) was developed in a master's thesis. The method is an attempt to numerically approximate the solution using several grids of variable resolution. The communication of data between the grids is designed to conserve energy. The generalization of the method to the full Navier-Stokes equation would lead to a dynamical subgrid model for high-Reynolds number turbulence that has a physical basis and is free of arbitrary parametrization. Such a subgrid model would greatly reduce the computational cost of simulations of turbulent drag and transport.