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Abstract
The accurate simulation of turbulent flows is a central computational challenge in many important applications, including global climate change estimation, environmental science, ocean and atmosphere dynamics, energy efficiency improvement and optimization of industrial processes. As an example, turbulence predictions are key to limiting damage of hurricanes (estimated to be hundreds of billions of dollars in 2017). These are fundamentally non-linear problems that probed in this thesis through numerical computations and supporting mathematical analysis. The accuracy of turbulence models depends on their turbulent dissipation. The dissipation is studied and it is utilized to validate results with the Statistical Equilibrium Law as the benchmark: 1. In Chapter 3, the energy dissipation in a turbulence model discretized on an underresolved mesh is delineated. This is the first connection between computational experience and mathematical analysis in this direction [63]. 2. It is rigorously proved in Chapter 4 that the over-dissipation (wrong accuracy) of a turbulence model can be corrected using van Driest damping [62]. This had been an open question (e.g. [5] p. 78) since 1963. 3. The temperature in natural convection is uniformly bounded in time. Although the problem has been studied for a long time, no better bounds for the approximate temperature than an exponential growth in time were obtained e.g. [78, 79, 87]. In Chapter 5, it is proved that the temperature approximation is bounded sub-linearly in time by introducing a new interpolation [19].
