Fast Calculation of Flow Ensembles

Computing Ensembles occurs frequently in the simulation of complex flows to increase forecasting skill, quantify uncertainty and estimate flow sensitivity. The main issue with ensemble calculation is its high demand of computer resources vs. the limited computer resources existing. Generally computing a large ensemble is prohibitive due to the high computational cost of numerical simulation of nonlinear dynamical systems. Moreover, to compute ensembles of moderate/small size, resolution is very often sacrificed to reduce computation time. In this thesis, we study an efficient ensemble simulation algorithm that can reduce the computing cost significantly making computing a large ensemble or an ensemble of high resolution possible. The motivation for the new algorithm is that for linearly implicit methods, the linear solve is a large contributor to overall complexity and it is far cheaper in both storage and solution time to solve linear systems with the same coefficient matrix than with different coefficient matrices. We present this algorithm with different ensemble time stepping methods. These methods are carefully derived and both theoretically and numerically investigated. Computing an ensemble simultaneously allows each realization to access ensemble data and the use of means and fluctuations in numerical regularizations for each realization. We put forth two ensemble eddy viscosity regularizations that remove severe timestep condition for high Reynolds number flows. The study of the ensemble eddy viscosity regularizations also suggests reconsidering an old but not as well developed definition of the mixing length. This mixing length vanishes at solid walls without van Driest damping, increases stability and improves flow predictions in our preliminary tests. The goal of conventional turbulence models is to produce a model that accurately predicts time averaged or ensemble averaged flow statistics. In this thesis, we develop a new family of ensemble based turbulence models and study their convergence by analyzing the evolution of model variance. For these new turbulence models from the calculated ensemble (at low cost), the kinetic energy in fluctuations can be directly calculated without additional modeling, reducing the computing cost while increasing the physical fidelity of the models.