Computational Modeling of Complex Flows

In this thesis we consider the following aspects of computational modeling of complex flows: (i) subgrid modeling, (ii) stability, (iii) a posteriori error estimation, and (iv) computational platform. <p />We develop a framework for adaptivity and error control for multiscale problems, in particular for turbulent flow, based on a posteriori error estimates. The a posteriori error estimates take the form of a space-time integral of residuals times dual weights, where discretization residuals relate to numerical errors from discretization, modeling residuals relate to modeling errors from subgrid modeling, and the dual weights govern the propagation of errors in space-time, and is given by solutions of the dual linearized Navier-Stokes equations. We compute approximate such dual solutions for both laminar and turbulent flows. <p />A framework for subgrid modeling based on scale similarity in a Haar Multiresolution analysis is developed. These models are shown to be effective in the case of convection-diffusion-reaction equations with fractal data, simulating turbulent data, and we use the models to estimate modeling residuals in turbulent flow. <p />A computational study of different mechanisms for transition to turbulence in shear flow is presented. <p />We develop the computational platform DOLFIN, an object-oriented library in C++ for finite element computation, where the methods in this thesis are implemented