Numerical methods in turbulence

Fluid motion and its richness of detail are described by the Navier-Stokes equations. Most of the numerical analysis existent to date is applicable for strong solutions (typically small body force and initial data). We prove that statistics of weak solutions are optimally computable in the simple but important case of small body force and large initial data. These estimates are used to predict drag and lift statistics, quantities of great interest in engineering. In the case of arbitrarily large body force and initial data, for shear flows, statistics of the computed solution are shown to behave according to the Kolmogorov theory.; Many times, in turbulent fluid flow, a direct numerical simulation becomes expensive. One alternative is Large Eddy Simulation (LES). It exploits the decoupling of scales, achieved via introduction of a filter, thus reducing the number of degrees of freedom in a simulation. A relatively new family of LES models is the Approximate Deconvolution Models (ADM). They have remarkable mathematical properties and perform well in computations. However, some reports claim that they are unstable for simulations with walls and require the addition of explicit stabilization.; We show that, given the right formulation, variational discretizations of the Zeroth Order Model, a member of the ADM family, are indeed stable. We present evidence that stability of one formulation is sensitive to the exact way in which filtering is performed and show some numerical results. An alternative formulation, which does not depend on the way filtering is performed, is also presented. In both cases we perform convergence studies. This is a first step in determining stable and robust discretizations for the whole family of ADM, as well as guidance for dealing with arbitrary geometries/domains that arise in practical applications.; Getting a prediction of a turbulent flow right also means getting the energy balance and the rotational structures correct, which means (in the large) matching the energy and helicity statistics. We apply similarity theory to the ADM and show that the model has a helicity cascade, linked to its energy cascade, which predicts the correct helicity statistics up to the cut-off frequency.