Numerical Methods in Turbulence

Fluid motion and its richness of detail are described by theNavier-Stokes equations. Most of the numerical analysis existent todate is applicable for strong solutions (typically small body forceand initial data). We prove that statistics of weak solutions areoptimally computable in the simple but important case of small bodyforce and large initial data. These estimates are used to predictdrag and lift statistics, quantities of great interest inengineering. In the case of arbitrarily large body force and initialdata, for shear flows, statistics of the computed solution are shownto behave according to the Kolmogorov theory.Many times, in turbulent fluid flow, a direct numerical simulationbecomes expensive. One alternative is Large Eddy Simulation (LES).It exploits the decoupling of scales, achieved via introduction of afilter, thus reducing the number of degrees of freedom in asimulation. A relatively new family of LES models is the ApproximateDeconvolution Models (ADM). They have remarkable mathematicalproperties and perform well in computations. However, some reportsclaim that they are unstable for simulations with walls and requirethe addition of explicit stabilization.We show that, given the right formulation, variationaldiscretizations of the Zeroth Order Model, a member of the ADMfamily, are indeed stable. We present evidence that stability of oneformulation is sensitive to the exact way in which filtering isperformed and show some numerical results. An alternativeformulation, which does not depend on the way filtering isperformed, is also presented. In both cases we perform convergencestudies. This is a first step in determining stable and robustdiscretizations for the whole family of ADM, as well as guidance fordealing with arbitrary geometries/domains that arise in practicalapplications.Getting a prediction of a turbulent flow right also means gettingthe energy balance and the rotational structures correct, whichmeans (in the large) matching the energy and helicity statistics. Weapply similarity theory to the ADM and show that the model has ahelicity cascade, linked to its energy cascade, which predicts thecorrect helicity statistics up to the cut-off frequency