12th LESLIE FOX PRIZE Dundee, 27th June 2005 WEAK UNIQUENESS OF THE NAVIER-STOKES EQUATIONS AND ADAPTIVE TURBULENCE SIMULATION

Abstract. We consider the problem of computational simulation of tur-bulence, where we study turbulent solutions to the incompressible Navier-Stokes equations. We construct approximate weak solutions using a stabilized Galerkin finite element method, here referred to as General Galerkin G2, for which we investigate uniqueness in output (or weak uniqueness) by solving an associated dual problem computationally, with data coupling to the particular output we are interested in. For simulation of turbulent flow we refer to the adaptive version of G2 as Adaptive DNS/LES, with part of the flow being re-solved in a Direct Numerical Simulation DNS, and part of the flow being left unresolved in a Large Eddy Simulation LES, with the stabilization in G2 acting as a dissipative subgrid model. We present computational results using Adap-tive DNS/LES, where we find that for the problem of simulating the turbulent flow past various bluff bodies we are able to compute mean value output, such as drag, using ∼10-100 times less degrees of freedom than in typical LES com-putations using ad hoc mesh refinement. We further use Adaptive DNS/LES to simulate the turbulent flow past a cylinder rolling along ground