A new approach to computational turbulence modeling

We present a new approach to Computational Fluid Dynamics CFD using adaptive sta-bilized Galerkin finite element methods with duality based a posteriori error control for chosen output quantities of interest. We address the basic question of computability in CFD: For a given flow, what quantity is computable to what tolerance to what cost? We fo-cus on incompressible Newtonian flow with medium to large Reynolds numbers involving both laminar and turbulent flow features. We estimate a posteriori the output of the com-puted solution with the output based on the exact solution to the Navier-Stokes equations, thus circumventing introducing and modeling Reynolds stresses in averaged Navier-Stokes equations. Our basic tool is a representation formula for the error in the quantity of inter-est in terms of a space-time integral of the residual of a computed solution multiplied by weights related to derivatives of the solution of an associated dual problem with data con-nected to the output. We use the error representation formula to derive an a posterori error estimate combining residuals with computed dual weights, which is used for mesh adap-tivity in space-time with the objective of satisfying a given error tolerance with minima