*RECRUITMENT TALK* We present a new approach to CFD and Computational Turbulence Modeling using adaptive stabilized Galerkin finite element methods with duality based a posteriori error control for chosen output quantities of interest, 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. We refer to our methodology as Adaptive DNS/LES, where automatically by adaptivity certain features of the flow are resolved in a Direct Numerical Simulation DNS, while certain other small scale turbulent features are left unresolved in a Large Eddy Simulation LES. The stabilization of the Galerkin method giving a weighted least square control of the residual acts as the subgrid model in the LES. The a posteriori error estimate takes into account both the error from discretization and the error from the subgrid model. A crucial observation from computational examples is that the contribution from subgrid modeling in the a posteriori error estimation can be small, making it possible to simulate aspects of turbulent flow without accurate modeling of Reynolds stresses. Using the a posteriori error estimates we further consider the question of uniqueness of weak solutions to the Navier-Stokes equations, where we give computational evidence of both uniqueness and non-uniqueness in outputs of weak solutions.