Predictive Turbulence Modeling by Variational Closure

We show that a variational implementation of probability density function (PDF) closures has the potential to make predictions of general turbulence mean statistics for which a priori knowledge of the incorrectness is possible. This possibility exists because of realizability conditions on effective potential functions for general turbulence statistics. These potentials measure the cost for fluctuations to occur away from the ensemble-mean value in empirical time-averages of the given variable, and their existence is a consequence of a refined ergodic hypothesis for the governing dynamical system (Navier–Stokes dynamics). Approximations of the effective potentials can be calculated within PDF closures by an efficient Rayleigh–Ritz algorithm. The failure of realizability within a closure for the approximate potential of any chosen statistic implies a priori that the closure prediction for that statistic is not converged. The systematic use of these novel realizability conditions within PDF closures is shown in a simple 3-mode system of Lorenz to result in a statistically improved predictive ability. In certain cases the variational method allows an a priori optimum choice of free parameters in the closure to be made.