PREDICTIVE TURBULENCE MODELING BY VARIATIONAL CLOSURE

We show that a variational implementation of probability density funct ion (PDF) closures has the potential to make predictions of general tu rbulence mean statistics for which a priori knowledge of the incorrect ness is possible. This possibility exists because of realizability con ditions on ''effective potential'' functions for general turbulence st atistics. These potentials measure the cost For fluctuations to occur away from the ensemble-mean value in empirical time-averages of the gi ven variable, and their existence is a consequence of a refined ergodi c hypothesis for the governing dynamical system (Navier-Stokes dynamic s). Approximations of the effective potentials can be calculated withi n PDF closures by an efficient Rayleigh-Ritz algorithm. The failure of realizability within a closure for the approximate potential of any c hosen statistic implies a priori that the closure prediction for that statistic is not converged. The systematic use of these novel realizab ility conditions within PDF closures is shown in a simple 3-mode syste m of Lorenz to result in a statistically improved predictive ability. In certain cases the variational method allows an a priori optimum cho ice of free parameters in the closure to be made.