An ensemble of Gaussian fields with multifractal statistics for turbulence

Abstract: Obtaining accurate field statistics continues to be one of the major challenges in turbulence theory and modeling. From the various existing modeling approaches, multifractal models have been successful in capturing intermittency in velocity gradient and increment distributions. On the other hand, superstatistical models from nonequilibrium statistical mechanics have shown the capacity to model PDFs of various statistical turbulent quantities as ensembles of simpler stochastic processes. Here, we present an approach that generates field statistics in the form of a characteristic functional by promoting a model for multifractal increment statistics to an ensemble of Gaussian fields. By carefully designing the correlation function and the corresponding weight of each subensemble, we are able to define a functional that exhibits multifractal two-point inertial-range and dissipation-range statistics, and that blends into realistic large-scale behavior. Additionally, the method is capable of producing multifractal statistics with any of the widely used singularity spectra. We characterize the fidelity of our approach through comparisons to literature results from direct numerical simulations. Overall, our framework thereby bridges between three different perspectives to turbulence: superstatistics, multifractals, and functional approaches to turbulence.