Constructing Turbulent Field Statistics with an Ensemble of Multifractal Gaussian Fields

Obtaining field statistics for fluid turbulence remains an outstanding challenge in turbulence theory and modeling. Multifractal models for velocity gradient and increment distributions successfully capture intermittency, but currently do not give easy access to more general multi-point statistics—i.e. field statistics. Here, we present a method 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. Careful construction of the correlation function and the corresponding weight of each sub-ensemble allows us to define functionals with multifractal two-point inertial statistics, gradient statistics (dissipation range), as well as a large-scale cut-off. Additionally, the method is capable of producing multifractal statistics with any of the widely used singularity spectra (log-normal, She–Lévêque, etc.).