On the formulation of turbulence field statistics with lognormal multifractal velocity increments

Obtaining field statistics for fluid turbulence remains an outstanding challenge in turbulence theory and modeling despite the availability of the linear, closed Hopf equation for the characteristic functional. Multifractal models, e.g. for velocity increment distributions, are successful in capturing intermittency, but they do not contain the full statistical information of turbulent velocity fields and thus do not enjoy closed equations that can be derived from first principles. Here, we present a method that generates field statistics, in the form of a characteristic functional, that reduces to the two-point lognormal multifractal statistical model in the inertial range of scales. The functional is constructed as a superposition of Gaussian characteristic functionals each defined by a scaling exponent. A parameter transformation recovers a length scale dependence for the resulting velocity increment distributions that is consistent with multifractal theory. Applications and statistics derived from the functional are discussed, and comparisons are drawn with related approaches.