On one-dimensional surrogacy of probability density function of enstrophy in isotropic turbulence

One-dimensional surrogates of fully three-dimensional (3D) quantities such as fluctuations of dissipation rate and enstrophy in turbulent flows have played an important role in many experimental and computational studies of intermittency. This paper addresses the connections between the probability density functions (PDFs) of the square of the magnitude of a fluctuating vector and that of a single Cartesian coordinate component under the condition of isotropy. A pair of rigorous relations between the PDFs of 3D enstrophy and its one-dimensional (1D) surrogate is derived mathematically. These relations are general, and independent of the functional form of the PDFs, degree of non-Gaussianity or the Reynolds number. The 1D surrogate is substantially more intermittent than its 3D counterpart. The functional forms of the respective PDFs are also examined using numerical simulation data, especially for regimes of very small and very large amplitudes. In the latter case, stretched exponential fits with different algebraic pre-factors but the same exponents at a given Reynolds number are found to agree very well with the data available.