Model of stretching vortex filaments and foundations of the statistical theory of turbulence

Although the statistical properties of small-scale velocity perturbations in homogeneous and isotropic hydrodynamic turbulence have been thoroughly studied experimentally and numerically, no definite theoretical explanation is available yet. The concept of breaking vortices, commonly accepted as the primary turbulent mechanism, not only fails to account for a number of facts but also is self-contradictory. In this review, we discuss an alternative concept according to which the stretching of vortices rather than their decay is the determining process. The evolution of stretching vortex filaments and their properties are derived directly from the Navier–Stokes equation. The model of stretching vortex filaments explains the power-law behavior of velocity structure functions and the intermittency of their exponents, thus imparting physical meaning to multifractal theory, which is based on dimensional considerations. The model of vortex filaments is the only theory that explains the observed differences between the scaling exponents of longitudinal and transverse structure functions.