Riemann’s wave and an exact solution of the main turbulence problem

Abstract

ABSTRACT The paper presents a review of the results that allowed us to find an exact analytical solution to the main problem of the turbulence theory consisting in a closed description of any moments and spectra of all random fields that are described by the Euler hydrodynamic equations for a compressible medium. This solution is based on an exact and explicit analytical solution to n-dimensional Euler equations in the limit of large Mach numbers (S. G. Chefranov, 1991). Based on the Dirac delta function theory, this solution gives an n-dimensional generalization of the well-known implicit Riemann (1860) solution to the one-dimensional Euler equations. In the one-dimensional case, the resulting solution exactly coincides with the explicit form of the Riemann solution for an arbitrary Mach numbers. We have obtained for the first time the exact value of the universal scaling exponent -2/3 for a spectrum of the turbulence energy dissipation rate corresponds to the exact analytical solution to fourth-order two-point moments of the velocity field gradient. We have noted a good agreement between this value and the observational data of turbulence intermittency in the surface atmosphere layer (M. Z. Kholmyansky, 1972) and with the findings of the well-known turbulence intermittency model by Novikov-Stewart (1964).