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

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).

KEYWORDS:

• Fluid flow
• Riemann wave
• turbulence
• compressibility
• viscosity

Acknowledgments

The author dedicate this paper to the memory of Valerian Il’ich Tatarskii (October 13, 1929–April 19, 2020), an outstanding man and science researcher. The author sincerely grateful to Valerian Il’ich for his care. I express my kind gratitude to Ya. G. Sinai for a detailed and friendly discussion at his seminar in Moscow on July 9, 2019 and further support of the works presented in this article. The author also grateful for the similar support and interest in the work to G. S. Golitsyn, L. P. Pitaevskii and U. Frisch. I thank E. A. Novikov, L. A. Ostrovsky and I. Procaccia for attention to the work and useful discussions, as well as M. Kholmyansky, V. Yakhot and S.N. Gurbatov for the articles sent and their analysis.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

Data available within the article or its supplementary materials.

Additional information

Funding

The study was supported by the Russian Science Foundation [grant number 14-17-00806P] and by Israel Science Foundation [grant number 492/18].