Towards a self-consistent Boltzmann's kinetic model of fluid turbulence

Abstract

A closure for the effective relaxation time of the Boltzmann–BGK kinetic equation for fluid turbulence is presented, based on a double-averaging procedure over both kinetic and turbulent fluctuations. The resulting effective relaxation time appears to agree with values obtained via a renormalisation group treatment of the Navier–Stokes equation only at low values of k/T, the ratio of turbulent kinetic energy to fluid temperature. For $k/T>0.1,$ the kinetic treatment delivers a significantly longer effective relaxation time.

Keyword:

• Kinetic theory of turbulence

Acknowledgments

The author owes a huge debt of knowledge to Hudong Chen, Viktor Yakhot and to the late SA Orszag, a wonderful master and a much-missed friend. This paper was prepared on the occasion of the Simons Symposium ‘Universality: Turbulence across vast scales’. The author is grateful to the Simons Foundation for financial support and great hospitality and to David Spergel and Phil Mocz for very stimulating discussions. This research has also received funding from the European Research Council, under the European Union's Horizon 2020 Framework Programme (No. FP/2014- 2020)/ERC Grant Agreement No. 739964 (COPMAT).

Disclosure statement

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

Additional information

Funding

The author is grateful to the Simons Foundation for financial support and great hospitality and to David Spergel and Phil Mocz for very stimulating discussions. This research has also received funding from the European Research Council, under the European Union's Horizon 2020 Framework Programme (No. FP/2014-2020)/ERC Grant Agreement No. 739964 (COPMAT).