Turbulence in Large-Scale Two-Dimensional Balanced Hard Sphere Gas Flow

Rafail V Abramov

doi:10.3390/atmos12111520

Abstract

In recent works, we developed a model of balanced gas flow, where the momentum equation possesses an additional mean field forcing term, which originates from the hard sphere interaction potential between the gas particles. We demonstrated that, in our model, a turbulent gas flow with a Kolmogorov kinetic energy spectrum develops from an otherwise laminar initial jet. In the current work, we investigate the possibility of a similar turbulent flow developing in a large-scale two-dimensional setting, where a strong external acceleration compresses the gas into a relatively thin slab along the third dimension. The main motivation behind the current work is the following. According to observations, horizontal turbulent motions in the Earth atmosphere manifest in a wide range of spatial scales, from hundreds of meters to thousands of kilometers. However, the air density rapidly decays with altitude, roughly by an order of magnitude each 15–20 km. This naturally raises the question as to whether or not there exists a dynamical mechanism which can produce large-scale turbulence within a purely two-dimensional gas flow. To our surprise, we discover that our model indeed produces turbulent flows and the corresponding Kolmogorov energy spectra in such a two-dimensional setting.

Highlights

In his famous work, Reynolds [1] demonstrated that an initially laminar flow of a liquid consistently develops turbulent motions whenever the high Reynolds number condition is satisfied
We examined the Fourier spectrum of the kinetic energy of the simulated flow, and found that its time average decayed with the rate of inverse five-third power of the wavenumber, which corresponded to the famous Kolmogorov spectrum
We investigate the ability of our model of a balanced compressible hard sphere gas flow [9] to produce turbulent motions from a laminar initial condition in a purely two-dimensional setting, which corresponds to the large-scale dynamics of the

Summary

Introduction

Reynolds [1] demonstrated that an initially laminar flow of a liquid consistently develops turbulent motions whenever the high Reynolds number condition is satisfied. Kolmogorov [2,3,4] and Obukhov [5,6,7] observed that the timeaveraged Fourier spectra of the kinetic energy of an atmospheric flow possess a universal decay structure, corresponding to the inverse five-thirds power of the Fourier wavenumber. The physics of turbulence formation in a laminar flow, as well as the origin of power scaling of turbulent kinetic energy spectra, remain unknown. In our recent work [9], we considered a system of many particles, each of mass m, interacting solely via a repelling short-range potential φ(r ), with the convention that φ(r ) → 0 as r → ∞. In the limit of infinitely many such particles, we obtained, via the standard Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) formalism [10,11,12], the following Vlasov-type equation [13] for the mass-weighted distribution density of a single particle f (t, x, v):

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Conclusion