Abstract
In recent works, we proposed a hypothesis that the turbulence in gases could be produced by particles interacting via a potential and examined the proposed mechanics of turbulence formation in a simple model of two particles for a variety of different potentials. In this work, we use the same hypothesis to develop new fluid mechanics equations that model turbulent gas flow on a macroscopic scale. The main difference between our approach and the conventional formalism is that we avoid replacing the potential interaction between particles with the Boltzmann collision integral. Due to this difference, the velocity moment closure, which we implement for the shear stress and heat flux, relies upon the high Reynolds number condition rather than the Newton law of viscosity and the Fourier law of heat conduction. The resulting system of equations of fluid mechanics differs considerably from the standard Euler and Navier–Stokes equations. A numerical simulation of our system shows that a laminar Bernoulli jet of an argon-like hard sphere gas in a straight pipe rapidly becomes a turbulent flow. The time-averaged Fourier spectra of the kinetic energy of this flow exhibit Kolmogorov’s negative five-thirds power decay rate.
Highlights
Reynolds1 demonstrated that an initially laminar flow in a long straight pipe develops turbulent motions whenever the high Reynolds number condition is satisfied
While the Newton law of viscosity and the Fourier law of heat conduction are typically used to truncate the velocity moment hierarchy emerging from the Boltzmann equation12–15, in our potential-driven system, we used the high Reynolds number condition to obtain the closures for the shear stress and heat flux
We develop and test a fluid mechanical model for a macroscopic turbulent flow, which is triggered by a short range interaction potential
Summary
Reynolds demonstrated that an initially laminar flow in a long straight pipe develops turbulent motions whenever the high Reynolds number condition is satisfied. We truncate the velocity moment hierarchy using the high Reynolds number condition, just as we did in our preceding work.11 This results in a system of fluid dynamics equations for the density, momentum, and pressure variables, which is markedly different from the conventional Euler equations. We use the same closure for the shear stress and heat flux, derived from the high Reynolds number condition, as we introduced in our recent work.11 This closure leads to a system of transport equations for the density, momentum, and pressure variables. To verify that our model simulates a real-world turbulence process, we compute the time averages of the Fourier transform of the kinetic energy of the simulated flow We observe that these averages decay with the rate of negative five-thirds power of the Fourier wavenumber, which is a famous Kolmogorov spectrum.
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