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Register a new userMacroscopic turbulent flow via hard sphere potential
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Authors
Rafail V. Abramov
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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 which 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 Kolmogorovs negative five-thirds power decay rate.
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For a large system of identical particles interacting by means of a potential, we find that a strong large scale flow velocity can induce motions in the inertial range via the potential coupling. This forcing lies in special bundles in the Fourier space, which are formed by pairs of particles. These bundles are not present in the Boltzmann, Euler and Navier-Stokes equations, because they are destroyed by the Bogoliubov-Born-Green-Kirkwood-Yvon formalism. However, measurements of the flow can detect certain bulk effects shared across these bundles, such as the power scaling of the kinetic energy. We estimate the scaling effects produced by two types of potentials: the Thomas-Fermi interatomic potential (as well as its variations, such as the Ziegler-Biersack-Littmark potential), and the electrostatic potential. In the near-viscous inertial range, our estimates yield the inverse five-thirds power decay of the kinetic energy for both the Thomas-Fermi and electrostatic potentials. The electrostatic potential is also predicted to produce the inverse cubic power scaling of the kinetic energy at large inertial scales. Standard laboratory experiments confirm the scaling estimates for both the Thomas-Fermi and electrostatic potentials at near-viscous scales. Surprisingly, the observed kinetic energy spectrum in the Earth atmosphere at large scales behaves as if induced by the electrostatic potential. Given that the Earth atmosphere is not electrostatically neutral, we cautiously suggest a hypothesis that the atmospheric kinetic energy spectra in the inertial range are indeed driven by the large scale flow via the electrostatic potential coupling.
We study the 6-dimensional dynamics -- position and orientation -- of a large sphere advected by a turbulent flow. The movement of the sphere is recorded with 2 high-speed cameras. Its orientation is tracked using a novel, efficient algorithm; it is based on the identification of possible orientation `candidates at each time step, with the dynamics later obtained from maximization of a likelihood function. Analysis of the resulting linear and angular velocities and accelerations reveal a surprising intermittency for an object whose size lies in the integral range, close to the integral scale of the underlying turbulent flow.
The ultimate goal of a sound theory of turbulence in fluids is to close in a rational way the Reynolds equations, namely to express the time averaged turbulent stress tensor as a function of the time averaged velocity field. This closure problem is a deep and unsolved problem of statistical physics whose solution requires to go beyond the assumption of a homogeneous and isotropic state, as fluctuations in turbulent flows are strongly related to the geometry of this flow. This links the dissipation to the space dependence of the average velocity field. Based on the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations, it has been recently observed that the closure problem is strongly restricted, and that it implies that the turbulent stress is a non local function (in space) of the average velocity field, an extension of classical Boussinesq theory of turbulent viscosity. The resulting equations for the turbulent stress are derived here in one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case the integral kernel giving the turbulent stress, as function of the averaged velocity field, takes a simple form leading to a full analysis of the averaged turbulent flow in the limit of a very large Reynolds number. In this limit one has to match a viscous boundary layer, near the walls bounding the flow, and an outer solution in the bulk of the flow. This asymptotic analysis is non trivial because one has to match solution with logarithms. A non trivial and somewhat unexpected feature of this solution is that, besides the boundary layers close to the walls, there is another inner boundary layer near the center plane of the flow.
Turbulence modeling is a classical approach to address the multiscale nature of fluid turbulence. Instead of resolving all scales of motion, which is currently mathematically and numerically intractable, reduced models that capture the large-scale behavior are derived. One of the most popular reduced models is the Reynolds averaged Navier-Stokes (RANS) equations. The goal is to solve the RANS equations for the mean velocity and pressure field. However, the RANS equations contain a term called the Reynolds stress tensor, which is not known in terms of the mean velocity field. Many RANS turbulence models have been proposed to model the Reynolds stress tensor in terms of the mean velocity field, but are usually not suitably general for all flow fields of interest. Data-driven turbulence models have recently garnered considerable attention and have been rapidly developed. In a seminal work, Ling et al (2016) developed the tensor basis neural network (TBNN), which was used to learn a general Galilean invariant model for the Reynolds stress tensor. The TBNN was applied to a variety of flow fields with encouraging results. In the present study, the TBNN is applied to the turbulent channel flow. Its performance is compared with classical turbulence models as well as a neural network model that does not preserve Galilean invariance. A sensitivity study on the TBNN reveals that the network attempts to adjust to the dataset, but is limited by the mathematical form that guarantees Galilean invariance.
We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely the number of quasi-stationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can neither be recovered using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasi-stationary states.
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