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Register a new userTurbulent energy spectrum via an interaction potential
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Authors
Rafail V. Abramov
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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.
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In a recent work, we proposed a hypothesis that the turbulence in gases could be produced by particles interacting via a potential - for example, the interatomic potential at short ranges, and the electrostatic potential at long ranges. Here, we examine the proposed mechanics of turbulence formation in a simple model of two particles, which interact solely via a potential. Following the kinetic theory approach, we derive a hierarchy of the velocity moment transport equations, and then truncate it via a novel closure based on the high Reynolds number condition. While standard closures of the velocity moment hierarchy of the Boltzmann equation lead to the compressible Euler and Navier-Stokes systems of equations, our closure leads to a transport equation for the velocity alone, which is driven by the potential forcing. Starting from a large scale laminar shear flow, we numerically simulate the solutions of our velocity transport equation for the electrostatic, gravity, Thomas-Fermi and Lennard-Jones potentials, as well as the Vlasov-type large scale mean field potential. In all studied scenarios, the time-averaged Fourier spectra of the kinetic energy clearly exhibit Kolmogorovs five-thirds power decay rate.
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.
We adress the problem of interactions between the longitudinal velocity increment and the energy dissipation rate in fully developed turbulence. The coupling between these two quantities is experimentally investigated by the theory of stochastic Markovian processes. The so-called Markov analysis allows for a precise characterization of the joint statistical properties of velocity increment and energy dissipation. In particular, it is possible to determine the differential equation that governs the evolution along scales of the joint probability density of these two quantities. The properties of this equation provide interesting new insights into the coupling between energy dissipation and velocity incrementas leading to small scale intermittency.
The present numerical investigation uses well-resolved large-eddy simulations to study the low-frequency unsteady motions observed in shock-wave/turbulent-boundary-layer interactions. Details about the numerical aspects of the simulations and the subsequent data analysis can be found in three papers by the authors (Theo. Comput. Fluid Dyn., 23:79--107 (2009); Shock Waves, 19(6):469--478 (2009) and J. of Fluid Mech. (2011)). The fluid dynamics video illustrates the complexity of the interaction between a Mach 2.3 supersonic turbulent boundary layer and an oblique shock wave generated by a 8-degree wedge angle. The first part of the video highlights the propagation of disturbances along the reflected shock due to the direct perturbation of the shock foot by turbulence structures from the upstream boundary layer. The second part of the video describes the observed block-like back-and-forth motions of the reflected shock, focusing on timescales about two orders of magnitude longer than the ones shown in the first part of video. This gives a visual impression of the broadband and energetically-significant peak in the wall-pressure spectrum at low frequencies. The background blue-white colouring represents the temperature field (with white corresponding to hot) and one can clearly appreciate why such low-frequency shock motions can lead to reduced fatigue lifetimes and is detrimental to aeronautical applications.
The mechanisms governing the low-frequency unsteadiness in the shock wave/turbulent boundary layer interaction at Mach 2 are considered. The investigation is conducted based on the numerical database issued from large-eddy simulations covering approximately 300 cycles of the low-frequency shock fluctuations. The evaluation of the spectrum in the interaction zone indicates that the broadband low-frequency unsteadiness is predominantly two-dimensional, and can be isolated via spanwise averaging. Empirically derived transfer functions are computed using the averaged flow field, and indicate the occurrence of a feedback mechanism between downstream flow regions and shock fluctuations. The transfer functions are also used as an estimation tool to predict the shock motion accurately; for the largest streamwise separation between input and output signals, correlations above 0.6 are observed between predicted and LES data. Computation of spectral proper orthogonal decomposition (SPOD) modes confirms the existence of upstream traveling waves in the leading spectral mode. Finally, the spectral modes obtained using selected flow regions downstream of the shock enable the reconstruction of a significant portion of the energy in the interaction zone. The current results shed further light on the physical mechanisms driving the shock motion, pointing towards a causal behavior between downstream areas and the characteristic unsteady fluctuations at the approximate shock position.
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