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Register a new userEnergy Spectra of Developed Turbulencein Helium Superfluids
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We suggest a minimal model for the 3D turbulent energy spectra in superfluids, based on their two-fluid description. We start from the Navier-Stokes equation for the normal fluid and from the coarse-grained hydrodynamic equation for the superfluid component (obtained from the Euler equation for the superfluid velocity after averaging over the vortex lines) and introduce a mutual friction coupling term, proportional to the counterflow velocity, the average superfluid vorticity and to the temperature dependent parameter $q=alpha/(1+alpha)$, where $alpha$ and $alpha$ denote the dimensionless parameters characterizing the mutual friction between quantized vortices and the normal component of the liquid. We then derive the energy balance equations, taking into account the cross-velocity correlations. We obtain all asymptotical solutions for normal and superfluid energy spectra for limiting cases of small/big normal to superfluid density ratio and coupling. We discuss the applicability limits of our model to superfluid He II and to $^3$He-B and compare the model predictions with available experimental data.
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A single-wavenumber representation of nonlinear energy spectrum, i.e., stretching energy spectrum is found in elastic-wave turbulence governed by the Foppl-von Karman (FvK) equation. The representation enables energy decomposition analysis in the wavenumber space, and analytical expressions of detailed energy budget in the nonlinear interactions are obtained for the first time in wave turbulence systems. We numerically solved the FvK equation and observed the following facts. Kinetic and bending energies are comparable with each other at large wavenumbers as the weak turbulence theory suggests. On the other hand, the stretching energy is larger than the bending energy at small wavenumbers, i.e., the nonlinearity is relatively strong. The strong correlation between a mode $a_{bm{k}}$ and its companion mode $a_{-bm{k}}$ is observed at the small wavenumbers. Energy transfer shows that the energy is input into the wave field through stretching-energy transfer at the small wavenumbers, and dissipated through the quartic part of kinetic-energy transfer at the large wavenumbers. A total-energy flux consistent with the energy conservation is calculated directly by using the analytical expression of the total-energy transfer, and the forward energy cascade is observed clearly.
A weakly nonlinear spectrum and a strongly nonlinear spectrum coexist in a statistically steady state of elastic wave turbulence. The analytical representation of the nonlinear frequency is obtained by evaluating the extended self-nonlinear interactions. The {em critical/} wavenumbers at which the nonlinear frequencies are comparable with the linear frequencies agree with the {em separation/} wavenumbers between the weak and strong turbulence spectra. We also confirm the validity of our analytical representation of the separation wavenumbers through comparison with the results of direct numerical simulations by changing the material parameters of a vibrating plate.
We address the phenomenon of drag reduction by dilute polymeric additive to turbulent flows, using Direct Numerical Simulations (DNS) of the FENE-P model of viscoelastic flows. It had been amply demonstrated that these model equations reproduce the phenomenon, but the results of DNS were not analyzed so far with the goal of interpreting the phenomenon. In order to construct a useful framework for the understanding of drag reduction we initiate in this paper an investigation of the most important modes that are sustained in the viscoelastic and Newtonian turbulent flows respectively. The modes are obtained empirically using the Karhunen-Loeve decomposition, allowing us to compare the most energetic modes in the viscoelastic and Newtonian flows. The main finding of the present study is that the spatial profile of the most energetic modes is hardly changed between the two flows. What changes is the energy associated with these modes, and their relative ordering in the decreasing order from the most energetic to the least. Modes that are highly excited in one flow can be strongly suppressed in the other, and vice versa. This dramatic energy redistribution is an important clue to the mechanism of drag reduction as is proposed in this paper. In particular there is an enhancement of the energy containing modes in the viscoelastic flow compared to the Newtonian one; drag reduction is seen in the energy containing modes rather than the dissipative modes as proposed in some previous theories.
We study the necessary condition under which a resonantly driven exciton polariton superfluid flowing against an obstacle can generate turbulence. The value of the critical velocity is well estimated by the transition from elliptic to hyperbolic of an operator following ideas developed by Frisch, Pomeau, Rica for a superfluid flow around an obstacle, though the nature of equations governing the polariton superfluid is quite different. We find analytical estimates depending on the pump amplitude and on the pump energy detuning, quite consistent with our numerical computations.
It is important to know the accurate trajectory of a free fall object in fluid (such as a spacecraft), whose motion might be chaotic in many cases. However, it is impossible to accurately predict its chaotic trajectory in a long enough duration by traditional numerical algorithms in double precision. In this paper, we give the accurate predictions of the same problem by a new strategy, namely the Clean Numerical Simulation (CNS). Without loss of generality, a free fall disk in water is considered, whose motion is governed by the Andersen-Pesavento-Wang model. We illustrate that convergent and reliable trajectories of a chaotic free fall disk in a long enough interval of time can be obtained by means of the CNS, but different traditional algorithms in double precision give disparate trajectories. Besides, unlike the traditional algorithms in double precision, the CNS can predict the accurate posture of the free fall disk near the vicinity of the bifurcation point of some physical parameters in a long duration. Therefore, the CNS can provide reliable prediction of chaotic systems in a long enough interval of time.
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