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Register a new userIdentification of Separation Wavenumber between Weak and Strong Turbulence Spectra for Vibrating Plate
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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.
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Variety of statistically steady energy spectra in elastic wave turbulence have been reported in numerical simulations, experiments, and theoretical studies. Focusing on the energy levels of the system, we have performed direct numerical simulations according to the F{o}ppl--von K{a}rm{a}n equation, and successfully reproduced the variability of the energy spectra by changing the magnitude of external force systematically. When the total energies in wave fields are small, the energy spectra are close to a statistically steady solution of the kinetic equation in the weak turbulence theory. On the other hand, in large-energy wave fields, another self-similar spectrum is found. Coexistence of the weakly nonlinear spectrum in large wavenumbers and the strongly nonlinear spectrum in small wavenumbers are also found in moderate energy wave fields.
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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.
We compare experimental data and numerical simulations for the dynamics of inertial particles with finite density in turbulence. In the experiment, bubbles and solid particles are optically tracked in a turbulent flow of water using an Extended Laser Doppler Velocimetry technique. The probability density functions (PDF) of particle accelerations and their auto-correlation in time are computed. Numerical results are obtained from a direct numerical simulation in which a suspension of passive pointwise particles is tracked, with the same finite density and the same response time as in the experiment. We observe a good agreement for both the variance of acceleration and the autocorrelation timescale of the dynamics; small discrepancies on the shape of the acceleration PDF are observed. We discuss the effects induced by the finite size of the particles, not taken into account in the present numerical simulations.
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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