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In this paper high resolution wave probe records are examined using wavelet techniques with a view to determining the sources and relative contributions of capillary wave energy along representative wind wave forms. Wavelets enable computations of conditional spectra and turn out to be powerful tools for the study of the development and propagation of capillary waves. They also enable the detailed analyses of the relative contributions to the spectrum of the wave peaks and troughs.
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In elastic-wave turbulence, strong turbulence appears in small wave numbers while weak turbulence does in large wave numbers. Energy transfers in the coexistence of these turbulent states are numerically investigated in both of the Fourier space and the real space. An analytical expression of a detailed energy balance reveals from which mode to which mode energy is transferred in the triad interaction. Stretching energy excited by external force is transferred nonlocally and intermittently to large wave numbers as the kinetic energy in the strong turbulence. In the weak turbulence, the resonant interactions according to the weak turbulence theory produces cascading net energy transfer to large wave numbers. Because the systems nonlinearity shows strong temporal intermittency, the energy transfers are investigated at active and moderate phases separately. The nonlocal interactions in the Fourier space are characterized by the intermittent bundles of fibrous structures in the real space.
Using various techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz and geometric singular perturbation theory, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of various traveling waves mediating the transition to turbulence. The most complicated behaviour can be found in an intermediate Reynolds number regime, where the traveling waves exhibit arbitrarily long periodic-like dynamics indicating the onset of chaos. Our analysis provides a systematic mathematical approach to identifying the transition to spatio-temporal turbulent structures that may also be applicable to other models arising in fluid dynamics.
An experimental procedure for studying soliton gases in shallow water is devised. Nonlinear waves propagate at constant depth in a 34,m-long wave flume. At one end of the flume, the waves are generated by a piston-type wave-maker. The opposite end is a vertical wall. Wave interactions are recorded with a video system using seven side-looking cameras with a pixel resolution of 1,mm, covering 14,m of the flume. The accuracy in the detection of the water surface elevation is shown to be better than 0.1 mm. A continuous monochromatic forcing can lead to a random state such as a soliton gas. The measured wave field is separated into right- and left-propagating waves in the Radon space and solitary pulses are identified as solitons of KdV or Rayleigh types. Both weak and strong interactions of solitons are detected. These interactions induce phase shifts that constitute the seminal mechanism for disorganization and soliton gas formation.
The dynamics of spherical laser-induced cavitation bubbles in water is investigated by plasma photography, time-resolved shadowgraphs, and single-shot probe beam scattering enabling to portray the transition from initial nonlinear to late linear oscillations. The frequency of late oscillations yields the bubbles gas content. Simulations with the Gilmore model using plasma size as input and oscillation times as fit parameter provide insights into experimentally not accessible bubble parameters and shock wave emission. The model is extended by a term covering the initial shock-driven acceleration of the bubble wall, an automated method determining shock front position and pressure decay, and an energy balance for the partitioning of absorbed laser energy into vaporization, bubble and shock wave energy, and dissipation through viscosity and condensation. These tools are used for analysing a scattering signal covering 102 oscillation cycles. The bubble was produced by a plasma with 1550 K average temperature and had 36 $mu$m maximum radius. Predicted bubble wall velocities during expansion agree well with experimental data. Upon first collapse, most energy was stored in the compressed liquid around the bubble and then radiated away acoustically. The collapsed bubble contained more vapour than gas, and its pressure was 13.5 GPa. The pressure of the rebound shock wave initially decayed $propto r^{-1.8}$, and energy dissipation at the shock front heated liquid near the bubble wall above the superheat limit. The shock-induced temperature rise reduces damping during late bubble oscillations. Bubble dynamics changes significantly for small bubbles with less than 10 $mu$m radius.
We present numerical simulations of the three-dimensional Galerkin truncated incompressible Euler equations that we integrate in time while regularizing the solution by applying a wavelet-based denoising. For this, at each time step, the vorticity filed is decomposed into wavelet coefficients, that are split into strong and weak coefficients, before reconstructing them in physical space to obtain the corresponding coherent and incoherent vorticities. Both components are multiscale and orthogonal to each other. Then, by using the Biot--Savart kernel, one obtains the coherent and incoherent velocities. Advancing the coherent flow in time, while filtering out the noise-like incoherent flow, models turbulent dissipation and corresponds to an adaptive regularization. In order to track the flow evolution in both space and scale, a safety zone is added in wavelet coefficient space to the coherent wavelet coefficients. It is shown that the coherent flow indeed exhibits an intermittent nonlinear dynamics and a $k^{-5/3}$ energy spectrum, where $k$ is the wavenumber, characteristic of { three-dimensional homogeneous isotropic turbulence}. Finally, we compare the dynamical and statistical properties of Euler flows subjected to four kinds of regularizations: dissipative (Navier--Stokes), hyperdissipative (iterated Laplacian), dispersive (Euler--Voigt) and wavelet-based regularizations.
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