Shock Waves and Noise in the Collapse of a Cloud of Cavitation Bubbles

Calculations of the collapse dynamics of a cloud of cavitation bubbles confirm the speculations of Morch and his co-workers and demonstrate that collapse occurs as a result of the inward propagation of a shock wave which grows rapidly in magnitude. Results are presented showing the evolving dynamics of the cloud and the resulting far-field acoustic noise

The Noise Generated by the Collapse of a Cloud of Cavitation Bubbles

The focus of this paper is the numerical simulation of the dynamics and acoustics of a cloud of cavitating bubbles. The prototypical problem solved considers a finite cloud of nuclei that is exposed to a decrease in the ambient pressure which causes the cloud to cavitate. A subsequent pressure recovery then causes the cloud to collapse. This is typical of the perturbation experienced by a bubble cloud as it passes a headform or the blade of a ship propeller. The simulations employ the fully non-linear, non-barotropic, homogeneous flow equations coupled with the Rayleigh-Plesset dynamics for individual bubbles. This set of equations is solved numerically by an integral method. The computational results confirm the early speculation of Morch and his co-workers (Morch 1980 & 1981, Hanson et al. 1981) that an inwardly propagating shock wave may be formed in the collapse of a cavitating cloud. The structure of the shock is found to be similar to that of the steady planar shocks analyzed by Noordij and van Wijngaarden (1974). The shock wave grows rapidly not only because of the geometric effect of an inwardly propagating spherical shock but also because of the coupling of the single bubble dynamics with the global dynamics of the flow through the pressure and velocity fields (see also Wang and Brennen 1994). The specific circumstances which lead to the formation of such a shock are explored. Moreover, the calculations demonstrate that the acoustic impulse produced by the cloud is significantly enhanced by this shock-focusing process. Major parameters which affect the dynamics and acoustics of the cloud are found to be the cavitation number, [sigma], the initial void fraction, [alpha-zero], the minimum pressure coefficient of the flow, [C Pmin], the natural frequencies of the cloud, and the ratio of the length scale of low pressure perturbation to the initial radius of the cloud, [D/A-zero], where D can be, for example, the radius of the headform or chord length of the propeller blade. We examine how some of these parameters affect the far field acoustic noise produced by the volumetric acceleration of the cloud. The non-dimensional far-field acoustic impulse produced by the cloud collapse is shown to depend, primarily, on the maximum total volume of the bubbles in the cloud normalized by the length scale of the low pressure perturbation. Also, this maximum total volume decreases quasi-linearly with the increase of the cavitation number. However, the slope of the dependence, in turn, changes with the initial void fraction and other parameters. Non-dimensional power density spectra for the far-field noise are presented and exhibit the [equation] behavior, where n is between 0.5 and 2. After several collapse cycles, the cloud begins to oscillate at its natural frequency and contributes harmonic peaks in its spectrum

Observations of Shock Waves in Cloud Cavitation

This paper describes an investigation of the dynamics and acoustics of cloud cavitation, the structures which are often formed by the periodic breakup and collapse of a sheet or vortex cavity. This form of cavitation frequently causes severe noise and damage, though the precise mechanism responsible for the enhancement of these adverse effects is not fully understood. In this paper, we investigate the large impulsive surface pressures generated by this type of cavitation and correlate these with the images from high-speed motion pictures. This reveals that several types of propagating structures (shock waves) are formed in a collapsing cloud and dictate the dynamics and acoustics of collapse. One type of shock wave structure is associated with the coherent collapse of a well-defined and separate cloud when it is convected into a region of higher pressure. This type of global structure causes the largest impulsive pressures and radiated noise. But two other types of structure, termed 'crescent-shaped regions' and 'leading-edge structures' occur during the less-coherent collapse of clouds. These local events are smaller and therefore produce less radiated noise but the interior pressure pulse magnitudes are almost as large as those produced by the global events. The ubiquity and severity of these propagating shock wave structures provides a new perspective on the mechanisms reponsible for noise and damage in cavitating flows involving clouds of bubbles. It would appear that shock wave dynamics rather than the collapse dynamics of single bubbles determine the damage and noise in many cavitating flows

Orbital-resolved vortex core states in FeSe Superconductors: calculation based on a three-orbital model

We study electronic structure of vortex core states of FeSe superconductors based on a t$_{2g}$ three-orbital model by solving the Bogoliubov-de Gennes(BdG) equation self-consistently. The orbital-resolved vortex core states of different pairing symmetries manifest themselves as distinguishable structures due to different quasi-particle wavefunctions. The obtained vortices are classified in terms of the invariant subgroups of the symmetry group of the mean-field Hamiltonian in the presence of magnetic field. Isotropic $s$ and anisotropic $s$ wave vortices have $G_5$ symmetry for each orbital, whereas $d_{x^2-y^2}$ wave vortices show $G^{*}_{6}$ symmetry for $d_{xz/yz}$ orbitals and $G^{*}_{5}$ symmetry for $d_{xy}$ orbital. In the case of $d_{x^2-y^2}$ wave vortices, hybridized-pairing between $d_{xz}$ and $d_{yz}$ orbitals gives rise to a relative phase difference in terms of gauge transformed pairing order parameters between $d_{xz/yz}$ and $d_{xy}$ orbitals, which is essentially caused by a transformation of co-representation of $G^{*}_{5}$ and $G^{*}_{6}$ subgroup. The calculated local density of states(LDOS) of $d_{x^2-y^2}$ wave vortices show qualitatively similar pattern with experiment results. The phase difference of $\frac{\pi}{4}$ between $d_{xz/yz}$ and $d_{xy}$ orbital-resolved $d_{x^2-y^2}$ wave vortices can be verified by further experiment observation

Shock Wave Development in the Collapse of a Cloud of Bubbles

A numerical simulation of the collapse of a cloud of bubbles has been used to demonstrate the development of an inwardly propagating shock wave which grows rapidly in magnitude. The fully non-linear nonbarotropic homogeneous flow equations are coupled with single bubble dynamics and solved by a stable numerical scheme. The computational results demonstrate the structure of the shock wave as well as its strengthening effect due to the coupling of the single bubble dynamics with the global dynamics of the flow through the pressure and velocity fields. This appears to confirm the speculation of Morch and his co-workers that such shock formation is an important part of cloud collapse

Application of NASTRAN for stress analysis of left ventricle of the heart

Knowing the stress and strain distributions in the left ventricular wall of the heart is a prerequisite for the determination of the muscle elasticity and contractility in the process of assessing the functional status of the heart. NASTRAN was applied for the calculation of these stresses and strains and to help in verifying the results obtained by the computer program FEAMPS which was specifically designed for the plane-strain finite-element analysis of the left ventricular cross sections. Adopted for the analysis are the true shape and dimensions of the cross sections reconstructed from multiplanar X-ray views of a left ventricle which was surgically isolated from a dog's heart but metabolically supported to sustain its beating. A preprocessor was prepared to accommodate both FEAMPS and NASTRAN, and it has also facilitated the application of both the triangular element and isoparameteric quadrilateral element versions of NASTRAN. The stresses in several crucial regions of the left ventricular wall calculated by these two independently developed computer programs are found to be in good agreement. Such confirmation of the results is essential in the development of a method which assesses the heart performance

Loss of purity by wave packet scattering at low energies

We study the quantum entanglement produced by a head-on collision between two gaussian wave packets in three-dimensional space. By deriving the two-particle wave function modified by s-wave scattering amplitudes, we obtain an approximate analytic expression of the purity of an individual particle. The loss of purity provides an indicator of the degree of entanglement. In the case the wave packets are narrow in momentum space, we show that the loss of purity is solely controlled by the ratio of the scattering cross section to the transverse area of the wave packets.Comment: 7 pages, 1 figur

Maximum Path Information and Fokker-Planck Equation

We present in this paper a rigorous method to derive the nonlinear Fokker-Planck (FP) equation of anomalous diffusion directly from a generalization of the principle of least action of Maupertuis proposed by Wang for smooth or quasi-smooth irregular dynamics evolving in Markovian process. The FP equation obtained may take two different but equivalent forms. It was also found that the diffusion constant may depend on both q (the index of Tsallis entropy) and the time t.Comment: 7 page