Bubble screens are a useful problem in understanding the fundamental physics of shock/bubble-cloud interactions and are used to prevent damage of submerged structures due to UNDEX (Domenico,
1982). Reflection and transmission of linear wave propagation through a bubble screen were formu- lated by Carstensen & Foldy (1947) and Commander & Prosperetti (1989). Here, shock/bubble- screen interaction is considered as an application example of the bubbly flow computations.
0 0.5 1 1.5 2 1 2 3 4 5 pl / pl0 σ = 0 0 0.5 1 1.5 2 1 2 3 4 5 x[ cm] pl / pl 0 σ = 0.7 t = 0.016 ms t = 0.11 ms t = 0.16 ms t = 0.20 ms t = 0.35 ms
Figure 4.7: Spatial evolution of the averaged liquid pressure for shock propagation through an air/water bubble screen ofα0 = 0.005 andRref
0 = 50µm. The screen is placed between the dotted lines. Att= 0, the precursory wave reaches a probe just downstream of the screen.
One-dimensional shock propagation through an air-bubble screen ofα0= 0.005 at STP in water is now computed with the initial void fraction distribution:
α= α0, if 0< x < L, ǫ, otherwise, (4.12)
where 0 < ǫ ≪ α0 and L = 2 cm. The equilibrium bubble size in the screen (0 < x < L) is lognormally distributed about Rref
0 = 50 µm and with σ = 0 and 0.7. The incident, right-going shock with strengthplH= 5pl0is initially placed atx <0.
0 0.1 0.2 0.3 0.4 0 1 2 3 4 5 t[ m s ] pl / pl 0 σ= 0 σ= 0.7
Figure 4.8: Evolution of the liquid pressure for the transmitted waves.
interface (x= 0), the incident shock reflects as a rarefaction wave and transmits as a shock due to the fact that the acoustic impedance of the screen is smaller than that of water (i.e.,ρc < ρlcl). The transmitted shock trapped in the screen keeps reflecting at the interfaces, and the pressure inside the screen eventually increases to the incident shock pressure. We see that the bubble size distribution smoothes out the oscillatory structure of the trapped waves.
The pressure just downstream of the screen is presented in figure 4.8. The precursory waves propagating with the sonic speed of water are measured at t = 0. Note that for the case with no bubble screen, the probe measurement would show an instantaneous jump to plH at t ≈0. The bubble size distribution withσ = 0.7 increases the amplitude of the precursory wave because the distribution decreases the attenuation of high-frequency waves (see figure 3.2). The transmitted shock waves leave the screen at late times, and the liquid pressure increases in a step-wise manner because of the reflections of the trapped waves in the screen. As expected, the distribution makes the averaged pressure evolution less oscillatory and broadens the averaged shock width. This implies that the polydisperse bubble screen may be capable of more effectively cushioning UNDEX impulsive loading than the monodisperse screen, but there is still a need to quantify the scattering effect in each realization in order to further investigate the practical implications.
4.6
Summary
One-dimensional shock propagation in bubbly liquids was simulated to quantify the effects of poly- dispersity on the averaged shock dynamics. The steady shock relations were derived and employed as the initial conditions. The comparison to the experiment ofKamedaet al.(1998) demonstrated that the present method is capable of resolving the oscillatory shock structure that appears in the monodisperse case. The numerical experiments revealed that the averaged shock structure becomes less oscillatory as the bubble size distribution broadens. If the distribution is sufficiently broad, the shock profile is practically monotonic as experimentally identified byBeylich & G¨ulhan(1990). Because the different-sized bubbles can oscillate with different frequencies, phase cancellations in a polydisperse mixture occur locally. For cases with the broad size distributions, the polydisperse cloud does not oscillate in volume (or in void fraction) due to the phase cancellations and can be considered to behave quasistatically, regardless of individual bubble dynamics. In this case, the effect of polydispersity dominates over the single-bubble-dynamic damping.
Chapter 5
Dynamics of cavitation clouds
This chapter concerns the dynamics of cavitation clouds caused by the structural interaction with an underwater shock. First, the classical UNDEX/FSI theories of Cole (1948) and Taylor(1950) are reviewed, and the present FSI model with which we simulate the experiment of Rajendran & Satyanarayana(1997) is presented and verified. One-dimensional cloud cavitation with monodisperse and polydisperse nuclei is then simulated to clarify the fundamental mechanism of the inception and collapse of cavitation bubbles.
5.1
Sequence of underwater shock events
The sequence of events of interest in UNDEX can be summarized as follows:
• A solid explosive such as trinitrotoluene (TNT) detonates and rapidly produces a gas bubble with very high pressure.
• The gas bubble expands and produces a strong shock that propagates spherically in the ambient water.
• The shock interacts with interfaces such as the solid target, the seabed and the sea surface.
• There often appear tension waves in water due to structural deformation during reflection or acoustic impedance mismatch.
• The resulting tension waves can cause cloud cavitation, and violent collapse of the cloud of cavitation bubbles may follow.
Particularly, we are now interested in the interaction of UNDEX shocks with the target and the subsequent cloud cavitation. The cloud collapse can account for erosion of the submerged structures (Brennen,1994,1995). In the following, we review Cole’s empirical formula for the shock evolution and Taylor’s fluid-structure interaction model. These classical theories are employed in the present simulations.