Table of Contents:
Free drop of a water column in a closed tank
This benchmark test was proposed by the liquid sloshing community at the ISOPE 2010 conference to exam numerical codes’ capability of handling multidimensional water impact problems involving air entrapment (Dias and Brosset, 2010). Figure 8 shows the setup for this problem. A rectangular water column (p2 = 1000 kg/m3) with width 10 m and height 8 m is initially at rest in the closed tank and air (pi = 1 kg/m3) fills the remainder of the tank. The initial pressure is p = 1 bar in the tank. Under gravity, the water column will drop and impact upon the bottom of the tank at around t = 0.64 ~ 0.65 s. The impact pressure at this moment is of particular interest to ship structural engineers for this type of problem, since it is fundamentally a key issue for the safety of liquefied natural gas carriers.
To obtain an accurate prediction of the pressure peak, use of a fine mesh is recommended by other researchers (Braeunig et ah, 2009; Guilcher et ah, 2010). Therefore, we equally distribute 1500 mesh cells in the vertical direction and 200 mesh cells in the horizontal direction. Figure 9 gives two snapshots of the pressure field in the tank just before impact and at the impact time t = 0.65 s. It is clearly shown that the highest pressure occurs at the bottom centre of the tank and the distribution of pressure is symmetric about the central section (y axis). In Figure 10, we present several snapshots of the volume fraction contours in the liquid tank. An interesting finding is that a small amount of air is trapped between the water body and the bottom surface of the tank. This body of air undergoes not only compression due to the liquid impact (at t = 0.65 s) but also expansion when the gas
Figure 8: Initial setup for free drop of a water column in a closed 2D tank (Units in metres).
Figure 9: Snapshots of the pressure contours in the closed tank. The black curve is the free surface (ft| = 0.5).
Figure 10: Snapshots of the volume fraction for the gas phase in the closed tank. The water column starts to deform upon impacting the bottom of the tank. A small amount of air is trapped between the free surface and the bottom of the tank, and the air undergoes compression and expansion.
Figure 11: Time history of the absolute pressure at the bottom centre of the tank. There is good agreement with the impact time t = 0.65 predicted by Guilcher et al. (2010), Braeunig et al. (2009) and the present work, while incompressible OpenFOAM gives an earlier estimate at t = 0.64 and a lower pressure peak.
phase pressure exceeds the environmental liquid phase pressure. At t = 0.75 s, the portion of trapped air pocket appears to be a very thin layer, then has a cylindrical shape at t = 1 s and a half-cylindrical shape at t = 1.2 s.
The time history of the absolute pressure at the bottom centre of the tank is plotted in Figure 11. Guilcher et al.’s results computed independently using an SPH code (Guilcher et ah, 2010) are represented by the red curve with “+” symbol. Braeunig et al.’s computation (Braeunig et ah, 2009) is represented by the blue dashed line. We use a black curve to illustrate the results obtained by the present compressible method. The green curve indicates the result obtained on a coarse mesh (200 x 700 cells) with the present method. We have also used the interFoam module from the open-source software OpenFOAM®2.1.1, which is based on an incompressible two-fluid finite volume method, to compute this problem on the same mesh and present these results with a green line. Comparing the impact time, the present compressible results agree well with Guilcher et al.’s solution and Braeunig et al.’s work (t = 0.65 s), while interFoam gives a slight!}' early prediction at t = 0.64 s and the lowest pressure peak at 8.7 bar. The others produce much higher predictions at over 20 bar. After the peak, the pressure begins to fall. The minimum pressure following the peak obtained by interFoam is about 2 bar; Braeunig et al. produced a non-physical negative pressure with some oscillations (Braeunig et al., 2009); Guilcher et al. gave a value of 1 bar (Braeunig et al., 2009); the present compressible method obtains much lower but positive values for the minimum pressure after the pressure peak. Only the present method permits the fluid to expand sufficiently far to be in tension.
Underwater explosion near a planar rigid wall
This case is chosen to investigate a shock wave interaction with a planar rigid wall in water with possible hull cavitation loading/re-loading on the wall. Figure 12 depicts the set up for this problem. At the initial state, an explosive gas bubble of unit radius is located at the origin (0,0) in water. The density of the air bubble is pg = 1270 kg/m3 , the pressure is pg = 8290 bar and the ratio of specific heats is 7g = 2. For the water, the density is pi = 1000 kg/m3, the pressure is pi = 1 bar, the polytropic constant is 7/ = 7.15 and the pressure constant is pc = 3 x 108.
The computational domain is a rectangular region of [—6,6] x [—6,3]. The horizontal rigid planar wall is located at у = 3. The domain is uniformly covered by 360 x 270 mesh cells which is the same as the grid used in Xie’s work (Xie, 2005). The problem is also solved on a fine mesh with 720 x 540 cells to ensure the convergence of the numerical solution. A numerical pressure gauge is placed at the centre of the planar wall (point P) to monitor the impact pressure. Around this gauge point, we also integrate the pressure in a square region of 4 x 4 to obtain the impact force or loading. The length of this region is along the planar wall, and the width is perpendicular to the x — y plane. The boundary conditions for this problem correspond a rigid upper wall and all other boundaries are transmissive (see Sections 4.2 and 5.2 of Xie (2005)).
Figure 13 shows several snapshots of the pressure contours in the domain computed on the coarse mesh. At t = 1.5 ms, the explosion generated main shock has been reflected from the planar wall. The reflected shock interacts with the expanding gas bubble undergoing a refraction process that results in a strong rarefaction wave propagating back towards the planar wall at t = 2 ms and a weaker reflected shock traversing the gas bubble. After the rarefaction wave impacts the wall and is reflected at t = 3 ms, a low pressure region forms near the planar wall causing a cavitation region to appear in this low pressure region at t = 4 ms. Simultaneously, the main shock continues to propagate through the computational domain.
Figure 12: Setup of an underwater explosion near a planar rigid wall problem. The top boundary is a solid wall, all the others are open boundaries. The radius of the gas bubble is 1 m. The domain is discretised by a uniform mesh with 360 x 270 cells.
Figure 13: Line contours of pressure and the exploding gas bubble (grey colour) computed on the mesh with 360 x 270 cells.
Figure 14: Comparison of the impact loading on the solid wall. Left: pressure time series at point P; Right: force acting on a 4m x 4m area of the solid wall. Red and green lines are the present results computed on the coarse (360 x 270) and fine mesh (720 x 540) respectively, black lines are Xie’s computation on a mesh with 360 x 270 cells (Xie, 2005).
At t = 5.5 ms, the cavitation pocket collapses from the centre of the cavitation zone, resulting in a water-jet forming directed towards the planar wall and a relatively strong secondary compression wave propagating towards the gas bubble. The secondary compression wave impacts the gas bubble and leads to a second rarefaction wave propagating back towards the planar wall at t = 7 ms in a cyclic process that subsequently weakens with time.
Figure 14 shows the pressure (at point P) and force (covering the blue region around point P) acting upon the planar rigid wall. Numerical solutions computed on the coarse and fine meshes are almost identical and this confirms the convergence of the results. R can clearly be seen from this figure that the structural loading consists of shock loading and subsequent cavitation collapse reloading. The peak pressure of the shock is much higher than the peak pressure of cavitation collapse, whilst the duration of cavitation collapse reloading is longer than that of shock.
The present work gives about 7% higher prediction of the shock wave peak pressure (7000 bar) compared to Xie’s numerical solution (6500 bar) (Xie, 2005). The start time of the cavitation collapse predicted by the present work is t = 5.05 ms, while Xie’s one-fluid solver prediction is t = 5.25 ms (Xie, 2005). The amplitude of the pressure at cavitation collapse computed by the present method is very close to Xie’s calculation. Considering the impact force, the present work and Xie’s solution give almost the same amplitude of the peak force at about 9.8 x Ю10 N. The rise time of the peak force calculated by Xie is slightly higher than our prediction. This can be attributed to small differences in the numerical dissipation present in the respective flow solvers.