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Plunging wave impact at a vertical wall

Here we perform an exploratory calculation to establish the viability and promise of the method for violent wave impact simulations involving air pockets and aeration. Figure 22 shows the setup for a plunging wave impact event. The length of the wave tank is 3 m and the height is 0.8 m. A step of 0.2 m height is placed in the bottom right part of the wave tank starting at x = 1.75 nr with a 45° slope to cause the wave to steepen and break. A piston type wave maker is placed at the left boundary of the domain to generate waves. The still water depth is d = 0.3 nr. The whole NWT is divided into two sub-donrains. A two-fluid NWT based on the incompressible Navier-Stokes equations developed in our previous work (Qian et ah, 2006) is used to deal with the left sub-domain. This incompressible solver, which is named AMAZON-SC, adopts an interface-capturing method to treat the free surface as a discontinuity in density. We utilise the present compressible flow model (14) to handle the right-sub domain, where an air pocket will be trapped or enclosed by the water body. The dashed line in Figure 22 indicates the interface between the incompressible and compressible flow solvers. Buffer zones are used near the interface to exchange flow information between the two solvers. Within the buffer zones, one or two layers of mesh cells for each component solver as required are placed on the opposite side of the interface. The flow information including density, velocity and pressure at these mesh cells is obtained from the companion solver domain through interpolation. More details of the coupling of component flow solvers will be reported separately in future work. A background uniform Cartesian mesh is used to overlay the flow domain, and the basic mesh cell is a square with size of li = 0.01 m. Solid boundaries not aligned with the Cartesian mesh in the left sub-domain are treated using the cut-cell method (Qian et al., 2006).

Setup of the plunging wave impact problem

Figure 22: Setup of the plunging wave impact problem. The numerical wave tank is divided into two sub-domains occupied by the incompressible flow solver (Qian et ah, 2006) and the present two-phase compressible flow solver.

Water surface elevations in the numerical wave tank (without structure) at a

Figure 23: Water surface elevations in the numerical wave tank (without structure) at a: =1 m and x = 2 m.

Before computing the plunging breaker impact problem, we first conduct a simple test to generate a solitary wave using the incompressible solver. The solid step is removed from the NWT and the right boundary of the domain is treated as an open boundary. The amplitude of the solitary wave is a = 0.09 m. The wave is generated by prescribing the paddle movement according to Rayleigh’s solitary wave theory (see Katell and Eric, 2002). Figure 23 shows the water surface elevations at x = 1 nr and x = 2 nr. The wave crest takes 0.52 s to travel between these two locations. Obviously, the phase speed of the solitary wave is equal to c* = 1.92 m/s. The theoretical phase speed for solitary waves can be calculated as c = Jg(d + a). We obtain c = 1.95 m/s for this test case so the relative error between the computed and theoretical wave phase speeds is less than 1.5%.

The integrated numerical wave tank is now used to solve the plunging wave impact problem. The paddle is used again to generate a solitary wave with height a = 0.2 nr. In addition to the integrated NWT, we also use the established standalone in-house incompressible two- fluid NWT AMAZON-SC to solve this problem for the purposes of comparison. According to field measurements and laboratory experiments, the first pressure spike in the form of a “church-roof” shape is a key to the safety to structures. Therefore, we focus on this phase of the impact in the current discussion. Computation of the subsequent wave evolution will be considered in future work.

Figure 24 shows the profiles of the free surface at different times. The red solid line represents the results with AMAZON-SC, the stand-alone incompressible solver, and the blue dashed line indicates the solution obtained with the compressible solver. At t = 2.13 s, the two solvers give almost the same profiles. We notice that an obvious discrepancy of the free surface profiles appears at t = 2.15 s. Although the wave crests are almost the same distance away from the vertical wall in the horizontal direction, the wave crest obtained by the incompressible solver is higher than the compressible solver. The wave trough obtained by the compressible solver moves upward along the wall faster than the incompressible solver. The free surface beneath the wave crest obtained by the compressible solver is closer to the vertical wall than the incompressible solver. At t = 2.16 s, we notice that the wave crest obtained by the incompressible solver moves upward significantly higher than the compressible solver, and this trend continues to t = 2.17 s when the waves almost impact the wall. The water continuously moves upward after the wave impacts the wall. From the figures, it is not difficult to observe that the trapped air pocket predicted by the compressible solver is much smaller than the incompressible solver. It would seem that compressibility effects play an important role in changing the shape of the air pocket and the

Snapshots of the free surface profiles

Figure 24: Snapshots of the free surface profiles. Mesh step size is li = 0.01 m. Incompressible and compressible solvers produce almost the same wave crest velocities in the ж-direction. Comparison of the time and spatial evolution of the air pockets between the two solvers show discrepancies due to compressibility effects. The volume of the air pocket predicted by the compressible solver is much smaller than the incompressible solver.

free surface. The incompressible assumption appears to lead an overestimate of the volume of the air pocket for this type of problem.

In Figure 25, we present several snapshots of the pressure distribution in the wave field at different times. The first row illustrates the results with the stand-alone incompressible solver AMAZON-SC and the second row corresponds to computations with the compressible solver. For the compressible solver, we can clearly see that the pressure in the air pocket increases dramatically, and the pressure rise travels downstream along the vertical wall and tank bottom. At t = 1.95 s, a second compression (pressure increase) in the air pocket is captured by the compressible solver. These phenomena are much less apparent in the predictions with the incompressible solver.

Quantitative comparison of pressures is made and presented in Figure 26. We gauge the pressures in the air pocket. For this problem, the size of the air pocket is relatively large as its diameter is around 10 cm. A significant amount of the wave energy is stored in the pocket through compression. Consequently, the air pressure rises significantly to about 114000 Pa as shown in the figure. Under the constant density assumption, the incompressible solver cannot deal with compressibility effects and only predicts a peak pressure of 104000 Pa. The rise time of the pressure spike for the incompressible solver is almost ten times that of the compressible solver. After the first peak, the air expands to a low pressure. The compressible solver predicts a fall to around 9500 Pa, whilst the incompressible solver predicts about 101000 Pa.

Snapshots of the pressure distributions in the numerical wave tank. From left to right, t = 2.155, 2.16, 2.165 and 2.195 s. The black curve represents the free surface. Top

Figure 25: Snapshots of the pressure distributions in the numerical wave tank. From left to right, t = 2.155, 2.16, 2.165 and 2.195 s. The black curve represents the free surface. Top: incompressible solver; Bottom: compressible solver. Pressure contour range: pmjn = 100000 Pa, Ртах = 112000 Pa, Ap = 400 Pa.

Time evolution of the pressure in the air pocket

Figure 26: Time evolution of the pressure in the air pocket. Sub-atmospheric pressures are captured by the compressible solver (solid line) but not the incompressible solver (dashed line). There is a significant decrease from the first pressure peak to the second peak with the compressible solver. Rise times and pressures obtained by the compressible solver are markedly different to those of the incompressible solver.

The pressure distributions along the vertical wall at different times computed by the compressible solver are shown in Figure 27. The pressure in the region (0.3 < у < 0.4) is strongly influenced by the trapped air pocket. The air pocket is continuously compressed from t =2.156 to 2.167 s. It is noted that at t =2.184 s the air pocket undergoes expansion and the pressure reduces accordingly to sub-atmospheric values. Thus, this local region is experiencing seaward (suction) force. These numerical findings confirm Bullock et al.’s work (Bullock et al., 2007) that negative gauge pressures indeed occur in violent wave impact

Pressure distribution on the vertical wall at t

Figure 27: Pressure distribution on the vertical wall at t = 2.156, 2.162, 2.167 and 2.184 s (computed by the compressible solver). Pq is the atmospheric pressure, h = 0.1 is the initial water depth before the vertical wall.

events and the resultant seaward force has the potential to cause the removal of blocks from masonry structures.

In their laboratory experiments of overturning wave breaking on structures, Lugni et al. observed that after the strongest first impact pressure peak, the pressure decreases to a value lower than atmospheric pressure and a subsequent second pressure peak is observed much lower than the first one (see Figures 4 and 5 of Lugni et ah, 2010). The numerical findings produced by the present compressible solver of a steep pressure spike followed by a negative gauge pressure and subsequent lower second pressure peak, etc. agree qualitatively well with Lugni et aids experiments (Lugni et ah, 2010), although the wave conditions are not exactly the same.

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