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Water entry of a rigid plate

4.4.1 Experiment and computation setup

The experimental study was carried out in the Ocean Basin at Plymouth University’s COAST Laboratory. The ocean basin is 35 m long by 15.5 m wide and has an adjustable floor allowing operation at different water depths up to 3 m. The falling block includes a rigid flat impact plate connected to two driver plates and its total mass can be varied from 32 kg (block 1) to 52 kg (block 2). The impact plate is 0.25 m long and 0.25 m wide with a thickness of 0.012 m. The impact velocity can vary between 4 m/s and 7 m/s by adjusting the initial position of the plate. Pressures were measured during impact by five miniature pressure transducers installed on the impact plate as illustrated in Figure 15. The actual impact velocity was integrated from the measured data recorded by an accelerometer mounted on the top of the flat plate.

In the laboratory, the drop height of the plate is calibrated to obtain impact velocity

5.5 m/s while the present numerical model is constructed in the Eulerian frame of reference consistent with a finite volume method. To solve the water entry problem, we fix the flat plate in the numerical mesh and cause the water to move upward with the prescribed impact velocity. This strategy is appropriate for a short-duration impact process (Ng and Kot, 1992). Since violent hydrodynamic impacts are usually inertia dominant, viscous effects are currently ignored in the numerical computation. A compatible no-penetration velocity boundary condition therefore should be applied at the flat plate.

Experiment setup. Left

Figure 15: Experiment setup. Left: a sketch of the facility; right: configuration of the instrumentation on the impact plate (Units in mm). Pi- P5 are pressure transducers, S1-S5 are the influence areas of the pressure transducers and A1 is the accelerometer.

Before the trapped air layer breaks into small bubbles, turbulence does not play a significant role in the impact process, thus the flow could reasonably be assumed to be symmetric about the two central sections of the plate. Consequently, we compute only a quarter of the whole flow field chosen as a cubic region [0, 0.4] x [0, 0.4] x [0, 0.4] (x — у — z, units in m) to cover a quarter of the plate located in the region [0, 0.125] x [0.2, 0.212] x [0, 0.125] (xуz, units in m). The thickness of the plate is 0.012 m. The initial calm water surface is set at у = 0.1 m and so the distance to the plate is 0.1 m in the hydro-code. Symmetry conditions are applied at the boundaries corresponding to the central sections of the plate. The numerical domain is discretised by 80 x 1200 x 80 mesh cells. The purpose of using large number of cells in //-direction is to capture as accurately as possible the peak loadings. The numerical results and experimental measurements for pure and aerated water entries are presented in the following, respectively.

4.4.2 Pure water entry

Figure 16 presents the time series of gauge pressures and total slamming force on the plate for an impact velocity v = 5.5 m/s. The phases of these data have been properly shifted to correlate the first pressure peak at Pi to time t = 0. It is clearly shown that the evolution of the impact pressure loading at the plate centre consists of distinct stages including shock loading, fluid expansion (low pressure) loading and secondary re-loading.

When the plate approaches the water surface, the pressure rises very sharply to the first and highest peak value in less than 1 ms and then drops very quickly to zero. The first and highest pressure spike could be considered as a shock loading (Mitsuyasu, 1966). The duration of the shock loading is less than 2 ms. Table 2 lists the peak loadings on the plate. At Pi, the maximum value of the shock load is 21.57 bar for the numerical simulation and is within 22.16 bar and 23.66 bar for the laboratory measurements. At the other four symmetric gauge points P2, 3, 4 and 5, the computed peak values are 10.44 bar; the laboratory measurement shows that the data varies from 6.53 bar to 11.42 bar for block 1 and it varies from 7.77 bar to 10.71 bar for block 2. The discrepancy between these pressure gauge measurements taken at symmetric positions on the plate may be due to plate flexure or minor variations in plate impact angle. Any of these factors may cause some transducers to contact the water slightly earlier than others. Regarding the total impact loading, the slamming force is positive (upward direction) with a peak value around 68.35 kN to 69.57 kN during this stage (see Table 2). Just after the first peak, the measurement data for Pi also shows a pressure spike up to 4.6 bar in the time series at around t=0.6 ms. This spike is not captured in numerical simulation. The underlying reason that caused this small secondary pressure spike might be due to the reflection of shock (pressure wave) from the bottom of the basin.

The impulses of shock loading on the plate obtained in experiments and numerical simulations are presented in Table 3. These values are calculated as temporal integrals of local pressures or overall forces through the interval [b>, ta]> which covers the lifespan of the shock load. The computed peak pressures at Pi and overall forces listed in Table 2 are close to the measurement for entry velocity v = 5.5 m/s , but the rise time is longer compared to experiment. Therefore the computed impulses are greater than experiment results as shown in Table 3.

After the shock load, the extensively compressed air layer trapped beneath the plate expands and induces the pressure to further decrease towards a vacuum. The lowest absolute pressure obtained in the numerical computation is 0.20 bar at Pi and 0.36 bar at P2 (see Table 4). The measured lowest absolute pressures show a variety of values including negative and positive readings. These negative values may be due to the acoustic noise generated by the shock in the ocean basin. However the observed trend in the measured data is close to the

Pure water slamming loads on the plate for

Figure 16: Pure water slamming loads on the plate for <.'=5.5 m/s. All the pressures presented here are gauge values. The horizontal dashed line represents a perfect vacuum pressure. Phases of all the results are adjusted to correlate the first pressure peak to time <=0. The masses of blocks 1 and 2 are 32 kg and 52 kg respective!}'.

Table 2: Peak gauge pressures and forces on the plate for pure water entry.

Instrument

PI (bar)

P2(bar)

P3(bar)

P4(bar)

P5(bar)

F(kN)

Block 1

22.16

6.53

8.46

9.43

11.42

68.35

Block 2

23.66

8.31

10.71

7.77

8.63

69.57

Numerical

21.57

10.44

10.44

10.44

10.44

68.41

numerical simulation results. After the lowest value, the pressure starts to increase towards zero with a second contraction of the trapped air. At Pi, the duration of the fluid expansion load is 5.2 ms for the numerical simulation and 3.5 ms ~ 3.7 ms for the experiment. At the four symmetric gauge points, the duration of the fluid expansion load is 3.7 ms for the numerical simulation and around 3.5 ms for experiment . The sub-atmospheric pressure on the lower surface causes the plate to experience negative (downward in direction) impact force for around 4 ms as shown in Figure 16. The computed minimum impact force on the plate is -3.54 kN, the measured data is -3.92 kN for block 1 and is -3.29 kN for block 2.

Table 3: Impulses of shock loading on the plate for pure water entry. Pressure and force impulses are calculated as P1 = ff* pdt and F1 = f d t respectively, where t, and fa are the times immediately before and after the shock loading.

Instrument

Pi (Pa-s)

Pi (Pa-s)

Pl(Pa-s)

Pi (Pa-s)

Pi (Pa-s)

F'(N-s)

Block 1

652.46

335.21

327.90

355.95

359.73

25.27

Block 2

719.71

381.59

359.25

388.14

402.53

28.55

Numerical

849.10

614.33

614.33

614.33

614.33

35.02

Table 4: Minimum absolute pressures and forces on the plate for pure water entry, t € [0,5ms],

Instrument

Pl(kPa)

P2(kPa)

P3(kPa)

P4(kPa)

P5(kPa)

F(kN)

Block 1

0.59

41.74

-62.19

27.86

24.05

-3.92

Block 2

-2.67

8.81

-6.37

3.86

22.69

-3.29

Numerical

20.03

36.15

36.15

36.15

36.15

-3.54

When the pressure recovers from sub-atmospheric to atmospheric values, the air layer does not stop contracting but is further compressed and leads to secondary re-loading of the plate. This re-loading is less severe compared to the shock loading. At Pi, the numerical simulation produces stronger re-loading (3.71 bar) compared to the laboratory measurement (1.5 bar ~ 2 bar). This may be due to the inviscid assumption adopted in the numerical simulation, which currently ignores the fact that part of the fluid kinematic energy would be transformed into turbulence and/or dissipated by viscosity. The amplitude of the entire re-loading (force) obtained from numerical simulation is slightly higher than experiments.

Snapshots of the loadings on the plate obtained by numerical simulation and/or experiment are given in Figure 17. The left side represents the solution at t= -0.035 ms, just before the occurrence of shock load peak. The right side indicates the result at f=2.365 ms, which is at a time very close to the trough of the fluid expansion load. The time t=0 corresponds to the first pressure peak at Pi. The top row illustrates contours of the computed pressure distribution on the lower surface of the plate. The middle row depicts the pressure distribution along the horizontal central section of the plate (lower surface). The bottom row shows the computed velocity field at centres of the mesh cells that are just beneath the lower surface of the flat plate. At t= 0.035 ms, the pressure is nearly axisymmetric on the plate apart from the four corners. The computed highest absolute pressure (21.30 bar) is found to be at the plate centre and decreases outwards. The measured pressure here is 21.54 bar for block 1 and 21.95 bar for block 2. The computed lowest pressures (5.78 bar) are at the four corners of the plate. The velocity vectors are directed outward from the plate centre. This means the fluid beneath the plate is being expelled away from the plate. At t=2.365 ms, the plate is experiencing non-axisymmetric sub-atmospheric pressure loading. The computed lowest pressure (0.20 bar) is at the plate centre and it increases outwards. Meanwhile, the measured pressure at the plate centre is 0.33 bar for block 1 and 0.21 bar for block 2. The computed highest pressures (1.04 bar) are at the four corners of the plate. The velocity vectors are directed inwards towards the plate centre. This indicates that ventilation may happen as the fluid is being drawn in under the plate.

Impact loadings on the flat plate at Г—0.035 ms (left) and T=2.365 ms (right) for c=5.5 m/s, pure water entry

Figure 17: Impact loadings on the flat plate at Г—0.035 ms (left) and T=2.365 ms (right) for c=5.5 m/s, pure water entry.

Some researchers have claimed that cavitation occurs in hydrodynamic slamming events as the pressure descends dramatically after the shock load (Faltinsen, 2000). In the present experimental study, some transducers provide negative minimum absolute pressures, the measured positive minimum absolute pressures are within the range of 0.59 kPa to 47.14 kPa. The present minimum pressures obtained in the numerical simulation at Pi is 20.03 kPa. Compared to the water saturation pressure of 1.70 kPa at 15°C, most of the valid/positive experimental data and all the numerical results are still above the cavitation condition except the lowest pressures 0.59 kPa measured at Pi of block 1. Consequently, we are currently not able to conclude that cavitation certainly occurs in the water entry of the square plate. However, it is obvious that the computed and measured low pressures are very close to the cavitation condition, and so the surrounding water is very likely to be in tension.

4.4.3 Aerated water entry

As the shock load is the most severe and dangerous force acting on the flat plate structure, here we focus on investigating effect of aeration on the first peak loading. Before presenting the results for aerated water entry, it is necessary to clarify the differences in introducing air bubbles into the water between the experiment and the numerical computations.

In the laboratory, the bubble generation system was placed at the bottom of ocean basin, which is filled with fresh water for a depth of 1 m. The air bubbles rise from the bottom of the ocean basin to the water surface and break up. The size of the bubbles significantly increase as they ascend. Therefore the aeration level near the water surface is notably higher than at the bottom of basin. Nevertheless the bubble aeration was measured at one water depth only (0.25 m away from the water surface) and assumed uniform throughout depth.

The numerical model does not treat the air bubbles explicitly in a one-by-one manner, but instead assumes the water to be uniformly mixed with air bubbles at a specified concentration level (and thus implied speed of sound). Thus, the aeration level is represented by the air volume fraction aq in Equation (14). As the change of aeration level with water depth ax (y) is currently not available, we apply the average representative value measured in laboratory to set up the initial conditions for the numerical computations and assume that ai (y) does not vary with the water depth. Obviously, this is a simplified and conservative treatment as the real aeration level near the water surface is notably under-estimated in the numerical model’s setup.

Figure 18 shows the time series of the gauge pressures on the plate under different aeration levels. The left column represents the pressures measured for block 1 (32 kg), the middle column indicates the data for block 2 (52 kg) and the right column illustrates the numerical results. From the top to bottom rows are pressures gauged/computed at locations Pi, 2, 3, 4 and 5. Note that all the time series have been shifted to correlate the first peak of Pi to time t = 0.

Close inspection of these figure shows that the peak impact pressures on the plate are significantly reduced by aeration. In the numerical computation, at all gauge points, the pressures are continuously reduced by increased aeration levels. The highest aeration level «1=1.6% reduces all the pressures by 49.4% ~ 50.7%. Looking at the measurement data, the reduction in pressure is greater when the same level of aeration is used. Here, we would like to emphasise again that the aeration level near the water surface is noticeably higher than the basin bottom. In the experiment, the aeration levels were measured at only one depth with a distance of 0.25 m away from the water surface. Since the real aeration near the water surface is higher than the measurement point, we only implement the significant average measurement value, which is rather conservative/under-estimated, in the numerical simulation.

Looking at the measured pressures at P2, 3, 4 and 5 shown in Figure 18, it is easy to spot that the occurrences of the pressure peaks are notably non-synchronous with Pi by phase shifts up to 4.2 ms. In the experiment, the water surface is not flat for the aerated cases due to the disturbance caused by rising bubbles, and the surface disturbance is greater for higher aeration levels. Therefore the pressure transducers installed on the plate are not in contact with the water surface at the same time. While in the numerical simulation, the surface is assumed to be flat. This leads to the significant difference in phase and amplitude of the pressure peaks. The details of peak pressures as well as impact forces measured/computed

Impact pressures on the plate in water with different aeration levels (ar=0~1.6%) for v =5.5 m/s. Left

Figure 18: Impact pressures on the plate in water with different aeration levels (ar1=0~1.6%) for v =5.5 m/s. Left: block 1 (32 kg); Middle: block 2 (52 kg); Right: numerical computation. From the top to bottom rows are pressures at gauge points Pi, 2, 3, 4 and 5. The computed results at P2, 3, 4 and 5 are identical.

under different aeration levels and impact velocities are listed in Table 5. The computed minimum absolute pressures and forces under different aeration levels are listed in Table 6.

Figure 19 shows the computed pressures on the plate near the occurrence of peak loading and trough loading in 0.8% aerated water. Similar to the pure water entry cases, at the time near the occurrence of peak loading, the pressure distribution on the plate is nearly axisym- metric apart from the four corners. At the time near the occurrence of trough loading, the pressure distribution is non-axisymmetric. The trapped air undergoes intensive compression and expansion. The pressure distribution for cases with higher aeration levels are similar to these two figures (except the amplitude of pressure), therefore they are not included here.

Table 5: Maximum gauge pressures and forces on the plate for aerated water entry.

Instrument

PI (bar)

P2(bar)

P3(bar)

P4(bar)

P5(bar)

F(kN)

Block 1

0.0

22.16

6.53

8.46

9.43

11.42

68.35

0.8

6.35

6.40

6.14

2.47

6.23

18.55

1.0

3.77

3.86

3.87

4.82

1.93

12.11

1.6

-

-

-

-

-

-

Block 2

0.0

23.66

8.31

10.71

7.77

8.63

69.57

0.8

2.89

7.50

3.94

6.08

6.27

21.69

1.0

3.67

3.97

4.03

4.12

2.92

15.87

1.6

-

-

-

-

-

-

Numerical

0.0

21.57

10.44

10.44

10.44

10.44

68.41

0.8

13.75

6.41

6.41

6.41

6.41

46.65

1.0

12.81

5.97

5.97

5.97

5.97

43.72

1.6

10.92

5.14

5.14

5.14

5.14

37.76

Table 6: Computed minimum absolute pressures and forces on the plate for aerated water entry.

a,(%)

Pl(kPa)

P2(kPa)

P3(kPa)

P4(kPa)

P5(kPa)

F(kN)

0.0

20.03

36.15

36.15

36.15

36.15

-3.54

0.8

8.67

28.79

28.79

28.79

28.79

-4.34

1.0

9.56

32.47

32.47

32.47

32.47

-4.21

1.6

14.61

44.72

44.72

44.72

44.72

-3.65

Figure 20 shows the peak pressures at Pi and total impact forces on the plate for pure and aerated water entries obtained in experiment and numerical simulation. All the numerical results indicate that both the pressure and force decline with increased aeration levels. When bubbles are generated to aerate the fresh water in laboratory, the water surface is greatly disturbed by the quickly rising and breaking bubbles. This causes some difficulty in measuring impact loadings and leads to obvious phase shifts of pressure peaks. Nevertheless the experimental data illustrates that aeration can dramatically reduce the peak loadings.

Figure 21 shows the impulses of shock loading on the plate for entry into pure and aerated water. These impulses are calculated as time integrals of pressure or force in the interval [<ь, ta], which covers the lifespan/duration of the shock load. Looking at the pressure impulse at Pi, the numerical simulation shows a declining trend with increased aeration level (but the reduction is mild compared to the reduction of peak pressures); the experiment does not clearly show this declining trend. Looking at the force impulse on the plate, neither numerical computation nor experiment shows a clear declining or rising trend: the force impulse experiences both rise and decline. The difference in shock load impulse between pure water and aerated water with highest aeration level is presented in Table 7. For pressure impulse at Pi, the maximum aeration level seems to be able to reduce it by 11.9%~19.1%. Meanwhile the maximum aeration level causes up to 17.2% variation of force impulse.

Computed contours of absolute pressures (Units in bar) on the plate for t>=5.5 m/s and aeration level 0.8%. Left

Figure 19: Computed contours of absolute pressures (Units in bar) on the plate for t>=5.5 m/s and aeration level 0.8%. Left: t =0.18 ms, just after the occurrence of peak loading at PI; Right: {=3.18 ms, near the occurrence of trough loading at Pi.

Aeration effects on the peak impact loadings for v=5.5 m/s. Left

Figure 20: Aeration effects on the peak impact loadings for v=5.5 m/s. Left: Peak gauge pressure at Pi; Right: Total impact force on the plate.

Aeration effects on the impulse of shock loadings for u=5.5 m/s. Left

Figure 21: Aeration effects on the impulse of shock loadings for u=5.5 m/s. Left: pressure impulse at Pi (Units in Pa-s); Right: Total force impulse on the plate (Units in N-s).

Table 7: Variation of shock load impulse between pure water impact and aerated water (with highest aeration level) impact.

Instrument

P1, (Pa-s)

F‘(N-s)

Block 1

652.4C

574.85

-11.9

25.27

24.13

-4.5

Block 2

719.71

582.58

-19.1

28.55

23.05

-17.2

Numerical

849.10

728.90

-14.2

35.02

33.08

-5.5

 
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