The Risk to the Undersea Engineering Ecosystem of Systems: Understanding Implosion in Confined Environments

Abstract

As humans continue to develop the undersea engineering ecosystem of systems, the consequences of catastrophic events must continue to be investigated and understood. Almost every undersea pressure vessel, from pipelines to sensors to unmanned vehicles, has the potential to experience a catastrophic collapse, known as an implosion. This collapse can be caused by hydrostatic pressure or any combination of external loadings from natural disasters to pressure waves imparted by other implosion or explosion events. During an implosion, high-magnitude pressure waves can be emitted, which can cause adverse effects on surrounding structures, marine life, or even people. The imploding structure, known as an implodable volume, can be in a free-field or confined environment. Confined implosion is characterized by a surrounding structure that significantly affects the flow of fluid around the implodable volume. Often, the confining structure is cylindrical, with one closed end and one open end. This work seeks to understand the effect of fluid flow restriction on the physics of implosion inside a confining tube. To do so, a comprehensive experimental study is conducted using a unique experimental facility. Thin-walled aluminum cylinders are collapsed inside a confining tube within a large pressure vessel. High-speed photography and 3D Digital Image Correlation are used to gather structural displacement and velocities during the event while an array of dynamic pressure sensors capture the pressure data inside the confining tube. The results of this work show that by changing the size of the open end, referred to as the flow area ratio, there can be a significant effect on the structural deformations and implosion severity. It also reveals that only certain configurations of holes at the open end of the tube play a role in the dynamic pressure pulse measured at the closed end of the tube. By understanding the consequences of an implosion, designers can make decisions about where these pressure vessels should be in relation to other pressure vessels, critical infrastructure, marine life, or people. In the same way that engineers design for earthquakes and analyze the impact their structures have on the environment around them, contributors to the undersea engineering ecosystem should design with implosion in mind.

1. Introduction

Deep underwater, there are a variety of pressure vessels that form an undersea engineering ecosystem of systems that play a vital role in our life on land. Undersea pipelines carry gas or oil, powering systems that create electricity. Undersea cables and the related infrastructure carry data communications. Remotely operated vehicles allow for the maintenance of systems, scientific exploration, or rescue. Undersea sensors provide us with information on how the world is changing around us. Humans occupy these undersea pressures to enable deep-sea exploration, which would otherwise be impossible for scuba divers [1]. As technology continues to advance, there is even the possibility of creating undersea habitats for humans [2]. The impact the undersea ecosystem of systems has on our everyday lives is enormous. As humans continue to design systems to survive the depths of the ocean, the creation of stable and sustainable undersea pressure vessels becomes increasingly important.

It is well known that these undersea pressure vessels are subject to a very harsh marine environment and can be vulnerable to numerous loadings from natural disasters like earthquakes to anchor impacts from fishing activities. These loadings, combined with the depth pressure, make some pressure vessels susceptible to implosion, the sudden inward collapse of the structure. Once the implosion occurs, a high-pressure wave can be emitted that can have a tremendous impact on everything in the vicinity. It is fundamentally important to understand the potential impacts of undersea pressure vessels and the waves they may emit on other systems within the engineering ecosystem, marine habitats or biology, or even personnel.

It has long been known that high-magnitude pressure waves affect marine creatures, from invertebrates like oysters [3] to charismatic megafauna like whales. These pressure waves can disrupt behavior or even kill marine creatures [4]. These pressure waves also have the potential to cause chain reactions, which could cause additional implosions. One such example of this occurred at the Super-Kamiokande neutrino detection facility, where one implosion caused a chain reaction of thousands more [5]. It is readily apparent that an implosion can affect everything from man-made systems to marine biology. Therefore, as important as it is to design to avoid implosion, it is equally important to understand the consequences of an implosion. By understanding the consequences of an implosion, designers can make decisions about where these pressure vessels should be in relation to other pressure vessels, critical infrastructure, marine life, or people. In the same way that engineers design for earthquakes and analyze the impact their structures have on the environment around them, contributors to the undersea engineering ecosystem should design with implosion in mind.

Implosion can be broadly categorized into occurring in two types of environments: free-field and confined. An implosion in a free-field environment is defined by the absence of nearby structures that could significantly affect the dynamics of the collapse. Free-field implosion has been studied by multiple authors [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] and is reasonably well understood for many different structures and situations. Confined implosion, by contrast, is characterized by an object or group of objects in close proximity to the collapsing pressure vessel that significantly affects the flow of fluid. The physics of the implosion and resulting pressure pulse can change drastically. In some cases, the energy of the implosion can be focused, resulting in pressure waves much larger than what would be seen in a free-field environment. In other cases, the lack of water to drive the collapse can stop the implosion from progressing [24,25].

Despite the importance that implosion can play in our everyday lives and the complexity of confined implosion, limited research has been performed. The work that has been conducted to date is limited to a cylindrical confining structure with one open end and one closed end. Gupta et al. [24,25] and Salazar et al. [26] performed implosion experiments in a small tank. These studies concluded that, due to the lack of available water, there is no pressure wave emitted. Other work, including [27,28,29,30], has shown that, with enough available water that can enter the system, the implosion pressure pulse could be even larger than free-field.

Overall, the amount of fluid that can enter the confined environment plays a dramatic role, but the amount of this flow as a function of the emitted pressure pulse has not been studied. This paper discusses the results of a comprehensive set of experiments intended to investigate the effect of flow restriction on the implosion of aluminum cylindrical volumes inside a confining tube. A unique experimental setup, described in Section 2, is used to record pressure and structural deformation information during the implosion process. In later sections, the results of the experiments are described in detail, including a comparison between in-tube and free-field implosion, the repeatability of in-tube experiments, the effect of flow restriction, and the effect of the number of holes.

2. Experimental Procedure

There are two separate experimental setups in this work: free-field and confined. The free-field setup is virtually identical to previous work and is discussed in detail by Gupta et al. [19]. The objective of the free-field experiment is to provide a direct comparison to previous work at the appropriate collapse pressure, specimen size, and material. The confined implosion setup is comprised of multiple separate structures: a large pressure vessel, a confining tube, and an implodable volume.

The implodable volumes are constructed from commercially available 6061-T6 aluminum seamless extruded tubing with a nominal outer diameter of 38.1 mm (1.5 in), unsupported length of 254 mm (10 in), and wall thickness of 0.89 mm (0.0349 in). The implodable volumes are sealed at both ends using solid aluminum endcaps and O-rings. All specimens have a collapse pressure between 2.58 MPa (374 psi) and 2.65 MPa (387 psi). The pressure plots in the rest of the paper are normalized by the collapse pressure to eliminate the effect of the hydrostatic pressure on the pressure history.

As shown in Figure 1, implodable volumes are placed concentrically inside (Figure 1C) the confining tube (Figure 1B) that sits in the center of the underwater pressure vessel facility (Figure 1A). The pressure vessel facility is a 2.13 m (84 in) diameter semi-spherical vessel. A schematic of the test setup is shown in Figure 2.

The confining tube is 1143 mm (45 in) in length and has an inner diameter of 178 mm (7 in). The confining tube consists of a closed end and an open, or partially open, end. There are three sections of the confining tube: two made up of 6061-T6 aluminum with a wall thickness of 25.4 mm (1 in) and one 152 mm (6 in) section made up of a transparent, acrylic window section. The acrylic section is used to record high-speed photography of the implodable volume during collapse. The interfaces between sections have O-rings in order to prevent leakage into the tube. A cross-section of the confining tube (Figure 2, Section A) shows the pressure sensor locations and their corresponding channel numbers.

The open end of the confining tube can be outfitted with restricting plates that reduce the available water flow area into the tube. The flow area ratio, or AR, is defined as the ratio between the area available for flow and the total available area of the open tube. For example, an area ratio of 100% indicates that the tube is fully open, while an area ratio of 5% indicates that only a small hole is available. The restrictor plates are 25.4 mm (1 in) thick aluminum. In the case where a restrictor plate is present, the confining tube is 25.4 mm (1 in) longer. No plate is required for the fully open case where the AR is 100%. The dimensions of the restrictor plates used are shown below in

Table 1. Restrictor plate information.

Plate No.	No. Holes	Hole Location	Area Ratio	Hole Dia. (mm)	Hole Dia. (in)
Plate 1	1	Center	5%	39.9	1.57
Plate 2	1	Center	10%	56.4	2.22
Plate 3	1	Center	22%	83.3	3.28
Plate 4	1	Center	30%	97.3	3.83
Plate 5	1	Center	59%	136.4	5.37
Plate 6	1	Edge	30%	n/a 1	n/a 1
Plate 7	16	Edge	30%	24.4	0.96
Plate 8	64	Center	30%	12.2	0.48

¹ Plate 6 is not a circle and therefore does not have a diameter. The radius of the removed material in plate 6 is 3.5 in (89 mm), and the largest width is 2.38 in (60.5 mm).

.

Restrictor plates 1 through 5 consist of a single hole in the center of the plate, as shown in Figure 3. Plates 6 through 9 have different hole configurations in order to understand the configurations’ effect on the implosion.

The inside of the confining tube is outfitted with dynamic, flush-mount pressure sensors (PCB Piezotronics model 113B22) along the length in order to measure the pressure inside the tube throughout the event. The free-field pressure sensors used outside the confining tube are PCB Piezotronics model 138A06 Tourmaline ICP Underwater Blast Sensors. In Figure 2, the flush-mount sensors are shown as yellow squares, and the free-field sensors are shown as green circles. The pressure signal is recorded using an Astro-med Dash 8HF-HS data recorder at a sampling rate of 2 MHz. The pressure data is filtered using a low-pass fourth-order Butterworth filter with a threshold frequency of 5 kHz.

High-speed photography is used to record the event through optical viewing windows in the pressure vessel facility. Two high-speed cameras (Photron SA1) record at 40,000 frames per second. The specimens are painted with a random black-and-white pattern in order to employ the technique of 3D Digital Image Correlation (DIC). Three-dimensional DIC is used to measure the displacements and velocities of the structure during collapse. Gupta et al. [30] have established this technique for this specific setup.

To conduct the experiment, the pressure vessel, and subsequently the confining tube, are filled with water. Note that the implodable volume is sealed with ambient pressure air inside. A small air gap is left at the top of the tank to reduce the pressure drop inside the tank once the implosion is initiated. Nitrogen is added to the air gap to pressurize the water until the specimen reaches critical pressure and collapses.

3. Results and Discussions

3.1. Fundamentals of In-Tube Implosion

In order to understand the effect of flow restriction, it is first critical to understand how implosion in a free-field environment differs from that in a confined environment. In this section, a comparison between a free-field case and an in-tube case with a fully open end (AR = 100%) is shown. In the following sections, this free-field case will be compared to in-tube, flow-restricted cases.

The differences between the two cases are illustrated in Figure 4. In Figure 4a, the pressure–time history is shown for the gauge that yields the highest pressure for each case. Therefore, for the field-field case, the gauge is 50.8 mm (2 in) from the center of the implodable, and for the in-tube case, the gauge is at the closed end of the confining tube [20,31]. The velocity at the center of the implodable volume is shown in Figure 4b.

For the case where the confining tube has one completely open end, the difference between in-tube and free-field implosion is stark. During a free-field implosion, the opposing walls of the specimen accelerate towards one another, causing a pressure drop around the implodable [6]. Once the walls make contact, their rapid change in velocity causes a change in momentum in the surrounding fluid, thus causing a high-pressure wave to radiate away from the implodable volume. For the in-tube case, the implodable volume begins to collapse as the opposing walls accelerate toward one another. This causes a pressure drop around the implodable. At the bottom of the confining tube, the water moves away from the closed end to compensate for the drop in pressure local to the implodable, causing a very significant drop in pressure at the closed end. This drop in pressure is referred to as the under-pressure region and is made up of the duration the pressure is lower than the hydrostatic pressure. Due to the lower pressure inside the confining tube, water begins to flow into the open end of the confining tube. When the water reaches the closed end of the confining tube, it decelerates and over-compresses, causing a large spike in pressure known as the over-pressure region [30]. This cycle of over-pressure and under-pressure continues inside the tube as the pressure waves oscillate axially.

Comparing the in-tube and free-field pressure histories in Figure 4a, there is a significant difference in the magnitude and time scale of the under-pressure and over-pressure regions. Fundamentally, the mechanisms that cause these pressures, the motion of the water, are the same for both cases. However, the confining structure significantly alters the fluid motion around the implodable volume. In order to continue collapsing, the implodable volume requires a driving force, like the hydrostatic pressure of the water. For the free-field case, the implodable volume can pull water from all sides. By contrast, the in-tube case can only draw water from the open end of the confining tube. This is what causes the differences in the pressure between the two cases. Because the in-tube case can only draw water from the open end, the size of that opening can become very important.

In terms of the center-point velocity of the implodable volume, shown in Figure 4b, there are significant differences between free-field and in-tube implosion. For the free-field case, the implodable reaches its maximum velocity and then rapidly decelerates to a velocity of zero, indicating wall contact. For the in-tube case, the maximum velocity is less than that of the free-field. After reaching maximum velocity, the in-tube case decelerates more slowly than the free-field case. This deceleration is because the hydrostatic pressure within the tube has decreased, and water has not yet compensated for the drop in pressure by rushing through the open end of the tube. This is readily apparent when comparing the under-pressure region of the two curves. The driving force inside the tube decreases, which affects the velocity of the implodable volume. This was first identified by Gupta et al. [30].

It can be seen that the fluid flow around the implodable volume plays an important role in the resulting implosion dynamics. It is obvious that the free-field and in-tube cases are different in terms of both pressure and velocity. These differences can be attributed to the availability of water. For the in-tube case, water can only be drawn into the tube through the open end. Therefore, the size of the open end will play a fundamental role in the implosion dynamics. In the next sections, this concept will be explored at length.

3.2. Repeatability of In-Tube Experiments

The level of repeatability of identical tests is fundamental to understanding the results of this work. Three different cases are repeated, each repeat using a single-hole restrictor plate of a different area ratio. For each case, the pressure history is shown at the bottom (a) and top (b) of the confining tube. The repeatability of these tests can be used to provide an understanding of what magnitude differences matter in experiments. The three repeated cases are for area ratios of 100% (Figure 5), 30% (Figure 6), and 5% (Figure 7).

Russel Comprehensive error [31] is used in order to quantify the repeatability of the experiments. Russel Comprehensive error takes into account both the magnitude and phase when comparing two transient signals. A lower value of error corresponds to a more closely related signal. The table below,

Table 2. Repeatability of in-tube implosion cases.

Case	RC Closed End	RC Open End	Peak Pressure	Over-Pressure Impulse	Collapse Pressure
AR = 100%	0.105	0.095	1.2%	5.0%	2.7%
AR = 30%	0.126	0.132	1.4%	6.1%	0.5%
AR = 5%	0.076	0.036	1.5%	0.8%	1.3%

, shows the Russel Comprehensive (RC) error at the closed end and open end of the confining structure for each repeat case. The table also shows the percent difference between peak pressures at the closed end, dynamic over-pressure impulse at the closed end, and collapse pressure of the repeat cases.

Overall, these cases are all extremely repeatable. The maximum difference for peak pressure is 1.5% and for over-pressure impulse is 6.1%. This also shows that small differences in the collapse pressure do not have a significant effect on the repeatability between tests in terms of pressure. The Russel Comprehensive (RC) error will be used in Section 3.4 to help describe the effect of hole configuration. The RC error shows that the case where the area ratio of 30% is the least repeatable of the three cases.

3.3. Effect of Flow Restriction

In order to understand the effect of flow restriction on the implosion dynamics, experiments conducted using plates 1 through 5 are compared in this section. Each plate has a single hole in its center, and the area ratio (AR) is between 5% and 59%. These cases of reduced flow area are compared to the fully open case, which has an area ratio of 100%. First, the deformation of the structure will be discussed. Next, the relationship between these deformations and the pressure inside the confining tube will be established.

3.3.1. Structural Response

The full-field, out-of-plane velocity contours of the implodable volume for two area ratios are shown in Figure 8. The comparison is between the fully open (AR = 100%) and heavily restricted (AR = 5%) velocities. As identified by Gupta et al. [30], the implodable volume deformation for in-tube implosion can be broken into the following phases: (1) acceleration, (2) deceleration, (3) contact initiation, and (4) buckle propagation. The acceleration phase is very similar for both cases, where the center of the tube begins to accelerate inward, which is shown in images 0 ms through 1.5 ms. After the acceleration phase, however, there are clear differences between the two cases. The deceleration phase lasts longer for the flow-restricted (AR = 5%) case. Wall contact, which marks the end of the deceleration phase, occurs at 2.2 ms for the 100% case and 2.7 ms for the 5% case. The buckle propagation phase also lasts longer for the restricted case. The specimen is fully flattened at 3.5 ms for the 100% case but 5.5 ms for the 5% case. The flow restriction causes the collapse rate of the specimen to decrease. During the buckle propagation phase, it has been noted by Gupta et al. [30] that there is an asymmetry in which the end of the implodable toward the open end deforms first because the pressure drop is less towards the open end.

In order to understand the effect of each flow restriction, two out-of-plane velocity plots are shown in Figure 9. Figure 9a shows the velocity at the center point of the implodable. Figure 9b shows the velocity at point P1, which is a point on the implodable volume along the center axis but toward the open end of the tube. Looking at the center-point velocity, all cases accelerate similarly to a maximum (negative) velocity at around 1.5 ms. There does not appear to be a relationship between maximum center-point velocity and the area ratio. During the acceleration phase, the flow restriction does not appear to play a significant role in the center-point velocity. Any contributions of flow restriction are likely within the error of the DIC or repeatability of the experiments. Next, the structure begins to decelerate and approach wall contact, which is designated by a velocity of 0 m/s. At the center point, the 100% and 30% cases appear to reach wall contact at very similar times, around 2.2 ms. The other cases do not reach wall contact until 2.5 ms or later. There is not a consistent relationship between area ratio and time to wall contact. This can be because the velocity is affected by many parameters, including the collapse pressure of the experiment and material imperfections. However, discounting the 30% case as an outlier, there is a clear effect of area ratio during the deceleration phase as all other cases clearly take longer to reach wall contact than the 100% case. Although the effect of each area ratio cannot clearly be resolved, there is an influence of area ratio on the collapse of the structure. In the next section, these structural deformations will be related to the pressure inside the confining structure.

Figure 9b shows the out-of-plane velocity at point P1. At point P1, closer to the open end of the tube, the velocity first begins to increase to a local maximum. As previously discussed, the initial acceleration phase (up to 1.5 ms) is very similar for all cases. As the acceleration phase ends, the velocity curves at point P1 begin to diverge. Each of the cases experiences a different maximum (negative) velocity. This maximum is likely a combination of experimental parameters such as the collapse pressure or material properties of the specimen as well as the flow restriction. After this maximum, the structure enters the deceleration phase and the velocity begins to decrease.

After the deceleration phase, wall contact occurs at the center of the specimen and point P1 enters the buckle propagation phase. For the 100% case, during the buckle propagation phase, the velocity again increases to a maximum before quickly decelerating to 0 m/s (wall contact). During the buckle propagation phase of the 5% case, the velocity continues to decrease until a velocity of 0 m/s (wall contact). Therefore, the area ratio has a clear effect on the velocity of point P1 when comparing these two cases. The other cases show velocity profiles that are between 100% and 5%. There is no clear relationship between area ratio and structural velocity. However, there is a clear effect of area ratio when comparing the 100% and 5% cases.

Overall, flow restriction has an effect on the structural velocities after the acceleration phase. However, there is not a clear relationship between the area ratio and the time to wall contact at the center point. At the center point, the similarity between most flow restriction cases indicates that the effect on velocity may not change significantly with the area ratio. In the following section, the relationship between these structural deformations and the pressure inside the confining tube will be established.

3.3.2. Pressure Response

The following section seeks to understand the effect of flow restriction on the pressure of the implosion. The dominant cause of the pressure spike for this in-tube case is the in-rushing water from the open end of the tube. The peak pressure magnitude is the largest at the closed end of the tube [25]. Figure 10a shows the pressure history at channel 2, at the closed end of the tube, and Figure 10b shows the pressure history at channel 5, near the open end of the tube.

At the closed end of the tube, Figure 10a, the initial drop in pressure, which occurs up to around 3 ms, is nearly identical for all cases. During this initial drop, water moves away from the closed end of the tube to support the decreasing pressure local to the implodable volume. This causes the pressure at the closed end of the tube to decrease significantly such that cavitation forms. As discussed in the previous section, the acceleration phase is very similar for all cases, so it is expected that the bottom gauge would be similar for all cases. The closed end is significantly far from the open end of the tube, and therefore the pressure waves created by the water flowing into the open end have not yet reached the closed end.

However, the initial under-pressure at the top of the confining tube, channel 5 in Figure 10b, is quickly affected by the water flowing into the open end. While the pressure at the closed end is similar up to 3 ms, the pressure near the open end is similar for less than 2 ms. The minimum pressure at 2 ms decreases with the area ratio, as the smaller area ratios have an increasingly difficult time compensating for the drop in pressure in the confining tube caused by the collapsing implodable volume. This is the first indication that the average pressure inside the confining tube decreases with area ratio, which equates to less pressure driving the collapse of the implodable volume. As discussed in the previous section, the time to wall contact changes when the flow is restricted. This can be attributed to the lower driving pressure around the implodable volume inside the confining tube.

After the initial drop in pressure, the water entering the open end increases the pressure throughout the confining tube. As the pressure begins to increase inside the confining tube, after 3 ms in Figure 10a, the flow restriction plays a dramatic role. From the structural deformations shown in Figure 8a, the 100% case has completed its deformation by 3.5 ms. This is because there was sufficient hydrostatic pressure in the region of the implodable volume to continue to deform the specimen throughout the event. Due to the fact that the implodable volume is completely flattened, the pressure at the closed end increases as more water enters the confining tube. However, the over-pressure region is different for the flow-restricted cases. Using the 5% cases as the extreme example, there is an increase in pressure at the closed end from 3 ms to 4 ms, a decrease from 4 ms to 5 ms, and another increase beyond 5.5 ms. From the structural deformations in Figure 8b, it is known that the implodable volume does not complete its deformation until 5.5 ms. Unlike the 100% case, the implodable volume does not experience enough driving pressure to fully collapse. Therefore, once the pressure inside the tube begins to increase as water enters, the pressure reaches a large enough value to deform the implodable volume again. The deformation of the implodable volume causes the drop in pressure seen at 4 ms as the water at the closed end moves towards the implodable volume to compensate for the change in volume. This second drop in pressure is seen for all cases with area ratios of less than 30%, and there appears to be a relationship between the area ratio and the magnitude of the drop in pressure. After this drop in pressure, the pressure inside the confining tube increases to its maximum or peak pressure.

Overall, the deformation of the implodable volume and the pressure inside the confining tube are inherently coupled. As the structure deforms, the pressure inside the confining tube decreases, which can lead to a decreasing rate of deformation. This decreasing rate of deformation can lead to less water flowing into the confining tube, potentially leading to a less severe implosion. This is why the area ratio has such a significant impact on the physics of the in-tube implosion: it changes the pressure driving the structure, which affects the entire in-tube implosion process.

The peak pressure at the bottom of the confining tube as a function of the flow area ratio is shown in Figure 11. The graph shown is normalized by collapse pressure; therefore, the hydrostatic pressure during collapse is shown as a value of one on the graph. As shown in Figure 11, the peak pressure at large area ratios is very minimally affected. For a large decrease in flow area ratio, from 100% to 59%, the resulting peak pressure only decreases by 3.9%. However, at an area ratio of 5%, the peak pressure is 41% less than the 100% case.

The relationship between peak pressure and area ratio can be represented using the following equation:

$$\mathrm{P}={{\mathrm{K}}_1\mathrm{e}}^{{\mathrm{K}}_2\left(\mathrm{A}\right)}+{{\mathrm{K}}_3\mathrm{e}}^{{\mathrm{K}}_4\left(\mathrm{A}\right)}$$

(1)

where P is the peak pressure, K₁, K₂, K₃, and K₄ are constants shown in

Table 3. Equation (1) constants and goodness-of-fit metrics.

K1	K2	K3	K4	R2	RMSE
1.885	0.002	−1.665	−0.227	0.9586	0.0692

, and A is the flow area ratio. This relationship is the best fit of the data, and the measure of the goodness of fit is shown in Table 3. The adjusted R² and Root Mean Square Error (RMSE) values in Table 3 provide an excellent correlation between the experimental data and Equation (1).

Equation (1) shows that the peak pressure as a function of area ratio can generally be split into two regions: the linear region and the decay region. The linear region is the region in which the relationship between peak pressure and area ratio is relatively linear over a large number of area ratios. In Figure 11, the linear region occurs for area ratios above around 22%. The decay region is the region in which the relationship between peak pressure and area ratio changes significantly over a small number of area ratios. In Figure 11, the decay region occurs for area ratios below around 22%. Equation (1) consists of two exponential functions, each with two constants. Although the two functions do not act independently, each function largely describes one region of the curve.

The term consisting of constants K₁ and K₂ describes the linear region. K₁ is a positive constant that describes the stable peak pressure. This is the magnitude of the peak pressure at which the linear region begins. At an area ratio of 22%, the peak pressure is 1.96, which is very close to the value given by K₁. Physically, this is a measure of the overall severity of the implosion. K₂ is a small value that can be either positive or negative and indicates the slope of the linear region. Physically, this is a measure of the sensitivity to changes in area ratio throughout the linear region. A negative value indicates a downward slope, and a positive value indicates a positive slope. A value of zero indicates a horizontal line at a constant value of K₁. The closer the value of K₂ is to zero, the lesser the contributions of the area ratio from 100% to where the decay region begins.

The term consisting of constants K₃ and K₄ describes the decay region. K₃ is a negative constant that is similar in magnitude to constant K₁. K₃ helps describe the shape of the “knee” in the curve. K₄ is a small, negative constant that describes the rate of decline of the decay region. Physically, K₄ is proportional to the area ratio that is the transition point between the two regions of the curve.

These constant values are likely a function of many parameters, including the implodable volume and confining structure geometry. Therefore, this set of constants is only valid for this particular set of parameters, including the material and size of the implodable volume.

The magnitude of the impulse of the normalized dynamic pressure is shown in Figure 12 for the first over-pressure region at the bottom of the confining tube (channel 2). Recall that the first over-pressure region is time where the pressure is above the hydrostatic pressure once collapse is initiated. The normalized dynamic impulse is in units of normalized pressure (unitless) multiplied by milliseconds. This value of impulse is for dynamic values of pressure. The impulse increases slightly as the area ratio decreases until the 10% case. In this case, there is a very sharp decrease in impulse.

The relationship between impulse and area ratio can be represented using the following equation:

$$\mathrm{I}={{\mathrm{K}}_5\mathrm{e}}^{{\mathrm{K}}_6\left(\mathrm{A}\right)}+{{\mathrm{K}}_7\mathrm{e}}^{{\mathrm{K}}_8\left(\mathrm{A}\right)}$$

(2)

where I is the over-pressure impulse, K₅, K₆, K₇, and K₈ are constants shown in

Table 4. Equation (2) constants and goodness-of-fit metrics.

K5	K6	K7	K8	R2	RMSE
1.763	−0.001	−1.930	−0.132	0.989	0.038

, and A is the flow area ratio. This relationship is the best fit of the data, and the measure of the goodness of fit is shown in Table 4. The goodness-of-fit metrics indicates a great correlation between the experimental data and Equation (2).

Equations (1) and (2) are in the same form but have a different set of constants. Identical to Equation (1), the impulse as a function of area ratio can also be described by two regions: the linear region and the decay region. Constants K₅ and K₆ describe the linear region, while constants K₇ and K₈ describe the decay region. The contributions of these constants to each region described for Equation (1) are valid for Equation (2). The linear region of the impulse curve is more stable than the linear region of the peak pressure curve, as K₆ is closer to zero than K₂. It is also interesting to note that while the peak pressure is increasing over the linear region, the impulse is decreasing. It is not understood why that is the case, but there is nothing to indicate that peak pressure and impulse should behave the same way.

For the free-field case shown in Figure 4, the peak pressure is 1.7, which is very similar to the case where the area ratio is 10%. The over-pressure impulse for the free-field case is 0.12, which is significantly less than all in-tube impulses shown. In terms of over-pressure impulse only, the in-tube cases of this work are all more severe implosions. Due to the relatively small impulse of the free-field case, it is expected that the contributions to the in-tube over-pressure impulse from the collapse of the implodable volume are small. This supports the fundamental idea of in-tube implosion, where the bulk of the large pressure wave is caused by the momentum of the water hitting the closed end.

3.3.3. Post-Experiment Results

The deformed, post-experiment implodable volume for each area ratio is shown in Figure 13. The right side of each specimen is the side closer to the open end of the confining tube. It can be observed that, for the 100% through 22% cases, there is fracturing on both ends of the specimen near the endcap. However, for the 10% case, there is no fracturing on the right side. For the 5% case, there is no fracturing on either side.

The percent of the volume deformed as a function of area ratio is shown in Figure 14, where a volume of 100% is the theoretical maximum deformation that the implodable volume can undergo. For area ratios between 100% and 22%, the volume deformed is between 91% and 94%. As the area ratio begins to decrease further, the volume deformed begins to decrease. At an area ratio of 5%, the volume deformed is as low as 77%.

Figure 13 and Figure 14 provide additional insight into the deformations of the implodable volume during its collapse. For area ratios from 100% through 22%, there appears to be little effect on the implodable volume once the experiment is complete. For these cases, the deformed volume and fracture locations are very similar for all cases. For the smaller area ratios, 10% and 5%, there appears to be an effect of area ratio in terms of both fracture and deformed volume. This is very similar to the relationship between implosion severity in terms of peak pressure and impulse and area ratio. There is a linear region, where the effect of area ratio is small, and a decay region, where the effect of area ratio is large. The post-test specimens provide insight that once the area ratio reaches a certain threshold, there is not enough pressure inside the confining tube to continue to deform the structures. The pressure inside the confining tube in these cases is not larger than the pressure required to overcome the plastic hardening of the implodable volume.

3.4. Effect of Hole Configuration

The previous discussions have used restrictor plates 1 through 5 that have a single hole in the center of varying size. In order to understand the effect that different hole configurations may have on the physics of in-tube implosion, plates 6 through 9 are used. Each of these configurations has an area ratio of 30%.

The Russel Comprehensive (RC) error for the comparison between two 30% single-hole cases is 0.126 at channel 3 (towards the bottom of the tube) and 0.132 at channel 5 (top of the tube). This error will not be used on its own to quantify the similarity between the two signals. Rather, only the relative RC error will be used. In other words, the error of each hole configuration case is compared to the error of the repeat case. If the error is less than or equal to the repeatability error, the hole configuration is considered to have no effect. The RC error is calculated for each hole configuration at the closed end (channel 2) and the open end (channel 5) compared to the 30% case test A. The pressure histories are shown in Figure 15, where the left plot is of channel 2 and the right plot is of channel 5. A summary of the error for each case is shown in

Table 5. The Russel Comprehensive error compared to the single-hole 30% case.

Case	Closed End	Open End
Single-Hole Repeat	0.126	0.132
Off-Center	0.091 (−)	0.139 (+)
16-Hole Ring	0.176 (+)	0.297 (+)
64-Hole Center	0.112 (−)	0.178 (+)

, where an error larger than the repeatability error is shown in red or with a “(+)”, while the error smaller than the repeatability error is shown in green or with a “(−)”.

For all cases, the hole configuration affects the pressure signal at the open end. It is not surprising that the flow behavior very close to the restricting plate changes as the configuration of the holes changes. At the closed end, there is no effect of hole configuration caused by the off-center or 64-hole cases. However, the 16-hole ring case shows that there may be some effect for this case at the bottom of the tube. This is readily apparent in Figure 15 as the signal begins to go out of phase with the others. In general, it appears that hole configuration does not have a significant impact on the pressure history at the bottom of the confining tube. However, specific hole configurations, such as the 16-hole ring, do show an effect on the pressure history that is larger than the typical experimental repeatability.

4. Conclusions

As humans continue to develop the undersea engineering ecosystem of systems, the consequences of catastrophic collapse must continue to be investigated. Almost every undersea pressure vessel, from pipelines to sensors to unmanned vehicles, has the potential to experience a catastrophic collapse, known as an implosion. This collapse can be caused by hydrostatic pressure or any combination of external loadings from natural disasters to pressure waves caused by external implosions or explosions. During an implosion, high-magnitude pressure waves can be emitted, which can cause adverse effects on surrounding structures, marine life, or even people.

A comprehensive experimental study is conducted to understand the effects of flow restriction on the physics governing in-tube implosion. The work utilizes a unique test facility, including a large pressure tank outfitted with viewing windows. High-speed photography, 3D Digital Image Correlation, and an array of dynamic pressure sensors are used to obtain a large set of data. The following key conclusions can be drawn from this work:

• There is a significant difference in the timing and magnitude of the pressure–time history between free-field and in-tube implosion for all instances of flow restriction. This highlights the differences in the fluid behavior between the two scenarios.
• The consistent nature of the collapse of aluminum implodable volumes correlates to a high level of repeatability for in-tube experiments. Using this metric of repeatability as a relative quantity, it was shown that the configuration of holes has an effect on the pressure at the open end of the confining tube. At the closed end of the confining tube, this effect is lessened, where only the 16-hole ring shows a Russel Comprehensive error larger than the baseline repeatability.
• The area ratio contributes more significantly to the deformation of the implodable volume in later phases of the deformation as the flow of fluid into the confining tube becomes a more important contributor to the implosion dynamics. The maximum difference in peak velocity between flow restriction cases is 20% during the acceleration phase, but the maximum difference in peak velocity between flow restriction cases is 65% during the buckle propagation phase.
• For this combination of implodable volume and confining structure, the effect of area ratio on the severity of an implosion can be estimated by a combination of exponential functions and separated into two regions: a linear region and a decay region.
• In the linear region, which occurs between area ratios of 100% and 22%, there is very little impact on key metrics. The maximum difference between the two points within that region is 16% for peak pressure, 6% for over-pressure impulse, and 4% for total volumetric deformation.
• In the decay region, which occurs between area ratios of 22% and 5%, there is a significant impact on key metrics. The maximum difference between the two points within that region is 32% for peak pressure, 56% for over-pressure impulse, and 18% for total volumetric deformation.

By understanding the consequences of an implosion, designers can make decisions about where these pressure vessels should be in relation to other pressure vessels, critical infrastructure, marine life, or people.