Investigation into the behaviors of ventilated supercavities in unsteady flow

Ventilated supercavitation, i.e. a special case of cavitation in which the cavitating objective is completely enclosed in a gas bubble generated by injecting gas into liquid, has gained considerable attention for its potential applications in high-speed underwater transportation.¹ It has been traditionally characterized by using non-dimensional parameters including the ventilated cavitation number, σ_C = 2(P_∞ − P_C)/(ρ_WU²); Froude number, $Fr=U/gdC$⁠; and air entrainment coefficient, C_Q = Q/(⁠$UdC2$⁠), where P_∞ and P_C refer to the test-section pressure upstream of the cavitator and the cavity pressure, respectively; ρ_W, U, and g correspond to the liquid density, free stream velocity in the test-section, and gravitational acceleration, respectively; d_C denotes the cavitator diameter; and Q is the volumetric air flow rate (note that in our study we use the volumetric air flow rate under a standard condition, i.e., Q_As, to define air entrainment coefficient C_Qs). Additionally, in the closed water tunnel experiments, the blockage effect from the cavity is parametrized with the blockage ratio, B = d_C/D × 100%, in which D is the hydraulic diameter of the closed tunnel test section.

In practical applications of using ventilated supercavitation for high-speed underwater transportation, the operation of a supercavitating object faces unavoidable unsteady conditions induced by the motions of the object as well as the variation of incoming flows due to changes in depth or surface waves. These unsteady effects significantly impact the operation of the supercavitating object shown in the previous investigations.^2,3 Specifically, Lee et al.² showed that the presence of a large force fluctuation caused by the interaction between the rear portion of the cavity and a supercavitating model during its dynamic pitching. This will vary according to the change of flow depth and surface wave characteristics. In an unsteady flow generated from a pair of flapping hydrofoils, Sanabria³ also demonstrated the substantial fluctuation of the planing force due to the variation of supercavity geometry associated with the flow unsteadiness.

However, to date, despite a large number of studies focusing on the control of ventilated supercavitation,^4–7 there is only a handful of studies investigating the behaviors of a ventilated supercavity under the unsteady flow. Logvinovich first proposed supercavity independency theory for the variation of cavity geometry in unsteady flows, i.e., every lateral section of a cavity expands relative to the trajectory of the body center almost independent of the prior or subsequent body movement, dependent only on pressure and velocity at the current moment.⁸ However, such theory has not been thoroughly tested over a broad range of unsteady conditions. Through both experiments and numerical simulation, Semenenko⁹ investigated the cavity behaviors under unsteady conditions, including randomly adding pulses into ambient pressure and varying submerged depth during the water entry process of a supercavitating object. Besides a report of qualitative observation of various supercavity behaviors (e.g., periodical variation of the length, closure mode change, and generation of waves on the gas-liquid interface of the cavity) under different unsteady conditions, the study did not provide quantitative analysis of the correlation between these behaviors and flow unsteadiness. Nouri et al.¹⁰ studied the behaviors of a supercavity generated by a wedge cavitator under unsteady flows induced by an oscillating plate attached onto the side of the test section. They showed that the cavity exhibits a maximum degree of periodic fluctuation when the oscillation frequency of the plate matches the natural frequency of the cavity which depends on the cavitation number and the cavity dimension. Recently, using the unsteady flow produced by a gust generator composed of two flapping hydrofoils, a series of studies at the Saint Anthony Falls Laboratory (SAFL) investigated the behaviors of a ventilated supercavity under surface waves.^11–13 Specifically, Lee et al.¹¹ used a 10-mm-in-diameter cavitator to generate a supercavity and studied its behavior under a range of unsteady flow conditions (i.e., angles of attack of the hydrofoil flapping motion were ±2°, ±4°, ±6° with the frequency varying from 1 Hz to 10 Hz). Lee et al.¹¹ showed that the dimension and cavitation number of the supercavity change periodically, with the frequency and the amplitude of the variation depending on the angle of attack and gust frequency. Furthermore, Karn et al.¹² compared the closure variation for the cavity generated with the same 10 mm cavitator under low and high flapping frequencies (i.e., 0.71 vs 7.1 Hz) at high angles of attack (±10°). It was found that the ventilated supercavity closure alternates periodically between a twin vortex mode (TV) and the re-entrant jet mode (RJ) in line with the periodic fluctuations of the cavitation number. In a study of the ventilation demand for formation and sustenance of a ventilated supercavity, Karn et al.¹³ showed that the flow unsteadiness, produced by the flapping foil with angles of attack at ±2°, ±4°, and ±6° and gust frequencies of 1, 3, and 5 Hz, can considerably increase the ventilation required to form and sustain a supercavity.

Although the above-mentioned studies provided some insights into the behaviors of the supercavity under the unsteady flow, there is still a dearth of investigations on the fundamental flow physics and general characteristics of a supercavity under systematically-controlled unsteady flow settings. Therefore, here we report a systematic investigation of the behaviors of the ventilated supercavity over a broad range of unsteady conditions generated from the similar setups employed in the prior studies at the SAFL. The present paper is structured as the following: Sec. II describes the experimental methods including the facility, experiment setup, and measurements under different flow conditions. In Sec. III, we first summarize a variety of distinct cavity states observed in the current investigation and explain their underlying physics. Subsequently, the variation of cavity states under different unsteady conditions is summarized in a cavity state map, and the change of cavity state maps according to the change of ventilated rate and cavitator size is further analyzed. Finally, the conclusion and further discussion of the experimental results are provided in Sec. IV.

The experiments are conducted in a recirculating cavitation water tunnel at the Saint Anthony Falls Laboratory (SAFL) shown in Fig. 1. The test section of the tunnel has a dimension of 1.2 m in length and a 0.19 m × 0.19 m cross section and has transparent windows at the sides and at the bottom for observation and applying optical-based measurements. The highest velocity in the tunnel test section is 20 m/s, and the turbulence level is approximately 0.3%. There is a large dome-shaped settling chamber upstream of the test section which provides for fast bubble removal ensuring continuous operation of cavitation and ventilation experiments. The settling chamber is vented to the atmosphere through an atmospheric valve during the course of the experiments. This tunnel has been used for a number of supercavitation^12–16 and hydrofoil aeration experiments¹⁸ in the recent years.

A forward-facing cavitator mounted on a ventilation pipe (FFM) is employed to generate a stable ventilated supercavity with a clear interface (Fig. 2). As shown in Fig. 2, the unsteady conditions are introduced using a gust generator consisting of two parallel NACA0020 flapping hydrofoils mounted upstream of the cavitator. The hydrofoils have a chord length of 40 mm, driven by an eccentricity flywheel out of tunnel via pivot arms. The pivot arms apply the periodic pitching motion to the hydrofoils, which allows the hydrofoils flapping in phase to generate a periodic flow inside the water tunnel (details provided by Korpriva¹⁸). The flapping motion of the hydrofoils provides a vertical perturbation propagating downstream with free stream speed and a test section pressure variation associated with varying blockage ratios of the hydrofoils. Depicted in Fig. 3, the perturbation produced from the gust generator can be characterized with its wavelength, λ = U/f_g, in which f_g is the rotation frequency of the flywheel and wave amplitude, ε = V_max/(2πf_g), where V_max denotes the maximum vertical speed of the gust flow under different angles of attack of flapping hydrofoils (AoAs) and f_g.^11,19,20 Time-varying velocity amplitudes of periodical gust flows of the current setup are measured with Laser Doppler Velocimetry (LDV) as described by Lee et al.¹¹ The LDV measurements show that the frequency of the periodical gust flow in the test section is equal to the oscillation frequency of the gust generator at each flow condition. In each gust cycle, there are two maximum perturbations induced by the hydrofoils passing their neutral position upwards and downwards, respectively. It is worth noting that for a gust of frequency f_g, the frequency of test section pressure variation associated with moving blockages of the hydrofoils occurs at 2f_g and 90° out of phase with respect to the maximum vertical perturbation induced by the moving hydrofoils. Shown with the LDV results,¹¹ V_max increases with the increasing angle of attack of the gust motion. Additionally, the increase of f_g attenuates the blockage effect from the moving hydrofoils due to the swifter transition of the hydrofoils from high AoA to low AoA in a gust cycle. Therefore, wave amplitude [i.e., ε = V_max/(2πf_g)] can also represent the test section pressure fluctuation associated with varying blockage of hydrofoils under different unsteady conditions in the current investigation.

Simultaneous pressure measurement and high-speed imaging are conducted in the current experiments. Two different pressure measurements are carried out through two pressure taps shown in Fig. 2, i.e., test section pressure P_∞ before the gust generator using an AP 10-50 absolute pressure sensor and pressure difference across the cavity termed as $ΔP=P∞−Pc$ through a test section pressure tap and another pressure tap on the FFM with a DP 15-38 differential pressure sensor. The plus side of the differential sensor is connected to the same port of the absolute pressure sensor measuring P_∞ during the course of the experiments. Both sensors are sampled at 1000 Hz with the uncertainty in the pressure measurement of ±100 Pa. For the cases with pressure data provided, we confirmed that the minus side of the differential pressure sensor (i.e., cavity pressure) was always located inside the cavity during the course of the experiments. The cavity pressure P_c is derived from the measurement of the two sensors above-mentioned with an uncertainty of ±150 Pa. An extra differential pressure sensor (Rosemount 3051S) is used to monitor pressure difference between the settling chamber and test section for the determination of flow speed. A Photron APX-RS high speed camera capable of maximum recording speed at 3000 fps for a full sensor size of 1 Megapixels is employed to capture the behaviors of the ventilated supercavity under different unsteady conditions. The amount of air injected into the back of the cavitator to generate and sustain the ventilated supercavitation is controlled by an Omega FMA-2609A mass flow controller with a full-scale reading of 55 SLPM (i.e., standard liter per minute) placed distant from the test section. The maximum uncertainty in the measurement of C_Qs, and Fr, and σ_C are all around 2%, estimated in accordance with those presented previously.¹² 

The ventilation and unsteady flow settings for each set of experiments in the present study are summarized in Table I. It is worth noting that the current investigation focuses on the general behaviors of a ventilated supercavity in the unsteady flow. Therefore, although the past studies have commented on the influence of the Reynolds number and Weber number on supercavity formation^13,15,16 and closure,¹⁴ the Froude number (Fr) is chosen as the dominant non-dimensional parameter to characterize the influence of gravity and corresponding deformation of the cavity. Additionally, for all the experiments, the water tunnel flow speed is kept constant and sufficiently high to assure negligible gravitational influence on the shape of the cavity under the steady flow condition (i.e., without the gust generator in operation), discussed by Lee et al.¹¹ and Karn et al.¹² For the present paper, the case with the cavitator diameter of 10 mm (i.e., B = 5%) and ventilation rate of 10 SLPM (i.e., C_Qs = 0.19) is chosen as the reference case. The influence of the ventilation rate and cavitator size on the supercavity behavior under unsteady conditions is then investigated through comparison with the reference case.

The high speed visualization and pressure signals reveal five distinct states of the supercavity under unsteady conditions, referred to as the stable state, wavy state, pulsating state I, pulsating state II, and collapsing state hereafter. The stable cavity is observed under low AoA and f_g. As shown in Fig. 4 and video S1 of the supplementary material, the supercavity only exhibits a very small amplitude of oscillation without appreciable deformation of the cavity surface. Correspondingly, the pressure difference across the cavity surface (⁠$ΔP$⁠) fluctuates irregularly at the magnitude close to the noise level of the pressure transducer [Fig. 5(a)]. The power spectrum density (PSD) of $ΔP$ does not show any prominent peaks corresponding to the gust cycle or its high order harmonics [Fig. 5(b)]. Overall, under low AoA and f_g, the perturbation induced by the gust generator is not strong enough to cause appreciable change of supercavity behavior.

The wavy state occurs under moderate AoA and high f_g. The supercavity displays a clear periodic wavelike undulation at its surface with a variation of closure types between the re-entrant jet (RJ) and a hybrid mode of the twin vortex (TV) and RJ (TVRJ) (see Fig. 6 and video S2 of the supplementary material). Accordingly, as shown in Fig. 7(a), $ΔP$ exhibits a periodic variation with double peaks in each gust cycle, evidenced by the prominent peaks appearing at both f_g and 2f_g, respectively, in the PSD of the $ΔP$ signal [Fig. 7(b)]. Particularly, the two peaks in P_C correspond to the two neutral positions [i.e., AoA = 0°, phases I and III in Fig. 8(a)] of the flapping foils with a slight delay of 0.014 s matching the time required for the vertical perturbation to propagate from the gust generator to the cavity pressure measurement port. Specifically, as the flapping foils start from their neutral position and move upwards at maximum vertical velocity in a gust cycle (phase I), the gust generator induces the highest vertical perturbation in a cycle. Due to the neutral position of the foils, the total tunnel loss at this phase is the lowest, which leads to the highest tunnel velocity and the corresponding local minimum of the test section pressure upstream of the gust generator (note that the settling chamber is vented to the atmosphere through an atmospheric valve). Accordingly, RJ closure emerges potentially due to the relatively strong adverse pressure difference at the closure region (a detailed analysis presented by Karn et al.¹²). As the foils move toward their maximum upward position (phase II), the tunnel loss gradually rises, associated with a decrease in tunnel velocity and an increase in the test section pressure. Correspondingly, the perturbation from the gust generator causes the cavity pressure to drop from its maximum and a transition of cavity closure from the RJ to TVRJ (due to increasing $ΔP$ as discussed by Karn et al.¹²). In addition, the cavity surface deformation and the bubble shed-off from the interface due to a Kelvin-Helmholtz (KH) instability can be clearly observed. When the foils move downward back to their neutral location (phase III), the tunnel loss decreases and the test section pressure decreases accordingly. The cavity pressure begins to increase and reaches the second peak in a cycle with the same time delay as the one mentioned before, and the closure shifts back to the RJ from the TVRJ. In the remaining phases of the gust cycle (III → V), the trends of pressure variation and closure change repeat those observed during the first half of the cycle (i.e., phase I to III) in general. Additionally, compared to the supercavity deformation during phase I to II, the wavelike undulation of the cavity surface appears in an opposite phase during the phase of hydrofoil moving downward (III to V).

Pulsating state I is observed under high AoA and low f_g. As demonstrated in Fig. 9 and the corresponding video S3 of the supplementary material, the supercavity at this state exhibits the periodic shed-off of gas pockets at the rear part of the cavity leading to a significant fluctuation in length, similar to the pulsation behavior described by Michel.²⁰ The ΔP of the cavity at this state exhibits the similar behaviors comparing to the previous case as illustrated in Fig. 10. This process is also accompanied with wavelets generating on the cavity surfaces and the variation of the closure type between the TV and RJ (Fig. 11). Unlike the wavy state, due to the relatively weak vertical perturbation associated with the gust generator operating at low f_g (i.e., large wavelength λ), the supercavity at this state does not show appreciable wavelike undulation on cavity surfaces. Moreover, $ΔP$ and the corresponding P_C exhibit periodic variation with double peaks in each gust cycle, and the two peaks in $ΔP$ (i.e., local minima of P_C) occur at the times of the shedding of bubble pockets when the flapping foils reach maximum AoA. In comparison to the wavy state, as the hydrofoils change in a gust cycle, both the test section pressure and supercavity closure yield similar trends. However, unlike the wavy state, the variation of $ΔP$ and P_C in this phase is primarily dictated by the pressure fluctuation associated with the periodic variation of the hydrofoil blockage. Therefore, little delay is observed between the phase change of hydrofoils and the signals of $ΔP$ and P_C since the propagation of such pressure fluctuation occurs at the speed of sound in water. Moreover, the increasing AoA of flapping foils (compared to the wavy state) leads to stronger fluctuations of $ΔP$ and P_C, which further induces a significant cavity length fluctuation. Particularly, during the transition from phases II to III and from phases IV to V [corresponding to II → III and IV → V in Fig. 11(a), respectively], the cavity length shrinks drastically, resulting in enhanced KH instability and consequently the formation of a series of wavelets on the cavity surface. It is also worth noting that the wavelet formation appears more prominent during II → III compared to IV → V when the gravity acts against the upward motion of the interface. In addition, at pulsating state I, the closure transitions from the RJ to TV, instead of the TVRJ in the wavy state, which may be the result of higher tunnel loss associated with the gust generator operated at higher AoA.¹² Noteworthy, although supercavity pulsating behaviors have been broadly reported in the literature,^9,16,20,21 the underlying mechanism of the supercavity pulsation reported here differs from some of the prior studies. Specifically, without introducing an external perturbation of the incoming flow, the pulsating behavior was observed with the cavity generated from a disk cavitator under relatively low flow speed (2.35 m/s) but a high ventilation rate (C_QS > 100).^20,21 Michel²⁰ attributed such cavity pulsation to the KH type instability developed at the gas-liquid interface at a high ventilation rate. Moreover, with the unsteady flow settings, the cavity has a pulsating behavior during the water entry process correlated with the variation of the submerged depth.⁹ Recently, supercavity pulsation was also reported in a study of ventilated supercavitation generated from a gas-jet cavitator.¹⁶ It was shown that the supercavity exhibits a strong oscillation at the front and intermittent shed-off of the gas pocket in the rear portion during the transition from a stable cavity to jet-type cavity potentially due to the turbulent fluctuation within the gas jet near the stagnation location at the front of the cavity.

With an increase in gust frequency (f_g) from pulsating state I, the cavity continues to pulsate periodically and the pressure signature shows distinct features compared to that of pulsating state I (see Fig. 12 and video S4 of the supplementary material). We refer to this state as pulsating state II. Specifically, $ΔP$ displays a sharp spike followed by a sequence of small wavelets in each gust cycle [Fig. 13(a)], corresponding to the spectral peak at f_g and a series of subsequent peaks at higher frequencies in the pressure signal [Fig. 13(b)]. Moreover, unlike pulsating state I, the spike in $ΔP$ displays a 0.13 s delay with respect to phase III of the hydrofoils when they start to move downwards from their neutral position. This signal delay matches the time required for the vertical perturbation generated from the hydrofoil to propagate to the location where the supercavity sheds off air pockets. In addition, the drastic variation of the cavity length during the pulsation [I → II in Fig. 14(a)] results in the abrupt drop of P_C (i.e., the spike in $ΔP$⁠). Therefore, we suggest that the cavity pulsation at this state is primarily dictated by the vertical perturbation induced by the hydrofoils rather than the pressure fluctuation associated with varying blockage in pulsating state I. It is worth noting that, unlike the previous state, the cavity under pulsating state II always initiates its pulsation from its upper surface in each gust cycle. We suggest that it might be related with that the upper surface of the cavity is more susceptible to the perturbation of the incoming unsteady flow since the inverse alignment of the density gradient with respect to the direction of gravity.

At the highest AoA (i.e., AoA = ±10°) and f_g above 2 Hz, the supercavity eventually collapses under a fixed ventilation rate and tunnel speed (see Fig. 15 and video S5 of the supplementary material). At this state, the unsteadiness induced by the gust generator is so strong and rapid such that the supercavity does not have sufficient time to recover through ventilation after its breakup. Note that the reliable pressure measurements cannot be obtained due to water splashing near the cavity pressure tap.

The different cavity states over a broad range of unsteady conditions are summarized in the cavity state maps. First, the cavity states’ transition under the reference case is summarized in Fig. 16. As shown in Fig. 16(a), with fixed f_g, the increase of AoA enhances the amplitude of the pressure fluctuation in a gust cycle, leading to the pulsating and eventually the collapsing of the supercavity. In comparison, increasing f_g amplifies the vertical perturbation under fixed AoA, which results in augmenting waviness in the cavity geometry (i.e., stable to wavy, pulsating I to II). To further elucidate the key trends in the supercavity state transition, the state map is also presented with half wavelength (λ/2) and wave amplitude (ε), normalized by the cavity length under the steady condition (L) and cavitator diameter (d_C) and shown in the logarithmic scale, respectively [Fig. 16(b)]. The non-dimensionalization of wave parameters is intended to compare the perturbation produced from the gust generator with the characteristic length scales of the supercavity. Remarkably, in the non-dimensionalized state map, the critical unsteady conditions demarcating where the cavity starts to lose its stability are located approximately on a linear curve with a slope of 1.0 and an intercept of 0.14 at the horizontal axis [the blue dashed line in Fig. 16(b)]. Above such a critical line (with increasing ε or decreasing λ or the combination of both), the supercavity exhibits pulsating behaviors or collapses. Here we propose a physical interpretation of this critical condition line using the schematic presented in Fig. 17. The schematic shows a ventilated supercavity with a length L, formed with a cavitator of d_C in diameter under steady conditions. The peak deformation on the cavity surface, depicted as $εmax′$ in the figure, is strongly correlated with the maximum vertical perturbation induced by the motion of the gust generator (ε). Accordingly, the distance between two adjacent peaks matches the half wavelength of the perturbation (λ/2). The shaded area characterizes the level of distortion (in comparison to its steady contour) that the supercavity experiences in the unsteady flow. Note that the supercavity length fluctuates around its length in the steady flow due to the variation of the air leakage caused by flow perturbation. We propose that the cavity loses its stability once the level of its distortion (characterized by the shaded area) exceeds the cavity dimension under the steady condition (characterized by the area of the 2D profile of the cavity). Such a critical condition can be expressed in a simple mathematical form below by assuming that the cavity has an elliptical shape of small curvature according to the theoretical work by Logvinovich and Serebryakov,²² 

Under the experimental settings of the current study, the wavelength of the flow perturbation (λ) satisfies 0 ≤ (2π/λ)L ≤ π (i.e., the cavity length is smaller than the half wavelength over all the unsteady conditions). Therefore, we could reduce the above Eq. (1) into the following form:

For L ≪ λ (applicable to most of the unsteady conditions in the current experiment), the above expression can be further simplified using the Taylor series of cos x and eliminate the higher order terms (O⁴),

Theoretically, for a cavity with cavitation number σ_C under the steady condition, its maximum diameter can be represented as the following:²² 

Thus, the criterion of the cavity to lose its stability becomes

To facilitate further comparison with Fig. 16(b), the logarithmic form of the above criterion is presented as follows:

The above expression represents a linear curve with a slope of 1 and the intercept of $log100.82(1+σC)/0.96σC/2$ in Fig. 16(b). It is worth noting that the blockage effect does not exist in the practice of underwater transportation. Therefore, to incorporate the current theoretical analysis results in the real scenarios, the cavitation number in the above-mentioned equations should be transferred to its equivalence in an unbounded flow (σ_∞) based on the equation $σ∞=(2σC2−σmin2)/2σC2$ proposed by Karlikov and Sholomovich.²³ σ_min in the above equation is the minimum cavitation number achievable in a closed-wall setup corresponding to the blockage ratio. Table II below shows the comparison of the bounded and unbounded cavitation numbers (calculated in the above equation) for the current experiments.

In the current case, for the 10 mm cavitator (i.e., B = 5%), σ_C measured with steady conditions is 0.13, leading to an intercept of 0.14. Additionally, beneath the critical linear curve, the cavity states can be further demarcated by a vertical line (red dashed line) into region I of the stable state and region II of the wavy state [Fig. 16(b)]. This vertical line, corresponding to a critical condition of (λ/2) ≈ L, indicates that the supercavity transitions to the wavy state when the half-wavelength of the perturbation decreases to be comparable to the cavity length.

The influence of the ventilation rate on cavity state transition is examined in Figs. 18 and 19. With increasing C_Qs, compared to the case at lower C_Qs, the state map exhibits similar distribution of different supercavity states and the critical conditions that separate different states (Fig. 18). However, we do not observe any collapsing state with this case, even under the highest angle of attack AoA = ±10° with f_g > 2 Hz due to the increased ventilation with sufficient gas momentum that enables the supercavity to recover from intermittent breakups. With the decrease of C_Qs, the momentum of the ventilation gas is no longer sufficient to recover drastic breakups of the cavity, leading to the transition of pulsating states observed in the cases of higher ventilation to the collapsing state (Fig. 19). Nevertheless, the cavity state maps still yield similar trends on the distribution of different states and the critical conditions governing the state transition.

Finally, the influence of the cavitator size on the cavity state map is summarized in Fig. 20. As shown in Eq. (5), increasing cavitator size leads to more stable cavities under the same set of unsteady conditions. Specifically, comparing to the reference case, the cavity generated with a 20 mm-in-diameter cavitator (i.e., B = 9%) does not collapse even under the highest angle of attack (i.e., AoA = ±10°) in our experiments. Instead, the supercavity exhibits different types of pulsating behavior in region III [Fig. 20(b)]. It is worth noting that in a closed-wall flow facility, varying cavitator size causes the change of blockage ratio (B) and σ_C, which shifts the critical condition line that determines the cavity starts to lose its stability according to Eq. (5). For the present case with σ_C = 0.20 under steady conditions for B = 9%, the intercept of the blue dashed line is calculated to be 0.04 using Eq. (5), which matches closely with the offset of the critical line illustrated in Fig. 20(b). As for the practice of underwater transportation in a real scenario, we suggest that the intercept of this critical line should match closely with the case of the smaller cavitator size with the non-existing blockage effect.

A systematic investigation of ventilated supercavity behaviors under a broad range of unsteady flows has been conducted at the SAFL high speed water tunnel. The study uses a ventilated supercavity generated with a forward facing model under varying ventilation rates and cavitator sizes. The unsteady flow is produced by a gust generator consisted of two periodical flapping hydrofoils with varying gust frequency (f_g) and angle of attack (AoA). Five distinct cavity states are observed through high speed imaging and pressure measurement, including the stable state, wavy state, pulsating state I and II as well as collapsing state. During the stable state occurring at low AoA and f_g, the cavity does not exhibit appreciable deformation on its surface. At the wavy state under moderate AoA and high f_g, the supercavity displays clear wavelike undulation on its surface with the periodic variation of closure modes. At the high AoA with low f_g, the supercavity shows significant fluctuations in length with the intermittent shed-off of gas pockets at its rear part (i.e., pulsating state I). With further increase in f_g, the cavity pressure shows a strong spike with an intermittent cavity breakup (i.e., pulsating state II). The cavity eventually collapses under the highest AoA (AoA = ±10°) and high f_g. The supercavity states under a broad range of unsteady conditions are summarized in the cavity state map. The maps demonstrate that the supercavity has transition (from stable state) to pulsating states with increasing AoA and to the wavy state with increasing f_g. Furthermore, the state map over the non-dimensionalized wavelength and wave amplitude of the perturbation shows that the supercavity loses stability and evolves to pulsating or collapsing when the level of its distortion induced by the flow unsteadiness exceeds the cavity dimension under the steady condition. With the change of ventilation condition or cavitator size, the general distribution of the supercavity states and the critical conditions for the transition among different states remain unchanged. However, increasing ventilation rate or cavitator size suppresses the occurrence of the collapsing state of the cavity under the same set of unsteady incoming flow conditions.

The present study is able to provide a comprehensive map of supercavity behaviors and establish the connection of these behaviors with the unsteady conditions of the incoming flow. Such information is critical for the operation of a supercavitating device in a practical environment. Specifically, the operation of a supercavitating vehicle should avoid the collapsing state and mitigate the negative impact of pulsating and wavy states with well-designed control strategies. Suggested by Sanabria et al.,²⁵ these variations of cavity geometry will result in a supercavitating vehicle penetrating cavity surface and the variation of the planing force exerted both on the vehicle and control surfaces. Our current study has suggested that such operational control can be achieved effectively through the proper design of ventilation strategy and cavitator geometry. Further investigations will incorporate such knowledge into the design of the control strategy for the supercavitating device operating under different unsteady conditions.

See supplementary material for different supercavity states. S1 is the high-speed video of a stable state supercavity captured with 250 fps. S2 is the high-speed video of a wavy state supercavity captured with 1500 fps. S3 refers to a pulsating state I supercavity captured by the high-speed camera operated with 250 fps. S4 refers to a pulsating state II supercavity captured by the high-speed camera operated with 1500 fps followed by video S5 of a collapsing state cavity captured with 1500 fps. All the videos are played back with 30 fps. The ventilated supercavities in the videos are generated under reference conditions.

This work is supported by the Office of Naval Research (Program Manager, Dr. Thomas Fu) under Grant No. N000141612755 and the start-up funding received by Professor Jiarong Hong from the University of Minnesota.