Experimental investigation on cavitating and self-sustained oscillating flow in a horizontal axisymmetric cavity Open Access

Cavity flows display diverse self-sustained oscillation features and a wide oscillating frequency range. Their appearance in aeronautics,¹ hydraulics,^2,3 and fluids engineering^4,5 has attracted the interest of scientists and engineers, who have studied their behavior in order to suppress or utilize them. Cavity flow has also been a fundamental problem in fluid mechanics. Rectangular cavity flow is the primary type of cavity flow, as shown in Fig. 1(a). The interactions between the separated shear layer flow and the cavity geometry generate self-sustained oscillations. A large number of studies have been carried out, using both experimental and numerical methods, to understand the self-sustained oscillation characteristics and mechanisms,^6,7 the flow structures and flow fields,^8–11 and the oscillation frequency by means of predictive models.^12,13

In axisymmetric cavity flow, a liquid or gas jet flows through an axisymmetric cavity, as shown in Fig. 1(b). Unlike rectangular cavity flow, axisymmetric cavity flow is wall-bounded, resulting in a confined jet flow.^14,15 Rockwell and Naudascher^16,17 have categorized cavity flows into three groups: fluid-dynamic, fluid-resonant, and fluid-elastic. They categorized the axisymmetric cavity flow into the fluid-dynamic group on the basis of the shear layer interaction with the impinging body. However, Morel¹⁸ and Geveci et al.⁹ investigated the gas flow inside an axisymmetric cavity and showed that the self-sustained oscillation was jointly induced by a fluid-dynamic feature (namely, the impinging shear layer) and cavity resonance.

For water jet flow through an axisymmetric cavity, additional flow phenomena have been described. Jiao et al.¹⁹ reported that low-frequency and large-amplitude (LFLA) self-sustained oscillations occurred under certain geometrical and working jet conditions, but they did not clearly address when and why LFLA oscillations occur. An axisymmetric cavity also induces high-frequency self-sustained oscillations and enhances water jet cavitation, boosting rock drilling, material erosion, or cutting processes.^4,5,20 The conditions and mechanisms that lead to the occurrence of these phenomena are different, and they are not completely understood. Investigating the internal flow inside an axisymmetric cavity can help understand the mechanisms behind these unsteady flow phenomena. As mentioned above and illustrated in Fig. 1(b), there are three fundamental flow phenomena inside the cavity: confined jet flow, cavitating water jet flow, and an impinging shear layer. The interactions between these flow phenomena generate complex dynamic flow features inside the cavity.

Conventional analysis methods for nonlinear flow systems are in the time domain or by linear approximation. Recently, nonlinear dynamic approaches have been applied to flow systems^21–24 to analyze numerical pressure or acoustic signals. The results of nonlinear analysis reveal the nonlinear dynamics of the flow phenomena under study. The nonlinear dynamic information can be reconstructed from the computed or measured pressure time series data by the nonlinear time series method,²⁵ and these results may lead to a better comprehension of flow phenomena.

Therefore, the present work experimentally investigates self-sustained cavitating oscillatory jet flow inside a horizontal axisymmetric transparent cavity. Specifically, the investigation focuses on the dependence of the flow features and self-sustained oscillation characteristics on the jet Reynolds number and the cavity length. Furthermore, the nonlinear time series method is employed to analyze the pressure signals and reveal the nonlinear oscillating characteristics of the system.

The cavity consisted of an inlet nozzle, an axisymmetric chamber, an impinging body, and an outlet nozzle and pipe; these components were clamped between two stainless steel disks by four stainless steel screws, as shown in Fig. 2. Two types of impinging body (with α = 120° and 180°) were employed in the tests. The cavity was made of polymethyl methacrylate (PMMA). Thus, the internal flow could be observed through the transparent cavity wall.

Pressure holes were set up on the cavity wall and the impinging body to measure the pressure at different locations. The geometrical parameters of the axisymmetric cavity are listed in Table I; they were all kept constant except for the cavity length. The inlet nozzle diameter d₁ = 11 mm; all the other parameters are nondimensionalized with respect to this value. The cavity length is defined for the impinging body characterized by α = 120°, as the distance between the inlet nozzle outlet and the inlet of the outlet nozzle.

A large number of tests were carried out for different geometrical parameter combinations within the ranges 1.5 ≤ d₂/d₁ ≤ 2.25, 9.5 ≤ d_c/d₁ ≤ 15, 3.75 ≤ L_c/d₁ ≤ 11.25 and two impinging body angles α = 120° and 180°. Similar dynamic characteristics were evidenced for the flow phenomena present in these tests, compared with the restricted parameter set in Table I, although the measured data were not exactly the same. In other words, the inception mechanism of self-sustained oscillations under unsubmerged and submerged jet flow conditions discussed in detail in Sec. III is representative of a wide range of geometric parameter combinations not presented here for the sake of brevity.

A schematic of the open horizontal loop test rig used is shown in Fig. 3; it consisted of a water supply pump, a pipe booster pump, an electromagnetic flow meter (0.3 Grade accuracy, 0–32 m³/h), the axisymmetric cavity, pressure transducers (0.5 Grade accuracy, 0–3 MPa for working pressure, ±100 kPa for others), a data acquisition system, and a high-speed camera. The discharge and working pressure p₀ were controlled by means of valve 5. The maximum velocity of the jet u₁ was 50 m/s. The flow rate, working pressure, and cavity pressure measurements were synchronized. The water temperature was around 22 °C. The reservoir surface was open to the atmosphere, and so the dissolved oxygen was kept at about 7.6 ppm and the dissolved oxygen saturation at about 92%. The outlet nozzle was connected to the atmosphere during experiments.

The calibrated results showed excellent linear behavior for all transducers, with a linear correlation coefficient R² > 0.99. The uncertainties of all pressure transducers were within ±0.50%. The uncertainties associated with the flow meter and the data acquisition (DAQ) were E_q = ±0.26% and E_d = ±0.10%, respectively. The electromagnetic flow meter was excited by one-half of the AC frequency of 50 Hz, i.e., the response frequency was 25 Hz, in order to measure the instantaneous flow rate accurately.

The frequency of data acquisition was set to 1500 Hz. Because we focused on low-frequency hydraulic oscillations (<5 Hz), a low-pass filter with a cutoff frequency of 40 Hz and a moving average filter were used to remove electrical noise signals and obtain smooth and undistorted signals.

A NAC GX-8 high-speed camera was used to record images at 1000 fps with a shutter speed of 2000 s⁻¹ and an image resolution of 1024 × 768 pixels. The recording time at this resolution reached 4.2 s. A common video camera was also used to record the flow patterns at 25 fps inside the cavity simultaneously.

On the basis of a large number of flow observations inside the axisymmetric cavity, two flow states were identified: unsubmerged and submerged jet flow. Figure 4(a) shows the unsubmerged flow state, in which the water jet flows through the air inside the cavity, and Fig. 4(b) shows the submerged flow state, in which the cavity is full of water and the jet flows through it. The likelihood of unsubmerged or submerged flow states depends on the cavity length and working jet conditions. Generally speaking, the unsubmerged flow state appears for shorter cavity lengths (L_c/d₁ < 8.55).

The jet velocity u₁ is calculated using the continuity equation according to the measured average and instantaneous flow rate. The kinematic viscosity is set to ν = 1.0 × 10⁻⁶ m²/s.

The time series of the pressure oscillation collected from different positions on the cavity wall are almost the same, except for a slight difference in amplitude. Therefore, the pressure data of pressure hole M1 (Fig. 2) are selected as the representative results to investigate the self-sustained pressure oscillations inside the axisymmetric cavity. Figures 5 and 6 present time series of the pressure oscillations at different Reynolds numbers for unsubmerged and submerged flow conditions, respectively. It can be seen that there are different flow characteristics at nearly the same Reynolds number, depending on whether the jet is submerged or not. For the unsubmerged flow condition, increasing the Reynolds number leads to a frequency increase accompanied by a decrease in the mean pressure and the oscillation amplitude. However, for the submerged jet flow state, the pressure fluctuations only appear beyond a Reynolds number threshold and subsequently decrease in amplitude. Overall, the pressure fluctuation amplitude is significantly larger for the unsubmerged cavity flow.

A representative pressure time series is shown in Fig. 7 for different cavity locations, where p₀ is the working pressure, p₁ is the pressure on the cavity wall, p₂ is the pressure at the outlet nozzle, and p_t is the stagnation pressure at the center of the outlet jet. All curves show regular LFLA oscillations. Although the pressure time series have different shapes, the dominant frequency of all oscillation curves is 0.37 Hz, suggesting that the self-sustained oscillations have a common generation mechanism. The interaction between the jet and the axisymmetric cavity is the source of this unsteady flow phenomenon. Therefore, in this and the following subsections, we analyze the flow features and oscillation characteristics on the basis of the measured pressure data inside the cavity. The relative pressure inside the cavity was negative.

As mentioned in Sec. III A, the LFLA self-sustained oscillation occurs only in the unsubmerged flow state, whose appearance depends critically on the Reynolds number and the cavity length. Figure 8 shows a pressure transient event inside the cavity resulting from the variation of the Reynolds number in the range of 0.69 × 10⁵–4.84 × 10⁵ for the following geometric parameters: d₁ = 11 mm, d₂/d₁ = 1.55, d_c/d₁ = 10.91, α = 120°, and L_c/d₁ = 4.03. In this case, the LFLA oscillation occurred in the range of Reynolds number Re = 2.5 × 10⁵–3.5 × 10⁵, as shown by the LFLA region of Fig. 8. Furthermore, the LFLA oscillations only appeared for a cavity length L_c/d₁ < 8.55.

In the following discussion, given that the cavity length (L_c/d₁ < 12.66) was relatively short and the working jet velocity was high (Re > 0.5 × 10⁵), the gravitational acceleration of the jet flow in the vertical direction is neglected.

To explain the LFLA oscillation occurrence mechanism, hereinafter, we discuss the jet flow inside the axisymmetric cavity on the basis of the averaged parameters and round-free turbulent jet flow theory. The jet spread angle in the near field depends on the ambient pressure and the density ratio. No theoretical model can accurately predict the variation of spread angle with Reynolds number. Still, the experimental results of Xie et al.²⁶ have shown that the spread angle increases slightly with increasing Reynolds number.

For a fixed cavity length, the Reynolds number range can be partitioned on the basis of the presence of LFLA self-sustained oscillation: Re < Re_cr1, Re_cr1 < Re < Re_cr2, Re > Re_cr2, where Re_cr1,2 are the threshold values where the phenomenon appears and disappears. Similarly, for a fixed Reynolds number, the cavity length can be partitioned as L_c < L_cr1, L_cr1 < L_c < L_cr2, L_c > L_cr2, where L_cr1,2 are the threshold cavity lengths.

Figure 9 presents sketches of the flow for the two critical Reynolds number conditions and the two critical cavity lengths. For the Re < Re_cr1 or L_c < L_cr1 condition, the jet spread angle θ is smaller than a critical spread angle θ₁ corresponding to Re = Re_cr1, where the jet cross-sectional area at the outlet nozzle is equal to the outlet nozzle cross-sectional area. For θ < θ₁, the axisymmetric cavity is connected to the atmosphere through the outlet nozzle; Thus, the ambient pressure and the surrounding gas inside the cavity remain stable and equal to the atmospheric pressure, as shown in Fig. 8, and the jet flow does not interact with the impinging body or the outlet nozzle wall. LFLA self-sustained oscillation does not occur for these ranges of Reynolds number and cavity length.

For Re > Re_cr2, the jet spread angle is larger than the critical spread angle θ₂, corresponding to Re = Re_cr2. The jet flow closes the cavity completely and entrains the gas trapped within the cavity toward the outside. This process results in a sustained low pressure inside the cavity and the disappearance of the LFLA self-sustained oscillation. Relatively small-amplitude pressure oscillations occur in this range.

In the Re_cr1 < Re < Re_cr2 range, there is a transition from open cavity flow to closed cavity flow. Although the spread angles θ in this range are larger than θ₁, the jet flow seems to close the cavity, as shown in the average flow sketch of Fig. 9(a). In fact, the spread angle is not steady, but increases with increasing ambient pressure²⁶ inside the axisymmetric cavity. Furthermore, the asymmetry of the space around the jet, a consequence of the water body below it, induces a swing of the jet in the radial direction. On the basis of these observations, the mechanism of LFLA oscillation can be described as follows. When θ > θ₁ and there is no deviation of the jet column in the radial direction, the jet flow closes the cavity and entrains gas from the inside toward the outside, leading to a pressure drop inside the cavity and an asymmetric pressure distribution around the jet. These two factors result in the spread angle decreasing and the jet column swinging radially, respectively. Then, the cavity is connected with the atmosphere through the outlet nozzle, and mass transfer occurs between the inside and the outside of the cavity. Consequently, pressure starts to increase, the jet flow becomes steady, the spread angle increases, and then the jet closes the cavity again. This process repeats periodically under specific geometrical and working conditions, generating an LFLA self-sustained oscillation. This can be observed from the enlarged p₂ curves shown in Fig. 7(b), in which the maximum value is the atmosphere value (0 Pa, open cavity flow state), while the minimum value is about −5000 Pa (closed cavity flow state).

The LFLA oscillation mechanism for different cavity lengths can be explained similarly. Figure 9(b) shows two critical cavity lengths (L_cr1, L_cr2) and their relationship. When the cavity length L_c exceeds L_cr2, the jet flow closes the cavity, and the cavity fills with water. In this submerged flow state, the oscillation amplitude is significantly smaller than for the LFLA self-sustained oscillation. When the cavity length L_c is shorter than L_cr1, a stable state inside the cavity is obtained because of its connection with the atmosphere through the outlet nozzle. LFLA self-sustained oscillation occurs at cavity lengths in the range L_cr1 < L_c < L_cr2, owing to a periodic connection between the cavity and the outside.

Although the jet spread angle varies with the Reynolds number, it remains limited to a certain range depending on the density ratio between the jet fluid and the surrounding fluid. Laboratory tests for a water free jet revealed that the spread angle is below 1°, not taking account of the mixing layer. According to the experimental critical cavity length L_cr2/d₁ = 8.55, the smallest averaged half-spread angle, taking account of the mixing process in the shear layer, can be evaluated as θ = arctan[(d₂ − d₁)/(2L_cr2)] ≈ 1.83°. This means that when the cavity length is greater than 94.05 mm (corresponding to L_cr2/d₁ = 8.55), a submerged jet flow state is always generated, and LFLA self-sustained oscillation does not occur. On the other hand, when the cavity length is shorter than L_cr2/d₁ = 8.55, the unsubmerged jet flow condition occurs, with the probability of its appearance increasing as the cavity length decreases. In the present study, LFLA self-sustained oscillation was revealed, but it is a great challenge to determine the thresholds, because these depend on cavity geometry and working conditions. This needs further investigation in the future.

This subsection illustrates the flow patterns inside an axisymmetric cavity in a LFLA self-sustained oscillation cycle (see video 1, supplementary material). Figure 10 presents a representative pressure time series of an LFLA oscillation cycle, while the images in Fig. 11, recorded with the high-speed camera at Re = 2.87 × 10⁵, depict the flow patterns inside the cavity at different times corresponding to the labels in Fig. 10.

The variation of the bubble number reveals a change in the flow pattern inside the cavity. According to the development and attenuation of the bubble population, the oscillation cycle can be divided into four stages.

The appearance of an easily identifiable flow pattern is defined as the beginning point of stage I and the initial time t = 0 s. The time series is shown in Fig. 10; the pressure at time t = 0 s is about −0.06 MPa. As shown in Fig. 11 (0 s), at this moment, the upper part of the cavity is filled with air, whereas the bottom part contains water. Some droplets of water–air mixture in shear flow are ejected toward the cavity wall with a certain velocity owing to the interaction between the jet shear layer and the impinging body. This process induces the growth of some air bubbles within the water body at the bottom of the cavity. These ejected droplets are gas nuclei, and they will separate out and grow into bubbles in the water when the pressure decreases afterwards.

As the pressure inside the cavity continues to drop, gas nuclei in the water grow to form bubbles, resulting in a significant increase in the number of bubbles inside the cavity (Fig. 11, 0.06–0.13 s). At t = 0.19 s, the bubbles occupy the entire cavity. This moment defines the beginning of stage II.

In stage II, the pressure decreases and then increases; the maximum number of bubbles is reached at t = 0.73 s, when the pressure is minimum according to the time series. The high bubble population, occupying the entire cavity, is sustained for 0.84 s (Fig. 11, 0.19–1.03 s). As the pressure rises, the number of bubbles starts to decrease gradually, leading to stage III, as shown in Fig. 11 (1.06–1.10 s). The number of bubbles reaches its minimum as the pressure increases beyond −0.06 MPa, marking the start of stage IV, which lasts for ∼1.8 s (Fig. 11, 1.20–3.16 s). The whole oscillation period is 3.1 s.

However, it is not possible to identify the pressure variation from the pattern of variation of the bubbles. As discussed above, the pressure drop is due to the jet flow closing the cavity and entraining gas toward the outside. When the pressure inside the cavity falls below −0.06 MPa, bubbles grow and gas separates from the water. The number of bubbles reaches its maximum at the minimum pressure value, as does the jet deviation, whereas the spread angle attains its minimum value. Therefore, the jet flow can no longer close the cavity; instead, the cavity connects with the atmosphere through the outlet nozzle. Consequently, the pressure inside the cavity begins to increase, bubble growth is gradually suppressed, and the number of bubbles falls to its minimum when the pressure increases beyond −0.06 MPa. The jet flow becomes steady as pressure increases and subsequently closes the cavity when the pressure reaches its maximum value; then, the LFLA oscillation begins its next cycle.

The spread angle of the jet flowing through the outlet nozzle increases with increasing number of bubbles, because the gas content in the jet increases, in agreement with the results of Wright et al.²⁷ Consequently, the spread angle of the jet shear layer in the cavity varies with the gas void.

Figure 12 shows the variations of pressure and cavitation number with Reynolds number in the range Re = 10⁵–5 × 10⁵. There are two inflection points in the pressure curve. Before the first inflection point lies a range defined as period I, whereas period II refers to the range between the two inflection points. The Reynolds number range after the second inflection point is divided into periods III and IV on the basis of the pressure stability situation inside the cavity. Figure 13 depicts the flow patterns of these periods.

In period I, the pressure and cavitation number inside the cavity decrease continuously, whereas the slope of the cavitation number is nearly constant, because the pressure decreases in magnitude with increasing Reynolds number. The pressure oscillates when the Reynolds number approaches the inflection point, owing to the unsteadiness of the jet flow. Figures 13(a)–13(c) present the flow patterns observed in period I. As reported by Gopalan et al.,²⁸ cavitation first occurs in the jet shear layer associated with vortex structures, and the bubbles generated are deformed (lighter points in the images) by the shear force. The cavitation type in this period is shear layer cavitation. As the Reynolds number increases, the number of bubbles increases until the inflection point.

In period II, the pressure decreases significantly with increasing Reynolds number, whereas the cavitation number slope stays almost constant as in period Ⅰ. Indeed, the cavitation number decreases significantly to a small value of ∼0.05 in a narrow Reynolds number range of 2.0 × 10⁵–2.7 × 10⁵ in this condition. Figures 13(c) and 13(d) show the flow pattern transition inside the cavity from period I to period II. As the pressure is reduced rapidly, the bubble shape changes from small, densely packed deformed bubbles [Fig. 13(c)] to large, sparsely packed spherical bubbles [Fig. 13(d)] (see video 2, supplementary material). The explanation for this behavior is that the cavitation type changes from shear cavitation to fully developed cavitation. The shear cavitation occurs only within the shear layer, whereas the fully developed cavitation occurs in the entire cavity, because the pressure falls to the vapor pressure.

The cavitation-type transition occurs in different ranges of Reynolds number, depending on whether the latter is increasing or decreasing. For decreasing Reynolds number, the transition value is smaller than in the case of increasing Reynolds number. In other words, hysteresis occurs, and this is more significant for greater cavity lengths.²⁹ 

The pressure reaches its minimum value at the second inflection point and then grows slowly in periods III and IV. However, the cavitation number continues to decrease, albeit asymptotically, i.e., the slope approaches zero. There are no pressure oscillations or cavitation number variations during period III. The cavitation in this period is the most developed, meaning that the cavity becomes filled with vapor bubbles. Because of the lower density of vapor inside the cavity, the flow pattern remains stable [Figs. 13(d) and 13(e)]. The spread angle of the outlet jet flow in period III is the largest among the four periods, owing to the higher vapor content. The Reynolds number range in period III is narrow, implying that the stable state is temporary.

The pressure experiences irregular oscillations in period IV, because the swing of the jet column in the radial direction and the variation of the spread angle are stochastic, which can be observed from the irregular variation of the outlet jet flow [Fig. 13(f)].

Figures 14 and 15 show time series of cavitation numbers and their spectra for different Reynolds numbers during periods III and IV. The time series for Re = 2.79 × 10⁵ is almost constant, corresponding to a condition in period III. The time series corresponding to period IV, however, exhibits irregular oscillations with much smaller amplitude than the LFLA oscillations. Although the time series oscillations are irregular, their average amplitude and dominant frequency are vary consistently with Reynolds number. The dominant frequency increases linearly with increasing Reynolds number, while the amplitude decreases linearly with it, as shown by the arrows in Fig. 15.

Figure 16 shows the cavitation number as a function of the Reynolds number in the range 1.6 × 10⁵–5 × 10⁵ for different cavity lengths. As discussed above, the dependence of the cavitation number on the Reynolds number experiences four periods, which are evidenced for every cavity length; the same flow characteristics are observed for the corresponding periods at each cavity length. Nevertheless, the cavitation number dependence exhibits some differences depending on the cavity length for different periods. First, the Reynolds number for the second inflection point increases linearly with increasing cavity length, which means that it is more difficult for fully developed cavitation to occur in longer cavities. Second, as the cavity length increases, the pressure and the flow pattern become increasingly unstable in period I, whereas they become increasingly stable in period IV. The dependence of the transition Reynolds number on cavity length for periods II and III can be expressed as Re_II–III = 25 742L_c/d₁ + 89 182, for which the goodness of fit is R² = 0.989.

It is difficult to determine the oscillation characteristics from data in the time and frequency domains. In this subsection, the nonlinear time series method is used to analyze the nonlinear characteristics of self-sustained oscillations induced by the axisymmetric cavity, on the basis of experimental pressure data. The oscillation characteristics of the system can be estimated from the shapes of the reconstructed attractors, namely, a limit cycle for periodic oscillation, a near-periodic orbit for semiperiodic oscillation, and a strange attractor for chaotic oscillations. A total of N = 3000 data series are extracted from the M1 pressure transducer with a sampling interval of 1/150 s, implying a total of 20 s of information on the system. The noise is filtered out during the smoothing process.

Figures 17 and 18 illustrate the three-dimensional reconstructed attractors for the unsubmerged condition Re = 2.87 × 10⁵ and the submerged condition Re = 2.85 × 10⁵, respectively. Using the C–C method, the embedding dimension and time delay for the unsubmerged condition are found to be m = 6 and τ = 62, while for the submerged condition, they are m = 7 and τ = 41. The calculated largest Lyapunov exponents are 0.0062 and 0.0012, and their corresponding time series curves are shown in Figs. 5 and 6, respectively.

For the unsubmerged condition, so-called LFLA periodic oscillations are observed. However, the attractor reconstructed using nonlinear time series analysis features a semiperiodic orbit, meaning that the LFLA oscillation is not strictly periodic, but is affected by stochastic factors. For the submerged condition, the reconstructed strange attractor indicates that the oscillation is of chaotic character. For the unsubmerged condition, all the reconstructed attractors feature a similar shape as that presented in Fig. 17, whereas they are strange attractors for the submerged condition, as in Fig. 18.

As was discussed in Sec. III B, the dominant mechanism of LFLA oscillations in the unsubmerged condition is the periodic connection between the inside of the cavity and the atmosphere. In the submerged condition, this mechanism does not exist, because the cavity is completely closed by the jet flow. The oscillation is induced by the stochastic variation of the jet spread angle and its deviation in the radial direction, which explains the chaotic characteristics indicated by the corresponding attractor shape. The stochastic oscillation amplitude induced by these two factors is significantly smaller than the LFLA oscillation. Therefore, it can be concluded that the LFLA oscillations are jointly induced by three factors, namely, the periodic connection between the inside of the cavity and the atmosphere, and the stochastic factors associated with the spread angle variation and the jet core radial deviation. These dominant periodic and minor small-amplitude stochastic factors jointly affect the generated semiperiodic LFLA self-sustained oscillation characteristics.

In this study, experiments have been carried out to reveal the generation mechanism of LFLA self-sustained oscillations, the oscillation characteristics, the nonlinear phenomena, and the cavitating flow patterns of a horizontal axisymmetric cavity flow.

First, two flow states, unsubmerged jet flow and submerged jet flow, were identified inside the cavity. The LFLA oscillations only occur under unsubmerged jet flow conditions. The cavity length dictates the likelihood of occurrence of LFLA oscillations: this likelihood has been shown to decrease as the cavity length increases. The LFLA oscillations feature semiperiodic characteristics, given that the dominant mechanism is a periodic connection between the cavity interior and the atmosphere, whereas the minor factors, namely, the jet spread angle variation and the jet radial deviation, are stochastic. Second, the oscillation amplitude in the submerged flow state is much smaller than that in the case of LFLA oscillations, and the oscillations are chaotic. The mechanism here is stochastic jet angle spread and its deviation in the radial direction. Third, the flow pattern inside the cavity can be divided into four stages according to the number of bubbles present. These bubbles separate from the water body or cavitate when the pressure decreases below a certain value. Cavitation occurs when the cavity is closed by the jet. Finally, the flow pattern inside the cavity in the submerged state can be partitioned into four different periods according to the pressure variations and the flow patterns observed.

Although this article has elucidated the generation mechanism of LFLA self-sustained oscillations, subsequent investigations are needed to establish a quantitative theoretical dynamic model. The present experimental results may help understand the self-sustained cavitating oscillation phenomenon occurring in the cavity, whereas a quantitative model may facilitate the suppression or enhancement of this phenomenon in fluids engineering applications.

The supplementary material provides two experimental videos. The first one displays an LFLA oscillation process in Fig. 11; the second one displays the flow pattern change from period Ⅱ to period Ⅲ in Fig. 13.

This research was supported by the National Natural Science Foundation of China (Grant No. 51509209).

The authors have no conflicts to disclose.

Yuchuan Wang: Writing – original draft (lead). Bifeng Li: Writing – review & editing (equal). Lei Tan: Conceptualization (equal); Software (equal). Chuanchang Gao: Methodology (equal); Project administration (equal); Supervision (equal). Sebastián Leguizamón: Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.