Insights into ventilation hysteresis shift due to flow unsteadiness in ventilated supercavitation
Kyungduck Yoon, Jiaqi Li, Siyao Shao, Ashish Karn, Jiarong Hong
HHighlights
Insights into ventilation hysteresis shift due to flow unsteadiness in ventilated supercavitation
Kyungduck Yoon,Jiaqi Li,Siyao Shao,Ashish Karn,Jiarong Hong• Formation ventilation correlated with behavior of cavitator recirculation zone• Recirculation position and size vary upon flow unsteadiness• Formation ventilation can decrease upon increasing Fr and flow unsteadiness• Instability of cavity interface correlated with collapse ventilation demand a r X i v : . [ phy s i c s . f l u - dyn ] F e b nsights into ventilation hysteresis shift due to flow unsteadiness inventilated supercavitation Kyungduck Yoon a,b , Jiaqi Li a,c , Siyao Shao a,c , Ashish Karn d and Jiarong Hong a,c , ∗ a St. Anthony Falls Laboratory, University of Minnesota, Minneapolis, MN 55414, USA b The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA c Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55414, USA d Department of Mechanical Engineering, School of Engineering, University of Petroleum and Energy Studies, Energy Acres, Bidholi, Dehradun,Uttarakhand 248007, India
A R T I C L E I N F O
Keywords :Ventilated supercavitationVentilation hysteresisForward facing model
A B S T R A C T
Understanding the air injection strategy of a ventilated supercavity is important for designing high-speed underwater vehicles wherein an artificial gas pocket is created behind a flow separation deviceto reduce skin friction. Our study systematically investigates the effect of flow unsteadiness on theventilation requirements to form ( 𝐶 𝑄𝑓 ) and collapse ( 𝐶 𝑄𝑐 ) a supercavity. Imposing flow unsteadinesson the incoming flow has shown an increment in higher 𝐶 𝑄𝑓 at low free stream velocity and lower 𝐶 𝑄𝑓 at high free stream velocity. High-speed imaging reveals distinctly different behaviors in therecirculation region for low and high freestream velocity under unsteady flows. At low free streamvelocities, the recirculation region formed downstream of a cavitator shifted vertically with flow un-steadiness, resulting in lower bubble collision and coalescence probability, which is critical for thesupercavity formation process. The recirculation region negligibly changed with flow unsteadinessat high free stream velocity and less ventilation is required to form a supercavity compared to that ofthe steady incoming flow. Such a difference is attributed to the increased transverse Reynolds stressthat aids bubble collision in a confined space of the recirculation region. 𝐶 𝑄𝑐 is found to heavily relyon the vertical component of the flow unsteadiness and the free stream velocity. Interfacial instabil-ity located upper rear of the supercavity develops noticeably with flow unsteadiness and additionalbubbles formed by the distorted interface shed from the supercavity, resulting in an increased 𝐶 𝑄𝑐 .Further analysis on the quantification of such additional bubble leakage rate indicates that the devel-opment and amplitude of the interfacial instability accounts for the variation of 𝐶 𝑄𝑐 under a widerange of flow unsteadiness. Our study provides some insights on the design of a ventilation strategyfor supercavitating vehicles in practice.
1. Introduction
Ventilated supercavitation refers to the formation of anartificial gas pocket in water flow created by air injection be-hind a flow separation device, i.e., cavitator in such a waythat the so-formed cavity is large enough to surround an im-mersed vehicle. This phenomenon has been broadly inves-tigated for its potential applications in the drag reductionfor high-speed operation of underwater vehicles [1]. Due tothe complex multi-phase interactions involved in cavitatingflows, which are sensitive to flow conditions, a significantamount of research has been conducted on characterizing thebehaviors of ventilated supercavities [2] as well as on me-chanical control strategies [3]. However, despite numerousresearch reported on the characterization of general behav-iors of ventilated supercavities [4], an area that has not hith-erto received significant attention, is the ventilation strategy,i.e. optimally controlling the ventilation rate for the super-cavity to be formed and sustained under various flow condi-tions.Compared to several investigations that report generalcavity behaviors, such as geometry, shape, and cavity clo-sure, only a handful of studies focus on the ventilation re- ∗ Corresponding author
[email protected] (J. Hong)
ORCID (s): quirements associated with the supercavity formation and itssustenance upon formation. For instance, Karn et al. [5],explored the ventilation hysteresis phenomenon in great de-tail and established that the ventilation demands to form andto sustain a supercavity may be significantly different, thelatter being much smaller than the former. In a followupwork, Karn et al. [6] investigated the ventilation demandsof the supercavity under various flow settings and provideda detailed explanation of the cavity formation and collapseprocesses, relating each with bubble coalescence proficiencyand pressure balance near the closure. Their research hasbeen conducted for a backward-facing model (BFM) withonly a disk type cavitator. However, there is an inherentlimitation with this type of cavitator configuration – it ne-glects the effects of the cavitator shape or the presence of themounting strut, both of which have been shown to noticeablyaffect the cavity behaviors [7–9], especially the supercavityformation process [10]. Recently, Shao et al. [11] studiedthe ventilation demands for a forward-facing model (FFM)with a variety of cavitator geometries (cone, disk, and non-axisymmetric) to consider both the cavitator shape and themounting strut effects. They have found out that the conetype cavitator required the least ventilation flow to form andsustain a cavity. They have also observed that the interactionbetween the mounting strut and the air-water interface leadsto a noticeable change in collapse ventilation demand com-
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Page 1 of 9 nsights into ventilation hysteresis shift due to flow unsteadiness in ventilated supercavitation pared to that of the BFM case. Their detailed analysis on themomentum balance between the air injection and the esti-mated re-entrant jet at the closure further supported that there-entrant jet governs the cavity collapse process. However,although it has been reported that the ventilation demanddepends crucially on the flow unsteadiness [6], which maysignificantly alter the operation of the supercavitating object[3, 12], the investigations on ventilation demand and venti-lation hysteresis to date have been limited to the steady flowconditions only. Therefore, to connect the lab-scale exper-iments with the practical situations of underwater vehiclesencountering surface waves, experimental investigations ex-ploring the role of different cavitator shapes and mountingstrut effects in unsteady flows is needed, not only to under-stand general cavity behaviors, but to investigate the under-lying physics with an express emphasis on the ventilationdemand and ventilation hysteresis.Though mostly limited to the general cavity behaviorsor the ventilation demands of the BFM supercavity, a fewrecent studies investigated ventilated supercavitation underunsteady flows by using a gust generator that consists of flap-ping hydrofoils [6, 12–14]. Such a setup was deployed tosimulate unsteady incoming flows by controlling either theangle of attack (
𝐴𝑜𝐴 ) or the flapping frequency ( 𝑓 𝑔 ) of thehydrofoils. In particular, it has been reported that for theunsteady flows the cavity dimensions and cavitation num-ber ( 𝜎 𝑐 ) periodically change [12] and closure variation is ob-served between twin-vortex and re-entrant jet [13]. Shao etal. [14] further classified FFM supercavity into five distinctstates (namely stable, wavy, pulsating 1, pulsating 2, andcollapsing states), characterizing each state based on the si-multaneous pressure measurement and high-speed imaging.They observed transitions across these states with a changein either 𝐴𝑜𝐴 or 𝑓 𝑔 and further proposed a stability criterionfor these state transitions. Karn et al. [6] studied the for-mation and collapse ventilation demand trends with respectto the change in 𝐴𝑜𝐴 and 𝑓 𝑔 for BFM and observed that allthese demands increase with higher flow unsteadiness. Theynoted that such flows impose a vertical perturbation to theindividual bubble movements that lead to the increased for-mation ventilation demand. They also commented on the in-ternal pressure fluctuation that leads to higher collapse venti-lation demand. However, their explanations of the observedtrends heavily relied on the implications of flow perturbationand pressure fluctuation but lacked visual evidence.Therefore, as a follow-up study of Shao et al. [11], wepresent a systematic examination of unsteady flow condi-tions, generated by the gust generator, on the formation andsustenance ventilation demand of FFM model (including themounting strut effect) with a cone type cavitator which hasshown to require the least ventilation demand to form andsustain the supercavity. The rest of the sections are as fol-lows: Section 2 provides detailed explanation on the exper-imental methods. The results of our study are presented inSection 3. Specifically, Section 3.1- 3.2 demonstrates theresults of cone cavitator with different flow regimes. Sec-tion 3.3 provides a quantitative estimation of change in the Figure 1:
High-speed cavitation water tunnel at the Saint An-thony Falls Laboratory (SAFL), University of Minnesota. Thisschematic is adapted from [11]. collapse ventilation demand due to the flow unsteadiness.Section 4 provides a summary of the current study.
2. Experimental Setup and Methodology
The experiments are conducted in the high-speed cavita-tion water tunnel at Saint Anthony Falls Laboratory (SAFL),University of Minnesota. As shown in Fig. 1, the flow fa-cility consists of a closed recirculating tunnel with a largevolume dome-shaped settling chamber located upstream ofthe test section, designed for fast bubble removal during theventilation experiments. The dimension of the test sectionis .
20 m × 0 .
19 m × 0 .
19 m (length, height, and width),and the bottom and the two side windows of the test sectionare made up of Plexiglas for optical access. During the ex-periments, the free stream velocity is calculated through aRosemount 3051 differential pressure sensor that measuresthe differential pressure between the settling chamber andthe test section. Correspondingly, the desired free streamvelocity in the test-section is regulated by feedback controlof the motor attached to the centrifugal pump, located at thebottom section of the water tunnel facility. In recent years,this facility has been used extensively for a number of ven-
Table 1
Experimental conditions illustrating the incoming flow un-steadiness. Froude number (
𝐹 𝑟 ) is calculated based on thecavitator diameter ( 𝑑 𝑐 ). In the table, 𝐹 𝑟 refers to low 𝐹 𝑟 and
𝐹 𝑟 refers to high 𝐹 𝑟 hereafter, which is demarcatedbased on the bubble concentration regime from previous stud-ies [6, 11].
𝐹 𝑟 = 𝑈 ∕ √ 𝑔𝑑 𝑐 𝐴𝑜𝐴 [ ° ] 𝑓 𝑔 [Hz]0 05 2 , ,
10 12 , ,
10 50 015 2 , ,
10 12 , ,
10 5
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Figure 2:
Test section with gust generator setup. Units are in mm . tilated partial- and supercavitation experiments [15–17].Similar to the setup from [14], the gust generator is in-stalled at the upstream of the test section (Fig. 2). Unsteadyflows are generated by the continuously flapping NACA0020hydrofoils (detailed in [18]) with various range of 𝐴𝑜𝐴 and 𝑓 𝑔 imposing a vertical perturbation in the flow that prop-agates downstream. During the experiments, supercavita-tion is generated by ventilating air behind the cavitator. Themass flow rate of the ventilated air is controlled by an OmegaEngineering FMA-2609A mass flow controller that has aproportional-integral-derivative algorithm for controlling ata constant desired flow rate up to 55 SLPM with the uncer-tainty within ±0 . 𝐹 𝑆 . A FFM cone-type cavitator witha diameter of 30 mm has been employed, which is the exactsame model as reported by [11] to facilitate the investigationof unsteady flow effects.Table 1 lists the conditions that are investigated in thecurrent experiment. Froude number (
𝐹 𝑟 ) is used to scalethe free stream velocity, with a characteristic length of thecavitator diameter ( 𝑑 𝑐 ). The experiments are conducted attwo different 𝐹 𝑟 , and in each case the
𝐴𝑜𝐴 and 𝑓 𝑔 are var-ied in a wide range to simulate various incoming flow un-steadiness. The underlying reasons for the choice of such Frwill be discussed shortly hereafter. The steady flow condi-tion for comparison refers to zero degree of 𝐴𝑜𝐴 and zerofrequency of 𝑓 𝑔 , i.e., foils parallel to the unperturbed flowdirection. During the experiments, the ventilation demands( 𝐶 𝑄 = ̇𝑄 ∕ 𝑈 𝑑 𝑐 ) are measured five times at each flow con-dition to ensure robustness from potential outliers. Specif-ically, the ‘Formation ventilation demand’ ( 𝐶 𝑄𝑓 ) refers toa critical ventilation rate at which the bubbles beyond thecavitator start to coalesce and form a stable supercavity witha transparent air-water interface. The ‘Collapse ventilationdemand’ ( 𝐶 𝑄𝑐 ) refers to the minimum ventilation rate re-quired, at which a supercavity can be sustained upon forma-tion, without collapsing. The measurements of these ventila-tion demands are recorded by changing the ventilation rateswith an increment or decrement less than . SLPM to en-sure a stable variation of ventilation rate and to accuratelycapture the critical value of 𝐶 𝑄𝑓 and 𝐶 𝑄𝑐 .The ventilation demands are investigated for 𝐹 𝑟 (low)and (high) that represents different regimes of bubble sizeand concentration [6, 11]. Particularly, the low 𝐹 𝑟 regime,also referred to as the low bubble concentration regime, con-
Figure 3:
Ventilation coefficients for low
𝐹 𝑟 regime under vary-ing flow unsteadiness modified by
𝐴𝑜𝐴 and 𝑓 𝑔 . Dotted horizon-tal lines indicate steady flow condition, dashed lines indicate 𝐶 𝑄𝑓 trends, dashed-dotted lines indicate 𝐶 𝑄𝑐 trends. Symbols: ■ , 𝐶 𝑄𝑓 at 𝐴𝑜𝐴 ° ; □ , 𝐶 𝑄𝑐 at 𝐴𝑜𝐴 ° ; ⧫ , 𝐶 𝑄𝑓 at 𝐴𝑜𝐴 ° ; ◊ , 𝐶 𝑄𝑐 at 𝐴𝑜𝐴 ° ; ▴ , 𝐶 𝑄𝑓 at 𝐴𝑜𝐴 ° ; △ , 𝐶 𝑄𝑐 at 𝐴𝑜𝐴 ° . sists of a relatively bigger bubble size distribution [19], andincreasing flow speed in this regime breaks up individualbubbles. Therefore, in the low Fr regime, increasing 𝐹 𝑟 re-sults in the growth of 𝐶 𝑄𝑓 since more ventilation is requiredto increase the size and number of bubbles to coalesce andform a supercavity [6]. High 𝐹 𝑟 regime, on the contrary,consists of a higher concentration of smaller bubble size dueto the dominant role of turbulence-induced bubble breakup.Consequently, in high
𝐹 𝑟 regime, an increase in flow speedfavors the bubble coalescence process and therefore 𝐶 𝑄𝑐 de-creases, since the bubble concentration in this regime is al-ready high enough to restrict the free space for bubble move-ment and eventually increase the chance of bubble collision[6]. A critical 𝐹 𝑟 that demarcates such different regimes isfound to be around 10 [6, 11].A Photron APX-RS high-speed camera is deployed tofurnish visual evidences on the observed trends of 𝐶 𝑄 . Thetime resolution for the high-speed imaging varies with 𝐹 𝑟 ,from (for low
𝐹 𝑟 regime) to (for high
𝐹 𝑟 regime) with a sensor size of
512 × 512 pixels. The high-speed images for 𝐶 𝑄𝑓 are taken at the foamy cavity state(terminology from [5]) with the ventilation rate slightly be-low 𝐶 𝑄𝑓 to infer the effect of flow unsteadiness in the bubblecoalescence process. For investigating 𝐶 𝑄𝑐 , the high-speedimages are acquired at the ventilation rate slightly above 𝐶 𝑄𝑐 to infer the effect of flow unsteadiness on the gas leakagemechanism and the cavity stability.
3. Results
𝐹 𝑟 regime
As shown in Fig. 3, 𝐶 𝑄 at low 𝐹 𝑟 regime are plottedfrom steady to various unsteady incoming flow conditions.
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Figure 4: (a) Sample high-speed image at the field-of-view (highlighted region in the schematics) is taken at 𝐶 𝑄 slightly below 𝐶 𝑄𝑓 . The region marked with the white dotted line in the sample image indicates the recirculation region with constantly lowintensity due to bubbles blocking the light path from the illumination source. Based on the image sequence, standard deviationsof image intensity for the (b) steady and (c) unsteady flows ( 𝐴𝑜𝐴 ° , 𝑓 𝑔 ) at low 𝐹 𝑟 are plotted at each pixel location.The brighter region indicates a higher fluctuation of the brightness throughout the recorded sequence. Regions marked with graydotted lines imply the size of the recirculation region for the steady flow case. The difference between steady and unsteady flowsare plotted in (d), indicating that the incoming flow unsteadiness imposes vertical fluctuation of the recirculation region.
In this regime, 𝐶 𝑄𝑓 trends (dashed lines) show a slight in-crease with 𝑓 𝑔 but are minimally influenced by 𝐴𝑜𝐴 . 𝐶 𝑄𝑐 plot shows a similar increasing trend but with an exception at 𝐴𝑜𝐴 of ° (low 𝐴𝑜𝐴 ), which shows a little disparity as com-pared to that of the steady flow condition (the dotted linebelow). The gap between 𝐶 𝑄𝑓 and 𝐶 𝑄𝑐 can be attributed tothe occurrence of ventilation hysteresis at each flow condi-tion, which decreases with 𝑓 𝑔 for 𝐴𝑜𝐴 ° (moderate 𝐴𝑜𝐴 )and
𝐴𝑜𝐴 ° (high 𝐴𝑜𝐴 hereafter).In an attempt to adduce a possible mechanism for the ob-served 𝐶 𝑄𝑓 trends, further investigation is conducted withhigh-speed imaging of the foamy cavity state slightly be-low the 𝐶 𝑄𝑓 for both steady and unsteady flow conditions(Fig. 4a). As shown from a sample image, the light sourcefrom the other side of the test section illuminates the bubblyflow, and the bubble movements result in intensity fluctu-ations in the recorded images. By taking the standard de-viation of the entire image sequence for a sufficiently longduration at each pixel location, it is possible to provide anestimate of the size of the recirculation region beyond thecavitator. The recirculation region consists of a high densityof bubbles that are optically thick (and therefore shows a rel-atively low standard deviation of the pixel intensity), and theremaining flow field exhibits large fluctuations in intensity atthe rear location. A timespan of four times the gust cycle isconsidered sufficient for the computation of standard devia-tion of intensity variations in the images.Figure 4 illustrates the methodology adopted to trace outthe recirculation region at the supercavity rear portion. Theschematic shown in Fig. 4a outlines the region of interestchosen for the analysis of the bubble images. Standard devi-ation values at each pixel location for steady (Fig. 4b) flows,unsteady (Fig. 4c) flow conditions, and the absolute differ-ence between the two (Fig. 4d) are presented. The gray dot-ted regions in Fig. 4b and c (representing the inferred bound-ary of the recirculation region for steady incoming flow case)indicates that the recirculation region for the steady flow ismore elongated than that for the unsteady flow. Stated dif-ferently, Fig. 4d suggests that the incoming flow unsteadi- ness imposes fluctuation of the recirculation region beyondthe cavitator. Therefore, a slight increment in 𝐶 𝑄𝑓 for un-steady flows may be attributed to the vertical fluctuations ofthe recirculation region, which adversely affects the bubblecoalescence efficiency, by decreasing the chance of bubblecollisions to form a larger bubble in the low 𝐹 𝑟 regime. Sim-ilarly, an increase in
𝐴𝑜𝐴 has shown to amplify the magni-tude of the vertical perturbation [14] of the incoming flow,and therefore may adversely impact overall bubble coales-cence efficiency. Likewise, for the same
𝐴𝑜𝐴 , a greater 𝑓 𝑔 means there is lesser time interval conceded per gust cyclefor the bubbles to coalesce and form a supercavity, and hence 𝐶 𝑄𝑓 grows with 𝑓 𝑔 .High-speed images acquired near the closure of the su-percavity also provide interesting visual evidence on the 𝐶 𝑄𝑐 trends as well. From Fig. 3, it is observed that 𝐶 𝑄𝑐 plotshows an increasing trend with 𝑓 𝑔 for moderate and high 𝐴𝑜𝐴 but has a minimal significance for low
𝐴𝑜𝐴 . Figure 5clearly demonstrates that the interface instability at the up-per surface near the closure is periodically developed at eachgust cycle when the flapping foil is heading upward. Notethat Shao et al. [14] attributed such phenomenon to the in-verse alignment of the density gradient with respect to thedirection of gravity. For low
𝐴𝑜𝐴 , a wavy pattern of theoverall shape of the supercavity is observed without any no-ticeable distortion of the air-water interface (Fig. 5a). Thisinstability, however, appears and becomes stronger for un-steady flows with higher
𝐴𝑜𝐴 (Fig. 5b-c), periodically shed-ding off a certain volume of gas near the closure which even-tually leads to a periodic fluctuation of cavity length due tothe additional gas leakage. As regards the effect of 𝑓 𝑔 on 𝐶 𝑄𝑐 trends, similar to the formation case, an increase in 𝑓 𝑔 essentially implies lesser time available per gust cycle forthe supercavity to recover its length caused by the periodicshedding of the gas pocket (for moderate and high 𝐴𝑜𝐴 ).High-speed imaging of the supercavity collapse process fur-ther suggests the role of interface instability that has not beendiscussed before in the explanations of its trend on BFM cav-itator provided by Karn et al. [6], where the change of 𝐶 𝑄𝑐 Yoon et al.:
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Figure 5:
High-speed imaging taken near the closure for low
𝐹 𝑟 regime with various
𝐴𝑜𝐴 , (a) ° , (b) ° , and (c) ° , respectively.Red dotted lines help visualize the air-water interface distortion due to the development of instability. Stronger instability isobserved for higher 𝐴𝑜𝐴 that leads to periodic additional gas leakage. For low
𝐴𝑜𝐴 , however, no additional gas leakage isobserved.
Figure 6:
Ventilation coefficients at high
𝐹 𝑟 regime undervarying flow unsteadiness modified by
𝐴𝑜𝐴 and 𝑓 𝑔 . Dottedhorizontal lines indicate steady flow condition, dashed linesindicate 𝐶 𝑄𝑓 trends, dashed-dotted lines indicate 𝐶 𝑄𝑐 trends.Symbols: ■ , 𝐶 𝑄𝑓 at 𝐴𝑜𝐴 ° ; □ , 𝐶 𝑄𝑐 at 𝐴𝑜𝐴 ° ; ⧫ , 𝐶 𝑄𝑓 at 𝐴𝑜𝐴 ° ; ◊ , 𝐶 𝑄𝑐 at 𝐴𝑜𝐴 ° ; ▴ , 𝐶 𝑄𝑓 at 𝐴𝑜𝐴 ° ; △ , 𝐶 𝑄𝑐 at 𝐴𝑜𝐴 ° . for unsteady flows was attributed only to the internal pres-sure fluctuation in a supercavity. 𝐹 𝑟 regime
In contrast to the observations in the low
𝐹 𝑟 regime, the 𝐶 𝑄 plots for the high 𝐹 𝑟 regime, as shown in Fig. 6, exhibit aslightly different trend, both qualitatively and quantitatively.In particular, for low and moderate
𝐴𝑜𝐴 , 𝐶 𝑄𝑓 plot showsa marginal reduction with a consequent rise in 𝑓 𝑔 , and theslopes become steeper for the higher 𝐴𝑜𝐴 . 𝐶 𝑄𝑐 plot stillshows a rising trend even for the lowest 𝐴𝑜𝐴 . However, akinto the low
𝐹 𝑟 case, the ventilation hysteresis gap reduceswith increasing
𝐴𝑜𝐴 and 𝑓 𝑔 .Next, in a manner analogous to the low 𝐹 𝑟 regime case,the recirculation regions are traced out at the supercavity rearportion by computing standard deviations of each pixel in-tensity in the recorded images of foamy for steady (Fig. 7a) and unsteady (Fig. 7b) flow conditions, to understand the 𝐶 𝑄𝑓 trends observed in the high 𝐹 𝑟 regime at low and mod-erate
𝐴𝑜𝐴 . Surprisingly, as shown from Fig. 7c, the recircu-lation region in the high
𝐹 𝑟 regime remained relatively un-changed with the incoming flow unsteadiness. In this regime,the inherent high-level turbulence near the wake region mayminimize the effect of vertical flow perturbation. Also, ithas been reported that the flow unsteadiness results in highertransverse Reynolds stress, or 𝜈 ′ 𝜈 ′ ∕ 𝑈 𝑀 , within the recircu-lation region beyond the backward-facing step or blunt body[20, 21]. The amplified vertical velocity perturbations areimposed on the bubble motion, enhancing the bubble col-lision probability that is crucial to the coalescence processwithin the confined recirculation region, which is alreadyconstricted by the presence of the mounting strut. Thus,in the situation of the presence of severe flow unsteadiness, 𝐶 𝑄𝑓 decreases in comparison to the steady flow case. Fur-ther, a greater 𝐴𝑜𝐴 entails imposing a consequently largervertical perturbation to the incoming flow [14] and it is pos-sible that the resulting vertically intensified bubble move-ments may be responsible for the reduction of 𝐶 𝑄𝑓 for higher 𝐴𝑜𝐴 . Higher 𝑓 𝑔 has a similar role in contributing to the in-creased vertical perturbation of the bubble movements forthe given low and moderate 𝐴𝑜𝐴 unsteady flow cases.The influence of flow unsteadiness on 𝐶 𝑄𝑐 for this spe-cific unsteady flow conditions (low and moderate 𝐴𝑜𝐴 ) arefurther plotted in Fig. 8a-b. As shown from the figure, evenfor the lowest
𝐴𝑜𝐴 , a certain gas volume sheds off from thesupercavity due to the developed interface instability. In fact,compared to the low
𝐹 𝑟 regime, all
𝐴𝑜𝐴 cases show strongerdistortion of the air-water interface. As a result, 𝐶 𝑄𝑐 re-quired for the supercavity to sustain itself off collapsing, in-creases owing to the additional gas leakage. The effect of 𝐴𝑜𝐴 and 𝑓 𝑔 on 𝐶 𝑄𝑐 seems to share the same mechanism asthat of the low 𝐹 𝑟 regime.For the high
𝐴𝑜𝐴 , the trends observed for both 𝐶 𝑄𝑓 and 𝐶 𝑄𝑐 exhibit a significantly disparate behavior as comparedto the rest of the experimental conditions. For instance, asshown from Fig. 6, 𝐶 𝑄𝑓 slightly diminishes at low 𝑓 𝑔 butthen surges again to a value that is greater than that for thesteady flow condition. In addition, 𝐶 𝑄𝑐 drastically growswith 𝑓 𝑔 and exceeds the steady 𝐶 𝑄𝑓 and finally coincides Yoon et al.:
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Figure 7:
Standard deviations of image intensity for the (a) steady and (b) unsteady flows at high
𝐹 𝑟 regime are plotted at eachpixel location. Bright region indicates fluctuation of brightness throughout the recorded sequence. Difference between steadyand unsteady flows are plotted in (c), which is not noticeable.
Figure 8:
Samples of high-speed images acquired near the closure for high
𝐹 𝑟 regime with various
𝐴𝑜𝐴 , (a) ° , (b) ° , and (c) ° , respectively. Red dotted lines help visualize the air-water interface distortion due to the development of interface instability.Stronger instability is observed for higher 𝐴𝑜𝐴 that leads to periodic additional gas leakage. with the 𝐶 𝑄𝑓 trend at higher 𝑓 𝑔 . Such a phenomenon sug-gests the absence of ventilation hysteresis for highly unsteadyflows with respect to 𝐴𝑜𝐴 and 𝑓 𝑔 . At this condition, asshown from the high-speed images (Fig. 9), a drastic changeof the cavity length is observed for every gust cycle. Inparticular, when the flapping hydrofoil reaches its ampli-tude during every gust cycle, the cavity length is truncatedto a very short one, due to the strong interface instability(Fig. 8c). This is followed by a recovery of cavity elonga-tion in sync with the augmentation in re-entrant jet intensity,from its weakest to the strongest. As presented in Fig. 9, there-entrant jet becomes strong enough to block the light pathfrom the illumination source located on the other side of thetest section. As a result, the optical thickness of the inter-nal cavity grows as the re-entrant jet strengthens, despite thepresence of the supercavity with a smooth surface. The ab-sence of the ventilation hysteresis at this flow condition maybe caused by this phenomenon, as the majority of the inter-nal cavity consists of water and there exists less volume ofgas inside. Such phenomenon is consistent with the obser-vation from Kawakami and Arndt [22] where they noted thatFFM configuration supercavity has less ventilation hystere-sis gap compared to that of the BFM supercavity, due to thepresence of the mounting strut that reduces the internal gasvolume for the FFM models. The alterations in 𝐶 𝑄𝑐 due to the flow unsteadiness isnot quantitatively understood yet. Such a difference may be attributed to various reasons such as internal pressure fluctu-ation, the re-entrant jet momentum change (similar to [11]),or the additional gas leakage induced by the interfacial insta-bility. However, we observed that the flow unsteadiness andits state (e.g. flapping foils heading upward or downward)has a negligible effect on the re-entrant jet-speed estimation(following an approach similar to [11]), indicating that there-entrant jet momentum does not govern the cavity collapseunder unsteady flows. It is quite likely, then to conjecture, ofthe predominant influence that the interfacial instability mayhave on 𝐶 𝑄𝑐 . Hence, to substantiate this hypothesis, high-speed images of unsteady flow are investigated by estimatingthe additional gas leakage from the observed interface insta-bility for both high and low 𝐹 𝑟 regimes.Figure 10 illustrates how the interface instability is char-acterized for calculating the individual volume of gas bub-bles shedding off from the supercavity. The interface in-stability is approximated as a sinusoidal pattern which ischaracterized by its wavelength, wave amplitude, and wavefrequency. Based on the observation that the instability isstronger when
𝐴𝑜𝐴 is higher, we further approximate thecross-sectional shape of the instability as sinusoidal since theinterface instability is associated with the normal interactionbetween the vertical component of the free stream velocityand the upper interface of the supercavity. Here, we assumethat the cross-sectional shape of the supercavity is circularunder unperturbed flow conditions. The volume of gas shed-ding off the supercavity is calculated based on such approx-imations. It is important to note that the additional shed-ding only occurs at every gust cycle when the flapping foil
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Figure 9:
Snapshots of high-speed imaging taken near the closure for high
𝐹 𝑟 at the highest
𝐴𝑜𝐴 ( ° ) and 𝑓 𝑔 ( ). As thecavity recovers its length from its shortest ( 𝑡 = 𝑡 ), the re-entrant jet becomes stronger. Such re-entrant jet blocks the light pathinside the supercavity and the internal region correspondingly becomes darker, and the highlighted transparent regions decreasecorrespondingly. Red arrows indicate the re-entrant jet. Figure 10:
Illustration of interface instability characterization.Interface instability is characterized by its wavelength ( 𝜆 ), waveamplitude ( ℎ ), and wave frequency. The interface instabilitynear the closure of the supercavity is assumed to be 2D alongthe perimeter of the upper half of the cross-section. is heading downward and also that the wave frequency men-tioned here is independent of the gust frequency. Thus, theadditional gas leakage per second induced by the interfaceinstability is calculated by multiplying the individual shed-ding volume with half of the wave frequency. The acquiredleakage rate is in volumetric terms, so an additional step isrequired to convert it into a mass flowrate. Assuming the in-ternal pressure is spatially uniform, cavitation number from[11] is used for mass flowrate estimation. We further assumethat the internal pressure is temporally uniform, since the in-ternal pressure fluctuation range has been reported to be lessthan [14] which yields a maximum uncertainty of lessthan when converting the volumetric flow rate into themass flow rate.An estimation of the gas leakage rates under differentexperimental conditions can now be obtained. Figure 11presents a bar plot depicting the variations of Δ 𝐶 𝑄𝑐 due toflow unsteadiness ( 𝐶 𝑄𝑐,𝑢𝑛𝑠𝑡𝑒𝑎𝑑𝑦 − 𝐶 𝑄𝑐,𝑠𝑡𝑒𝑎𝑑𝑦 ), across a rangeof 𝑓 𝑔 and 𝐴𝑜𝐴 both in the low and high
𝐹 𝑟 regimes. It isworth noting that in this depiction, instances of extreme un-steadiness are avoided such as
𝐴𝑜𝐴 ° cases at low 𝐹 𝑟 whenthe interface instability does not lead to any additional gasleakage. Likewise, for
𝐴𝑜𝐴 ° cases at high 𝐹 𝑟 , the super-cavity truncates its length so much in each gust cycle thatit precludes any reasonable estimation of wave characteris-tics. The red bars in the figure are the actual increment of
Figure 11:
Increments in collapse ventilation demand due to in-coming flow unsteadiness for (a) low and (b) high
𝐹 𝑟 regimes.Red bars indicate the recorded data from Fig. 3 and Fig. 6 andblue bars indicate the calculated differences. 𝐶 𝑄𝑐 in SLPM recorded in the current study, correspondingto the 𝐶 𝑄𝑐 data points in Fig. 3 and Fig. 6. The blue barsindicate the calculated additional gas leakage rate inducedby interface instability. As shown from the figure, the calcu-lated leakage rates show an overestimation for the low 𝐴𝑜𝐴 cases and an underestimation for high
𝐴𝑜𝐴 cases. Such adifference may be attributed to the sinusoidal assumption ofthe interface profile. In reality, the shape of the surface in-stability varies from cnoidal to breaking waves depending onthe wave amplitudes. Also, interface instability characteri-zation is based on the high-speed imaging near the closure,and uncertainty in the estimation of the wave amplitude ofthe weak instability is tantamount to greater uncertainties inthe calculations of gas leakage rate. Nevertheless, consider-ing that
10 cm ∕s at standard conditions is roughly equiva-lent to . SLPM, our estimation shows a reasonable matchwith the experimentally measured values. Thus, our resultsindicate that the additional gas leakage induced by the inter-face instability indeed governs the increase in 𝐶 𝑄𝑐 observedfor unsteady flows. Yoon et al.:
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4. Summary and Conclusion
In this study, we have investigated the ventilation char-acteristics of a supercavity generated by a forward-facingcone-type cavitator under unsteady flows. Flow unsteadi-ness is adjusted by changing either the angle of attack (
𝐴𝑜𝐴 )or the frequency ( 𝑓 𝑔 ) of the flapping foils located upstreamof the test section. At a lower free stream velocity, the for-mation ventilation demand ( 𝐶 𝑄𝑓 ) and collapse ventilationdemand ( 𝐶 𝑄𝑐 ) both increase with flow unsteadiness exceptfor the collapse demand at the lowest 𝐴𝑜𝐴 . At a larger freestream velocity however, the observed trends are found tobe markedly different. With increase in flow unsteadiness,a gradual rise in 𝐶 𝑄𝑐 is observed up to an 𝐴𝑜𝐴 of °. But,at a higher 𝐴𝑜𝐴 of °, the collapse demand shows a drasticescalation and coincides with the formation demand trend,implying that the ventilation hysteresis no longer exists inhighly unsteady flows. High-speed imaging reveals that thechange in the recirculation region behind the cavitator withflow unsteadiness is responsible for the change in 𝐶 𝑄𝑓 . At alower free stream velocity, the recirculation region after thecavitator shifts vertically in response to the incoming wavesinduced by the gust generator. At a higher free stream ve-locity, the recirculation region shows a negligible disparitybetween the steady and unsteady flow conditions. An in-creased transverse Reynolds stress may be ascribed to thedecreased 𝐶 𝑄𝑓 due to the higher collision and coalescenceprobability of the dispersed bubbles within the recirculationregion. For 𝐶 𝑄𝑐 , it is believed that the increasing trend withincoming flow unsteadiness is due to the interface instabil-ity developed at the upper surface of the supercavity near theclosure, periodically shedding off the bubbles from the su-percavity. Subsequently, the additional gas leakage rate isestimated based on the high-speed imaging of the interfaceinstability. The estimated additional gas leakage rate showsa reasonable match with the measurement, suggesting thatthe interface instability governs the increase in 𝐶 𝑄𝑐 for un-steady flows.Our measurements provide a detailed explanation of thechange of 𝐶 𝑄 with incoming flow unsteadiness, which shedssome light on the ventilation strategy of cavitating vehicles.Specifically, the ventilation demand trends under unsteadyflows show some difference with Karn et al. [6] where theymeasured the ventilation demands with a backward-facingstep cavitator, which bears significantly less semblance tosupercavitating vehicles that have a solid body inside the su-percavity. Our study suggests that depending upon the flowunsteadiness, the underlying physics that govern the forma-tion and collapse of a supercavity are distinctly different.Further, by approximating the shape and profile of the inter-face instability as sinusoidal and assuming the cross-sectionof the supercavity to be circular, an estimate of the shed-offgas volume rate from the supercavity is calculated. Theseestimations however may yield a higher uncertainty whenthe wave amplitude increases. In reality, the equations ofthe curvilinear supercavity profile are more complex, partic-ularly with the continuously varying amplitude. In addition,in the current study, we have only investigated one flow con- dition in both low and high 𝐹 𝑟 regime. For supercavitatingvehicles in practice, different ventilation strategies should beapplied as the ventilation demand trends vary with vehiclespeeds. Therefore, it would be of great interest to further ex-amine the change of the ventilation demands with varyingfree stream velocity to investigate the
𝐹 𝑟 effect, in order tocomprehensively understand the underlying physics.
Declaration of Competing Interests
The authors declare that they have no known competingfinancial interests or personal relationships that could haveappeared to influence the work reported in this paper.
Acknowledgements
This work is supported by the Office of Naval Research(Program Manager, Dr. Thomas Fu) under grant No. N000141612755.
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