High-speed visualization and PIV measurements of cavitating flows around a semi-circular leading-edge flat plate and NACA0015 hydrofoil

International Journal of Multiphase Flow 60 (2014) 119–134

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International Journal of Multiphase Flow j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / i j m u l fl o w

High-speed visualization and PIV measurements of cavitating flows around a semi-circular leading-edge flat plate and NACA0015 hydrofoil A.Yu. Kravtsova a,b, D.M. Markovich a,b, K.S. Pervunin a,b,⇑, M.V. Timoshevskiy a,b, K. Hanjalic´ b,c a

Kutateladze Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia Department of Physics, Novosibirsk State University, Novosibirsk, Russia c Department of Chemical Engineering, Delft University of Technology, The Netherlands b

a r t i c l e

i n f o

Article history: Received 2 June 2013 Received in revised form 14 December 2013 Accepted 18 December 2013 Available online 31 December 2013 Keywords: Cavitation Partial cavities Cavitation pattern 2D hydrofoils High-speed imaging PIV

a b s t r a c t Cavitating flows around a flat plate with semi-circular leading edge and a NACA0015 hydrofoil at attack angles ranging from 0° to 9° and with varying cavitation number are investigated using high-speedimaging visualization (HIV) and particle-imaging velocimetry (PIV). Several known types of cavitation common to both foils, but also some different patterns, were observed. At small angles of incidence (less than 3°), cavitation on the plate begins in the form of a streak array (bubble-band) whereas on the hydrofoil as traveling bubbles. For the regimes with developed cavitation on the NACA0015 hydrofoil, the scattered and discontinuous bubble streaks branch and grow but subsequently merge into bubble clouds forming a remarkably regular lattice pattern. Once the incidence angle increased to 9°, the cavitation on the hydrofoil changed to a streaky pattern like that on the plate at small attack angles, whereas the regime on the plate showed no significant changes. The PIV method proved to be usable for measuring the instantaneous velocity also in the gas–vapor phase, albeit with reduced accuracy that was evaluated and accounted for on the basis of the effective (validation-surviving) number of imaging samples. The time-averaged velocity and turbulence moments show that the incipience of cavitation is governed by the development of the carrier-fluid flow around the foil leading edges, but the subsequent flow pattern depends strongly on the cavitation regime displaying markedly different distributions compared to the non-cavitating case. The main cavitation parameters: the maximum cavity length, the cloud cavity streamwise dimensions and the cloud shedding Strouhal number are analyzed and presented in function of the cavitation number and the attack angle in different scaling. The measurements confirm qualitatively the trends reported in the literature, but show also some quantitative differences, notably between the two foils considered. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Cavitation encountered in various machineries and devices operating with liquids are by far the major cause of damages that seriously impair the equipment availability, reliability and durability. The problem is in particular challenging in large-scale hydropower plants where the efficiency and longevity are the key requirements for successful and safe operation. Flow in complex configurations of real power systems and their prime components – hydroturbines, supply ducts and draft tubes is usually highly turbulent, unsteady and featured by large-scale vortical structures (LVS) and passive secondary motions of different origin and types. Often and especially at variable off-design loads, one unavoidably ⇑ Corresponding author at: Kutateladze Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia. Tel./fax: +7 3833356684. E-mail address: [email protected] (K.S. Pervunin). 0301-9322/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijmultiphaseflow.2013.12.004

encounters cavitation and the consequent undesirable two-phase phenomena. The dispersed phase (bubbles, clouds) is governed by turbulence and the LSV dynamics (turbulent dispersion), while the bubble motion and the vapor–gas phase in general significantly changes properties of turbulent fluctuations in the carrier phase (turbulence modulation) due to a number of physical mechanisms. Nonlinear interactions of LSV with gas–vapor dispersed phase lead often to strong flow pulsations, which may come into resonance with the structures natural frequency and cause excessive vibrations and, as a consequence, failure of hydraulic equipment or, at worst, even emergencies. The quest for improving the hydromachinery efficiency, durability and safety has necessitated ever-new studies of cavitation aimed at reducing its negative effects. More recently, these studies take the advantage of modern measuring techniques and computer simulations and are directed more towards gaining a deeper insight into physical mechanisms and processes, various regimes of cavitation and its interaction with flow of the carrier fluid. These

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methods make it possible to investigate simultaneously the spatial distribution and the time-dynamics of cavitation-generated gas– vapor dispersed phase, the vortical and turbulence structures of the carrier fluid and their interactions. This, in turn, should lead to better understanding and the ability to forecast cavitation and the consequent erosion, as well as to provide guidance for their control. It is noted, however, that the methods for reducing the negative effects of cavitation have to satisfy the requirement for power loss minimization. Despite extensive research, detailed quantitative information for cavitating flows required for design and optimization, as well as for verification of the existing and development of advanced mathematical models, is still lacking and, to a large extent, limited to simplified conditions and usually 2D geometries. Experiments on full-scale objects are very difficult or can even be impossible, but the laboratory investigations are also costly and require expensive laboratory setups. Thus, most publications in the literature report on research in simple geometries (Venturi nozzles, scaled-down shaped bodies, e.g., NACA series hydrofoils, etc.) though at varying conditions and with focus on diverse issues of the cavitation incipience and development of different regimes. The acquisition of systematic experimental data for various cavitation regimes is still an issue. Depending on flow conditions (foil shape and surface texture, velocity and pressure fields, overall flow configuration and others), one can distinguish a variety of cavitation patterns (bubbles, vortex or cloud cavitation, supercavitation, etc.), but practically all spatial cavitation patterns can be grouped into two generic types – partial cavities and supercavities. Partial cavities occur when a cavity closes on the cavitating surface, while a supercavity closes downstream in the wake of the cavitating object. The partial cavities can further be classified into closed and open cavities, depending on the flow in the cavity closure region (Laberteaux and Ceccio, 2001). A typical example of closed cavities is the sheet cavitation whereas that of the open cavities is the cloud cavitation. A cloud cavitation arises when a large portion of vapor-filled cavity underlined with re-entrant fluid breaks off, forming a cavitation cloud. Experimental studies report on visual observations and regime maps with qualitative descriptions (e.g., Franc and Michel, 1985; Kjeldsen et al., 2000; Callenaere et al., 2001), point pressure (Kubota et al., 1989; Leroux et al., 2005; Cervone et al., 2006) and velocity measurements (Kubota et al., 1989; Astolfi et al., 2000). In the last two decades particle image velocimetry (PIV) has been used by several research groups (e.g., Tassin and Ceccio, 1998; Gopalan and Katz, 2000; Foeth et al., 2006; Dular et al., 2007; Huang et al., 2013), proving that PIV can serve as a suitable tool to study cavitating flows even in strongly unstable regimes. Admittedly, the seeding particles can act as the cavitation nuclei, but Tassin et al. (1995) showed that, at certain conditions and with the proper choice of particles, this problem can be alleviated. Measurements of the local volume fraction of vapor phase within a cavity and in the wake past a foil were also reported (e.g., Coutier-Delgosha et al., 2006). The impact of thermal effects on the onset of various forms of cavitation and instabilities was studied by Cervone et al. (2006). Much of the work published in the literature deals with the experimental investigation of cloud cavitation (e.g., Kawanami et al., 1997; Callenaere et al., 2001). As widely acknowledged, when a sheet cavity reaches a certain dimension, periodic oscillations of cavity appear. This process is accompanied by a cavity (in the form of clouds) shedding downstream. Kubota et al. (1989) showed that a cloud cavity consists of a large-scale vortex and a cluster of small vapor bubbles situated in the core of the vortex. In various experiments (e.g., George et al., 2000) it was revealed that cloud cavity pulsations occur at similar Strouhal numbers. Callenaere et al. (2001) and Kawakami et al. (2008) proposed correlations between separation of a cloud cavity

and the re-entrant jet generated in the cavity closure region due to the adverse pressure gradient. This paper reports on the investigating of cavitation in flows over two generic foils: a round-nosed flat plate and a symmetric NACA0015 hydrofoil at different attack angles and cavitation numbers. High-speed images and movies are analyzed to detect and interpret the cavitation regimes, interactions of the flow hydrodynamics and cavitation, partial cavity initiation and detachment, and cloud cavity shedding. The characteristic integral cavitation parameters, deduced from visual observations and measurements, are correlated in nondimensional forms in terms of the cavitation number and attack angle. The PIV-measurements at various flow conditions provide insight into the velocity and turbulence fields and their modulations in various cavitation regimes. 2. Experiment 2.1. The cavitation rig Experiments were carried out in the Cavitation tunnel of the Institute of Thermophysics SB RAS, which represents a closed hydrodynamic circuit. The setup is equipped with two centrifugal pumps, ultrasonic flowmeter, temperature and pressure sensors (Fig. 1). The dimensions of the tunnel are L  H  W = 8.4  2.2  1.1 m. Its working section consists of a heat exchanger, honeycomb, nozzle, test section and diffuser. The temperature of the working liquid is maintained constant by means of a control system consisting of the temperature sensor and electromagnetic valve driven by a PID control. The 500 mm long honeycomb cells have a rectangular cross-section of 25  25 mm. The nozzle of 790 mm in length is shaped by the fourth-power polynomial profile, with the area contraction of 16. The test section is a 1.3 m long channel of a  b = 80  250 mm rectangular cross-section. The flat parallel sidewalls are equipped with transparent windows for visual observations. The divergence angle of diffuser sidewalls adjoining the smaller sidewalls of the test section is 3.5°, while those abutting upon the larger sidewalls are parallel. A special shaft is located above the heat exchanger in order to regulate the static pressure inside the circuit. The reverse channel of the setup includes a swivel elbow of 180° with 750 mm radius and 250  250 mm cross-section, a straight pipeline of 300 mm diameter, a flow-measuring part and a duct guiding working liquid to the pumps. The maximum flow rate through the rig is 1147 m3/h, which corresponds to the maximum free flow velocity of 15.93 m/s. The flow rate is adjusted by varying the pump rotation speed by frequency convertors. 2.2. Experimental conditions and measurement technique We investigated the spatial structure and dynamics of partial cavities and measured the velocity and turbulence intensity in flows around a 2D plate with semi-cylindrical nose and sharp-cut end (100 mm chord and 15 mm thickness), and a NACA0015 series hydrofoil (100 mm chord) at four attack angles a = 0°, 3°, 6° and 9°. The foil inclination was determined by using an optical quadrant with measurement precision of 3000 that allowed to adjust the attack angle manually with an uncertainty of 60 . The test objects were made of brass with roughness level of about 1.5 lm. The working liquid was distilled water. The amount of dissolved air in liquid was estimated by Henry’s law under the normal conditions to be approximately 0.023 g of air in 1 kg of water. Admittedly, the results here presented (especially for the cavitation incipience) may differ from those obtained with deaerated water, but in view of the fact that practically all real hydrodynamic equipment operate with natural water with varying amount of dissolved air and impurities, distilled water can serve as a

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pressure gauge

pressure tank

air filter

precision reducer

121

laser

mirror

reducer

PID-controller pressure sensors

CCD-camera with lens

coolant loop flowmeter

synchronizer

computer

pumps

Fig. 1. Sketch of the cavitation tunnel in Institute of Thermophysics SB RAS: P, T – pressure and temperature transducers, respectively.

well-defined and reproducible reference fluid. The liquid temperature was kept at 30 °C with uncertainty of ±0.1 °C and the overpressure in the setup was fixed to 0.5 bars. The flow regime is defined by the cavitation number, r ¼ ðPin  PV Þ=ðqU 20 =2Þ, where Pin is the static pressure at the test section inlet, PV is the water vapor pressure, U0 = Q/(a[b  h(a)]) is the bulk flow velocity, Q is the volume flow rate measured by a flowmeter, and h is the flow blockage due to the foil which depends on a, ranging from h = 15 mm (equal to the foil thickness) to 100 mm (the foil length) for the incidence angles a = 0° and 90°, respectively. In order to observe various cavity patterns, the cavitation number was adjusted by varying the dynamic pressure, qU 20 =2, in the range from 0.5 to 5.5. Under the present conditions, ~ 0 =U 0 , was lower PV = 0.044 bar. The initial turbulence intensity, u than 1%. The Reynolds number, Re, based on U0 and the chord length, C, was in the range between 0.9  106 and 1.7  106. The Strouhal number for the nondimensional cloud shedding frequency is defined in terms of the cloud streamwise dimension, Dcl, and the cloud convection velocity, Ucl. The uncertainties in defining of cavitation number and estimating the Reynolds and Strouhal numbers were 6%, 3% and 4%, respectively. The dynamics and spatial structure of gas–vapor cavities was analyzed by high-speed visualization using Photron FASTCAM SA5 camera at a frame rate of 20 kHz taken from above and from the side. The cavity length was evaluated from the high-speed image analysis. For quasi-steady cavities, it was treated as the averaged position of the cavity trailing edge close to the flow centerline. For unsteady cavitation, the cavity length was regarded as the cavity maximum length near the flow centerline before the cloud shedding. The cavity oscillation frequency was also evaluated from visual data. The camera time resolution (the time interval between two consecutive frames) was 50 ns. Thus, in the worst case (see Table 1) the uncertainty of the oscillation frequency measurement was about 1%. The velocity fields were measured by a ‘‘PIV-IT’’ PIV-system consisting of a double-pulsed Nd:YAG Quantel EVG00200 laser (wavelength 532 nm, repetition rate 15 Hz, pulse duration 10 ns, pulse energy 200 mJ), a CCD-camera (10 bits per pixel, matrix resolution 2048  2048 pixels) equipped with Nikon AF Nikkor 50 mm f/1.4D lens and an optical low-pass filter (bandpass edge at 570 nm), as well as a synchronizing processor. The PIV-system was operated via a computer using ‘‘ActualFlow’’ software (Akhmetbekov et al., 2006). The current measurements were carried out at the sampling rate of 4 Hz. The thickness of the laser light sheet formed by a cylindrical lens to illuminate tracer particles was about 0.8 mm in the measurement plane that coincided with the central longitudinal plane of the test section parallel to its larger sidewalls. The distance between the camera and the laser sheet was 527 mm. The size of the measurement area was approximately

124  124 mm. Since in a cavitating flow the micro-size vapor bubbles can act as tracers, their contaminating effect on the PIV measurements was reduced by adding fluorescent tracers (average size 10 lm, wavelength range 550–700 nm) into the working liquid. In order to enhance the quality of the registered images, the data were subjected to a two-step pre-processing. First, the mean two-frame intensity field over 5000 initial image pairs was subtracted from each image pair. This allows to compensate the nonuniformity of the image exposure, independently for the first and second frames, as well as to remove glares and shadows in the images. As a result, the intensity of both frames becomes quite similar, improving the accuracy of the velocity vectors calculation in further processing. Secondly, all image pairs were masked to remove from the calculations the areas corresponding to the foil and the shadow. The velocity fields were calculated using the iterative cross-correlation algorithm with a continuous window shift and deformation and 75% overlap of the interrogation windows. In addition, during the correlation step the local particle image concentration was accounted for in the correlation analysis only in those interrogation areas where the particle concentration level is above the given number of particles. The threshold value for seeding particles concentration was set to be 5 tracers per 32  32 pixel area. The tracers were regarded as the convolution of a Gaussian mask with 1 pixel radius and initial image over 5  5 pixels window, with the correlation threshold being 0.7. The sub-pixel interpolation of a cross-correlation peak was performed over three points, using one-dimensional approximation by the Gaussian function. In order to have a relatively large dynamic range (the span between the maximum and minimum velocity), the initial size of the interrogation window was chosen to be 64  64 pixels, but it was subsequently reduced so that the final interrogation window was 8  8 pixels, which provided high enough spatial resolution. Westerweel (1997) and Sciacchitano et al. (2013) argued that for a 32  32 pixels interrogation area the typical measurement error of the crosscorrelation procedure does not exceed 0.1 pix. Accordingly, the estimated relative velocity errors were equaled to 1% and 4% for the particle displacements of 8 and 2 pixels, respectively. The obtained instantaneous velocity vector fields were validated with the three procedures applied successively: the peak validation with the threshold 2.0, the adaptive median filter over 7  7 nodes (Westerweel and Scarano, 2005) and the cluster validation with the coefficient of 50. The Cluster Validation consists of searching for simply connected domains of vectors considered as valid at a current step of processing. Thereafter, all detected domains of vectors are subjected to the following scrutiny. If the number of vectors in a simply connected domain is less than a certain threshold then all

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Table 1 Characteristic cavitation parameters for different regimes at three attack angles for the plate and the NACA0015 hydrofoil.

a (°)

Round leading edge flat plate

NACA0015 hydrofoil

r

f (1/s)

LC (m)

Dcl (m)

U0 (m/s)

r

f (1/s)

LC (m)

Dcl (m)

U0 (m/s)

Ucl (m/s)

3

1.53 1.38 1.26 1.14

106 80 50.1 28.7

0.03 0.05 0.07 0.11

0.041 0.042 0.043 0.042

11.33 11.67 11.99 12.34

9.8 7.8 8.77 8.98

– – 0.93 0.9

– – 119 19

– – 0.066 0.11

– – 0.016 0.045

– – 12.61 12.65

– – 13.3 12

6

2.27 2.06 1.86 1.69 1.54 1.4

239 114 95.3 87.1 98.8 57.3

0.025 0.034 0.04 0.061 0.083 0.109

0.013 0.033 0.047 0.032 0.052 0.048

9.87 10.23 10.62 10.98 11.34 11.71

9.2 8.8 8.4 8.03 8.3 6.9

– 1.46 1.37 1.21 1.08 0.97

– 119 66 18 18 18

– 0.045 0.057 0.083 0.104 0.114

– 0.013 0.027 0.085 0.067 0.071

– 11.37 11.51 12.01 12.4 12.77

– 10.1 9.8 9.4 10.3 10

9

2.78 2.53 2.31 2.1 –

188 128 103 80.8 –

0.018 0.019 0.032 0.046 –

0.028 0.039 0.033 0.041 –

9.2 9.52 9.84 10.2 –

– 8.14 7.78 11.5 –

2.41 2.21 2 1.81 1.6

130 120 98 76 68

0.024 0.033 0.042 0.05 0.061

– 0.033 0.033 0.051 0.052

9.62 9.9 10.25 10.64 11.1

– 9.2 7.6 8.97 7.9

(a)

Ucl (m/s)

[m/s] 10

(b)

7.5 5 2.5 0 Fig. 2. An example of the final vector field of the instantaneous velocity around the plate for cloud cavitation (r = 1.55) at a = 3°: (a) the whole velocity field (every tenth and every fourth vector is shown in the horizontal and vertical direction, respectively); (b) the zoomed area of the velocity field depicted by the dashed rectangle in the left image (every fifth and every second vector is shown in the horizontal and vertical direction, respectively).

those vectors are treated as fallacious. If the number of vectors is higher than the threshold, all the vectors retain their status. This procedure is necessary to remove from further consideration those vectors which cannot be verified at the previous steps of validation. It is required particularly in cases when there is a relatively high probability of appearance of relatively large areas with quite low concentration of tracers in images and, consequently, invalid vectors in the final vector fields. For instance, this is typical for dispersed two-phase flows, particularly for cavitating flows during the cloud shedding process. The areas of invalid vectors can also contain a number of vectors with a valid status, although they can actually be non-physical and thus invalid. The other validation procedures including Adaptive Median Filter often fail to sift such vectors. Therefore, the application of the Cluster Validation procedure for data verification in the configurations as here considered is regarded as justified. Presumably, the Cluster Validation filter can sometimes remove also the valid vectors, but here this issue is not critical since the number of such vectors in the total statistics is very small. Cavitation bubbles do not affect significantly the accuracy of the results because the used optical filter transmits only the light emitted by the fluorescent particles and suppresses the reflected light from the micro-bubbles. However, the bubble clouds surely influence the number of measured instantaneous velocity vectors (sample range) taken into consideration to calculate the mean velocity and all statistical moments. For example, the areas occupied by cavitation clouds in an instantaneous velocity field, as shown in Fig. 2, do not contain velocity vectors and, thus, are totally removed from consideration in the subsequent processing steps. In practice, the sample range decreases substantially within the region of an attached cavity location and becomes almost zero near

the foil surface. The detailed analysis of the statistics effect on the accuracy of turbulence characteristics calculation is given in Appendix A. 3. Results The results of the visualization and field measurements (velocity and turbulence statistics) that follow (Sections 3.1 and 3.4) are given only for a = 3° and 9°, considered as sufficient for both the qualitative and quantitative illustration of the flow and turbulence features. In all plots in Section 3.4, the reference point coincides with the leading edge of a foil in the measurement plane. The integral parameters (Section 3.3) show results for other values of the attack angle considered. 3.1. Visualization Fig. 3 shows typical patterns of partial cavities occurring on the suction side of the plate and NACA0015 foil. In the case of plate, cavitation inception occurs at r  2.72 as a bubble-band – an array of bubble-filled streamwise streaks – which differs from the common sheet attached cavity1 found sometimes behind the leading edge of the foil. Similar streaks were registered by Brandner et al. (2010). The cavity length is LC/C = 0.07. The typical distance between the streaks is about 1 mm, or 1.0% of the chord length, as readily seen in Fig. 5-a. This cavitation pattern persists until approximately r = 2.25 (Fig. 3-a.1). As the cavitation number is decreased, the 1 Sheet cavity represents a gas–vapor film of a relatively stable length with clear interface. In the region of the sheet cavity closure, the main flow reattaches to the solid surface of the cavitating object.

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123

Fig. 3. Instantaneous images of partial cavities (top view) on the suction side at a = 3°: (a) the plate when (a.1) r = 2.25, LC/C = 0.07, (a.2) r = 1.70, LC/C = 0.23, (a.3) r = 1.39, LC/C = 0.5, St = 0.43, (a.4) r = 1.14, LC/C = 0.91, St = 0.13 and (b) the NACA0015 hydrofoil when (b.1) r = 1.14, (b.2) r = 1.06, LC/C = 0.54, (b.3) r = 0.93, LC/C = 0.66, (b.4) r = 0.90, LC/C = 1.06, St = 0.07.

streaks become thicker and tend to interact and merge, forming what now may be qualified as an attached streaky-sheet cavity (Fig. 3-a.2). Here it should be noted that the cavity interface is frothy and, in contrast to results of other papers in the literature (e.g., Franc and Michel, 1985), in the present work the sheet cavity is accompanied by shedding of small (up to 0.15C) clusters of bubbles at the cavity closure region. This is suspected to be due to either a relatively high level of roughness of the hydrofoil surface or a relatively large (compared to deaerated water) amount of dissolved air in the

water. The higher the roughness is, the earlier transition to turbulence of a boundary layer occurs and, consequently, the more gas– vapor phase is trapped by vortices, with other conditions being the same. Owing to the amount of dissolved air, the local dispersedphase fraction within the cavity is considerably higher than it would be in the case with deaerated water. As well known, a decrease of the cavitation number leads to the growth of the streamwise extent of the cavity and even to transition to another cavity pattern when a certain threshold is

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exceeded. When the cavitation number is decreased to r = 1.7 (Fig. 3-a.2), the cavity changes to be of transitional type.2 At this regime, the spatial structure of the cavity is essentially 3D. The cavity length increases up to LC/C = 0.23. Downstream, an increase in local pressure leads to the collapse of micro-bubbles and, consequently, to a drop of the dispersed-phase concentration. A further decrease of cavitation number results in cloud cavitation3 (Fig. 3-a.3). It is proved that separation of cloud cavity is accompanied by liquid circulation, which leads to the formation of a large-scale vortex around the cloud cavity (Kubota et al., 1989). The characteristic Strouhal numbers for the unsteady cavitation regimes are given in the caption of Fig. 3. In addition, downstream of the plate, one can observe thin streamwise and spanwise structures filled up with vapor and gas. Their origin seems to be connected with the generation of strong vortices past the trailing edge of the plate (wake vortices). Inside the cores of those vortices, the local pressure most probably falls below the vapor pressure, leading to water vaporization. When r = 1.14, the cavity extent grows up to LC/C = 0.91 (Fig. 3-a.4). However, the periodic separation of single large clouds is no longer visible while the attached cavity remains unsteady. Simultaneously, the Strouhal number decreases about four times in comparison with the cloud cavitation case presented in Fig. 3-a.3. Cloud cavities shed at the attached cavity closure region become smaller like those shown in Fig. 3-a.2. The clouds are not stable and vanish very quickly when convected downstream (do not reach the position of one chord length behind the plate trailing edge); only quite rarefied agglomerations of micro-bubbles are observed at that position. All these facts imply that the cavitation type changes again. Under these conditions, system instabilities (not backward underflow) determine the cavity behavior (for more detailed analysis browse Section 3.3). Nevertheless, it should be noted that sometimes separation of the whole cavity takes place. This regime at r = 1.14, therefore, can be considered as intermittent between these two types of unsteady cavitation. In the case of NACA0015 hydrofoil, flow pattern differs significantly. Initially, when r = 1.14 separate micro-bubbles arise at a distance of about 0.03C downstream from the leading edge of the hydrofoil (Fig. 3-b.1). These bubbles are convected downstream with the local fluid velocity of 17.1 m/s. The bubble size grows at a rate of about 5.8 m/s. Such dynamics of the bubble growth and movement can be observed until the bubbles collapse further downstream along the collapse line (curved due to the presence of the channel sidewalls) shown in Fig. 3-b.1. As suspected, at this threshold location the magnitude of the local pressure reaches the value that is sufficient to cause the bubble collapse. This cavitation pattern is referred to as the traveling bubble cavitation. At this regime, the cavity is not formed yet. When the cavitation number decreases further (Fig. 3-b.2 and b.3), a transitional cavity type can be observed. As in the previous case, the micro-bubbles initiate from the leading edge of the cavity and extend roughly to the hydrofoil half-length. The bubble size varies from zero up to d/C = 0.13. It should be noted that the cavity leading edge is quite difficult to determine. The trailing part of the cavity is frothy. At the cavity closure region, the bubbles collapse and separate into horseshoe-shaped clouds immediately downstream. These clouds are likely to be situated in the vortex cores and replicate their forms. When r is about 0.9 (Fig. 3-b.4), the transition to unsteady cavitation governed totally by the system 2 This cavity type is intermediate between the sheet and cloud cavities. In general, at the closure region transitional cavity is accompanied by continuous shedding of relatively small clouds and larger horseshoe-shaped ones consisting of a bulk of gas– vapor micro-bubbles at different parts of the cavity closure region. 3 As widely acknowledged, cloud cavities are characterized by detachment of the whole cavity from the foil surface and quasi-periodic cloud shedding process due to the appearance of re-entrant jet. A detailed description of cloud cavitation can be found, e.g. in Callenaere et al. (2001).

instabilities occurs (more details are given in Section 3.3). The cavity length grows rapidly: for a small variation of r in the range from 0.93 down to 0.9, it increases by 60% up to 1.06C. The cloud cavitation regime was not observed. However, as in the case with transitional cavity, one can observe horseshoe-shaped clouds together with the large-scale ones. Fig. 4 shows the length variation of unsteady cavities in time for both foils. The process of attached cavity separation and growth as well as cloud cavity formation and downstream convection is illustrated in Fig. 4-a for the plate (this regime is also shown in Fig. 3-a.3). As seen at the start time (Fig. 4-a.1), the cavity initially detaches over the full span of the plate. However, the detachment line is broken. It is straight over 0.3C along the spanwise direction in the central part (wetted surface) of the plate. However, the lateral cavities remain at both sides of the plate, as one can see in Fig. 4-a.1. At the sides, the detachment line turns out to be inclined to the walls of the test section so that the angle between the normal to the line created by the trailing edges of the lateral cavities and the test section walls is 21°. This seems to be caused by the near-wall velocity gradient like in the case with the collapse curve for the traveling bubbles cavitation (Fig. 3-a.1). After 1.35 ls from the cavity separation (Fig. 4-a.2), the detached cloud slightly shifts downstream; its form becomes more spherical. At the same time, the attached lateral cavities grow. The wetted part of the foil surface practically disappears and the two lateral cavities join to form a full-span attached cavity. The inclination angle of the edges of the newly formed cavity sides augments up to 28°. Thereafter, when 3.3 ls passed from the cavity separation (Fig. 4-a.3), the cloud cavity becomes also already formed (no cavity tears void of air–vapor mixture exist) with a characteristic dimension of 0.41C. The attached cavity extends up to 0.1C along the foil centerline. The inclination angle of the cavity sides takes its maximum of 41°. The attached cavity continues to grow until it reaches the maximum length of 0.5C. Simultaneously, the cloud cavity moves gradually downstream. Shortly after this moment (Fig. 4-a.5), the backward underflow reaches the cavity leading edge and causes it to break. The typical speed of the attached cavity growth was estimated to be approximately 3.66 m/s. The distance between the two successive clouds increases from 0.85C at zero time to 0.95C at 8.55 ls and eventually can be estimated to be roughly 1.26C. Finally, the cloud cavity is accelerated up to 4.2 m/s by the main flow. The frequency of the cloud cavity passage, thus, is assessed to be 71 Hz. Additionally, convection of cavitating cores of the wake vortices take place at frequency of 195 Hz that is about 2.5 times higher than that for cloud cavities. Moreover, as can be observed in Fig. 4-a.4 and a.5, the border effect leads to the separation of the attached cavity sides due to the development of the near-wall re-entrant flow. The cavity sides separation occurs at an inclination of 24° to the channel sidewalls. The underflow moves at the cavity trailing edge from both sides directly to the cavity center at the leading edge. In the case of unsteady cavitation on NACA0015 hydrofoil (Fig. 4-b), cavitation never disappears totally. Initially at 0 ls (Fig. 4-b.1) it occurs in the form of traveling bubbles as for the regime in Fig. 3-a.1. However, the bubbles are about 15% smaller but their number increases (Fig. 4-b.2). They grow downstream and merge forming a cavity as in the case shown in Fig. 3-b.2. The cavity spreads downstream with a speed of 3.7 m/s. The cavity sides are inclined at 37° to the channel sidewalls. About 12.7 ls after the cavity begins its extension, the frothy part starts to develop at the cavity trailing edge. After 34.2 ls (Fig. 4-b.3), the cavity reaches its maximum length. The frothy part occupies now the aft half of the cavity. Distorted vortex rings filled with air-vapor mixture are shed downstream with a frequency of 19 Hz. Subsequently, the cavity starts to decay. A number of relatively small clouds appear (see Fig. 4-b.4 and b.5).

A.Yu. Kravtsova et al. / International Journal of Multiphase Flow 60 (2014) 119–134

flow direction

flow direction

α = 3°

125

z x

leading edge -------------------------

trailing edge -------------------------

(a.1) 0 µs

flow direction

(a.4) 5.3 µs

(a.3) 3.3 µs

(a.2) 1.35 µs flow direction

α = 3°

(a.5) 8.55 µs z

x

leading edge -------------------------

trailing edge -------------------------

(b.1) 0 µs

(b.2) 12.7 µs

(b.3) 34.2 µs

(b.4) 44.4 µs

(b.5) 49.3 µs

Fig. 4. Time-series of growth-reduction cycle of unsteady cavity length pulsations (top view) on the suction side of (a) the plate when r = 1.39, LC/C = 0.5, St = 0.43, T = 12.5 ls and (b) the NACA0015 hydrofoil when r = 0.90, LC/C = 1.06, St = 0.07, T = 52.15 ls. a = 3°.

3.2. Incipience of cavitation on the round-leading-edge plate: the near-wall structures We return now to Fig. 3-a.1 to discuss in more details the incipience of cavitation on the plate, which, as mentioned above, appears in form of a bubble band – a regular array of uniformly spaced bubble streaks (see also Fig. 5-a). The pattern seems to be related to the transition to turbulence immediately downstream from the round leading edge. Namely, it is well known that a single-phase (thus non-cavitating) laminar boundary layer developing around a shaped leading edge of a thick plate or a foil separates shortly after the edge. The transition to turbulence is triggered by and occurs within the separation bubble (separation-induced transition, as illustrated in Fig. 5-c by computational streamlines and turbulent kinetic energy of Hadzˇic´ and Hanjalic´ (1999)). Moreover, the transition is known to begin with spanwise sinusoidal variation of the mean velocity followed by creation of streamwise counter-rotating vortical streaks very close to the wall associated with fluid sweeps and ejections (e.g., Alam and Sandham, 2000),

as sketched in Fig. 5-b. The streaks contain concentrated vorticity that, in turn, creates low pressure at their cores, which can be sufficient to initiate the incipience of cavitation bubbles. In singlephase flows, streaks were found to have a universal spacing when normalized by wall-units, i.e., at Dz+  100 (e.g., Smith and Metzler, 1983) where Dz+ = zUs/m and Us is the local friction velocity. With the inflow velocity of 10 m/s this would imply that the friction coefficient is about 0.00013. Unfortunately, we did not measure the wall shear stress: the streaks are believed to appear within or immediately after the recirculation bubble where the wall shear is indeed very small. The RANS computations of flow around a similar plate (but at a higher free-stream turbulence, FST) by Palikaras et al. (2002) suggest a friction factor of 0.002 in the separation bubble, which would result in Dz+  350. A higher value of Dz+ in a cavitating boundary layer is expected, as the bubbles inside the streaks force them to expand laterally and to merge, thus increasing their size and spacing. However, irrespective of whether the streak dimensions and spacing comply or not with those in non-cavitating

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y

z 1cm

x

x

Leading edge

(b)

z

Δzz Δ

Δz

(a)

(c)

Fig. 5. (a) A blowup of bubble streaks on the plate at zero incidence angle (like those shown in Fig. 3-a.1) with indicated streak spacing of about 0.8–1.2 mm (Re = 1.17  106, r = 2.27); (b) a sketch of streak formation; (c) computed streamlines and turbulent kinetic energy contours in the T3L3 case (FST = 2.5%) of the Coupland and Brierley (1996) experiment (from Hadzˇic´ and Hanjalic´ (1999)).

flows, the bubbly-streak pattern suggests that the cavitation incipience in the case considered is governed by the boundary layer transition, local separation and streaks formation. The above conjecture is supported (albeit indirectly) by the measurements of the streamwise velocity for the cavitating and non-cavitating flows at the similar conditions (Figs. 6 and 7). First, the results are compared with the wind tunnel measurements by Coupland and Brierley (1996), though these experiments were performed with a somewhat thinner plate (10 mm), lower cross-section blocking and at a substantially higher FST. The wall-normal distance y was normalized with the boundary layer thickness d estimated as the position at which the mean velocity reaches roughly the far-from-the-wall value U1, as the evaluation of the momentum (or displacement) thickness is unreliable for the present measurements due to a very small thickness of the boundary layer and the insufficient optical resolution of the PIV system close to the foil surface. Despite the differences in the configurations mentioned above and substantially different free-stream turbulence, the velocity profiles show good agreement for air and water, supporting the credibility in the present PIV measurements. Although none of the measurements resolves the separation bubble, the earlier computations of the Coupland and Brierley flow with a second-moment RANS closure model integrated up to the wall by Hadzˇic´ and Hanjalic´ (1999) reproduced very well the measured velocity profiles and the second-moment turbulence statistics and showed clearly the existence of a tiny separation bubble and the transition to turbulence within the bubble, Fig. 5-c. The satisfactory agreement of the present measurements with those by Coupland and Brierley leads us to expect the same scenario (laminar separation bubble, transition to turbulence within the bubble and formation of downstream streaks), as detected in Fig. 5-a (see also Alam and Sandham, 2000).

2.5

y

δ

2

x C = 0.39

x C = 0.225

x C = 0.091 2.5

In order to gain a better insight into the effect of cavitation on the near-wall mean velocity distribution, Fig. 7 shows a blow-up of the velocity profiles around their peaks for the same crosssections shown in Fig. 6. At x/C = 0.091 the velocity maximum (Umax/U0)loc = 1.42 is highest for the case of streak cavitation due to the augmented effective flow blockage compared with the non-cavitating conditions ((Umax/U0)loc = 1.33), which corroborates the theoretical results on the separation bubble for this cavitation regime. The vertical position y/C of the velocity maximum is roughly 0.02. A decrease of the cavitation number r (an increase of the mean flow velocity U0) results in a suppression of the peak, so that (Umax/U0)loc = 1.28 (see the transitional cavitation regime in Fig. 7 for x/C = 0.091). The maximum eventually decreases down to (Umax/U0)loc = 1.24 for the cloud cavitation regime; its vertical position, however, does not change. At x/C = 0.225 (Fig. 7 for x/C = 0.225), the velocity maximum is lower, (Umax/U0)loc  1.16, for both the non-cavitating and streak cavitation cases. Now the transitional cavitation leads to a growth of the peak, (Umax/U0)loc = 1.25, in contrast to the previous case at x/C = 0.091. These values are smaller than the highest ones in section x/C = 0.091 because of the lateral expansion of the accelerated flow downstream. The peak for the cloud cavitation conditions becomes somewhat higher, (Umax/U0)loc = 1.27. In this section, the peak shifts a little away from the plate surface up to y/C = 0.05 with cavitation development. Further downstream, at x/C = 0.39 (Fig. 7 for x/C = 0.39), the local velocity maxima coincide for all regimes, so that (Umax/U0)loc = 1.11–1.13. However, the velocity maximum is located a little further away from the plate surface: at y/C  0.02 for cavitation-free case and at 0.1 for cloud cavitation. The above findings can be explained in the following way. As known, a decrease in the cavitation number results both in transition to another cavity pattern and, for the current cavity type,

2.5

y

δ

2 1.5

1.5

1

1

1 0.5

0.5 U / U0

0 0

0.4

0.8

1.2

δ

2

1.5

0.5

y

U / U0

0 0

0.4

0.8

1.2

U / U0

0 0

0.4

0.8

1.2

Fig. 6. Streamwise components of the mean velocity over the semi-circular leading edge flat plate at zero incidence angle in three cross-sections. Present measurements: O no cavitation, 4 streak cavitation (the same regime as in Fig. 5-a); + Coupland and Brierley (1996), T3L3 case.

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A.Yu. Kravtsova et al. / International Journal of Multiphase Flow 60 (2014) 119–134 0.5

y C

0.4

0.5

x C = 0.091

y C

0.4

0.5

x C = 0.225

0.4

0.3

0.3

0.3

0.2

0.2

0.2

0.1

U U0

0 1.1

1.2

1.3

x C = 0.39

0.1

0.1

U U0

1.4

U U0 0

0

1

y C

1

1.1

1.2

1.3

1

1.1

1.2

Fig. 7. A blow-up of streamwise mean velocity profiles over the semi-circular leading edge flat plate at zero incidence angle in three cross-sections (the same as in Fig. 6): j cavitation-free flow; streak cavitation; transitional cavitation; cloud cavitation.

extension of the cavity and a growth of cavity clouds that detach from the foil. This behavior is accompanied by pulsations of the cavity at its closure. Consequently, pulsations at the cavity trailing edge lead to an increase in liquid velocity fluctuations (see Fig. 13), more intensive mixing and lateral momentum transfer, which, in turn, reduce the velocity peaks close to the leading edge and raise them downstream. In general, it can be concluded that the cavitation incipience changes significantly the mean velocity distributions close to the foil surface. The developed cavitation suppresses the mean velocity maximum close to the foil leading edge and enhances it downstream as seen in Fig. 7 for x/C = 0.091 and 0.225. The development of cavitation makes the local mean velocity distributions somewhat smoother along the foil surface. Additionally, the velocity peak shifts away from the foil surface with the cavitation evolution. Moreover, the velocity maximum moves closer to the foil leading edge with an increase of the attack angle. 3.3. Integral cavitation parameters In order to summarize the visual observations of the cavitation patterns, the major integral parameters – the cavity characteristic dimension and the frequency of its pulsations – are presented as a function of the cavitation number and incidence angle for the range of 0–9° for both test objects. A summary of the cavity and regime parameters considered is given in Table 1 for both foils. The maximum length of the attached cavities (observed visually from high-speed photographs) normalized by the foil chord length is presented in Fig. 8-a and b as a function of the cavitation number r. The same results are plotted also versus the ratio of the cavitation number and the double angle of attack, r/2a (Fig. 8-c and d). Both sets of figures show of course the same trend, except that the plot versus r/2a, as argued by Brennen (1995), brings the LC/C data for various attack angles closer to each other. For the NACA hydrofoil, all data irrespective of the cavitation regime virtually collapse into a steep (almost vertical) line at r/2a = 0.17, which agrees reasonably well with results by Kjeldsen et al. (2000) and Callenaere et al. (2001). In Fig. 8 the measured values corresponding to different regimes are separated by thin lines. As reported by Callenaere et al. (2001), relatively short partial cavities are almost insensitive to external pressure fluctuations and, therefore, are stable enough. This is confirmed in Fig. 8. On the contrary, long cavities are extremely susceptible to external perturbations and their behavior is quite unstable. Brennen (1995) derived analytically a power low for LC/C in function of (r/2a)2 for a flat plate at supercavitation regimes. Our measurements plotted in the LC/C versus r/2a diagram, Fig. 8, while showing a similar trend, do not collapse into one curve and thus do not follow the above scaling. However, the data in Fig. 8-a and b can well be approximated as LC =C ¼ Ar3 þ B for the flow over the plate and by LC =C ¼ Ar1 þ B for the NACA hydrofoil, where A and B are constants, admittedly different for each angle of attack. The values of these constants are given in Table 2.

To characterize and compare the cloud-cavity size, we evaluated the cloud streamwise dimension Dcl defined as the length of the cloud along the foil suction-surface centerline at the time instant when the cloud center was at the foil trailing edge. In the case of unsteady cavities governed by system instabilities, the cloud cavity size was determined as the longitudinal dimension of the final (just before the attached cavity collapse) conglomerate of micro-bubbles shed downstream at the same time moment. This cluster consisting of a number of smaller clouds can be seen in Figs. 3-b.4 and 4-b. Fig. 9 shows Dcl evaluated from visualizations versus the cavitation number. As seen in Fig. 9-a, for the plate the cloud dimension is almost constant for all r and a considered and equals to 0.3–0.5 of the plate length. However, for the NACA0015 hydrofoil, Dcl appears to be very sensitive to r and varies in the range from 0.1C up to almost the foil length but depends also considerably on the attack angle. Thus, for a = 3° Dcl changes from 0.15 to 0.45C over a relatively narrow range on the cavitation number (0.9–1), see also Fig. 3. According to Callenaere et al. (2001), we conjecture that such a different behavior of cloud cavities is mainly caused by different pressure gradient distributions on the foil suction sides when cloud cavities are formed and is also connected with the attached cavity length and thickness. The frequency of cloud-cavity shedding, usually expressed in terms of the Strouhal number, St, is an important parameter of interest especially in practical applications. However, different characteristic dimensions and velocities have been used by various researchers to define it. Leroux et al. (2005) and Kawakami et al. (2008) used the foil chord length and freestream velocity, while Coutier-Delgosha et al. (2007) employed the mean cavity length and the mean flow velocity. The most common definition is, however, based on the maximum attached cavity length and mean velocity (e.g., Laberteaux and Ceccio, 2001, and Leroux et al., 2005). In the present work, we consider St = fDcl/Ucl, where Dcl is the cloud streamwise dimension introduced earlier and Ucl is the cloud convection velocity estimated from the measurements of the cloud center velocity when passing the foil trailing edge.4 In this region, cloud cavities were observed to have a nearly spherical shape. Figs. 10 and 11 show the dependence of the Strouhal number on r/2a and the Dcl. For the plate, the Strouhal numbers differ significantly for various cavitation numbers and lie in the range of 0.1–0.6 almost regardless of a, with r/2a varied between 0.1 and 0.25. The cloud dimension is, on the other hand, almost constant for various Strouhal numbers at the same incidence angles. This implies that, if the cloud dimension is fixed and the mean flow velocity monotonously increases to reduce the cavitation number at a certain attack angle, the frequency of cloud shedding should decrease (see Table 1). For the NACA hydrofoil, the Strouhal number takes values in the range 0.05–0.2 for small and moderate 4 Admittedly, it has been remarked (e.g., by Kubota et al. (1989)) that cloud cavity progressively accelerates and grows downstream.

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Semi-circular leading edge flat plate 1.5

NACA0015 hydrofoil 1.5

LC C

(a)

1.25

(b)

supercavitation

1.25

unsteady cavities

1

LC C

1

unsteady cavities

0.75

0.75

0.5

0.5

transitional cavitation streak cavitation

0.25

0 1.5

1

2

3

4

traveling bubbles/ streak cavitation

0.25

σ

0

transitional cavitation

σ

0 0

5

1

1.5

LC C

(c)

1.25

LC C

4

5

α = 0° α = 3° α = 6° α = 9°

unsteady cavities

1

0.75

3

(d)

1.25

unsteady cavities

1

2

0.75

0.5 0.25

σ / 2α

0 0.2

0.4

0.6

0.8

traveling bubbles/ streak cavitation

0.25

streak cavitation 0

transitional cavitation

0.5

transitional cavitation

σ / 2α

0 1

0

0.2

0.4

0.6

0.8

1

Fig. 8. Dependence of the maximum cavity length on the cavitation number r for the plate and NACA0015 hydrofoil. The lines fitted through the symbols in plots (a) and (b) are obtained from the correlation LC =C ¼ Arn þ B discussed above and coefficients are given in Table 2.

Table 2 The values for the constants A and B in approximating curves LC =C ¼ Ar3 þ B for the case with plate and by function LC =C ¼ Ar1 þ B for the case with NACA hydrofoil for different attack angles.

0.8

St

0 3 6 9

Semi-circular leading edge flat plate

NACA0015 hydrofoil

A

B

A

B

0.6629 1.4346 2.3663 2.7158

0.2167 0.0779 0.0279 0.0099

2.3998 2.2172 1.96 1.908

2.4737 1.568 0.8366 0.5586

α = 3° α = 6° α = 9°

St

0.6

0.6

a (°)

NACA0015 hydrofoil

Semi-circular leading edge flat plate 0.8

(a)

0.4

(b)

0.4 0.2

0.2 Dcl C

Dcl C

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

Fig. 11. Dependence of Strouhal number on normalized cloud cavity dimension.

Semi-circular leading edge flat plate Dcl C

1 0.8

NACA0015 hydrofoil Dcl C

1 0.8

0.6

α = 3° α = 6° α = 9°

0.6

0.4

0.4

(a)

0.2 0

1

2

(b)

0.2

σ

0

σ

0 3

0

1

2

3

Fig. 9. Cloud cavity streamwise dimension at the trailing edge of the foils against cavitation parameter.

Semi-circular leading edge flat plate 0.8

(a)

0.6

NACA0015 hydrofoil 0.8

St

0.6

0.4

St

(b)

α = 3° α = 6° α = 9°

0.4

0.2

0.2 σ / 2α

0 0

0.1

0.2

0.3

0.4

σ / 2α

0 0

0.1

0.2

0.3

0.4

Fig. 10. Strouhal number versus the ratio of cavitation number to double attack angle.

incidence angles (66°), and is slightly above 0.4 for a = 9° (which is very close to the values found in the classic form of cloud cavitation (Table 1) when r/2a is diminished from 0.15 down to 0.07. However, unlike in the case of the plate, the Strouhal number for the NACA0015 hydrofoil remains rather constant for various Dcl (Fig. 11). Callenaere et al. (2001) demonstrated that, in the case of thick cavities, a re-entrant jet was formed and moved upstream, not interacting with the cavity interface until it reached the cavity leading edge and cut the cavity interface. However, when the cavity thickness was of the order of the re-entrant jet thickness, a strong interaction between the cavity interface and the re-entrant jet occurred throughout its upstream movement. Moreover, the reentrant jet instability was shown to affect only relatively short partial cavities. Therefore, we can conjecture that, in the case of the plate, the cavities are primarily controlled by the typical re-entrant jet instability since, in most of the present conditions, the cavities are relatively short (about or less than half of the plate length) and thick (not shown here) not to impede the re-entrant jet formation and movement. For the NACA foil, the dependence of Dcl on r is substantially more complicated (see Fig. 9) as the cloud shedding seems to be governed by different instabilities and affected by adverse pressure

A.Yu. Kravtsova et al. / International Journal of Multiphase Flow 60 (2014) 119–134

gradient. It is very likely that for relatively high angles of incidence,5 the cavity becomes thick enough for the re-entrant jet to play the key role in cloud shedding. However, at small and moderate attack angles, short cavities on the NACA0015 hydrofoil are subjected generally to system instabilities, associated with an interaction of the cavity itself and, for instance, other components of the hydraulic system (e.g., inlet or outlet pipelines). However, long cavities on the NACA0015 hydrofoil undergo mainly the surge-type instabilities that appear at the limit between partial cavitation and supercavitation due to their small width and relatively large length (Callenaere et al., 2001).

3.4. Mean velocity and turbulence characteristics around inclined foils Based on the ensemble of N0 = 5000 instantaneous velocity fields in the vertical midplane, the distributions of the mean velocity, turbulent kinetic energy, second- and third-order statistical moments of turbulent fluctuations were calculated over the suction side of the two foils considered. Fig. 12 shows some profiles of the streamwise component of the mean velocity excess over the bulk flow velocity, (U  U0)/U0, for the plate at a = 3° and the NACA0015 foil at a = 3° and 9° for three different cavitation conditions. The first figures in each case (Fig. 12-a.1, b.1 and c.1) correspond to cavitation-free flow, the second (Fig. 12-a.2, b.2 and c.2) to streak and traveling bubbles cavitation, and the last ones (Fig. 12-a.3, b.3 and c.3) to transitional cavitation regimes. In the front of the leading edge (not shown here), the flow locally decelerates approaching the stagnation point and the region of slowdown shifts to the pressure side of the foils in all the cases. The mean velocity profiles at x/C = 0 are almost identical for all the regimes and, therefore, also not presented. Due to the flow deflection and acceleration around the leading edge, the mean velocity reaches soon its maximum above the upper surface of the foils. For the plate at the initial (highest) cavitation number r = 3.02, the velocity reached the value Umax/U0  1.63 at the point of x/C = 0.07 and y/C = 0.12, whereas, for the NACA hydrofoil at the same incidence angle at the initial r = 1.42, it is Umax/U0  1.42 at x/C = 0.12; y/C = 0.12. For the NACA0015 foil at a = 9°, Umax/U0  1.72 at x/C = 0.05; y/C = 0.14. However, when r decreases, the velocity maximum firstly increases but then gradually decreases and moves gradually downstream, with the y-position almost unchanged. In the case of plate, the reverse shift of the maximum along x-axis, noticed on the hydrofoil, does not occur. For illustration, the values of the maximum velocities and their locations for three cavitation numbers are listed in Table 3. Downstream from the region of higher mean velocity along the suction side of the hydrofoils, the mean velocity peaks decrease and eventually disappear, except for the plate at the higher cavitation number (Fig. 12-a.1) where at the plate end, x/C = 1.0, a mild velocity peak of 1.2U0 appears again due to the vanishing wall shear at the edge separation. No separation on any of the foils has been detected apart from the trailing edge of the plate. In the case of transitional cavitation on the plate (Fig. 12-a.3), the transverse dimension of the wake (not shown in the figures) is significantly larger owing to its bluff shape and reaches 0.35C at the distance of 1C from the foil trailing edge. In case of the NACA foil, it is 0.07C at a = 3° and 0.17C at a = 9°. This is caused by generation of large-scale vortex structures in the shear layers over the cloud cavity. Here it should be noted that these observations are only for the averaged properties, as the separation points (lines) are not fixed but pulsate around the mean value. 5 The variation of the attack angle leads simultaneously both to redistribution of the pressure gradient on the foil surface and to a change in the cavity thickness (Callenaere et al., 2001).

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In all three cases, the velocity profiles along the foil surfaces are quite similar for the same cavitation types except for the foil trailing part. At x/C = 1 for the plate (Fig. 12-a.1), velocity exceeds U0 everywhere for y/C > 0.02 and falls very quickly when approaching the surface. This is true for each regime on the plate (cf. Fig. 12-a.1, a.2 and a.3). For the NACA foil at the same attack angle, the velocity profile is substantially different. For the non-cavitating case (Fig. 12-b.1), the mean velocity coincides with U0 for y/C > 0.05. However, it becomes smaller than U0 for y/C < 0.05, with the minimum 0.5U0 located approximately at y/C = 0.023. Right behind the foil trailing edge, the mean velocity is equal to 0.75U0. Unlike in the flow over the plate, this region of velocity slowdown is caused by the formation of a turbulent wake past the foil that affects upstream flow in the neighboring zone due to the sharp-pointed trailing edge. With a decrease of cavitation parameter, the region of slowdown gradually ascends to the level of y/C = 0.16 for the transitional type of cavitation (see Fig. 12-b.2 and b.3). However, the streamwise location and the magnitude of the minimum show almost no change. At the higher incidence (Fig. 12-c.1), the velocity dip shifts away from the foil trailing edge to roughly y/C = 0.031 and its magnitude drops down to 0.25U0. Cavitation slightly smoothes this profile so that this minimum becomes slightly greater than 0.4U0 but the region of flow slowdown extends up to y/ C = 0.28. An intriguing issue is the relation between the hydrodynamic flow separation and the cavitation cloud detachment from the foil surfaces. No doubt, the two phenomena are interlinked and mutually dependent, but at this stage, it is difficult to portray a clear scenario of this nonlinear interaction. For incipient and mild cavitation, flow separation can be considered as being fully controlled by the hydrodynamics of the carrier fluid, i.e., the adverse pressure gradient on the trailing part of the foil. However, cavitation is always followed by expansion of the gas–vapor phase and the consequent flow acceleration, which will tend to delay flow separation. On the other hand, detachment of cavity clouds has been observed in non-separating flows, though in most cases it is accompanied with re-entrant jet flow underneath, which is indicative of flow separation. It seems plausible, however, that cloud detachment will trigger and promote flow separation and affect its behavior. In general, one can conclude that the velocity fields for the three different cavitation regimes show visible differences illustrating the strong effect of the cavitation in the carrier fluid. Especially the developed cavitation on both foils (Fig. 12-a.3, b.3 and c.3) shows conspicuous suppression of the velocity peaks over the front part of the foil surfaces for the plate by 15 % and for the NACA foil at a = 3° by 20 % . However, due to intensive mixing in the attached cavity and promotion of early flow separation associated with cloud detachment, this suppression for the NACA foil at a = 9° is not so large and reaches only 11 % . The turbulence statistics show, as expected, also very strong influence of the cavitation regime, the shape of the test foil and the attack angle. Although all second and third moments of the streamwise and lateral velocity fluctuations are available, the analysis of the turbulence field is confined at present to the streamwise turbulence intensity field and the dominant Reynolds shear stress. Fig. 13 shows the profiles of the r.m.s. of the streamline fluctua~ , around the two bodies for the same cavitation numbers tions, u as in Fig. 12. As seen in Fig. 13, the streamwise velocity fluctuations are higher on the plate for x/C < 0.3 (the location of the maximum thickness of the NACA0015 foil) compared with those on the NACA0015 hydrofoil at both attack angles (cf. Fig. 13-a.1, b.1 and c.1). When ~ becomes slightly larger near the surface of the NACA x/C > 0.3, u ~ distributions change drastically again at foil. However, the u ~ increases up to 0.28U0 at y/C = 0.0025 for the plate x/C = 1. The u and 0.2U0 at y/C  0.0125 for the NACA0015 foil at the same a.

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Fig. 12. Downstream evolution of the streamwise component of the mean velocity for (a) the plate at a = 3° when (a.1) r = 3.02 (cavitation-free flow), (a.2) r = 2.48, LC/C = 0.06 (streak cavitation), (a.3) r = 1.7, LC/C = 0.23 (transitional cavitation), (a.4) a typical side-view photo corresponding to a.3; (b) the NACA0015 hydrofoil at a = 3° when (b.1) r = 1.42 (cavitation-free flow), (b.2) r = 1.14 (traveling bubbles), (b.3) r = 0.93, LC/C = 0.66 (transitional cavitation), (b.4) a typical side-view photo corresponding to b.3 and (c) the NACA0015 hydrofoil at a = 9° when (c.1) r = 3.55 (cavitation-free flow), (c.2) r = 2.61, LC/C = 0.17 (streak cavitation), (c.3) r = 2.17, LC/C = 0.3 (transitional cavitation), (c.4) a typical side-view photo corresponding to c.3. The x-axis is directed downstream along the foil suction side. Filled symbols show reliable values, while empty ones correspond to values with higher uncertainty (see details in Appendix A). Solid thin lines depict profiles for the single-phase case shown in a.1, b.1 and c.1. Dashed lines denote the time-averaged interface contours of the attached cavity.

Table 3 Maximum of the mean velocity and its location for the representative cavitation numbers. Regime

Cavitation-free flow Streak/bubbles cavitation Transitional cavitation

Round leading edge flat plate

NACA0015 hydrofoil

a = 3°

a = 3°

a = 9°

r

Umax/U0

x/C

y/C

r

Umax/U0

x/C

y/C

r

Umax/U0

x/C

y/C

3.02 2.48 1.70

1.46 1.47 1.40

0.10 0.11 0.17

0.13 0.12 0.14

1.42 1.14 0.93

1.42 1.43 1.35

0.12 0.26 0.23

0.12 0.13 0.13

3.55 2.61 2.17

1.72 1.77 1.64

0.05 0.12 0.14

0.14 0.16 0.17

~ ¼ 0:27U 0 at y/C = 0.0225. For For the higher a, the maximum is u the two higher cavitation numbers corresponding to the non-cavitating flow and streaky-sheet or traveling bubbles cavitation in all three cases, the turbulence is confined within the attached boundary layer, the outer stream being unaffected. However, for the lower cavitation numbers corresponding to the transitional cavitation in all the cases, the flow separation and cavitation clouds detachment generate intensive velocity fluctuations that spread much beyond the immediate regions around the foils.

Kubota et al. (1989) reported that there is usually a region of high vorticity inside a cloud cavity, and the small-scale incoherent velocity fluctuations are not uniformly distributed over the cavity but are concentrated near its boundary. Overall, a decrease in the cavitation number leads to a rise in liquid velocity fluctuations. ~ profiles This effect is initially mild: the difference between the u for the cavitation-free and streak/bubbles cavitation cases (cf. Fig. 13-a.1, a.2, b.1, b.2, c.1 and c.2) is generally less than 10%. For lower cavitation numbers, this difference becomes more

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Plate with semi-cylindrical nose at

(a.1) y/C 0 0.05

0 0.15

0.25 0

0 0.25

= 3°

0.25 0

u / U0 0.25

NACA0015 hydrofoil at = 9°

NACA0015 hydrofoil at = 3°

(b.1) y/C 0 0.05

0 0.15

0.25 0

0 0.25

0.25 0

u / U0 0.25

(c.1) y/C 0 0.05

0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

0

0 0 0.1 0.25

(a.2) y/C 0 0.05

0 0.15

0.5

0.75

1

x/C

0.25 0

0 0.25

0.25 0

u / U0 0.25

(b.2) y/C 0 0.05

0 0.15

0.5

0.75

1

x/C

0.25 0

0 0.25

0.25 0

u / U0 0.25

0 0.1 0.25

(c.2) y/C 0 0.05

0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

(a.3) y/C 0 0.05

0 0.15

0.5

0.75

1

x/C

0.25 0

0 0.25

0.25 0

u / U0 0.25

0 0.1 0.25

(b.3) y/C 0 0.05

0 0.15

0.5

0.75

1

x/C

0.25 0

0 0.25

0.25 0

u / U0 0.25

(c.3) y/C 0 0.05

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

0

0 0.5

0.75

1

x/C

0 0.15

0 0.1 0.25

0.04

0 0.1 0.25

0 0.25

0.25 0

u / U0 0.25

0.5

0.75

1

x/C

0.25 0

0 0.25

0.25 0

u / U0 0.25

0.5

0.75

1

x/C

0.25 0

0 0.25

0.25 0

u / U0 0.25

0.5

0.75

1

x/C

0

0 0 0.1 0.25

0.25 0

0 0 0.1 0.25

0

0 0.15

0 0.15

0 0 0.1 0.25

0.5

0.75

1

x/C

0 0.1 0.25

Fig. 13. Downstream evolution of the streamline turbulence intensity (root-mean-square) for (a) the plate at a = 3° when (a.1) r = 3.02 (cavitation-free flow), (a.2) r = 2.48, LC/C = 0.06 (streak cavitation), (a.3) r = 1.7, LC/C = 0.23 (transitional cavitation); (b) the NACA0015 hydrofoil at a = 3° when (b.1) r = 1.42 (cavitation-free flow), (b.2) r = 1.14 (traveling bubbles), (b.3) r = 0.93, LC/C = 0.66 (transitional cavitation) and (c) the NACA0015 hydrofoil at a = 9° when (c.1) r = 3.55 (cavitation-free flow), (c.2) r = 2.61, LC/C = 0.17 (streak cavitation), (c.3) r = 2.17, LC/C = 0.3 (transitional cavitation). The x-axis is directed downstream along the foil suction side. Filled symbols show reliable values, while empty ones correspond to values with higher uncertainty (see details in Appendix A). Solid thin lines depict profiles for the single-phase case shown in a.1, b.1 and c.1. Dashed lines represent time-averaged interface of the attached cavity.

Table 4 The maximum r.m.s. of the velocity fluctuations and their locations for the representative cavitation numbers. Regime

Cavitation-free flow Streak/bubbles cavitation Transitional cavitation

Round leading edge flat plate

NACA0015 hydrofoil

a = 3°

a = 3°

a = 9°

r

~ max =U 0 u

x/C

y/C

r

~ max =U 0 u

x/C

y/C

r

~ max =U 0 u

x/C

y/C

3.02 2.48 1.70

0.08 0.22 0.32

0.06 0.16 0.25

0.12 0.10 0.12

1.42 1.14 0.93

0.17 0.19 0.23

0.48 0.48 0.35

0.09 0.09 0.11

3.55 2.61 2.17

0.19 0.29 0.36

0.02 0.09 0.3

0.13 0.14 0.14

pronounced (see Fig. 13-a.3, b.3 and c.3) reaching 20%, 15% and 25% compared to the cavitation-free case for the plate, the NACA foil at a = 3° and the NACA foil at a = 9°, respectively. In the transitional cavitation, the turbulence is spread all around the plate (r = 1.7) and the NACA foil at a = 9° (r = 2.17) with the maximum intensity ~ =U 0 ¼ 0:32 at the point (0.25; 0.12) for the plate and of u ~ =U 0 ¼ 0:36 at the point (0.3; 0.14) for the hydrofoil. u Qualitatively a similar pattern is observed also for the NACA hydrofoil at a = 3°, though generally the peaks of turbulence intensity appearing in the thin boundary layer on the foil surface are milder than those in the case of plate and at the higher attack angle. The highest turbulence intensity peaks for all cavitation numbers appear near the trailing edge, which in the case of transitional cavitation (Fig. 13-b.3) far exceed the peaks in other two regimes. Again, the difference between the non-cavitating flow and traveling bubble cavitation (Fig. 13-b.1 and b.2) is minor, the latter showing somewhat higher turbulence level especially along the trailing part,

presumably enhanced by cavitation. In contrast, just as in the case of plate, the transitional cavitation generates much higher turbulence which spreads fast laterally almost up to y/C = 0.4. Thus, comparing the fields in Fig. 13, one can conclude that transition to the developed cavitation regimes leads to a significant modification of the turbulence level and its distribution. For the ~ =U 0 for the cavitation-free case appears plate, the maximum of u at x/C = 0.06 and y/C = 0.12. It moves downstream and increases from 0.08 up to 0.32. However, for the NACA0015 hydrofoil at a = 3°, the maximum of u~=U 0 shows the same growth dynamics with a decrease in the cavitation number, but occurs slightly downstream and shifts upstream compared to the plate. Besides, the increase of the maximum is substantially less compared to the plate: from 0.17 up to 0.23 (see Table 4). The dynamics of ~ maximum for the NACA foil at 9 is very similar to that for the u ~ =U 0 increases from 0.19 up to 0.36 and shifts from the plate: u 0.02 downstream to 0.3.

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Plate with semi-cylindrical nose at (a.1) y/C 0.05

0

0.006 0

0 0.01

= 3°

0.01 0

NACA0015 hydrofoil at (b.1)

0.01

0 y / C 0 0.006 0.05

0.01 0

0 0.01

= 3° 0.01 0

NACA0015 hydrofoil at (c.1) 0.01

0 y / C 0 0.006 0.05

0 0.01

0.01 0

0.01

0.5

0.75

1

x/C

0.01 0

0 0.01

0.01 0

0.01

0 0.1 0.25

0.5

0.75

1

x/C

0 y / C 0 0.006 0.05

0.01 0

0 0.01

0.01 0

0.01

0.5

0.75

1

x/C

0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

0

0 -0.25

(a.2) y/C 0.05

0 0.1 0.25

0

0.006 0

0.5

0.75

0 0.01

0.01 0

0 0 0.1 0.25

x/C

(b.2) 0.01

0 y / C 0 0.006 0.05

0.5

0.75

1

x/C

0.01 0

0 0.01

0.01 0

0.01

0 0.1 0.25

(c.2)

0 y / C 0 0.006 0.05

0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

0

0

-0.25

(a.3) y/C 0.05

0 0.1 0.25

0

0.006 0

0.5

0.75

0 0.01

0.01 0

0 0 0.1 0.25

0.5

0.75

1

x/C

0 y / C 0 0.006 0.05

0.01 0

0 0.01

0.01 0

0.01

x/C

(b.3) 0.01

(c.3)

0.04

0.04

0.04

0.03

0.03

0.03

0.02

0.02

0.02

0.01

0.01

0.01

0

0

-0.25

0 0.1 0.25

0.5

0.75

x/C

= 9°

0.01 0

0 0 0.1 0.25

0.5

0.75

1

x/C

0 0.1 0.25

Fig. 14. Downstream evolution of the Reynolds shear stress huti for (a) the plate at a = 3° when (a.1) r = 3.02 (cavitation-free flow), (a.2) r = 2.48, LC/C = 0.06 (streak cavitation), (a.3) r = 1.7, LC/C = 0.23 (transitional cavitation); (b) the NACA0015 hydrofoil at a = 3° when (b.1) r = 1.42 (cavitation-free flow), (b.2) r = 1.14 (traveling bubbles), (b.3) r = 0.93, LC/C = 0.66 (transitional cavitation) and (c) the NACA0015 hydrofoil at a = 9° when (c.1) r = 3.55 (cavitation-free flow), (c.2) r = 2.61, LC/C = 0.17 (streak cavitation), (c.3) r = 2.17, LC/C = 0.3 (transitional cavitation). The x-axis is directed downstream along the foil suction side. Filled symbols show reliable values, while empty ones correspond to values with higher uncertainty (see details in Appendix A). Solid thin lines depict profiles for the single-phase case shown in a.1, b.1 and c.1. Dashed lines represent time-averaged interface of the attached cavity.

The distribution of the primary Reynolds shear stress huti for the non-cavitation case shown in Fig. 14-a.1, b.1 and c.1 follow the usual pattern for a plate and foils. As seen in Fig. 14-a.1, huti starts to develop right behind the leading edge of the plate and eventually reaches the minimum value of 0:014U 20 at the plate trailing edge. For the NACA0015 hydrofoil, huti begins to grow past the maximum foil thickness at 0.3C. At the foil trailing edge, the stresses reach their minimum and maximum values: 0:003U 20 and 0:003U 20 at a = 3°; 0:005U 20 and 0:003U 20 at a = 9°. Cavitation onset leads to a growth of the Reynolds shear stress right behind the attached cavity, whereas they turn out to be unchanged within the cavity (Fig. 14-a.2, b.2 and c.2). Their values increase significantly so that humi ¼ 0:004U 20 at x/C = 0.25 (right behind the cavity) on the plate, humi ¼ 0:004U 20 at x/C = 0.5 on the NACA foil at a = 3° and humi ¼ 0:011U 20 at x/C = 0.25 on the NACA foil at a = 9°. The boundary layer thickness raises in all cases. However, at the trailing edge, huti increases twice for the plate but remains practically uninfluenced for both cases of the NACA0015 hydrofoil. Once cavitation is developed (Fig. 14-a.3, b.3 and c.3), huti-distributions alter significantly: huti become positive near the leading edge of the cavity in all the cases (see section x/C = 0.1) and its amplitude increases substantially in the central part of the cavity and right behind it.

4. Conclusions Visual analysis of high-speed images and PIV measurements of the velocity field and its fluctuations in flows around a flat plate

with semi-cylindrical nose and a NACA0015 hydrofoil, both at 3° and 9° incidence angles, have been performed for a series of cavitation numbers, corresponding to different cavitation regimes. Despite a relative similarity of the size and shape of the two foils considered, the cavitation patterns are different. On the plate, the cavitation begins with a bubble band (bubble streak array), arguably governed by flow instability, transition to turbulence and formation of near-wall streamwise vortical streaks with lowpressure core. With a decrease in the cavitation number, streaks merge and transform into a cloud which detaches downstream. In contrast, the process around the NACA hydrofoil begins with traveling bubbles, which, when the cavitation number is decreased, coalesce creating clusters of larger bubbles. The bubble formation takes subsequently a remarkably regular lattice pattern before collapsing into a cloud. The integral cavitation parameters follow the general trends reported in the literature (e.g., Brennen, 1995; Callenaere et al., 2001), though showing notable quantitative differences in the two flow cases considered. The maximum cavity length LC decreases with the cavitation number, but more rapidly for the plate (/ r3) than for the NACA foil (/ r1). In both cases the curves come closer to each other for different attack angles when plotted against r/2a. The cloud streamwise length is almost independent of r for the plate. In contrast, for the NACA foil it shows a very high sensitivity to r even within the narrow range of values considered, but also depends notably on the attack angle. It was demonstrated that for the plate at each attack angle considered and the NACA0015 foil at relatively high angles of incidence, the cavities are mostly controlled by the re-entrant jet instability. In these

A.Yu. Kravtsova et al. / International Journal of Multiphase Flow 60 (2014) 119–134

cases the cavities are relatively short (less than half of the plate length) and thick. However, the cavities on the NACA0015 hydrofoil at small and moderate angles of attack (no more than 6°) are subjected generally to system instabilities for short cavities and surge-type instabilities for long cavities due to their small width and relatively large length. The velocity and turbulence fields for the three different cavitation regimes show visible differences illustrating the strong effect of the cavitation on the carrier fluid. Especially the developed cavitation on both foils show a notable suppression of the velocity peaks over the front part of the foil surfaces due to intensive mixing in and by clouds and the promotion of early flow separation associated with cloud detachment. The hydrodynamic flow separation due to adverse pressure gradient and the cavitation-cloud detachment with re-entrant jet underneath are closely related: the separation enhances the cavitation and creation of the cloud, which in turn feeds back into the carrier fluid flow. The progressive growth of gas–vapor cavity promotes early separation by shifting the separation point upstream, as well as more intensive recirculation. The cloud cavities lead to formation of an intense turbulent wake behind the cavity closure region, significantly extended in the transverse direction, as well as to a significant intensification of turbulent fluctuations due to strong shear at the cloud edges and generation of large-scale vortical structures over the vapor cavity. Thus, the transition to unsteady cavitation leads to global modification of the flow and turbulence and significant changes in the flow pattern as compared to the cavitation-inception case. It is recognized that for a more complete analysis of the cavitation phenomena, simultaneous measurements of the velocity of both phases as well as the concentration of the gas–vapor phase would be much desired. Such studies could be performed by using some novel advanced imaging techniques (such as Planar Fluorescence for Bubbles Imaging technique described in detail in Akhmetbekov et al. (2010) and Dulin et al. (2011)). Acknowledgements The work was partially supported by RFBR (Grants 13-0801411-a and 14-08-31590-mol_a) and the Government of the

133

Russian Federation (Grant 11.G34.31.0046, Lead scientist K. Hanjalic´, Novosibirsk State University). The authors are also grateful to Dr. Mikhail Tokarev for programming of the iterative cross-correlation algorithm and validation procedures, as well as for his valuable contribution to clarifying the applicability of the PIV technique in cavitating flows. Appendix A The accuracy of the measured turbulence properties depends significantly on the number of samples (‘‘sample range’’) used for gathering the turbulence statistics. Vectors in the near-wall region are often removed during the validation steps of processing because of insufficient tracer concentration, low signal-tonoise ratio and other reasons. This inevitable elimination of the contaminated and invalid data requires to estimate the effective sample range for each interrogation window and to apply the adequate correction. Fig. A1 shows profiles of the sample range N normalized with N0 = 5000 (the number of measured instantaneous velocity fields) for non-cavitating flow (a) and for a developed cavitation (b) near the surface for both foils at the same sections as in Figs. 12–14. As seen, the sample range reduces sharply when the foil surface is approached and becomes almost zero at the surface. For non-cavitating flow, Fig. A1-a.1 and b.1, the upper border of this region is located at 2 mm from the surface. For the bubble and streak cavitation (not shown here), the N/N0 profiles coincide everywhere with those for the non-cavitating flow except in the near-wall region, but even here the difference does not exceed 10%. For developed cavitation (Fig. A1-a.2 and b.2), the N/N0 profiles deviate significantly over the foil surface compared to the non-cavitating flow. This occurs due to the passage of cloud cavities through those areas in the initial PIV-images. Bubble clusters cover the tracer particles and attenuate their intensity significantly. Therefore, the number of the tracers becomes occasionally insufficient to calculate velocity vectors correctly for a certain interrogation window. Moreover, the attached cavity overlaps most number of the tracers and, consequently, the sample range is reduced substantially within the attached cavity.

Fig. A1. Sample range used to calculate the turbulence characteristics at different cross-sections. (a) plate at a = 3° when (a.1) r = 3.02 (cavitation-free flow), (a.2) r = 1.7, LC/C = 0.23 (transitional cavitation); (b) NACA0015 hydrofoil at a = 3° when (b.1) r = 1.42 (cavitation-free flow), (b.2) r = 0.93, LC/C = 0.66 (transitional cavitation). Solid lines in a.2 and b.2: non-cavitating flow shown in a.1 and b.1. Dashed lines: time-averaged interface of the attached cavity.

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Table A1 ~ and huti over the basic uncertainty e0 for different sample The r.m.s.-values e for U, u ranges. N

eU/e0

eu~ =e0

ehuti/e0

10 100 250 500 1000 2500 5000

2.79 1.32 1.08 1.13 1.05 1.02 1

7.92 2.35 1.55 1.33 1.18 0.97 1

13.76 4.57 2.76 2.07 1.47 1.03 1

3 2.5 2 1.5 1 0.5 0.5

1

1.5

2

2.5

3

3.5

4

Fig. A2. Dependence of the measurement uncertainty e on the sample range N used ~ and huti. for calculation of U, u

In order to estimate how the sample range influences the measurement uncertainty, we calculated the r.m.s.-values of each quantity for different sample ranges over a cross-section upstream of the foil, in this case at x/C = 1 from y/C = 0.8 up to y/C = 0.8, where the flow can be considered undisturbed. In this crosssection, the values of each quantity are assumed to be uniformly dispersed, with the mean values being constant. The mean and r.m.s.-values were computed for several sample ranges, N, specified in Table A1. The r.m.s. can be regarded as the measurement uncertainty, e. Its distribution is shown in Fig. A2 in terms of e/e0, where e0 is the basic uncertainty of the PIV-measurement computed for N0 = 5000. As seen in the graph, the uncertainty grows when N is decreased, but for large N it converges for each quantity to a certain value (e.g., cf. the points for N = 2500 and 5000 in Fig. A2). However, the growth rate of the uncertainty differs for various quantities. The steepest inclination is for huti-correlation, whereas the most flat one is, obviously, for the mean velocity. We used the following arbitrary criterion to treat the uncertainty acceptable or not. If the ratio of e/e0 is less than 1.5, the uncertainty was considered quite reasonable. This means that N ~ and huti, respectively. must be more than 80, 300 and 1000 for U, u Therefore, the values of every quantity which were calculated on the basis of the appropriate sample range are regarded as reliable and shown by filled symbols in all profile plots in Figs. 12–14. The values for which e/e0 lies between 1.5 and 2 (these both thresholds are depicted in Fig. A2 by dashed horizontal lines) are treated as having still acceptable uncertainty and displayed by empty symbols. This turns into the following range for N: ~ and 600 < N < 1000 for huti. 30 < N < 80 for U, 160 < N < 300 for u The ones with e/e0 > 2 are removed from consideration because of too high uncertainty and not portrayed in the above graphs.

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