Y. Ito, K. Seto, T. Nagasaki, “Periodical shedding of cloud cavitation from a single

hydrofoil in high-speed cryogenic channel flow”, Journal of Thermal Science, 18,

pp.58-64 (Jan 2009)(Springer)

Yu ITO: Assistant Professor

Periodical shedding of cloud cavitation from a single hydrofoil in high-speed cryogenic channel flow

Yu ITO, Koichi SETO, Takao NAGASAKI

Tokyo Institute of Technology, G3-33, 4259 Nagatsuta-cho, Midori-ku, Yokohama, 226-8502, Japan

In order to explain criteria for periodical shedding of the cloud cavitation, flow patterns of cavitation around a plano-convex hydrofoil were observed using a cryogenic cavitation tunnel of a blowdown type. Two hydrofoils of similarity of 20 and 60 mm in chord length with two test sections of 20 and 60 mm in width were prepared. Working fluids were water at ambient temperature, hot water and liquid nitrogen. The parameter range was varied between 0.3 and 1.4 for cavitation number, 9 and 17 m/sec for inlet flow velocity, and −8° and 8° for the flow in-cidence angle, respectively. At incidence angle 8°, that is, the convex surface being suction surface, periodical shedding of the whole cloud cavitation was observed on the convex surface under the specific condition with cavitation number and inlet flow velocity, respectively, 0.5, 9 m/sec for liquid nitrogen at 192°C and 1.4, 11 m/sec for water at 88°C, whereas under the supercavitation condition, it was not observable. Periodical shedding of cloud cavitation occurs only in the case that there are both the adverse pressure gradient and the slow flow region on the hydrofoil.

Keywords: cloud cavitation, cryogen, visualization, hydrofoil

Introduction

Cavitation is one of the difficult problems to overcome in the development of liquid fuel rocket engines, because high performance engines require higher combustion pressure so that their turbo-pumps have to operate at the higher rotating speed. Especially, cryogenic cavitation is complex, because cryogens such as LH2, LOX and LN2 have different thermodynamic features from ordinary fluids like H2O. This is called “thermodynamic effect.” Therefore it is not a good way to rely upon existing abundant empirical database of the ordinary fluid pumps in the design for cryogenic ones. Consequently, visible experiments that take the thermodynamic effect into ac-count are required in order to understand the fundamental features and the structure of the cryogenic cavitation in more detail.

Regarding visualization of cavitation around a hydro-foil by using water as working fluid, an experimental observation on flows past a plano-convex hydrofoil was carried out by Wade et al. [1] in 1966. Their photos of cavitation on the hydrofoil are very useful to recognize the cavity oscillation, however, their major aims were to measure lift and drag coefficients of the hydrofoil, there-fore pressure on the hydrofoil surface was not measured. Le et al. [2] noted the difference of cavitation profiles with various angle of attacks (AOA) and cavitation numbers (σ) in visible experiments using a plano-convex hydrofoil. Kjeldsen et al. [3] classified types and length of cavitation by a combination of AOA and σ, and showed the range of periodical shedding of the cloud cavitation by using Strouhal number. Franc [4] showed a relationship among cavitation compliance K, rate of cavity volume change and rate of local pressure change around a hydrofoil. The

Nomenclature AOI angle of the flow incidence [degree] Subscripts b blockage ratio [%] in inlet of the test section C chord length [mm] 1 60 mm upstream of the hydrofoil center D hydrofoil thickness [mm] 2 3 or 9 mm upstream of the hydrofoil center on

the hydrofoil suface d distance between the hydrofoil surface and the channel wall [mm]

e cavity thickness [mm] 3 3 or 9 mm downstream of the hydrofoil center

on the hydrofoil suface

f frequency [Hz] p pressure [Pa]

L left side at 60 mm downstream of the hydrofoil center

T temperature [K] u flow velocity [m/sec]

R right side at 60 mm downstream of the hydro-foil center

W channel width [mm] back outlet of the test section Greek letters

σ cavitation number

frequency of periodical oscillation can be estimated by the cavitation compliance K and the channel geometry. The relationship he pointed out seems to be useful in considering the channel wall confinement effect on peri-odical shedding of the cloud cavitation. Callenaere [5] used a diverging step in a water tunnel to clarify a re-entrant jet. He revealed an adverse pressure gradient plays an important role to trigger the re-entrant jet, and the interaction between the re-entrant jet and the cavity causes periodical shedding of the cloud cavitation.

Meanwhile, some cryogenic experiments have been reported for various flow configurations. Hord [6] carried out experimental studies with flow visualization on a cavitating flow around a 2 dimensional symmetric thick hydrofoil by using LN2 and LH2 in 1973. Thereafter, in order to investigate a faster chocked flow with cavitation in LN2, LOX and LCH4, an experimental study using three different shape nozzles was carried out by Si-moneau and Hendricks [6] in 1979, who made no effort to visualize the flow patterns. Several U.S. groups [8-11] re-ported on a subsonic flow of LHe, however measure-ments were only for pressure or mass flow rate, not elaborating flow visualization. There have been very few studies to visualize cryogenic cavitation, because of dif-ficulty in the experiments. Hori et. al. [12] carried out visualized experiments on the cavitating flow of LN2 by employing the same nozzle profile as Simoneau and Hendricks. Ishii and Murakami [13] studied the flow of He I and He II in a small nozzle incorporating flow visuali-zation successfully.

As mentioned above, for the cavitation around hydro-foils there have been several experiments in the case of water flow, but very few in cryogenic flow. One of the objectives in the present study has been therefore to in-vestigate the cavitation around a hydrofoil in a compara-

tive way of flow visualization between the cryogen and the ordinary fluids to elucidate the distinction due to the thermodynamic effect. The authors [14] reported flow pat-terns of cavitation around a hydrofoil by using a cryo-genic cavitation tunnel as shown in Fig. 1(a). Periodical shedding of the whole cloud cavitation on the plano- convex hydrofoil was observed in the case of positive angle of the flow incidence. In this study more compre-hensive experiments were performed for various channel widths and hydrofoil sizes, as well as flow velocities and incidence angles in order to make clear the fundamental nature of cavitation around a hydrofoil.

Experimental Arrangements

The cryogenic blowdown tunnel as shown in Fig. 1 was employed. Figure 2 shows test sections and the hy-drofoil geometry. To prevent the failure due to thermal stress, it was made of stainless steel except the view window made of polycarbonate. The test section was 288 mm long, 20 or 60mm wide and 20mm deep rectangular channel, wherein the hydrofoil was set at the center. The hydrofoils had two types. First one was 20 mm in both chord and span with its shape of the same plano-convex as in the literature [14], i.e. one surface being plane and the other convex with a radius of 26 mm. Second one was 20 mm in span length but 60 mm in chord length with similarity of the first one, i.e. the radius of the con-vex surface being 78 mm. Pressure-taps, P1, P2, P3, Pback, and RTDs, T1, TL, TR, were also installed. Pback was lo-cated at the lower tank.

Experimental scheme

In order to explain criteria for periodical shedding of

60

P1 & T1

18

P2

P3 60

TL TR

60

P1 & T1

6

P2

P3

60

TL TR

(a) Channel width W = 20 mm (b) Channel width W = 60 mm

Fig. 1 Cryogenic cavitation tunnel of the blowdown type

(a) plano-convex hydrofoil (C = 20 mm, W = 20 or 60 mm)

(b) plano-convex hydrofoil (C = 60 mm, W = 60 mm)

Fig. 2 Hydrofoil profiles and arrangement

the cloud cavitation, effects of wall confinement, hydro-foil size and flow velocity were investigated. Firstly, to clarify the effect of distance d between the hydrofoil sur-face and the channel wall, a comparison of the results between channel widths W = 20 and 60 mm was per-formed for the same chord length hydrofoil with C = 20 mm. Secondly, to elucidate the effect of hydrofoil size C,a comparison of the results between hydrofoil sizes C =20 and 60 mm in the case of the same blockage ratio b =18.1 % was carried out. Thirdly, to reveal the effect ofvelocity, a comparison of the results between inlet veloc-

ity uin = 9.4 and 17.7 m/sec was conducted. All the com-parisons were performed with cold and hot waters and liquid nitrogen as the working fluids. Finally, criteria and the mechanism of periodical shedding of cloud cavitation were discussed.

Results and Discussion

Effect of angle of the flow incidence AOI Figure 3 shows the cavitation patterns at various AOIs

in the case of W = 60 mm, C = 60 mm and D = 6 mm.

2

φ 1

1

20

5 10

1φ 1

1

3

φ 1

3 20

6

φ 1.5

3

20

10 10

3

φ 1.5

3

9

φ 1.5

9

60

AOI = +8° +4° 0° −4° −8°

AOI = +8° +4° 0° −4° −8°

The values of b and d varied depending on AOI. At AOI = 8˚, a cavity was formed from the leading edge on the suction convex surface, and its length was much larger than the chord length. At AOI = 4˚, a cavity was formed upstream of the hydrofoil center on the suction convex surface, and its size was smaller than that at 8˚. At AOI = 0˚, a cavity was formed around the hydrofoil center on the convex surface and another small cavity was formed from the leading edge on the plane surface. The sizes of both cavities were the smallest among all angles. At AOI = −4˚, a cavity was formed from the leading edge on the suction plane surface and another small cavity was formed downstream of the hydrofoil center on the pres-sure convex surface. They joined below the trailing edge of the hydrofoil. At AOI = −8˚, two cavities were also formed on both surfaces. Their sizes were the largest in all angles.

Table 1 shows the cavitation occurrence in each working fluid and AOI. C, W and uin were set at 20 mm, 20 mm and 10 m/sec, respectively. Here, σs were recal-culated by using instantaneous velocity, pressure and temperature when the snapshots were taken instead of the average velocity, pressure and temperature during each experiment as shown in literature 14. In the case of cold water, σ = 2.5, small cavitation near the leading edge of the hydrofoil was observed to exist at AOI = −8˚ and 8˚, whereas there was no cavitation at the other AOIs. On the other hand, in the cases of hot water, σ = 1.38, and liquid nitrogen, σ = 0.58, cavitation occurred except at AOI = 0˚. Especially at positive AOIs, periodical shedding of cloudcavitation was observed. Figure 4 shows the temporal change of periodical shedding of the cloud cavitation at AOI = 8˚. According to the results, periodical shedding of cloud cavitation occurred at the positive angle of the flow incidence. The angle of the flow incidence was set at 8° hereafter without annotation.

Table 1 Cavitation pattern (W=20mm, C=20mm, 10inu ≈ m/sec)

AOI

+8° +4° 0° −4° −8°

Cold water 15°C 2.5 X X X X X

Hot water 90°C 1.38 P P X S S Liquid nitrogen

−193°C 0.58 P P X S S

P: Periodical shedding of cloud cavitation S: Steady cavitation (partial cavity or supercavitation) X: No cavitation

Effect of distance d By comparing cavitation patterns among these two test sections and two hydrofoil types, criteria of periodical shedding of cloud cavitation was investigated taking the wall confinement effect into account. In the condition of Fig. 4, distance d was 8.8 mm. Figure 5 shows cavitation patterns in the case of W = 60 mm, C = 20 mm, D = 2 mm, AOI = 8˚ and d = 28.8 mm. A cavity on the whole hydrofoil surface was formed and periodical shedding at just the trailing part of the cavity occurred. The shedding size was smaller than that in the case of W = 20 mm. Liquid flow swerved aside from the hydrofoil and the cavity on its surface, and accelerated because the effec- tive width of the channel became narrow. Therefore, local pressure P3 became lower than the back pressure Pback, then an adverse pressure gradient, i.e. (P3 -Pback)/7 mm, was formed on the downstream portion of the hydrofoil surface. The adverse pressure gradients in Fig. 4 were −6.4 and −3.0 MPa/m for hot water and liquid nitrogen,respectively, however, those in Fig. 5 were −3.4 and −1.7MPa/m.

Generally there is a very slow flow region due to separation on the suction surface. When there are both adverse pressure gradient and slow flow region, a reverse

AOI = −8˚ (σ = 0.56) −4˚ (0.52) 0˚ (0.37) 4˚ (0.39) 8˚ (0.53)

Fig. 3 Supercavitation at various angles of flow incidence (Nitrogen, W = 60 mm, C = 60 mm, D = 6 mm)

t = 0.00 msec 2.25 msec 4.50 msec 6.75 msec 9.00 msec

(a) Hot water at σ = 1.38, uin = 11.4 m/sec, T1 = 88.4 ˚C = 361.6 K, P1 = 152 kPa, P3 = 80 kPa, Pback = 125 kPa, f = 111 Hz, Re = 6.87x105

t = 0.00 msec 2.79 msec 5.57 msec 8.36 msec 11.1 msec

(b) Nitrogen at σ = 0.58, uin = 9.4 m/sec, T1 = -192 ˚C = 81 K, P1 = 177 kPa, P3 = 124 kPa, Pback = 145 kPa, f = 89.7 Hz, Re = 1.05 x106

Fig. 4 Periodical shedding of the whole cloud cavitation from the hydrofoil (W = 20 mm, C = 20 mm, D = 2 mm, AOI = 8˚, b = 18.1 %, d = 8.8 mm)

flow occurs on the suction surface. The reverse flow tears the cavity from the hydrofoil surface and the main liquid flow washes the cavity out downward, that is, shedding of the cloud cavitation occurs. Thereafter there is no cav-ity on the hydrofoil surface transiently. Next cavity is generated near the leading edge again, and this process is repeated periodically.

In the case of strong wall confinement effect because of d = 8.8 mm, shedding of the cloud cavitation occurred on a large scale as shown in Fig. 4, however, in the case of weak wall confinement effect because of d = 28.8 mm, shedding of the cloud cavitation occurred partially as shown in Fig. 5.

Effect of hydrofoil size C Figure 6 shows the cavitation pattern in the case of

triple size hydrofoil, that is, W = 60 mm, C = 60 mm, D = 6 mm, AOI = 8˚ b = 18.1 % and d = 26.3 mm. A cavity was also formed on the whole hydrofoil suction surface and periodical shedding of the cloud cavitation occurred only on the trailing part of the cavity.

Firstly, the results are compared with those in Fig. 4 because of geometrical similarity with the same blockage

ratio b, which was the ratio of the streamwise projection area of the hydrofoil to the cross-sectional area. Adverse pressure gradients, i.e. (P3 − Pback)/21 mm, were −1.6, −0.6 MPa/m for hot water and liquid nitrogen, respec-tively. They were much smaller than those in Fig. 4. Dueto the smaller adverse pressure gradient, the adverse flowregion was limited to just the trailing part of the cavity. Inaddition, the cavities sizes were thicker and longer thanthose in Fig. 4 due to large C. Consequently, the adverseflow could not affect the whole cavity except the rear halfbecause the adverse flow region was smaller than thecavity size.

Secondly the results shown in Fig. 6 are compared with those in Fig. 5 because of the same channel width W = 60 mm. As above mentioned, the cavity size for C = 60 mm was larger than that of C = 20 mm. However, be-cause the distances ds were sufficiently larger than the hydrofoil thicknesses D and the cavity thicknesses e, the channel wall confinement effects were insignificant, and the cavity was stable.

Effect of velocity uin Figure 7 shows the cavitation pattern in the case of the

t = 0.00 msec 1.00 msec 3.00 msec 4.00 msec 5.00 msec 6.00 msec

(a) Hot water at σ = 0.60, uin = 13.2 m/sec, T1 = 92.2 ˚C = 365.4 K, P1 = 126 kPa, P3 = 82 kPa, Pback = 106 kPa, Re = 8.23x105

t = 0.00 msec 1.00 msec 2.00 msec 3.00 msec

(b) Nitrogen at σ = 0.33, uin = 9.2 m/sec, T1 = −194 ˚C = 79 K, P1 = 136 kPa, P3 = 107 kPa, Pback = 119 kPa, Re = 9.69 x105

Fig. 5 Cloud cavitation shedding of the trailing part of the cavity (W = 60 mm, C = 20 mm, D = 2 mm, AOI = 8˚, b = 6.0 %, d = 28.8 mm)

t = 0.00 msec 2.00 msec 4.00 msec t = 0.00 msec 2.00 msec 4.00 msec (a) Cold water at σ = 1.20, uin = 13.9 m/sec, T1 = 18 ˚C = 291 K, (b) Hot water at σ = 1.30, uin = 11.3 m/sec, T1 = 90 ˚C = 363 K,

P1 = 111 kPa, P3 = 12 kPa, Pback = 107 kPa, Re = 7.84 x105 P1 = 148 kPa, P3 = 75 kPa, Pback = 108 kPa, Re = 2.06 x106

t = 0.00 msec 2.00 msec 4.00 msec 6.00 msec 8.00 msec 10.00 msec

(c) Nitrogen at σ = 0.53, uin = 9.2 m/sec, T1 = −194 ˚C = 79 K, P1 = 144 kPa, P3 = 110 kPa, Pback = 123 kPa, Re = 2.87 x106

Fig. 6 Cloud cavitation shedding of the trailing part of the cavity (W = 60 mm, C = 60 mm, D = 6 mm, AOI = 8˚, b = 18.1 %, d = 26.3 mm)

same geometry as results in Fig. 4(b), however, uin = 17.7 m/sec bein=g much faster than uin = 9.4 m/sec in Fig. 4(b). Supercavitation was formed. The adverse flow re-gion became thin because the boundary layer along a hydrofoil was thin due to large uin.

Nitrogen at σ = 0.52, uin = 17.7 m/sec, T1 = −193 ˚C = 80 K, P1 = 203 kPa, P3 = 111 kPa, Pback = 124 kPa, Re = 1.97x106

Fig. 7 Supercavitation (W = 20 mm, C = 20 mm, D = 2 mm, AOI = 8˚, b = 18.1 %, d = 8.8 mm)

Conclusion

In order to explain criteria for periodical shedding of the cloud cavitation, a comparison of the cavitation pat-terns among two test sections and two types of hydrofoils were performed.

In the case of C = 20 mm, periodical shedding of the cloud cavitation occurred at W = 20 mm, uin = 9.4 m/sec and d = 8.8 mm, however, it did not occur at W = 60 mm, uin = 9.2 m/sec and d = 26.3 mm because velocity outside of the cavity did not accelerate significantly due to large d and local pressure at W = 60 mm decreases less than that at W = 20 mm for back pressure. Therefore the ad-verse pressure gradient was smaller than that at W = 20 mm and an adverse flow did not occur.

By comparison of similar geometries, i.e. W = C = 20 mm and W = C = 60 mm, periodical shedding of the cloud cavitation occurred in the case of 20 mm, however, in the case of 60mm it did not occur because the cavita-tion thickness was larger than the adverse flow region, so the adverse flow did not significantly affect periodical shedding of the cloud cavitation.

In the case of W = C = 20 mm, periodical shedding of the cloud cavitation occurred at uin = 9.4 m/sec, however, it did not occur at uin = 17.7 m/sec. Cavitation thickness were almost the same, but slow flow region on the hy-drofoil became thin due to high velocity, therefore the adverse flow region existed inside the cavity, so the ad-verse flow did not significantly affect periodical shedding of the cloud cavitation.

According to the experimental results on the plano-

convex hydrofoil, it is concluded that periodical shedding of the cloud cavitation occurs only in the case that there are both the adverse pressure gradient and the slow flow region, namely in the case that the adverse flow region can be formed thicker than the cavity.

The authors would like to thank Professor Toshio Nagashima at the University of Tokyo for his advice and guidance.

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