Embed Size (px)
DESCRIPTION
Multiphase Flow
Citation preview
1
Cav03-OS-7-008 Fifth International Symposium on Cavitation (CAV2003)
Osaka, Japan, November 1-4, 2003
EXPERIMENTAL STUDY OF A VENTILATED SUPERCAVITATING VEHICLE
Martin Wosnik
St. Anthony Falls Laboratory University of Minnesota, USA
Travis J. Schauer Goodrich Corporation Burnsville, MN, USA
Roger E.A. Arndt St. Anthony Falls Laboratory University of Minnesota, USA
ABSTRACT Supercavitating vehicles need to be supplied with an artificial cavity through ventilation until they accelerate to conditions at which a natural supercavity can be sustained. A study has been carried out in the high-speed water tunnel at St. Anthony Falls Laboratory to investigate some aspects of the flow physics of such a supercavitating vehicle. Digital strobe photography images were taken to qualitatively describe the cavity shape and wake details. In addition, the amount of ventilation gas required to sustain an artificial cavity at different velocities was investigated. It was found that the strut shape critically affects air demand through cavity-strut wake interaction. The wake of ventilated cavities was then characterized quantitatively using Particle Image Velocimetry (PIV). Since two-camera, filtered PIV was prohibitively costly due to the large amount of fluorescent particles required to seed the tunnel, a new grayscale technique was developed to measure the void fraction of gas to liquid in the wake of the supercavitating body. Further PIV measurements are currently underway to explore this grayscale technique in bubbly wakes. INTRODUCTION
Supercavitation occurs when a submerged, moving body is enveloped by a large, continuous cavity. It provides a means of significantly reducing the drag of an underwater body, thus enabling a dramatic increase in maximum speed. Small cavitation numbers, σc=2(p∞-pc)/ρU∞
2 < 0.1, are required for supercavitation to occur. There are different ways of achieving this: (1) by increasing the free stream velocity, also referred to as natural supercavitation (this requires U∞>45m/s at sea level, and increases with submersion depth, or p∞), (2) by decreasing the ambient pressure, p∞ (only feasible in closed-circuit water tunnels), or (3) by increasing the cavity pressure, pc, through ventilation of the cavity (artificial or ventilated supercavitation). Reichardt [1] first showed that it was possible to create and study supercavitation by artificially ventilating the flow around a body.
Even for vehicles designed to travel at naturally supercavitating conditions, drag must be reduced by artificial supercavitation to enable them to accelerate to conditions at
which a natural supercavity can be sustained. These ventilated supercavities are the subject of this experimental study.
Fundamental similarity parameters for ventilated supercavities are the cavitation number, σc, and the Froude number, Fr = U∞/(gd) 1/2. When the Froude number is low, gravitational effects are important relative to other forces acting on the supercavitating body. One of the main problems has been to determine the amount of ventilation gas required to sustain a supercavity, quantified by the dimensionless air entrainment coefficient, Q = Q/U∞d2. As found by previous studies, gas loss from the supercavity occurs by two distinct mechanisms of cavity closure: when σc Fr > 1, cavities are characterized by the shedding of toroidal vortices and the occurrence of non-stationary re-entrant jets, whereas when σc Fr < 1, cavities are dominated by gravity and ventilation gas is lost through two stationary vortex tubes (Semenenko [2]).
While a large amount of research has been conducted to determine the air entrainment coefficient in the twin vortex regime, much less data are available in the case of the re-entrant jet regime. This flow regime occurs, e.g., when a supercavitating torpedo is initially launched and accelerating to its final, steady-state velocity. One of the goals of the current research was to quantify the air injection coefficient under a variety of different cavitation and Froude numbers corresponding to the re-entrant jet regime (σc Fr > 1). Further, characteristics of the cavity surface and wake details were examined using digital strobe photography and Particle Image Velocimetry.
NOMENCLATURE d diameter of cavitator [m] Fr = U∞ /(gd) ½, Froude number [-] g gravitational acceleration [m/s2] kl correction factor for light sheet intensity variation [-] L length of cavity [m] m& mass flow rate of gas [kg/s] p pressure [N/m2] pc cavity pressure [N/m2] pv vapor pressure of liquid [N/m2] p∞ freestream static pressure [N/m2]
2
Q = Q/U∞d2, air entrainment coefficient [-] Q volumetric flow rate of gas at cavity pressure [m3/s] r radial coordinate [m] S Strouhal number [-] U velocity in wake of test body [m/s] Umax maximum velocity downstream of test body [m/s] U∞ freestream velocity [m/s] x streamwise coordinate [m] δu =(Umax-U), velocity defect in wake [m/s] η void (volume) fraction of gas to liquid [-] µ dynamic viscosity of fluid [kg/m s] ρ fluid density [kg/m3] ρg gas density [kg/m3] σ =2(p∞-pv)/ρU∞
2, cavitation number based on vapor pressure [-] σc =2(p∞-pc)/ρU∞
2, cavitation number based on cavity pressure [-] DESCRIPTION OF EXPERIMENT
The experiments were carried out in the high-speed water tunnel at St. Anthony Falls Laboratory (SAFL). This water tunnel is a recirculating, closed-jet facility with absolute pressure regulation and is capable of velocities in excess of 20 m/s.1 The test section measures 0.19m (W) x 0.19m (H) x 1m (L), and is fitted with observation windows on three sides. A special design of the tunnel allows for the removal of large quantities of air during ventilation experiments. A schematic of the facility is shown in Figure 1.
Figure 1: Schematic of water tunnel.
The ventilated flow study was carried out using a test body with a sharp edged disk at the nose and special ventilation ports. It is equipped with static pressure ports and miniature pressure transducers for measurement of unsteady pressure in the cavity. It is sting-mounted, and can be fitted with interchangeable disks and rotated in the pitch plane. Careful consideration was given to cavitation choking. Compared to the unbounded case, the
1 Unfortunately, for the work reported here water speeds were limited to 10
m/s due to an aging motor and controller. The water tunnel has recently undergone renovations that included replacement of the old 150HP Direct Current Motor with a modern 75HP Alternating Current Motor. As of this writing, more experiments on ventilated supercavitation are under way.
cavity length will increase when a body is mounted in the test section, which is bounded on all sides (tunnel walls are fixed streamlines). Eventually, the cavity length will become infinite at a cavitation number greater than zero, which is known as the choking condition, c.f. Tulin [3]. Once the flow is choked, further reductions in cavitation number are not possible. For the tunnel used in the current research, a 1 cm disk causes choking to occur at a cavitation number of approximately 0.1. Therefore, the body was designed so that it would be fully contained within a cavity generated by a 1 cm disk at cavitation numbers below approximately 0.14. A drawing of the test body is shown in Figure 2.
Figure 2: Cross-section of test body. Diameter of sharp-edged disk cavitator (on left) is 1 cm.
The general setup for the PIV measurements is shown in Figure 3 (PIV System: TSI, 12-bit dynamic range cameras, 2048x2048 pixel resolution). The test body was mounted to one of the side windows to facilitate optical access from two planes at a right angle (front and bottom). For measurements in the non-cavitating case, 1-3 µm diameter titanium dioxide particles were used for seeding (Stokes number St=0.002).
Typically for a two-phase (cavitating) flow, the system can be set up as a 2-camera, filtered PIV to obtain the velocity fields of both the gas and liquid phase. For this setup, the liquid phase would be seeded with fluorescent particles so that the light scattered by the gas phase (same wavelength as that of the incident light of the pulsed YAG laser) and liquid phase (Laser Induced fluorescence) would be at different wavelengths. PIV imaging of the flow field of both phases could then be accomplished with optical filters, c.f. Khalitov and Longmire [4]. However, due to the size of the water tunnel used in the current research, fluorescent particles could not be used because of cost considerations. Therefore, in the cavitating case only the gas velocity was measured. Details of these measurements will be discussed below.
Flow
Dual Nd:YAG lasers and optics
Laser sheet
Image field
Camera
Figure 3: General PIV setup for bubbly wake.
3
QUALITATIVE DESCRIPTION OF CAVITY AND WAKE Initially, a strut with circular (cylindrical) cross-section was
used, so that its shape and flow disturbance would be the same for all angles of attack. However, the cylindrical strut had many negative effects, including vibration at a Strouhal number typical of bluff-body wakes, which induced a disturbance to the cavity walls, and increased ventilation demand. Therefore, a strut with an elliptical cross-section was designed and fitted over the cylindrical strut to alleviate these problems.
Typical pictures of supercavities with cylindrical and elliptical struts are shown in Figures 4 and 5, respectively. A closer examination of the opaque regions (see detail zoom) reveals that the cavity surface is not smooth in these areas. This is due to re-entrant jets. As the re-entrant jet surges upstream, it loses momentum due to friction. Gravity then causes water from the unsteady re-entrant jet to fall from the test body surface and impact on the cavity boundaries.
Note that the re-entrant jet in Figure 5 (elliptical strut) has surged further upstream than in Figure 4 (cylindrical strut). A possible reason for the difference could be the smaller influence of the elliptical strut on the cavity boundaries. It appears that smoother boundaries may delay re-entrant jet breakup compared to the cylindrical strut case.
Figure 4: Cavity shape and re-entrant jet interaction with cylindrical strut. Arrow in detail zoom points to re-entrant jet.
VENTILATION GAS REQUIREMENTS In Figure 6 it can be seen that the strut shape critically
affects air demand through cavity-strut wake interaction. First, the cylindrical strut caused large distortions to the cavity boundary. In addition, large amounts of ventilation gas were entrained behind the cylindrical strut due to the large pressure drop in the wake of the strut. The amount of gas entrained by the strut is greatly reduced behind the elliptical strut due to its more aerodynamic shape. (The pressure in the strut wake is lower, compared to the elliptical strut shape, causing larger amounts of injected ventilation air to be entrained.)
Two methods to measure unsteady cavity pressure were attempted in the current research: miniature Entran pressure transducers (first two holes downstream of the cavitator in
Figure 5: Cavity shape and re-entrant jet interaction with elliptical strut. Arrow in detail zoom points to re-entrant jet.
Figure 2) and static pressure ports (last two small circumferential holes on the right in Figure 2). The miniature transducers had small thermal mass and self-heated due to operating power. Changing flow over the transducer changed the sensor’s equilibrium temperature, leading to a large error in the measurement which could not be corrected for. The static pressure port measurements were affected by the re-entrant jet and the repeatability was poor. Thus neither pressure measurement was reliable, and the cavitation number based on cavity pressure had to be computed in an alternative way.
Figure 6: Distortion of cavity shape due to cylindrical (left) and elliptical (right) struts. U∞ and p∞ are constant. Q increases from
bottom to top, while constant in each row of images.
The cavity half-length was measured from a large number of digital images and the cavitation number was determined from en empirical relation for cavity length by Waid [5]. Once the cavitation number was known, the cavity pressure and the volumetric flow rate of the injected gas at the cavity pressure of the injected gas could be obtained. The air entrainment coefficient versus cavitation number for a 1cm disk with the
4
elliptical strut is shown in Figure 6. Note that below a cavitation number of about 0.14, the cavity length grows beyond the length of the test body.
0.00
0.10
0.20
0.30
0.40
0.50
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45Cavitation Number
Air E
ntra
inm
ent C
oeffi
cent
Figure 7: Digital images of cavity with 1cm diameter cavitator disk and elliptical strut.
Next, the calculation of the cavitation number was refined using numerical results from Brennen [6], which give a correction for the cavity length in a bounded flow. The data for cavitator disks of 1cm and 1.5cm diameter, for both the cylindrical and elliptical struts, are shown in Figure 8. The Froude number for each set of data ranges from about 20 to 40. Note how the air entrainment coefficient at small cavitation numbers is significantly lower for the elliptical strut when compared to the cylindrical strut data. This plot clearly shows how the location of the steep increase in the air entrainment coefficient (ventilation gas requirements) depends on cavitator size. However, the cavity length where the air entrainment coefficient increases dramatically is approximately the same for both cavitator diameters. This length corresponds to the point where the cavity length approximately equals the length of the test body. This suggests that once a supercavity extends beyond the length of the body it is enveloping, a significant increase in ventilation gas is necessary to further increase the length of the cavity.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55
Cavitation Number
Air E
ntra
inm
ent C
oeffi
cien
t
Cylindrical Strut, 1 cm Disk
Elliptical Strut, 1 cm Disk
Cylindrical Strut, 1.5 cm Disk
Elliptical Strut, 1.5 cm Disk
Figure 8: Air entrainment data for both disks and struts,
cavitation number computations were refined using data of Brennen [6].
WAKE MEASUREMENTS WITH PIV Measurements of the near-wake were performed in both
non-cavitating and cavitating (ventilated) regimes. A schematic of the wake with applicable coordinate system and nomenclature is shown in Figure 9. The measurements reported here were performed with a camera and lens setup that resulted in a pixel resolution of approximately 70µm/pixel. An interrogation area of 32x32 pixels was used, giving a system resolution of about 2.25mm. Standard cross-correlation and post-processing (filter) techniques were used. Note that all PIV results presented here were obtained with the cylindrical strut, and should therefore be considered preliminary pending further measurements currently being carried out.
U∞
x
r Umax
u(r)
Figure 9: Schematic of wake behind body.
Velocity data in the wake for the non-cavitating case is shown in Figure 10, plotted in similarity variables for the axisymmetric near-wake (high Reynolds number, based on velocity deficit). The data collapses reasonably well, except for the upper portion of the wake, which is attributed to flow non-uniformity in the tunnel. The mean, free-stream flow velocity was 6.36 m/s. The PIV measurements were taken with a single camera, and 500 particle image pairs were averaged.
-3
-2
-1
0
1
2
3
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12
δu/x-2/3
r/x1/
3
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 10: Power law relationships in non-cavitating wake for U∞ = 6.36 m/s, with δu=(Umax-U). Average of 500 vector fields.
For measurements in the ventilated cavitating wake, fluorescent particles could not be used to distinguish the phases due to cost considerations, as previously mentioned. Instead, a new technique was developed to determine void fraction from PIV measurements of the cavitating flow, where air bubbles were used as particles.
5
Standard PIV images were recorded from reflections of the air bubbles. Due to the relatively high void fraction of the gas phase enough bubbles were present in the interrogation regions in the bubbly wake to provide a usable signal for cross-correlation. However, the percentage of validated velocity vectors was lower for the cavitating case (25% to 70% at the center of the bubbly wake, decreasing with downstream distance) than for the seeded, non-cavitating case (90%). This was due to bubble reflection and refraction and varying bubble density. The number of validated vectors also dropped off fairly rapidly moving radially outward from the center of the wake.
Bubble velocity data in the cavitating wake are shown in Figure 11. The cavitation number for this case was approximately 0.15. The data were taken at the same free-stream conditions and for the same field of view as the data in the non-cavitating regime shown in Figure 10. Again, 500 image pairs were acquired and used to calculate an average velocity field. Only the points for interrogation areas which had validated vectors in at least 15% of the 500 images were plotted. Therefore, each data point represents, at a minimum, an average of 75 vectors. Since not as many vectors were averaged in the cavitating regime, the data are not nearly as smooth as the data in the non-cavitating regime. However, the data at least show good qualitative results, if not quantitative, as discussed later. As can be seen, the data show the velocity in the wake tending towards the free-stream velocity as the downstream distance increases.
-3
-2
-1
0
1
2
3
3 3.5 4 4.5 5 5.5 6 6.5 7
Bubble Velocity in Wake (m/s)
r (cm
)
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 11: Bubble velocity in cavitating wake for U∞ = 6.36 m/s
and σc = 0.15.
Since the data in Figure 11 were taken at the same free-stream conditions as the data in Figure 10, a comparison between the two data sets can be easily made. Velocity data in the wake for both the cavitating and non-cavitating regimes are shown in Figure 12. Note that liquid velocities for the non-cavitating case and bubble velocities for the cavitating regime are shown for the same downstream locations. It can be seen that there is good agreement in the velocity data between the two regimes. This is true even though a fairly small number of vectors were validated for the cavitating regime compared to the non-cavitating regime. The observed agreement between the liquid and bubble velocities is expected if it is assumed that there is little to no slip between the liquid and gas phases. The
good agreement between the data at the same free-stream conditions lends support to the quantitative validity of the PIV results in the cavitating regime.
-3
-2
-1
0
1
2
3
3 3.5 4 4.5 5 5.5 6 6.5 7
Velocity in Wake (m/s)
r (cm
)
x = 0.5 cm, cavitatingx = 3.0 cm, cavitatingx = 6.0 cm, cavitatingx = 9.0 cm, cavitatingx = 0.5 cm, non-cavitatingx = 3.0 cm, non-cavitatingx = 6.0 cm, non-cavitatingx = 9.0 cm, non-cavitating
Figure 12: Non-dimensional velocity data in non-cavitating
wake for U∞ = 6.36 m/s. Average of 500 vector fields The first step in calculating the void fraction in the wake
was to analyze the grayscale levels of the PIV images. The average grayscale value for each pixel was calculated from a series of PIV images. Once the average grayscale value for each pixel was determined, the background noise was subtracted, so that a grayscale value of zero corresponds to a void fraction of zero. Next, the assumption was made that the intensity of the reflected light was proportional to the void fraction. With this assumption, the general shape of the void fraction distribution versus radial position was obtained by plotting the grayscale value for each pixel. Up to this point, the curve is only qualitatively correct as the magnitude of the void fraction must still be determined. However, calculating the void fraction is a simple process since the velocity of the gas phase can be measured with PIV and the air injection rate can be measured with a flow meter, in our case a rotameter.
The void fraction of gas to liquid for, η, was determined by solving the equation
dA k U(r)ηρm lA g∫=& , (1)
where kl is a correction factor for light sheet intensity variation explained below. The density of the gas, ρg, was estimated in the following manner: First, the pressure of the gas was assumed to be equal to that of the surrounding fluid, with the pressure being a constant for a given downstream location. Next, the pressure was calculated using Bernoulli’s equation between free-stream conditions and liquid flow outside the cavity at a given downstream position with velocity Umax. Once the pressure was known, the density of the gas could be calculated from the ideal gas law. The temperature of the gas was assumed to be equal to the temperature of the injected air, which was approximately the same as the water temperature in the tunnel. Finally, since the shape of the void fraction curve is known from the PIV images, the magnitude can be determined by numerically integrating equation (1).
6
The above procedure for determining the void fraction in
the wake will now be illustrated with experimental results. The same procedure for cross-correlating the PIV images as in the non-cavitating flow was used in the cavitating flow. In addition, the same post processing tools were used to eliminate spurious vectors.
The average grayscale values for six downstream locations are shown in Figure 13. The data were normalized such that a value of zero corresponds to black and a value of one corresponds to pixel saturation (white). Here, the grayscale levels are seen to be highest for downstream locations greater than six centimeters. This may seem odd since the void fraction is highest closest to the body (where the wake has not had a chance to spread and the velocity is lowest) and it was assumed that the void fraction is proportional to the grayscale level. However, the laser was centered at a downstream location of approximately seven centimeters for these measurements, and corrections for light sheet intensity variation need to be made. Since the laser is brightest at the center and decays in intensity from the centerline of the sheet, the bubble reflections are the most intense at the center of the sheet. Figure 13 thus only shows the relative shape of the void fraction curves, and corrections for the light sheet intensity are needed to determine the magnitude of each curve. Note also that the curves do not go to zero outside of the wake due to background noise in the images.
-3
-2
-1
0
1
2
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Normalized Grayscale Level
r (cm
)
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 13: Normalized grayscale levels in the wake for U∞ =
6.36 m/s and σc = 0.15. Average of 500 images. Figure 14 shows the grayscale levels at various downstream
locations after the average background noise was removed. Note how the curves now approach a grayscale level of zero, indicating a void fraction of zero percent. Still, the curves only show the general shape of the void fraction variation, not the absolute magnitude.
The final void fraction results are shown in Figure 15. The void fraction was calculated by using equation (1) and the midpoint rule for numerically approximating the integral. The velocity data used to determine the void fraction were shown in
-3
-2
-1
0
1
2
3
-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16
Normalized Grayscale Level
r (cm
)
x = 0.5 cmx = 3.0 cmx = 4.5 cmx = 6.0 cmx = 7.5 cmx = 9.0 cm
Figure 14: Normalized grayscale levels after removing
background noise. U∞ = 6.36 m/s, σc = 0.15.
Figure 11. Note that the velocity data do not extend to the edge of the wake since the number of validated vectors there was low. Therefore, as an approximation the velocity values at the outer edges of the profiles at each downstream location were extended radially outward and kept constant so that the void fraction could be determined. The correction factor for intensity variation of the light sheet, kl, was approximated to be only a function of downstream position and independent of radial position r.
-3
-2
-1
0
1
2
3
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14
Void Fraction, η
r (cm
)
x = 0.5cmx = 3.0cmx = 4.5cmx = 6.0cm
Figure 15: Calculated void fraction in wake for U∞ = 6.36 m/s
and σc = 0.15. Average of 500 vector fields and images.
The maximum void fraction is decreasing with downstream distance in Figure 15, which, at the very least, is the qualitatively correct result. Continuity for the gas phase is satisfied - the mass flow rate of the gas is a constant at each downstream location. Note from Figure 11 that since the velocity in the wake increases with downstream distance, the void fraction must decrease. Also, since the velocity data for the gas phase agreed well with the liquid velocity in the non-cavitating case, the magnitudes of the void fraction curves are assumed to be correct.
7
The somewhat surprising result that the void fraction
distribution in this dimensional plot does not spread with downstream distance needs more investigation. Most likely it is attributable to simply extending the radially limited gas phase measurements in the cavitating case outward at constant velocity values, as discussed above.
SUMMARY AND OUTLOOK A study was carried out in the high-speed water tunnel at St. Anthony Falls Laboratory to investigate some aspects of the flow physics of such a ventilated supercavitating vehicle. Cavity shape and re-entrant jet interaction were described qualitatively using digital strobe photography. Ventilation gas requirements to sustain an artificial cavity were studied at different velocities. It was found that the strut shape critically affects air demand through cavity-strut wake interaction. Velocity measurements were made in the both the non-cavitating and the ventilated cavitating wake. A new grayscale technique was developed to measure the void fraction of gas to liquid in the wake. The technique was found to show promise as an inexpensive means to measure void fraction in two-phase flows. Further measurements are currently underway to explore this grayscale technique in bubbly wakes over a greater range of flow conditions. Also, alternate methods to directly measure fluctuating cavity pressure are currently being explored.
ACKNOWLEDGMENTS The authors gratefully acknowledge support from the
Office of Naval Research, program manager Dr. Kam Ng. M. Wosnik gratefully acknowledges support from the
Minnesota Supercomputing Institute (MSI).
REFERENCES [1] Reichardt, H. The Laws of Cavitation Bubbles at Axially Symmetrical Bodies in a Flow. Ministry of Aircraft Production (Great Britain), Reports and Translations No. 766, 1946. [2] Semenenko, V.N. Artificial Supercavitation. Physics and Calculation. RTO AVT Lecture Series on “Supercavitating Flows,” Brussels, Belgium, 2001. [3] Tulin, M.P. Supercavitating Flows. In Streeter, V. (ed.), Handbook of Fluid Dynamics, McGraw-Hill, New York, pp. 12-24 to 12-46, 1961. [4] Khalitov, D.A., Longmire, E.K. Simultaneous two-phase PIV by two-parameter phase discrimination. Experiments in Fluids, 32, pp. 252-268, 2002. [5] Waid, R.L. Cavity Shapes for Circular Disks at Angles of Attack. Cal. Institute of Technology. Report No. E-73.4, 1957. [6] Brennen, C. A numerical solution of axisymmetric cavity flows. J. Fluid Mechanics, 37, pp. 671-688, 1969.
