Abstract

This experimental study examines the mechanisms causing cavitation breakdown in an axial waterjet pump. The database includes performance curves, images of cavitation, measured changes to endwall pressure as the blade passes, as well as velocity and pressure distributions inside the blade passage, the latter estimated using Bernoulli's Equation in the rotor reference frame. They show that cavitation breakdown is associated with a rapid expansion of the attached cavitation on the blade suction side (SS) into the blade overlap region, blocking part of the entrance to this passage, increasing the velocity and reducing the pressure along the pressure side (PS) of the blade. Initially, expansion of the SS cavitation compensates for the reduced PS pressure, resulting in a slight increase in performance. Further reduction of the inlet pressure causes a rapid decrease in performance as the SS pressure remains at the vapor pressure, while the PS pressure keeps on decreasing. In addition, during the breakdown, entrainment of the cloud cavitation by the tip leakage vortex generates the previously observed perpendicular cavitating vortices (PCVs) that extend across the passage and reduce the through-flow area in the tip region. Tests have been repeated after installing circumferential casing grooves aimed at manipulating the tip leakage flow and reduce the formation of PCVs. These grooves indeed reduce the tip region blockage during early phases. However, they have a small effect on the performance degradation by cavitation breakdown, presumably owing to their limited effect on the attached SS cavitation and tip region cloud cavitation.

1 Introduction

Cavitation breakdown refers to the dramatic performance degradation of axial pumps when their inlet pressure falls below some critical level. In spite of considerable efforts, the mechanisms causing this phenomenon are still not fully understood. Theoretical analysis by Jakobsen [1] concludes that the cavitation breakdown is related to a condensation shock caused by a rapid decrease in the speed of sound within the cloud cavitation. Based on experimental observations, Pearsall [2] shows that the performance breakdown occurs when the attached cavitation on the suction side (SS) surface extends into the blade overlap region, “choking” the flow. This overlap region starts when a line aligned perpendicularly to the blade chord touches the leading edge of the adjacent blade and continues to a perpendicular line extending from the chord of the adjacent blade to the trailing edge of the first one. Using Reynolds-averaged Navier–Stokes (RANS) simulations, Lindau et al. [3] attribute the breakdown to choking that occurs when the attached cavitation reaches the blade trailing edge. Kim and Schroeder [4] claim, based on RANS results, that the rapid loss of net positive pressure on the pressure side (PS) of the blade caused by the growing cavity on the adjacent SS is the reason for the performance loss. More recently, Tan et al. [5] show that the breakdown is related to the formation of large-scale perpendicular cavitating vortices. These structures develop as the tip leakage vortex (TLV) entrains part of the cloud cavitation and reorients it in a direction that is perpendicular to the blade SS. When this interaction occurs in the blade overlap region, and perpendicular cavitating vortices (PCVs) extend from the SS of one blade to the PS of the adjacent one, the passage is blocked, causing performance degradation. Subsequently, the PCVs have been observed experimentally in other turbomachines and confirmed by numerical simulations [68]. Based on this perceived mechanism, preventing the formation of PCVs could presumably alleviate the rapid degradation during the breakdown. Hence, part of the research described in this paper involves an attempt to use circumferential casing grooves (CGs) to alter the trajectory of the TLV and disengage it from the location of the cloud cavitation. CGs have already been used successfully to delay stall [9,10] and to suppress the instabilities associated with cavitation in inducers [11]. However, attempts by Choi et al. [12] to use axial casing grooves to delay cavitation breakdown in an inducer have had limited success.

In this paper, performance tests, high-speed visualizations of cavitation, and Stereo-PIV (SPIV) measurements in the noncavitating part of the passage are carried out to elucidate the changes to the flow field during cavitation breakdown. The measured velocity upstream and in the passage is used for estimating the pressure distribution in the wetted part of the blade passage, including the blade PS, to evaluate the changes to blade loading as the cavitation on the SS surface expands with decreasing cavitation index. The results elucidate the chain of events leading to a decrease in blade loading as the attached cavitation along the SS expands into the blade overlap region, and entrainment of the cloud cavitation by the TLV blocks the tip region. They show that while tip blockage induced by PCVs and cloud cavitation affect the performance, the loss of blade loading involves the entire passage, as the SS is maintained at the vapor pressure by the attached cavitation, and the PS pressure decreases with decreasing inlet pressure. Hence, while the casing grooves alter the structure of cavitation in the tip region, they have a very limited effect on the conditions for cavitation breakdown. The experimental setup, which is introduced in Sec. 2, is followed by a presentation of the experimental results for the untreated endwall, and by a detailed discussion, which includes prior data, aimed at elucidating the processes involved. The paper concludes with a presentation of results and a discussion on the effects of the CGs.

2 Experimental Setup

The axial waterjet pump (AxWJ-2) used in the current study has been designed by Michael et al. [13] and used already in several studies. Figure 1(a) provides a sketch of this pump, and Table 1 summarizes the relevant parameters. This pump has six rotor blades with a constant outside diameter of 305.2 mm and eight stator blades, which taper to a 213.4 mm nozzle. Detailed descriptions of the pump and cavitation phenomena occurring in it can be found in Refs. [5] and [1316]. When installed in the Johns Hopkins University refractive index matched facility [17], the measured tip clearance without grooves is 0.7 mm. The setup includes a half-filled tank located above the loop and connected to a supply of high-pressure gas and a vacuum pump for controlling the mean pressure in the facility. Cooling jackets surrounding some of the pipe sections of the main loop are used for controlling the liquid temperature. The working fluid is a concentrated aqueous sodium iodide solution, whose specific gravity is 1.8, its kinematic viscosity is 1.1 × 10−6 m2 s−1 at the current temperatures (22–24 °C), and its vapor pressure, 1.2 kPa, is slightly lower than that of the pure water [5,18,19]. The refractive index of this solution matches that of the acrylic pump housing, allowing us to perform optical measurements in the rotor passage without distortions. However, the present rotor is made of Aluminum, the very same rotor discussed in Refs. [5] and [14], owing to the large unsteady loading associated with cavitation breakdown (breaking the acrylic rotor). It should be noted that this rotor has been designed to delay cavitation breakdown by establishing a nearly uniform pressure distribution along the blade SS [13]. Circumferential casing grooves at varying locations are created by installing a combination of 6.35 mm thick transparent acrylic inserts in a slot surrounding the rotor, as illustrated in Figs. 1(a) and 1(b). Three CG configurations have been tested, with CG1 centered at the blade leading edge (LE), CG2 located near the midchord, and CG3 located close to the trailing edge (TE). All grooves have the same width of 30.4 mm and depth of 6.35 mm.

Fig. 1
(a) The AxWJ-2 with casing groove (CG3) installed and the coordinate system for SPIV measurement; and (b) the configurations of circumferential grooves
Fig. 1
(a) The AxWJ-2 with casing groove (CG3) installed and the coordinate system for SPIV measurement; and (b) the configurations of circumferential grooves
Close modal
Table 1

Relevant rotor geometric data

Number of rotor blades6
Number of stator blades8
Tip profile chord length (c)274.3 mm
Tip profile axial chord length (cA)127.4 mm
Rotor radius (Rr)151.9 mm
Casing radius (R)152.6 mm
Casing diameter (D1)305.2 mm
Circumferential groove width30.4 mm (0.11c)
Circumferential groove depth6.35 mm (9.1h)
Outflow section diameter (D2)213.4 mm
Pipe inner diameter downstream of the pump (D)304.8 mm
Tip clearance (h)0.7 mm
Tip clearance ratio (2hD−1)4.6 × 10−3
Tip profile pitch (ζ)159.1 mm
Tip profile solidity (−1)1.72
Tip profile stagger angle (γ)27.7 deg
Rotor angular velocity (Ω(n))94.2 rad s−1 (900 rpm)
Tip speed (UT)14.3 ms−1
Tip profile Reynolds number (Rec)3.6 × 106
Number of rotor blades6
Number of stator blades8
Tip profile chord length (c)274.3 mm
Tip profile axial chord length (cA)127.4 mm
Rotor radius (Rr)151.9 mm
Casing radius (R)152.6 mm
Casing diameter (D1)305.2 mm
Circumferential groove width30.4 mm (0.11c)
Circumferential groove depth6.35 mm (9.1h)
Outflow section diameter (D2)213.4 mm
Pipe inner diameter downstream of the pump (D)304.8 mm
Tip clearance (h)0.7 mm
Tip clearance ratio (2hD−1)4.6 × 10−3
Tip profile pitch (ζ)159.1 mm
Tip profile solidity (−1)1.72
Tip profile stagger angle (γ)27.7 deg
Rotor angular velocity (Ω(n))94.2 rad s−1 (900 rpm)
Tip speed (UT)14.3 ms−1
Tip profile Reynolds number (Rec)3.6 × 106

High-speed images of cavitation are recorded by a PCO® dimax high-speed camera at 1800 frames per second, corresponding to 20 frames per blade passage period when the rotor is operating at a constant speed of 900 rpm. The setup for SPIV measurement is shown in Fig. 2. The images are recorded by a pair of 2048 × 2048 pixels PCO® 2000 interline transfer CCD cameras located on both sides of the laser sheet. Optical distortions are minimized by viewing the sample area through prisms with an outer surface aligned perpendicularly to the lens axis. The flow is seeded with 13 μm, silver-coated, hollow spherical particles that have a specific density of 1.6, slightly less than that of the fluid. To characterize the effect of cavitation on the pump performance, the SPIV measurements focus on the flow along PS inside the passage and the noncavitating parts of the tip region. The laser sheet is orientated to minimize the detrimental effects of light scattered by the cavitation. This sheet is almost perpendicular to the SS surface at the entrance to the blade overlap region. To define the orientation and location of this sheet, we use a general coordinate system (r, θ, z) that has an origin located at the pump center, and coincides with the plane of the LE of the blade. The coordinate system associated with the sample plane is (x̃, ỹ, z̃), where x̃ and ỹ are located within the illuminated area, and z̃ is inclined by β = 40 deg to the z direction (Figs. 1(a), 2(a), and 2(b)). The origin of (x̃, ỹ, and z̃) is located at r/R =0.995 (R =152.6 mm is the casing radius), θPIV = −37.5 deg (θ = 0 is the vertical direction), and z =0. Figures 2(c) and 2(d) look at the sample area from the back. By maintaining the laser sheet in the same location, but recording data for different blade orientations, one can obtain data in different planes relative to the blade LE. The measurements have been performed in four planes. For each one, 200 images pairs have been recorded for a series of cavitation indices.

Fig. 2
(a) and (b): Top and section (A–A) views of the SPIV setup; (c) a perspective view of the AxWJ-2 rotor looking from behind showing the location of the PIV laser sheet at the LE; and (d) a sample map of |Uz|/UT superimposed on the rotor image
Fig. 2
(a) and (b): Top and section (A–A) views of the SPIV setup; (c) a perspective view of the AxWJ-2 rotor looking from behind showing the location of the PIV laser sheet at the LE; and (d) a sample map of |Uz|/UT superimposed on the rotor image
Close modal

Calibration of the SPIV system follows a two-step process described by Wieneke [20]. As discussed in Ref. [21], the initial coarse calibration step is performed by raising the entire optical system and traversing a target in a small chamber containing the same fluid. The second, the so-called self-calibration procedure, is carried out using particle images acquired in the actual sample area, after lowering the system back. Image preprocessing consists of applying background removal and the application of a modified histogram equalization algorithm [22] to enhance particle traces. The FFT-based cross-correlations algorithm for calculating the velocity followed by universal outlier removal [23] are performed using the commercial software package, lavision Davis. The sample area size is 75.3 × 114.8 mm2, and the vector spacing is 0.41 mm for 24 × 24 pixels2 interrogation windows with a 50% overlap. Due to obstruction by the TLV cavitation, only part of the PIV image is available, but it still allows us to examine the flow and pressure distributions along the PS of the blade, from the casing, r/R =1.0, down to r/R =0.66, which represents 50% of the rotor blade span. As shown in Fig. 2(c), the “triangular” field of view (FOV) is bounded by the blade PS to the left and the casing wall on the top. The SPIV measurements have been performed for the baseline case without grooves, and the CG3 case, with the groove located near the blade trailing edge.

All the velocity components have been transformed from the laser sheet coordinate system (ũ, ṽ, and w̃) into the global cylindrical system (ur, uθ, and uz), with the corresponding ensemble-averaged components denoted as (Ur, Uθ, and Uz). The coordinates and vectors transformations are shown in Table 2, with the definitions of variables indicated in Table 1 and in the Nomenclature section.

Table 2

Coordinate transformation

r=(x̃+RrcosθPIV)2+(ỹcosβ+z̃sinβRrsinθPIV)2
θ=tan1ỹcosβ+z̃sinβRrsinθPIVx̃+RrcosθPIV
z=ỹsinβ+z̃cosβ
ur=ũcosθ+(ṽcosβ+w̃sinβ)sinθ
uθ=ũsinθ+(ṽcosβ+w̃sinβ)cosθ
uz=ṽsinβ+w̃cosβ
r=(x̃+RrcosθPIV)2+(ỹcosβ+z̃sinβRrsinθPIV)2
θ=tan1ỹcosβ+z̃sinβRrsinθPIVx̃+RrcosθPIV
z=ỹsinβ+z̃cosβ
ur=ũcosθ+(ṽcosβ+w̃sinβ)sinθ
uθ=ũsinθ+(ṽcosβ+w̃sinβ)cosθ
uz=ṽsinβ+w̃cosβ

A sample distribution of axial velocity magnitude (|Uz|/UT) prior to cavitation breakdown, but with limited attached cavitation on the blade SS, is presented in Fig. 2(d). It shows the expected axial velocity increase with distance from the PS at midspan and the deficit near the tip. The latter occurs due to combined effects of the casing boundary layer and blockage induced by a cavitating TLV (discussed later, see also Ref. [5]). In this case, the sample plane coincides with the leading edge of the blade tip. However, most of the data presented in this paper correspond to a plane that intersects with the blade tip at s/c =0.065, i.e., inside the passage. Here, s is the distance from the leading edge along the blade tip chord, and c is the blade tip profile chord length.

3 Results

3.1 Cavitation Performance.

Figure 3(a) shows the flow rate-head curves for the pump without casing grooves (baseline). The test results reported by Chesnakas et al. [14] using the very same rotor, but in a different facility, are also included for comparison. The flow rate coefficient is defined as
φ=QnD3
(1)
Fig. 3
(a) The pump performance curve without casing grooves under noncavitating conditions; (b) and (c) the variations of the flow coefficients and total head with decreasing cavitation number starting from φ = 0.75
Fig. 3
(a) The pump performance curve without casing grooves under noncavitating conditions; (b) and (c) the variations of the flow coefficients and total head with decreasing cavitation number starting from φ = 0.75
Close modal
Here, Q is the volumetric flow rate, n is the rotor angular speed in revolutions per second, and D is the diameter of the inlet. The flow rate is calculated by integrating the velocity profile acquired by translating a Pitot tube in the radial direction far downstream of the pump. The total head rise coefficient is defined as
ψ=ps,2ps,1+ρ2[(QA2)2(QA1)2]ρn2D2
(2)

Here, as illustrated in Fig. 1(a), the subscripts 1 and 2 refer to the planes where the pressure taps are located, A is the through-flow area, and ps is the measured static pressure. The static head rise, Δps = ps,2ps,1, is measured directly by a differential pressure transducer connected to the pressure taps. To account for variations of the flow downstream of the stator, the transducer side measuring ps,2 is connected to five circumferentially distributed pressure ports with different locations relative to the stator blades. The uncertainties associated with the head rise and flow rate measurements are around 1.2% and 1.7%, respectively. The performance increases with decreasing flow rate until φ < 0.61, when the head rise starts to drop rapidly, indicating the onset of stall. According to the measurements by Chesnakas et al. [14], this pump has a peak efficiency of 89% at φ = 0.76.

The changes in flow rate and total head rise coefficients with decreasing cavitation numbers are shown in Figs. 3(b) and 3(c), respectively. The inlet pressure-based cavitation number (index) is defined as
σ=ps,1pv0.5ρUT2
(3)

where ps,1 is the absolute pressure measured at the pump inlet (Fig. 1(a)), pv is the vapor pressure of the working fluid, and UT is the rotor tip speed. The data shown in this section are obtained starting at an initial flow rate of φ = 0.75, i.e., very close to the peak efficiency point, and then gradually reducing the mean pressure in the facility. During the experiment, the loop is solely driven by the pump, i.e., no efforts are made to compensate for changes in flow rate. For each point, the pump performance is measured after running the machine at the same condition for more than 40 s. Both φ and ψ remain unchanged between 0.18 < σ < 0.78 but decrease abruptly at σ < 0.17. Just before the breakdown, around σ = 0.17, ψ increases slightly while φ hardly changes. In the following discussion, we refer to this point as “breakdown onset.” It should be noted that this very abrupt breakdown curves in Figs. 3(b) and 3(c) are not unique to this pump, and that similar curves have also been observed in heavily loaded inducer pumps [11,24]. In addition, the tests conducted in a water tunnel using the very same rotor at 2000 RPM [14] and CFD simulations by Lindau et al. [3] show a similar sharp decrease in head at cavitation breakdown. Owing to the significant role that expansion of the cavitation into the blade overlap region in the present case, it is likely that the extent of blade overlap might play a role in the rate of performance degradation. Milder breakdown curves have been observed for pumps with little overlaps, such as marine propellers [25,26]. Additional results for a lower starting flow rate and changes associated with the casing grooves are discussed in Sec. 3.5.

3.2 Appearance of Cavitation.

Figure 4 contains a series of images showing the progression of cavitation in the rotor passage with decreasing σ without casing grooves. The corresponding σ and performance parameters are indicated in each plot. Figures 4(a) and 4(b) correspond to conditions before breakdown, Fig. 4(c) shows the cavitation at the breakdown-onset point, and Figs. 4(d) and 4(e) demonstrates the extent of cavitation once breakdown occurs. At σ = 0.526, the attached cavitation appears only along the SS leading edge. As σ is lowered to 0.186, still before breakdown, the sheet cavitation expands toward but does not reach the blade overlap region. Tip leakage cavitation, which starts in the tip gap, forms a bubbly sheet that extends into the passage up to the cavitating TLV. At breakdown-onset condition, the SS attached cavitation propagates into the blade overlap region. At this stage, the area covered by the attached cavitation starts oscillating in large amplitudes, from the beginning of the overlap region to about 80% of the tip chordlength. A sample time sequence demonstrating these oscillations at the same operating condition is presented in Fig. 5. While a small area is covered by cavitation at t = t0 and t = t0+3Δt (Figs. 5(a) and 5(d)), most of the lower blade is covered at t = t0t and t0+2Δt (Figs. 5(b) and 5(c)).

Fig. 4
Evolution of cavitation in the rotor passage with decreasing cavitation number for the baseline case. The corresponding operating conditions are indicated in each image: (a) high pressure with limited LE cavitation, (b) before breakdown, (c) breakdown onset, and (d) and (e) after cavitation breakdown.
Fig. 4
Evolution of cavitation in the rotor passage with decreasing cavitation number for the baseline case. The corresponding operating conditions are indicated in each image: (a) high pressure with limited LE cavitation, (b) before breakdown, (c) breakdown onset, and (d) and (e) after cavitation breakdown.
Close modal
Fig. 5
Rapid variation in the SS area covered by attached cavitation at the onset of breakdown. Here, Δt corresponds to two blade passing periods (Δt = 0.022 s). Blue lines define the boundary of the attached cavitation.
Fig. 5
Rapid variation in the SS area covered by attached cavitation at the onset of breakdown. Here, Δt corresponds to two blade passing periods (Δt = 0.022 s). Blue lines define the boundary of the attached cavitation.
Close modal

When σ is reduced further to 0.160 (Fig. 4(d)), corresponding to a 2 kPa decrease in the absolute inlet pressure from σ = 0.171, the head rise decreases by more than 5% (8 kPa). The attached cavitation stops oscillating and remains expanded over a broad area covering a large portion of the blade SS, reaching the trailing edge near the tip, but not at midspan. Furthermore, a perpendicular cavitating vortex (PCV) appears near the SS trailing edge of one blade and extends into the passage toward the PS of the neighboring blade. As discussed in Ref. [5], the PCV develops as the TLV entrains the cloud cavitation that develops downstream of the attached cavitation on the blade SS. The performance keeps deteriorating as the cavitation index is reduced to σ = 0.155 (Fig. 4(e)). Here, the attached cavitation covers large fractions of the SS, and the PCVs grow in size and extend deeper into the passage, covering much of the tip region near the trailing edge.

3.3 Velocity and Pressure Distributions.

The SPIV measurements have been conducted at five different cavitation indices, starting from φ = 0.75 and σ = 0.77, which is used as the baseline case, and then at four representative lower pressures. Figure 6 shows the distribution of axial velocity magnitude (|Uz|/UT), in a plane that intersects with the blade tip chord at s/c =0.065. The five conditions are: (i) high pressure when cavitation is suppressed (σ = 0.77); (ii) close to, but before breakdown (σ = 0.180); (iii) breakdown onset, when the attached cavitation oscillates and the head rise peaks (σ = 0.170); (iv) “early breakdown” when the performance of the machine already deteriorates (σ = 0.166); and (v) “deep breakdown” (σ = 0.161), when the head coefficient is 9% lower than that without cavitation. For convenience, each velocity distribution is accompanied by the corresponding point on the head coefficient plot.

Fig. 6
Evolution of |Uz|/UT with decreasing σ at s/c = 0.065 for the baseline case: σ = 0.75—without cavitation; σ = 0.180—just before breakdown; σ = 0.170—breakdown onset; σ = 0.166—early breakdown; and σ = 0.161—deep breakdown. Dotted line in σ = 0.77 indicates the boundary of the field of view for σ = 0.180–0.161. A head rise curve is included in most figures to indicate the corresponding operating condition.
Fig. 6
Evolution of |Uz|/UT with decreasing σ at s/c = 0.065 for the baseline case: σ = 0.75—without cavitation; σ = 0.180—just before breakdown; σ = 0.170—breakdown onset; σ = 0.166—early breakdown; and σ = 0.161—deep breakdown. Dotted line in σ = 0.77 indicates the boundary of the field of view for σ = 0.180–0.161. A head rise curve is included in most figures to indicate the corresponding operating condition.
Close modal

At σ = 0.77, |Uz| is low near the endwall casing and along the PS, increasing with decreasing distance from the blade SS. The reduced velocity along the casing is associated with the casing boundary layer as well as other phenomena, such as tip leakage flows, TLV formation, etc. (e.g., Refs. [2729]). The lower velocity along the PS is inherent and expected. The fields of view for the other σ are smaller due to a partial visual blockage by the cavitation in the tip region, as indicated by the dotted line. At σ = 0.180, |Uz| increases slightly everywhere. Further reduction in pressure to σ = 0.170 causes a substantial increase in velocity over almost the entire sample area, including the vicinity of the PS, but not in the tip region. For example, along the PS and at x̃/R = −0.35, there is a 6% increase. Since the overall flow rate in the machine remains essentially unchanged, the increase in axial velocity along the PS is likely to be a result of (cavitation-induced) blockage in other regions of the rotor passage, predominantly along the SS. Recall that the visual observations (Figs. 4(c) and 5) indicate that under similar conditions, the attached cavitation reaches the blade overlap region, oscillates, and expands rapidly into the aft part of the passage. Hence, they support the claim that the increase in velocity along the PS is associated with cavitation-induced blockage. The slight reduction in pressure to σ = 0.166, when the flow rate and the head rise start to drop, causes a decrease in axial velocity in the outer parts of the PS and along the tip region, but an increase along the inner parts, i.e., at x̃/R < −0.2. In spite of the overall decrease in flow rate, the reduced passage still accelerates the flow in the inner part of the passage. At σ = 0.161, there is a reduction in axial flow over the entire sample area, including the PS, but especially along the endwall casing. The latter is presumably associated with the large PCVs, the cavitating TLV, and the cloud cavitation near the trailing edge (Figs. 4(d) and 4(e)).

Figures 7(a) and 7(b) show the evolution of circumferential and radial velocity components for three of the abovementioned values of σ, respectively. The magnitude of Uθ is much smaller than that |Uz|, but it decreases by about 50% at breakdown onset (σ = 0.170) and keeps on decreasing as the cavitation index is reduced further. Note that a decrease in Uθ implies an increase in velocity in the rotor reference frame (ΩrUθ). During the transition from σ = 0.180 to σ = 0.170, both |Uz| and (ΩrUθ) increase, resulting in a nearly unchanged flow angle relative to the blade (not shown). Hence, the phenomena occurring during breakdown onset do not involve substantial changes to the flow angle. As the pressure is reduced further, (ΩrUθ) increases slightly, together with the axial velocity in the inner part of the passage. Hence, the flow angle is still maintained at a very similar level. In contrast, in the outer parts of the passage, the angle increases. The radial velocity components (Fig. 7(b)) are very low over the entire sample area and decrease slightly with decreasing cavitation index.

Fig. 7
Evolution of (a) Uθ/UT, and (b) Ur/UT with decreasing cavitation number for the baseline case: σ = 0.180—just before breakdown; σ = 0.170—breakdown onset; and σ = 0.166—early breakdown
Fig. 7
Evolution of (a) Uθ/UT, and (b) Ur/UT with decreasing cavitation number for the baseline case: σ = 0.180—just before breakdown; σ = 0.170—breakdown onset; and σ = 0.166—early breakdown
Close modal
Knowledge of the averaged velocity distribution upstream and within the rotor passage enables us to use Bernoulli's equation to estimate the pressure distributions along the blade PS. Neglecting effects of viscous and Reynolds stresses, and assuming a steady flow in the rotor reference frame, along a streamline
pρ+W22(Ωr)22=const
(4)
where W is the total velocity magnitude in the rotor reference frame [30,31]. This constant is usually referred to as rothalpy in turbomachines [27]. This equation relates the change in fluid energy to the energy added/subtracted by the blade rotation in the rotor reference frame. In the current case, considering that Ur is very small in the field of view (Fig. 7(b)), the radial displacement of the streamlines is assumed to be negligible. Hence, the (Ωr)2/2 term remains the same for points located along each streamline upstream and within the field of view, where, as shown before, the radial velocity is low. Consequently, the time-averaged pressure in the passage is
p(r,θ,z)=pin+0.5ρ(Uz,in2+(ΩrUθ,in)2)0.5ρ((ΩrUθ)2+Uz2)
(5)
where Uz,in and Uθ,in are the measured velocity components at z/R =0.65, well upstream of the rotor LE [32]. Since the inlet conditions have only been recorded at φ = 0.76, the Uz,in profiles at other flow rates are scaled by assuming the same radial distributions. Since the present analysis focuses on the effect of cavitation on the performance of the machine, in addition to σ, we define the local cavitation index
σlocal=p(r,θ,z)pv0.5ρUT2
(6)

Its distribution along the PS of the blade will be used as a representative of the pressure difference across the blade when the SS is covered by the attached cavitation. The distributions σlocal are shown in Fig. 8. As is evident, the slight decrease in σ from 0.18 to 0.17 (breakdown onset) causes an order of magnitude larger decrease in σlocal everywhere in the passage. This drastic reduction in pressure is associated with the corresponding increase in axial velocity magnitude. A further reduction to σ = 0.166 causes an additional milder decrease in pressure along the inner part of the PS. In the rest of the passage, especially in the tip region, the pressure hardly changes. Between σ = 0.166 and 0.161, the latter corresponding deep breakdown, the pressure distribution changes very little near the tip but decreases slightly deeper in the passage.

Fig. 8
Evolution of σlocal with decreasing cavitation number for the baseline case: σ = 0.180—just before breakdown, σ = 0.170—breakdown onset, σ = 0.166—early breakdown, and σ = 0.161—deep breakdown
Fig. 8
Evolution of σlocal with decreasing cavitation number for the baseline case: σ = 0.180—just before breakdown, σ = 0.170—breakdown onset, σ = 0.166—early breakdown, and σ = 0.161—deep breakdown
Close modal

3.4 Discussion on the Causes for Cavitation Breakdown.

In this section, we combine the present and previous findings of cavitation breakdown to introduce a plausible explanation for the mechanisms involved. Starting with a brief summary, the precursor for cavitation breakdown, namely, the breakdown-onset condition, occurs when the attached cavitation on the blade SS expands into the blade overlap region, in agreement with Pearsall [2], and fluctuates (Fig. 5). At this condition (σ = 0.17), the flow rate does not change, and the head increases slightly. The latter trend is consistent with simulations performed for the same pump geometry [3] and for different machines [33,34]. Yet, the SPIV measurements near the entrance to the overlap region show that the mean axial velocity increases by 5–6%, and the mean pressure decreases by more than 25% along the PS of the blade. The increase in axial velocity without a change in flow rate is presumably associated with a reduction in the through-flow area caused by the attached cavitation along the SS of the neighboring blade. The decrease in pressure over the entire entrance area might affect the rapid expansion of the attached cavitation to the aft part of the passage.

While the decrease in PS pressure concurrently with an increase in total head appears to be puzzling, it can be explained as follows: Measurements of pressure distributions on the surface of a cavitating 2D hydrofoil by Shen and Dimotakis [35] show that the pressure inside the cavitation area on the SS is very close to the vapor pressure, and recovers to the fully wetted values only downstream of the cloud cavitation. Hence, in parts of the SS covered by cavitation, the pressure is lower than that in a fully wetted flow. For this isolated foil, the pressure along the PS is not affected significantly by the cavitation, at least as long as the SS cavitation does not reach the trailing edge of the foil. Consequently, partial cavitation can actually increase the lift force. In the current pump, although the PS pressure drops, it is accompanied by a rapid expansion of the SS cavitation and the area where the pressure is equal to the vapor pressure. Hence, the blade loading might still increase in spite of the decreases in PS pressure. Evidence supporting this postulate can be obtained from the endwall casing pressure measurements in the same machine performed by Tan et al. [5] using two flush-mounted piezo-electric transducers (see Fig.9(a)), the first (Transducer 1) located near the LE (s/c =0.175), and the second (Transducer 2), at the midchord (s/c =0.488). Relevant results are presented in Fig. 9(b), which shows the phase-averaged pressure of transducer 2, where Cp = p(θ)/0.5ρUT2 for different σ. Here, θ refers to the orientation of the blade relative to the sensors, with the sharp decrease in pressure occurring as the blade passes by the transducer, peaking on the PS, and having a minimum value along the SS. The difference between them, namely, ΔCp = Cp,PSCp,SS, is used here as a representative for the pressure difference across the blade, i.e., the local tip loading, although it does not occur at the same chordwise location. Note the sharp decrease in SS pressure at σ = 0.171, the breakdown-onset condition.

Fig. 9
(a) Illustration of the endwall pressure transducer locations; (b) phase-averaged casing wall pressure measured by transducer 2 for different cavitation indices; (c) pressure differences across the blade at the casing wall for the baseline case, superimposed on the pressure and flow coefficients stating from φ = 0.7 (adapted from Tan et al. 2015 [15])
Fig. 9
(a) Illustration of the endwall pressure transducer locations; (b) phase-averaged casing wall pressure measured by transducer 2 for different cavitation indices; (c) pressure differences across the blade at the casing wall for the baseline case, superimposed on the pressure and flow coefficients stating from φ = 0.7 (adapted from Tan et al. 2015 [15])
Close modal

Figure 9(c) presents the effect of reducing σ on ΔCp for both transducers together with the corresponding pump performance. To facilitate the discussion, sketches showing different stages of cavitation, corresponding hypothetical pressure distributions along the chord, together with the location of the pressure transducers, are presented in Fig. 10. It should be noted that the area covered with vapor, hence having a vapor pressure, expands beyond the zone expected based on the pressure distribution of the noncavitating flow. As is evident, ΔCp1 decreases with decreasing σ, with ∂ΔCp/∂σ ∼ 1 at 0.175 < σ < 0.25, i.e., before breakdown onset. Under these conditions, the SS attached cavitation already expands to transducer 1 (Fig. 10(a)), hence the SS pressure is expected to be nearly constant and close to the vapor pressure (Fig .10(a) pressure distribution). The slope of the ΔCp1 then implies that the PS pressure decreases at almost the same rate as the pressure upstream of the rotor. Since the corresponding performance and flow rate remain nearly unchanged, the decrease of blade loading near the LE (decreasing ΔCp1) must be compensated by an increase in loading somewhere else along the blade. According to Shen and Dimotakis [35], the decrease in airfoil loading near the leading edge is compensated by an increase in pressure difference across the blade at higher chord fractions, where the pressure in the cavitation-covered SS remains low. A similar sketch is provided in Fig. 10(a), which shows the decrease in blade loading near the LE is compensated by the increase of loading downstream of the blade chord. Indeed, at the breakdown-onset conditions, as the SS cavitation expands rapidly to cover a large fraction of the SS, including the location of transducer 2 (see illustration in Fig. 10(b)), ΔCp2 increases, while ΔCp1 decreases rapidly (Figs. 9(c) and 10(b)). In support of this claim, Fig. 9(b) indicates that at σ = 0.17 the increase in ΔCp2 is mostly attributable to the decrease in the SS pressure. Hence, it appears that the rapid expansion of the SS cavitation results in a slight increase in head rise just before breakdown, a phenomenon observed in numerous turbomachines before [3,33,34], in spite of the decrease in loading near the LE. Although the cavitation-induced blockage reduced the PS pressure near the leading edge, the corresponding rapid expansion of the SS cavitation appears to make up for it.

Fig. 10
A sequence of sketches showing different stages of cavitation development along with illustrations of the corresponding blade pressure distributions (−σlocal, as defined in Eq. (6)): (a) before breakdown; (b) at breakdown onset; and (c) during deep breakdown. In the pressure profiles, the solid lines represent the hypothetical pressure distribution with SS cavitation, while the dashed lines show the would-be distribution without cavitation.
Fig. 10
A sequence of sketches showing different stages of cavitation development along with illustrations of the corresponding blade pressure distributions (−σlocal, as defined in Eq. (6)): (a) before breakdown; (b) at breakdown onset; and (c) during deep breakdown. In the pressure profiles, the solid lines represent the hypothetical pressure distribution with SS cavitation, while the dashed lines show the would-be distribution without cavitation.
Close modal

A few comments should be made before concluding this discussion. First, in Ref. [5], the decrease in ΔCp1 at breakdown onset is linked to the formation of PCVs, which extend from the SS to the point where transducer 1 is exposed to the PS. However, the present pressure calculations near the LE (Fig. 8) indicate that the decrease in PS pressure extends to large fractions of the blade span well below the tip region. Hence, while the PCVs may contribute to the pressure reduction in the tip region, the current data suggest that the reduction in blade loading is associated with cavitation-induced blockage. Second, we cannot provide a substantiated explanation for the oscillation in the area covered by cavitation at breakdown onset, but one could still postulate about the possible mechanism. First of all, the breakdown is not associated with instabilities related to the onset of stall, which as Fig. 3(a) indicates, occurs at φ = 0.61, i.e., at a significantly lower flow rate than those of the presently investigated cases. A plausible explanation involves the increase in performance, i.e., the pressure in the vicinity of the blade trailing edge, at breakdown onset, as the SS cavitation expands rapidly. This increase in pressure is likely to suppress the expansion of the SS cavitation in the aft part of the blade. Therefore, the extent cavitation there decreases, resulting in a decrease in blade loading and trailing edge pressure, hence an expansion of the SS cavitation. In the proposed mechanism, the oscillations of SS cavitation is a result of the (short-lived) increased performance with decreasing cavitation index at breakdown onset. Third, while the improvement in performance owing to cavitation-induced changes to the flow near the leading edge is also possible, the present values of ΔCp1 and ΔCp2 indicate that at least in the tip region, the increase in blade loading occurs at midchord, not at the leading edge. Finally, since the test loop is maintained at the same conditions during the test, the simultaneous slight increase in head and decrease in flow rate at breakdown onset might seem contradictory. However, the associated change to the loop resistance, namely, ψ/φ2, is smaller than 1%. It might be associated with cavitation-induced changes to the flow structure and/or flow oscillations occurring at the plane where the exit pressure is measured (plane A1 in Eq. (2)).

Upon further reduction in cavitation index to σ = 0.166, the head rise and flow rate drop sharply, consistent with the decrease in ΔCp1, the PS pressure at midspan, and ΔCp2. With a further decrease to σ = 0.161 (deep breakdown), the attached cavitation near the tip region reaches to the blade trailing edge (Figs. 5(d), 5(e), and 10(c)) and stops oscillating, while ΔCp1 (Fig. 9(c)) and the pressure along the LE pressure-side plateau (Figs. 8 and 10(c)). The plateau is related to reduced flow rate and inlet pressure at the same time, which keeps the PS pressure nearly unchanged (Fig. 10(c)). Hence, at this stage, the reduction in performance is no longer associated with the leading edge, but with a decreased blade loading at midchord, as eluded to from the reduction in ΔCp2. In Tan et al. [5], the latter trend is attributed to the cavitation-induced blockage as the entire tip region becomes occupied by the PCVs and cloud cavitation (Figs. 5(d) and 5(e)). The increased impact of the tip blockage is consistent with the decrease in |Uz| near the LE tip, as shown in Fig. 6. Near the entrance, while ΔCp1 and pressure in the entire outer part of the passage change very little (Fig. 8), the pressure deeper in the passage keeps on decreasing slightly, i.e., the blade loading decreases away from the tip.

3.5 Effects of the Casing Grooves.

Figure 11 shows the performance curves for all the present circumferential casing grooves. For φ = 0.65–0.8, the casing grooves cause a 1%–2% reduction in total head rise compared to the baseline untreated endwall. For the baseline case, the performance slope becomes positive at φ < 0.61, i.e., showing evidence of stall. The present CG1 and CG2 do not seem to delay the onset of stall and cause a 2% loss of performance. Installing CG3 crate a performance plateau at φ < 0.65, but in this case, the slope does not become positive. It should be noted that multiple other studies have shown that circumferential grooves, sometimes multiple ones, delay the onset of stall in axial turbomachines [9,10,36]. The minimal effect in the present study might be related to their shallow depth.

Fig. 11
Performance curves for the baseline and CG cases under noncavitating conditions
Fig. 11
Performance curves for the baseline and CG cases under noncavitating conditions
Close modal

The effect of cavitation number on performance for all the cases is summarized in Fig. 12. The tests have been carried out at two different initial flow rates, the first starting from φ = 0.74–0.75, and the second from φ = 0.67–0.68. The first corresponds to BEP, as discussed before, and the second is still well above stall conditions. For the two flow rates of the present study, early signs of cavitation along the blade leading edge for the baseline case appear at σ ≈ 0.65 for φ = 0.68 and at σ ≈ 0.53 for φ = 0.75. The observed increase in cavitation inception index as the flow rate is decreased below design, but above stall, conditions are consistent with the trends reported by Schiavello and Visser [37]. For all cases, the total head drops abruptly at σ = 0.17, indicating that the CGs do not alter the breakdown cavitation index noticeably. Furthermore, the breakdown cavitation index is not affected significantly by the flow rate, hence the blade load distribution. Both trends are similar to those obtained in tests performed by Kang et al. [11] for an inducer with a deeper CG installed near the LE. Hence, the inability to delay the breakdown is unlikely to be associated with the depth of the groove. However, that study shows that the CGs suppress other cavitation-induced instabilities at pressures above breakdown.

Fig. 12
Evolution of: (a) flow rate coefficient, and (b) total head coefficient with decreasing cavitation numbers for two different starting flow rates and for several circumferential casing grooves
Fig. 12
Evolution of: (a) flow rate coefficient, and (b) total head coefficient with decreasing cavitation numbers for two different starting flow rates and for several circumferential casing grooves
Close modal

Detailed descriptions of the evolution of cavitation for each of the CGs are provided in Ref. [16]. Here, we present only a few examples that are relevant to the discussion about cavitation breakdown. Figure 13 compares the cavitation phenomena for CG1 to the baseline at two different cavitation indices. At σ > 0.3, CG1 entrains the TLV and aligns it with the downstream end of the groove (Fig. 13(a)). Consequently, although the shedding of cloud cavitation on the SS persists for both cases (Figs. 13(a) and 13(b)), the CG1 groove decouples the TLV from the cloud cavitation. Hence, the formation of a PCV is prevented. However, with decreasing pressure to σ < 0.2, i.e., still before breakdown (Fig. 13(c)), significant parts of the triangular cavitating area and the TLV appear downstream of the groove. Furthermore, the PCVs resulting from the interaction of the TLV with the cloud cavitation develop in a location that is very similar to that of the baseline case (Fig. 13(d)). For both cases, at this pressure, the PCVs develop upstream of the blade overlap region, hence it does not affect the cavitation breakdown. With further reduction in pressure down to breakdown (not shown), although a fraction of the TLV remains trapped within the CG1, the phenomena downstream of the groove do not appear to be different from those of the baseline (Figs. 3(c)3(e)). Similar phenomena happen for CG2 as well, although the TLV trapping occurs further downstream. In summary, while both CG1 and CG2 trap a fraction of the TLV, phenomena occurring downstream of the grooves, where part of the TLV “escapes,” do not appear to be significantly different from those of the baseline.

Fig. 13
(a) A sample image demonstrating the entrainment of the TLV by CG1 at σ = 0.312, prior to breakdown compared to (b) a baseline sample at similar conditions. (c) Part of the vortex escapes from CG1 at σ = 0.176 compared to (d) a baseline case under similar conditions. Solid red lines indicate the boundaries of CG1, and dashed lines show the same location, but without grooves.
Fig. 13
(a) A sample image demonstrating the entrainment of the TLV by CG1 at σ = 0.312, prior to breakdown compared to (b) a baseline sample at similar conditions. (c) Part of the vortex escapes from CG1 at σ = 0.176 compared to (d) a baseline case under similar conditions. Solid red lines indicate the boundaries of CG1, and dashed lines show the same location, but without grooves.
Close modal

The CG3 groove does not appear to have a noticeable impact on the appearance of cavitation prior to breakdown, including the formation of PCVs upstream of the overlap blade region. When the sheet cavitation expands rapidly during early phases of breakdown, the PCVs also form upstream of the CG3 groove as well. Once cavitation reaches this groove, a comparison between Figs. 14(a) and 14(b) demonstrates that CG3 suppresses the formation of PCVs near the TE of the blade and restricts the attached cavitation to regions located upstream of this groove. Yet, in spite of this positive effect, CG3 has a minimal effect on the conditions for cavitation breakdown. As the data embedded in Fig. 14 indicates, the only noticeable effect is a slight (a few per cent) improvement in flow rate and head rise at similar cavitation numbers. This observation challenges the claim that the PCVs play the primary roles in the cavitation breakdown, as proposed by Tan et al. [5]. Upon further minor reduction in cavitation index to σ ∼ 0.15, both the flow and head coefficients for CG3 decrease rapidly, the PCVs expand to the blade trailing edge, and the difference between them and those of the baseline diminish. In summary, the present grooves do not have a significant impact on the conditions for cavitation breakdown.

Fig. 14
A comparison between cavitation near the trailing edge of: (a) CG3, and (b) baseline cases during deep breakdown. The formation of PCVs is suppressed and the chordwise extent of cavitation is reduced by CG3 during this phase of cavitation breakdown.
Fig. 14
A comparison between cavitation near the trailing edge of: (a) CG3, and (b) baseline cases during deep breakdown. The formation of PCVs is suppressed and the chordwise extent of cavitation is reduced by CG3 during this phase of cavitation breakdown.
Close modal

The next discussion examines the effect of CG3 on the changes to axial velocity as the cavitation number is reduced. Figure 15 compares the changes to |Uz| as the cavitation index is reduced from prebreakdown to breakdown-onset conditions. For the baseline case, we use (|Uz,σ=0.170|−|Uz,σ=0.180|), and for the CG3 cases, (|Uz,σ=0.173|−|Uz,σ=0.187|). The differences in the exact values of σ are caused by slight (<1 kPa) changes in the inlet pressure during the tests. The CG3 data also have a smaller field of view because of particles trapped between the insert and the pump casing limiting the view, hence part of the baseline view is partially masked and the discussion focuses on matched locations. For both cases, the breakdown onset is characterized by an increase in axial velocity and a decrease in pressure along the blade PS. The differences between the two cases along the PS are quite small, but the increase in |Uz| near the blade tip for CG3 is lower than that of the baseline case. Figure 16 provides a similar comparison, but this time the cavitation index changes from prebreakdown to deep breakdown conditions. Here, negative values indicate a decrease in |Uz|, i.e., an increase in blockage. As is evident, the tip region of the baseline flow has a broader area with an increased blockage, and the magnitude of the velocity decrease is also higher. These trends might be associated with the suppression of the PCVs further downstream, as shown in Fig. 14.

Fig. 15
Changes in |UZ|/UT from before breakdown to breakdown onset for: (a) the baseline case, (|Uz,σ=0.170| − |Uz,σ=0.180|)/UT; and (b) the CG3 case, (|Uz,σ=0.173| − |Uz,σ=0.187|)/UT. The partially masked area in (a) is not available in (b).
Fig. 15
Changes in |UZ|/UT from before breakdown to breakdown onset for: (a) the baseline case, (|Uz,σ=0.170| − |Uz,σ=0.180|)/UT; and (b) the CG3 case, (|Uz,σ=0.173| − |Uz,σ=0.187|)/UT. The partially masked area in (a) is not available in (b).
Close modal
Fig. 16
Changes in |UZ|/UT from before to deep breakdown for: (a) the baseline case, (|Uz,σ=0.161| − |Uz,σ=0.180|)/UT; and (b) the CG3 case (|Uz,σ=0.158| − |Uz,σ=0.187|)/UT)
Fig. 16
Changes in |UZ|/UT from before to deep breakdown for: (a) the baseline case, (|Uz,σ=0.161| − |Uz,σ=0.180|)/UT; and (b) the CG3 case (|Uz,σ=0.158| − |Uz,σ=0.187|)/UT)
Close modal

4 Discussion and Conclusions

Performance tests, high-speed imaging, and SPIV measurements have been carried out to study the flow structures and pressure within the rotor passage of an axial waterjet pump during cavitation breakdown, with and without circumferential casing grooves. The results enable us to answer several questions. First, consistent with Pearsall [2], the onset of breakdown occurs when the attached cavitation on the blade SS reaches the blade overlap region at all flow rates as well as with or without casing grooves. Just before the rapid performance degradation, the attached cavitation expands rapidly with large oscillations in the SS area covered by it. Cavitation-induced blockage to the through-flow area at the entrance to the overlap region accelerates the flow in the remaining space, causing a significant (∼25%) reduction to the pressure along the blade PS. This pressure reduction presumably causes further expansion of the attached cavitation on the SS. At this stage, the performance is not degraded yet, and in fact, the pump head even increases slightly, consistent with RANS simulations of the same pump performed by Kim and Schroeder [4] and Lindau et al. [3] as well as numerous studies performed in other machines [3,33,34]. From the measurements of pressure difference across the blade tip at midpassage, it appears that reduction in the SS pressure as the attached cavitation expands rapidly compensates for the reduction in PS pressure, causing an increase in blade loading at midspan, and presumably an increase in overall performance. This elevated performance implies an increase in pressure near the trailing edge of the blade, hence should limit the expansion of the SS cavitation there, pushing it upstream. Upon contraction of the resulting SS cavitation, the blade loading would decrease, causing an expansion of the attached cavitation, and the observed high amplitude oscillations in the cavitating area.

A further slight reduction in cavitation index causes a rapid deterioration in flow rate and head, increased tip region blockage, but only a slight increase in velocity and decrease in PS pressure at midspan of the blade overlap region. The oscillations in SS attached cavitation diminish, and the covered surface expands to the vicinity of the blade trailing edge, at least in the tip region. With the SS pressure now maintained at the vapor pressure, and the PS pressure forced to decrease everywhere by the inlet pressure, the pump performance inherently deteriorates. These observations are consistent with the measured decrease in pressure difference across the blade tip at midchord. In the tip region, PCVs extending from the SS of one blade to the vicinity of the PS of the neighboring one occupy most of the blade overlap area. Cloud cavitation also appears under the PCVs, but only near the SS. Both structures contribute to an increase in the tip region blockage, which is evident also from the velocity measurements at the beginning of the blade overlap region. Hence, two phenomena appear to affect the breakdown, namely, reduction in blade loading over a broad area and blockage in the tip region.

A prior study [5] in our lab has attributed the cavitation breakdown only to the formation of PCVs, an observation supported by several other recent studies [68]. Hence, in current experiments, we have tried to use circumferential grooves placed in several axial locations to manipulate the tip leakage flow and trajectory of the tip leakage vortex, hoping to affect the PCV formation. The results show that when the CG is located near the blade trailing edge (CG3), the PCV formation is indeed delayed, and the velocity measurements confirm that the tip blockage is reduced. However, while CG3 causes a slight improvement in pump performance after breakdown, it has a minimal effect on the conditions for cavitation breakdown, in agreement with previous studies involving inducers [12]. With a further slight reduction in cavitation index, the PCVs reappear, and the performance deteriorates rapidly, reaching conditions that are similar to those of the machine without grooves. These trends indicate that while the PCV formation might contribute to the degradation in performance, it is not the primary reason. As noted above, the global deterioration is associated with a decrease in PS pressure over substantial fractions of the blade, which is imposed by the inlet conditions, while the SS pressure remains at the vapor pressure level.

Acknowledgment

This project is sponsored by the Office of Naval Research. Ki-Han Kim is the Program Officer. The authors would like to thank Yury Ronzhes for his contributions to the construction and maintenance of the test facility.

Funding Data

  • Office of Naval Research (ONR) (Grant Nos. N00014-09-1-0353 and N00014-18-1-2635; Funder ID:10.13039/100000006).

Nomenclature

     
  • A =

    through-flow area

  •  
  • c =

    rotor blade tip chord

  •  
  • Cp =

    pressure coefficient

  •  
  • D =

    diameter of the inlet

  •  
  • h =

    width of the rotor blade tip gap

  •  
  • n =

    rotor angular speed in revolutions per second

  •  
  • ps,1, pin =

    static pressure at pump inlet

  •  
  • ps,2 =

    static pressure at stator outlet

  •  
  • pv =

    vapor pressure of NaI solution

  •  
  • Q =

    volumetric flow rate

  •  
  • R =

    casing radius

  •  
  • Rr =

    rotor radius

  •  
  • r, z, θ =

    radial, axial and circumferential coordinates

  •  
  • s =

    rotor blade chordwise coordinate

  •  
  • ũ, ṽ, w̃ =

    velocity components in the laser sheet coordinate system

  •  
  • ur, uz, uθ =

    radial, axial and circumferential velocity

  •  
  • Ur, Uz, Uθ =

    ensemble-averaged radial, axial and circumferential velocity

  •  
  • UT =

    rotor blade tip speed

  •  
  • Uz,in, Uθ,in =

    axial, circumferential velocity at inlet

  •  
  • W =

    fluid speed in the rotor reference frame

  •  
  • ũ, ỹ, z̃ =

    laser sheet coordinate system

  •  
  • θPIV =

    circumferential location of the laser sheet coordinate system

  •  
  • ρ =

    NaI solution density

  •  
  • σ =

    cavitation index

  •  
  • σlocal =

    local cavitation index

  •  
  • φ =

    flow coefficient

  •  
  • ψ =

    total head rise coefficient

  •  
  • Ω =

    rotor angular velocity

References

1.
Jakobsen
,
J. K.
,
1964
, “
On the Mechanism of Head Breakdown in Cavitating Inducers
,”
ASME J. Basic Eng.
,
86
(
2
), pp.
291
305
.10.1115/1.3653066
2.
Pearsall
,
I. S.
,
1973
, “
Design of Pump Impellers for Optimum Cavitation Performance
,”
Proc. Inst. Mech. Eng.
,
187
(
1
), pp.
667
678
.10.1243/PIME_PROC_1973_187_060_02
3.
Lindau
,
J. W.
,
Pena
,
C.
,
Baker
,
W. J.
,
Dreyer
,
J. J.
,
Moody
,
W. L.
,
Kunz
,
R. F.
, and
Paterson
,
E. G.
,
2012
, “
Modeling of Cavitating Flow Through Waterjet Propulsors
,”
Int. J. Rotating Mach.
,
2012
, pp.
1
13
.10.1155/2012/716392
4.
Kim
,
S.
, and
Schroeder
,
S.
,
2010
, “
Numerical Study of Thrust-Breakdown Due to Cavitation on a Hydrofoil, a Propeller, and a Waterjet
,”
28th Symposium on Naval Hydrodynamics
, Pasadena, CA, Sept. 12–17, pp.
630
643
.
5.
Tan
,
D.
,
Li
,
Y.
,
Wilkes
,
I.
,
Vagnoni
,
E.
,
Miorini
,
R.
, and
Katz
,
J.
,
2015
, “
Experimental Investigation of the Role of Large Scale Cavitating Vortical Structures in Performance Breakdown of an Axial Waterjet Pump
,”
ASME J. Fluids Eng.
,
137
(
11
), p.
111301
.10.1115/1.4030614
6.
Zhang
,
D.
,
Shi
,
L.
,
Shi
,
W.
,
Zhao
,
R.
,
Wang
,
H.
, and
van Esch
,
B. P. M.
,
2015
, “
Numerical Analysis of Unsteady Tip Leakage Vortex Cavitation Cloud and Unstable Suction-Side-Perpendicular Cavitating Vortices in an Axial Flow Pump
,”
Int. J. Multiphase Flow
,
77
, pp.
244
259
.10.1016/j.ijmultiphaseflow.2015.09.006
7.
Cao
,
P.
,
Wang
,
Y.
,
Kang
,
C.
,
Li
,
G.
, and
Zhang
,
X.
,
2017
, “
Investigation of the Role of Non-Uniform Suction Flow in the Performance of Water-Jet Pump
,”
Ocean Eng.
,
140
, pp.
258
269
.10.1016/j.oceaneng.2017.05.034
8.
Zhang
,
D.
,
Shi
,
W.
,
van Esch
,
B. P. M.
,
Shi
,
L.
, and
Dubuisson
,
M.
,
2015
, “
Numerical and Experimental Investigation of Tip Leakage Vortex Trajectory and Dynamics in an Axial Flow Pump
,”
Comput. Fluids
,
112
, pp.
61
71
.10.1016/j.compfluid.2015.01.010
9.
Takata
,
H.
, and
Tsukuda
,
Y.
,
1977
, “
Stall Margin Improvement by Casing Treatment—Its Mechanism and Effectiveness
,”
J. Eng. Power
,
99
(
1
), pp.
121
133
.10.1115/1.3446241
10.
Fujita
,
H.
, and
Takata
,
H.
,
1984
, “
A Study on Configurations of Casing Treatment for Axial Flow Compressors
,”
Bull. JSME
,
27
(
230
), pp.
1675
1681
.10.1299/jsme1958.27.1675
11.
Kang
,
D.
,
Arimoto
,
Y.
,
Yonezawa
,
K.
,
Horiguchi
,
H.
,
Kawata
,
Y.
,
Hah
,
C.
, and
Tsujimoto
,
Y.
,
2010
, “
Suppression of Cavitation Instabilities in an Inducer by Circumferential Groove and Explanation of Higher Frequency Components
,”
Int. J. Fluid Mach. Syst.
,
3
(
2
), pp.
137
149
.10.5293/IJFMS.2010.3.2.137
12.
Choi
,
Y.-D.
,
Kurokawa
,
J.
, and
Imamura
,
H.
,
2007
, “
Suppression of Cavitation in Inducers by J-Grooves
,”
ASME J. Fluids Eng.
,
129
(
1
), pp.
15
22
.10.1115/1.2375126
13.
Michael
,
T. J.
,
Schroeder
,
S. D.
, and
Becnel
,
A. J.
,
2008
, “
Design of the ONR AxWJ-2 Axial Flow Water Jet Pump
,” West Bethesda, MD, Report No. NSWCCD-50-TR-2008/066.
14.
Chesnakas
,
C. J.
,
Donnelly
,
M. J.
,
Pfitsch
,
D. W.
,
Becnel
,
A. J.
, and
Schroeder
,
S. D.
,
2009
, “
Performance Evaluation of the ONR Axial Waterjet 2 (AxWJ-2)
,” West Bethesda, MD, Report No. NSWCCD-50-TR-2009/089.
15.
Tan
,
D.
,
Li
,
Y.
,
Chen
,
H.
,
Wilkes
,
I.
, and
Katz
,
J.
,
2015
, “
The Three Dimensional Flow Structure and Turbulence in the Tip Region of an Axial Flow Compressor
,”
ASME
Paper No. GT2015-43385. 10.1115/GT2015-43385
16.
Chen
,
H.
,
Li
,
Y.
,
Doeller
,
N.
,
Koley
,
S. S.
,
Keyser
,
B.
, and
Katz
,
J.
,
2016
, “
Effects of Circumferential Grooves on the Cavitation and Performance of an Axial Waterjet Pump
,”
31st Symposium on Naval Hydrodynamics
, Monterey, CA, Sept. 11–16.
17.
Tan
,
D.
,
Li
,
Y.
,
Wilkes
,
I.
,
Miorini
,
R.
, and
Katz
,
J.
,
2015
, “
Visualization and Time Resolved PIV Measurements of the Flow in the Tip Region of a Subsonic Compressor Rotor
,”
ASME J. Turbomach.
,
137
(
4
), p.
041007
.10.1115/1.4028433
18.
Patil
,
K. R.
,
Tripathi
,
A. D.
,
Pathak
,
G.
, and
Katti
,
S. S.
,
1991
, “
Thermodynamic Properties of Aqueous Electrolyte Solutions. 2. Vapor Pressure of Aqueous Solutions of Sodium Bromide, Sodium Iodide, Potassium Chloride, Potassium Bromide, Potassium Iodide, Rubidium Chloride, Cesium Chloride, Cesium Bromide, Cesium Iodide
,”
J. Chem. Eng. Data
,
36
(
2
), pp.
225
230
.10.1021/je00002a021
19.
Bai
,
K.
, and
Katz
,
J.
,
2014
, “
On the Refractive Index of Sodium Iodide Solutions for Index Matching in PIV
,”
Exp. Fluids
,
55
(
4
), pp.
1
6
.10.1007/s00348-014-1704-x
20.
Wieneke
,
B.
,
2005
, “
Stereo-PIV Using Self-Calibration on Particle Images
,”
Exp. Fluids
,
39
(
2
), pp.
267
280
.10.1007/s00348-005-0962-z
21.
Chen
,
H.
,
Li
,
Y.
,
Tan
,
D.
, and
Katz
,
J.
,
2017
, “
Visualizations of Flow Structures in the Rotor Passage of an Axial Compressor at the Onset of Stall
,”
ASME J. Turbomach.
,
139
(
4
), p.
041008
.10.1115/1.4035076
22.
Roth
,
G. I.
, and
Katz
,
J.
,
2001
, “
Five Techniques for Increasing the Speed and Accuracy of PIV Interrogation
,”
Meas. Sci. Technol.
,
12
(
3
), pp.
238
245
.10.1088/0957-0233/12/3/302
23.
Westerweel
,
J.
, and
Scarano
,
F.
,
2005
, “
Universal Outlier Detection for PIV Data
,”
Exp. Fluids
,
39
(
6
), pp.
1096
1100
.10.1007/s00348-005-0016-6
24.
Yamamoto
,
K.
, and
Tsujimoto
,
Y.
,
2009
, “
Backflow Vortex Cavitation and Its Effects on Cavitation Instabilities
,”
Int. J. Fluid Mach. Syst.
,
2
(
1
), pp.
40
54
.10.5293/IJFMS.2009.2.1.040
25.
Lindau
,
J. W.
,
Boger
,
D. A.
,
Medvitz
,
R. B.
, and
Kunz
,
R. F.
,
2005
, “
Propeller Cavitation Breakdown Analysis
,”
ASME J. Fluids Eng.
,
127
(
5
), pp.
995
1002
.10.1115/1.1988343
26.
Boswell
,
R. J.
,
1971
, “
Design, Cavitation Performance, and Open-Water Performance of a Series of Research Skewed Propellers
,” David W Taylor Naval Ship Research and Development Center, Bethesda, MD, Report No. NSRDC-3339.
27.
Khalid
,
S. A.
,
Khalsa
,
A. S.
,
Waitz
,
I. A.
,
Tan
,
C. S.
,
Greitzer
,
E. M.
,
Cumpsty
,
N. A.
,
Adamczyk
,
J. J.
, and
Marble
,
F. E.
,
1999
, “
Endwall Blockage in Axial Compressors
,”
ASME J. Turbomach.
,
121
(
3
), pp.
499
509
.10.1115/1.2841344
28.
Stauter
,
R. C.
,
1993
, “
Measurement of the Three-Dimensional Tip Region Flowfield in an Axial Compressor
,”
ASME J. Turbomach.
,
115
(
3
), pp.
468
476
.10.1115/1.2929275
29.
Bindon
,
J. P.
,
1989
, “
The Measurement and Formation of Tip Clearance Loss
,”
ASME J. Turbomach.
,
111
(
3
), pp.
257
263
.10.1115/1.3262264
30.
Lyman
,
F. A.
,
1993
, “
On the Conservation of Rothalpy in Turbomachines
,”
ASME J. Turbomach.
,
115
(
3
), pp.
520
525
.10.1115/1.2929282
31.
Lakshminarayana
,
B.
,
1996
,
Fluid Dynamics and Heat Transfer of Turbomachinery
,
Wiley
,
New York
.
32.
Tan
,
D.
,
2015
, “
Common Features in the Structure of Tip Leakage Flows
,” Ph.D. thesis,
The Johns Hopkins University
, Baltimore, MD.
33.
Guinard
,
P.
,
Fuller
,
T.
, and
Acosta
,
A.
,
1953
, “
An Experimental Study of Axial Flow Pump Cavitation
,” California Institute of Technology Hydrodynamics Laboratory, Pasadena, CA, Report No. E-19.3.
34.
Stripling
,
I. R.
, and
Acosta
,
A. J.
,
1962
, “
Cavitation in Turbo Pumps-Part 1
,”
ASME J. Basic Eng.
,
1
(
61
), pp.
1
13
.10.1115/1.3657314
35.
Shen
,
Y.
, and
Dimotakis
,
P. E.
,
1989
, “
The Influence of Surface Cavitation on Hydrodynamic Forces
,”
Proceedings of the 22nd American Towing Tank Conference
, St-John's, NF, Aug. 8–11, pp.
44
53
.
36.
Houghton
,
T.
, and
Day
,
I.
,
2011
, “
Enhancing the Stability of Subsonic Compressors Using Casing Grooves
,”
ASME J. Turbomach.
,
133
(
2
), p.
021007
.10.1115/1.4000569
37.
Schiavello
,
B.
, and
Visser
,
F. C.
,
2009
, “
Pump Cavitation—Various NPSHR Criteria, NPSHA Margins, and Impeller Life Expectancy
,”
Proceedings of the 25th International Pump Users Symposium,
Houston, TX, Feb. 23–26, pp.  
113
143
.