# Interaction between Uneven Cavity Length and Shaft Vibration at the Inception of Synchronous Rotating Cavitation

Authors: Y. Yoshida; Y. Kazami; K. Nagaura; M. Shimagaki; Y. Iga; T. Ikohagi
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

Asymmetric cavitation is known as one type of the sources of cavitation induced vibration in turbomachinery. Cavity lengths are unequal on each blade under condition of synchronous rotating cavitation, which causes synchronous shaft vibration. To investigate the relationship of the cavity length, fluid force, and shaft vibration in a cavitating inducer with three blades, we observed the unevenness of cavity length at the inception of synchronous rotating cavitation. The fluid force generated by the unevenness of the cavity length was found to grow exponentially, and the amplitude of shaft vibration was observed to increase exponentially. These experimental results indicate that the synchronous shaft vibration due to synchronous rotating cavitation is like selfexcited vibrations arising from the coupling between cavitation instability and rotordynamics.


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## Body

## 1. Introduction

For a turbopump, an inducer is placed upstream of the
main impeller to obtain high suction performance, and thus it is one of the key
components of a liquid propellant rocket engine. Unfortunately, under some
operating conditions, cavitation instabilities such as rotating cavitation or
cavitation surge are observed [1, 2]. In
addition to these cavitation instabilities, synchronous rotating cavitation
with uneven cavities on each blade is often observed in three-bladed inducers
at relatively low cavitation numbers. Synchronous rotating cavitation results in severe synchronous shaft vibration and causes degradation of suction performance due to an asymmetric cavity
pattern [2–5].Previous studies on the relationship between synchronous
rotating cavitation and shaft vibration have been conducted by Hashimoto et al.
[6] and Kobayashi [5], who found that there is a significant correlation
between the amplitude of the shaft vibration and the onset of synchronous
rotating cavitation. In addition, Fujii et al. [7] examined such an unsteady
phenomenon at lower cavitation numbers through measurements of the pressure fluctuation
and blade stress fluctuation in which the pattern of the cavity on each blade
changes irregularly while the overall asymmetric pattern is maintained.The interaction between
uneven cavity length and shaft vibration under synchronous rotating cavitation,
however, is poorly understood. To clarify the interaction and the origin of
this complicated instability, we investigate the growth of the cavity and the
shaft vibration in detail with a focus on the inception of synchronous rotating
cavitation. We also estimated the fluid force on the inducer based on
measurement of unsteady pressure distributions and compared it with the
amplitude of the shaft vibration.


## 2. Experimental Facility and Apparatus

### 2.1. Experimental Facility

The experiment was conducted in the Cryogenic Inducer
Test Facility (CITF) at Kakuda Space Center
of the Japan Aerospace Exploration Agency (JAXA). Figure1 shows a
schematic of the test facility. This
facility has two tanks, one upstream (run-tank) and one downstream (catch-tank). The temperature of liquid
nitrogen as a working fluid is 74 K measured at the inducer inlet.Figure 1
Schematic diagram of the cryogenic inducer test facility.The test inducer has three blades with sweep cutback
at the leading edge. Its solidity is about 2.1. The diameter is about 160 mm,
and the blade geometry is modified helix. The
inducer is driven by an electric motor with an epicyclic gear at a rotation speed of 18 300 rpm, equal to that of an
actual turbopump, and the flow rate is set to remain constant atQ/Qd=1.06. The rotor is supported by self-lubricated angular
contact ball bearings arranged in two spring-preloaded pairs at the inducer
side and motor side.


### 2.2. Experimental Apparatus

To investigate the behavior of cavitation, it is
better to observe the cavitation directly. However, it is difficult to visualize
the cavitation at the inducer in cryogenic fluid, that is, nitrogen. Therefore,
to estimate the cavitating state of the inducer indirectly, eight pressure
sensors (pos. 1-pos. 8) are installed
along the blade from the leading edge to the trailing edge on the casing as
shown in Figures2 and 3
[8].Figure 2
Schematic diagram of the test section showing the locations of pressure sensors and shaft displacement sensors.Figure 3
Development view of the inducer showing locations of pressure sensors along the inducer blade.Pressure sensors
detect the cavitation near the blade tip, that is, tip leakage vortex
cavitation. However, these sensors are unable to detect the cavitation in the
backflow upstream of the inducer, that is, backflow vortex cavitation. From
previous studies, it is well known that the cavitation near the blade tip
considerably affects suction performance and cavitation instability. In
addition, the blade cavitation at mid-span seldom appears. Thus, it is
reasonable that the cavity length near the blade tip is regarded as an “indication”
of cavitation.Using these
pressure sensors at each location, contour plots describing the cavitation
state in the blade channel are obtained. Figure4 was drawn by rearrangement of
the wave forms of all the sensors.
When a cavitation occurs, the pressure reaches at the saturated vapor pressure as blue region in
Figure 4. Therefore, the blue area in these figures shows “estimated cavity
region.” Using these figures at each cavitation number, the cavity length can
be observed by this indirect visualization. We defined the length from the leading edge to
trailing edge of the estimated cavity region as “cavity length
(Lc).”Figure 4
Example of estimated cavity region obtained from the measured pressure distribution.Two shaft  displacement sensors forX-Y directions are
installed at the backend of the rotor as shown in
Figure 2. The sensor for
shaft vibration should be installed near the inducer. However, there is no
space around the inducer due to the limitation of the pump structure.


## 2.1. Experimental Facility

The experiment was conducted in the Cryogenic Inducer
Test Facility (CITF) at Kakuda Space Center
of the Japan Aerospace Exploration Agency (JAXA). Figure1 shows a
schematic of the test facility. This
facility has two tanks, one upstream (run-tank) and one downstream (catch-tank). The temperature of liquid
nitrogen as a working fluid is 74 K measured at the inducer inlet.Figure 1
Schematic diagram of the cryogenic inducer test facility.The test inducer has three blades with sweep cutback
at the leading edge. Its solidity is about 2.1. The diameter is about 160 mm,
and the blade geometry is modified helix. The
inducer is driven by an electric motor with an epicyclic gear at a rotation speed of 18 300 rpm, equal to that of an
actual turbopump, and the flow rate is set to remain constant atQ/Qd=1.06. The rotor is supported by self-lubricated angular
contact ball bearings arranged in two spring-preloaded pairs at the inducer
side and motor side.


## 2.2. Experimental Apparatus

To investigate the behavior of cavitation, it is
better to observe the cavitation directly. However, it is difficult to visualize
the cavitation at the inducer in cryogenic fluid, that is, nitrogen. Therefore,
to estimate the cavitating state of the inducer indirectly, eight pressure
sensors (pos. 1-pos. 8) are installed
along the blade from the leading edge to the trailing edge on the casing as
shown in Figures2 and 3
[8].Figure 2
Schematic diagram of the test section showing the locations of pressure sensors and shaft displacement sensors.Figure 3
Development view of the inducer showing locations of pressure sensors along the inducer blade.Pressure sensors
detect the cavitation near the blade tip, that is, tip leakage vortex
cavitation. However, these sensors are unable to detect the cavitation in the
backflow upstream of the inducer, that is, backflow vortex cavitation. From
previous studies, it is well known that the cavitation near the blade tip
considerably affects suction performance and cavitation instability. In
addition, the blade cavitation at mid-span seldom appears. Thus, it is
reasonable that the cavity length near the blade tip is regarded as an “indication”
of cavitation.Using these
pressure sensors at each location, contour plots describing the cavitation
state in the blade channel are obtained. Figure4 was drawn by rearrangement of
the wave forms of all the sensors.
When a cavitation occurs, the pressure reaches at the saturated vapor pressure as blue region in
Figure 4. Therefore, the blue area in these figures shows “estimated cavity
region.” Using these figures at each cavitation number, the cavity length can
be observed by this indirect visualization. We defined the length from the leading edge to
trailing edge of the estimated cavity region as “cavity length
(Lc).”Figure 4
Example of estimated cavity region obtained from the measured pressure distribution.Two shaft  displacement sensors forX-Y directions are
installed at the backend of the rotor as shown in
Figure 2. The sensor for
shaft vibration should be installed near the inducer. However, there is no
space around the inducer due to the limitation of the pump structure.


## 3. Rotating Cavitation

Rotating cavitation is cavitation instability, in which the cavity lengths
become uneven on the three blades. There are two types of such cavitation,
namely, “super-synchronous” and “synchronous” rotating cavitation. In the first
type, as shown in Figure5(a), the pattern of cavity unevenness propagates from
a blade to an adjacent blade, while in the latter type as shown in Figure 5(b),
the cavity remains on synchronous rotation. The super-synchronous rotating
cavitation is an unsteady phenomenon in which unevenness of the cavity propagates
at a speed 1.1-1.2 times higher than that of inducer rotation. On the other
hand, synchronous rotating cavitation is a steady asymmetric phenomenon in
which unevenness of the cavity does not propagate.(a) Examples of estimated cavity region under super-synchronous rotating cavitation for every rotation. (b) Examples of estimated cavity region under synchronous rotating cavitation for every rotation.
(a)(b)


## 4. Experimental Risults

### 4.1. Head Coefficient, Cavity Length, and Shaft Vibration

In a previous work [9], the relation of the cavitation
performance, cavity length, and shaft vibration has been examined. The diamond-shaped
symbols (⋄) in Figure 6
show variations of nondimensional cavity length (Lc/h) normalized by the blade
spacing (h) for each channel (i.e.,
three channels for the three-bladed inducer). The circles (o) in Figure 6
show variations of the inducer head coefficient (ψ/ψ0) normalized by the reference
head coefficient (ψ0). In
addition, the triangles (▴ and  Δ) in Figure 6 show the amplitude
of the fluctuating cavity length, and the crosses
(×) show the time averaged cavity length under
the super-synchronous rotating cavitation. The average cavity
length increases with decreasing of cavitation number. The middle panel of Figure 6 shows shaft vibration detected by the displacement sensors. The crosses (×) in the bottom
of Figure 6 show the propagation speed ratio
(ω/Ω) of rotating
cavitation.Figure 6
Cavitation performances, cavity length of each channel (top), shaft vibration (middle), and variations of frequency of shaft vibration (bottom).As the cavitation number (σ) decreases,
super-synchronous rotating cavitation occurs at a cavity length of about Lc/h=0.5(atσ/σ0=0.85). Under such cavitation, cavity length fluctuates on three blades with a certain
phase lag, as shown in Figure 7(a). As a result, it can be seen that the unevenness
of cavity length propagates in the same direction as the rotor with a speed
ratio of ω/Ω=1.1-1.2, and that super-synchronous
shaft vibration (ω/Ω=1.1-1.2) is generated due to the unevenness
of the cavity length, as shown in the bottom of Figure 6.(a) Fluctuations of cavity length under super-synchronous rotating
cavitation. (b) Fluctuations of cavity length under synchronous
rotating cavitation.
(a)(b)In contrast, synchronous rotating cavitation occurs at
a cavity length ofLc/h≅0.9(atσ/σ0=0.55)
and disappears at a cavity length of Lc/h≅1.5. Therefore,
the trigger of super-synchronous/synchronous rotating cavitation must be
governed only by the cavity length, which is inferred to be one type of fluid
dynamic instability with cavitation. This result agrees with the theoretical
analysis of the alternate blade cavitation calculated by Horiguchi
et al. [10].
They reported that the alternate blade cavitation commences when the cavity
length approaches Lc/h≅0.65 in an even
number of blade cascades. Under synchronous rotating cavitation, the unevenness
of cavity length appears clearly as three lengths (short, long, and long) in
Figure 7(b). The cavity pattern remains uneven on
each blade. Unevenness of cavity length under synchronous rotating cavitation is greater
than that under super-synchronous
rotating cavitation.In addition, the circles (∘) in Figure 6
show the variations of the inducer head. Within the range of the occurrence of
super-synchronous rotating cavitation, the inducer head is not affected by this
cavitation instability. However, the inducer head temporarily decreases within the range of
the occurrence of synchronous rotating cavitation. After breaking away from the
synchronous rotating cavitation, cavity lengths become completely equal near Lc/h=1.5, and then the head break
begins gradually. This means that the inducer head is strongly dependent on the
cavity length.


### 4.2. Behavior of
Synchronous Rotating Cavitation

As mentioned above, the synchronous rotating cavitation
causes inducer head loss and synchronous shaft vibration. We now focus on the
growth of the cavity length at the inception of synchronous rotating cavitation
in detail to examine its mechanism.Figure8 shows the behavior of uneven cavity length at
the inception of synchronous rotating cavitation. This figure corresponds to
the region bounded by red dashed-line in Figure 6. The diamond-shaped symbols (⋄) in Figure 8
show variations of nondimensional cavity length (Lc/h) of each channel.
To
estimate the unbalanced fluid force acted on the inducer, the pressure
distribution measured by pressure sensors on the casing is integrated.
Therefore, this estimated fluid force is regarded as only an indication of
fluid force affecting the inducer. The
triangles (Δ) in Figure 8 show the fluid force (F) normalized by the reference fluid
force (F0). Figure 9 shows
estimated cavity region near inception of synchronous rotating cavitation. In
Figure 9, the phase of fluid force is indicated by arrows. Under synchronous rotating
cavitation, the phase of fluid force is the direction of the channel in which
short cavity occurs. The bottom of Figure 8 shows the
variation of the shaft vibration during the same time.Figure 8
Cavity length of each channel, fluid force (top), the amplitude of shaft vibration (middle), and the Lissajous  figures showing the orbit of shaft vibration (bottom).Figures of estimated cavity region near inception of synchronous rotating cavitation (SRC).
(a)(b)(c)Before 48.05 seconds,
cavity length, fluid force, and shaft vibration fluctuate slightly, so it
should be considered that the synchronous rotating cavitation does not yet
occur. After 48.05 seconds, the unevenness of cavity length commences and fluid
force also begins to increase, indicating that the onset of synchronous
rotating cavitation as cavitation instability is at 48.05 seconds. After 48.22 seconds,
the degree of unevenness of cavity length and the unbalanced fluid force is rapidly stop increasing, indicating
that the synchronous rotating cavitation develops fully at 48.22 seconds. The rapid
growth time (from 48.05 seconds to 48.22 seconds) of the cavity unevenness and
the shaft vibration is 0.17 seconds, that is, about fifty rotations.Under the steady
condition of the synchronous rotating cavitation (after 48.22 seconds), cavity
lengths of channels 1 and 2 are stable and longer(Lc/h>1.5), but that of channel 3 is unstable and shorter caused by interaction
with the longer cavity in channels 1 and 2 (Lc/h<0.8).In the bottom of Figure8, the Lissajous figures show the orbit of the
shaft vibration from the measurement of the X-Y displacement sensors. The
amplitude is small, and the orbit is random prior to the synchronous rotating
cavitation (before 48.05 seconds), while after 48.22 seconds, the orbit of the
shaft vibration can be visualized as a large limit circle under the full
developed synchronous rotating cavitation. It seems that the shaft vibration
caused by synchronous rotating cavitation is like self-excited vibrations.


## 4.1. Head Coefficient, Cavity Length, and Shaft Vibration

In a previous work [9], the relation of the cavitation
performance, cavity length, and shaft vibration has been examined. The diamond-shaped
symbols (⋄) in Figure 6
show variations of nondimensional cavity length (Lc/h) normalized by the blade
spacing (h) for each channel (i.e.,
three channels for the three-bladed inducer). The circles (o) in Figure 6
show variations of the inducer head coefficient (ψ/ψ0) normalized by the reference
head coefficient (ψ0). In
addition, the triangles (▴ and  Δ) in Figure 6 show the amplitude
of the fluctuating cavity length, and the crosses
(×) show the time averaged cavity length under
the super-synchronous rotating cavitation. The average cavity
length increases with decreasing of cavitation number. The middle panel of Figure 6 shows shaft vibration detected by the displacement sensors. The crosses (×) in the bottom
of Figure 6 show the propagation speed ratio
(ω/Ω) of rotating
cavitation.Figure 6
Cavitation performances, cavity length of each channel (top), shaft vibration (middle), and variations of frequency of shaft vibration (bottom).As the cavitation number (σ) decreases,
super-synchronous rotating cavitation occurs at a cavity length of about Lc/h=0.5(atσ/σ0=0.85). Under such cavitation, cavity length fluctuates on three blades with a certain
phase lag, as shown in Figure 7(a). As a result, it can be seen that the unevenness
of cavity length propagates in the same direction as the rotor with a speed
ratio of ω/Ω=1.1-1.2, and that super-synchronous
shaft vibration (ω/Ω=1.1-1.2) is generated due to the unevenness
of the cavity length, as shown in the bottom of Figure 6.(a) Fluctuations of cavity length under super-synchronous rotating
cavitation. (b) Fluctuations of cavity length under synchronous
rotating cavitation.
(a)(b)In contrast, synchronous rotating cavitation occurs at
a cavity length ofLc/h≅0.9(atσ/σ0=0.55)
and disappears at a cavity length of Lc/h≅1.5. Therefore,
the trigger of super-synchronous/synchronous rotating cavitation must be
governed only by the cavity length, which is inferred to be one type of fluid
dynamic instability with cavitation. This result agrees with the theoretical
analysis of the alternate blade cavitation calculated by Horiguchi
et al. [10].
They reported that the alternate blade cavitation commences when the cavity
length approaches Lc/h≅0.65 in an even
number of blade cascades. Under synchronous rotating cavitation, the unevenness
of cavity length appears clearly as three lengths (short, long, and long) in
Figure 7(b). The cavity pattern remains uneven on
each blade. Unevenness of cavity length under synchronous rotating cavitation is greater
than that under super-synchronous
rotating cavitation.In addition, the circles (∘) in Figure 6
show the variations of the inducer head. Within the range of the occurrence of
super-synchronous rotating cavitation, the inducer head is not affected by this
cavitation instability. However, the inducer head temporarily decreases within the range of
the occurrence of synchronous rotating cavitation. After breaking away from the
synchronous rotating cavitation, cavity lengths become completely equal near Lc/h=1.5, and then the head break
begins gradually. This means that the inducer head is strongly dependent on the
cavity length.


## 4.2. Behavior of
Synchronous Rotating Cavitation

As mentioned above, the synchronous rotating cavitation
causes inducer head loss and synchronous shaft vibration. We now focus on the
growth of the cavity length at the inception of synchronous rotating cavitation
in detail to examine its mechanism.Figure8 shows the behavior of uneven cavity length at
the inception of synchronous rotating cavitation. This figure corresponds to
the region bounded by red dashed-line in Figure 6. The diamond-shaped symbols (⋄) in Figure 8
show variations of nondimensional cavity length (Lc/h) of each channel.
To
estimate the unbalanced fluid force acted on the inducer, the pressure
distribution measured by pressure sensors on the casing is integrated.
Therefore, this estimated fluid force is regarded as only an indication of
fluid force affecting the inducer. The
triangles (Δ) in Figure 8 show the fluid force (F) normalized by the reference fluid
force (F0). Figure 9 shows
estimated cavity region near inception of synchronous rotating cavitation. In
Figure 9, the phase of fluid force is indicated by arrows. Under synchronous rotating
cavitation, the phase of fluid force is the direction of the channel in which
short cavity occurs. The bottom of Figure 8 shows the
variation of the shaft vibration during the same time.Figure 8
Cavity length of each channel, fluid force (top), the amplitude of shaft vibration (middle), and the Lissajous  figures showing the orbit of shaft vibration (bottom).Figures of estimated cavity region near inception of synchronous rotating cavitation (SRC).
(a)(b)(c)Before 48.05 seconds,
cavity length, fluid force, and shaft vibration fluctuate slightly, so it
should be considered that the synchronous rotating cavitation does not yet
occur. After 48.05 seconds, the unevenness of cavity length commences and fluid
force also begins to increase, indicating that the onset of synchronous
rotating cavitation as cavitation instability is at 48.05 seconds. After 48.22 seconds,
the degree of unevenness of cavity length and the unbalanced fluid force is rapidly stop increasing, indicating
that the synchronous rotating cavitation develops fully at 48.22 seconds. The rapid
growth time (from 48.05 seconds to 48.22 seconds) of the cavity unevenness and
the shaft vibration is 0.17 seconds, that is, about fifty rotations.Under the steady
condition of the synchronous rotating cavitation (after 48.22 seconds), cavity
lengths of channels 1 and 2 are stable and longer(Lc/h>1.5), but that of channel 3 is unstable and shorter caused by interaction
with the longer cavity in channels 1 and 2 (Lc/h<0.8).In the bottom of Figure8, the Lissajous figures show the orbit of the
shaft vibration from the measurement of the X-Y displacement sensors. The
amplitude is small, and the orbit is random prior to the synchronous rotating
cavitation (before 48.05 seconds), while after 48.22 seconds, the orbit of the
shaft vibration can be visualized as a large limit circle under the full
developed synchronous rotating cavitation. It seems that the shaft vibration
caused by synchronous rotating cavitation is like self-excited vibrations.


## 5. Discussion

### 5.1. Growth of
Synchronous Rotating Cavitation

During the growth of the synchronous rotating cavitation in Figure8, it is
observed that the fluid force caused by
the unevenness of the cavity length develops exponentially, and that the amplitude of shaft vibration also increases exponentially.
The exponential fit curves along the fluid force and the amplitude of shaft
vibration are shown in Figure 8 within 48.05–48.22 seconds. To
compare the growth behavior of the fluid force with that of the shaft
vibration, logarithmic decrements (δ) are estimated
by  (1)  using these fit curves:(1)ε=ε*×e−δε(t−tinc)/T×cos(Ω(t−tinc)),F=F*×e−δF(t−tinc)/T.   The logarithmic decrement of the shaft vibration,
δε, is  −0.03, and that of the fluid force, δF, is  −0.046. Both logarithm decrements are negative
with almost equal values.Figure10 shows the amplitude of shaft vibration as a
function of the fluid force during the inception of the synchronous rotating
cavitation. This figure presents that the amplitude of shaft vibration increases
linearly with fluid force. When the amplitude of synchronous shaft vibration
increases, the clearance gap between the tips of the rotating inducer and the
casing wall varies at each blade. The degree of the tip leakage cavitation
strongly depends on the tip clearance [1]. Hence, it is considered
that this synchronous shaft vibration is acted upon by the synchronous fluid
force through the nonuniform tip leakage cavitation on each blade, that is, synchronous rotating cavitation.Figure 10
Increase of the amplitude of shaft vibration as a function of the fluid force.From these results, this synchronous shaft vibration is indicated to be like
self-excited vibrations with a coupling between the synchronous rotating
cavitation and the rotor vibration, although rotordynamic fluid forces caused
by cavitation have long been known to be the self-sustaining force of
rotordynamics [3].


### 5.2. Hydrodynamics-Rotordynamics
Coupling Mechanism

From the relation of
cavity length, fluid force and shaft vibration mentioned above, the estimated mechanism
of the development of synchronous rotating cavitation is presented in Figure11. First, when the cavity length reaches
a critical length, cavitation instability occurs as flow field dynamics
[10]. The equal cavity length of the three blades becomes
uneven, that is, this is the start of the synchronous rotating cavitation.
Nonuniform pressure and velocity distribution around the inducer caused by the
uneven cavity length generate the fluid force acting on the inducer. Second,
this fluid force results in rotor displacement and whirl motion in rotordynamics.
This whirl motion creates nonuniform tip clearance, which results in the acceleration
of cavity unevenness associated with leakage flow in the nonuniform tip clearance.
As this process is repeated, synchronous rotating cavitation grows. This hydrodynamics
(with cavitation)—rotordynamics coupling mechanism in the inducer is similar to
the aerodynamic-rotordynamic coupling mechanism in an axial compressor
[11] with regard to the effect of nonuniform tip clearance.
This mechanism is shown as a picture in Figure 12.Figure 11
Hydrodynamics (cavitation)—rotor-dynamics coupling mechanism.Figure 12
Figure of hydrodynamics (cavitation)—rotordynamics coupling mechanism.On the other hand, Bhattacharyya et al. [12] experimentally
investigated the rotordynamic forces in a whirling cavitating inducer,
and d’Agostino et al.
[13] also analyzed it. Their results showed the occurrence of large destabilizing
peaks in the tangential force acting on the whirl orbit for a positive
frequency ratio (ω/Ω≅0.2) at a certain cavitation condition. However, there have
been no reported results concerning the whirl frequency ratio at ω/Ω=1.0, that is, synchronous rotating cavitation.
Therefore, to more fully clarify the mechanism of synchronous rotating
cavitation, an extension of those experiments is underway using Ebara Rotordynamic
Test Stand (EBARTS) [14], which will add to our understanding
of the hydrodynamics (with cavitation)-rotordynamics coupling mechanism.


## 5.1. Growth of
Synchronous Rotating Cavitation

During the growth of the synchronous rotating cavitation in Figure8, it is
observed that the fluid force caused by
the unevenness of the cavity length develops exponentially, and that the amplitude of shaft vibration also increases exponentially.
The exponential fit curves along the fluid force and the amplitude of shaft
vibration are shown in Figure 8 within 48.05–48.22 seconds. To
compare the growth behavior of the fluid force with that of the shaft
vibration, logarithmic decrements (δ) are estimated
by  (1)  using these fit curves:(1)ε=ε*×e−δε(t−tinc)/T×cos(Ω(t−tinc)),F=F*×e−δF(t−tinc)/T.   The logarithmic decrement of the shaft vibration,
δε, is  −0.03, and that of the fluid force, δF, is  −0.046. Both logarithm decrements are negative
with almost equal values.Figure10 shows the amplitude of shaft vibration as a
function of the fluid force during the inception of the synchronous rotating
cavitation. This figure presents that the amplitude of shaft vibration increases
linearly with fluid force. When the amplitude of synchronous shaft vibration
increases, the clearance gap between the tips of the rotating inducer and the
casing wall varies at each blade. The degree of the tip leakage cavitation
strongly depends on the tip clearance [1]. Hence, it is considered
that this synchronous shaft vibration is acted upon by the synchronous fluid
force through the nonuniform tip leakage cavitation on each blade, that is, synchronous rotating cavitation.Figure 10
Increase of the amplitude of shaft vibration as a function of the fluid force.From these results, this synchronous shaft vibration is indicated to be like
self-excited vibrations with a coupling between the synchronous rotating
cavitation and the rotor vibration, although rotordynamic fluid forces caused
by cavitation have long been known to be the self-sustaining force of
rotordynamics [3].


## 5.2. Hydrodynamics-Rotordynamics
Coupling Mechanism

From the relation of
cavity length, fluid force and shaft vibration mentioned above, the estimated mechanism
of the development of synchronous rotating cavitation is presented in Figure11. First, when the cavity length reaches
a critical length, cavitation instability occurs as flow field dynamics
[10]. The equal cavity length of the three blades becomes
uneven, that is, this is the start of the synchronous rotating cavitation.
Nonuniform pressure and velocity distribution around the inducer caused by the
uneven cavity length generate the fluid force acting on the inducer. Second,
this fluid force results in rotor displacement and whirl motion in rotordynamics.
This whirl motion creates nonuniform tip clearance, which results in the acceleration
of cavity unevenness associated with leakage flow in the nonuniform tip clearance.
As this process is repeated, synchronous rotating cavitation grows. This hydrodynamics
(with cavitation)—rotordynamics coupling mechanism in the inducer is similar to
the aerodynamic-rotordynamic coupling mechanism in an axial compressor
[11] with regard to the effect of nonuniform tip clearance.
This mechanism is shown as a picture in Figure 12.Figure 11
Hydrodynamics (cavitation)—rotor-dynamics coupling mechanism.Figure 12
Figure of hydrodynamics (cavitation)—rotordynamics coupling mechanism.On the other hand, Bhattacharyya et al. [12] experimentally
investigated the rotordynamic forces in a whirling cavitating inducer,
and d’Agostino et al.
[13] also analyzed it. Their results showed the occurrence of large destabilizing
peaks in the tangential force acting on the whirl orbit for a positive
frequency ratio (ω/Ω≅0.2) at a certain cavitation condition. However, there have
been no reported results concerning the whirl frequency ratio at ω/Ω=1.0, that is, synchronous rotating cavitation.
Therefore, to more fully clarify the mechanism of synchronous rotating
cavitation, an extension of those experiments is underway using Ebara Rotordynamic
Test Stand (EBARTS) [14], which will add to our understanding
of the hydrodynamics (with cavitation)-rotordynamics coupling mechanism.


## 6. Conclusions

By experimental
investigation of
the relationship of the cavity length, fluid force,
and shaft vibration at the inception of synchronous rotating cavitation, the
following points were clarified.(1)
The growth time
of the cavity unevenness and the shaft vibration at the inception of
synchronous rotating cavitation is only fifty rotations.(2)
Fluid force
caused by the unevenness of the cavity length develops exponentially, as does the amplitude of shaft vibration.(3)
The logarithm decrement
of the amplitude of shaft vibration is almost equal to that of the fluid force.(4)
The synchronous
shaft vibration due to the synchronous rotating cavitation seems to be like
self-excited vibrations, which is coupled hydrodynamics (cavitation instability)
with rotordynamics in Figure11.



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*Source: 218978-2008-11-03.xml*