|
Items in cart [0] |
Questions? Call 888-624-8373 |
| Copyright © 2009. National Academy of Sciences. All rights reserved. Terms of Use and Privacy Statement |
Below are the first 10 and last 10 pages of uncorrected machine-read text (when available) of this chapter, followed by the top 30 algorithmically extracted key phrases from the chapter as a whole.
Intended to provide our own search engines and external engines with highly rich, chapter-representative searchable text on the opening pages of each chapter.
Because it is UNCORRECTED material, please consider the following text as a useful but insufficient proxy for the authoritative book pages.
Do not use for reproduction, copying, pasting, or reading; exclusively for search engines.
OCR for page 599
Instability of Partial Cavitation:
A Numerical/Experimental Approach
R.E.A.. A ndt', C.C.S. Song', M. Kjeldsen2., J. He', A. Keller
('Saint Anthony Falls Laboratory, University of Minnesota, ~Norwegian University of
Science and Technology, 3Technical University of Munich)
AflSTltACT
Sheet es itation md fhe t msiti m to cloud
cavitstion on hyd of oils md marme propellers results m
s highly umtable flow thst cm ind ce significmt
fluct stions in lift, thmst md tcrque in ord r to gsin s
better under t mdi g of fhe complex physics mvolved,
m i teg sted mmmericsFexperimental ir~stigstion wss
carried out A D NACA OO I S hyd of oil was selected for
tudy, be mse of its previous use by several
mve tigstors around fhe world The simubtion medhod
ology is based on krge eddy simuhtion LES), usmg s
barohopic phase mod I to couple fhe contimity md
mome tum qusti ms The c mpl mentary experiments
were carried st two different scales in two differe t wster
tum is Tests st fhe Ssmt A fhony Falls Lsb omtory
(SAFL) w re carried out m s I 9 cm square wster t m I
md s geom h ically scaled up series of te ts w ~e carried
out m th 30 cm sq me wster tmm I st fhe
Versuchsmstalt fur Wasserbm (VAO) m Obemach,
Germ my The te ts were designed to complement each
other md to capitali e on the sp cisl fetmes of each
facilit
INTRODUCTION
Marme propellers md hyd of oils must of
operste m fhe cavitstmg regime Various typ s of cavi
tstion c m be found in p~actice, mcludmg bubble
es itation, shet ca vitation, cloud cavitstion, vortex tube
cavitstion md vertex sh et cavitation, dep ndmg upon
how the low pressure regions me genemted b pite of
considemble ~esearch, fhe~e are still m my festmes of th
problem thct have not ben properly e plored For
example, mcepti m st dies me based m f lly wetted fl w
properties, i e pressme dish ibuti m, t rbulff~ce level et
in fhe absence of cavitstion On the other h md,
chssical mod is of developed cavitation consid r mly
cavitstion mmmber as fhe primary varisble Whst has not
been given sd quste sttention is s class of partislly
cavitating flows in which thee is on int roction
betwen Jluid turbulence and cavitohon For enample,
vort :x genemtion st the h sill g edge of shet cavitstion
is s m mifestati m of fne cavitation itself Ve et sl, 1997)
This is m import mt fmdmg since t rbulff~ce is nommally
sttributed to bemg s factor m fhe incepti m process, but
es itation ss s mech ml m for t rbulence generstion has
beff~ given s mt sttention
Cuvitation is slso k ow to produce sir bubbles
due to incondff~sible gas commg out of solution m low
pressme (supersstmsted regi ms of fhe fl w Th
production of bubbly flows in hyd mlic equipme t m
have insidious effects on th stability of op rstion md
on vibmtioa There me s variety of ~efe~ences in th
literstme to th i tenehtion betweff~ cavitation per
formance md dissolved sir dstmg back ss early as 50
years sgo How ver, s quantitoLve und r tmdmg of
the intenektion betwen dissolved gas md cavitstion
phffmmerur is still bey md our g s p
A particukr ly import mt fcrm of cavitation from
s techmical pomt of vi w is sttach d es itation on th
smface of liftmg su faces At typical mgles of sttack
this takes the fcrm of s shet, of ff~ terminsted st th
t~ailmg edge by s highly dynamic fomm of cloud
es itation Vortex cavitstion is often observed m fhe
cloud which is msed by vo ticity shed i to fhe flow
field These cavitstmg micro shuct res me highy
energetic md me re ponsible fcr signifi mt levels of
noise md erosioa Lsbomtory experimentation indicstes
thst s variety of cavitstmg fl w pstterm are possible
wifhm fhe sigma~ mgle of sttsck (. · ) pkme ~jeldsen, et
a1, 1999)
In spite of m my excellent st dies, fhe detsils
concernmg the t msition of sheet cavitation to cloud
es itation is till not understood From s d sign poi t of
view, cavitati g flows must be modeled over s given
perfcrmance envelope m fhe · · plane m ord r to
OCR for page 600
accurately predict performance at off-design conditions
and to assess the potential for noise and erosion. This
requirement is still far from being realized at the present
time. For example, it is well known that the modeling of
partial, time-averaged cavities is not simple, due to the
inverse character of the flow representation in the
vicinity of the cavity and its wake. In addition, partial
cavity models cannot explain the breakup of sheet
cavitation at the trailing edge into detached cavitation
clouds. The process is inherently unsteady even for
steady free stream conditions. Within a certain envelope
of and the process is also periodic (Kjeldsen, et
al,l999~. This creates a modulation of the trailing cloud
cavitation that is highly erosive and very noisy (Arndt
et al,l9974.
Although considerable research has gone into
avoiding cavitation, little effort has been made to
understand the complex physics associated with
operation in the partially cavitating regime. The
occurrence of sheet cavitation and the transition to
cloud cavitation results in a highly unstable flow that
can induce significant fluctuations in lift, thrust and
torque. In order to gain a better understanding of the
complex physics involved, an integrated
numerical/experimental approach was used. A 2D NACA
0015 hydrofoil was selected for study, because of its
previous use by several investigators around the world.
The simulation methodology is based on large eddy
simulation (LES), using a barotropic phase model to
couple the continuity and momentum equations. These
simulations were carried out in an interactive way with
the experimental efforts. For example, early simulations
indicated that pressure side cavitation can occur under
special conditions. This was experimentally confirmed
and has led to an exhaustive investigation of cavitation
induced lift oscillations.
The LES approach was adopted to resolve space
wise variation of the flow due to large scale eddies. The
weakly compressible flow approach proposed by Song and
Yuan (1988) is used to resolve the rapidly changing flows
in the liquid phase and, at the same time, to speed up the
computation as compared with a typical incompressible
flow approach. To overcome the difficulty presented by
the unsteady and multiple free surfaces, Kubota et al
(1992) proposed a "bubble two-phase flow model" based
on an assumption that the fluid consists of a bubble-water
mixture and the bubble volume change is governed by a
modified Rayleigh-Plesset equation. More recently Song
et al (1997) assumed the fluid to be barotropic and the
density is a continues function of pressure in the entire
liquid-vapor phase.
The pioneering works of Helmholtz (1868) and
Kirchhoff (1869) assumed the cavitating flow to be a
steady state free-surface potential flow of the ideal fluid.
They completely ignored the dynamics of the gas flow
and neglected the viscosity and com-pressibility of the
liquid phase. An obvious penalty of neglecting viscosity
is the inability to simulate flows having strong velocity
gradient. Less known fact is that the compressibility
effect is important if pressure changes rapidly with time,
even when Mach number is very small (see Song, 19964.
By ignoring gas dy-namics, the numerical treatment of
free surface boun-dary may become untenable for highly
unsteady and multiple cavity cases. Unsteady cavity
flows about hydrofoils are time wise and space wise
highly variable two-phase flows, making it necessary to
account for compressibility, viscosity as well as the gas
dynamics.
The experiments were carried at two different
scales in two different water tunnels, at the Saint
Anthony Falls Laboratory (SAFL) and at the Versuch-
sanstalt fur Wasserbau (VAO) in Obernach, Germany.
The tests were designed to complement each other and
to capitalize on the special features of each facility.
THEORETICAL CONSIDERATIONS
Compressibility boundary layer theory
The importance of compressibility in small
Mach number flows has long been overlooked. It is
generally known that the compressibility effect is
negligible if the flow is steady and Mach number is
small, M 0.2. However, Song (1996) has shown that the
compressibility can't be ignored if the flow is highly
unsteady even when M is very small. The equation of
state of water in liquid phase can be accurately
represented by the following equation for a very wide
range of pressure change.
2
p pa aO ~
o) (1)
In the above equation "a" is the sound speed and the
subscript "o" represents a reference condition. By
eliminating from the equation of continuity using Eq. 1
the following equation is obtained.
V p ~ V O
t 1 O (2)
OCR for page 601
The second term of Eq.2 is proportional to M2 and
negligible when M is small. But the first term can by
neglected if the pressure changes rapidly with time. This
situation is analogous to the case that some viscous
terms in the equation of motion need to be retained in
viscous boundary layers of high Reynolds number
flows. The following equation of continuity for weakly
compressible flow is obtained by dropping the second
term:
t
Oat V o (3)
For an incompressible fluid aO and the equation
reduces to the following classic equation.
V 0 (43
Eq.3 is valid as long as M is small but Eq.4 is valid only
when M is small and the flow is not highly unsteady.
Using the impulsively started flow about a
circular cylinder as an example, Song (1996) showed that
the weakly compressible flow equations, Eq.3 plus the
equations of motion of an incompressible fluid, gives the
solution for the entire flow development process. The
flow is transient and compressibility dominates during an
initial time period of , which is called the com-
pressibility (time) boundary layer thickness. It is defined
as the time required for the steady flow to be 99 %
established. For non-viscous fluids, the fully established
flow is the steady incompressible flow, which is
independent of aO assigned in the computation. For a
cylinder of diameter D in an infinite domain, the
compressibility boundary layer thickness, , was
determined, numerically as well as theoretically, to be
U 5M `53
D
where U is the ambient velocity. In other words, the
weakly compressible flow equations contain the
hydraulic transient flow solution, which continuously
evolves into the steady incompressible flow. While the
transient flow is compressibility dependent the con-
verged outer solution is independent of compressibility
or M. Eq.5 also implies that the steady flow is 99%
established when the starting pressure wave travels 5D
distance away from the cylinder. But, if the cylinder
changes its velocity within = 5 D/a, then the com-
pressibility affects the flow all the time.
Song and Chen (1996) further analyzed the same
impulsively started flow problem for the case of viscous
fluid at modest Reynolds number. In this case the time
required for the regular vortex-shedding flow to be
established is much greater than , because the speeds
of convection and diffusion are much smaller than the
sound speed for small M flow. The outer solution or the
established flow is an unsteady vortex shedding flow of
essentially incompressible fluid except that there is also
an acoustic component in the solution. The acoustic
component is relatively weak and can be filtered out to
yield the purely hydrodynamic solution. It was shown
that the solution is identical to the solution of the
incompressible flow equations in every detail, both
space wise and time wise. But more importantly, the
outer solution is independent of M assumed during the
solution process. Thus, it is possible to select an
arbitrarily large M to speed up the computation. It was
shown that, by using M = 0.4, the speed of computation
was about 100 times that of the best incompressible flow
method known to the authors.
A single-phase flow approach for cavitating
flows
A cavitating flow contains the liquid phase, which is
essentially incompressible and the gas phase, which is
highly compressible. Cavitation is such a violent
phenomenon due to radical density change around the
critical pressure, its full resolution is not practical. The
method proposed herein is to regard the liquid-vapor as
a single-phase fluid whose equation of state may be
approximated by a continuous function which is linear in
the liquid phase but non-linear in the gas phase.
Consistent with the weakly compressible flow approach,
Eq.1 is used when the pressure is above the critical
value, Pc This equation is joined smoothly at the critical
pressure by :
p Ai I for
PE P PC (6)
and joined by the following equation at the lower end:
p B
, for O p PE (7)
The coefficients are so adjusted that the resulting
pressure-density curve has a desirable shape.
OCR for page 602
The equations of continuity and motion may be written in
a conservative form as follows.
U F S. F iE jF kG (8)
t
where S is a source, which may or may not exist and i, j,
k are the unit vectors in x, y, and z directions,
respectively. The functions U. E, F. G and S are defined
in the following.
U by,
E ~ a2u, U2 p
U. V, ~
1 (9)
XX ~ UV .Xv 7 UW xz1 T
(10)
F ~ a2 v, up ,C,,, V 2 p yy7 vw IT
(11)
G ~ a2w, uw xz, vw yz7 W2 p
S ~ V a, O. O. o1T
(12)
(13)
The source term S is identically equal to zero for the
liquid phase.
Numerical Approach
The finite volume approach with MacCormack' s
(1969) predictor-corrector method has been used with
good results. Eq. 2 is first averaged over a computational
finite volume. In the process, the volume integral of
divergence is converted to the surface integral by
applying the divergence theory. The resulting equation
iS
U ~ FndA S (14)
A
In the above equation the superscript bar means volume
averaged quantity, A is the surface area, and n is the
unit normal vector on the surface of integration. Since
the quantities to be calculated are always the volume
averaged values, the superscript bar will be dropped
hereafter. The above equation is a simple statement of
the law of conservation that the rate of increase of U in
a volume is equal to the net influx of F plus the source
S. Thus, Eq.14 can be discretized with any standard
finite difference scheme. Yuan (1986) describes detailed
description of the process used herein.
Since a finite volume approach can resolve only
the quantity of scale larger than the finite volume used,
it is necessary to use a closure model for turbulent flow
simulation. A simple Smagorinsky (1963) subgrid scale
turbulence model is used. This model assumes that the
shear stress tensor ij can be calculated by adding an
eddy viscosity ~ to the kinematic viscosity
Smagorinsky proposed the following equation for the eddy
viscosity.
t
1 U. u.
(C )2 1 J (15)
2 x; x;
In the above equation C is a coefficient to be determined
and is size of the finite volume. Experience indicates
that a C distribution, which is 0 on the wall and increases
to 0.12 outside of the viscous boundary layer generally
gives satisfactory results.
The size of finite volume used is often too large
to adequately resolve the sharp velocity gradient near a
solid wall where the viscous boundary layer is thin. For
this reason a wall function or a partial-slip condition
(becomes no-slip when boundary layer is thick) is used:
see He and Song (1991) for a description of this
condition. Because cavity is assumed to be a continuous
extension of the liquid, no free surface boundary
condition is necessary. Only the hydrodynamic quantities
are of interest (acoustics are ignored) and, therefore, an
arbitrarily large value of M= 0.4 is chosen for the liquid
part of the flow to speed up the computation. Recall that
the outer solution of the weakly compressible flow
equations is independent of M and represent the
incompressible flow. Although much larger M can be
used, the gain in computational speed diminishes as M is
further increased.
EXPERIMENTAL METHOD
In order to gain a better understanding of the
complex physics involved, a simple geometry was
studied. A 2D NACA 0015 hydrofoil was selected
because of its previous use by several investigators
around the world. The experiments were carried out at
OCR for page 603
two different scales in two different water tunnels. Tests
at SAFL were made in a 19 cm square high speed water
tunnel. A complementary set of tests using a geo-
metrically similar configuration was carried out in the 30
cm square water tunnel at the Versuchsanstalt fur
Wasserbau (VAO) in Obernach, Germany using a
hydrofoil having a 128 mm chord length.
The SAFL test setup is shown in Figure 1.
Because of the wide range of tests envisioned in the
program, two identically shaped hydrofoils were used.
One foil was mounted on a lift balance for force
measurements. A second foil was highly instrumented
with an array of static pressure ports and an array of
piezoelectric film transducers, quartz crystal transducers,
and miniature accelerometers. The SAFL facility was
also equipped for acoustic studies, Laser Doppler
Anemometry for the study of wake characteristics and
Phase Doppler Anemometry for measure of bubble size
distribution in the wake.
The larger foil used in the Obernach tests was
mounted at each end in specially designed force
balances utilizing Kistler quartz crystal sensors as shown
in Figure 2. These force balances have very high
frequency response ~ 3 KHz), which was found to be an
important factor in the success of the program.
LabView data acquisition software was used in
both facilities. This allowed for compatibility of the data
sets between the US and Germany. This was of
immeasurable benefit in coordinating measurement
activities at two different laboratories. from each facility.
An added benefit was the ability to carry out on-line
comparisons.
Measurements at Obernach were complemented
with normal and high speed video at 4,500 frames per
second. At the same time, observations at SAFL were
made utilizing a 35 mm camera and strobe lighting
triggered by specially conditioned pressure signals
measured on the foil. By capturing a sequence of photos
at different time delays, equivalent framing rates as high
as 10O,000 per second can be achieved.
LIFT OSCILLATIONS
Excellent agreement was obtained between the
numerical simulations and the data obtained at SAFL and
Obernach. As an example, the measured cavity length at
various angles of attack is presented in Figure 3 in the
form of 1/c vs /2 . These data are compared with
previously collected data for an NACA 0015 hydrofoil
with an elliptic planform of aspect ratio 3 by Arndt et al
(1997~. The comparison is made by adjusting /2 for
the 3D data to equivalent 2D values, using standard lift
line theory. Also plotted is the partial and super-
cavitation theory of Watanabe et al (1998~.
~ .. ... ~
~ V ~
I:
:::. i~:~.~1::~. :~
Figure 1. Schematic of sheet/cloud cavitation test set up
in the SAFL High Speed Water tunnel.
Figure 2. View of hydrofoil and lift balance setup at
Obernach
Courtesy of Professor Watanabe who
applied his cavitating flat plate theory to our test
setup. The effects of blockage are apparently minimal
since the theory agrees well with the theory of Acosta
(1955) for an unconfined flat plate.
OCR for page 604
U: ~ ~ i tots
IJ-3r.~.
I ~ ~ 2r.~.
(,$~A
0,4: -A
.
)~ in: ~ ~
~:~:^
3~.$
~ ' <
:i$~ o~4~.~$
~:!~:~63~
~ ': [~'’$
dally ~ Wa~.~:.~,
Li
.
~ -4 $~
. .
Or ~
Figure 3. Cavity length data. The purple and green data
are from the Obernach experiments.
All the data agree except for the 2 data. The
discrepancy at 2° is not unexpected since careful
inspection indicates that bubble cavitation occurs at this
angle of attack. Considering the assumptions made in
the theory, the agreement with experiment is quite good.
It is also noted that agreement between the 2D foil and
the elliptic planform foil is quite satisfactory,
suggesting that a quasi-2D theory might be a good
estimate for cavity length for elliptically loaded foils.
The process was found to be highly dynamic.
Cavitation induced lift oscillations have spectral
characteristics that vary considerably over a range of 1.0
/2 8.5, where is the cavitation number end is
the angle of attack, as shown in Figure 4. The amplitude
of the fluctuations can exceed 100% of the steady state
lift and are associated with the periodic shedding of
vertical clouds of bubbles into the flow. Three types of
oscillatory behavior are noted:
O D.25 0~5 0.75 ~u
Figure 4. The measured lift oscillations for a cavitating
hydrofoil have complex spectral characteristics. This is a
collection a spectra collected over a range of cavitation
number and flow velocity. The angle of attack, , is 8
degrees. Very similar results occur at other angles of
attack.
predicted by Watanabe et al (19984. They refer to type II
modes as Vc 0.75.
The amplitude of the fluctuations can exceed
100% of the steady state lift and are associated with the
periodic shedding of vertical clouds of bubbles into the
flow. Amplitude data were normalized in the form:
Normalized Amplitude L rms
1.0 /2 4: A strong spectral peak exists at
a Strouhal number, fc/U, of about 0.15 that is
independent of cavitation number.
4 /2 6: A higher frequency, albeit weaker
spectral peak dominates. The frequency of this
peak is almost a linear function of cavitation
number and corresponds to a constant Strouhal
number, based on cavity length, of about 0.3.
6 /2 8.5: Bubble/patch cavitation can
occur. This induces a distinct, very low
frequency, spectral peak.
The transition that occurs at /2 = 4 corresponds to a
relative cavity length, 1/c, of about 0.75. This behavior is
u2sc (16)
where LrmS is the root mean square lift, is density, U is
velocity, S is the span and c the chord length. Data
obtained at two different scales with varying velocity and
angle of attack were found to collapse very well. This is
shown in Figures 5a and 5b.
Two competing mechanisms are found for the
induced shedding of cloud cavitation. At high values of
/2, reentrant jet physics dominate, with sheet cavity
oscillations at a frequency, based on cavity length, of
fl/U 0.3. Atlow values of /2 ,bubbly flow shock wave
phenomena dominate with a constant Strouhal number
based on chord length of fc/U 0.2. A significant effect
on the wake structure is also noted.
Frequency data collected from high speed video
of the flow are shown in Figure 6. These data agree very
well with the lift data shown in Figure 4. Note that at
approximately /2 4, there is a sharp transition from one
type of frequency trend to the other. In fact, Joint
OCR for page 605
1 ~ i i 1 1-
2 4
ma,
Is
~ dens. TJ=S~Omis
- 7 dens. U=10.4mis 1.5
- ~ dogs. IJ=10.~7mis
W - ~ ~ ~
:~(~^oA.
Us': '):4,?3~i'^
S mats
* to: - s
1 1
to
Figures 5a and b. Variation of rms lift with /2 . The
upper figure is from the SAFL tunnel where lift is sensed
by the pressure difference across the foil (Kjeldsen et al,
1999)
Frequency Time Analysis (JTFA) clearly visualizes this
process. This is shown in Figure 7. Lift spectra are
shown for /2 = 3.82 and 3.34. In both cases there are
two frequency peaks. At the higher value of /2 the
higher frequency peak has a larger amplitude. At the
lower value of /2 the lower frequency peak has a larger
amplitude. Only through the use of JTFA can it be
found that both mechanisms do not occur
simultaneously. As shown in figured, when the lower
frequency is dominant, it also occurs for a greater
portion of time. When the higher frequency is dominant,
it persists over a greater portion of time. This leads us
to believe that two different mechanisms with different
frequency characteristics are possible in this range of
/2
COMPARISONS WITH SIMULATIONS
Bubble/Patch Cavitation
When the attack angle is small, the boundary layer
of non-cavitating flow is thin and attached to the foil
every where except at the trailing edge. If the cavitation
~~U vs ~~ (Half)
1 .4
1 .3
1 .2
1 .1
0.9
0.8
0.7
0.6
n ~
~ . ~
O .4 -
O .3 -
O .2 -
0.1
O -
~ g ~ ~ ~
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
| Duo = 10mis1
| BUD= 8m~ |
l
6.5 7 7.5
Figure 6. Measured frequency of oscillation from
high speed videos. (VAO data)
number is gradually decreased, then a bubble
cavity will appear first near the nose where the pressure
is minimum. The expansion due to bubble cavity in-
duces boundary layer separation as indicated by an
instantaneous vorticity field shown in Fig. 8. The
corresponding pressure field shown in Fig. 9 indicates
that the bubble is located at the middle of this large
separation eddy. The computed pressure at three points
on foil surface and lift coefficient as functions of time
are shown in Fig. 10. The lift coefficient curve exhibit
a saw-teeth form indicating that the time of bubble
cavity formation coincides with the time of maximum
lift. The bubble slides along the foil and collapsing
near the trailing edge as the lift decreases to the
minimum. As the collapsed cavity in the form of a large
eddy is being transported in the wake, the boundary
layer gradually recovers its original form and the lift
increases to a maximum. A new bubble cavity is formed
when the lift is maximum and the pressure is minimum
again. The computed Strouhal number for this case is,
S = fc/U= 0.1 which corresponds to Type 3 oscillations.
Sheet-Cloud Cavitation (partial cavitations
In the midrange of /2 an attached sheet
cavity will form near the nose. This cavity is highly
unsteady; it periodically grows and breaks off to form a
cloud cavity. Fig.ll shows vorticity fields at several
instants within one cycle of vortex shedding or cloud
cavity formation. This figure suggests that the cavity
break off is due to the reentrant jet striking the cavity
surface (reentrant jet is a thin positive vortex sheet). It
also suggests that the cloud cavity is a large eddy con-
taining many small eddies. Comparison with a phase
OCR for page 606
:~
:v~v ~
:
: ~
1 tJ~ If ~0 JUG
.:: :. :.- i: :::v:.g I:
?:: ?' :?~:?~
4.-~
___4
/2 =3.34
O.s
as .,
/2 = 3.89
300 NEW
.::: ?~.:~. .
w
to'':
2.~:~:?~:
Figure 7. Comparison of conventional lift spectra and JTFA analyses for two values of /2
locked photograph of the process indicates how well
the simulation captures the essential physics of the
process.
Typical computed vorticity, pressure and
density fields at one instant are shown in Fig.12. This
figure reveals, among other things, that a large number
of bubble cavities reside inside the large shed vortex
where pressure is low to form a cloud cavity. It is also
interesting to note that the reentrant jet just broke the
sheet cavity represents a high pressure zone, as to be
expected. A short sheet cavity is located near the nose
and a large non-cavitating vortex is located at mid-
chord. The pressure field shown here is essentially the
hydrodynamic pressure field that is transported mainly
by convection. For the condition shown in Fig. 11 the
sheet cavity breaks off near mid-chord and the cloud
cavity collapses near the trailing edge.
Some typical spectrum of lift oscillation for at
various cavitation numbers are shown in Fig.13. For
the condition shown in Fig. 11, there are two distinctive
peaks in the spectrum. The first peak corresponds to
the frequency of cloud cavity generating vortex
shedding. As experiment has shown, this frequency is
roughly inversely proportional to the cavity length.
Fig. 13 also shows a second peak having twice the
frequency of the first peak. The source of the second
harmonic is not clear at this time, but may be due to the
existence of the non-cavitating vortex shedding as
shown in Fig. 12.
Sheet-Cloud Cavitation (full cavitations
There is a significant change in the flow
pattern when the maximum sheet cavity length exceeds
the chord length. This is because the cavity terminates
on the foil part of the time and terminates away from the
foil the rest of the time. In other words, the cavity
oscillates between partial cavity condition and
supercavitating condition. Fig. 14 shows six instan-
taneous vorticity fields during one period of
sheet/cloud cavitation. Observe that the sheet cavity
start at the nose (see picture a), grows to equal the chord
length (see picture c), and eventually exceeds the chord
~ see pictures d and e). The super cavity terminates
where the positive vorticity extending from the pressure
side meets the negative vorticity that is the cavity
surface on the suction side.
OCR for page 607
Figure 8. Vorticity field for bubble/patch cavitation, /2
= 6.44
Figure 9 Pressure field during bubble/patch cavitation,
~$,. vim, .,.,:.~.:~..} 5$~,~, {; ~ ~ ~~ ,~.~ .i,~. ',~e ~i.,. fib I,, - .;~, ~~
~~ .$ ~ ~ ~6 Y..u ~3 ~ it. r~ a. ~ .~.~ ~~ Aft. ~~
· ~~ ~ -; Y ~ · -:. ::: : 9: :~::::::~: :::: .:: ::: ~::~ so :::::
~.$ ~ -I.: ..~
~ 1, A. §) .~ :~ apt, .-~.e 0.~ ~ 5 )~s >~ .~;
: 5 ~ .4
gas :~ ~ I ~~ ~4 ~ r.~6 ~~H .~ ~ 22 ~~ ,~
~~:.°:~f"~ ~
Figure 10. Numerical simulations of Type III oscil-
lations due to bubble/patch cavitation. The lower curve
displays lift fluctuations. The upper three curves are
pressure at varius values of x/c.
Figure 11. Numerical simulations at / = 4.3 are compared with a phase-locked photo of the process. Flow is from left
to right. Shown is the variation over one cycle of oscillation. Red denotes positive vorticity.
OCR for page 608
Figure 12 Simulation snapshot for /2 = 4.1
From picture (e) the maximum sheet cavity
length is estimated to be about 1.3 c. This long sheet
cavity is about to be broken off by the reentrant jet as
shown by picture (e). Pictures (f,a,b) show the
convection of the large eddy or the cloud cavity after it
is broken off. The cloud cavity collapses further
downstream, which is outside of the region shown in the
figure. Fig. 15 is an illustration of the pressure field at
the same time increments as for the vorticity. This figure
also shows evidence of shock wave produced by the
collapsing cloud cavity traveling upstream. As observed
experimentally, the shock wave can destroy the sheet
cavity near the nose for short period of time as shown
by picture (f) of Figure 14 (very thin boundary layer
indicates no cavitation.) This was a surprise when first
observed experimentally.
In addition, Figure 13 shows that the lift
oscillations also contain two peaks. In this case the
primary peak is that of the sheet/ cloud cavity
oscillation, which is dependent on the chord length
rather than the cavity length. Also in agreement with
experimental observations.
Wake Characteristics
Cavitation has a profound effect on the wake
characteristics behind a hydrofoil. During the current
study, LDV measurements were made in the wake of the
foil in cavitating and non-cavitating conditions. Velocity
profiles were obtained a several cross-sections down-
stream of the trailing edge of the foil. Mean velocity
data were obtained and analyzed for a variety of
conditions ~ and ). To date, two different types of
analysis have been performed. These analyses provide
not only insight into understanding the periodic
phenomena cited above but also provide insight into the
problem of predicting bubbly wakes due to cavitation.
A , : rj .
~.5; :'..~..
Figure 13. Computed variation in spectral characteristics
withincreasing . /2 varies from 1.75 to 5.73
Figure 14. Computed vorticity field for /2 = 2.15
Figure 15. Numerical simulation of the pressure field
during extensive cavitation, /2 = 2.15
OCR for page 609
The basic idea for both methods is an observed increase
in measured data rate when bubbles are passing through
the LDV measurement volume. This increase is, in
general, greater than 10 times that for non-cavitating
conditions. The wake flow behind the hydrofoil was
also investigated for both cavitating and non-cavitating
conditions.
Using LDV measurements, it is found that self-
similarity of the wake shape exists quite close to the
trailing edge (x/c=0.12), for both cavitating and non-
cavitating flow. Mean velocity profiles for cavitating and
non-cavitating conditions are presented in Figure 16.
The data are presented in a form suggested by the
similitude of turbulent wakes:
U,tf ~ [~C] (17)
where x is distance from the trailing edge, y is the
distance normal to the flow measured from the center of
the wake c is the chord length and Uref is the velocity at
the edge of the wake. It is clearly evident that the rate of
spreading of the wake is significantly larger under
cavitating flow conditions. This effect has been also
reported by other workers in the field (Kubota et al,
19894. When cloud cavitation is present, large vertical
structures containing numerous bubbles are observed to
be shed into the wake. These clouds of bubbles extend
much further in the cross stream direction than the
viscous wake associated with non-cavitating flow.
Much of the important physics in the process
are obscured by the averaging process. While the non-
cavitating LDV signal resembles a typical turbulence
signature, the cavitating signal is skewed towards lower
velocity. The strong negative fluctuations in velocity
are due to the imprint left by the periodic passage of
vortices from the cavitation process that extends much
further from the wake centerline than a typical viscous
wake. This is graphically illustrated in the numerical
simulations shown in Figure 11. In fact there is excellent
agreement between animated simulations and high
speed videos of the vertical clouds of bubbles.
The numerical simulations shown in Figure 11
are for a case close to the maximum amplitude of lift
oscillations. Note the strong upwelling close to mid-
span that results in strong oscillations in lift and surface
pressure. It should also be pointed out that averaged
numerical data also agree very well with the experi-
mental data shown in Figure 16 for both cavitating and
::
: ~0
a:
~ O~:1
,,
~ 0.~4
3
0']5 I"''''" 7,,
~ 13 2'
.. ~ I.
i. . .~ ’~ a.
~ : ~ i 5
~~N'~' -
;. :' -
£~g :~1~w :~^
ARABS Deg~
A..- ~ ~ ~
. , . ~0 ~ ~
~ ~--~ x~.~5
i---------- ~ ~~_O,:~
:~ - ~0,.~2
~ i
~o cS . ~.~ ~ , ~ ~ ~ <
:~i ~`,8 ~076 ~0,4: '-~2 ~ 0,> '0,4 0~6 0~8 1
....... ~ : ~ ~ , ; : ~ : ~ ,3
Non o~t'j[~g FroW:
1
.~3 Dog'
'''"""'''""'' - ~:0 ~ ~ ~
.:.: :.::.. ^~:c,~5
+-'''''''''''d ~.~7
47 ~, - ~.0 62
,'.A, ~ L_
O'er' I'm
.:
I.
_ 'a
~ 05 . 3 = ~~ v A . :~ ~ ~ ~ 3 ~ _~ 7 _ _ 3 . £ .
~1 ~70~8 ~6 ~14 -0j ~ (~ Or ~ ~ Hi; ~ ~ 0~ ~ ~
~:~x c1i^~5
Figure 16 Experimental (above) and numerical (below)
wake data with/without cavitation compared. y = 0 is
taken to be that of maximum velocity deficit for the non-
cavitating case.
non-cavitating flow. In the case of cavitating flow,
it is important to note that the numerical simulations fit
the data close to the trailing edge, but deviate from the
measurements further downstream. It is conjectured that
this is an effect of dissolved gas that has come out of
solution. The current numerical model considers only
vapor that will condense downstream, while gas bubbles
will persist in the wake.
It is also natural to conclude that the gas
OCR for page 610
content in the wake will show a cyclic behavior. By also
considering the fact that bubbles are counted more
efficiently than naturally occurring LDV seeds in the
flow, the average velocity will contain more weight from
the shed bubbles. This has an effect on the LDV
measurements that still needs to be resolved. However,
the observation that the wake spreading is significantly
more pronounced with cavitation is still qualitatively
correct.
It was also found that a plot of data rate versus
time could give additional information regarding
cavitation dynamics. The data rate is defined as the time
elapsed between two acquired valid samples by the LDV
system. In general the best agreement is found for the
two measurements in the intermediate domain between
performance breakdown and super-cavitation. In
comparing the LDV frequency domain fingerprint with
the corresponding lift, similarities are found. This is
shown in Figure 17.
It is noted that higher harmonics show up in
both the spectra of the lift fluctuations and the data rate.
It can be conjectured from the spectrum analysis and
from inspection of the data rate time series alone, that
the bubble shedding process occurs over a limited
fraction of the total lift cycle. Since the lift and data rate
spectra are similar, it can be inferred that the lift
oscillations are also a result of the relatively rapid sheet
cavity collapse process.
In concluding this section the following items are
underscored:
The LDV data rate is a useful diagnostic.
Spectral analysis of these data indicate a link
between the shedding of bubble clouds into
the wake and the observed lift oscillations.
The observed self similarity of the mean
velocity data for both cavitating and non-
cavitating conditions when compared with the
numerical simulations suggest that dissolved
incondensible gas plays an important role in
the wake dynamics.
In interpreting averaged velocity data from
cavitating flows care must be taken because of
the bias induced by slower moving bubble
clouds that produce a higher than average data
rate.
:~’
: ~
~ .:
By: ,.
:1
~~:j.:5,~,-.-9 ~~..)
... .
I'd.
Figure 17. Comparison of lift dynamics with the FFT of
data rate in a cavitating flow.
NOISE AND SURFACE PRESSURE
CHARACTERISTICS
Surface pressure and noise data are additional
diagnostics that provide further insight into this complex
flow process. Figure 18 is a plot of surface pressure
measured at various positions along the span of the foil.
Note that the peak amplitude corresponds to about /2
= 3 at about x/c = 0.75. This correlates well with the
inEnrm~tinn Aicnl~f~A in Fi~llr`~ I I
~~ ~~ ~~~ ~~ ~ ~~ ~ ,~ ~ ~ . The peak rms
pressure is about 15 times the dynamic pressure. This
value could be even higher because of size effects in
piezoelectric film measurements (Arndt et al, 1997~. As
was already shown in Figure 5, the rms lift peaks at a
slightly larger value of /2 , but a direct comparison is
not expected since the lift oscillations are due to the
integrated effect of the pressure oscillations over the
entire foil.
OCR for page 611
As would be expected, the spectral
characteristics of the surface pressure in the region of
maximum intensity should be similar the to the spectral
characteristics of the surface pressure. This is shown in
Figure 19 that is similar to Figure 4. Note that a similar
bifurcation in the peak frequency occurs at about /2
= 4. In analogy with the lift data displayed in Figure 7,
spectra at two values of /2 are shown for x/c = 0.74 are
shown in Figure 20. Two dominant peaks are evident,
with the higher frequency peak being dominate at the
higher value of /2 . When comparing these spectra
with JTFA analyses of the same data, it is also noted
that either one frequency or the other occurs, but not
simultaneously.
The duration of each frequency is proportional
to the amplitude of that frequency component in the
conventional spectrum. With lift oscillation and
pressure data as a guide, the spectral characteristics of
the noise can be analyzed The spectra are normalized in
the following manner:
by,
where Pa is the mean square acoustic pressure and f
is the frequency bandwidth.
Noting that the spectral characteristics of the
lift oscillations vary considerably with /2, it is
interesting to compare how the spectral characteristics
of the noise vary over the same range. This is shown in
Figure 21 where spectra at different values of /2 are
compared over the same range shown in Figure 4 for lift
oscillations. It is very clear that the noise maximizes in
the same region as the lift oscillations are a maximum.
The noise also persists to much higher frequencies.
COMPARISON WITH NON-CAVITATING
SEPARATED FLOWS
Numerous experiments and numerical studies
of flows over a foil in a non-cavitating condition have
been carried out for several decades. The general flow
patterns at different attack angles are well described
based on both experiments and com-putations. For the
purpose of comparison and for better understanding of
the mechanics of cavity flows, this numerical model has
also been used to simulate non-cavitating flows about
NACA0015 foil. A non-cavitating condition is simulated
by assigning a very large value to the cavitation number
while the
Figure 18. RMS surface pressure. The intensity of the
S ~ fc) Pa2 color denotes the amplitude normalized to dynamic
U4 fc (17) pressure.
U
Norm. alized Surface Pressure Spectra, xfc = 0.74
U=10.1 mJ5
U = 8 m..fs
~ ~ ~ mix
fug
Figure 19. Surface pressure spectra at Vc = 0.74
OCR for page 612
OCR for page 614
OCR for page 615
Representative terms from entire chapter:
surface pressure
Soo
400
300
200
Ann
2 4
CHICO
~ 8
Nonnalizecl
hmplibide
gg.g337
-1 03.671
- 1 07.409
-11 1.146
-1 14.884
-1 18.621
- 1 22.359
- 1 26!096
- 1 29.834
- 133.57 1
- 1 37.309
- 141 .046
-144.784
-148.521
- 1 52.259
Figure 21 Spectral characteristics of radiated sound.
v== , .
.% -as. i.,
-is' .
i: , . ::
,= A s ~ s z~ so ..
~a~s~.~x.~ i. , -
:s.~ sit ~ . . ~ ..
.s s~.~.s:~. . ::
~~.~:s 1' .L
. ~ : ~ ,
... ~
-g s A: ~ A. lo: son,
<~ lo. (..~. #.
· . -
.~
-: ad,. ::,: jc
..... . ~ A .
-a {~ ~~ ....... ~
770) ~ :: ~ :: 777~-: ,j. ~,3: .'
~~.~
/2 = 3.49
~.~,,,~,.~.,.!
. TT~ T:~
i..
is- i-. ~
.. .. ~
i,
..... , ,4.~ . i.
__ (~ ~ ~ -
T
G
_~,
_.
: Fat
/2 =3.91
i ........
,.. i. .,c::~
· ~ ~ >~ s ~.:
-my ~ ~
if: ~
the foil. These measurements were made using a new
method utilizing piezoelectric film.
Figure 22. Simulation of non-cavitating flow at = 12
is dominated by a periodic growth and detachment of a
boundary layer separation bubble. The downstream part
of the elongated separation bubble is a large coherent
eddy dominated by negative vorticity. But there is a thin
layer of positive vorticity near the foil surface shown in
red. This reverse flow with positive vorticity is a
reentrant jet that is responsible for the breaking away of
the sheet cavity. The way that the reentrant jet terns
upward and wraps around the detached large eddy can
be observed in both cases.
SUMMARY AND CONCLUSIONS
An extensive investigation of sheet/cloud
cavitation was carried out using both numerical and
experimental techniques. Properly normalized, the data
agree in both amplitude and spectral characteristics.
Data for lift, surface pressure and noise are remarkably
consistent. Three flow regimes are noted roughly
demarcated by the parameter /2 . A strong bifurcation
in the data exists at /2 where the maximum amplitude
in lift, noise and pressure is noted.
Excellent agreement with numerical sim-ulations
was achieved. This leads to a very optimistic view of the
simulation model used. Steps have been taken to
significantly reduce computation time. The authors feel
that this combined numerical/experimental approach
shows great promise for further research.
It is important to underscore the fact that /2
is a parameter whose genesis is found in linearized
inviscid theory for a thin flat plate. It was a surprise that
this parameter was even an approximate description of
cavitating flow. Detailed examination does show that the
vertical extent of the cavity flow is less at lower angle of
attack even though Vc and /2 are constant.
The acoustic radiation from different cavitation
patterns on a NACA 0015 hydrofoil were studied. In
addition, the noise characteristics were correlated with
measurements of unsteady pressure on the surface of
The noise characteristics are very similar to the
characteristics of the lift oscillations. Numerical and
experimental work have identified three regions of
cavitating flow. Cavitation noise spectra were also found
to vary significantly with the type of cavitation, e.g.
bubble/patch, sheet/cloud or fully developed super-
cavitation. The best correlating parameter is /2 . This
is parameter is only truly valid for a thin flat plate
(Acosta, 19554. However even for the relatively thick foil
studied here, the overall features of the flow are
determined by this parameter.
The acoustic pressure peaks at about /2 .=
3.5. The spectra persist to high values of frequency in
the range 2.5 /2 5.0. Reasonably good collapse of
the data was found by normalizing in the form
tt ~~,~ ( U)
U
The surface pressure technique shows good
promise for detecting regions of intense cavitation
activity. The observed pressure characteristics are in
good qualitative agreement with the numerical
simulations and the measured lift oscillations. The array
of techniques developed to study this complex flow
show good promise for surface pressure/acoustic
radiation and lift/acoustic radiation correlations.
ACKNOWLEDGMENTS
This work was sponsored by the National
Science Foundation and the Office of Naval Research.
Mr Michael Levy provided invaluable assistance in
obtaining the noise and pressure data. Mr. Fayi Zhou
and Dr. G. Wang have also provided some help at the
early stage of the program. The computer time has been
provided by the Minnesota Supercomputer Institute,
University of Minnesota through its grant program.
REFERENCES
Acosta, A.J. (1955) "A note on partial cavitation of flat
plate hydrofoils" Calif. Inst. of Tech. Hydro Lab, Rep.E-
19.9
Arndt, R. E. A. (1981), "Cavitation in fluid machinery and
hydraulic structures", Ann. Rev. Fluid Mech., Vol. 13.
AnPdt, R. E A , Voi g t, R L Jr, S mctai, J. , Ro do igloo, P.
md Feneim, A, (1989) "Cavitation E osion m
Hydrot rbim s,", Joun~ol of Hydraulic Erg n~nrg,
ASCE, October
Appdt, R. E A, :' ml, 5 md t II., C R. (1997) "Applicatipm
of piezoelechic them in cavitation less ch" Journal of
HvdEmlicE meermz,Vol l'i no 6, Jmle
Doligalski, T L, Smith, C R. and Walker, ~ D A,
1994, Vortex interactions with walls, Ann Rev Fluid
Mech, vol 26, pp 573 616
Helmholt, H (1868), On Discontinuous Movements
of Fluid -:1 Mag, 36, No 4, 337 346
Izumida, Y. Tamiya, S. and Kate, H. 1980, The
relationship between characteristics of partial cavitation
and f ow separation, Froc 10th IAHR Symp, Tokyo,
pp 168 181
Kjeld en, M, AnPdt, R. E A md Effert, M, (1999)
"Ipve tigation of Un lead Cz itation Phenomena"
Procedmgs of She 3~ ASM JSME Fluids E ginering
Conferee , 5 m Prancisco, CA, July
Kirchhoff, G (1869), Zur Theorie Freier
FlussigLeitsstrahlen, Jr reine u angew, Math, 70,
289 298
Kiya, M ,, and Sasaki, K, 1985, Structure of large
scale vortices and unsteady reverse f ow in the
reattaching zone of a turbulent separation bubble
Journal of Fluid Mechanics, vol 154, pp 463 491
Kubotz, A, Kato, H. Yzmagn hi, H. md Mzedz, M,
'Un teady sh p tme mesm ment of cioud cavitation om
z foil sectiom using condbtional sampli g tech ique" J.
Pluid E g,lil,204210
MacCormack, R W. (1969), The effect of Viscosity
in Hypervelocity Impact Cratering, A A Faper 69
354, Cincinnati, Ohio
Riabouchinsky, D (1919), On Steady Fluid Motion
with Free Surface, Froc London Math Soc 19, 206
215
Song, C 5 (1965)"'Two dimensional Supemz itating
ptate Osciltati g Under z P'ee Surface", Jourpal of Ship
Resemch Vol. 9, No 1, 40 55
Song, C C S and Yuan, M (1988), A weakly
Compressible Flow Model and Rapid Convergence
Methods, Journal of Flmds Engmeenng, Vol 110, 441
445
Song, C C S (1996), Compressibility bounda y Layer
Theory and its Significance in Computational
Hydrodynamics, Journal of Hydrodynamics, Series B.
Vol 8, No 2, 92 101
Song C C S and Chen X (1996), Compressibility
Bounda y Layer and Computation of Small Mach
Number Flows, Hydrodynamics, Theo y and
Applications, Froc 2nd International Conf on
Hydrodynamics, Hong Kong, 815 820
Song, C C S. He, ], Zhou, F and Wang, G (1997),
"Numerical Simulation of Cavitating and Non cavitating
Flows over a Hydrofoil, " St A thony Falls Laborato y,
Frolect Report No 402, University of Minnesota
Tulin, M F (1958), New Development in the Theo y
of Supercav~tahng Flows Froc Second Symp on
Naval Hydrodynamics, ONR/ACR 38, pp 235 260
Watanab 5 TsuJimoto Y. Fr mc JP md Michet
Procedmgs of Third I terpational 5 mPmmm on
Cz itation'Gr noble, Prance, April
Wu, T Y (1956), A Free Streamline Theo y for Two
dimensional Fully Cavitated Hydrofoils, Journal of
Mathematics and Fhysics, 35, 236 265
Ye, Z. Gopat m, S md Kztz, J. (1998) "On fhe Plow
Stmct ~e md Vo ticity P od ction due to Sheet
Czvitatipm" Procedmgs of 1998 ASME Pluid
E gin ering Division Summer Meetmg, Pzper No
PEDSAM98 5301
Zaman, K B M Q. McKinzie, D ], and Rumsey, C
L 1989 A natural low frequency oscillation of the
f ow ove~ an airfoil near stalling conditions, Journal of
Fluid Mechanics, vol 202, pp 403 442
DISCUSSION
L Bri mc m-Marjollet
Bcssm d'Esscis des Csrenes
I cm not sure to well under t Ed what you
consider c bubble patch cavitation In my
opinion, you use c density/pressure curve m your
model everywhere in the fluid So, you did not
take mto account statistical point of view for
bubble cavitation So, what is the accuracy of
the lif time dependence curve (fig 10) for
bubblepat hcavitation?
AUTHOR'S REPLY
Bubble cavity in this paper refers to c Urge
detached cavity, which is generated near the nose
of She foil Ed convected away m periodic
maimer it usually occurs when the attack Ogle
is small Ed the cavitation mmmber is only
slightly below She critical value its occurrence is
quite predictable Ed deterministic So the
phenomenon described h re is quite different
from the cavitation gas mmclei which occurs
r mdomly Bec mse the size of She bubble cavity
is quite large, it c mses She Ifft to oscillate It the
Strouhal mmmber based on the cord length of
cutout 0 I he computed lift oscillation agrees
very well with the experimental data
