NAVY DEPARTMENT
THE DAVID W. TAYLOR MODEL BASIN
WASHINGTON 7, D.C.
CAVITATION OF SURFACE ROUGHNESSES
KAVITATSIYA NEROVNOSTEY POVERKHNOSTI
by K.K. Shalnev Translated by R.D. Cooper
ARCHIEF
de
December 1955 Translation 259CAVITATION OF SURFACE ROUGHNESSES
(Kavitatsiya Nerovnostey Poverkhnosti)
by
K.K. Shalnev
Zhurnal Tekhnicheskoy Fiziki, USSR, VoI. 21, No. 2, 1951, pp. 206-220
Translated by R.D. Cooper
TABLE OF CONTENTS
Page
ABSTRACT
i
EROSION DUE TO CAVITATION OF SURFACE ROUGENESSES
i
THEORETICAL EXAMINATION OF TEE CONDITIONS FOP TEEINCEPTION OF THE CAVITATION OF SURFACE ROTJGHNESSES 3
EXPERIMENTAL METHODS WITH CAVITATING SURFACE ROUGHEESSES 7
Experimental Methods with the Surface Roughness Models
8Experimental Methods with Models of Slits
il
EXPERThNTAL RESULTS AND THEIR DISCUSSION 12
Structure of the Cavitation of a Roughness
12
Similitude in Development of the Cavitation of a Roughness.
.16
Influence of the Relative Height of the Roughness
on the Development of Cavitation of
Rouginesses
19Influence of the Boundary Layer on the Development
of the Cavitation of Rouglmesses
21
Concerning the Cavitation of a Hydrofoil with a
Smooth and with a Rough Surface
22
CONCLUSIONS 23
2-ABSTRACT
This paper presents the results of experimental inves-tigations of the effect of surface roughnesses on cavitation. The investigations consisted of water-tunnel tests of roughness models in which standard techniques were used to investigate the region of cavitation. By means of high-speed motion pictures of the region of cavitation behind roughness models, the periodicity
of cavitation formation with respect to time in the initial stages of development is demonstrated. The periodicity of the develop-ment of cavitation is shown to conform to the Strouhal law. An hypothesis is formulated with regard to the inception of cavita-tion on the axis of the vortices in the region of separated flow behind the roughness at the moment the vortex attains its maximum
diameter. As evolved on the basis of this hypothesis, conditions for the inception of cavitation for various surface roughness pro-files are confirmed by experiments with models of slits, which are encountered in hydraulic machinery having rough surfaced walls. It is demonstrated that a rough surface may be the cause
of premature cavitation while, for the same flow parameters but with a smooth surface, cavitation depeMing only on the body form need not,necessarily and usually does, not occur. The effect of the boundary layer on the danger of excitation of the cavita-tion of surface rougbnesses is explained.
EROSION DUE TO CAVITPTION OF SURFACE ROUGENESSES
In the practical operation of hydraulic machinery, cases o± ero-sion are encountered on parts o± the machinery behind surface roughnesses which were formed, for instance, by inaccurate assembly of the parts, by defective machining, and by accidental damage to the surface of the parts.
From an investigation of turbines and pumps damaged in normal operation, we will present several examples of erosion due to the cavitation of roughnesses on the walls of a rotor housing. In the first case, Figure la, erosion was observed ahead of and behind a projection which was formed at the point of transition from the cylindrical to the spherical part of the cast-iron housing of a rotary-blade turbine. The projection was the result of defective finishing of the walls of the housing. In the second case, Figure lb, erosion was observed ahead of a projection formed by a stainless steel liner in a cast-iron housing because the liner sheets were thicker than the recess machined in the housing for them. In the third case, Figure ic, erosion of the steel liner of a turbine housing was dis-covered behind the heads of rivets and behind crests of roughnesses result-Ing from defective machining of the housing. Although the rivet heads were
ai O alU) CH U) OU) Q)
H
Q)-Po
OaJ
H -P (J) ,-4 OHO
H ai Q) Q) ,-1HJ
E-1 I aiHq
Q)1References ere listed on page 2.
cut off and finished flush with the liner, after a short period of turbine operation the joint between the liner sheets and the housing became loose and the steel rivet heads projected beyond the surface of the liner. As a fourth example, we present the case of erosion on the suction side of the stainless steel blades of a rotary-blade pump. On gne blade erosion was discovered behind ridges, not more than 0.7 mm in height, which resulted from scratches made by a sharp cutting tool; on another blade erosion was discovered behind small mounds with gradual slopes to such a small height that they could scarcely be detected by hand.
The short time in which erosion develops and the direction in which the pits caused by erosion are distributed appear as characteristic
features of erosion due to the cavitation of surface roughnesses.
Cases are known in which the cavitation of the roughnesses was able to produce considerable erosion.1 For example, cavitating roughnesses of about 0.# mm in height at the joint of the upper and lower halves of the rotor housing of a rotary-blade turbine resulted in considerable erosion of the cast-steel housing which extended over a belt about 100 mm in width after three months of turbine operation.
A detailed investigation of the flow in the interblade space and in the gap between the blade tip and the rotor housing shows that the dis-tribution of the erosion pits on the walls of the housing coincides with the direction of the absolute flow in the gap. The distribution of the erosion pits on the blade coincides with the direction of relative flow along the blade. This propertythe direction in which the pits caused by erosion due to the cavitation of surface roughnesses are distributedcan be used, for instance, to resolve the question concerning the causes of hous-ing erosion, viz, whether blade cavitation or cavitation of surface rough-nesses on the housing walls is the cause of the erosion on the housing walls.
We will present the results of investigations of the cavitation of surface rougbnesses. The results have application mainly to the cavita-tion of roughnesses on the walls of the slit which is formed as a result of the structural clearance between the blade tip and. the rotor housing of hydraulic machinery.
TUEORETICAL PXtNATI0N 0F TUE CONDITIONS FOR TRE INCEPTION
0F TEE CAVITATION 0F SURFACE ROtJGHNESSES
The flow in a slit about a surface roughness in the form of a sill will be considered as the fluid ef flux from e.0 orifice formed by the partial restriction of the stream from one side by the sill. Corresponding to the notation of Figure 2, we have the following relations;
1. The coefficient of restriction of the slit by the sill
'////,////,','/
/
M
,/,,,,)r,,,,,J/,,fl,
M
Mst
Figure 2a - Flow about a Roughness of Triangular Profile
the coefficient of con-striction of the jet by the sill
= a/(a51 - a);
the coefficient of con-striction of the flow in the slit
= a./a , from which we find
,jsl j si
a
= a
a.
jsl
jThe flow ahead of the slit undergoes a partial restriction from one side, which is evaluated by the coefficient
m=F ¡F =a b ¡ab =a /a
si
sisi
si oowhere b and b are the width of
si
the slit and of the channel ahead of the slit, and b = b
si
We will use the following characteristics of flow in slits, which are either assumed or known empirically.
1. The flow in slits and out of them appears to be uniform and steady, with the exception of areas of perturbation on the
walls
of the slit and in the regions adjacent tothem.
The region of perturbation or separation of the flow about a roughness consists of periodically formed vortices which are swept down-stream by the flow, as is evident in experiments with the flow about sill-like projections on the side walls of a channel.2 During the interval of time in which the vortex is growing to maximum size, but bas not yet left the region of perturbation, the flow is everywhere uniform and steady with the exception of the region of perturbation.
According to experimental data from a study of the flow structure Uhind poorly streamlined bodies3'4 and according to theoretical concepts with regard to the structure of the vortex pattern,5 the rotation of fluid particles within the vortex occurs at constant angular velocity. The
pres-sure on the axis of the vortex is given by the formula Figure 2b - Flow Figure 2c -Flow
about a Roughness about a Step of Segmental
Profile
Figure 2 - Schematic of the Flow about Surface Roughnesses
in
a
Slit M / ¿ / MMsj
Io
o'
MJ 7,? u,p p u
o k k
y - y 2g
where and uk are the pressure and rotational velocity of the particles at the radius k, respectively,
y is the weight per unit volume of the fluid, and
g is the acceleration of gravity.
1. By changes in pressure and velocity in the flow through the slit,
kinematic velocity similitude and pressure similitude at similar points of the flow are maintained up to the very onset of cavitation.
Cavitation arises at any point in the fluid where the absolute static pressure is reduced to a value equal to the vapor pressure of water at the given temperature.
According to experiments with the cavitation of a circular cylin-der3 and to experiments, to be described later, with the cavitation of rough-ness models, the cavitation which is developed in the region of separated flow behind poorly streamlined bodies consists of periodically formed
cavi-ties which are swept doistream by the flow. In this case, the cavities
arise soute distance behind the body.
The minimur pressure in the region of separated flow behind a
roughness occurs on the axis of rotation of the vortex at that time in its development when it attains its largest diameter which causes the greatest constriction of the jet in the slit.
We will write the energy equation of the flow from the section Moe to the section M for the case in which the surface roughness has the form of a sill of triangular profile, Figure 2a:
2 2 fr)
p_ V_
p _jsl
/ i
nt sì-+-=
+1
+ + + y 2g y(2
2 ent 2g \jsl
ajsl
In which p and y are the pressure and velocity, respectively, in the section M ahead of the slit where the influence of the slit on the flow is not manifested,
[1]
f2]
is the velocity in the free stream section of the slit,
V
si
V
=v
¡cijsl
si jsl
and are the resistance coefficients of the entrance to the
en en
slit nd of the constriction of the jet in the slit by the sill,
is the frictional resistance of the path from M to
referred to q = y2 /2g.
s] si
We will obtain the pressure in the constricted section behind the sill from Equation [21:
(i
+ tent 2
+ +
-ml--V Y
\a.
2 ent frJ2g
sl
where m = v/v1. Then the pressure on the axis of the vortex, if we take
Uk
= V.5l
and k =.51
in [i], will bePo
(2+
_Y -.
---=--1
+ +-I 2 ent fr
a
j sl
We will call the relation
P -P
00 V
X = -'
sI yq
si
the cavitation coefficient of the slit, and ve will designate its critical
value, i.e.,
the value correspording to the onset of cavitation, b3 X1 Let us agree to add to the index si the indices tr, seg, and st which, depending on the form of cross section of the roughness, indicate,respectively, a sill of triangular section, a spherical segment, or a step. Equation [Lt} represents the formula relating Àitr for a slit in the
pre-sence of roughnesses of the first type on its walls to the parameters which chnracterize the construction of the slit, the suriace roughness on its walls and the flow in the slit. According to
[5]
the value ofX'j.tr
can be obtained from experiment, if p00 and v are measured for such conditions of cavitation. According to[3
the value ofAl.tr
can be obtained from the characteristic parameters of the slit and of the flow in it, which are knon In the absence of cavitation.If the sill has a profile of ovoid form (segment), then
aj
i
and = in Formula [li-]. Then the condition for the inception of the
cavitation of a sill of segmental profile is given by the formula 2 + ' * ent 2 x = + +
-I5
[6]
slseg
a2
ent fr 2 V psl_ min
p00-p
y2+'
ent 2 = - + +-m
sltr
2 ent fra
jsl
Setting mjn = where p is the vapor pressure of water, at the moment of inception of the cavitation, we obtain from [] the condition for inception of the cavitation of the sill in the silt in the nondimensjon-al form
[5]
T
If instead of a sill the surface roughness has the form of a step, then setting Cnt = O and ci = 1, we obtain the condition for the inception of the cavitation of a step which is given by the formula
2
X
=2
+,
-m
slst
ent irFor the case in which we mow the values of and v1 in the
section M51, where the influence of a roughness on the flow is not
manifest-ed, then, after writing the energy equation from the section M51 to the sec-tion M, we arrive by arguments analogous to the previous ones at the formu-las relating the cavitation coefficient of a roughness,
psi
-X
-r yq
si
to the parameters characterizing the flow in the slit. They can be obtained from [u], [6], and [7] by setting m = 1.0 and tent - O. Adding to À in place
of the index r the indices tr, seg, st, we obtain
2 +
*
ent À = +-1
[9] tr 2 fr cijsl
2 +*
ent À = +-1
seg 2 fr ci*
À=l+
st frThe latter formulas relating to the cavitation coefficients of roughnesses show that, if the resistances nt and
fr be neglected, then the values of Àr and
e depend only on the coefficients of constriction
of the jet in the slit ans. the value of the cavitation coefficient
ap-pears as a constant equal to unity. From Formulas [9] to [il) it follows also that with increasing slit height, for which a4l.0 and cij
-*
1.0, the condition for the inception of the cavitation of rougbnesses does notde-pend on the form of these roughnesses, since À* 4
À* + À* 4 1.
tr seg st
ixPERINENTAL METHODS WITH CAVITATING SURFACE ROUGETSSES
Experimental investigations of the cavitation of surface rough-nesses were performed on roughness models and on models of a slit whose walls were artificially roughened. These tests were conducted in a small
[T)
[8]
water tunnel (Moscow Hydrodynarnic Tunnel of the Pan-Soviet Institute of Hydro-Mechanics).6 The test section of the MET consisted of
interchange-ahle casings which differed from one another in cross-sectional area. The velocity and pressure in the test section were regulated independently by varying the speed of the tunnel pumps and by force pumping, i.e., by
ex-hausting the air from the space above the free surface of the water in the highest part of the MHT.
EXPERIMENTAL THODS WITH T SURFACE R0UGESS MODElS
For the experiments with surface roughness models, casing No. 1 was used; it was located immediately behind the orifice-type water meter
of the MET. The test section of the casing, which was 4OO mm long and 70.0 mm wide, had a variable height which was regulated. by the upper and lower internal glass plates of the casing. During experiments for the de-tection of the noise of cavitation the glass bottom of the casing was re-placed by a bronze plate to which a piezo-quartz transmitter was affixed; the transmitter was located at a distance of li5 mm from the crest of the
sill. The special measuring equipment, used. for investigating the physical
structure of cavitation, consisted. of high-speed motion picture apparatus and instruments for detecting the noise of cavitation.
The high-speed motion picture apparatus was constructed on a prin-ciple utilizing periodic spark flashes from the discharge of condensers across an air gap between two electrodes. The periodicity of the spark flashes was obtained because the discharge of the condenser across a spark gap occurred more rapidly than Its charge across a resistance. Air was blown between the electrodes of the spark gap. Actually, the photographic apparatus consisted of a drum 1000 mm in diameter, which was placed in a
light-proof housing, and. an object glass of lens power 1:3.7, which was built into one of the walls of the housing and which had. an automatic
shutter (Kompur). Film was attached to the outer surface of the drum. The drum was rotated by a 2.0 hp electric motor through a belt drive.9 In
experiments concerned. with photographing the region of cavitation, the sill was attached to the internal glass bottom of the casing so that the image being photographedthe sill and its region of cavitationlay between the
object lens of the apparatus and the spark gap (Figure
3).
With such an arrangement, a silhouetted representation of the cavitation was obtained on the film because the region of cavitation, filled with a mixture of vapor bubbles and liquid., is less transparent than the noncavitating flow.-
85 2159
175
90
Figure
3 -
Schematic of the Experiments with High-Speed Motion Pictures of the Region of CavitationThe instruments for recording the noise of' cavitation consisted of the following devices: (1) a piezo-quartz transmitter, (2) an amplifier,
(3)
an oscillograph, and (L) photographic equipment.10 Externally the piezo-quartz transmitter had the form of a plug, li-0 mm in diameter, and wasthread-ed for screwing into the bottoni of the casing. The plug had an internal chamber in which the quartz crystal was located. The quartz crystal was clamped in the chamber between a diaphragm and a lower cover plate which hermetically sealed the chamber. The diaphragm, which was 5.0 min in diame-ter, was connected to a piston 1.0 to 2.0 mm in diameter that was subjected to stream pressure. The space between the upper cover plate and the diaphragm was filled with a viscous mixture of rosin and wax.
The direct-current amplifier consisted of three cascading
ampli-fiers. The first cascade of the amplifier, which consisted of a CT-6 tube in a brass hood in order to prevent interferences from electromagnetic fields, was located immediately adjacent to the transmitter. During opera-tion, a cathode-type oscillograph with an electronic-beam tube made by the Svetian factory was used (the screen diameter was 110 mm). The light spot on the screen of the oscillograph, which oscillated proportionately to the variations in the sound pressure, was recorded on moving film attached to the thum of the high-speed movie equipment.
Brass models of the surface rougi-messes, i.e., of the sills, at-tached to the bottom of the casing, had the cross-sectional form of -criangles
or of spherical segments. The section of the sills attached to the glass bottom of the casing had a more complex form which consisted of a triangle with a rectangular base.
Variations in the free stream sections of the tunnel test section, in which the roughness models were tested, and the conventional notations of the order of the experiments are presented in Table 1. The cross-sectional profiles of the roughness models are shown in Figure -I-.
h1
8H
T-2 I 8-C-3 T-4, T-5, T-6 -r C-IFigure )4 - Profiles of the Models of Surface Roughnesses
TABLE i
To classify the conditions of cavitation established, during the experiments, the pressure at various points in the casing channel, the pressure drop in a water gage and the temperature of the water were measured and the external effects of cavitationthe length of the
cavitation region and the noise of
cavitationwere recorded. In some
experiments high-speed motion pic-tures of the cavitation region were taken and the moment of inception of
cavitation was determined by its noise, using the piezo-quartz trans-mitter. The length of the cavity was measured in units of a and repre-sented. by kg, or in millimeters and represented by 1k
We have shom in these ex-periments with what care the onset
of cavitation must be determined when using only visual methods,
be-cause in the initial stage of cavita-tion the rapid growth and
disappear-ance of' the cavities cannot always
No. Designation of Test Type of Test Number of Tests as1 V51 rn/sec percent k g Re x l i T-1 Motion 14 2)4.0 5.2- 7.14 2.11-2.57 -- 14.5-io 37 -2 T-2 Pictures 1 214.0 9.14 1.92 -- 14.0- 6.0 33.0 3 T-3 Cavitation 114 20.0 14.0-11.2 0.97-2.23 -- 2.5-19.0 27.0-7)4.1 14 T-14 Observation 6 10.6 3.6-10.8 2.11-2.30 3-6 13.0-15.0 13.14-140.5 5 T-5 of Forrns of 6 20.14 5.6-13.3 1.314-1.55 10-7 13.0-15.0 17.5-142.8 6 T-6 Cavitation 5 30.5 5.2-11.6 1.9)4-2.58 13-15 0 15.2-35.3 7 T-6 - 5 30.5 5.2-11.6 1.12-1.314 io-8 13.0-15.0 15.2-35.3 8 C-1 1 20.2 12.8 1.79 -- 0 58.2 9 C-1 " 5 20.2 6.0-13.7 0.73-0.99 16f'24 10.0-11.0 27.3-614.0 10 C-2 6 20.2 10.2-13.6 1.71-2.36 15-17 0 61.8-814.5 11 C-2 9 20.2 6.9-114.8 0.88-0.97 8-
---
142.0-92.0 12 C-3 5 20.2 5.2-10.1 3.82-14.140 6-8 0 714.3-114.5 13 C-3 5 20.2 5.2-10.1 2.814-3.06 3-5 5-6 714.3-114.5 T-3 C-2p si
V
li
be detected by eye. However, the characteristic sound of cavitation which accompanies the formation of the cavities can be heard even above the other extraneous noises in the laboratory.
The velocity of the flow over the models y51 was found from the formula sl = Q/F51, where Q is the discharge, determined by means of the tunnel's calibrated orifice-type water meter, end F51 is the area of the free-stream section of the casing. The pressure Psi/Y for calculating Xr was found from the formula
- (15.6
h +12.6
h) lo - ha fr
where h is the atmospheric pressure measured by a mercury barometer,
a
h is the difference in the tube readings of a mercury-water
manometer which was connected to a tube in the section M , and
si
h is the frictional resistance on the part from the section M up
fr si
to the roughness model, obtained by measurements made with the casing in the absence of the model.
The following method was used for interpreting the motion-picture films of the cavitation region and for determining the Strouhal number S. By measuring the distance i between those frames which correspond to
suc-cessive inceptions of cavitation and by measuring the number of drum revo-lutions per minute, n, the period r = 1/u = 60 /irDn was found, where D is the diameter of the drum. Then the frequency of inception of cavitation is N =
i/F
and the Strouhal number is S = Na/v, where y is the velocity in thesection restricted by the sill. The Reynolds number was calculated from the formula Re = av/ii, where y is the coefficient of kinematic viscosity.
EPERThNTAL MB'THODS WTTH MODELS OF SLITS
In experiments with models of slits, casing No. 2 having a 70 by 70 mm free-stream section was used. The flow was constricted to a slit 70 by
5.3
mm in cross-section and 100 mm long by a brass insert, which wasplaced transverse to the channel so that the eff lux from the slit occurred with a partial restriction of the flow from one side ahead of the slit. This Insert or constricting wall of the slit simulated a blade tip, while
the guide wall opposite it simulated the wall of the rotor housing of a
-
hydraulic nichine. The edge of the constricting wall was rounded off atthe entrance of the slit to a radius of 20 mm to prevent the appearance of forms of cavitation other than the cavitation of the sill. Surface rough-nesses in the form of three projections with triangular cross sections were
attached to the constricting wall of the slit immediately behind the rounded-of f edge at the entrance to the slit at a distance of 5.0 imu from
each other. Four slit models 51 to Si4 were tested, each of which bad identical heights and. widths, but differed from one another in the height of the sill; a = 0.0; 0.3 0.6; and. 1.3 mm. During the experiments with the slit models, the discharge and. the pressure distribution on the guide wail were measured and observations of the development of cavitation were made. The onset of cavitation of the sills was determined. by ear from the noise of the cavitation. From the pressure measurements and the discharge, the cavitation coefficient XS1 was calculated by Formula [i] and. the dis-charge coefficient was calculated by the formula = v5j1J/2gH where H is the pressure difference across the slit. The pressure for the calculation
of was determined 'by the formula
p
--= (13.6 h + 12.6 h) io
y a
where h is the difference in tube readings' o a mercury-water mcnometer in the section M
PERIENTAL BESUIPS AND TTh DISCUSSION
STRtJCTUBE 0F TI CAVITATION OF A ROUGEESS
From visual observations, the development of avitation behind sills occurs in the following sequence. The initial stage of cavitation is characterized by the appearance of rare, brief flashes of fog some dis-tance donstream from the sill and by short, faint clicking sounds. With a decrease in the cavitation coefficient the flashes of fog occur so frequently that the region of cavitation already appears as a contnuous whitish cloud, gradually expanding both in the direction of flow and. against it.
High-speed motion pictures of the region of cavitation give a de-tailed picture of the development of cavitation behind a sill and a repre-sentation of its structure. When observed with the naked eye, the region of cavitation behind a roughness appears as a comparatively stable fog, but actually consists cf cavities that form periodically, grow to some limit and then are swept away 'by the stream.
The set of experiments T-3, in which both the various stages of cavitation and the various speed regimes were studied with only a slight variation in kg, gives a visual representation of the structure of the cavi-tation of a roughness in the initial stage. In these experiments it was
possible to ascertain the initial stage of the development of cavitation behind a sill of triangular profile by means of motion pictures with a
speed of 1lOOO_5OOO frames per second. The condition for cavitation, approxi-mately at the moment of its inception, is determined as Xtr =
2.33.
In the photographs (Figures 7a and 6a) we see the formation of the cavities behindFigure 7a - Initial Stage of the Development of Cavitation
Ii
s.
/
Figure 7b - Intermediate Stage of the Development of Cavitation
, :.'
I4
'/
fY :?
.:J
¿....
...J..d
41
Figure5c
-Cavitation Condition Close to the Separation Stage
Figure
5
-Motion Pictures of the Cavitation of the Sill T-5 at a Speed of 14.00O5OOO Frames
per Second
Location of the inception of cavitation and the direction of flow are shown by
arrows.
¼
Figure 6a - Enlarged. Frames of Figure 5a
N
Figure 6h - Enlarged Frames of Figure 5h Figure 6
Locations of the inception of cavitation are showa by arrows.
the sill at distances ranging from 2.5a to 2.75a. In visual observation, such a cavitation condition is characterized by flashes of faintly percep-tfble fog and occasional clicking sounds. In the other photographs of more advanced stages of the development of cavitation (Figures 5h and 6b) the point at which the cavity forms is closer to the sill.
The photographs of cavitation at kg 18-20 do not show period.icity in the development of cavitation, at least, not on the forward and central portions of it. The rear portion of the cavitation, which appears to the naked eye as faint flashes of fog, is not completely showa on the photographs (Figure
5c).
Because the edge of the sills was located, normal to the optical axis of the object glass during the experiments, the photographs of cavita-tion show the forms of the cavities (vortices) in side view, and not in end view. If it he assumed that the cavitation cavities are generated at the axis of rotation of the vortices which are torn away from the edge of the sill, then the photographs indicate an irregular pressure distribution along the axis of rotation of the vortices with the minimum pressure located at the center of the axis of the vortices. In the intermediate stages of the development of cavitation the cavities have a spindle-shaped form with axis
15
sometimes rectilinear and parallel to the axis of the sill and sometimes as though arbitrarily bent.
The fact, revealed by high-speed ¡notion pictures of the cavitation region behind a roughness, that the cavitation arises some distance behind the body that induces it in the initial stage of its development and then, as the cavitation coefficient decreases, the position of its inception moves toward the body, is used in the present paper in the derivation of formulas
for evaluating A. This characteristic of cavitation, which is displayed in regions of separated flow with the formation of vortices, was also de-tected by the author in an experiment with a circular profile.3
It should be noted that in the search for cavitation criteria, up to the time of the last mentioned investigation no distinction was made in the character of the stream flow about the body, for which such a criterion was to be found.3 For instance, in an analogous case, Thomson assumed that the process of formation of an evacuated vortex in the cavitation 0±' a sphere begins with the formation of an evacuated cavity on the surface of the
sphere at the point of minimuii pressure.11
Let us examine the characteristic properties of the periodic de-velopment of the cavitation of roughnesses using the set of experiments T-1, which were conducted at a low film speed of 1600 frames per second. The stage of cavitation development is characterized in these experiments by kg 1ì. For these conditions the irregularity in the periodicity of
cavita-tion development is clearly shown (Table 2). In some cases the development of cavitation was observed with a period 2 to 2.5 times shorter than the a'erage period deter:ained over the complete duration of a given test. Such a distinct difference in the short period (we will call it the complementary period) and the fundamental, longer one cannot be explained as experimental error and must be a natural phenomenon associated with the stream flow about a sill. Petrov and Steinberg in their experiments on the investigation of
TABLE 2 j No. T X iO sec No. T X sec No. T X sec No.
T x 10
sec 1 251- 8 292 15 24-2 22 216 2 292 9 229 16 280 23 229 3 178 lO 216 17 267 24 153 4r8
II
114 18 318 25 305 5 305 12 153 19 191 26 242 6 267 13 369 20 305 27 229 7 254 lJ 280 21 369 28 318the flow behind poorly streamUned bodies'2 also observed the existence of two periods of pressure and velocity pulsation. In the subsequent examina-tion of our experiments we will always distinguish between the fundamental period and frequency and the complementary period and frequency.
The number of separated cavitation cavities per second under vari-ous conditions of flow in the set of experiments
T-3
are presented inTable
3.
According to the data presented, a graphical representation of N = f(v) will have a linear characteristic for the various states of cavi-tation within the limits of kg from 1.5 to 10.0.TABLE 3
SIMILITUDE IN DEVELOFT OF TI
CAVITATION 0F A ROUGHNESSThe experiments using high-speed motion pictures of the cavitation region of rougbnesses and the experiments for the determination of values of the cavitation coefficients corresponding to the various stages of cavi-tation make it possible to evaluate the kinematic similitude of the cavita-tion of roughnesses with respect to the periodicity of the phenomenon and
the similitude with respect to the development of the forms of cavitation. Let us examne the kinematic similitude o± the cavitation of roughness by means of a criterion expressed by the formula for the Strouhal number S. According to the set of experiments T-1, S fluctuates within narrow limitE in the initial stages of cavitation, but a tendency for the fluctuation to subside at larger values of kg is noted (Figure
7).
For fully developed cavitation at kg 10, the Stroulial number decreases from S = 0.20-0.23 -to S = 0.15. This decrease in S cannot be attributed to ex-perimental error, but seems to be the result of the reduced periodicity in the development of cavitation in its later stages.The set of experiments T-2 were carried out at one velocity and for one stage of cavitation only. A comparison of the results of the sets of experiments T-1 and T-2 shows the influence of the height of the sill on the Strouhal number S, namely, S was decreased by a factor of 2.0-2.3 as the height of the sill decreased from
6.o
to3.0
mm.In the set of experiments
T-3
with kg varying from O to 19, simili-tude of the periodicity of the inception of cavitation is fixed at S = 0.095-0.125. In comparison to the set of experiments T-1, a small increase in S was noted in these experiments with decreasing X, which is easily attributedy, m/sec
.3
8.0 8.2 8.9 11.2T, cavities/sec 100.0 219.0 217.O 24-9.O 322.0
17
to deficiencies in the test (a small 0.30 number of periods were recorded)
rather than adherence to a physical s 0.20
law. 0. 0
Raving shown that the
Strou.hal number S is independent of 4.0 the development of cavitation in its
initial stages, ve will examine the
'tr20
dependence of the Strouhal number Son the Reynolds number Re in the o24
range of values of Xtr from 1.83 to
l.1landkg from.5to 10.0.
If Figure7 -
Strou.hal Number S for thethe value of S = 0.087, which was Cavitation of Surface Roughnesses
determined by the measurement Gf as a Function of Reynolds
only two periods, be excluded from Number Re, Xr Constant
consideration, then the average value
of S determined from the data of nine experiments is equal to
0.107,
and the relative deviations of the individual experiments lies with the limits= ± 10 percent (Figure
7).
Such deviations can be attributed to experi-mental error, in consequence of which it must be concluded that in our ex-periments the value of S for the cavitation of roughesses is independentof Re.
The average value of S for the set of experiments T-3 occupies an intermediate position between the values of S for the sets of experiments T-1 and T-2 and shows a tendency toward a decrease of S with a decrease in the height of the sill.
We will compare the results of our determination of the Strouhal number S from the data of the set of experiments T-1 with the data of the experiments with plates made by Petrov and Steinberg12 and with data fron the deteimination of the periodicity of formation of vortices behind a
wedge'3 (Table f1-). TABLE li-o o O o
.
.#
S oT-I
QT-2
T-3
e o s.
..
SNo. Test Body Re x S Remarks
i Plate12 10.0-30.0 0.23 Air, uncorrected for
tunnel wall effects
2 Sill T-1 L.o.0_60.o 0.23 Water, cavitation
3 Wedge'3 5.8-20.2 0.25 Water
32 40 48 56 64 72 80
24
20
4
O 2 4 6
PV/y
q , Measured in the Jet
Figure
8 -
Similitude in the Develop- certain values of kg. In the setment of the Cavitation of experiments
T-3
(Figure 9) theof Sills T and C limiting value is kg = 17-20, because
Thus, from the above it follows that in our experiments the periodicity in the development of the cavitation of roughnesses obeys the Strouhal law in the same way as in the case of streamlined bodies in the absence of cavitation and free from wall effect. The numerical value of the Strouhal criterion for the cavitation of roughnesses depends on (1) the
stage of cavitation, and (2) the height of the roughnesses, if S be calcu-lated using the average velocity in the channel section and not the velocity of the ,jet at the crest of the roughnesses.
We will establish similitude for the development of the forms of cavitation of roughnesses by examining the properties of the cavitation coefficients of the roughnesses which were obtained in our experiments.
The independency, obtained in our experiments, of the values of the cavitation coefficients and Xseg on the velocity v1 can be investi-gated on plots of (p
- )/y = f(q51), as is shown in a typical plot for
the set of experiments T-6 and C-3 (Figure
8).
The experimental points on such plots fall along a straight line, inclined toward the ordinate axis,which indicates that the ratio (p - p)/vq51 or the average values of
and Xseg are constant. In order to show the degree of correspondence of
the values of and Xseg for each experiment with the average value, the
deviations A= (k
av - ' )/Xav were
calculated (Table 1). The magnitudes of the deviations presented in
Table i lie within the limits of accuracy for the determination of
'tr and Aseg both for the initiai stages of cavitation and for fully developed cavitation. Consequently,
the
valuesof
Xsegfound in
the experiments can serve as criteria of cavitation for its various stages. In connection with Table i we should point out that in the sets of experi-ments T-6 and C-15 the beginning of cavitation was determined visually, and in the set of experiments C-2 by the noise of cavitation.
Eowever, the evaluation of similitude in the forms of fully de-veloped cavitation by means of the cavitation coefficients of rough-nesses is limited, beginning with o o
k9:0
k9 = 12-14 c-3A
r
Figure
9 -
Cavitation Coefficient of Sills T as a Function of Length of Cavitation 19 o 0.08 0. 6 0.24 0//0st
Figure lo - Effect of Constriction of Height of Slit by the Sill on
the Cavitation Coefficient of Sills of Triangular
and Segmental Profile for these values of kg the
coeffi-cient. Atr is constant and equal to Xtr
0.87.
An analogous "loss ofquality' of the cavitation coeffi-cients of roughnesses is also ob-served in the case of cavitation coefficients in experiments with a
streamlined body, which is free of wall effects, and can be attributed to the influence of the pressure gradient in the direction of the longitudinal axis of the water tunnel.
INFLUENCE OF TBE RELATlV. EEIGHT OF TBE ROUGHNESS
ON T
DEELOPNT OF
CAVITATION OF ROUGUNESSESWe will consider the question of the influence of a/a51 on at the inception of cavitation for kg = L4. and kg = 10114. (Figure lO). We
will compare the experimental values of Xr with the values calculated
by
Formula [9], taking nt = 0. We will take the values of required for
these calculations from the experiments
by
Sbaumian.14 These experiments were conducted in the free flow from a shielded opening, the flow beingdependent on the degree of the one-sided constriction introduced by the wall of the shield. The experimental values of aT obtained by him are close to the values of ci (Reference 17), calculated according to the theory of
flow from an infinitely long slit given by Joukowsky,16 and. close to the values of obtained in experiments with flow from a circular orifice in a thin wall.17 However, the direct use of the experimental data of Shauinian
in our calculations would be erroneous because as a result of the periodicity of the development of vortices behind surface roughnesses, the coefficient
Co r i Iculo ted Test Direct I o
T-I
I T-2 I A T-3 T-4H
T-5 e T-6 o C-I I I 0 C-2 I C-3 ______40
r'
+ S i t Mod. O k9:12I4 + 4.0 3.2 2.4 .6 0.8 8.0 6.4 4.8 o. g) o, .6 T-I T-3 o o-0 4 8 2 6 20 k9 0.32 0.40does not appear constant in value. Based on the experiments with motion pictures of the region of cavitation behind roughnesses, described here, and. on our other experiments with motion pictures of the cavitation of a
circular profile,3 we will take into account the experimental values
which are related to the moment of maximum contraction of the flow from the wall behind the roughness. For calculations of Xr in such a case we will use = (1
+ a)/2
assuming that these values of more than all theothers correspond to the average values of p and v1 measured in our
experiments.
We will make the calculations of Àeg for sills of segmental shape with the assumption that aj = 1.0, i.e., with the assumption that the jet of flow about sills with a rounded edge does not experience additional constric-tion behind them.
We will represent the results of the calculations of and
on a plot Of4rseg - Í'(a/a51) (Figure lO). As was expected from
con-sideration of the Formulas
19J - Eli],
the curves of and eg convergewith decreasing values of a/a31 and meet at a/a51 0.
On the plot are also shown experimental points determined from measured pressures and. velocities for the initial state of cavitation. In each case the experimental values of 4r and Xeg are determined in the following way. In the set of experiments T-6 the onset of cavitation was
determined by a visual method and is taken as an average of five
obser-vations. As seen from the plot, the experimental value of Xtr is larger than the theoretical by 17 percent. The value of from the set of ex-periments T-1 is found by extrapolating the curve of 'tr = Í'( kg) (Figure
9).
Based on the accuracy of the methods applied in determining Atr, we can explain the deviations in the experimental and theoretical values of and eg as the probable errors in the methods used. Consequently, those values of the cavitation coefficient of a surface roughness which can be determined by calculation from
[9] -
[li] should serve as a criterion for the inception of cavitation of these surface roughnesses.The curves of Àtr = f(a/a51) for developed cavitation are con-structed from experimental points for sills of triangular profile.
Extra-po1atirt then to a/as1 = 0, ve find values of close to one another. It
should be noted that the development of cavitation of the sill In the set of experiments T- occurs differently from the development of cavitation which corresponds to a sill of triangular profile and is closer to the character of cavitation development 0±' the sill of segmental profile. Such a difference, possibly, is explained by the effect of too high a. base
of the sill which eliminates the constriction of the jet behind the sill. The fact, that the curves of Xtr,seg = f(a/a51) converge for de-creasing values of a/a51 and meet at a/a51 = O both for the initial stage of cavitation and for the fully developed stage of' cavitation, indicates that in channels of great height or for very small roughness heights in
21
slits, when a/a51 is vanishingly small, the conditions for the development of cavitation do not depend on the form of the surface roughnesses. The
maximum values of and. Xeg which correspond to the onset of cavitation
are both equal to unity. After inception, cavitation quickly turns into fully developed cavitation for small fluctuations of the pressure or velocity.
INFLUENCE OF TEE BOUTDARY LAYER ON TEE DEVELOPMEM
OF Ti± CAVITATION OF ROUGENESSES
In experiments with slit models in the absence of an artificially created roughness, the formation of cavitation on the opposing wall of the slit was often observed; this resulted from the accidental formation of surface roughnesses. Having located the sill on the constricting wall of the slit, it was necessary to investigate the influence of the boundary layer thickness and its characteristic velocity distribution on the forma-tion of cavitaforma-tion,18 inasmuch as a laminar layer of considerable thickness (as was detected In experiments with nozzles)19 or even a layer separated from the wall was expected on the constricting wall in tuis part of the
slit. Actually, for a height a =
0.3
(Model Si2) cavitation of the sills was not detected and the discharge coefficient of the slit remained invariable in comparison with Model Si1 (Table7).
For sill heights a = 0.6 xm cavitation was easily detected. Cavitation first appeared be-tween the first two sills which were located closer to the entrance where the boundary layer thickness vas less and the velocity at the crest of thesills was greater. The discharge coefficient of this model was less than the discharge coefficient of Model Si1.
TABLE 7
The cavitation of the sills of height a =
1.3
rmn in Model Sl4was very noticeable, but it also first arose at the first sills and. for a
larger value of Xl than in Model Si3. If the value of obtained
in the experiments be reduced to units in accordance with Formulas [6]
-[8)
and [9) - [11], then this value can be compared to the values X forNo. Model e. M si
1 Sl1 O
0.97
l.1402 512 0.3
--3 Si3
o.6
0.89
1.97
the roughness models. Such a comparison shows (Figure 10) that the value of A1.tr for Model Si4 is closer to the curve for Ar = f(a/a51) than the value of Asl.tr of Model Si3. In the latter case the considerable differ ence in Xsl.tr and si.tr fOr an identical value of a/a51 is explained as the effect of the boundary layer thickness which is more significant for a model with sills of smaller heights.
A comparison of the values of X1 for slits with a smooth surface and for slits with a rough surface shows that cavitation arises sooner in the second case than in the first, i.e., it arises sooner than the so-called "slotted tail cavitation" in slits of Type C,2° as is seen in the first line of Table 5 where = l.1.0.
CONCERNING TEE CAVITATION 0F A HYDROFOIL WITH A SMOOTH AJD WITh A ROUGE STJBFACE
The energy equation of the flow over the surface of a hydrofoil profile, free of the effects of adjacent hydrofoils and of tunnel wall ef-fects so that it can be assumed that a/a51+0, can be written in the form
00 V
+ =
,U -
pvwhere the index 'I,u" designates the lower or upper surface of the profile. We find that the formula relating the cavitation coefficient of the
hydro-foil profile Apr = (c'i - p)/vq, to the cavitation coefficient of a roughness Xr = (Po - p)/vq,is
(A
+l)q
=(À +l)q
pr ¿,u r
or
A
=x(1-
)-_
pr r l,u l,u
where Pl,u = i - q1/qis the pressure coefficient. We designate by that value of the cavitation coefficient of the profile for which cavi-tation on the profile, which is determined 'by its form, arises and, in
addition, we assume that A'r = Then, the difference
A* _A'*X
(l-f
)-
+ppr pr r 1,u 1,u ruin
makes it possible to compare the danger of the inception of cavitation of a hydrofoil with a smooth and a rough surface under the condition that the boundary layer is identical for both cases.
25
Taking for such a comparison identical points on both hydrofoils, i.e., assuming that
=
.,
we find that the difference Àr - is
positive for all value of r>l This indicates that the inception of
cavitation for a hydrofoil with a rough surface occurs sooner than for one with a smooth surface.
CONCLUS IONS
The inception and. development of the cavitation of individual surface roughnesses is analogous to the inception and development of the separated cavitation of poorly streamlined bodies. In particular, in the initial stages of the development of cavitation its periodicity is also characterized by the Strouhal number.
If the crests of the surface roughnesses are located outside of the boundary layer, then Formulas
[9) -
[11] can be used. for an approximate estimate of the danger of cavitation of the rougbnesses. If the crests of the roughnesses are located within the boundary layer, then for suchcalcu-lations it is necessary to add the coefficient x =
v51/v.1 to the first
term of the Formulas[9) -
[fi where v.5l can be taken equal to theveloc-ity at the level of the crests of the roughness with a certain approximation, which in general requires special investigation.
5.
In slit-like channels the danger of the inception of cavitation and the danger of erosion depend on the relative height of the roughness and. the thickness and velocity distribution of the boundary layer.1 In channels where the height of the roughnesses is vanishingly small in comparison to the height of the channel, the danger of the incep-tion of cavitaincep-tion and erosion depends only on the structure of the bound-ary layer in the flow about the parts of the hydraulic machine. The part of a blade profile with a turbulent boundary layer wïil be more subject to cavitation and erosion than the part of the profile with a laminar boundary layer or the part with separated flow.
The author wishes to express his appreciation to A.M. Rurniantzev, B.M. Frad.kin, A.I. Markov, and H.F. Shuravin for their cooperation in the studies of the cavitation damage of turbines and pump installations.
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Pan-Soviet Scientific Received In the
Research Institute of Ed.itor1s Office
NAVYOPPO PRNCWAH DC.
25
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