3 .EP. 98k
ARCE
SYMPOSIUM ON
"HYDRODYNAMICS OF SHIP AND OFFSHORE PROPULSION SYSTEMS"
HØVIK OUTSIDE OSLO, MARCH 20. - 25., 1977
"A THEORY FOR THE VIBRATORY FORCES ON A FLAT PLATE ARISING FROM
INTERMITTENT PROPELLER BLADE CAVITATION"
By John P. Breslin
Stevens Institute of Technology Davidson Laboratory Hoboken, New Jersey
SPONSOR: DET NORSKE VERITAS
Lab. v. Scheepshouwkund
Technische
Hogschoa
ABSTRACT
Theoretical studies of the effect of intermittent cavitation on propeller blades have been focusedon the prediction ofpressures at any
field point. This paper addresses the calculation of the force induced
by intermittent blade cavitation when the cavity geometry as a function of blade position is provided by either theory or experiment. A ship
of small draft and flat buttock lines in way of the propeller is replaced by a flat plate lying on the surface of a stream. The force induced on
the plate by the expansion and contraction of the cavitation on each
blade of an m-bladed propeller is found without determining the detailed
distribution of induced pressures, much of which makes no contribution
to the force. It is found that the force is given by a sum of many terms which reflect the effects of the simultaneous cavity motion and blade
rotation. A simple formula for the force arising from what is expected
to be the dominating term which depends on the second time derivative of the cavity volume is evaluated using cavity volume data reported by
Sontvedt and Frivold. The results show large values at the higher multi-ples of blade harmonics. The largeness of the force amplitudes at the higher harmonics may well be an artifact of the abruptness of the
vanish-ing of the cavity for which the data is, unavoidably, lackvanish-ing in precision. Further studies are indicated, perhaps in the use of the theory for a
compressible medium in the neighbornood of those values of time at which
the cavity collapses.
-1-INTRODUCTION
As fs well known, the increased loading on conventionally con-figured propellers has led to the frequent occurrence of intermittent propeller blade cavitation in the region between ±60° about the 12 o'clock position on tankers as well as on high-speed container ships.
Measure-ments of hull pressures on models in variable pressure facilities, as well as on ships, have shown these cavity-induced pressures to be large
not only at blade frequencies, but also at higher harmonics. While
there have been theories1'2 put forth for the radiated pressure field about blades with transient cavities, there have not been any analyses which display the roles of the cavity geometry, the harmonics of cavity
kinematics and propeller-boundary geometry on the various harmonics of
blade frequency hull forces.
The analysis which follows reveals that the dominant cavity-induced
force on a flat-bottomed shallow draft ship is expressible in a very simple form when the cavity volume as a function of blade position is
known (as it is in at least one case from stereo photographs reported
by Sontvedt3). The theory also reveals a host of additional terms aris-ing from couplaris-ing of the cavity harmonics of all orders with the spatially generated harmonics arising from varying blade aspect and distance from
the boundary during rotation. In addition, one is able to perceive how the higher harmonics of the force (twice and three times blade rate) can be greater than that at blade frequency. It is found that the larger
contributors to the force are very slowly changing functions of tip clear-ance for equal amplitudes of cavity volume harmonics. Thus, in spite of the simplicity of the idealized ship boundary, this theory reveals parametric dependence which is expected to be valid for at least certain
-3-In what follows, the theory is outlined and the need for numerical evaluations demonstrated. Unfortunately, it has not been feasible to carry out a complete numerical study at this time.
ANALYSIS
The ship hull is idealized to one which is locally flat in way of
the propeller and which has small draft in that vicinity and is,
conse-quently, replaced by a flat plate which extends to x = ±cc in the fore
and aft direction and has a fixed beam of 2b. The plate lies on the sur-fact of a spatially varying stream. The propeller axis is located d units directly below the plate and has a radius of r0.
The time or blade-positional"dependent cavity can be considered to be generated by a source distribution extending over the blade
sur-face covered by the cavity. We at first consider the effect of a single concentrated source M and later generalize to a distribution of sources
of density m' whose value can be expressed directly in terms of the total cavity boundary velocity. The condition on the free surface is one of
zero pressure and negligible water surface deformation and, hence, a condition of zero velocity potential (the usual high frequency free-surface condition which is certainly applicable here). To meet this condition, the source is imaged by an equal sink at any instant above
the water surface as indicated in Figurel.
If we wish to find only the force induced by the source and its image, we can do so by solving a simpler problem than that of finding the detailed pressure distribution on the plate. Thus, if 4 is the velocity potential of the plate in the presence of the source and its negative image, then the linearized pressure at any point in the fluid
is given by p = PU
where is the potential
of
the sourceis the pQtential
of
its negative imageOn the plate + 4'. = 0 and /x (. + = 0, so the force on the
plate from a single rotating, time-Varying source is simply
cob
Z =
f
J
p(x,y,d)dydx-m -b
Z=-p J
f
4'dydx- -b
One may observe that the convective pressure plays no role in the force,
regardless
of
the length of the plate and the location of its edges, say at x = x1 and x = x2, relative to the location of the propeller since-bx
dx
= -b2
- 4'(x1,y))dy= 0 as4'(x2,y)
4'(x1,y) E
0 The form of (3) suggests that, to find the force, we need onlyde-termine the integral
b
I
4'(y,o,t)dy -b where (y,z,t) =f
(x,y,z,t)dx X=_mrather than 4'. As 4' depends only on two space variables, we can reduce
our problem to one in two dimensions. We need only determine the
conver-sion of the conditions on 4' to provide the conditions that 4' must meet.
Thus 4' must satisfy
Thi..s method has been kindly pointed out by Prof. F. Ursell, consultant to Davidson lzboratory.
-2w(y,d) = 0
II >
±0
-5-b, z = d (10) b, z = d (11)z<d
(12)v2=0
(5)--=
-2w(x,y,d) b, z = d,aH x
(6) = 0I)'i
> b, z = d,aH x
(7)±O
r-- (8)where w is the vertca1 velocity induced by the source at the plate.
It is easy to show that must satisfy
+ = 0 (9)
2 az2
The problem posed by (9) through (12) can be solved to determine as an operation on w. This involves a series of rather involved integrations.
It seems more adroit to go directly to the force by finding only
b.
I
c(y,d;t)dy -bby employing Green's theorem, in a manner similar to that of Vorus.
If we introduce an harmonic function ji(y,z) such that
P/z
1.0on jy b, z = d; p = 0,
I'
> b, z = d; ij --0, /2
= z2 and consider+ s +
---
ds (13)taken-around the area between the line z = d, a large semi-circle of
all functions in (13) are harmonic in that area arid satisfy the
radia-tion condiradia-tions imposed, we find from Green's theorem that
b
f
(y,d;t)dy = -Mi-b
and now the force due to a solitary source is
z1 = p {M(t)
ip(s,o(t)))
(15)
where s,O are the polar coordinates of the source location. Thus, the
force can be found by determining the function 4' and evaluating at the source coordinates.
One procedure for finding 4' is to distribute z-directed dipoles in
< b and z = d and then determine the distribution of dipole strength by solving the singular integral equation posed by requiring 4'IBz =
along z = d and y in -b to b. Thus, we can find that
(z-d) /b2 - 2 d
4'
--b
)2+ (zd)2 (16)
and this can be expressed as the vertical velocity induced by a source distribution of density 2/b2 - r2, i.e.
b
1 c
4'--b
/b2 - 2.n "(y-fl)2 + (z-d)2dii
This observation is crucial as it enables us to express its harmonic con-tent Ln terms of the source angular variable by using the Fourier expan-sion of £.n /r2 + 2
-
2rs cos(O-y) to obtain, after some manipulations,b
2 n-i /b2 -r2 cosn-y
dii cos(n-i)8 (18)
f
n=l -b (n2d2)fl"2
This is simply a Fourier series which can be expressed as
*
See Formulas and Theorems for the Functwns of Math. Phys., W. Mag'nus,
F. Oberhettinger, Chelsea Pub. Ca., N.Y., p. 77)
r = A(s)cosnQ n=o A = 2d /b2 - r d = '2(/b2 d2 -d) ¶ 2 b lb2 -t2 cos(n+l)yd
A(s)=
-L
n+1
fl iT(2+d2)
2y = -tan (-) (see Figure 2).
To account for an entire distribution of sources with the instan-taneous boundaries of the cavitating region, we replace M by m(s,O')sdO',
as in Figure 3, and the local source density m is related to the local cavity ordinate r(s,O') by
a V a
m
=
-The force due to all sources is, upon using the dummy angle
a = - 0 + c: r 0
a(s)a
ava
= s'(0) A ( £.(s,0) - .--} (s,a+0 -o)Isdads n n=o 0 Swhere a(s) is the angular position of the leading edge of the cavity measured from the mid-chord of the blade; is the arc length to the terminus of the cavity from section mid-point.
Since 0 all along the leading edge of the blade and along the terminating curve of the cavity boundary, we may extract the time deriva-tives without contribution from the derivaderiva-tives of the limits and obtain,
after some reductions,
-7-with
r o
ct(s)
Z - -
f
sA (s)f
ç(s,a+o-o)cosn(ct+e-o)dcids tno s(0)
s,O) S (25) a(s)r
-
P -- L nf
A (s) (V - ws)f
r(s,a+6 -)sinn
(+o -c)dctds
s
(o)
'0
S
Now we can observe that the Umits s(0) and 2(s,5)/s are cyclic
functions of blade angle. We may now define
r o ai's) (A cos'8+8 snvO) =
f
sAf
V $(e)
"Z/s V 0s,a+O -a)cosn (cc-a)dcxds (26)
n-V=±qm
,q=0,l,2,3
(33)and
r a
(A'cos'+sinv)
=-f sA
f
C(s,cx+O -cl)sinn (ct-a)dctds (27) S Li'sr
0 cx
(CcosvO+V'sinv)
= f(v-ws)A
f
(s,cto -o)sinn (ct-a)dads
(28)
S L/s
0 r
0 ci
(C cos'O+V sinO) =
f
(V-ws)A
f
r cosn(ct-adadsV v fl
0 Then
a2
Z =
p
{(A cosvO+8
sinvO)cos nO + (A'cosv& +8'sinvO)sin nO)at2noV=o
' V Va
-
P n{(C'cosVB+V'sinvO)cosnO+(C cosVO+V sinv8)stnn}
V V
n0
\)=o()
The total force for an m-bladed propeller is obtained by shifting the
blade angle to 0 - Zirk/m and summing over k from 0 to m - 1. Then the
only terms which contribute are those for which
n+V=qm
,q0,l,2,3
(32)where q is the blade frequency harmonic order number. The sum of these terms will produce m times the harmonic coefficients of the force from
a single blade.
We shall not write out all this plethora of terms, being content here to evaluate what is probably the dominating term. We may expect the second term of (25) to be weak because
- ws = /j2 + (rw)2 - ws '. 0
out near the blade tip. In the first term of (25), we focus only on the
term n = 0 and note from (20) that A is independent of s. Then we have,
0 for a single blade,
32
ZpA (e)
0where v(o) is the instantaneous cavity volume.
If we express the cavity volume as a Fourier series and sum over all blades as described above, we will obtain the following formula for
this contribution to the force.
32
Z1 '.' pmA -
(acosqme+b
sinqmO)°
3t2 q=o qm qm
Since 32/Z3t2 = w232/a02, we find
Z1 ".' -pw2m3A q2(a cos qmO + b
sin qmo) qm
qo
2ir 121T where a-
f
V(o)do ; a 0 2n qm= -
f
V(0)cosqmode 0 2ir b =!
V(6)sinqmode qm ir 0 (3L1)-9-We may see by inspection that the higher harmonic contributions,
viz. q = 2,3,4.. .will be buoyed upit by the factor q2. Thus, for a
5-bladed propeller, even if the amplitude of the 15th harmonic of the
cavity volume is of second order with respect to the 5th, the term
32a15 will be of the same order as a5, and we may see that it is
possi-ble to obtain important contributions to the force at frequencies which
are multiples of the blade number.
EVALUATION
Drs. Sontvedt and Frivold3 have reported determinations of the
cavity volume on blades of a twin screw tanker. The variation of cavity volume with blade angle from these stereo photographs is given in Figure 1
The harmonic coefficients, as defined by (36) and (37), are listed
in Table 1. Here we can see that the modulus Ia2 + b2 decreases very
slowly with increased order v.
To evaluate the modulus of the force due to the term defined by
(35) with A taken from (20), i.e.
Z11 '2pm3(q)2 [1d2+b2-dJ v'a2 +bz
qm qm
qm
we use the diameter of the 5-bladed propeller (m = 5) from the 224,000 ton tanker of Ref. 3 (10.9 ft.) and place it beneath our plate of semi-width bdiameter (i.e., 21.Bft.wide) with 3ft. tip clearance. The
propel-ler revolutions were approximately 114/min., giving w = 11.94 rad/sec. The values of the forces at three frequencies are seen in-Table 2 to be large ad increase remarkably wIth order number q. To put these in
per-spective, the forces on this boundary must be calculated from the propel-ler operating in the ship wake without cavitation. It is planned to do
this using data kindly provided by Frivold.
CONCLUDING RARKS
The foregoing calculations show that the blade frequency forces arising from the leading term of this theory for the effects of inter-mittent cavitation are large, particularly those at twice and three
times blade frequency. It is possible that the neighboring terms which
involve weghted integrals of the Cavity ordinate distribution coupling with the higher harmonics of the boundary function A will also be sig-nificant. tt is regretted that evaluations of these, as well as the forces generated by the non-cavitating propeller, could not be effected
in time for this paper. Finally, it is to be appreciated that the forces
arising from the pressure jumps on the blade while cavitating have not
been evaluated, This can be done only by solving the entire boundary
value problem. The effect of hull curvature and finite hull length, particularly aft of the propellers, will be sought in the future by applying analogous techniques to a semi-submerged spheroid.
REFERENCES
Johnsson, C.A. and Sontvedt, T.: "Propeller Excitation and Response
of
230,000 t.d.w. Tankers," Ninth Symposiwn on Naval Hydrodynamics, Paris, August 1972Noordzij, L.: "Pressure Field Induced by a Cavitating Propeller" International Shipbuilding Progress, Volume 23, April 1976, No.
260
Sontvedt, T. and Frivold, H.: "Low Frequency Variation
of
the Sur-face Shape of Tip Region Cavitation on Marine Propeller Blades andCorresponding Disturbances on Nearby Solid Boundaries," Eleventh Symposium on Naval Hydrodynamics, University College, London, April
1976
Vorus, W.S,: "A Method for Analyzing the Propeller-Induced Vibratory Forces Acting on the Surface
of
a Ship Stern," Trans. SHAME, Vol.82, 1974
1-the Presence of a Flat Plate in 1-the Water Surface at z = d
Fig. 2 - Geometry
of
Plate and Rotating, 0-Dependent SourceFig. 3 - Geometry
of
Cavity and Propeller BladeFig. 4 - Cavity Volume on a Propeller Blade as a Function of Blade Angle
PLATE OF BEAM 2b
IN WATER SURFACE
zd
d
b
PROPELLER AXIS
\\v(r,y)
SHIP AXIAL
AND TANGENTIAL
WAKE
FIG. I.
SCHEMATIC OF ROTATING,TIME-DEPENDENT
SOURCE
AND IMAGE SINK IN THE PRESENCE OFA FLAT PLATE
IN THE WATER SURFACE AT z:d
y
y',z')
FIG. 2.
GEOMETRY OF PLATE AND ROTATING,
CAVITY BOUNDARY
s(8)
z
BLADE REF LINE
DEFINED BY CENTER
OF SHAFT AND MID
POINT OF CHORD
AT HUB
LOCUS OF MIDPOINTS
OF CHORDS
3.10:
I2.10
LU-J
0
>
>-1.10
I-0
-J
0
I.-SUM OF FIRST
18 HARMONICS
0
-60°
-40°
_200
0
200
BLADE POSITION ANGLE 6 FROM WAKE PEAK
FIG.4.
CAVITYVOLUMEONAPROPELLER BLADE
AS A FUNCTION OF BLADE ANGLE
(FROM REF.3)
Modulj and Phases
of
Harmonics of BladeCavity Volume for 224,000-ton Tanker
Modulus Rarinonw 2 + bz Phase Order (lag) (millions of mm3) in degrees Moduli of Vertical Multiples of Blade Order No. q qm Forces at Frequency
Tqm
lbs. 10.3818027
-11.3111414635 20.3557183
-22.26421486 3 0.31.59653 -32.27141799 14 0.2675181-40.7517552
5 0.2162931 -146.83171475 6 0.16814358 -149.30214097 7 0.12966014 -146.7627969 8 0.1041561 -38.84753914 90.092264
-28.3595955
100.0890505
-19.7582870
110.0885036
-14.7535855
12 0.08671471 -12.146785141+ 13 0.0821411+0 -11.14711399 140.0756513
-10.14750327 15 0.06714952 -8.36114244 16 0.05941485 170.0529024
1.765141493 18 0.014838658.8696036
TABLE 2 TABLE 2 5 6428.. 2 10 1 01400: 3 15 1 6600"These large values may well be artifacts of the application of Fourier analysis to the very rapidly changing cavity volume as it passes to zero where the curvature of(0) becomes
discon-tinuous. Thismay require adiffer-ent type
of
mathematical analysis to deal with this behaviorism.For example, it is easy to show that, although the Fourier series
representation for a trpezoidally shaped Y(e) converges, the second derivative of this series will
