Proceedings of the Symposium on High Powered Propulsion of Large Ships, Part 2, Publication No. 490 of the NSMB, Wageningen, The Netherlands

O

1

I

I

P1974-8

SYMPOSIUM ON

HIGH POWERED PROPULSION OF LARGE SHIPS

PART 2

P1974-8

PART 2

DECEMBER 10, 11, 12 AND 13, 1974 WAGENINGEN, THE NETHERLANDS

TECHNISCHE UNIVERSITEIT

Scheepshydramechanica

Archief

Mekelweg 2, 2628 CD Delft

Te1:015-786873/Fax:781836

PUBLICATION NO. 490

latwadodunn vocr

Scheepshydromechankm

Awohlef

Thursday, December 12, 1974.

_{lAekehm 24 2828 CD DM}

TA:0115-7=73.Fac0111.7108

Afternoon session.

PROPELLER HULL INTERACTION; EXCITATION FORCES.

P

&-2

Chairman : Dr. L.A. van Gunsteren, Lips B.V., Drunen,

The Netherlands.

Method to calculate the pressure field induced by a XVI

cavitating propeller,

by L. Noordzij, Netherlands Ship Model Basin, Wageningen, The Netherlands.

Cavitation induced hull pressure; some recent develop- XVII

ments of model testing techniques,

by E. Huse, Skipsmodelltanken, Trondheim,

Norway.

Vibratory response of hull and line shafting to hydro- XX

dynamic excitations,

by G. Volcy and J.C. Masson, Bureau Ventas, Paris, France.

The problem of influence of solid boundaries on pro- XVIII

peller-induced hydrodynamic forces,

by M. Garguet and R. Lepeix, Chantiers de l'Atlantique, Saint Nazaire, France.

Propeller-induced hydrodynamic hull forces on a Great XIX

Lakes bulk carrier. Results of model tests and full scale measurements,

Propeller Hull Interaction; Excitation Forces

A METHOD TO CALCULATE THE PRESSURE FIELD INDUCED BY A CAVITATING PROPELLER

by

L. NOORDZIJ

NETHERLANDS SHIP MODEL BASIN, WAGENINGEN, THE NETHERLANDS

ABSTRACT

From experiments it is known that a cavitating propeller can

prod-uce large pressure fluctuations on a ship's hull. This is for the greater part due to the volume variation of the sheet cavity on the propeller blade. It is of importance to find a description of the cavity volume as a function of the angular position of the blade. In this paper a method is presented to calculate this

volume. The method is based on a well-known linearized theory about cavity flow past a hydrofoil. When this volume is known the pressure field can be found.

1. INTRODUCTION

From full-scale as well as from model experiments it appeared that a cavitating propeller can induce large pressure fluctuations on the ship's afterbody. The amplitude can be an order of magnitude larger than would occur with a non-cavitating propeller. For the latter the blade harmonic component determines for the greater part the pressure field. In the case of a cavitating propeller the second and even higher blade harmonics are of equal importance. The phenomenon of pressure amplification by cavitation, as in-dicated by Van Manen /1/, was systematically investigated by Huse /2/ and Van Oossanen and Van der Kooij /3/. Some full-scale experiments were conducted by Johnsson and SOntvedt /4/.

It appeared that the volume variation of the sheet cavity on the propeller blade plays a dominant role in the enhancement of the pressure amplitude of a non-cavitating propeller. To gain insight Huse /2/ started a theoretical investigation in order to cal-culate the pressure fluctuations. Also Johnsson and SOntvedt /4/

presented a first approximation of the pressure amplitude. As re-viewed by Noordzij /5/, the investigations have the disadvantage

that the volume of the cavity is estimated from experiments. The problem encountered in the experiments is the determination of the cavity thickness. The cavity extent can be measured rather accurately. So, due to the inaccurate value of the cavity thickness, large deviations appear between the experimental and theoretical pressure amplitude /2/. To overcome this a theoretical

approximation for the cavity must be found. When this volume is known the field pressure resulting from volume variations can be

calculated.

In this paper we therefore focus attention to a theoretical approximation of the cavity geometry. For this much will be used

from linearized theory for cavitating hydrofoils based on a free

streamline model. This is a two-dimensional theory for steady

flow which gives a relation for the cavity sectional area as a

function of cavitation number, angle of attack, blade camber and cavity extent. This can be found in, among others, Geurst /6/, /7/, Geurst et al /8/ and Hanaoka /9/. The theory of Geurst gives

results for partially and fully cavitating two-dimensional hydro-foils. As a first approximation we used this instead of the most general three-dimensional unsteady flow model. We approximate the

flow by a quasi-steady strip model. The propeller blade section _at

radius r and angular position

Y, Fig. 1, is

to

two-dimensional hydrofoil in uniform incompressible flow. At that particular station we assume the inflow velocity, cavitation number, angle of attack and camber to be known. One assumption is, as

mentioned, two-dimensional strip theory. The other is the steadiness of the problem. The error introduced by this approximation is found in the neglect of vortex shedding due to the non-uniformity of the wake and phase differences in chordwise direction. In a forthcoming paper we shall estimate the effect on the pressure field.

As shown in Fig. 1 the cavity extent L (chordwise) is smaller than the chord length c for r<rt and larger for r>rt. This is usually found in the experiments. So in the theory solutions for partially and fully cavitating hydrofoils will be used. When employing the theory of Geurst this will introduce a further problem, since the linear theory fails for cavity lengths approaching chord length. For the partially cavitating hydrofoil (r<rt) this break-down in

Propeller Hull Interaction; Excitation Forces

Fig. 1 Propeller blade with sheet cavity.

linear theory occurs for L/c >.75. For fully cavitating hydrofoils the failure occurs for L/c smaller than about 1.15. This is shown in Fig. 2 for a flat plate. In this figure L/c is given as a function of the ratio of the cavitation number a and angle of attack a for

a flat plate.

A solution to the instability problem, as shown in

Fig.

dashed curve when L/c-,-1, is to change the closure condition as used by Geurst, who considered the cavity rear to be closed. A possible

solution is to consider a re-entrant jet model for the cavity. The problem of cavity closure and the coupled failure at the trailing edge was treated by Hanaoka /9/. He assumed the cavity to be open and used an empirical expression for the thickness at the cavity rear. The results of Hanaoka are also shown in Fig. 2. The drawn curve clearly demonstrates the suppression of the in-stability. However, as yet the results of Hanaoka are not in a form useful for computational purposes and have not been fully analyzed at this time. Since we are interested in a direct first approximation and to keep computational procedures simple, the

/ c

2

0.75

XVI, 4

o 5 10 _/ 15

Fig. 2 The ratio of the cavity chordwise extent L and the chord length c is given as a function of the ratio of the cav-itation number a and the angle of attack a for a flat plate.

Geurst /6/, /7/.

Hanaoka /9/.

present method is based on Geurst's results. The problem for

.75<L/c<1.15 is solved by smoothing the curves in a linear manner. In the theory of the following sections we suppose the wake in

which the propeller operates to be known. Angle of attack ot,

camber A and cavitation number a are calculated from Van Oossanen /10/ for a given blade geometry in a particular angular position and radial station at the blade. We assume a parabolic-arc profile for the camber line. Then the cavity geometry on a blade section can be calculated. Since a measured wakefield is used, negative angles of attack and with this pressure side cavitation might appear. In the present theory this is left out of account. In the numerical

4

Propeller Hull Interaction; Excitation Forces

analysis care is taken of that. With the cavity geometry known for different angular positions the pressure field due to the varying geometry (volume) can be calculated. Actually the propeller rotates

in the vicinity of the hull. So the free field _{pressure must be}

corrected and even the cavity geometry. The latter correction is not carried through. With respect to the pressure some remarks

will be made. To calculate the pressure, taking into _{account the}

hull and the free surface, a diffraction problem must be solved. Since in general we are interested in vibratory forces, the

pressure is integrated _{over the hull. This is a rather tedious}

problem. When we are interested in the force on the ship, a more

direct approach should be used without need for solving the

diffraction problem. It can be done by using the so-called _Chertock

formulas (Ogilvie, /11/). An example of this method can be found

in, among others, Vorus /12/. In this paper we do not calculate the

force on the ship. To find out whether the presented approximation

of the cavity is reasonable we calculated the _{pressure to compare}

it with experiments. For the sake of illustration we approximated the diffraction problem by a so-called solid boundary factor of 2. The factor by which the pressure on an infinite flat wall must be

multi-plied.

In the present theory we do not account for the _{source located on}

a solid wall (propeller).

2. BASIC EQUATIONS FOR CAVITATING HYDROFOILS

In this section we first pay attention to the results _{for a}

partially cavitating hydrofoil.

_For

reader is referred to /6/ and /8/. A propeller blade _section

is placed in the flow of an incompressible non-viscous _{liquid. The}

liquid velocity at infinity is U, the angle of _{attack a(Fig. 3).}

In the theory the profile is represented by the camber _line.

Thick-ness effects are not taken into account. For this the reader is referred to Wade /13/.

If the pressure at suction side decreases below vapour pressure pv, we assume a cavity to occur. A measure for this is the cavitation

number

, Poo

-

PU4

XVI, 5

CAVITY (AREA S)

CAMBER LINE

Fig. 3 Schematic representation of a propeller

blade section with camber X placed in a uniform stream U making an anglea with U. where pc. is the pressure at infinity and p is the density of the liquid. For not too small values of a and a the cavity will appear on a portion of the suction side of the propeller. In the analysis of Geurst cavity length L, lift, drag and cavity areas S are calculated as functions of a, a and X. In this analysis a and X are assumed to be small with respect to unity. The boundary of the cavity is taken as a free streamline. This leads to the following

non-dimensional expressions for cavity length L (=sin2 6) and area S

respectively: a(l+cos6)sin6-2(1-cos6)+ l'cos6sin36 = 0, 2 2

and

(1-cos(5) (l+cosö) (-1-cos6+4cos36sin6)),

where 0 < 6 < .

(2.2) and (2.3) are valid for a parabolic arc profile. In general we are not dealing with parabolic arc profiles. However, from a computational point of view we assume (2.2) and (2.3) represen-tative for the class of profiles under consideration. In the case of the partially cavitating profiles we use a, a and X from /10/. X is the maximum camber. For given a, a and X, L can be calculated with help of (2.2). Once this is done the cavity sectional area S can be calculated with (2.3). A curve for the cavity length is shown in Fig. 4 for a given value of a and À.

XVI ,6

(2.2)

1.00

075

11

0.50

0.25

Propeller Hull Interaction; Excitation Forces

XVI, 7

(2.4)

0.02 0.04

a/o 0.06

0.08 1 00

Fig. 4 Cavity length L versus a10 for a given value of a and A(Geurst /8/).

This curve takes its maximum at 6=7/3 or L=.75. The value of

(a/'''max for a given value of A and a is

4/3 (a/a)

-max 72+9 X/a

So it might be possible to obtain two cavity lengths for a certain

value of a/a. However, as follows from Geurst, the linearized theory breaks down when L is near unity. Then only stable solutions can be expected for L=<.75.

Now we shall pay attention to the results for a fully cavitating hydrofoil. For the present theory we make another approximation. We shall use the results of a supercavitating flat plate. So

A is set equal to zero. This makes no essential difference with AO except for a lot of numerical problems. The case X=0 is con-sidered to be a satisfactory approximation. In a forthcoming paper the effect of camber on the pressure field will be investi-gated for a fully cavitating hydrofoil.

From /7/ the relations for a supercavitating flat plate are

a

and

7 a cotan 6

S

16 (1-sin6)2

where L = (cos26)-1 and 0<6<7/2.

S can be calculated from (2.5) and (2.6). Also for the fully cavitating case we take the values of a and a following from Van Oossanen /10/.

When 6.0, S.... This indicates the breakdown of the linear theory for L.1. So we are faced with unknown values of S for .75<L<1.15. This problem is simply solved by a linear matching of a partially and fully cavitating hydrofoil.

With the sectional cavity areas (S) known the cavity volume can

be calculated. The volume of the cavity is obtained by integrating

the cavity sectional area S over the blade radius (Fig. 5).

CENTER LINE MIDCHORD XVI, 8 LINE SOURCE SECTION AT r MIDCHORD LINE SOURCE

(2.6)

Fig. 5 Volume variation of sheet cavity represented by a number

dS

of line sources of strength -- Ar.

Propeller Hull Interaction; Excitation Forces

The pressure field induced by the volume variations of the cavity are found from the potential associated with this source behaviour of the cavity. To this end the volume variation of the cavity is represented by a line source at midchord of the cavity /2/.

The strength of the line source follows from the variation in time of the infinitesimal volume SAr. The infinitesimal velocity potential

AcP becomes

dS Ar dt

Acp

-4, DM

where Dm is the distance from the infinitesimal line source on a

blade to a field point (R,E,c) _{(Figs. 5 and 6).}

PORT STARBOARD

DIRECTION OF ROTATION

Fig. 6 Designation of coordinates of field point by cylinder

coordinates (R, E,

E),

In formulating (2.7) COSK is taken equal unity (< is rake). So as in the preceding and following sections no difference is made

between the radius along the blade and r. The expression _{for Dm}

is given in the appendix. The potential

(1) for the pulsating cavity

on one blade is found from (2.7) by integrating AcI) between rr and

In a frame moving with the ship the material derivative d/dt is approximated by 27119/9y. n is the number of shaft revolutions per

sec.; (2.8) becomes rr n 2 DS/9y dr. rm DM

3. PRESSURE FIELD INDUCED BY A CAVITATING PROPELLER

The instantaneous free space pressure p at a certain field point

(R, &, E) can be obtained from Bernoulli's equation

p + 1/2pU2 + = constant,

at

where Ut is the total velocity at the considered point. In this paper we assume Ut to be the sum of the ship velocity V and the induced velocity by the cavity. It will appear that the largest contribution to p stems from p3/9t.

From the appendix it follows that the pressure becomes, straight-forward linearizing (3.1),

p = 27np¡SP_ + pV

9y AF (3.2)

For the pressure we obtain from (2.9) and (3.2)

p =

2f t aS/aY

This represents the contribution of one propeller blade to the

pressure field. For a Z-bladed propeller we obtain

rt PN = E (7n2p 9y

f

I

So far we did not include the influence of the hull and the free

surface. In this paper we do not take into account the effect of

the free surface. Allowance has been made for the hull by

intro-ducing a boundary factor of 2. This factor follows from the exact

XVI, 10

(2.9)

(3.1)

(3.3)

Propeller Hull Interaction; Excitation Forces

solution for a source near an infinite flat boundary. Besides of this the effect of the wall on the cavity geometry is left out of account. Finally we obtain for the pressure on the hull

rt pt = - E (27n2

p TT

For comparison with experiments we also use the dimensionless version of (3.5):

k - pt P

pn2D"

where D is the propeller diameter. With (3.6), (3.5) yields rt k = -

(1;

S/'Y

Since the cavity gecmetry is not given explicitly as a function of

Y, (3.7) must be evaluated numerically. Before describing this

numerical process some remarks will be made for the pressure field generated by a cavitating propeller.

In the presented theory we considered the effect of a fluctuating cavity on the pressure field. This is a source type behaviour of the cavity. On the other hand there is the thickness effect of the cavity. This is a dipole type behaviour. As shown in /2/, the pressure field is dominated by the source behaviour. It follows that the ratio of the potential resulting from the source and the potential from the cavity thickness is of order (BS/aY)/S.

In the case where the cavity volume varies strongly during a propeller rotation this ratio can exceed unity considerably. This usually occurs for a propeller operating in a non-uniform

flow.

From (3.7) it follows that the ratio of the first and the second term, between the brackets, in the right-hand side is of order TrnD/V So in most cases the first term delivers the largest contribution to

XVI,11

(3.5)

(3.6)

For a propeller we have after a rotation over an angle 27/Z that each blade recovers the same pattern. Thus the fundamental frequency in the pressure field is the blade frequency, i.e., shaft frequency times blade number. Also for a fluctuating source strength this can be shown analytically. The pressure field can be given as a Fourier series with frequencies which are integer multiples of the blade

frequency. The pressure coefficient can be represented by a Fourier

series

CO

k (y) = E

Kv sin (my+4)m). m=l -m

Following the discussion above for a Z-bladed propeller this becomes

k (y) = E K sin (mZY4-qt,mz). m=l PmZ

It is our aim to calculate k and to determine KpmZ and cpmZ

4. NUMERICAL PROCEDURE

The numerical procedure consists of three parts.

First we we have to determine cavity extent and cavity sectional

area for both partial and full cavitation on each radial and angular

position. To this end we used the criterion that for 0/0,10-2

sheet cavitation will occur (for a<C) no cavitation will appear).

We also test whether this cavitation is full or partial. For partial

cavitation we have to solve (2.2) by an iterative method in order

to find the extent and sectional area. This is done with help of

Newton's method. When (a/a) is greater than

_(0/0)max,

(2.4) supercavitation is assumed. So for (a/u)>(a/G)max we jump

to the supercavitation curve (Fig. 2). This means that a too large

cavity area is introduced in a certain range of a/a. For the

super-cavitation case L and S are found from (2.5) and (2.6). By

sub-stituting (5 from (2.5) into (2.6), one should be aware of the

fact

that S takes its minimum for (1/L)=1/2/3 as shown in Fig. 7. In this

figure S is given as a function of L according to (2.6). So for

L<2/13 and L-.1 S becomes infinite. In the computational scheme we set S equal to the minimum value

10

5

o

Propeller Hull Interaction; Excitation Forces

I I I I

0.2 0.4 0.6

1/1

Fig. _{7 Cavity area S is given as a function of cavity}

extent according to (2.6) _{(Geurst /7/).}

Secondly with the cavity area known for the _{different radial and}

angular positions, we can calculate the pressure coefficient from

(3.7). These computations require both _{differentiation and}

inte-gration. This is carried through with help _{of second- and}

third-order polynomials. In that case one should be aware of the fact

that the number of intervals in angular _{direction corresponds}

with an integer multiple of the blade number. The integrals in

(3.7) are evaluated with an adapted _{Simpson's rule.}

Finally harmonic analysis is carried out numerically. Since we

XVI, 13

0.8 1/2 1.0 15

use 30 angular positions, we are able to calculate the ampltitude for the first 15 harmonic components.

5. DESCRIPTION OF HULL AND PROPELLER

In order to compare the results of our method with experimental results, we give hull and propeller geometry for a particular case. We use the geometry as given in Oosterveld et al /14/. Oosterveld et al presented results for the pressure amplitude measured at the hull of the ship for different propellers. The pressure is measured at different stations at the hull. In Fig. 8 these stations are indicated.

GENERATOR 2200 200 1700 1 2 3 O o o

0

Fig. 8 In this figure the stations where the pressure transducers are

mounted are given with respect to and R equal 0 (lengths

are given in mm). The ship model was made to a scale of 1:33.25.

Propeller Hull Interaction; Excitation Forces

In Table 1 the values for R, E and E are given for the various

positions at the hull.

CENTER LINE 7 5°

Table 1

With help of the theory in /10/ the values of a,a and X are calculated at the various radial and angular positions for the wake measured at the location of the propeller. The wake can be

found in /14/. The propeller (no. 4563) particulars are shown in

Fig. 9. GENERATOR LINE MIDCHORD

op-1s

AI

_oseurAl

*vim

(I°

For this propeller the pressure field is calculated. We also

determined the pressure field for a propeller with smaller camber

and larger pitch distribution (propeller no. 4768). Figure 10 shows

the distribution of pitch and camber as a function of the radial

distribution. 0.06 0.05 0.04 fm/c 0.03 0.02 0.01 o P/D 0.9 0.8 0.7 0.6 02 03 04 05 _06r/R07 0.8 09 10

RADIAL DISTRIBUTION OF MAXIMUM CAMBER

PROP No. 4768 _ 4563 0.5 02 03 04 05 0 6 ,0 7 08 09

RADIAL DISTRIBUTION OF PITCH

Fig. 10 Radial distribution of pitch and maximum camber of propellers no. 4563 and no. 4768.

The two propellers have the following general particulars:

number of blades Z = 5,

diameter D = 10.226 m,

hub/diameter ratio = 0.199.

The propeller rotates with n=1.3317 revolutions per second. The

ship velocity V=8.086 m/sec.

Hp 0 03 0.02 001 o -0.01 -002 -0.03 -0.04 0.004 0.003 0.002 Kp 0.001 -0.001 -0.002 -0.003 -0.004

Propeller Hull Interaction; Excitation Forces

For the propellers the cavity geometry is calculated at 7 radial and 30 angular positions. The 7 radial positions are: 2r/D=.4,

.5, .6, .7, .8, .9, .975.

6. RESULTS OF HULL PRESSURE CALCULATIONS; COMPARISON WITH EXPERIMENTS In Fig. 11 the pressure coefficient kp from (3.7) is given for

the various hull positions as a function of y (angular position).

HULL POSITIONS

0.04

XVI,17

Fig. 11 k (3.7) is given as a function of y for

different hull positions (see Fig. 8).

In Tables 2 and 3 the results of harmonic analysis are presented. Also the blade frequency component of the experiments is given.

In both tables K andmZ are the amplitudes and phase angles

PmZ

respectively according to (3.9).

Table 2 Propeller no. 4768; pressure amplitude and phase angle.

Table 3 Propeller no. 4563; vressure amplitude and phase angle. In Fig. 12 the observed as well as the calculated cavitation pattern is shown for the propellers no. 4563 and no. 4768.

XVI, 18

pattern

//

Propeller no. 4563

Fig. 12a Observed cavitation Calculated cavitation

pattern on model

position harmonic components experiment (model)

hull K

P5 05

K

P10 010 Kp/5 015

blade harmonic ampl.

1 .02 229° .003 163° 0 0 .04

2 .026 226o .006 164o 0 0 .07

3 .027 227o .006 166o 0 0 .08

4 .016 222o .004 152o 0 0 .03

position harmonic components experiment (model)

hull K 05 K cp K

0

P5 P10 10 p15 15 blade harmonic ampl.

1 .003 233o .001 338o 0 0 .02

2 .0037 230o .0011 334o 0 0 .04

3 .004 2310 .0011 335o 0 0 .04

Propeller Hull Interaction; Excitation Forces

XVI, 19

Propeller no. 4768

Fig. 12b Observed cavitation 'Calculated cavitation

pattern on model pattern

7. DISCUSSION AND CONCLUSIONS

The aim of this paper is to find a description of the geometry _of

the sheet cavity in order to calculate the excitation force _{on the}

hull. To verify the presented method we calculated the pressure for a configuration for which also experiments exist. It follows that the calculated pressure is systematically below the measured

pressure (tables 2 and 3). For propeller no. 4563 showing the least cavitation the pressure is considerably below the experimental pressure. This is not surprising, since in theory the contribution

of the non-cavitating propeller (the dipole type contribution) _was

left out of account. This contribution is of the order 10-2 (/2/,

/3/). This explains the deviation between theory and experiment at

least partially. Furthermore the cavity is located on a propeller.

The effect is to increase the pressure.

In theory we approximated the cavity by a line source of finite

length and variable strength. On the other hand the cavity can be

represented more exactly by an infinite number of _sources

represent-ing the volume variation of the cavity. The

order of the ratio of the chord length and the distance from the source to a point where we want to know the pressure. Considering the assumptions made this error is acceptable.

We conclude that the presented method to calculate the cavity geometry is recommendable since experimental determination of the cavity volume is avoided.

Some final remarks can be made about future work.

To calculate the force on the hull the source strength can be used in the so-called Chertock formulas. To calculate unsteady forces the effect of unsteadiness of the flow on the pressure field needs further analysis. Also the closed streamline model for the cavity needs further attention to avoid closure problems near the trailing

edge of the blade.

Acknowledgement. The author would like to thank Mr. D. Golden

(N.S.R.D.C.) for his invaluable help with the numerical analysis.

Propeller Hull Interaction; Excitation Forces 8. REFERENCES

Manen, J.D. van, "The Effect of Cavitation on the Interaction Between Propeller and Ship's Hull", Intern. Shipb. Progr. 19, No. 209, pp. 3-20, 1972, NSMB Publ. no. 397.

Huse, E., "Pressure Fluctuations on the Hull Induced by Cavitating Propellers", Norwegian Ship Model Experiment Tank Publication, No. 111, March 1972.

Oossanen, P. van and Kooij, J. van der, "Vibratory Hull Forces Induced by Cavitating Propellers", Trans. Royal Inst. Naval Arch., 115, pp. 111-144, 1973, NSMB Publ. No. 404.

Johnsson, C-A. and SOntvedt, T., "Propeller Excitation and Response of 230,000 TDW tankers. Full Scale/Model Experiments and Theoretical Calculations", DnV Report No. 79, 1972.

Noordzij, L., "A Note on Cavitation-Induced Pressure Fluctuations" Internal NSMB Publ. 1974.

Geurst, J.A., "Linearized Theory for Partially Cavitated

Hydrofoils", Intern. Shipb. Progr., 6, No. 60, pp. 369-384, 1959. Geurst, J.A., "Linearized Theory for Fully Cavitated Hydrofoils",

Intern. Shipb. Progr. 7, No. 65, pp. 17-27, 1960.

Geurst, J.A. and Verbrugh, P.J., " A Note on Camber Effects of a

Partially Cavitated Hydrofoil", Intern. Shipb. Progr. 6, No. 61, pp. 409-414, 1959.

Hanaoka, T., "Linearized Theory of Cavity Flow Past a Hydrofoil of Arbitrary Shape", Selected papers from J.S.N.A. (in English),

Vol. 3, pp. 56-74, 1969.

Oossanen, P. van, "Calculation of Performance and Cavitation Characteristics of Propellers Including Non-Uniform Flow and Viscosity", Doctor's Thesis, Delft Technical University 1974. NSMB Publ. No. 457.

Ogilvie, T.F., "The Chertock Formulas for Computing Unsteady Fluid Dynamic Force on a Body", Zeitschr. Angew. Math. Mech. 53, pp. 573-582, 1973.

Vorus, W.S., " A Method for Analyzing the Propeller-Induced Vibratory Forces Acting on the Surface of a Ship Stern",

Paper No. 4 presented at the Annual Meeting Soc. Naval Arch. Mar. Eng., Nov. 1974, New York.

Wade, R.B., "Linearized Theory of Partially Cavitating Plano-Convex Hydrofoil Including the Effects of Camber and Thickness",

J. of Ship Research 11, No. 1, pp. 20-27, 1967.

Oosterveld, M.W.C., Verdonk, C., Kooij, J. van der and Oossanen, P. van, "Some Propeller Cavitation and Excitation Considerations for Large Tankers", Paper presented at West European Conference

on Marine Technology, May 1974, The Hague.

Landau, L.D. and Lifshitz, E.M., "Fluid Mechanics", Vol. 6, Pergamon Press, 1959.

Propeller Hull Interaction; Excitation Forces APPENDIX

Distance D from Line Source to Field Point (R, C, e).

The coordinate system for the propeller is shown in Fig. 6. Field

points are indicated with cylindrical coordinates R, C, E. A

point on the blade is given with the cylindrical coordinates r, x, y. The sheet cavity is represented as a line source of finite

length defined with respect to midchord of the blade. In Fig. 5

the geometry of the cavity is shown. Some quantities must be known before calculating Dm. First there is the angle between the location of the line source and midchord, 6 say.

t1

-2r cos 8,

where t t2' L and 8 are defined in Fig. 5. Secondly, there is the

angle between midchord and center line, 8m. The center line is the line through the axis and the tip of the blade. em follows from the blade geometry as given in Fig. 9.

The distance between the infinitesimal line source and a field point

is Dm = ((C-P8/27+rsinK)2 + r2+R2 -2 1 2Rr cos(e-y-8+8 m 7 - --(M-1))) 1/2 Z

where K is the rake angle, z the blade number and M = 1(1)Z.

Free Field Pressure. The free field pressure is

2 9clo,

p = const 1/2pUt -Dt

We assume Ut to be the sum of the ship velocity and the induced

velocities by the cavity. The latter are the elements of grad (1).

These elements are assumed to be small with respect to V. From linearization we obtain for the fluctuating part of the

pressure

it

P = P_Dt + P V

When the source rotates steadily with n revolutions per second in a frame moving with the ship (A.5) yields

- -27n

9t e

With help of (A.6), (A.4) becomes

p = 27np _ay + _DE (A.7)

'

XVI, 24

Propeller Hull Interaction; Excitation Forces

CAVITATION INDUCED HULL PRESSURES, SOME RECENT DEVELOPMENTS OF MODEL TESTING TECHNIQUES

by E. HUSE

THE SHIP RESEARCH INSTITUTE OF NORWAY

TRONDHEIM , NORWAY

ABSTRACT

The pressure signal induced on the hull by cavities on the propel-ler is the sum of three contributions due to different physical effects. The most important contribution is the one due to cavity volume variation. In order to get this contribution correctly reproduced for all types of ships by model tests in a cavitation laboratory, it is necessary to test the propeller in a wake dis-tribution where the axial component has been corrected for scale effect. The tangential component is not so important.

Cavitation tests have been carried out in both model wake and a wake corrected according to the Sasajima method. The latter test shows best agreement with corresponding full scale measurements. The possibility of wall effects during such measurements in a cavitation tunnel is discussed. A new technique to measure cavity geometry by photogrammetry is briefly described.

INTRODUCTION

One of the most important aspects of propeller cavitation today is its influence on the total vibratory excitation forces induced

by the propeller. The prediction of such vibratory forces by _means

of model tests is therefore one of the main fields of interest of most cavitation laboratories.

The first experimental investigation of the pressure fluctuations

induced by a cavitating propeller was published in 1967 in Refer-ence /1/. This American investigation indicated that cavitation on the propeller could increase the pressure amplitudes on the hull by a factor of the order of 2. Further experimental investi-gations supported by theoretical calculations were published in a Norwegian publication in 1971, /2/. The conclusion from this publication was that propeller cavitation could be of paramount

importance to the pressure fluctuations induced on the hull. It showed experimentally as well as theoretically that an increase of the pressure amplitudes by a factor of the order of 10 might well occur. Due to specific phase relationships it further showed that this

factor of 10 in amplitude increase might well lead to an increase by a factor of the order of 100 of the total integrated force on

the afterbody. The next publication in this field was also a Norwegian work, /3/. This report included a complete theory for calculating the pressure signal induced on the hull by cavities of known geometry. More experimental data then appeared in a Dutch paper, /4/. The next publication in this field, a Norwegian and Swedish investigation, /5/, included model measurements, full scale measurements and theoretical calculations. After that time

a number of different reports on this subject have been published,

showing the considerable interest in this field today.

THEORETICAL CONSIDERATIONS

For theoretical considerations the pressure signal induced by the cavities on a propeller can conveniently be split up into three

parts:

A contribution due to the motion of cavities,

a contribution due to the volume variation of the cavities,

a contribution from the tip vortex cavity.

In /3/ complete formulae for the theoretical calculation of the

different contributions for cavities of known geometry have been

developed. Figs. 1, 2 and 3, taken from /3/, are included here to

show some important features of the pressure fluctuations induced

by cavitating propellers. The results in these figures have been obtained by theoretical calculations based on three different

Propeller Hull Interaction; Excitation Forces

cavitation patterns of given geometry. Cavitation Pattern No. 101

is shown in Fig. 1. The main feature of this cavitation pattern is that the cavitation on each propeller blade is constant during the whole revolution of the propeller, corresponding to a complete-ly even wake distribution. Cavitation extent from the 0.6 radius to the blade tips has been assumed. Tmax/C0.7 gives the assumed maximum cavity thickness in percent of chord length of the 0.7

radius. For Cavitation Pattern No. 101 this maximum thickness is assumed to be 10 percent of this chord length. This maximum thick-ness of the cavity is assumed to be at the 0.8 radius decreasing parabolically to zero at the blade tips and the 0.6 radius. The cavity thickness distribution over the chord is also assumed to be parabolic. A tip vortex cavity of constant diameter 1.2 percent of the propeller diameter has also been assumed for the

calcula-tions.

Now let us consider different propellers with three, four, five and six blades and let us for simplicity assume that the blade outline is equal for all four propellers, in fact equal to the outline of the propeller shown on the sketch of the cavitation pattern. The four propellers are assumed to operate at equal number of revolutions.

The result shown in the diagram of Fig. 1 is presented in the form

of a dimensionless amplitude defined by dividing the amplitude by

p n2d2

where pw = mass density of water,

n = shaft number of revs, per sec.,

d = propeller diameter.

Cml' Cm2 and Cm3 are the blade frequency non-dimensional pressure amplitudes due to cavity motion, cavity volume variation and tip vortex cavity, respectively. Ctot is the blade frequency amplitude

of the resulting pressure signal, i.e. the sum of Cmi, Cm2 and _Cm31

phase angles taken into account.

The calculations have been done at a field point vertically above the propeller, in the same axial position as the propeller and at a tip clearance of 60 percent of the propeller diameter.

10-2 2.0 1.6 1.2 0.8 0.4 o 100 90 45 30 15 0 345 330 315 Cavitation pattern No 101

XVII,4

Fig. 1 Theoretical single amplitude of blade-frequency component

of pressure fluctuation induced on the hull by Cavitation Pattern no. 101 for propellers of various blade number.

3 4 5 6

12 10 8 6 4 2 o

Propeller Hull Interaction; Excitation Forces

Cm1

Cm2

Ctot

79

45

Cavitation pattern No.102

9.5 10.0 9.5 30 15 79 0 345 330 315

XVII,5

411P

Fig. 2 Theoretical single amplitude of blade-frequency component

of pressure fluctuation induced on the hull by Cavitation

Pattern no. 102 for propellers of various blade _number.

Figure 2 shows corresponding results of calculations for the same propellers operating at the same number of revolutions as in Fig.1,

but in this case we have assumed, as shown by Cavitation Pattern

102, that the cavitation on each blade starts at a = 300°, i.e.

600 before top position increasing to a maximum value at the top

position and then decreasing again to zero at 600 after top

posi-tion. The variation of the cavity as a function of time or angle

is assumed to be such that the volume varies according to a

sine-curve.

Cavitation Pattern No. 103 shown in Fig. 3 starts at 30° before

top position and ends at 300 after top position, again varying

according to a sine-curve. Cavitation Patterns 102 and 103 are

at top position exactly equal to the Cavitation Pattern 101.

Studying the results shown in Figs. 1, 2 and 3 now reveals some

interesting features. From Fig. 1 we see that the contribution

from the tip vortex cavity is very small compared to the

contri-bution from cavity motion. In Fig. 1 we have of course no

contri-bution due to volume variation since the cavity on each blade is

for this cavitation pattern assumed to be constant during the

revolution. In Fig. 2 we see that the contribution from volume

variation has become the most important, and the total pressure

amplitude is now about four times as high as the one shown for Cavitation Pattern No. 101 in Fig. 1. For the Cavitation Pattern 103

we see from Fig.3 that the contribution from cavity volume

varia-tion has become even more important compared to the effect of

motion. For instance for the six-bladed propeller we see that

Cavitation Pattern 103 gives about four times as high pressure

amplitude as Cavitation Pattern 102 and about twelve times as high

as Cavitation Pattern No. 101.

From the numerical calculations shown above and from other calcu-lations performed in /2/ and /3/ the following general conclusions

may be drawn:

a. At positions on the hull above and ahead of the propeller the contribution from the tip vortex cavity will generally be far smaller than the signal induced by the non-cavitating propeller.

10-2.14 12 10 6 4 2 O

Propeller Hull Interaction; Excitation Forces

1 1 1 1 1 1 1 1 1

Cavitation pattern No.103

XVII,7

.104.0

Fig. 3 Theoretical single amplitude _{of blade frequency component}

of pressure fluctuation induced _{on the hull by Cavitation}

Pattern no. 103 for propellers of various _{blade number.}

4

3 ₅

6

The contribution from the motion of the cavity is physically equivalent to an increased blade thickness. Thus the cavitation

must be quite extreme if this contribution shall become of the

same magnitude as the signal from the non-cavitating propeller. The contribution fromvolume variation of the cavities may easily

become one order of magnitude higher than the contribution due

to motion of the same cavity.

So far we have only considered the pressure amplitudes. The main

parameter determining the resulting hull vibration is of course

the total excitation force obtained by integrating the pressure

over the hull, taking phase angles into proper account. For the

result of this integration process, a feature of great importance is that the pressure signal due to volume variation occurs with

equal phase angles all over the afterbody while the contribution

from cavity motion occurs with different phase angles over the

afterbody. In the latter case a cancelling effect occurs during the integration. Thus, if the pressure amplitude due to volume variation is found to be one order of magnitude higher than the

pressure amplitude due to cavity motion, it may well be that the

difference in resulting excitation force may be two orders of

magnitude.

The main conclusion from the above considerations is therefore

that for a ship with reasonable propeller clearances the pressure fluctuation due to cavity motion will generally not be able to induce any significant hull vibration. The contribution from cavity volume variation, however, may induce large excitation forces and should be paid all possible attention when designing and testing propellers. In the following sections of this paper we shall mainly concentrate on this very important contribution from cavity volume variation.

Those who are interested in more details in connection with the

theoretical calculations discussed above, should consult /3/ where

the complete theory and computer programs are described. Since

such complete calculations require the availability of a large

computer it may be useful to have a simpler formula to estimate the main contribution of the pressure signal, i.e. the contribution

Propeller Hull Interaction; Excitation Forces

from volume variation of the cavities. In the appendix of this paper the following formula has been derived:

2 2

27pwn Z Vbf

Pbf - _a

where

Pbf = blade frequency single amplitude of pressure due

to cavity volume variation, = number of blades,

Vbf = blade frequency single amplitude of Vtot,

Vtot (t) = sum of all cavity volumes as a function of time,

a _{= distance from field point on the afterbody to}

blade tips in upper position.

The blade frequency content Vbf of the _{cavity volume variation can}

never become more than half the maximum cavity volume _{occurring on}

each blade as it passes through the high wake peak in upright

position. Thus the above formula may be useful in indicating an

upper limit of the pressure amplitude when the cavity volume _is

estimated. In most cases, at least when _{running cavitation tests}

in a model wake, subsequent blades will overlap each other with

respect to cavitation as a function of time in the _{upper part of}

the propeller disc, so that when one blade _{starts to cavitate at}

entering the wake peak the previous blade is _{still cavitating as}

it

is

consider-ably less than half the maximum cavity volume _{on each blade.}

Other interesting features revealed by the _{above formula are that}

the pressure signal due to cavity volume variation decreases with the first power of the distance a. Furthermore, the signal may increase with increasing blade number, which is quite opposite to the usual result from considerations of a non-cavitating propeller. IMPORTANCE OF AXIAL WAKE DISTRIBUTION

The main conclusion to be drawn from _{the above theoretical}

con-siderations is that the cavity volume variation as a function of time is a most important parameter in determing the pressure

amplitudes on the hull. From the viewpoint of model testing tech-niques it is therefore very important that cavitation tests in-cluding pressure measurements are carried out in a correct wake distribution. And it should be noticed that a correct axial wake distribution is far more important than having correct tangential velocities in the propeller disc.

AV,

Aaaw wR

wR

XVII, 10

AVt

Aatw ilVt tan p ;11

Aa AV _Ta

13

Aaow

Fig. 4 Importance of axial compared to

tangential wake distribution.

Fig. 4 is meant to illustrate this fact. We see in the upper part of the figure the change of angle of attack Aaaw due to a change

of axial inflow velocity AV. The lower part of the figure shows

correspondingly the change in angle of attack due to a tangential

Q

Propeller Hull Interaction; Excitation Forces

velocity change AVt. In accordance with Fig. 4 the relative

im-portance of the two contributions to the angle of attack may be

expressed:

Aatw AVt

Av . tan $

Aaaw a

At the blade tips, which is the most important radius in this

con-nection, tan _{(3 will for most propellers be of the order of}

0.1.

For most types of ships the variation in axial velocity over the propeller disc is generally higher than the variation of tangen-tial velocity. From the above formula we therefore conclude that correct simulation of the tangential wake during the cavitation tests is of minor importance compared to a correct simulation of

the axial wake.

SCALE EFFECT OF WAKE DISTRIBUTION

It is well known that due to different Reynolds number in model and full scale, the boundary layer will be relatively much thicker on the model than it is on the ship. In /6/ is suggested a

prac-tical method to correct the model wake distribution to obtain the full scale wake distribution. This technique has been applied in a number of cases at The Norwegian Ship Model Tank. There are two cavitation tunnels in operation at The Norwegian Ship Model Tank. The larger one has a circular test section of 120 cm diameter. The

standard test procedure in this tunnel is to fit a dummy model,

whose aft part is made to an accurate scale. The wake distribution is simulated by fitting wire meshes to this afterbody model at some distance ahead of the propeller plane. Only the axial wake

is correctly simulated. The tangential wake is generally _somewhat

smaller than for the ship, due to the total width of the

after-body model being less than a corresponding complete hull _model.

However, as shown in the previous section the effect of the

tangential wake upon angle of attack variation, and thus cavitation pattern, is of minor importance compared with the axial wake.

Until recently it has been our standard procedure in this _tunnel

to simulate the wake distribution that is measured during towing

the complete hull model in the towing basin. Pressure _transducers

fitted flush with the surface of the afterbody model are used to

measure the propeller induced pressure amplitudes on the hull.

This technique of measuring pressure amplitudes induced by the cavitating propeller has been commercially used in our tunnel for several years now, and we have got some correlation data from full scale measurements of pressures on the corresponding ships. It is our experience that in some cases, particularly where there is relatively little cavitation on the propeller, or where each blade has been cavitating almost constantly during the whole rev-olution the correlation with full scale measurements has been quite satisfactory. In other cases, however, the tunnel

measure-ment has underestimated the full scale pressure amplitudes by a

considerable factor. The explanation of the discrepancy seems to be the scale effect of the wake distribution.

During the first half of 1974 we have, in order to investigate this, carried out cavitation tests in model wake and also in a wake distribution corrected according to the Sasajima method.

We shall here show such comparisons for two ships where we also

have full scale measurements available. Figs. 5 and 6 show the

model wake and the wake distribution corrected according to

Sasajima for ship No. 1. Fig. 7 shows the blade frequency pressure

amplitudes for the two cases. The full scale measurement, which has in this case been conducted by The British Ship Research Association, is of very good quality, taking into account even

a slight correction for the vibration of the afterbody of the

ship. Unfortunately, the model measurements in model wake and corrected wake are not directly comparable because the test in model wake was done with the tunnel pressure adjusted to give

correct cavitation number at the height of the shaft centerline,

while the test in corrected wake was done with the pressure

ad-justed to give correct cavitation number at the height of the

blade tips in upper position. Part of the difference is therefore

due to this difference in cavitation number. In order to obtain

direct comparison we intend, as a matter of fact, to carry out

one more test in model wake with the cavitation number adjusted

to the blade tips at top position. Anyhow, the test in corrected

wake shows excellent correlation with full scale.

Propeller Hull Interaction; Excitation Forces

XVII, 13

Fig. 5 Ship no. 1, model wake diStribution.

Figs.8 and 9 show the model wake and corrected wake, respectively,

of ship no.2. Fig. 10 shows the corresponding _{pressure amplitudes.}

We see

in

only slightly different from the cavitating condition in model

wake. In corrected wake, however, this _{difference has been}

in-creased by nearly one order of magnitude for _{transducers 1, 2 and}

4, which are all fitted on the starboard side. For transducer 3, fitted on the port side, certain phase relationships have led to

a reduction.

Fig. 6 Ship no.1, wake distribution corrected for scale effect.

900

o

Propeller Hull Interaction; Excitation Forces

XVII, 15

model wake,

_

corrected wake,

non-cavitating _measured

corrected wake, on ship cavitating

smodel wake, ,..4

:11 cavitating A

1L1

1L

1L1

w w

Transducer position no. -1.

Fig. 7 Ship no.1, blade frequency pressure amplitudes on hull.

Note: Test in model wake and corrected wake have here been done with cavitation number adjusted to height of propeller center and tips in top position, respectively.

P. 600

o

2'en

9.

Fig. 8 Ship no.2, model wake distribution.

Propeller Hull Interaction; Excitation Forces

Fig. 9 Ship no.2, wake distribution

corrected for scale effect.

300 200 E 4. -o cs 100 A o model wake, non-cavitating Emodel wake, cavitating Transducer pOsition

Fig. 10 Ship no.2, blade frequency pressure amplitudes on hull. Tests have been done at equal cavitation number and advance

coefficient. (based on thrust identity).

One more case, here denoted by ship no.3, is shown in Fig. 11.

This figure has been taken from /7/. The correction of the wake distribution has here been more arbitrarily chosen, and is not according to any specific method such as the Sasajima method. The result, however, shows that a pressure amplitude correlating better with full scale measurement is obtained by running the cavitation

test in a wake pattern with a more narrow wake peak than the model

wake.

The reason for the change in pressure amplitude obtained by cor-recting the wake distribution should be clear from the theoretical considerations at the beginning of this paper. It is all a question of obtaining correct cavity volume variation as a function of time, or blade angular position.

XVII, 18

2 3

corrected wake, non-cavitattog

0.6 0.4 0.2 O 270 0.06 0.04 0.02

Propeller Hull Interaction; Excitation Forces

W

/

\

\

\

_/

/

/-,

/

\

\\ \

/

\---\

_\

calscaleulated Cr ...-r- ,--fu m lscale asured . mod model llscale" wake model witiout avitation 16 18 20 22 24 26 28 30 32 X 103 PD ( HP)

Fig. 11 Ship no.3, pressure amplitudes in model wake, modified wake and measured on ship. Figure taken from reference 7.

XVII, 19

300 330 360 30 60 a 90

0.10

C p

The experimental results described above show only the pressure amplitudes. However, from the viewpoint of total excitation force on the hull it is very important to be aware of the phase relation-ships for the various contributions to the pressure field. The correction of the wake distribution mainly affects the contribution from cavity volume variation. Thus, if this correction of the wake distribution increases the pressure amplitude by for instance a factor of two, the excitation force will increase by a much larger factor. If the cavitation tests are done with a larger number of pressure transducers in order to integrate over the afterbody to obtain the total excitation force, the result may thus be complete-ly wrong unless the wake distribution and thereby the cavity volume variation is correctly modeled, taking the scale effect of the wake distribution into account.

WALL EFFECT IN CAVITATION TUNNELS

When measuring propeller induced pressures in a cavitation tunnel the pressure transducers in the hull model will record not only the pressure wave being transmitted directly from the propeller, but also the pressure waves reflected from the tunnel walls. This may give rise to a wall effect leading to erroneous results. At The Norwegian Ship Model Tank extensive theoretical and experi-mental investigations of this effect have been carried out. The results are described in /8/. This wall effect has been measured for the two cavitation tunnels at The Norwegian Ship Model Tank. The conclusions are that in the larger of the two tunnels the error due to this wall effect is within the measurement accuracy

with which it has been determined, i.e. less than 10 percent.

In the smaller one of the two tunnels, however, the wall effect gives such an error that the tunnel cannot be used for measuring propeller induced pressure fluctuations at all.

PHOTOGRAMMETRIC STUDY OF CAVITY GEOMETRY

For detailed studies of cavity geometry, for instance for the purpose of correlating theoretical calculations of cavitation in-duced pressures on the hull, it is necessary to have some means of measuring cavity thickness in the cavitation tunnel accurately.

Propeller Hull Interaction; Excitation Forces

A possible method to obtain this is by a photogrammetric technique. This technique consists in making simultaneous photographs of the cavitating propeller with two cameras. The photographs are then analysed in an apparatus called an "autograph". This is the type of instrument used for making three-dimensional maps from aerial photographs. Development of computer programs, photographing techniques etc. necessary to apply the photogrammetric technique in a cavitation tunnel, was done by The Institute of Geodesy and

Photogrammetry at The Technical University in Trondheim in

co-operation with The Norwegian Ship Model Tank during the first half of 1972. The investigations at that time were done in the smaller cavitation tunnel. Under the most favourable observation conditions one was then able to measure the thickness of the cavity on the

model propeller with an accuracy (standard deviation) of down to

0.2 mm.

So far, this technique has not been used for commercial cavitation testing. However, if anyone is interested, the possibility is there.

It requires a cavitation testing facility where the _{cameras can be}

placed very close to the model propeller. Furthermore, it requires

exceptionally good illumination and observation conditions. In fact,

the photographs must be so bright and clear that the various

de-tails of the surface of the cavity can be clearly seen.

As an illustration of the photogrammetric technique in general,

Fig. 12 shows two pairs of photographs taken simultaneously in

the cavitation tunnels of The Norwegian Ship Model Tank. _{By a}

Fig. 12 Stereo photographs from cavitation tunnels at The Norwegian

Ship Model Tank.

Propeller Hull Interaction; Excitation Forces

CONCLUSIONS

From theoretical considerations as well as experiments it can be concludedthatfor a propeller with reasonable tip clearance the pressure amplitude induced on the hull by the motion (displacement effect) of the cavities can hardly become sufficiently high to induce serious vibrations. The pressure due to cavity volume variation, however, is far more important. There are two reasons for this, first the pressure amplitudes may become much higher, secondly, the pressure occurs with equal phase all over the

after-body.

This volume variation of the cavities as a function of time is mainly determined by the axial wake distribution. The width of the wake peak in the upper part of the propeller disc is

particu-larly important.

Due to different Reynolds number the wake peak of the model is about twice as wide as that of the ship. In order to obtain reli-able hull pressures from cavitation tests it is therefore neces-sary to run the tests in a wake distribution which has been cor-rected for scale effect.

Experimental results already obtained at The Norwegian Ship Model Tank in Trondheim and at KMW Marine Laboratory, Kristinehamn, Sweden, confirm that running the cavitation tests in a wake

dis-tribution corrected for scale effect results in much better cor-relation with full scale measurements.

REFERENCES

I. Denny, S.B., "Comparison of Experimentally Determined and Theoretically Predicted Pressures in the Vicinity of a Marine Propeller", Naval Ship Research and Development Center, Report

2349, 1967.

Huse, E., "Trykkimpulser fra kaviterende propell", paper pre-sented at "Nordisk Skipsteknisk mOte", bo, Finland, 1971.

Huse, E., "Pressure Fluctuations on the Hull Induced by Cavi-tating Propellers", Norwegian Ship Model Tank Publication

No. 111, March 1972.

Oossanen, P. van and Kooij, J.v.d., "Vibratory Hull Forces Induced by Cavitating Propellers", RINA Spring Meeting 1972.

Johnsson, C-.A. and SOntvedt, T., "Propeller Cavitation and Response of 230,000 TDW Tankers", 9th Symposium on Naval Hydro-namics, September 1972.

Sasajima, H. and Tanaka, I., "On the Estimation of Wake of Ship",

llth ITTC, Tokyo, 1966.

BjOrheden, O., "Dynamic Forces Due to the Interaction between Propeller and Hull", ISME, Tokyo, 1973.

Huse, E., "Effect of Tunnel Walls upon Propeller-Induced Pres-sures in Cavitation Tunnels", IME Cavitation Conference,

Edinburgh, 1974.

Propeller Hull Interaction; Excitation Forces APPENDIX

Pressure Amplitude due to Volume Variation of Cavity

Let us consider a system of N cavities of volume Vi, the distance

of cavity no. i from the field point (on the hull) being ai. Representing the volume variation of each cavity by a point source one can show that the velocity potential at the field point will be (sea for instance Ref. /2/):

N V.

= E (

i=1 4,ai

From Bernoulli's equation we obtain, by neglecting the velocity-dependent term, the instantaneous pressure:

N

P(t)--Pw3tE'4Tra9t

i=1 .

Let us now assume that

the cavities are stationary in space,

the cavitation mainly occurs at the blade tips

intheirverticaluPdardsPositionsothata-is

for all cavities approximately equal to the tip

clearance a.

Multiplying by a "solid boundary factor" of 2.0 (Ref. /3/) to account for the image effect of the hull we thus obtain:

p(t) = 27a

where Vtot is now the sum of all cavity volumes.

The blade frequency content of this pressure signal is obtained by expressing Vtot as a Fourier series. The blade frequency single amplitude of the pressure thus becomes:

where

= shaft revs, per sec. = blade number

Vbf = blade frequency content of Vtot.

THE PROBLEM OF INFLUENCE OF SOLID BOUNDARIES ON PROPELLER-INDUCED HYDRODYNAMIC FORCES

by

M. GARGUET and R. LEPEIX

CHANTIERS DE L'ATLANTIQUE, SAINT NAZAIRE, FRANCE

1. INTRODUCTION

The possibility of reducing the risks of heavy vibrations onboard ships relies more and more on the possibilities of determining, at the early stage of the project, the magnitude of the excitations

that may be expected, especially those due to the rotation of the

propeller.

Concerning the so-called "surface forces", coming from the "fanning" action of the blades passing nearby the hull, the calculations of

the pressure fluctuations around a marine propeller are in con-tinuous development (/1/ to /6/).

Unfortunately, these calculations are at present limited by two main restrictions.

It must always be assumed that the velocities induced by the propeller are much lower than the ship speed. This enables

(linearization) the summing up of the various pressure fluctuations, coming from different causes (static loads and thickness of the blades, dynamic loads due to the non-homogeneous wake, volume and fluctuations of the possible cavitation zones).

The possible reflections on the boundaries, in the vicinity of the propeller (hull, rudder) cannot, in general, be taken exactly

into account.

These two restrictions make the application of the calculations very doubtful, for almost the totality of the present ships. The first limitation (linearization) seems difficult to overcome, since the non-linear theoretical calculations are at the present time insurmountable.

Propeller Hull Interaction; Excitation Forces

The second limitation (reflections) is extensively investigated, into several directions:

Studies in connection with the propeller-rudder interaction /7/, /8/, giving interesting results, but not directly connected to the pressure calculations on any hull forms.

A method /9/ based on the representation of the ship by means of sources, some being constant, and others being of variable intensity, to represent the flow surrounding the ship, i.e. the total motion and the local velocities induced by the propeller.

A semi-empiric method, consisting of the multiplication of the pressure calculated in open water (without hull, but assuming the real wake) by a factor S, called "Solid Boundary Factor", acting for the presence of the hull. This factor S must be derived

from comparisons between calculations and full scale measurements. The use of the empiric factor S is, actually, the only way

permitting valuable predictions for the pressure fluctuations, without needing a too long computing time.

Unfortunately, the variations of the factor S are still not very well known, and its use is then very limited.

The necessity appears to develop this knowledge by comparing as often as possible the full-scale pressure measurements and the calculations in open water.

Naturally, these comparisons must take care of different disturbing pressures, coming for instance from the general or local vibration of the afterbody.

Naval architects are very much interested in such a practical approach, and our aim in this paper is to bring some elements to contribute to its development.

2. THEORETICAL METHODS FOR THE PREDICTION OF SURFACE FORCES 2.1 Historical

Up to 1968 the use of the factor S was most often limited to the multiplication of the pressure in the vicinity of the hull by a

factor 2.

Calculations concerning the forces applied by the propellers upon the solid boundaries could,however,be performed under the following

assumptions.

2.1.1 An infinite plane parallel to the rotation axis (Fig. 1). The total vibratory force onto the plane is zero for any correctly

balanced propeller having two blades,at least in a homogeneous

wake /10/, /11/. For a single blade propeller, or in a non-homogeneous field, this force is equal and opposite to the

component of the force (applied onto the propeller) which creates the torsional force into the shaft. Under these circumstances, the factor S is always taken as being equal to 2 which is realistic in linear theory (see Appendix 1).

Fig. 1

2.1.2 An infinite strip parallel to the rotation axis (Fig. 2). The calculations made by the Davidson Laboratory /12/ are of

interest but still use S = 2 which renders the results quite approximative for the widths 1 of strips smaller than the propeller diameter.

2.1.3 A cylinder with its axis parallel to the rotation axis

(Fig. 3).

Breslin /13/ has calculated the image of the source in the cylinder using the functions of Bessel. He inferred from this -by integration - the image of all of the singularities representing a propeller blade. The result obtained is that the force applied

Propeller Hull Interaction; Excitation Forces

to the cylinder is equal for the external sources and their internal images; this force must therefore be only duplicated to take into account the perturbations of the velocity-field due to the presence of the cylinder.

Fig. 2

Fig. 3

2.1.4 A few simple forms (infinite wedges, etc....)

Pohl /14/ has calculated the forces produced by the propeller on

various simple forms, but assuming always that the presence of

these forms does not perturb the velocity-field.

Pressure calculations on the infinite wedges by Reed /15/ are not easy to deal with. They show most often S values between 1 and 2, sometimes higher than 2.

2.2 Present Possibilities.

It seems at this time that there are 3 different ways to approach the study of the pressures created by a propeller onto its

surrounding solid bodies; all these various approaches use a linear theory of pressure calculations.

2.2.1 The exact calculation(except the linear approximation) of the velocity field surrounding the propeller and the solid boundaries, by the research of the source images in relation to the boundaries. The pressure is obtained in any point, as in the other methods, using the linearized relation of Bernouilli. This method presents quasi-insurmountable difficulties as soon as the simple forms defined under the preceding paragraph are deviated

from.

2.2.2 The substitution of a continuous distribution of sources

for the hull:

constant sources generating the ship hull at a constant speed alternative sources allowing the flow to follow exactly the afterbody in the presence of the variable velocity field produced by the propeller.

For this, it is necessary to have a program available, with which

the velocity, in any point surrounding the propeller, in open water, can be calculated.

These sources are distributed over the surface of the hull and their symmetricals in relation to the free-surface must be added to keep a constant pressure on this free surface. The symmetricals must also be added to the propellers.

The calculation of the density of the sources /16/ is made