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Hull Vibration Excitation by Propeller Sources: a Link between Hydrodynamics and Marine Acoustics R. Kinns, (Dept. of Marine Technology, University of Newcastle, U.K.) N. Peake and 0. Rath Spivack (DAMTP, University of Cambridge, Cambridge, U.K.) Abstract Acoustic boundary element models are used to solve the Helmholtz equation, in order to explore the na- ture of fluctuating hull pressures due to propeller sources, when the wavelength of underwater sound is comparable to hull dimensions. Sources are rep- sources. resented by stationary monopoles and dipoles near a Considerable effort has been devoted to the pre- rigid hull. Results for submerged and floating bodies diction of hull pressure fluctuations near ship pro- are described. The hullsurface is assumed to tee rigid pelters, due to cavitating and non-cavitating pro- throughout. pelter sources. The principal focus has been on peri- A simple ellipsoidal representation of submerged and odic pressure fluctuations at multiples of bpf. Above floating bodies is used first to aid understanding of cavitation inception, these arise primarily from cyclic how hull pressure distributions are affected by the lo- cavitation due to rotation of the propeller in a spa- cation and frequency of propeller sources. The sea tially non-uniform wake field, but the effects of fluc- surface is represented using image technioues. to en- tuating forces, rotating steady forces and blade thick- ness can all be significant. There is increasing suc- cess in predicting random fluctuations by appropriate scaling of data from cavitation tunnels and towing tanks. Progress in prediction of periodic and random pressure fluctuations near the propeller, has, however outpaced the capability to predict hull vibration ex- citation. which can be particularly obtrusive, as well as sin- gle frequency components at multiples of propeller blade passing frequency (bpf). Satisfactory vibration prediction requires estimation of the distribution of fluctuating pressure over the whole of the hull sur- face, not just fluctuating pressure near the propeller sure zero pressure there when the body is floating. Solid boundary factors are used to indicate the prin- cipal effects of source location, frequency and the sea surface. Finally, results from an independent model of a cruise liner hull are used to illustrate the com- bined effects of diffraction, interference and flotation on hull forces, with explicit modelling of the free sur- face. It has been known for many years that procedures which focus on a limited area near the propellers can . give poor accuracy in prediction of hull vibration ex- intrO]UCtlOn citation. For example Cox, Vorus, Breslin and Rood (1978) show how significant forces due to cavitation Increasing emphasis on prediction of hull vibration can extend more than 15 propeller diameters forward due to propeller sources stems from the need to of the propeller itself. The results of that work are meet demanding requirements for passenger comfort. described by Breslin and Andersen (1994), who point These now focus on broadband random vibration, out that the surface of the sea can have a critical ef-
feet, which is not reproduced correctly in cavitation tunnels. It is, nevertheless, still common to limit at- tention to an area that extends only a few diameters forward of the propeller position and to assume that only hull forces on the same side of the ship as the propeller are significant in a twin-screw ship. The potential errors in these approximations are further increased by the effects of compressibility. It has almost always been assumed that there is no need to take account of the compressibility of wa- ter in prediction of hull vibration excitation at low multiples of bpf. The effects of retarded time are in fact significant at even low frequencies. The reason is that the wavelength of underwater sound is typically half the hull length at bpf and comparable to the hull beam at only four times bpf. This has various related effects. Firstly, variations of phase with distance from the source lead to interference between forces on dif- ferent parts of the hull. Secondly, hull diffraction depends on frequency, so that the amplitude of pres- sure on the hull due to a source of constant strength varies with frequency. Both effects are present for submerged as well as floating bodies. Thirdly, the quenching effect of the free surface changes with fre- quency, which has a particularly marked influence on hull forces due to sources that are near the stern waterline of a surface ship. None of these features are reproduced correctly in a numerical analysis that solves Laplace's, rather than the Helmholtz equation. The hull vibration excitation problem has at least as much in common with the prediction of underwater radiated noise (Ross, 1987) as prediction of hull pres- sures in the immediate vicinity of a propeller. , . . . . .. . . . . . The work described in this paper has its origins in the design of cruise liner propulsion systems for low vi- bration. It has been demonstrated at full scale (Kinns & Bloor, 2000) that the choice of propeller rotation direction can have a very marked effect on cavitation- induced vibration. The selection of rotation direction can influence both the nature of cavitation patterns and the spatial location of the principal sources. If reduced source strength is associated with increased distance of the source from the sea surface, then the hull excitation might be reduced by a smaller amount that expected from observation of cavitation sources. Acoustic boundary element modelling techniques have allowed the effect of source location to be ex- plored for both ellipsoidal bodies and real hull shapes (Bloor, 2002~. In that work, the surface of the sea was modelled throughout as a pressure release sur- face. The hull surface was assumed to be rigid in most of the exploration, so that only the hull shape, rather than its dynamic response which depends on internal structure, was modelled. Following initial investigation of the modelling approach using simpli- fied shapes (Bloor & Kinns, 2000), the effect of source position and frequency has been studied in depth for a typical cruise liner hull shape (Kinns & Bloor, 2002a). The work has been extended to include the effects of dipole sources (Kinns & Bloor, 2002b), which can represent fluctuating propeller forces, fi- nite blade thickness and the effects of fluid flow, in addition to monopole sources that represent the prin- cipal effects of unsteady cavitation. Two approaches can be used to determine the effects of the free surface of the sea. If the surface is as- Mach number is not replicated in model-scale tests sumed to be flat, then it is possible to use image that replicate Fioude number. If a 1/s scale model is techniques, whereby the hull is reflected in the sur- used, then the wavelength ~ is ~ too large in rela- face and the underwater sources are reproduced with tion to hull dimensions. The effects of compressibil- ity are therefore underestimated. Thus, the effects of the free surface and the relative phases of pres- sure on different parts of the hull are not the same at full and model scales, for a given multiple of bpf, or any other component in the frequency spectrum. It also means that the acoustic far fields of higher order opposite sign above the surface. This ensures that the pressure is zero everywhere on the notional sea sur- face. This is the approach used by Vorus (1974) for solution of Laplace's equation. It is used here to solve the Helmholtz equation. An alternative approach is to model the sea surface explicitly, which allows the effects of surface gravity waves to be explored. This sources, such as axial dipoles that represent fluctuat- approach was favoured by Bloor (2002~. ing thrust, are more significant at full than at model In this paper, we demonstrate first that results ob- scale. tained using boundary element models agree closely 2
with those derived from analytical expressions for the submerged sphere. We then show how fluid compress- ibility influences fluctuating pressure, solid boundary factor and hull force distributions on ellipsoidal bod- ies, where the sources are monopoles and dipoles close to the body. The body can be either submerged, or floating. We explore dependence on frequency, by varying the wavelength in relation to hull dimensions over ranges of practical interest. Finally, we show how results for a floating ellipsoid are related to those for a real cruise liner hull shape, using results in Kinns & Bloor (2002a,b). Formulation of the problem and fundamental equations We consider here the calculation of the pressure field (and related quantities) generated by an acoustic source ensonifying a submerged body. The assump- tion is made that the source is harmonic, which al- lows calculations to be carried out in the frequency domain. The governing equation for this problem is the Helmholtz equation, which can be written as: V2~(r) + k24(r) = 0, where <;b is the field (e.g. the acoustic pressure) and k = w/c is the wavenumber, with w the frequency of the source and c the speed of sound in the fluid. The field is related to surface pressure (see e.g. Morse and Feshbach, 1953) by the integral: otr)= ~ Gin dS/~o~'ndS, (2) where G is the Green's function, S the scatterer sur- face, and n is the outward normal on the surface. To solve equation 2 appropriate boundary conditions must be imposed. We have considered a scatterer with a perfectly reflecting surface, and therefore used Neumann boundary conditions with the derivative of the field vanishing on the surface: ~ = 0 ~ which reduces equation 2 to: Is In (3) The time harmonic dependence can be reintro- duced explicitly by using the full form of the field ¢(r) exp(ixt), to take compressibility into account. Numerical solution using bound- ary elements Here we summarise the numerical treatment of the boundary integral equation 2 to find the surface fields (the field or its normal derivative for hard or pres- sure release surfaces respectively) on a surface, for an arbitrary source. The integral equation can be dis- cretized, giving rise to a large matrix system which can be inverted using a 'black box' inversion routines such as NaG or Lapack routines. Let us consider a medium with axes x, y, z where z is the vertical axis directed upwards, x is the 'direction of propagation' or the axis of the scatterer, and y is the transverse horizontal coordinate. The surface is denoted by S(x, y, z). `1y The field can be expressed as the sum of incident and scattered components. Denoting the incident field with Hi, and assuming Neumann boundary con- ditions, i.e. a rigid surface, the governing boundary integral is given by: 4'~' ~ ~ ' ~ogr'jdr' (4) iLrJ = Berg / JS where r = (x, y, z) and r' = S(x', y', z'), say. G(r, r') is the free space Green's function, given by: eiklrr'I G(r r')= By taking the limit as: r ~ rS we obtain: Oitrs)=<btr5),( ~ ~ )~(r')dr' where now rS = S(x, y, z). 3 (5) (6)
In most cases of interest in the project the surface can be expressed as a function of x and a radial angle it. In order to treat this equation numerically it is therefore convenient to convert the integrations to be with respect to x, 6. We thus obtain: jirlctrs) = otrS),/ ,/ ~ 6~ )~(<r') ~y(r' jdxdb (7) where the known factor fly arises through the change of integration. Discretization is implemented as follows. Choose step-sizes /\x, /\§, and assume N. M points in the x, ~ coordinates respectively, so that we define: X X1, ·--, ON, = 61,---,~M, . . . An = rl/\x em = m/\~. Denote the discretized surface values by: °inc(<Xn, ~m) = anm <;b (Xn ~ em `) = (11) bum' (12) (13) Now we can discretize 7 by writing, for each point En' {em M N anm = ~~ Anmij bij (14) j=1 i=1 where anm represent the known incident field at an, ~m' b the unknown surface field, and Ar2mij represents the discretization of the rest of the integral in 7. This defines a set of (M x N) equations. It there- fore leads to a matrix equation, which we rewrite for convenience as: c= Bb where we have written the vectors in bold. Here b denotes the solution vector ¢~(x,§), c represents the incident field, and B is the (MN) x (MN) matrix whose elements are given by Anmnj We thus require: b = B-ic (16) This is evaluated by inversion of B using a NaG or Lapack routine. This solution can then be used in the integral above to evaluate the scattered near or far field. The solution was found to be stable with respect to change in discretization. For higher frequency problems, discretization is determined by wavelength (roughly 10 points per wavelength), but by geometry for lower frequencies. For a sphere it was typically found that 60 x 32 points gave acceptable accuracy. Tests on an ellipsoid shape also confirmed that the results were insensitive to an increase in the number (a) of surface elements for the frequency range of interest here. At higher frequencies, corresponding for exam- ple to several hundred Hz and a scatterer about 30m (10) long, it becomes necessary to increase the number of elements. On a standard 600MHz PC using NaG routines this took around 3 minutes. Lapack routines reduced this by a factor of 3, and on the fast Pentium 4 machine the same calculation took 10 seconds. Computa- tion time rises rapidly with N and M, approximately o<N3M3y Once the solution of Equation 7 has been found, the integration 4 to obtain the far field is relatively straightforward. Acoustic sources and physical quanti- ties We will present results for the total acoustic pressure on the surface of the scatterer: fairs) = Pi (rS) + ~SC(rs) (17) and for the Solid Boundary Factor (SBF), given by the ratio of the total and the incident field magni- (15) tudes at a point on the surface: ~ ¢(rs) ~ / ~ Tiers) The propeller will be represented as a monopole and as a dipole source. 4
The incident fields due to a monopole and dipole re- spectively are: ikr hi = A, r and Vi =D l i ~ eikr _ COSOL- kr r ' where or is the angle between the dipole axis and r, and A and D are source strengths (see for example Kinns & Bloor, 2002b). In addition to the total pressure on the hull surface, it is interesting to calculate the Solid Boundary Factor (SBF), which can vary considerably over the curved hull surfaces. In particular its magnitude can fluctu- ate widely where high multiple scattering occurs. Another quantity which is of practical interest to ship builders is the total force on the hull, which can be calculated by integrating the pressure over the sur- face. The cumulative force, which is the integral over part of the surface for increasing longitudinal extent forward of the stern, shows how excitation of different parts of the hull contribute to the total. Validation for the submerged sphere The numerical code which implements the boundary elements solution described in the previous section has been validated by comparison with an analytical solution. Analytical solutions exist for simple bodies ensonified by a monopole. In particular, we have used the an- alytical solution (e.g. Shelton and James, 1997) for the total pressure on a rigid spherical shell due to a monopole source. The expression for the total pres- sure at a point (r, §) on the surface of the shell can be written as car, 6) = Pier, 6) + User, §), (20) where the subscripts refer to incident and scattered term respectively. If the monopole source is located at (Ro, 0), each term can be expressed as an infinite 2.5~ sum of Bessel and Hankel functions and their deriva- tives, which depend on the size of the sphere a and on the position and frequency of the source through aim the arguments ha and kRo. The surface pressure on a perfectly reflecting sphere with diameter d has been calculated with a monopole `19y source located at a distance d/2 from the surface of the sphere for a range of frequencies, using both the analytical and the numerical Boundary Element code. The amplitude and the real and imaginary part of the field have been compared and show excellent agreement. Figure (1) for example shows the ana- lytical and numerical results for the real part of the surface pressure ~ = d/4. Other cases show similar agreement. Surface pressure (roar pan) on sphere for monoDole. `=d/4 I 1 analytical ~ bou~daryebment I 1.5 :.~.5 . o EL O - -1.S Hi Jit \~ 1 0 4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Distance along source direction Figure 1 Results for a submerged ellipsoid Having validated the numerical code, we then applied it to the case of a generic axisymmetric ellipsoid with a ratio 1:5 of diameter to length. Discretization of the hull surface was carried out as described earlier, using 32 points in the radial dimension, and 60 along the length of the hull, i.e. a total of 1920 surface elements. We present below results for different types of source, located at varying distances from the stern on the principal axis of the ellipsoid, and for a range of fre- 5
quenches. Monopole excitation We have calculated fields due to a monopole source located at 0.25d for four different wavelengths: ~ = 8d,4d,2d,d. 2 1.8t 1.6t 1.4i 1.2 0.8 0.6 0.4 0.2 Solid boundary factor for monopole at d/4 _ , , , , , . X=Bd - A=4d k=2d X=1d ~~ ,'4 ~~ __ _ _:;~~,< t 0 1 2 3 4 5 6 7 8 9 10 O- Figure 2 The results are shown in Figure 2, where the SBF is plotted against the distance along the x-axis, and has been calculated along the line on the top surface of the ellipsoid defined by the intersection with the plane [yZ]- SoLd boundary factor for monopoly at distance d 1.8 I1.6 1.4 1 1 0.8 0.6 0.4 0.2 _X=&d ;~=4d X=2d - - X=1d O- 0 1 2 ~ . ~ i~ ~ 1{ . ll is 3 4 5 6 7 8 9 10 Figure 3 The same set of results has also been obtained for the case where the position of the source is further from the ellipsoid, at x = d. These are shown in Figure 3. Changes with frequency are small until the wave- length falls towards the diameter of the ellipsoid. Then, diffraction causes a substantial increase in SBF near the bow. A surprising feature of the results is the local minimum in SBF just forward of the stern, which is almost independent of frequency in the range considered. We intend to explore this further. The SBF results are only influenced slightly by the separation of the source and the stern of the ellipsoid. This applies also to results for the axial and trans- verse dipoles, which have the directivity indicated by equation 19. Dipole excitation Calculations have been carried out also for transverse and axial dipoles. In the case of an axial dipole the distribution of the fields (and consequently of the Solid Boundary Factor) on the hull is axisymetric, as was the case for a monopole source. If the source is a transverse dipole, this axisymmetry is lost. We illustrate this in Figure 4, where the distribution of the acoustic pressure due to an axial and transverse dipole with ~ = d at distance d from the hull are com- pared. On the grey scale, lighter grey corresponds to higher values of the pressure. Surtace pressure due to transverse dipole Surtace pressure due to axial dipole Figure 4 6 B
The SBF distributions due to a transverse dipole at distance d from the hull, with its axis along the zaxis, are shown in Figure 5, where the SBF is plot- ted for the same chosen contour as before and for the same four frequencies. Solid boundary factor for trensvase dipole at distance d Figure 5 1.8 1.6 1.4 1.2 0.E 0.e O.~. O.; o I~ ~ =~ Figure 6 Results for a floating ellipsoid Our numerical code can also be applied to the case of a floating body, by implementing the image method (see for example Breslin 1994) to represent the source. We present here results for a half-submerged ellip- soid ensonified by a monopole, and show the effect of changing frequency and source depth. The monopole source is placed below the ellipsoid at typical propeller position, at d/10 and d/5 forward of stern waterline. Figures 6 is equivalent to 5, but for the case of an SBFforsourccto~rdd/1O.aideD~dl1ObeowFoatinchuI' axial dipole. Again, changes with frequency are small until the wavelength is reduced to about the the diameter of the ellipsoid. There are, however, substantial dif- ferences between the SBF distributions for the two dipole orientations and the monopole. In the case of the transverse ellipsoid, the SBF remains almost constant over most of the length. It rises increasingly sharply toward the bow when the wavelength is re- duced below twice the diameter. In the case of the axial dipole, the SBF changes with frequency nearer the stern when the wavelength is similarly reduced. 1.',~ ,.; 0.~ 0.( 0.4 0.2 7 ash Figure 7 `+-1~
For the case where the source is at d/10, we only show results for a source depth of d/10 below the hull surface (Figure 7~. In the other case, we show results for two different depths, namely d/10 and d/5 (Figures 8 and 9~. 2.5 . 2 .c ~ As8d -A=4d A=2d A=1d i/: ! I 0 1 2 3 4 5 6 7 8 9 10 distant along hull Figure 8 f 7g2°~°°- ) a. distance along hull Figure 9 The results for the floating ellipsoid show the pow- erful effect of the free surface of the sea in changing the dependence of SBF on frequency. As expected, the SBF is close to 2 for locations on the hull near the source, at any frequency in the range of interest where the source to hull separation is also a small fraction of a wavelength. The change with frequency elsewhere on the hull surface can be very large, espe- cially towards the bow, showing why it is important to include the effects of a finite speed of sound in surface ship analysis. Cumulative hull forces can be calculated simply by summation of forces on hull elements up to a given station forward of the stern, taking account of the surface orientation and dealing with the in-phase and out-of-phase components separately. The cumulative force magnitude can be derived from the two com- ponents at the specified summation limit. This can also be done for port and starboard sides separately, when the source is positioned to one side of the hull, in order to explore whether the force is dominated by hull excitation on the same side as the assumed propeller source. Figure 10 shows how the cumulative vertical force changes with distance along the ellipsoid, when the source is d/5 forward of the stern, d/10 below the hull and has frequencies such that the underwater sound wavelength is 8, 4, 2 and 1 times the ellipsoid diameter. The effects of increasing SBF are countered by interference between forces on different parts of the hull surface. Thus, the cumulative force oscillates with larger relative amplitude as frequency increases, and decays more slowly relative to its mean, while the total force tends to fall. 12 10 2; ~ x=8d A=4d A=2d =1d ~ _ 1 2 3 4 5 6 7 8 9 10 distance song hut Figure 10 The principal effects come from interference between the source and its negative image above the sea sur- 8
face. In effect, this is related to the ratio of the pres- the analysis at 12 and 24 Hz, but almost identical sure due to a vertical dipole with its elements sepa- results can be obtained using other extents and el- rated by twice the source immersion, to that due to ement distributions that ensure convergence for the the submerged source in isolation. selected values of A. The boundary element model The pressure ratio is close to unity for locations near for analysis at 12 and 24 Hz (Kinns & Bloor 2002a,b) to the submerged source, but varies greatly with fre- uses 600 elements to describe the hull, with an addi- quency elsewhere when the wavelength is not large in tional 1,176 elements to describe the sea surface. The relation to ellipsoid dimensions. Significant changes hull elements are distributed between 25 longitudinal with frequency are still apparent when the wave- segments, with 24 elements around the hull in each length is twice the length of the ellipsoid. It is only when the wavelength is very large indeed in rela- tion to hull dimensions that the SBF distribution ap- proaches the solution obtained using Laplace's equa- tion. These results for the ellipsoid show why there are such large changes in SBF distributions for surface ship hulls with increasing frequency, as described by Kinns and Bloor (2002a). The following cumulative force distributions are derived using their model. Results for a cruise liner hull A numerical description of a real cruise liner hull was used in Kinns & Bloor (2002a,b) to illustrate applica- tion of the modelling techniques to a real hull shape. The same hull description is used here. The hull is typical of modern twin-screw cruise liners having a single skeg. It has a nominal waterline length of 251 metres and a beam of 32.2 metres. The draught is 8.3 metres in level trim. Results are presented here for frequencies of 12, 24 and 48 Hz. These are the maximum values of one, two and four times propeller blade passing frequency (bpf) in the ship selected for detailed analysis. The underwater sound wavelength ~ is respectively 125, 62.5 and 1.25 metres. The beam is then close to 0.25N Figure 11 0.5A and ~ respectively. The principal difference be- Figure 12 shows the selected locations of sources in tween the numerical models for the floating ellipsoid the plane of the propeller disc, which is 8.4 metres and the cruise liner is that the sea surface was mod- forward of the stern waterline. Figure 11 also shows elled explicitly in the latter case, while image moth- hull sections near to the propeller disc. The monopole oafs were used for the ellipsoid. source is on the periphery of the propeller disc, 40° in- The sea surface is defined for a specified extent ei- board of top dead centre. This is close to the position ther side of the hull, as far forward as the bow. The of maximum cavitation volume for the selected ship surface extends by the same distance in the stern di- with inward rotating propellers. The dipole sources rection. An extent of 300 metres has been used for are positioned at the hub of the propeller, on the star- segment. Figure 11 shows the surface element distri- bution on the hull itself, for the starboard side only. The hull, but not the assumed excitation, is sym- metric. For clarity, elements are shown for alternate longitudinal stations, specified according to distance from the stern waterline. The sea surface extent is reduced to 100 metres for analysis at 48 Hz, where ~ is only 31.25 metres. Also the numbers of hull and sea surface elements are in- creased to 984 and 1,560, because the element sizes would otherwise be too large in relation to A. 3 6 9 12 Tranevsras co~oroinatc (metres) l 15 18 _35- ·-~--&1_ _tZ8 _ .._..174 _Z31 - ·-~--318_ _433 _ ·540~ 7&0 _ ·· O··~35m _~438_ · O ll02 _ 9
board side only. Sources on the port side are set to Results for the dipole can be presented as the ratio zero. of the magnitude of the cumulative hull force to the dipole force. The results are then valid for any ship having the same geometry and relative source posi- tion, regardless of scale. O -,-3 64 -12 _ ~ 0 3 I + ) 15 18 6 9 12 Tranever" coordinate (metros) Sechon a S.em 5~ als.'m S~ at 10.4 Par be. - .4~pd . ban + U$30- _~_ Figure 12 0 50 100 150 200 Distance from stern waterilne (mobas) _ 48 Hz source .~. 24Hz source 12Hz source t.0 _ o' . ~ 0.8 - :P O 0.6- _ ~ 0.4- => 0.2- o.o . v ~ ~ r I ~ 0 50 100 150 200 Distance from stern watarilna (matr - ) _ Vowel dim - I Tnnewrso dlpob Axial dipole Figure 14 Figure 14 shows the cumulative vertical force ratio distributions for 12 Hz sources at the propeller hub. The force in the vertical direction tend to be larger than in other directions, because of the shape of the hull. The force due to the vertical dipole reaches a maximum at about 15 metres forward of the stern, while the maximum is only reached between about 60 Figure 13 and 80 metres forward of the stern for the transverse Figure 13 shows the cumulative vertical force on the and axial dipoles. hull at frequencies of 12, 24 and 48 Hz for a monopole source having a maximum rate of change of mass flux equal to 1 n4 kg/sec2 The effects of the ch~n~?s in SBF and interference between forces on different parts of the hull surface can be seen clearly, with the initial peak moving aft as frequency increases. The scaling of results with ship size is discussed by Kinns and Bloor (2002a), who also show how forces change with source position and are distributed between port and starboard sides. These results reflect the charac- teristics described for the floating ellipsoid. 10 The distribution for the axial dipole shows the ef- fect of the inverted pressure field aft of propeller hub. There is a low secondary maximum at the propeller position, which is only a few percent of the overall maximum. The effects of the sea surface at the stern reduce its value substantially. The cumulative force ratio falls with increasing integration extent before rising again, so that the ratio for a distance up to about 12 metres forward is close to zero.
Conclusions 1.0 - o ° 0.8- ° 0.6- s o ~ 0.2- 3 of ~ o +vo~ ~ I Traneve~ de - ~ -bold dada J ~ 50 100 150 200 Distance from stern waistline (mebas) Figure 15 Figure 15 shows the effect of increasing the hub source frequency from 12 to 24 Hz. The cumulative ratios for the dipole sources are similar at 12 and 24 Hz up to about 12 metres forward, but then change with frequency. The effects of phase interference are more marked at 24 Hz, so that the cumulative force ratios for the axial and transverse dipoles reach max- imum values nearer the stern. The changes in the cumulative hull force distribution due to the verti- cal dipole show the increased influence of its acoustic field at 24 Hz. The maximum cumulative force is now reached at about 60 metres forward. These changes are caused primarily by the increased dipole acous- tic pressures for the same near-field pressure mag- nitudes, compounded by diffraction and free surface effects that vary with frequency. These results echo an important analysis by Cher- tock (1965), who showed that the ratio of the force on the outside of a submarine hull to the force trans- mitted by the tailshaft is almost constant, regardless of hull shape and the precise propeller force distribu- tion. Chertock's analysis did not include the effects of a finite speed of sound, nor was there any need to represent the remote sea surface. The force ratio is much larger for a cruise liner because the propeller is under, rather than behind, the hull. Also, the present results are influenced significantly by the longitudinal location of the propeller relative to the stern water- line. Acoustic boundary element models have been used to explore the nature of fluctuating hull pressures due to propeller sources. The Helmholtz equation has been solved, so that the effects of a finite speed of sound are included. For the present study, these sources were represented by stationary monopoles, represent- ing the principal effects of cavitation, and by station- ary dipole sources in different directions, which repre- sent fluctuating forces at the propeller. The hull sur- face was assumed to be rigid, with a pressure-release surface to represent the surface of the sea. In the first boundary element model, we used a simple ellipsoidal representation of submerged and floating bodies to support understanding of how hull pres- sure distributions are affected by the location and frequency of propeller sources. The model was ver- ified by comparing results for a submerged sphere with analytical solutions that are available for that case. The sea surface was represented using image techniques, to ensure zero pressure there. This ap- proach has been used previously to obtain solutions using Laplace's equation. We have concentrated on calculation of solid bound- ary factors (SBFs), which represent the effect of the rigid hull and pressure-release surface on the pressure that would be measured on a virtual hull surface for the same source in an unbounded sea. In the case of the submerged ellipsoid, having an aspect ratio of 5:1, we found significant diffraction effects when the wavelength was reduced towards the minor diameter, for a source lying aft of the ellipsoid along its longitu- dinal axis. These caused significant departures from the solution obtainable using Laplace's equation. At wavelengths that are not much larger than the length, these are compounded by interference between forces on different parts of the hull surface. Results have been presented for monopoles, longitudinal dipoles and axial dipoles. Much stronger dependence on frequency is observed when the source is below the floating ellipsoid, with the sea surface in its plane of symmetry. Then, the sea surface has the effect of transforming a monopole into a vertical dipole, causing interference between the pressure fields due to the submerged monopole 11
and its negative image above the sea surface. This causes SBF distributions to depend on frequency in the range where the wavelength is not much larger than the hull length, compounding the effects Of Cox, B D, Vorus, W.S., Breslin, J.P. and diffraction and interference that are present for a Rood, E P. "Recent theoretical and experi- submerged body. For the surface ship, this causes departures from SBF distributions calculated using Laplace's equation at frequencies that are well below typical propeller blade passing frequency. Finally, we used an independent boundary element model, with explicit representation of the sea surface, to show how hull forces depend on the nature and frequency of submerged sources for a cruise liner hull with twin screws. Features in the results are clearly related to those for the floating ellipsoid. Acknowledgements The work described in this paper forms part of on- going research to improve understanding of hull ex- citation and underwater sound radiation due to pro- pellers. The authors are grateful to BAE SYSTEMS, QinetiQ, Dstl and PRO Princess Cruises for their support and encouragement. They are also grateful to Meyer Werft of Papenburg, Germany, for provid- ing data that describe the hull of a modern cruise liner. References Bloor, C.D., "A Study of the Acoustic Pressures on a Ship's Hull due to its Propellers", Cambridge Uni- versity PhD thesis, 2002. Bloor, C.D. and Kinns, R., "Development of Acous- tic Boundary Element Models for the Prediction of Fluctuating Hull Forces due to Propeller Cavitation", Proceedings of NCT'50, Newcastle, April 2000, pp. 247-262. Breslin, J.P. and Andersen, P. Hydrodynamics of Ship Propellers, Cambridge University Press, 1994 Chertock, G., "Forces on a Submarine Hull Induced 12 by the Propeller", Journal of Ship Research, Septem- ber 1965, pp 122-130. mental developments in the prediction of pro- peller induced vibration forces on nearby bound- aries", Proceedings of Twelfth Symposium on Naval Hydrodynamics, 1978, pp 278-299. Kinns, R. and Bloor, C.D., "The Effect of Shaft Rotation Direction on Cavitation-Induced Vibration in Twin-Screw Ships", Proceedings of NCT'5O, New- castle, April 200O, pp. 231-246. Kinns, R. and Bloor, C.D., "Fluctuating Hull Forces due to Propeller Cavitation", to be published in RINA Transactions 2002. 7 Kinns, R. and Bloor, C.D., "Hull Vibration Excita- tion due to Monopole and Dipole Propeller Sources", to be published in Journal of Sound and Vibration, 2002. Morse, P.M. and Feshbach, H., "Methods of Theoret- ical Physics", McGraw-Hill, New York, 1953. Ross, D., "Mechanics of Underwater Noise", Penin- sula Publishing, Los Altos, California, 1987. Shelton, E.A. and James, J.H., "Theoretical Acous- tics of Underwater Structures", Imperial College Press, London, 1997. Vorus, W.S., "A Method for Analysing the Propeller- induced Vibratory Forces Acting on the Surface of a Ship Stern" Transactions SNAME Vol.82 1974 , . . . pp.186-210.
DISCUSSION H.B.Clausen,G.M.Keith,and U. M. Rasmussen 0degaard & Danneskiold-Samsee A/S, Denmark We congratulate the authors on this thorough and detailed investigation, and we thank them and the organisers of the 24th symposium for the opportunity to contribute to this discussion. It has often, and for a long time, been taken for granted that compressibility plays no significant role in the calculation of propeller-induced hull- pressure fluctuations, and a systematic evaluation of its effects is very welcome. The authors are very persuasive about the importance of compressibility in the accurate calculation of solid boundary factors at distances from the propeller comparable to the acoustic wavelength of the pressure pulsation. From their analysis it can not be doubted that frequency dependent diffraction effects have a significant influence on the hull-pressure fluctuations at these distances. We agree with all the authors' conclusions regarding the effects of compressibility on hull-pressure fluctuations. However, in order to evaluate compressibility effects on the vibration response of the ship, analyses such as those presented by the authors must be viewed in the light of vibration analyses of the ship's structure, which are usually expressed in terms of its structural vibration eigenmodes. It is the interaction of the distribution of the magnitude of the local pressure fluctuations with the eigenmode shapes, rather than the cumulative hull forces that determines the vibration response of the ship. A relative estimate of the strength of the local pressure fluctuations may be acquired from the article by differentiating the cumulative force distributions, given for example in figure 13. Quite clearly, the strongest local pressure fluctuations occur close to the propeller, and it is our experience that this region of farce magnitude local pressure forcing alone determines the vibration response of the ship. In the frequency range in which compressibility begins to play a noteworthy role, the vibration response becomes highly localised. In this case, the vibrational response away from the propeller is dominated by the structural transmission of the forcing from the large pressure fluctuations near the propeller. The structural transmission is considerably more efficient than the spherically divergent hydrodynamic transmission. Our comments are based on our experience with vibration analysis of ships. In general it is difficult to make statements about the relevance or otherwise of compressibility on fluid-structure interaction problems without conducting a detailed analysis of the structural vibration response. Evaluation of the coupled problem is, of course, contingent on being able to predict the compressible pressure fluctuations, and in this respect, the authors have made a significant contribution to the debate. AUTHORS' REPLY We thank Henrik Clausen and his colleagues for their constructive observations. Our aim has been to develop an analysis tool that allows the effect of compressibility to be explored, in terms of both its influence on hull vibration excitation and on underwater sound radiation from sources in the presence of the hull surface. The analysis is designed to expose effects at low to medium frequencies, where the wavelength of underwater sound is comparable to hull dimensions. Our consideration of cumulative forces, for different integration extents, was designed to show how large the area of integration should be to capture hull excitation in its entirety. We think that previous emphasis on maximum hull pressure has sometimes distorted design of propulsion arrangements for minimum hull vibration, because it places too much emphasis on dimensions such as clearance between the propeller and the hull, when the real emphasis should be on reduction of acoustic source strengths. We hope that our approach will allow a clearer distinction between different effects in the future, and thereby lead to improved design optimization. We agree that the actual effect of the complete hull pressure distribution on vibration can only be determined by considering the nature of the eigenmodes that govern hull response at a given frequency. Sometimes, these eigenmodes will exhibit antipodes near the position of maximum hull pressure, as well as being close to resonance. This is the situation of greatest concern to the structural designer, where maximum hull pressure will tend to determine the vibration characteristics of the ship. Our analysis suggests that the hull pressure at
relatively large distance from the propeller may sometimes have a significant influence on vibration, especially if the maximum near the propeller is close to a node in the principal vibration eigenmode. We agree that the significance of compressibility in these cases can only be determined for a given ship design using a coupled analysis of the structural vibration response. DISCUSSION Stephane Cordier Bassin d'Essais des Carenes, France Our experience shows that resultant forces are strongly dependent on the nonstationary nature of the source position (propeller rotation). Can you elaborate on the possibility of your code to model moving sources? AUTHORS' REPLY We elected to look at simple stationary sources first, in order to identify the underlying nature of hull force distributions and their dependence on source frequency. We are intending to generalize the code to include moving sources. DISCUSSION Merle Lucie Bassin d'Essais des Carenes, France Could you give some comments on the dimensional results of the floating hull case compared to the cruiser liner hull case? AUTHORS' REPLY The graph showing the cumulative vertical force for the floating hull case (Figure 10 in the paper) was derived for a monopole source with amplitude 4~ kg/sec2. The minor diameter of the ellipsoid was 2 metres and the vertical scale represents the cumulative force amplitude in N for that case. The cumulative force for any ellipsoid diameter d (metres) can be obtained by multiplying the hull forces in Figure 10 by did, for a specified value of Lid. Forces are directly proportional to source strength. The comparison with results for the cruise liner hull in Figure 13 must be limited to the qualitative trend of these curves for different wavelengths, because the hull shape and source location parameters are different. In particular, the source is beneath the keel in Figure 10 while it is offset to starboard for the results in Figures 12 to 14. Also, the keel depth of the ellipsoid is large in relation to that of the cruise liner. In a related paper (Kinns and Bloor, 2002b), hull forces for the cruise liner have been presented in the non-dimensional form: FIMob as a function of xib, where Fis the cumulative hull force, b is the beam, Mo is the monopole source strength and x is the distance forward from the stern waterline. The non-dimensional hull force due to a dipole is FIF`I, where F`' is the dipole source strength.
