Firs Inernaional Symposium on Marine Propulsors smp 9, Trondheim, Norway, June 9 Compuaion of hull-pressure flucuaions due o non-caviaing propellers Frans Hendrik Lafeber, Erik van Wijngaarden, Johan Bosschers Mariime Research Insiue Neherlands (MARIN), Wageningen, The Neherlands ABSTRACT This paper describes a new coupling procedure of wo boundary elemen mehods aimed a compuing propellerinduced hull-pressure flucuaions. The firs mehod compues he incompressible poenial flow of a propeller operaing in a ship wake field. The second mehod compues he propeller-radiaed pressures by replacing he incompressible-flow soluion on he roaing propeller blades by a se of rings of saionary sources. This source sysem is used as inpu in an acousic scaering analysis of he weed ship hull and undisurbed free surface based on he Kirchhoff-Helmholz inegral equaion. Thus, hull pressures are obained in he frequency domain on he basis of a ime-domain source descripion. hull pressures for non-caviaing condiions are presened for a conainer vessel ha was successively fied wih wo differen propellers, one being a six-bladed propeller designed for he vessel, he oher a wo-bladed propeller. Comparisons are made wih scale-model ess performed in a owing ank. For he wo-bladed propeller, he compued resuls are in good agreemen wih he experimenal resuls. The six-bladed propeller shows somewha larger variaions in correlaion. Keywords Hull-pressure flucuaions, non-caviaing propellers, boundary elemen mehods, validaion experimens. INTRODUCTION Compuaion of he pressure field on he hull due o a propeller operaing behind a ship is used o assess he exciaion forces on he afship in an early design sage. This hull-pressure field conains componens due o he passing blades as well as caviaion and is influenced by he diffracion effec of he weed ship hull and he free surface. For he compuaion of he oal pressure field all of hese conribuions should be aken ino accoun. Here, only he compuaion of he non-caviaing conribuion is considered using a few validaion experimens. The advanage of considering only he non-caviaing propeller is ha is flow is less difficul o compue han ha of he caviaing propeller, and ha he variabiliy in he measuremen daa is much smaller. The predicion of he diffracion effec can hus be assessed a larger accuracy han for a caviaing propeller. Anoher imporan issue is ha wih he reducion in shee caviy exens on modern propellers, he non-caviaing par becomes of greaer pracical imporance. Ofen, only he ip vorex is caviaing and he hull-pressure ampliude a he firs blade-passage frequency (BPF) is dominaed by he conribuion from he non-caviaing propeller. In his paper he hull-pressure flucuaions are compued using a new coupling procedure beween wo boundary elemen mehods (BEM). The compuaional approach is described in Secion. Secions 3 and 4 discuss he experimenal and compuaional procedures. Resuls of he validaion sudy are described in Secion 5. THEORY In his secion, he boundary elemen codes are briefly described. The mehod of coupling he wo codes by replacing he roaing sources represening he propeller by a se of rings of non-roaing sources is reaed in deail. The applicaion of Bernoulli s equaion for he deerminaion of he pressure flucuaions is discussed.. Propeller flow analysis For he analysis of he flow pas a propeller, use is made of a ime-domain BEM ha solves he incompressible poenial flow equaions for lifing surfaces, see e.g. Vaz & Bosschers (6) and Bosschers e al. (8). The mehod, designaed PROCAL, has been developed wihin MARIN s Cooperaive Research Ships (CRS) for he unseady analysis of caviaing propellers operaing in a prescribed ship wake. The code is a low order BEM ha solves for he velociy disurbance poenial using he Morino approach. A shee caviaion model was implemened in which he non-linear kinemaic and dynamic boundary condiions are ieraively solved assuming ha he caviy surface coincides wih he body surface. In he compuaions presened, he pich and conracion of he propeller wake geomery are prescribed using empirical formulaions, while he srengh of he wake is compued using an ieraive pressure Kua condiion. The srengh of he monopoles and dipoles for each panel on he body and wake is wrien o file for one revoluion.. Hull-pressure analysis For he analysis of he diffraced hull pressures induced by he propeller, use is made of a frequency domain BEM ha solves he Kirhhoff-Helmholz inegral equaion for compressible fluids (van Wijngaarden, 6). The
mehod, designaed EXCALIBUR, has originally been developed in MARIN s background research program and was recenly furher developed wihin CRS. I is a low order BEM code ha solves for he acousic velociy poenial using he Buron & Miller approach. Thus, no main flow is aken ino accoun and he free surface is modeled hrough a negaive-mirror-imaging procedure. The diffracion problem is solved for locaions on he hull on he basis of a prescribed se of sources. The pressure flucuaions are hen obained by applicaion of Bernoulli s equaion discussed in Secion.4..3 Combining PROCAL and EXCALIBUR In he pas, hull pressures would have been obained from a simple combinaion of he wo codes described above. EXCALIBUR compued a so-called solid boundary facor (SBF) in he frequency domain. PROCAL would hen compue he pressure in he free-field a he locaion of he hull. By muliplying his free field pressure by he SBF, he pressure on he hull was obained. However, as discussed by Bosschers e al. (8), his mehod has he drawback of modeling he hull-diffracion effec by one monopole only, whereas he source sysem acually consiss of a large se of monopoles and dipoles. A new mehod is presened here by which he complee soluion from PROCAL is used as inpu for EXCALIBUR. This new mehod is based on replacing he ime domain flow soluion on he propeller by a saionary se of rings of monopole and dipole sources in he frequency domain as suggesed by Brouwer (5). The propeller s elemenal sources of noise (i.e. monopoles and dipoles) are roaing around he propeller cenre line. This can also be said of he helicoidally shed voriciy; he propeller wake, which consiss of only dipoles. Assuming non-uniform, bu saionary propeller inflow, he source field becomes periodic in ime a BPF, ω, given by ω = ZΩ, wih Z he number of propeller blades and Ω he propeller revoluion rae in rad/s. Time, T, needed for one revoluion is T = π Ω. Likewise, T = π ZΩ = π ω denoes he blade-passage period. In he following, a procedure is presened for he Fourier ransformaion of he roaing sources o a se of saionary sources pulsaing a BPFs in he frequency domain. To ranslae he ime-domain soluion of sources moving wih he propeller from PROCAL o a frequency-domain se of saionary sources for EXCALIBUR, we assume he propeller blades and wake surfaces o be approximaed by Nb + Nw panels wih panel-collocaion poins, xi ( ), for i =.. Nb + Nw. During one revoluion of period T, N snapshos are made of he propeller a equidisan inervals of Δ = T /. The se of ime samples in a revoluion is indexed j = jδ wih j =.. N. Each of he Nb + Nw collocaion poins assumes N posiions, xi( j) = xij. Also, each of he panels can be associaed wih a poin source of monopole ype and srengh, σi( j) = σij, plus one of surface normal dipole ype and srengh, μi( j) = μij. Noe ha he railing vorex shees consis of only dipole sources. For breviy of noaion, we denoe source srenghs μ i ( ) or σ i ( ) a a fixed poin by f( ). This is he produc of he insananeous srengh, which we denoe f ij, and a sep funcion ha is one during he ime sep when he panel s collocaion poin coincides wih he source posiion, and zero a oher imes. As he number of ime seps is always aken as a muliple of he blade number, = Z, he sep funcion acually becomes one a Z ime seps during one propeller revoluion. Now, he Nb + Nw roaing sources are replaced by ( Nb + Nw) saionary ones. Then, f( ) can be developed ino a Fourier series based on an inerval covering one blade passage period, T, i n f() Re ( cne ω = ) () n in which j + T + iωn cn = f() e d T () j T wih ωn = nω for he n h harmonic and n. I follows, j +Δ fij + iωn cn = e d T j Δ Δ iω sin ( Δ ) n ω j n = fij e (3) T ωn Δ Z iω n Z j = fij e sincπ n and hence, Z Z iω ( ) n j f() = fij Resincπ n e (4) n wih sinc( x) = sin ( x) x. For a cerain harmonic order, n, he complex ampliude f n of source ij is given by Z Z + i π nzj/ N f = sincπ n fij n e (5) The sources wih complex ampliudes given by Eq. (5) form he inpu for EXCALIBUR. The scaering effec of he hull is hen compued for harmonics of he BPF. The resuling pressure flucuaions on he hull are given as a real ampliude and phase lead relaive o a sine funcion..4 Applicaion of Bernoulli s equaion Using he unseady Bernoulli equaion, he pressure disurbance p is given by φ p = iρωφ ρu ρ φ (6) x Where ρ is he densiy of he fluid, ω is he frequency,φ is he disurbance poenial and U is he free sream velociy in he direcion of he posiive x-axis, which is parallel o he ship cener line and posiive owards he bow. Assuming ha he square of he disurbance velociies is small, Eq. (6) is reduced o
p = iρωφ ρu φ (7) x When he convecive erm is negleced as well, we ge p = i ρωφ (8) Eq. (8) is currenly applied in EXCALIBUR o compue he complex ampliude of he pressure flucuaions on he hull for harmonics of BPF. 3 EXPERIMENTS A series of experimens was conduced o provide daa o validae he combinaion of he compuer codes PROCAL and EXCALIBUR. For he experimens, a model of a large conainer vessel was fied wih differen propellers, and bollard pull and owing ess were performed. During he bollard pull es, he model was fied wih a large wobladed sock propeller leading o a ip clearance of 4% of he propeller diameer. The pich was reduced such ha i generaed a minimal hrus (P.7 /D =.5). This way, only he hickness conribuion of he propeller plays a role. For he owing ess, he model was fied wih he sixbladed propeller designed for his ship. The ip clearance of his propeller was 33% of he propeller diameer. The model was owed a various speeds and wih differen propeller roaion raes, see Table, where N m is he model propeller roaion rae, V m he model speed, K T he dimensionless hrus coefficien and D m he propeller diameer. Figure : Two-bladed propeller fied o he ship model equipped wih flush-mouned pressure sensors Table : Model es condiions Tes Z [-] N m [RPM] BPF [Hz] V m [m/s] K T [-] D m [mm] A 6... 34 B 6 579 57.9.37.4 6 C 6 37 37..37 -. 6 During he ess, he pressure on he hull was measured by means of sensors, which were mouned flush in he area above he propeller, see Figure Figure. The resuls of he hull pressures are presened as he ampliude and phase of he pressure a he firs four harmonics of he BPF. Only he resuls of he nine pressure sensors close o he propeller are presened: sensors P4-P6, P9-P and P4-P6, see Figure. The pressure ampliudes a he oher sensors were no significan (i.e. smaller han. kpa full scale equivalen) and have been omied. Figure : Locaion of pressure sensors wih he propeller and rudder locaion indicaed (dimensions on model scale) 4 COMPUTATIONS PROCAL compuaions have been performed a he same propeller roaion rae and hrus coefficien as he model ess. For he bollard pull ess, seady compuaions were made using uniform inflow. For he owing-es condiions, unseady compuaions were performed using he measured ship wake, which was made effecive wih a force-field mehod. One propeller revoluion is composed of ime seps. In he compuaions presened, 3 panels have been used beween he railing and leading edge of he wo-bladed propeller and 4 panels in he case of he six-bladed propeller. In boh cases, panels were used beween roo and ip. Caviaion feaures have no been considered for he curren invesigaion. The PROCAL resul was hen used as inpu for he EXCALIBUR compuaions, as described in Secion.3. Only he af par of he ship was modeled wihou he rudder and he draugh was increased by he dynamic rim. A oal of 3 panels was used o describe he hull geomery. All compuaions were performed a model scale. Figure 3 gives an example of a combined PROCAL and EXCALIBUR soluion.
propeller wih almos no inflow and a low hrus are very good. This indicaes ha he conribuion of he propeller hickness o he diffraced hull-pressure field is properly capured. 45 4 Figure 3: Example of compuaional resul: propeller wih ship wake, propeller wake (one blade) and pressure field on he hull. Dos on he hull indicae measuring locaions. 5 RESULTS A comparison beween he compuaions and model es resuls can now be made for all hree ess presened in Table. The measured and compued pressure ampliudes are shown for he nine pressure sensors as saed in Secion 3. Of he compued phase, only he error is shown. The error in phase (lead) is given by ε phase = αcompued αmeasured (9) By dividing his error by he harmonic order n and he number of blades Z, he error relaive o he angular blade posiion in he ship wake is obained, ε phase ε relaive = () nz Using he phase and ampliude of he firs four harmonics, a ime series can be reconsruced of boh he compuaional and experimenal daa, 4 n n= ( ) p () = A sinnω+ α () A ime = he propeller s generaor line goes hrough op dead cener. 5. Tes A The resuls of he wo-bladed propeller (diameer 34 mm) show a good agreemen beween compuaion and experimen. As can be seen in Figure 4, he ampliude a he firs harmonic is prediced well, especially for he ransducers in he propeller plane. For he higher harmonics of es A (no shown), he error in ampliude increases slighly. In Figure 5 i can be seen ha he phase of he firs harmonic is compued quie accuraely as well. The small error is almos consan across he sensors. The relaive error remains of he same order of magniude (~7 ) for all harmonics. The ime series of one propeller revoluion is shown in Figure 6 for pressure sensor P, which is locaed sraigh above he propeller. The shapes of he ime series are very similar from which i can be concluded ha he nd, 3 rd and 4 h harmonics are compued accuraely as well. The small consan offse in phase is also visible. Summarizing, he resuls for he wo-bladed n Pressure ampliude [Pa] Y [mm] 35 3 5 5 5 4 5 6 9 4 5 6 Sensor number Figure 4: and measured ampliudes for s harmonic of es A 5 5-5 - Pressure phase difference [deg], es: A s harmonic, frequency: Hz 7 5 9-5 3 4 X [mm] Pressure [Pa] 3 Figure 5: Error in phase, s harmonic, es A 4-4 -6-8 - Time series of pressure a sensor P Tes A, model scale values 9 5 5-5 - -5 9 8 7 36 - Blade posiion [deg] Figure 6: Time series reconsruced from he firs four harmonics, es A
5. Tes B For he six-bladed propeller (diameer 6 mm) a he design K T, he compued and measured ampliude and phase show a reasonable agreemen direcly above he propeller a he firs harmonic, see Figure 7 and Figure 8. Y [mm] Pressure ampliude [Pa] 3 5 5 5 4 5 6 9 4 5 6 Sensor number Figure 7: and measured ampliudes for s harmonic of es B 5 5-5 - Pressure phase difference [deg], es: B s harmonic, frequency: 57.9 Hz -7-34 -9-5 3 4 X [mm] Pressure [Pa] 4 3 - -3-4 -4-4 -39-3 Figure 8: Error in phase, s harmonic, es B Time series of pressure a sensor P Tes B, model scale values 5 5-5 - -5 3 6 9 - Blade posiion [deg] Figure 9: Time series reconsruced from he firs four harmonics, es B A he higher harmonics, all sensors show insignifican ampliudes. This can also be seen in he reconsruced ime series which is an almos perfec sine (see Figure 9 in which only wo blade passages have been ploed). The error in ampliude and relaive phase error remain almos consan for he higher harmonics. 5.3 Tes C In es C, he hrus coefficien of he six-bladed propeller was reduced o almos zero by reducing he roaion rae of he propeller. Again, he larges errors are seen a he sensors wih very low ampliudes, see Figure. Direcly above he propeller, he ampliude error is small: +3% of he measured value. The disribuions of he pressures for he upcoming and downgoing blades are no prediced well. Y [mm] Pressure ampliude [Pa] 9 8 7 6 5 4 3 4 5 6 9 4 5 6 Sensor number Figure : and measured ampliudes for s harmonic of es C 5 5-5 - Pressure phase difference [deg], es: C s harmonic, frequency: 37 Hz 57 33-59 -7-5 3 4 X [mm] -7-88 -33 3 Figure : Error in phase, s harmonic, es C 5 5-5 - -5 In es C, he error in he phase is larger han in es B (Figure ). As was seen before, he sensors wih a low pressure ampliude also give he larges errors in he compued phase. When reconsrucing he ime series for es C, he errors in ampliude and phase become clearly visible, see Figure. Especially he pressures a he higher harmonics are no compued accuraely.
Pressure [Pa] 8 6 4-4 -6-8 - - Time series of pressure a sensor P Tes C, model scale values - 3 6 9 Blade posiion [deg] Figure : Time series reconsruced from he firs four harmonics, es C 5.4 Discussion of he resuls The fac ha he wo-bladed propeller showed more accurae resuls han he six-bladed migh be explained by he omission of he convecive erm in Bernoulli s equaion; compare Eqs.(7) and (8). A firs esimae of his erm revealed ha i can influence he compued ampliude by % o 5% and he phase by 5 o degrees. For he case of he wo-bladed propeller, including he convecive erm will no change he resuls since he inflow velociy U is almos zero. Anoher possible cause of deviaions beween he compuaions and he measuremens is he fac ha he flow voriciy is only coarsely modeled in he BEM. Leading edge vorices as well as he roll-up of he ip vorex are no properly capured. This may explain he less accurae predicion of he six-bladed propeller where in he zero load condiion, large pressure peaks a he leading edge on he face of he propeller were observed in he compuaions. These pressure peaks sugges he presence of a leading edge vorex in he measuremens. 6 CONCLUSIONS A new coupling procedure has been presened of wo boundary elemen mehods aimed a compuing propellerinduced hull-pressure flucuaions. The firs mehod compues he incompressible poenial flow of a propeller operaing in a ship wake field. The second mehod compues he propeller-radiaed pressures by replacing he incompressible flow soluion on he roaing propeller blades by a se of rings of saionary sources. Thus, hullpressures are compued in he frequency domain on he basis of a ime-domain source descripion. The mehod has been evaluaed using experimenal daa obained in a owing basin for a single-screw ship hull equipped wih a wo-bladed and a six-bladed propeller, boh operaing in non-caviaing condiions. For he wobladed propeller, he compued hull-pressure flucuaions show good agreemen wih he experimens. In he case of he six-bladed propeller a he design loading, he compued resuls are reasonable o good. However, when he loading is reduced o a value of almos zero he differences beween compuaions and experimens increase. The general conclusion is ha he combinaion of he wo boundary elemen mehods gives a reasonable o good predicion of he pressures on he hull of a ship due o a non-caviaing propeller. Furher improvemens are expeced by including he convecive erm in Bernoulli s equaion. The influence of he propeller wake and ip vorex model used in he boundary elemen mehod needs o be furher sudied, especially for off-design condiions. ACKNOWLEDGEMENTS The presen work was parly funded by he Cooperaive Research Ships (CRS). The graphical user inerface (used for Figure 3) was developed wihin CRS by DRDC Alanic, wih Dave Heah as main developer. The auhors also like o acknowledge he conribuions made by Herman Beeksma of MARINs sofware developmen deparmen. REFERENCES Bosschers, J., Vaz, G., Sarke, A.R. & Wijngaarden, E. van (8). Compuaional analysis of propeller shee caviaion and propeller-ship ineracion, Marine CFD 8, Souhampon, Unied Kingdom Brouwer, J. (5). Ship propeller-induced noise and vibraions, M.Sc. Thesis, Universiy of Twene, Enschede, The Neherlands Vaz, G., Bosschers, J., (6). Modelling hree dimensional shee caviaion on marine propellers using a boundary elemen mehod, CAV 6 Sixh Inernaional Symposium on Caviaion, Wageningen, The Neherlands Wijngaarden, E. van (6). Deerminaion of propeller source srengh from hull-pressure measuremens, Proceedings of he nd inernaional ship & noise conference 6, London, Unied Kingdom