Calculation of pressure fluctuations induced by the marine

propeller with a boundary element method

Miguel Onofre dos Santos Sardinha Gomesmiguel.onofre.gomes@tecnico.ulisboa.pt

Instituto Superior Tecnico, Universidade de Lisboa, Portugal

November 2016

Abstract

In this work the application of a tridimensional boundary element method to the computationof pressure fluctuations induced by the propeller is studied. The method is applied to a case studywhich consists in the determination of pressure fluctuation amplitudes and phases induced by a marinepropeller on a flat plate aligned with the undisturbed flow. The propellers DTMB 4118 and DTMB4119 are considered in the calculations and the influence of the advance coefficient and distance betweenpropeller tip and plate in the numerical solution is studied. The code PROPAN which implements alow order panel method is used to compute the flow velocity. The velocity is then used to determinepressure values with Bernoulli equation and an harmonic analysis is carried to compute the fluctuationamplitudes and phases. The convergence of the solution with discretization level is studied and thenumerical results are presented for open water diagrams, pressure fluctuation amplitudes and phases ona flat plate. Numerical results are compared with experimental values and results from other methodspresented in the literature.Keywords: Pressure fluctuation, Marine propeller, Boundary Element Method, Potential flow, Shipvibration.

1. Introduction

With the increasing demand for larger and fasterships, the topic of structural vibration affecting thiskind of structures becomes more important on theirdesign process [1]. Vibration on ships may causediscomfort to passengers and crew, it may impairmachine performance, increasing its maintenancecosts, and in the most critical cases may even lead tofatigue failure of main structural components jeop-ardizing the safety of the structure and the peopleon board [2]. There are three main sources of vi-bration on a ship: sea waves, ship’s main engineand propeller generated forces. The latter excitesthe ship’s structure either through the shaft or inthe form of pressure fluctuations induced on thehull. The fluctuating forces transmitted throughthe propeller shaft correspond to fluctuating thrustand moments caused by the non-uniform flow inwhich the propeller operates. The pressure fluctu-ations induced on the hull are caused by the ro-tating pressure field generated and carried by thepropeller and it can represent up to 90% of the ex-citation forces originated in the propulsion systemif cavitation occurs [3]. The objective of this workis to study the application of a panel method to thecalculation of such pressure fluctuations induced by

the propeller. To achieve that goal a case study isanalysed which consists in determining the ampli-tudes and phases of pressure fluctuations inducedon a flat plate near the propeller. No cavitationeffects are considered in this study.

2. Mathematical FormulationConsider a propeller of radius R with K blades sym-metrically placed around an axisymmetric hub, ro-tating with constant angular velocity Ω in a flowof incompressible ideal fluid. A Cartesian coordi-nate system (x0, y0, z0) fixed in the domain and aCartesian coordinate system (x, y, z) rotating withthe propeller are used to describe the problem. Thex and x0 axes coincide with the propeller rotationaxis and point towards the downstream direction ofthe undisturbed flow. The y0 and z0 axes are con-tained in the plane of the propeller with y0 pointingupwards and z0 completing the right-hand system.The y axis is coincident with the reference line ofone of the blades and z completes the system bythe right-hand system. The cylindrical coordinatesystems (x0, r0, θ0) and (x, r, θ) are related to theCartesian systems by

y0 = r0 cos θ0 , z0 = r0 sin θ0 , (1)

y = r cos θ , z = r sin θ . (2)

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The relation between the two reference frames is

x0 = x , r0 = r , θ0 = θ − Ω t , (3)

where t is the time variable. The undisturbed flowon the fixed reference frame is steady but not nec-essarily uniform and is called ~Ue(x0, y0, z0). In therotating coordinate system the undisturbed flow ve-locity is time dependent and is given by

~U∞(x, r, θ, t) = ~Ue(x, r, θ − Ωt)− ~Ω× ~x , (4)

with ~x = (x, y, z).

2.1. Potential flowThe flow velocity in any field point may be seenas the sum of the undisturbed flow velocity witha perturbation velocity ~v(x, y, z, t) and if the flowis irrotational then this perturbation velocity maybe written as the gradient of a scalar perturbationpotential,

~v(x, y, z, t) = ∇φ(x, y, z, t) , (5)

which satisfies Laplace equation

∇2φ(x, y, z, t) = 0 . (6)

The perturbation potential must verify the bound-ary condition at infinity

∇φ→ 0 , if |~r| → ∞ (7)

and the impermeability condition on the solidboundaries of the domain, which consist of the pro-peller blades surfaces SB and hub surface SH ,

∂φ

∂n≡ ~n · ∇φ = −~n · ~U∞ on SB ∪ SH , (8)

where ∂/∂n represents differentiation along the nor-mal to the surface and ~n is the unit vector normal tothe surface pointing to the exterior of the body. Theexistence of circulation around the blades requiresa vortex sheet to be shed from the blade trailingedge forming a wake surface SW . On this surfacetwo boundary conditions must be verified: that theflow velocity on the surface is continuous and equalto the surface velocity

~V + · ~n = ~V − · ~n = ~Vw · ~n on SW , (9)

and the pressure is continuous across the surface

p+ = p− on SW , (10)

where ~V is the flow velocity, ~Vw is the wake surfacevelocity, p is the pressure and the superscripts + and− represent values on each side of the wake surface.

In order to uniquely define the problem it is nec-essary to enforce a Kutta condition which boundsthe velocity on the blade’s sharp trailing edge tofinite values.

2.2. Wake modelRepresenting the wake surface by Sw(~x, t) = 0,equation (9) implies that

∂Sw

∂t+ ~V + · ∇Sw =

∂Sw

∂t+ ~V − · ∇Sw = 0 . (11)

Pressure values in the flow field are computed byBernoulli equation

p

ρ+

1

2|~V |2 +

∂φ

∂t=p∞ρ

+1

2|~U∞|2 , (12)

where p∞ is the pressure in the undisturbed flowand ρ is the fluid’s density. Applying Bernoulliequation to each side of the wake surface and sub-tracting, we obtain

∆p

ρ= −∂(∆φ)

∂t− 1

2

(|~V +|2 − |~V −|2

), (13)

where ∆p = p+ − p− and ∆φ = φ+ − φ− are thepressure jump and the potential jump across thesurface, respectively. Boundary equation (10) forcesthe pressure jump across the surface to be zero, re-sulting

∂(∆φ)

∂t+ ~Vm.∆~V = 0 , (14)

where ~Vm = 12 (~V + + ~V −) is the mean flow velocity

on the wake surface.Solving the velocity field requires the simultane-

ous solution of the vortex sheet location, given by(11), and the intensity of the vortices shed, givenby (14), which is generally difficult to achieve. Oneway to simplify the problem is to assume that themean flow velocity in the wake surface is equal tothe circumferential average of the undisturbed flowvelocity Ue. The solution of the wake surface loca-tion is of the form

Sw = x− xTE −Ue

Ω(θ − θTE) = 0 , (15)

corresponding to a helicoidal surface and the shedvortices intensities are determined by

∆φ(r, θ, t) = ∆φ

(r, t− θ − θTE

Ω

), (16)

with the boundary condition on the blade trailingedge,

∆φ(r, θTE(r), t) = ∆φTE(r, t) , (17)

and the initial condition,

∆φ(r, θ, 0) = ∆φTE(r, 0) . (18)

2.3. Integral equationApplying Green’s second identity and definingφ = 0 in the interior of the propeller bodies, the

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perturbation potential in a point P on the surfaceis determined by

2πφ(P, t)−∫∫

SB∪SH

φ(Q, t)∂

∂nQ

(1

R(P,Q)

)ds =

=

∫∫SW

∆φ(Q′, t)∂

∂nQ′

(1

R(P,Q′)

)ds +

+

∫∫SB∪SH

(~n · ~U∞(Q, t)

) 1

R(P,Q)dS ,

(19)

where R(P,Q) is the distance between the point Pand a point Q on the surface.

2.4. Velocity, pressure and forcesThe differentiation of the potential on the surfaceof the propeller yields the velocity field on that sur-face. Pressure values are computed by Bernoulliequation and presented in the form of pressure co-efficient

Cp =p− p∞12ρU

2∞

= 1− |~V |2

|~U∞|2− 2

|~U∞|2∂φ

∂t. (20)

The propeller thrust force and torque are computedthrough the integration of the pressure values onthe surface and are presented in the form of thrustcoefficient and torque coefficient, respectively,

KT =T

ρN2D4, KQ =

Q

ρN2D5, (21)

where N is the propeller revolution rate and D isthe propeller diameter.

3. ResultsPressure fluctuation amplitudes and phases inducedon a flat plate were calculated for propellers DTMB4118 and DTMB 4119. Three advance conditionswere considered corresponding to the design ad-vance condition, a heavy loaded advance conditionand the zero-thrust advance condition. In order toassess the influence of propeller tip clearance, theflat plate aligned with the undisturbed flow wasstudied in two different positions: the first one giv-ing a distance between the plate and propeller tip of10% of propeller radius and the second one givinga distance of 30% of propeller radius. The calcula-tions were made for several points along a longitu-dinal line belonging to the surface of the plate withcoordinates:

−1.5 6 x0/R 6 1.5 , y0/R = 0 , z0/R = −1.10 ,(22)

and

−1.5 6 x0/R 6 1.5 , y0/R = 0 , z0/R = −1.30 ,(23)

Calculated results are compared with experimen-tal values and the results from other methods pre-sented in [4].

3.1. Open Water DiagramsFirst, results of the open water performance dia-grams calculations are presented. Figure 1 showsthrust coefficient and torque coefficient calculatedfor several advance conditions ranged form J = 0to J = 1.160 for propeller DTMB 4118 and the re-spective experimental results. The same results forpropeller 4119 are presented in figure 2.

Figure 1: Open water diagrams for propeller DTMB4118 - Thrust coefficient and Torque coefficient.

Figure 2: Open water diagrams for propeller DTMB4119 - Thrust coefficient and Torque coefficient.

The results are in good agreement with the exper-imental values except for torque coefficient of pro-

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peller DTMB 4119 where a small discrepancy be-tween calculated and experimental values is seen.This discrepancy is minimal near the design ad-vance condition J = 0.833 and may be explainedby the fact that no viscous corrections were consid-ered in this model.

3.2. Design advance condition - J = 0.833

Results from amplitude and phase of pressure fluc-tuations calculated with propeller DTMB 4118 fordesign advance condition, J = 0.833, with a pro-peller tip clearance of 10% are shown in figures 3and 4, respectively. Results calculated for a tipclearance of 30% are shown in figures 5 and 6.

Figure 3: Pressure fluctuation amplitude. PropellerDTMB 4118, J = 0.833, z0/R = −1.10.

Figure 4: Pressure fluctuation phase. PropellerDTMB 4118, J = 0.833, z0/R = −1.10.

Calculated amplitudes overpredict the experi-mental ones in both cases. In the region down-stream of the propeller plane, fluctuation ampli-tudes fail to diminish, contradicting the experimen-tal results. This effect is more intense in the 10% tipclearance situation. Phase values behaviour agreeswith experimental values behaviour along the axialdirection for the 10% tip clearance but not in the30% tip clearance case.

Figure 5: Pressure fluctuation amplitude. PropellerDTMB 4118, J = 0.833, z0/R = −1.30.

Figure 6: Pressure fluctuation phase. PropellerDTMB 4118, J = 0.833, z0/R = −1.30.

Results obtained with calculations made for pro-peller DTMB 4119 are presented in figures 7 to 10.

Figure 7: Pressure fluctuation amplitude. PropellerDTMB 4119, J = 0.833, z0/R = −1.10.

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Figure 8: Pressure fluctuation phase. PropellerDTMB 4119, J = 0.833, z0/R = −1.10.

Figure 9: Pressure fluctuation amplitude. PropellerDTMB 4119, J = 0.833, z0/R = −1.30.

Figure 10: Pressure fluctuation phase. PropellerDTMB 4119, J = 0.833, z0/R = −1.30.

Similar conclusions may be drawn from these re-sults about the overprediction of amplitude valuesand the fact that they fail to approach zero in thedownstream region, noting that this propeller withthicker blades gives rise to larger amplitude fluc-tuations compared to the previous one. Again,phase results follow the evolution of experimentalvalues in the z0/R = −1.10 location but not on thez0/R = −1.30.

3.3. Heavy loaded avanced condition

Figures 11 to 14 present values calculated with anadvance coefficient J = 0.600, corresponding to anincrease in propeller loading compared with the de-sign condition.

Figure 11: Pressure fluctuation amplitude. Pro-peller DTMB 4118, J = 0.600, z0/R = −1.10.

Figure 12: Pressure fluctuation phase. PropellerDTMB 4118, J = 0.600, z0/R = −1.10.

Figure 13: Pressure fluctuation amplitude. Pro-peller DTMB 4118, J = 0.600, z0/R = −1.30.

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Figure 14: Pressure fluctuation phase. PropellerDTMB 4118, J = 0.600, z0/R = −1.30.

An increase in propeller loading is seen to leadto larger fluctuation amplitudes both in the maxi-mum value and in the region of downstream flow.Predicted values overestimate experimental ones forboth tip clearances. Results for propeller DTMB4119 are presented in figures 15, 16, 17 and 18.

Figure 15: Pressure fluctuation amplitude. Pro-peller DTMB 4119, J = 0.600, z0/R = −1.10.

Figure 16: Pressure fluctuation phase. PropellerDTMB 4119, J = 0.600, z0/R = −1.10.

Figure 17: Pressure fluctuation amplitude. Pro-peller DTMB 4119, J = 0.600, z0/R = −1.30.

Figure 18: Pressure fluctuation phase. PropellerDTMB 4119, J = 0.600, z0/R = −1.30.

3.4. Zero-thrust advance condition

Results presented in figures 19 to 22 show the am-plitudes and phases calculated with an advance co-efficient J = 1.160 corresponding to the zero-thrustadvance condition for both propellers.

Figure 19: Pressure fluctuation amplitude. Pro-peller DTMB 4118, J = 1.160, z0/R = −1.10.

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Figure 20: Pressure fluctuation phase. PropellerDTMB 4118, J = 1.160, z0/R = −1.10.

Figure 21: Pressure fluctuation amplitude. Pro-peller DTMB 4118, J = 1.160, z0/R = −1.30.

Figure 22: Pressure fluctuation phase. PropellerDTMB 4118, J = 1.160, z0/R = −1.30.

The results calculated with this advance coeffi-cient still overestimate the experimental values butsucceed in the representation of the diminishingpressure fluctuations on the downstream region ofthe plate. Computed phase values evolution alongthe plate agree well with experimental values forboth plate positions. Figures 23 trough 26 presentresults for propeller DTMB 4119 operating at zero-

thrust condition. The same behaviour as in pro-peller DTMB 4118 values may be seen in these re-sults for both amplitudes and phases.

Figure 23: Pressure fluctuation amplitude. Pro-peller DTMB 4119, J = 1.160, z0/R = −1.10.

Figure 24: Pressure fluctuation phase. PropellerDTMB 4119, J = 1.160, z0/R = −1.10.

Figure 25: Pressure fluctuation amplitude. Pro-peller DTMB 4119, J = 1.160, z0/R = −1.30.

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Figure 26: Pressure fluctuation phase. PropellerDTMB 4119, J = 1.160, z0/R = −1.30.

3.5. Wake surface contraction

Here wake contraction is considered. Four differentwake surface geometries are used to compute pres-sure fluctuation amplitudes and phases for the pro-peller DTMB 4118, with the three advance condi-tions and two plate positions. Each wake geometryhas a different exterior radius ru on the downstreamregion but all have the same radius on the bladetrailing edge Rte = 0.98. Results are presented infigures 27 to 30 for the design advance condition,figures 31 to 34 for the heavy loaded condition andfigures 35 to 38 for the zero-thrust condition.

Figure 27: Pressure fluctuation amplitude withwake contraction. Propeller DTMB 4118, J =0.833, z0/R = −1.10.

Figure 28: Pressure fluctuation phase with wakecontraction. Propeller DTMB 4118, J = 0.833,z0/R = −1.10.

Figure 29: Pressure fluctuation amplitude withwake contraction. Propeller DTMB 4118, J =0.833, z0/R = −1.30.

Figure 30: Pressure fluctuation phase with wakecontraction. Propeller DTMB 4118, J = 0.833,z0/R = −1.30.

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Figure 31: Pressure fluctuation amplitude withwake contraction. Propeller DTMB 4118, J =0.600, z0/R = −1.10.

Figure 32: Pressure fluctuation phase with wakecontraction. Propeller DTMB 4118, J = 0.600,z0/R = −1.10.

Figure 33: Pressure fluctuation amplitude withwake contraction. Propeller DTMB 4118, J =0.600, z0/R = −1.30.

Figure 34: Pressure fluctuation phase with wakecontraction. Propeller DTMB 4118, J = 0.600,z0/R = −1.30.

Figure 35: Pressure fluctuation amplitude withwake contraction. Propeller DTMB 4118, J =1.160, z0/R = −1.10.

Figure 36: Pressure fluctuation phase with wakecontraction. Propeller DTMB 4118, J = 1.160,z0/R = −1.10.

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Figure 37: Pressure fluctuation amplitude withwake contraction. Propeller DTMB 4118, J =1.160, z0/R = −1.30.

Figure 38: Pressure fluctuation phase with wakecontraction. Propeller DTMB 4118, J = 1.160,z0/R = −1.30.

Results show that wake contraction leads tosmaller fluctuation amplitudes downstream of thepropeller plane with greater differences in the heavyloaded condition and no influence in the zero-thrustcondition. Phase values seem to be almost insensi-ble to wake contraction in the 10% tip clearanceposition, being only affected in the near propellerdownstream region for the cases with 30% tip clear-ance.

4. ConclusionsPressure fluctuations induced on a flat plate werecalculated with a panel method. It was seen thatthicker blades, heavier propeller loads and smallertip clearances all lead to larger fluctuation am-plitudes and that the computed values react wellto this parameters. Fluctuation amplitude resultsoverestimated experimental values in every condi-tion studied. Phase values calculated agree wellwith the evolution of experimental values along theplate for a tip clearance of 10% of propeller ra-dius. In the region downstream of the propellerplane, calculated results fail to predict the decay of

fluctuation amplitudes with axial distance seen inthe experimental values. This effect was seen to beproportional to propeller loading and is attenuatedwith contraction of the wake surface.

References[1] Carlton, J., Vlasic, D. (2006), Ship vibration

and noise: Some topical aspects, Lloyd’s Reg-ister Techincal Papers.

[2] Lee, S., Liao, M., Wang, S. (2006), Propeller-Induced Hull Vibration Analytical Methods,American Bureau of Shipping.

[3] Asmussen, I., Menzel, W., Mumm, H. (2001),Ship Vibration, GLs ”Technology” n5/2001.Germanischer Lloyd.

[4] Denny, S. (1967), Comparisons of Experimen-tally Determined and Theoretically PredictedPressures in the Vicinity of a Marine Propeller,David Taylor Model Basin Report 2349.

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