Tone Noise Prediction of a Propeller Operating in Nonuniform Flows

Tone Noise Prediction of a Propeller Operatingin Nonuniform Flows

J. Xie∗ and Q. Zhou†

Huazhong University of Science and Technology, 430033 Wuhan, People’s Republic of China

and

Philip F. Joseph‡

University of Southampton, Southampton, England SO17 1BJ, United Kingdom

DOI: 10.2514/1.J050423

This paper examines the tone noise generation by propellers operating in a nonuniform mean flow. A numerical

approach is proposed, based in the frequency domain, for predicting far-field propeller noise, which requires the

aerodynamic sources to be integrated over the actual blade surface, rather than over the mean-chord surface. The

prediction of the radiated far-field tonal noise for a four-bladed propeller operating in a nonuniform mean flow is

performed. Validation of the method is presented through a detailed comparison between numerical predictions of

the far-field noise and the measured data, in which excellent agreement is obtained, both in magnitude and

directivity. This paper concludes that for first harmonic noise, matching the number of the main inflow harmonic of

nonuniform flow with blade number creates a dipole-type symmetric directivity pattern, and mismatching appears

as an asymmetric directivity pattern. Effects of a single inflow harmonic on the far-field radiated tonal noise are also

quantified in this paper, which shows that when the matching condition is satisfied, the main inflow harmonic

contributes most of the total radiated sound pressure, except at locations normal to the propeller axis. When the

matching condition is not satisfied, however, the total radiated sound pressure is mainly caused by the dominant

inflow harmonic and the inflow harmonic that equals the blade number.

Nomenclature

A = complex amplitude of inflow harmonicB = blade numberc = chord length of blade sectionc0 = speed of soundE = R�M (y1 � x1)G = Green function in the time domain�G = Green function in the frequency domainH0 = zero-order Hankel functionH1 = first-order Hankel functionJ = Bessel function�kn;r0 = nondimensional wave number of the nth velocity

harmonic at radius r0�kx = nondimensional wave numberl = radiated mode, indicating the order of blade-passing

frequencyM = Mach number of the mean flown = unit normal vector inward from the blade surface

�n1; nr; n��n = order of inflow harmonicP = pitch distanceR = mean-flow corrected distanceRS =

���������������������x21 � �2r2

pr = radial coordinate of observation point in the cylindrical

coordinate systemr0 = radial coordinate of source point in the cylindrical

coordinate systemt = time associated with arrival of sound wave at the

observation point

U = inflow velocityU0 = total incoming velocity at a propeller radiusu = axial inflow velocity�u = circumferentially averaged axial mean velocity~u = axial disturbance velocityVx = absolute amplitude of inflow harmonicx = coordinates associated with the observation point;

�x1; r; �0� for cylindrical coordinates and �x1; x2; x3� forthe rectangular coordinate system

�x = nondimensional chord coordinatey = coordinates associated with the source point; �y1; r0; �00�

for cylindrical coordinates and �yb1 ; yb2 ; yb3� for therectangular coordinate system

� = attack angle at a blade radius section, � � �H� =

����������������1 �M2p

�H = hydrodynamic pitch angle� = geometric pitch angle� = Dirac delta function� = azimuthal angle of observation point in the blade-fixed

cylindrical coordinate system�0 = azimuthal angle of source point in the blade-fixed

cylindrical coordinate system� = acoustic wave number related to the observation

frequency !, !=c0� = �=�2

�0 = density of unsteady oncoming flow = time shift between the moving reference frame and the

blade-fixed coordinate system = cos�1�x1=RS�� = propeller rotational speed! = observation frequency!0 = source frequency in the blade-fixed coordinate system

I. Introduction

P ROPELLERS usually operate in nonuniform mean flows inwhich large-scale inflow distortions are present in addition to

small-scale turbulence. The interaction of the blades with incominginflow distortions creates a periodically fluctuating pressure on theblade surface,which then leads to the radiation of discrete tones at the

Received 26 January 2010; revision received 15 September 2010; acceptedfor publication 22 September 2010. Copyright © 2010 by the AmericanInstitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies ofthis paper may be made for personal or internal use, on condition that thecopier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/11and $10.00 in correspondence with the CCC.

∗Ph.D. Student, Ship Science Department, 717 Jie Fang Road, Hubei.†Professor, Ship Science Department, 717 Jie Fang Road, Hubei.‡Professor, Institute of Sound and Vibration Research, Highfield.

AIAA JOURNALVol. 49, No. 1, January 2011

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blade-passing frequency and higher harmonics [1]. The interactionof the blades with inflow turbulence, on the other hand, leads to arandom blade loading that then radiates as broadband noise. In mostcases, tonal noise from propellers contains a greater sound powerlevel than the broadband noise component.

Since the 1970s there has been considerable interest in theprediction of aerodynamic noise using both empirical and theoreticalmethods [2–6]. Lowson and Ollerhead [7] presented a closed-formnear-field and far-field solutions for the unsteady pressure due tosource singularities in arbitrary motion. In addition, they providedanalytical solutions for the far-field directivity of a propelleroperating in nonuniform flows.Wright [8] examined the influence ofdiscrete frequency noise radiation due to periodic variations in themean flow by employing a modal approach. He concluded thatdifferent types of wakes can produce tonal noise radiation; mean-while, their amplitudes and directivities could have great discrep-ancies. The characteristics of the nonuniform flow upstream of apropeller are therefore critical to the accurate prediction of propellertone noise radiation.

Heller and Widnall [9] analyzed the rotor–stator interaction noiseof an axial compressor. By representing the unsteady blade forceswith equivalent dipoles, they developed an analytical solution forrotor tonal noise prediction in which good agreement was obtainedbetween experiment and theory. Blake [10] proposed a method forpredicting the radiated tonal noise from a rotating propeller andconcluded that at low-tip-speed Mach numbers the only inflowharmonics n that radiate sound are those for which n� lB, where l isthe radiated mode, and B is the blade number. Thus, the propellerresponds principally to those inflow harmonics that are multiples ofthe number of propeller blades. However, their models are restrictedto integration of the source surface distribution on the mean-chordsurface; the actual blade geometry is therefore neglected.

Recently, Seol et al. [11] numerically examined the thicknessnoise and loading noise of a marine propeller operating in non-uniform flows by using time-domain acoustic analogy. Theyconcluded that monopole noise due to blade thickness is very smallcompared with the dipole noise. However, they did not reveal therelationship between nonuniform flows and unsteady blade surfacepressure, and they did not discuss the effects of inflow harmonicnumber on the radiated noise.

As pointed out by Tyler and Sofrin [12], discrete tones are ofgenerallymuch higher sound pressure level than the broadband noisecomponent in nonuniformmeanflows. Therefore, the prediction, andhence control, of propeller tone noise is important. Two importantaspects must be addressed: one is to characterize the relationshipbetween the nonuniform mean velocity field and unsteady bladepressure, and the other is to quantify variant effects of inflowharmonics.

Following the acoustic analogy, the classical approach inaeroacoustics is to represent the acoustic sources as an equivalentdistribution of acoustic monopoles, dipoles, and quadrupoles. Thestrength of these sources is usually determined either experimentallyor analytically. For most frequency-domain prediction methods, thesource area of integration is mostly taken to be either on theprojection disk or the blade mean-chord surface (e.g., Lowson andOllerhead [7], Blake [10], Hanson [13], and Wojno et al. [14,15]).

The objective of the current research is to develop a frequency-domain numerical method for predicting far-field tonal noise ofpropellers operating in nonuniform flows by the use of the Greenfunction solution to the acoustic analogy equations and that includesmean inflow effects. The new formulation and the numerical methodpresented aremore important in situations inwhich theMach numberand the aerodynamic reduced frequencies are relatively small: say,marine propellers operating in nonuniform flows. The formulationprovides clear insight into the mechanism of the generation ofpropeller tone noise. This prediction approach has the advantage thatthe source integration is performed on the actual blade surface, ratherthan on the projected disk or blade mean-chord surface. Numericalpredictions of the tone noise level and directivity of a propelleroperating in various nonuniform flows will be validated againstexisting experimental data. The various effects of a single inflowharmonic are quantified in this paper.

II. Coordinate Systems

The analysis presented here is formulated in a moving referenceframe, which moves with the propeller at constant velocityU� �U; 0; 0�, as shown in Fig. 1. In the moving reference frame, thecoordinates of the observation point and the source point in acylindrical coordinate system are denoted by x� �x1; r; �0� andy � �y1; r0; �00�, respectively. The relationships between therectangular coordinate system and the cylindrical coordinatesystems are

x1 � x1; x2 � r cos �0; x3 � r sin �0 (1a)

y1 � y1; y2 � r0 cos �00; y3 � r0 sin �00 (1b)

In both the translating and rotating blade-fixed coordinatesystems, the observation point and the source point are denoted byxb � �x1; r; �� and yb � �y1; r0; �0�, respectively. The trans-formations between the moving coordinate system and the blade-fixed coordinate system are

x �x1; r; �0� � xb�x1; r; � ��� (2a)

Fig. 1 Relation between the moving coordinate system and the blade-fixed coordinate system.

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y �y1; r0; �00� � yb�y1; r0; �0 ��� (2b)

where is the time y shift between themoving coordinate system andthe blade-fixed coordinate system, and � is the angular velocity ofthe blade rotating in the opposite direction of �00, as shown in Fig. 1.

III. Green Function Solution

The starting point for the prediction of propeller tonal noise is theGreen function solution of the forced wave equation derived fromthe acoustic analogy. This is the fundamental equation governing thegeneration of aerodynamic sound in the presence of solid boundaries,whose Green function solution takes the form (for example,Goldstein [16])

p�x; t� �ZT

�T

ZZ��

Z@2G

@yi@yjT 0ij�y; � dy d

�ZT

�T

ZZS��

@G

@yifi dS�y� d �

ZT

�T

ZZS���0V

0n

DG

DdS�y� d (3)

where fi ��ni�p � p0� � njeij is the ith component of the forceper unit area exerted by the boundaries on the fluid, T 0ij is Lighthill’sstress tensor for isotropic flow in a region v��, V 0n is the normalvelocity on the blade surface S��, the primes on T 0ij and V

0n indicate

the quantities observed in the earth-fixed reference frame, T is somelarge but finite interval of time, and G�G�x; t; y; � is the Greenfunction of the wave equation relating to a medium with uniformmean flow. This Green function in an unboundedmedium is given by(for example, Zhou and Joseph [17])

G�x; t; y; � � 1

4�R�

� � 1

�2c0�R�M�y1 � x1�� � t

�(4)

where �2 � 1 �M2, M�U=c0 is the Mach number of the meanflow, and R is the mean-flow corrected distance:

R���������������������������������������������������������������������������������������������������������y1 � x1�2 � �2�r2 � r20 � 2rr0 cos��0 �� � �0��

q(5)

We shall make use of the Green function in the frequency domain,which takes the form of

�G�x; y; !� �Z 1�1G�x; t; y; �ei!�t�� dt� 1

4�Rei�E (6)

where ! is the angular frequency, E� R�M�y1 � x1�, �� �=�2,and �� !=c0 is the acoustic wave number related to the observationfrequency !.

IV. Nonuniform Velocity Field

The characteristics of the noise due to the interaction of a propellerwith mean inflow distortions are highly sensitive to the details ofinflow velocity field upstream of the propeller; therefore, itsdescription is extremely important. Experiments have shown that theinflow mean velocity at the propeller plane varies along the bladeradius [18], which usually takes the smallest value at the hub and thehighest value at the tip. Apart from the mean inflow velocity, thereexist velocity disturbances in three directions. It may be easilydemonstrated that the axial velocity component is most important indetermining the radiated noise. It affects not only the noise level anddirectivity, but also the mechanism of the radiated noise.

The axial inflowvelocity at point �r0; �0� in themoving coordinatesystem can be described by the sum of a mean velocity componentand an axial disturbance velocity of the form

u�r0; �0� � �u�r0� � ~u�r0; �0� (7)

where �u�r0� is the circumferentially averaged axial mean velocity atradius r0, defined by

�u�r0� �1

2�

Z2�

0

u�r0; �0� d�0 (8)

and ~u�r0; �0� is axial disturbance velocity, which can be representedin terms of a Fourier series:

~u�r0; �0� �X1n�1

An�r0�e�in�0 (9)

where An�r0� is the nth harmonic complex velocity amplitude atradius r0.

V. Unsteady Blade Surface Pressure

Model a given radial section of the propeller as a linear cascade ofairfoils, located at particular radius r0, moving tangentially withrespect to the freestream flow direction. If we consider a cylinder ofradius r0 cut through the propeller blade with cylinder axis coaxialwith the propeller axis, the blade-section profilewill closely fit on thecylinder surface. Unwrapping the cylinder surface onto a flat planeforms a right-angled triangle, as shown in Fig. 2. The adjacent side ofthe triangle is the circumference 2�r0, of the cylinder circular end.The opposite side is the advanced distance (pitch distance)P, parallelto the cylinder axis that a point would cover moving along thehelicoidal chord line during one revolution. The hypotenuse of thetriangle forms part of the blade-section chord line. The angle formedbetween the adjacent and the hypotenuse sides is referred to as thegeometric pitch angle �. Another right-angled triangle arises fromthe velocity triangle formed from the propeller forward velocity Uand the blade-section rotating velocity �r0. The total incomingvelocity U0 is therefore given by

U0 ����������������������������U2 � ��r0�2

p(10)

From this velocity triangle, as shown in Fig. 2, the hydrodynamicpitch angle can be obtained by

�H � tan�1�U=�r0� (11)

and the attack angle at the blade section is given by

�� � � �H (12)

The blade section samples each harmonic wave spatially andtemporally as it spins. A fluctuating pressure field around thepropeller is therefore produced. To fully account for the pressuredistribution on the blade section, strip theory will be used. Theunsteady blade response to each velocity harmonic will be estimatedusing the Sears unsteady aerodynamic theory.

However, it must be noted that the Sears analysis assumes theinflow disturbance to be normal to the blade section, in the blade-fixed reference frame. To apply Sears analysis to more general flowconditions, the normal component of the disturbance must beobtained. For a blade section, the unsteady pressure is antisymmetric[19]; that is, the pressures on the pressure and suction side of theblade are equal in magnitude, but � out of phase. Therefore, theunsteady pressure distribution on one side of the blade section can beexpressed as [10]

Fig. 2 Geometric pitch triangle and hydrodynamic velocity triangle.

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p�r0; �0; � � �0U0�r0� cos ��r0�X1n�1

An�r0�e�in�00Sears� �kn;r0�

(13)

Note that the pressure distribution is the summation ofcontributions made by all harmonic waves, where �r0; �0� representsthe local coordinates evaluated in the blade-fixed reference frame,��r0� represents the geometric pitch angle at radius r0, U0�r0�represents the mean inflow velocity at radius r0, as shown in Fig. 2,and �kn;r0 is the nondimensional wave number of the nth velocityharmonic normalized by half-chord at that particular radius r0.

Note that Sears� �kn;r0� in Eq. (13) is the product of Sears transferfunction and the chord distribution, defined by

Sears � �kx� � Se� �kx�������������1 � �x

1� �x

r(14)

where �kx is the nondimensional wave number (reduced frequency),and �x represents the nondimensional chord coordinates, which takesthe value of ��1; 1� from the leading edge to the trailing edge. The

classical Sears aerodynamic transfer function Se� �kx� for a two-dimensional vertical gust in an incompressible fluid is defined as

Se � �kx� �2i�kx�

�1

H�2�1 � �kx� � iH�2�0 � �kx�

�(15)

whereH0 andH1 represent the zero-order and the first-order Hankelfunctions, respectively.

VI. Tonal Noise Radiation

Note that the first term in Eq. (3) represents the quadrupole sourcecontribution and the third term represents the volume displacementsource. These terms are only found to be important at relative flowspeeds close to the sound speed [20]. Therefore, when consideringthe sound radiation due to the interaction between rotating blades andlarge-scale nonuniformities, only the second term of Eq. (3) is ofimportance, which is denoted by

p�x; t� �ZT

�T

ZZS��

@G

@yjfj dS�y� d (16)

Substituting the Green function of Eq. (6) into Eq. (16) andFourier-transforming the result with respect to t, one obtains theradiated pressure for a single blade surfaceSi at observer frequency!of the form

p̂�x; !� � 1

2�

ZT

�T

ZZSi

fj�y; �ei!@

@yj�G�x; y� dS�y� d (17)

To simplify the analysis, the following far-field approximations

are made: r0 and y1 � Rs, where Rs ����������������������x21 � �2r2

pis the flow

corrected distance from the origin of the blade-fixed reference systemto the observation point. It is natural to express the observation pointin the moving reference frame, whereas the source point is expressedin the blade-fixed coordinate system; hence, only the time-shiftvariable appears inside the expression of R and Rs. Expanding theexpression of R and ignoring terms of second order gives

R Rs � �y1 cos � �2r0 sin cos��00 � �0�� (18)

where cos � x1=Rs and sin � r=Rs. Note that is not ageometric angle; in terms of the small value of Mach number, it canbe regarded approximately as an azimuthal angle measured from thex1 axis. With the approximate expression for R of Eq. (18) and thederivatives @=@yi of the Fourier transform of the Green functionsubstituted into Eq. (17), one obtains

p̂�x; !� � �i�8�2

ei�Rs

Rs

ZT

�T

ZZSi

f�fT�y; ���y1 � x1�=Rs �M�

� fR�y; ��2��r0=Rs � sin cos��00 � �0��� fD�y; ��2 sin sin��00 � �0�g

ei!ei�M�y1�x1�e�i��y1 cos ��2r0 sin cos��00��0 �� dS�y� d (19)

where fT , fR, and fD denote the forces per unit area exerted by thepropeller blades in the y1, r0, and �0 directions, respectively. Here,the usual far-field approximation is made, whereby 1=R is replacedby 1=Rs. The forces in Eq. (19) are related to the blade surfacepressure of Eq. (13) by(

fTfRfD

)��

(n1nrn�

)p�r0; �0; � (20)

wheren� fn1; nr; n�g is the normal vector pointing inward from theblade surface.

Substituting the unsteady blade surface pressure distributionp�r0; �0; � of Eq. (13) into Eqs. (19) and (20) gives

p̂�x; !� � �i��08�2

ei�Rs

Rs

ZT

�T

ZZSi

�n1

�y1 � x1Rs

�M�

� nr�2

�� r0Rs� sin cos��00 � �0�

�� n��2 sin sin��00 � �0�

�

�X1n�1

An�r0�e�in��0���U0�r0�Sears� �kn;r0�

cos ��r0�ei!ei�M�y1�x1�e�i��y1 cos ��2r0 sin cos��0

0��0 �� dS�y� d

(21)

Ignoring the effects on the radiated sound field from scattering ofsound by adjacent blades, summing over B blades, and following asimilar procedure to [21], the final expression for the total radiatednoise is obtained of the form

p̂�x; !� � �i�B�04�Rs

ei��Rs�Mx1�

X1l��1

X1n�1

ei�lB�n���0��2���!� lB��

ZZSi

�n1

�y1 � x1Rs

�M�

� nr�2

�� r0Rs� i sin

�lB� n�r0 sin

� JlB�n�1��r0 sin �JlB�n��r0 sin �

��

� n��2lB� n�r0

�JlB�n��r0 sin �An�r0�U0�r0�

cos ��r0�ei��M�cos �y1e�ilB�0Sears� �kn;r0� dS�yb� (22)

where l represents the number of radiated harmonic noise order,which takes all integer values, n is the number of inflow velocityharmonic order, and ��!� lB�� denotes a Dirac delta functionevaluated at frequencies shifted by the blade-passing frequencieslB�. Note that Eq. (22) requires integration to be performed over theactual blade surface S�yb�, rather than over the mean-chord surface.The term of n1��y1 � x1�=RS �M� in Eq. (22) is a very importantparameter. This component accounts for the influence of theunsteady axial force on the radiated noise; furthermore, it can beeasily reduced to� cos �M in the far-field condition, where y1 issufficiently small comparedwith the observation abscissa x1. Thus, itcan be used to evaluate the radiated far-field noise directivity. Inaddition, the prediction equation (22) includes the effects due toradial force and tangential force.

VII. Comparison of Tonal Noise Predictions withExperimental Data

In this section the theory developed above is compared againstexperimental data reported by Subramanian and Mueller [22]. Tonal

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noise radiation measurements were made in an anechoic wind tunnelfor a four-bladed propeller developed by the David Taylor ShipResearch and Development Center. The propeller has an expandedarea ratio of 0.451, a tip diameter of 0.2504 m, a hub-to-tip ratio of0.2, and is coded as Prop 3714 in the U.S. Naval propeller listings.The main geometric parameters of the full-scale propeller can befound in [22]. The nonuniform flow was provided by with twoseparate flow generators, which produced a three or four-cyclesinusoidal velocity distribution. The measurements were made atpropeller rotational speeds n of 3000 and 3600 rpm at freestreamspeedsU of 12:7 m=s. A single hot-wire sensor, mounted on a three-dimensional traversing unit, was employed for quantifying thenonuniform variation of the axial flow velocity. An array ofmicrophones was used to measure the radiated far-field tonal noisedirectivity at a radius of 1.9 m on both sides of the propeller axis.

From the measured axial velocity data relating to the three-cycleand four-cycle flows, the harmonic variations Vx along a propellerradius given by Eq. (9) were computed. Figures 3 and 4 illustrate themain inflow harmonic variations along the propeller radius for thetwo different nonuniform flows.

Figure 3 shows the dominance of the third harmonic along theentire propeller radius. Although other velocity harmonics are alsopresent in the flowfield, they are significantly weaker in magnitude.The amplitude of the third harmonic linearly increases with radiusand reaches a value of 0.08 near the propeller tip. In Fig. 4, theamplitude of the fourth velocity harmonic is much higher than theother two harmonics. The maximum amplitude of the fourthharmonic is about 0.07 near the blade tip.

A numerical scheme has been developed for the efficientcomputation of the radiated tonal noise at different observationpoints. The blade surface is meshed into triangle elements with amaximum dimension of 8–10 times less than the mean-flow reducedwavelength. This allows analytic integration over each element facet,whose integrand involves the oscillating terms in Eq. (22). Figure 5shows a pressure-side mesh used to perform the numericalintegration required in Eq. (22).

The unsteady blade loading, which constitutes the aerodynamicsound sources, is predicted using Eqs. (13–15). Substituting theunsteady blade loading into the tonal noise prediction equation (22)allows predictions of the radiated tonal noise to be obtained. Aspointed out by Subramanian and Mueller [22], the first harmonicnoise (l� 1) is much greater than the noise at higher harmonics [22];thus, only the first harmonic noise is considered for comparisonpurposes. For the sake of comparison, the radiated tonal noisecomputed by the analytic solution proposed by Blake [10] is alsopresented.

Figure 6 shows a comparison of the first harmonic noise directivity(l� 1) for three-cycle flow at a propeller rotational speed 3000 rpm.The numerical prediction using the method proposed above istypically 2 dB greater than the measured sound pressure level at the50� position, and between angles from 90 to 130�, the differences areless than 1 dB. A trend of smooth decrease in the tonal sound levelfrom downstream to upstream is demonstrated. The predictionresults of Blake’s [10] analytic solution, however, deviate from themeasured results, with a maximum difference of approximately11 dB at 50�, which then diminishes as the observer angles increase.The downstream-to-upstream decreasing trend is not accuratelycaptured by the Blake model.

Figure 7 shows a comparison of the first harmonic noise directivityfor three-cycle flow at a propeller rotational speed 3600 rpm. Thenumerical predictions are observed to be even closer in level to the

Fig. 3 Three-cycle flow: radial variation of themain inflow harmonics:

△harmonic 1,□harmonic 2;○harmonic 3; * harmonic 4; +harmonic 5.

Fig. 4 Four-cycle flow: radial variation of the main inflow harmonics:

△ harmonic 1, □ harmonic 2, ○ harmonic 3, * harmonic 4, and +

harmonic 5.

Fig. 5 Mesh for numerical calculation of 3714 propeller, pressure side.

Fig. 6 Comparison of predicted and measured first harmonic direc-

tivities for three-cycle flow at propeller rotational speed 3000 rpm:

○ predicted, □ measured, and * Blake’s analytic solution.

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measured data at all nine measured points than those at the rotationalspeed of 3000 rpm. The maximum differences at this speed are nomore than 1.5 dB. The variation in noise between downstream-to-upstream observer angles is again well-captured. The prediction ofBlake [10] at this speed is in even poorer agreement, however, with adifference of 11.5 dB at 50�, and at 130�, it is slightly greater than themeasured values, by 0.5 dB. It also fails to capture the downstream-to-upstream trend of diminishing level.

Figure 8 illustrates a comparison of the first harmonic noisedirectivity for four-cycle flow at a propeller rotational speed3000 rpm. The numerical predictions show an excellent agreementwith the measured data at angles between 50 to 90�, and at anglesgreater than 90�, the numerical predictions are lower than themeasured values, with a maximum difference of about 6.3 dB at anangle of 100�. The minimum value is observed to be in the planelocated at 90� and represents a plane of symmetry. Although theprediction obtained using Blake’s [10] analytic solution also hassymmetric features, it has aminimumvalue of up to 31.2 dB,which istypically 13.5 dB lower than the measured value at an angle of 90�.

As shown in Fig. 9, the directivity at 3600 rpm has symmetricfeatures. The numerical predictions are even closer to the measuredvalues than those at 3000 rpm. The differences between thenumerical predictions and the measured data are within a scatter of2 dB at all spherical locations, except at 100�, where the numericalprediction is about 3.2 dB less than themeasured values. Blake’s [10]analytic solution also predicts the same characteristics as those at3000 rpm, with a minimum value of 33.8 dB, which is about 13.2 dBlower than measured values.

Comparing the sound pressure level directivity at the firstharmonic for the three-cycle flow and the four-cycle flow at apropeller rotational speed of 3000 rpm, as illustrated in Figs. 6 and 8,a clear feature is that the radiated noise levels for four-cycle flow arehigher than those of three-cycle flow, except in the direction normal

to the propeller axis. This is most likely a result of the differentcoupling effects of themain inflow harmonic and the rotating blades.

Note that the numerical prediction, the measured results, and thepredictions using Blake’s [10] analytic solution for four-cycle flowappears as an axial dipole-type symmetric far-field directivitypattern. This radiation symmetry appears to be a direct result of thesymmetry in the four-cycle mean velocity profile. From the tonalnoise prediction formulation of Eq. (22) and the properties of theBessel function, it can be deduced that the strongest inflow harmonicthat couples with the rotating blades must satisfy the matchingcondition lB� n� 0. Therefore, in confining analysis to the firstradiated harmonic noise (l� 1), one has n� 4. When the matchingcondition is satisfied (that is, the fourth inflow harmonic componentis dominant in nonuniform flows), the influence due to radial forceand tangential force can be neglected, and the axial force componentn1��y1 � x1�=RS �M� can be reduced to � cos ; thus, theprediction formula of Eq. (22) can be approximated as

�p̂�x; !��l�1 ��i�B�04�Rs

ei��Rs�Mx1�

X1n�1

ZZSi

n1 cos J0��r0 sin �An�r0�U0�r0�

cos ��r0�ei��M�cos �y1e�iB�0Sears� �kn;r0� dS�yb� (23)

The zero-order Bessel function and the cosine function are evenfunctions; therefore, the above approximation formula can beconsidered as an even function in terms of azimuthal angle . Thiswill lead to the result of an axial dipole-type symmetric directivity,due to strong coupling between the fourth inflow harmonic and therotating blades, as illustrated in Figs. 8 and 9. As to Blake’s [10]analytic solution, when thematching condition is satisfied, it can alsobe approximated as an even function in terms of azimuthal angle ;therefore, the corresponding result is a symmetric feature. Formismatching conditions (lB� n ≠ 0), where the fourth harmonic isnot significant, Eq. (22) cannot be approximated as an even functionin terms of azimuthal angle ; consequently, the result is anasymmetry in directivity, due to weak coupling, as illustrated inFigs. 6 and 7.

The sound pressure level of each radiated harmonic noise is thesumof the interactions between all inflowvelocity harmonics and therotating blades. Therefore, quantifying various effects of a singleinflow velocity harmonic on the radiated noise is important to theunderstanding of the mechanism of propeller tone noise, and thismay further assist in the reduction of propeller noise.

The sound pressure level for a single inflow velocity harmonic isnow investigated. Figure 10 shows the comparison of the predictedand single inflow harmonic directivities for the three-cycle flow at apropeller rotational speed of 3000 rpm. It is clear that the soundpressure level for inflow harmonic 3 makes the greatest contributionto the total radiated at angles between 40 to 130� and tends to zeroalong the propeller axis. At angles below 40� and above 130�,

Fig. 7 Comparison of predicted and measured first harmonic

directivities for three-cycle flow at propeller rotational speed

3600 rpm: ○ predicted, □ measured, and * Blake’s analytic solution.

Fig. 8 Comparison of predicted and measured first harmonic

directivities for four-cycle flow at propeller rotational speed 3000 rpm:

○ predicted, □ measured, and * Blake’s analytic solution.

Fig. 9 Comparison of predicted and measured first harmonic

directivities for four-cycle flow at propeller rotational speed 3600 rpm:

○ predicted, □ measured, and * Blake’s analytic solution.

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however, the noise directivity for inflow harmonic 4 is closer to thetotal radiated noise than that for harmonic 3. Note that the inflowharmonic 3 is the main inflow harmonic under three-cycle flowconditions, as shown in Fig. 3. Therefore, it may be concluded thatfor mismatching conditions (lB� n ≠ 0), the total radiated soundpressure level is mainly caused by two inflow harmonics: the maininflow harmonic and the inflow harmonic that satisfies n� lB.Furthermore, the sound pressure level for the main inflow harmonicdominates over most locations, except at locations near the propelleraxis, and the inflow harmonic that satisfies n� lB dominates atlocations near the propeller axis.

Figure 11 shows the comparison of the predicted and single inflowharmonic directivities for four-cycle flow at a propeller rotationalspeed of 3000 rpm. The sound pressure level for harmonic 4 isextremely close to the total radiated atmost of locations, except in thedirection normal to the propeller axis. A symmetric directivity is alsodemonstrated. By contrast, the sound pressure levels for harmonic 3and harmonic 5 are comparatively smaller at all locations and eventend to zero along the propeller axis. Note that in this symmetric flow,the inflow harmonic 4 is highly dominant everywhere, and thus thematching condition is satisfied. Therefore, for matching conditions,the total radiated sound pressure level is mainly determined by theinflow harmonic that satisfies n� lB, and a dipole-type symmetricdirectivity is obtained.

VIII. Conclusions

Anumerical approach in the frequency domain has been describedfor predicting propeller far-field tonal noise in nonuniform meanflows. The integration of the surface sources for computation of theradiated sound field is evaluated on the actual propeller surface,rather than over the mean-chord plane.

An extensive comparison between the predicted sound pressurelevel with the measured data shows generally excellent agreement,both in magnitude and directivity. The numerical prediction shows asymmetric directivity for four-cycle flow and an asymmetricdirectivity for three-cycle flow. A numerical scheme aimed atpredicting the sound pressure level for a single inflow velocityharmonic shows that when the matching condition n� lB issatisfied, the main inflow harmonic contributes very dominantly tothe total radiated sound pressure level, except at locations normal tothe propeller axis. When the matching condition is not satisfied,however, the total radiated sound pressure level is mainly caused bythe main inflow harmonic and the inflow harmonic that satisfiesn� lB. The sound pressure level for the main inflow harmonicdominates over most of locations, except at locations near thepropeller axis, and the inflow harmonic that satisfies n� lBdominates at locations near the propeller axis. Acoustic predictionswere also performed using a classical approach. The proposedformulation is shown to more closely match the experimental data,both for matching and mismatching conditions. The method will beused for parameter research of marine propeller noise due tointeraction between the rotating blades and the nonuniform flowscaused by upstream appendages.

References

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Fig. 10 Comparison of predicted and single harmonic directivities forthree-cycle flow at propeller rotational speed 3000 rpm: □ total

predicted, △ harmonic 3, * harmonic 4, and○ harmonic 5.

Fig. 11 Comparison of predicted and single harmonic directivities for

four-cycle flowat propeller rotational speed 3000 rpm:□ total predicted,

△ harmonic 3, * harmonic 4, and○ harmonic 5.

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C. BaillyAssociate Editor

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