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Steady and Unsteady Characteristics of a Propeller Operating In a Non-Unifo~m Wake: Comparisons Between Theory and Experiments F. Genoux, R. Baubeau (Bassin d'Essais des Carines, France) A. Bruere, M. DuPont (Office National des Etudes et Recherches Adrospatiales, France) ABSTRACT The predictions of the steady and unsteady cha- racteristics of a propeller operating in a non-uni- form wake has been a task of R&D for the past de- cades, seeking to meet the increasingly demanding requirements of acoustical discretion in the design of propellers for ships. The present paper exposes the latest work conducted at the Bassin d'Essais des Carenes in the theoretical and numerical fields to produce a numerical code able to answer this need, as well as the results issued from an original technology developed at ONERA to give access to the fluctua- ting pressure field on a blade. The code issued from these efforts is based on a linearized lifting surface theory and is fitted for low and moderate loadings. Its originality lies in its ability to solve either the inverse problem or the analysis one with the same numerical schemes. Its formulation is adequate for the computation of both steady and unsteady characteristics of a propeller operating in a non-uniform incoming flow. The convergence tests are commented to give an idea of the robustness of the code. The numeri- cal results are compared to experimental data available in the open litterature and to measure- ments derived from experiments conducted with the technology of thin film pressure transducers. INTRODUCTION The prediction of the steady and unsteady cha- racteristics of a propeller operating in a non-uni- form wake aims to meet the increasingly deman 607 ding requirements of acoustical discretion in the design of propellers for ships. These predictions rely on both theoretical and numerical develop- ments able to match the designer's need for a re- liable and accurate tool. The validation of the ap- proach requires extensive and accurate experi- mental data, thus motivating highly complex and heavy tests. From a designer stand-point, the knowledge of steady and unsteady loadings is nee- ded to compute the levels of fluctuating forces transmitted to ship through the shafts and the hull, and thus to optimize the dimensioning of shaft sup- ports and bearings for fatigue. A good prediction of radiated noise induced by the propeller as well as the evaluation of unsteady cavitation are also conditionned by an accurate computation of fluc- tuating pressure fields on propeller blades. Despite the clear need for predictive tools, the progress have been slowed for a long time due to the lack of computational power and the release of new gene- ration computers during the two last decades have certainly contributed to the improvments in the area of propeller computing. Although an extensive review of all the theore- tical and experimental works conducted on the prediction of steady and unsteady characteristics of propellers is out of the scope of the present paper, it is interesting to underline the main steps in both theories and documented experiments in the development of tools for the computation of unsteady loadings. The first consistent and perti- nent tools from an engineering point of view were based on lifting line method, ranging from quasi- steady approach (1, 2, 3) to two-dimensional uns- teady method (4, 5, 6~. Some refinements were brought by combining quasi-steady approach and two-dimensional unsteady method (7). The limita

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tions of these methods are precisely identified - a good review of their failures can be found in the reference (8~. Theoretical attempts (9), using matched asymptotic method, have been made to solve the most obvious and severe restrictions, such as neglecting three-dimensional effects - span, skew. Despite the save of computational time in com- parison to heavier methods, the remaining inaccu- racies - specially for low Expanded Area Ratio propellers - as well as the increasing computatio- nal power of computers have motivated the deve- lopment of new codes based on linearized lifting surface theory. The reference (10) summarizes the different steps taken in that direction for the last twenty years. The codes developed within this theory lead to a clear improvement of the accura- cy and reliability of the numerical results (111. From a designer's stand-point, the use of the li- nearized lifting surface theory to solve the inver- se problem - computing a propeller geometry mee- ting propulsive requirements - is proposed in (12), where the presence of an axisymetric body is taken in account. As opposed to the important number of theore- tical efforts, there are very few well documented experiments in the open litterature. Therefore, the possibilities of validation are limited and nu- merous numerical results are compared to the measurements referenced in (8) and (13) publi- shed in 1968, more than twenty years ago. This is partly due to the difficulties of gathering all the data needed for a reliable validation. Besides the access to the fluctuating forces transmitted to the shaft, one has to accurately measure the flow field feeding the propeller in its presence, and de- termine a proper procedure to deduce the effects of the suction from the wake inhomogeneities. Furthermore, there is no experimental data in the open litterature that is known delivering informa- tion on fluctuating pressures at the blade surface of a propeller. The lack of adequate sensors ful- filling the requirements for such measurements has clearly limited the knowledge of pressure fluctuations related to a propeller to measure- ments on adjacent walls. Unfortunately, no unified tool based on the li- nearized lifting surface theory was available to allow the computation of both steady and unsteady characteristics of a propeller operating in a non- uniform flow, within the solving of the inverse problem or of the direct one. The present paper exposes the latest work conducted at Bassin d'Essais des Carenes de Paris in the theoretical and numerical fields to develop a numerical code able to eliminate some of the previously mentioned restrictions, as well as the experimental results obtained in its facilities from an original technolo- gy developed at the Office National d'Etudes et de Recherches Aerospatiales aimed at giving access to the fluctuating pressure field on a blade. The numerical results are compared to experimental data found in the open litterature and to the ones obtained at Bassin d'Essais des Cardnes. The expe- riments are documented as carefully as possible to allow comparison. MODEL AND SOLUTION PROCEDURE The numerical code is based on a linearized lif- ting surface theory and is fitted for low and mode- rate loadings. The starting core was developed in the late seventies and limited to the solving of the inverse problem (12~. Implemented at Bassin d'Essais des Cardnes, the code has gone through many evolutions and is now stabilized in its matu- re form. Its originality lies in its ability to solve the in- verse problem - determination of pitch and camber laws for a given shaft power - as well as the di- rect analysis - computation of thrust and torque for a given geometry - within the same formula- tion. Such a feature allows the validation of the code used in its inverse mode - the important mode for the designer - by checking the accuracy of the results computed in its direct mode on geo- metries of reference propellers. The code has been extended to permit the calcu- lation of unsteady forces due to the interactions of a propeller with a non-uniform steady incoming flow. This calculation remains possible in both in- verse and direct modes, thus allowing skew opti- misation in the design process. The Figure 1 shows the geometry of the pro- blem. The propeller, which is represented by its geometry H. operates behind an axisymetric body C whose geometry is given by the equation of its meridian. The body's advance speed, 608

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Vs. is supposed constant and the single screw pro poller rotates at a constant rotation speed n. The plane of reference used to describe the propeller geometry is named [I. The helicoids emitted by the blades are noted I;. In both inverse and direct modes, the diameter, D, of the propeller, its number of blades, Z. the spanwise laws of skew, rake, maximum thickness and chord length, as well as the chordwise law of thickness are given. The spanwise pitch and cam- ber laws and the chordwise camber laws are known in the direct mode and are unknown in the inverse mode. The effective wake is supposed to be known in the propeller plane. right-handed propeller /~/ Y ~ ~\l ar V x V (4) ~(1 ) ~ I _-~ ~ (3) (2) Fig. 1 Geometry of the problem (1) Propeller H (2) Helicoids (3) propeller plane r] (4) Body C The geometry of the propeller H is described in the system of polar coordinates of axis the body axis. Following ITTC standards, one defines a right hand orthogonal system of Cartesian coordinates with the origin O coinciding with the centre of the propeller. The longitudinal axis x coincides with the body axis, positive downstream; the trans- verse axis, positive part; the third axis z positive upward. One uses a cylindrical system with origin O and longitudinal axis x. The effective wake field is described by the three components, Vr, Via, Vx, of the velocity vector, Or, 0), in the propeller plane, written in cylindrical coordinates. The three components are known by their harmonica! amplitudes, Aj k' and phases(p; k' for k varying from 0 to infinity: Vj = Vs ~ I, Aj, Or ~ cos [k ~ +(Pj, k] k=0 (1 The fluid is assumed to be incompressible and the flow irrotational. No presence of cavitation is considered within this work. Therefore, the abso- lute velocity field derives from a potential A, which satisfies the Laplace equation: Ad>=0 (2) Steady case In the steady case, the only amplitudes of ve- locity components that are not equal to zero are Ar 0 and Ax o The phases Hi 0, are equal to zero. The absolute potential ~ can be split in two terms: Her, E, x) =Vsx+~(r', E', x) `3' where the first term takes in account the body advance velocity and the second term is the rela- tive velocity potential written in the polar coordi- nate system, (O. r', 8', x), rotating with the pro- peller. The boundary conditions are written on the body C and on the propeller, H. including its hub. If n is the normal to the boundary, these conditions can be written as: - on the body C : Vt n-\si.n 2n n R where As is the advance ratio of the propeller, - on the propeller H: at , = al ~ (,8- B) ^\/ ~ ~ AX, o(r)\s ~ (5) where hi- and hi+ are the positions of back and face of the blade sections at the reduced radius, it, 609

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Vs. is supposed constant and the single screw pro peller rotates at a constant rotation speed n. The plane of reference used to describe the propeller geometry is named n. The helicoids emitted by the blades are noted A. In both inverse and direct modes, the diameter, D, of the propeller, its number of blades, Z. the spanwise laws of skew, rake, maximum thickness and chord length, as well as the chordwise law of thickness are given. The spanwise pitch and cam- ber laws and the chordwise camber laws are known in the direct mode and are unknown in the inverse mode. The effective wake is supposed to be known in the propeller plane. right-handed propeller y) 2 . dC (4) (1) :12) Fig. 1 Geometry of the problem (1) Propeller H (2) Helicoids ~ (3) propeller plane rat (4) Body C The geometry of the propeller H is described in the system of polar coordinates of axis the body axis. Following ITTC standards, one defines a right hand orthogonal system of Cartesian coordinates with the origin O coinciding with the centre of the propeller. The longitudinal axis x coincides with the body axis, positive downstream; the trans- verse axis, positive part; the third axis z positive upward. One uses a cylindrical system with origin O and longitudinal axis x. The effective wake field is described by the three components, V r, Via, Vx, of the velocity vector, Or, 8), in the propeller plane, written in cylindrical coordinates. The three components are known by their harmonica! amplitudes, Aj k' and phases(pj k' for k varying from 0 to infinity: t Vj = Vs ~ >, Al, k(r ) cos [k ~ +(Pi, k]) k_0 t1 ) The fluid is assumed to be incompressible and the flow irrotational. No presence of cavitation is considered within this work. Therefore, the abso- lute velocity field derives from a potential A, which satisfies the Laplace equation: ~ ~ = 0 (2) Steady case In the steady case, the only amplitudes of ve- locity components that are not equal to zero are Ar 0 and Ax o. The phases Hi 0, are equal to zero. The absolute potential ~ can be split in two terms: O(r, 8, x) = Vs x+~(r', B', x) `3' where the first term takes in account the body advance velocity and the second term is the rela- tive velocity potential written in the polar coordi- nate system, (O. r', B', x), rotating with the pro- peller. The boundary conditions are written on the body C and on the propeller, H. including its hub. If n is the normal to the boundary, these conditions can be written as: - on the body C : (4) where As is the advance ratio of the propeller, - on the propeller H: at,+ r _ 5~ !1 = ae + (A B) 3/ ~ ~ Ax' o(r)\s (5) where q~ and q~ are the positions of back and face of the blade sections at the reduced radius, in, 610

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duced by the interactions between the propeller and the hull. It can be split into two terms: - q1 strength of the source due to the sources distributed on the projected propeller H', - q2 strength of the source due to the doublets distributed on the projected propeller H' and the helicoids E. The intensity of sources located on the projec- ted propeller H' is directly related to the thick- ness law of the blade profile at the considered ra- dius, according to the relation: 14~+ - n-)N ~ q an M _ 2~nR M R (10) It should be underlined that it is not necessary to compute the strength of the source induced by the flow arond the body without the propeller. The use of equation (8) and the Kutta-Joukowski condition allow to derive the equation relating camber, pitch and source strength: Vs rl) R A r, of r ) O AM AX,o(r) 11 qP V [I ] OM dSP 1 || qP y ~3 nM dSp 4 a|| R2 R V :~ . nM dSp I1 ~ 6Tr. Ed R V ~rlP.PMl . Be dSp 4 ~ Jr ~ R2 L ~3 J Ha+ + art-) ~ an AM 2 R , + (-By)\/ ~ +Ax,O(r)\s M (11) In the direct problem, the unknowns 6, and 5tTr Ed can be directly computed from the known values of pitch and camber. In the inverse problem, the designer chooses the normalized circulation law and the perfor- mances - either thrust or torque to be by the pro- peller. Thus, the circulation law is defined with an unknown multiplicative constant Fmax In both modes, the discretization of the equation (1 1 ) produces a linear system with a predomi- nently diagonal matrix. The resolution of the sys- tem does not raise any particular difficulty. The forces are computed using the Joukowski theorem. In the inverse mode, the shock-free entrance condition supresses the suction force at the lea- ding edge of profiles. The integration of forces and moments produces a second degree equation with the unknown Fmax Reference (12) details the ap- proach. In the direct mode, the leading edge suction force has to be taken in account and is calculated with the method described in (14) and (15~. The suction effects due to the potential effects of the propeller on the body can be computed by integrating the efforts on its surface. These ef- forts are directly calculated using Lagally theo- rem. At last, the linearization of the equations with respect to As allows to compute the performances at off-design conditions close to the design point. Unsteady case In the presence of non-uniformities in the flow feeding the propeller, the wake field velocity vec- tor can be split into two parts: Vj(r, D) - Vs Al, O(r) + Vj(r, 8) (12) with: 00 ~ Vj(r' D) = Vs At, Al, k(r)COS[k ~ +9j, k] k-1 611 (13)

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~ where V' are the three components of the ve locity fluctuations encountered by the propeller blades during the rotation. Besides, it is assumed that the geometry of the helicoids, I, is not affected by the inhomogeneities of the incoming flow. Therefore, the solution of the potential is split into three terms: ~ O = vs x + ~(r'' D'? x)+ (rl, D', x) (14) With the mentioned assumption, the lineariza- tion of the problem eases considerably its solving, for the two first terms are solution of the-steady problem. As a further simplification, the unsteady interactions between the propeller and the body are neglected and the indetermination between sources and doublets is solved by assuming that the fluctuations of potential are only related to the doublets. Thus, the only boundary condition remaining ap- plies to the projected propeller surface, H': Vet n _ O In R (15) is: The integral equation associated to the problem ~ R 1 ~ V r(r' Ok) ~ V~(r, Ok) ~ Vx(r' Ok)/ 1 ~ 6P R V: P-l nMdSP lIOR2 Lm3J SAP R V Up.PM1 nM dSp ~3 ~ () (1 6) The angle Ok is the sum of the angle due to the propeller rotation and the phase shift from blade to blade. Thus, it is time dependent: Ok-2n ant (k-~) ( 1 7) The Kutta-Joukowski is implicitly satisfied at the trailing edge of the blades but, in the unsteady case, the value of the potential jump, 6, is not uniform on the helicoids and is not constant in time at the trailing edges of the blades. To solve the problem raised by the time-depen- dency of the potential, the following procedure has been implemented: - the value of the doublet associated to the jump ~ of potential 6 at the trailing edge at a given time is obtained from the integral equation (16), - the doublet element associated to the potential jump is convected downstream on the helicoid at a speed equal to the steady component of the local at-infinity velocity, - the computation is actually conducted on the first blade only, due to the blade-to-blade periodi- city of the solution. Nevertheless, the other blades are taken in account in the calculation, - the computation, which can be seen as a tran- sient approach, is stopped when the periodicity of the unsteady circulation is achieved. Numerical procedure The mesh used to solve the problems is based on a collocation method described in (14) and (15~. This method allows to use non-planar surface ele- ments which are required to get accurate results in the case of highly skewed propellers. In the unsteady case, the calculation of the uns- teady pressures is required. Thus, the time deri- vative of the potential is deduced from the analy- tical computation of the potential. Special care is to be taken for the doublet locations in the vicinity of the trailing edge. Therefore, second degree dou- blets have been used to insure the continuity of doublet intensity and of its first derivative at the borders of the panels adjacent to the trailing edge. Moreover, the time-step is chooser according to the Shannon rule to allow consistent harmonica! analysis. The numerical tests show that high en- ough discretizations in space and time give good stability of the results. 6~2

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COMPARISON TO E}OSWELL EXPERIMENTS This section presents and comments the nume- rical results when simulating the experiments re- forenced in (8) and (13~. These experiments consisted in measuring the characteristics of a se- ries of three-blade propellers in open water condi- tions and behind two wake screens - a three-cycle wake screen and a four-cycle screen. Four propel- lers were designed with the same diameter, thrust, speed of advance and rotational speed. The first three propellers have different Expanded Aspect Ratio: 0.3 for the Propeller NRSDC N 4132, 0.6 for Propeller NRSDC N 4118 and 1.2 for Propeller NRSDC N 4133. The fourth propel- ler, Propeller NRSDC N 4143 has a highly skew of 120 and an Expanded Aspect Ratio of 0.6. In the trials behind the wake screens, besides the monitoring of fluctuating resulting forces and mo- ments, the measurements included averaged thrust and torque measurements and the wake surveys. The computations were made on an ALLIANT FX- 80 machine equiped with 8 processors and 256 Mbytes of RAM. Both open water and behind wake generators conditions were numerically tested. In the open water tests, the design point plus two off-design points were computed. The mesh used in all cases was 1 1 points spanwise and 15 points chordwise. The durations of each case were in the order of 5 mn CPU for the steady cases and 3 mn CPU for the unsteady cases. Figures 2, 3, 4 and 5 show the comparison bet- ween computational and experimental results in the open water case. The thrust and torque coeffi- cients are plotted versus the advance coefficient. These variables are defined as: KTo= T p n2 D4 KQ0= Q p n2 D5 JO= VO n D (18) The analysis of the curves shows a satisfactory agreement between computation and measurement for the four tested propellers. The worst results are found with the smallest EAR propeller while the highly skewed propeller's performances are well predicted. For all propellers, the computed slopes of thrust and torque coefficients curves versus advance coefficient are very close to the experimental ones on a reasonably wide range of advance coefficient. Figures 6 and 7 show the comparisons between computations and experiments in the unsteady cases. As in reference (13), only the non-skewed propellers have been represented. The fluctuating force and torque coefficients are defined as: KTi _ Tj, z p n2 D4 for i = x, y, z KQi = i' Z for i = x, y, z - p n2 D3 (1 9) ~ ~ where Tj z and Q; z, i = x, y, z are respecti vely the axial, horizontal and vertical components of the force and moment fluctuating at the blade rate frequency of the propeller. The results obtained in the two experimental configurations - three-cycle and four-cycle wake screens are gathered on these two figures. The three-cycle screen generates the axial fluctuating force and torque, while the four-cycle screen ge- nerates the fluctuating transverse forces and ben- ding moments. With figure 6, one can see that the axial forces are well predicted. On the other hand, the trans- verse forces seem to be overestimated by the computation although the trend of the coefficients with EAR is respected. Figure 7 shows that the predictions of fluctuating moments are good in am- plitude as well as in trend versus EAR. BASSIN D'ESSAIS DES CARENES EXPERIMENTS In the prediction of unsteady efforts, one of the expected advantage of the linearized lifting surfa- ce methods over lifting line theories is to give ac- cess to the fluctuating force amplitudes as well to the fluctuating pressure field with a better accu- racy. This last hope is a requiroment for a better computation of sheet cavitation in non-uniform 613

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flow conditions. The lack of experimental data al- lowing the validation motivated new experiments consisting in equiping a propeller with pressure thin film transducers. Thin film pressure transducer technology The objective of the thin film pressure transdu- cers is to give accurate measurements of the fluc- tuating pressures at specified points located on the surface of a propeller blade with the smallest al- terations of the flow around the profiles. Besides, the sensitive part of the captor has to be as small as possible. To meet these requirements, a new kind of transducers -thin film pressure transdu- cers - have been developed (16~. Principle The pressure gauge uses capacitive variations. The sensitive element is made of a flexible dielec- tric foil metallised on both sides, thus forming a capacitor (see figure 8~. Pressure Fluctuations L 5 2 6 7 1 `,~ it/ 4~3 Or..; ~ ., ~.~.~.~ ~ - 3 Fig. 8 View of pressure thin film transducer (1 ) Sensitive electrodes (2) Glue layers (3) Profile (4) Dielectric sheets (5) Guard ring electrodes (6) Preamplifier (7) Polarisation source The thickness of the dielectric foil varies under the action of a pressure fluctuation. That produces a relative variation of the capacitance which is li- nearly proportional to the pressure fluctuation. The coefficient relating the pressure fluctuations to the capacitance variations is directly related to the mechanic characteritics of the foil - Elastic modulus and Poisson coefficient. The variation of capacitance is transformed into a voltage varia- tion using a polarization source V connected to the first electrode of the sensitive element and to the preamplifier connected to the second electrode. Thus, the output voltage Vs is proportional to the pressure fluctuation. In order to reduce the leakage effects on the sensor sensitivity, guard rings are mounted on both sides of the sensitive connection, which links the sensitive electrode to the input of the pream- plifier. The guard rings are driven by the output voltage of the preamplifier whose gain is one. Therefore, no current flows through the sensitive electrode into guard electrode capacitor and the leakage capacitors are completely cancelled. The basic transducer is composed of three 12.5 ,um thick dielectric sheets made of Kapton: - the first sheet contains the connections, the upper and lower sensitive electrodes, the upper guard ring, - the second sheet isolates electrically the lower sensitive electrode and the lower guard, - the third sheet contains the lower part of the guard ring. The electrodes and the connections are manu- factured by metal vacuum deposit which is 0.2 Am thick. The three sheets are bonded together and at the profile surface with 3 to 5 Am glue layers. On 0 0.30 m propellers, 3 to 6 transducers can be mounted on the same blade in the vicinity of high curvature area such as leading edge. The inte- gration on the blade itself is done using bonding technique and does not require special equipment. The preamplifiers are located as close as possible to the transducer, in the blade root. From the preamplifier exit, the signal is sent through co- axial cables to a rotating electronics feeding a slip ring. The same ring is used to bring the DC power to the preamplifiers and the probes. Underwater experiments require special water leakproof cau- tion. A fourth 50 ,um dielectric sheet made of Kapton metalled on one side is bonded on the three other ones. A special epoxy paint is coated all over the blade surface. Experience shows that the life time of the transducers is more than a month, 6~4

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even in deep water. Performances The main specifications of thin film pressure transducers, given under a 100 V polarization level, are: - sensitivity: - frenquency range: - defectivity at 1 kHz: - sensitive standard area: - thickness': - temperature range accepted: 2.5 10- 8 V/Pa 1 to105Hz 2 Pa/Hz1 /2 4 mm x 6 mm 170 ,um 0C to 60C The microelectronics volume required to mount the microelectronics associated to three preampli- fiers is less than 10 mm x 7 mm x 1.7 mm Experimental program The experimental program focused on the mea- surement of fluctuating pressures on the blades of a skewed seven-blade propeller operating in the wake of a submarine model. It also included the wake survey by Laser Doppler Velocimeter. The LDV used at Bassin d'Essais des Carenes gives ac- cess to the three components of the velocity field in the propeller plane. The propeller was equiped with six thin film pressure transducers, three on the back and three on the face. They were located mid-chord at three different radii, respectively 0.5R, 0.7R and O.9R. The model was mounted under the plateform of the towing tank Bassin 111 (length 220 m) and driven at constant speeds. All the presented tests were done at the same advance coefficient, Js, equal to 0.63. Pressure signals were monitored, processed and stored. A fast Fourier analysis was applied after several successive acquisitions to improve the si- gnal to noise ratio and to check the stationnarity of the signal. With a power spectal-type analysis, the pressure levels for the harmonics at the Shaft Rate frequency were extracted. Tables 1, 11 and 111 give all the informations rela- tive to the geometries of both the propeller BA No 2515 and the model. The symbols used are the ones recommended by the ITTC. The back and face ' including the waterproof protection coordinates are given in the usual frame in per- centage of the projected chord length on the blade mean line. The harmonica! content measured by LDV is given in Tables IV, V and Vl, which contain the ele- ven first harmonica! half amplitudes and phases for ten dimensionalized radii ranging from 0.0265 m to 0.1165 m and for the axial, orthoradial and radial components of the velocity vector. The coefficients Aj k introduced in formula (13) are obtained for k ranging from 1 to 11 by multiplying by 0.002 the values read in the tables. The coeffi- cients Ax 0 are given in Table 1. The experimental measurements are presented in Table Vll. The eight first harmonics at the Shaft Rate frequency are given for three model speeds that were selected during the tests. The reference is 1 pPa at O dB. Numerical results The computations were conducted with the code tested on the Boswell experiments. Figure 9 shows the results obtained in the steady case (harmonic O of the wake only) and the unsteady case (com- plete wake) when the propeller operates in the model wake. The reference advance coefficient is taken at the design point, i.e. Js equal to 0.63. There is a good agreement between calculations and measurements. Figures 10 and 1 1 give the amplitudes of the harmonics of fluctuating forces and moments ac- ting on one blade. No dynamical balance allowed a comparison to experiments. The pressure calculated versus time on the mesh points, analysed with a fast Fourier algo- rithm at the Shaft Rate frequency, and finally in- terpolated at the locations of the transducers. Table Vlil gives the power pressure levels in dB computed at the model speed of 1 m/s. Discussion The fluctuating pressure induced by the flow in- homogeneities can be written as: 615

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oo P= 2, Pk ei2nknt k= 1 (20) ~ where the coefficients Pk are complex numbers containing amplitude and phase informations for the harmonica! order k. The power spectrum ana- lysis gives the coefficients ok versus k, realated ~ to the complex Pk according to: * 1/2 ~ ~ Ok = Pk Pk 2 (2 1 ) ~ * ~ where Pk is the conjugate of Pk. One introduces a pressure coefficient at the harmonica! order k defined as: Kp,k - 20 109 Ok - 40 109 Vs (22) where the coefficients ok are the values co ming from either the experiment -Table Vll - or the computation - Table Vlil. Assuming that the wake harmonica! content is not affected by Reynolds effects, one finds that the pressure coefficients should remain constant except the harmonic 1 which is affected by Froude effects. Unfortunately, the analysis of the experi- mental data shows that the coefficients Kp i are far from being constant. These discrepancies can be attributed to the weak signal-to-noise ratio ob- tained at the velocity of 1 m/s besides the Reynolds and Froude effects affecting the wake content. Nevertheless, it should be underlined that experiments conducted on that model geometry in various test facilities - wind tunnel and towing tank - did not show a strong influence of the Reynolds number on the wake harmonica! content. For these reasons, only values of pressure coefficients obtained at the two highest model speeds have been averaged. This operation was also applied to the harmonic SRI despite the pre- ceeding remark. Figures 12 to 17 show the compa- risons between the calculated and averaged pres- sure coefficients. The analysis of the results underlines a good agreement between computation and experiment. The only problems encountered are mainly for the harmonic 1 and, on figure 17, for the harmonic 5. The discrepancies for harmo- nic 1 can be associated to the absence of correc- tion for the immersion variations in the averaging of pressure coefficients measured at different ve- locities. For the second problem, there is no clear explanation: the experiment as well as the compu- tation can be suspected at this stage. Nevertheless, it should be underlined that these results are quite encouraging. The differences bet- ween calculations and experiments are in most cases less than 3 dB. Such results are accurate enough to allow noise radiation prevision and give credit to the results obtained for the amplitude of fluctuating forces and moments. CONCLUSIONS The code that was developed at Bassin d'Essais des Carenes offers a good reliability, partly due to the use of new collocation techniques insuring the consistency of the method. The comparisons to ex- periments found in the open litterature and to measurements realized at Bassin are quite rewar- ding, both in steady and unsteady operating condi- tions. Nevertheless, as a dampening to this optimism, it should be remembered that a difference of 6 dB on levels is associated to a ratio of 100% on the linear values. Further experimental tests are un- derway to increase the base of experimental data. Specifically, measurements will be carried out in the vicinity of the leading edge and LDV acquisi- tions have already been realized in a plane adjan- cent to a propeller operating in a non-uniform flow. Such measurements are required to confirm the absence of propeller influence on the wake harmonica! content. As future possible developements, the method has the potential to take in account the unsteadi- ness of the flow due to the presence of low frequency turbulent structures in the wake. Besides, prediction of fluctuating sheet cavitation seems to be feasible because of the confidence in the accuracy of computed pressure field. 616

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FE~NCES (1 ) Lewis, F. M., Tachmindji, A. J., "Propeller Forces Exciting Hull Vibrations," Transactions of SHAME, Vol. 62, 1954 (2 ) Ritger, P. D., Breslin, J. P., "A Theory for the Quasi-Steady and Unsteady Thrust and Torque of a Propeller in a Ship Wake," Experimental Towing Tank Report 686, Jul 1958, Stevens Institute of Technology, Hoboken, New Jersey, USA (3 ~ MacCarthy, J. H., "On the Calculation of Thrust and torque Fluctuations of Propellers in Nonuniform Wake Flow," Report 1533, Oct 1961, Naval Ship Research and Development Center, Washington, D.C., USA (4) Lewis, F. M., "Propeller Vibration Forces," Transactions of SNAME, Vol. 71, 1963 (5) Sears, W. R., "Some Aspects of Nonstationnary Airfoil Theory and its Pactical Application," Journal of Aeronautical Sciences, Vol. 8, N 3, Jan. 1941 (6) Sevik, M., "Measurements of Unsteady Thrust in Turbomachinery," American Society of h/lechanical Engineers, Paper 64-FE-15, Mar. 1 964 (7) Krohn, J. K., "Numerical and Experimental Investigations on the Dependence of Transverse Force and Bending Moment Fluctuations on the Blade Area Ratio of Five-Bladed Ship Propellers," proceedings of the Fourth Symposium on Naval Hydrodynamics, Office of Naval Research, Aug. 1 962 (8 ~ Denny, S.B., "Cavitation and Open-Water Performances Tests of a Series of Propellers Designed by Lifting-Surface Methods," Report 2878, Sept. 1968, Naval Ship Research and Development Center, Washington, D.C., USA. (9) Guermond, J.-L., ~Thdorie Asymptotique Instationnaire de la ligne Portante. Application a l'Helice Marine en Sillage Non Uniformed, These de Docteur-lng~nieur, 1985, University Pierre et Marie Curie, Paris, France (10) Kerwin, J. E., Marine Propellers," [annual Review of Fluid Mechanics, Vol. 18, 1986, pp. 367-403. (11 ) Kerwin, J. E., "Prediction of Steady and Unsteady Marine Propeller Performance by Numerical Lifting-Surface Theory,. Transactions of the Society of Naval Architects and Marine Engineers, Vol. 86, 1978, pp. 218-253. (12) Luu, T. S., Dulieu, A., "Calcul de l'Helice Fonctionnant en Arribre d'un Corps ~ Sym~trie Axiale," Proceedings of Association Technique Maritime et Adronautique, 1977, pp. 301-317. (13) Boswell, R.J., Miller, M.L., "Unsteady Propeller Loading-Measurement, Correlation with Theory, and Parametric Study, Report 2625, Oct. 1968, Naval Ship Research and Development Center, Washington, D.C., USA ( 1 4 ) Guermond, J.-L., "About Collocation Methods for Marine Propeller Design," Proceeding of Propeller '88 Symposium, Paper N8, Sept. 88 (15) Guermond, J.-L., "Collocation methods and lifting-surfaces,. European Journal of Mechanics, B/Fluids, Vol 8, n4, 1989, pp. 283-305 (16) Portat, M., Bruere, A., Godefroy, J.-C., Helias F., "ON ERA developed thin film transducers and their applications," Proceedings of Sensors Symposium, Jan. 1982 617

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Fig. 15 Propeller BA N 2515 0.7R- Face- Midchord 200 A ._ Cal ._ a) o A ._ ._ o Q y a) ._ ._ ID o C' 100 o 200 100 O 200 180 160 140 120 100 Harmonic Order Fig. 16 Propeller BA N 2515 0.9R- Back- Midchord 5 Harmonic Orde Fig. 17 Propeller BA N 2515 0.9R- Face - Midchord ld43] ~ ~ ~' ~~l ~ d ~ ~ - ~ ~ ~ ~ ~ ~ - - . ~ ~ ~ ~ ~ ~ ~ ~ 1 2 3 . 4 5 6 Harmonic Order 622 ~ Kp computed |g Kp averaged HI Kp computed 1~11 Kp averaged HI Kp computed 88 Kp averaged 7 8

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Table I - Propeller BA NO 2515 and model geometries Propeller BA NO 2515 geometry Diameter D = 0.235 m Hub diameter d = 0,047 m Number of blades Z = 7 Rig ht- handed C/D H/D US iT 100*Re/C as/d x Ax ~ o 0.200 0.1603 0.2960 0.0000 0.000 4.7203 0.0105 0.4183 0.250 0.1702 0.3801 0.8673 0.0019 3.8411 0.0334 0.4289 0.300 0.1799 0.4698 2.7244 0.0060 3.1369 0.0587 0.4376 0.350 0.1900 0.5515 4.7791 0.0110 2.5618 0.0805 0.4513 0.400 0.2007 0.6158 7.1098 0.0172 2.0907 0.0950 0.4714 0.450 0.2110 0.6617 9.7462 0.0248 1.7157 0.1006 0.4883 0.500 0.2195 0.6924 12.6561 0.0333 1.4238 0.0974 0.5016 0.550 0.2257 0.7108 15.7396 0.0423 1.1871 0.0891 0.5220 0.600 0.2300 0.7198 18.8603 0.0513 0.9809 0.0818 0.5566 0.650 0.2323 0.7209 22.0084 0.0603 0.7993 0.0772 0.6033 0.700 0.2322 0.7150 25.2566 0.0693 0.6431 0.0736 0.6364 0.750 0.2284 0.7019 28.6280 0.0781 0.5086 0.0691 0.6513 0.800 0.2189 0.6808 32.0666 0.0863 0.3967 0.0638 0.6754 0.850 0.2009 0.6490 35.3733 0.0931 0.3188 0.0629 0.7074 0.900 0.1702 0.6014 38.7550 0.0976 0.2470 0.0479 0.7315 0.950 0.1274 0.5306 42.4303 0.0993 0.2136 0.0082 0.7535 0.975 0.0931 0.4826 44.0882 0.0980 0.2200 0.0055 0.7657 1.000 0.0429 0.4270 45.4417 0.0946 0.2466 0.0299 0.7774 is the reduced radius, r/R C is the chord length H is the geometrical pitch, US is the skew angle expressed in a, iT is the total axial displacement. Rake is considered positive downstream, Re is the curvature radius of the blade at the leading edge, ds/dx is the slope of the mean camber line versus the profile mean line at the leading edge Model geometry Model length LM= 6.9 m Model Midsection diameter DM= 0.625 m The meridian equation is given as: if O < x ~ 0.045914 LM, then r / RM= 5.7532*x if 0.045914 < x < 0.40171, then r / RM= Yaft((x-0.005439)*0.6/0.39627) if 0.40171 < x < 0.73582, then r / RM= 1. if 0.73582 < x < 1., then r / RM= Yfore((x-0.33955)*0.6/0.66045) where x is the longitudinal axis described in the paper and Yaft and Yfore are functions of x defined by: Yaft = x*sqrt(23.9927-91.82582*x+161.51997*x;~-140.33319*x;~+46.64635*x4) Yfore = x*sgrt(6.44272+12.57402*x-68.57971 *x2+81 ~ 66653*x3+32.1 0355*x4) 623

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DISCUSSION Hajime Maruo University of California at Santa Barbara, USA The most important feature of the unsteady propeller characteristics is the change of efficiency when the propeller is operating in the non- uniform wake. It is expressed by the relative relative efficiency. It is widely recognized that the relative rotative efficiency is lightly higher than one. Another feature of unsteady characteristics is the phase shift of the fluctuating thrust and torque. This is due to the term associated with B/3t in the pressure equation. In the present paper, data concerning the above quantity are not given. The computation of a propeller characteristics, when the propeller is operating in circumferentially varying wake, has been carried out by us several years ago (lSth ONR Symposium, Hamburg, 1984). According to our experience, the simple linearized lifting surface theory is not able to provide results which show satisfactory agreement with experiment, and the nonlinear deformation of the trailing vortex sheet in the slip stream must be taken into account. AUTHORS' REPLY The following Table gives the values of the steady components Tio and Qio' i = x, y' z resulting from the interactions between the propeller N 2525 and the non-uniform wake at the nominal design point Is equal to 0.63. Pig. 9 shows the comparison between the axial thrust and torque TO and Qxo' and the thrust and the torque computed in the uniform wake which nevertheless takes in account the radial gradient of axial velocities. V Ti o 10*Qi,o 0.170 x 0.142 0.0019 Y z 0.0021 0.0133 0.0257 In our opinion, the relative relative efficiency is not affected as much by the circumferential non-uniformities as by the radial gradient of axial velocity field. In the ideal case of a purely axisymmetric body ending with a large aft conicity angle, the relative rotative efficiency would be substantial while, in the absence of fins, rudders, ..., the unsteady periodical efforts would remain null. In regard to the question relative to phase shift, it appeared that phases are not important to know with accuracy in terms of their practical meaning. Therefore, the results are not given. At last, a theory mixing linearized lifting surface and nonlinear deformation of vortex sheddings, as suggested by the discussion, does not seem consistent. Moreover, the theory developed in the present paper gives numerical results in good agreement with the experimental measures available at the present time, thus, no extension in the suggested direction has been considered. 631

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