Hydrodynamic analysis of cycloidal propulsors

Hydrodynamic Analysis of Cycloidal

Propulsors

Terry Brockett, Member, Arete Associates, Arlington, VA

ABSTRACT

A phenomenologica]. description of the flow about, and performance of, cy-cloidal propulsors is extracted from available literature. An approximate three-dimensional analysis, employing elements of a lifting-line model and an actuator-disk model, is assembled to predict the hydrodynamic loads on an arbitrary-geometry cycloidal propulsor. Both local blade-oriented and total circumferentially-averaged thrust and torque loads are computed. Comparisons with experimental data and other analy-sis procedures show that the numerical implementation is generally adequate for preliminary design. Cavitation-incep-tion properties of the blade secCavitation-incep-tions are determined and an empirical predic-tion of tip-vortex cavitapredic-tion inceppredic-tion is included. Noise levels are estimated for operation with extensive cavitation on the blade. _{Several blade-pitch} mo-tions are evaluated and It is found that various forms of modified cycloidal pitch control are advantageous for in-creased thrust from a given unIt with little change in hydrodynamic efficiency or cavitatiOn onset. Suggestions are given for future work.

NONENCLATtJRE

Amplitude adjustmbnt for amplified cycloidal blade motion, aspect ratio of blade

Drag coefficient Lift coefficient

Non-dimensional torque coefficient Non-dimensional thrust coefficient Mean chord of blade

Pressure coefficient

Orbit diameter of blade motion Eccentricity; effective value if s.tbscript

Camber, frequency

Correction to induced velocity; adjustment of noise levels for cavitation nimbers

Non-dimensional circulation along blade span A CD cL CK0

ci

Cp p0 e f F G

THE SOCIETY OF NAVAL ARCF4rrEcTS AND MARINE ENGINEERS 601 PAVONIA AVENUE, JERSEY CITY, NJ 07306

Paper to be pmsented at the Propellers/Shafting 91 Symposium Virginia Beaoh, Virginia, September 17-lB. 1991.

* Present Address: Arete Associates, P.O. Box 16287, Arlington, VA 22215

Advance coefficient k _{Reduced frequency}

R0 _{Traditional torque coefficient}

1 _{Traditjoal thrust coefficient}

L Total lift of a single blade, func-tion of angular posifunc-tion

L>

Blade span or acoustic source lev-el, depending on subscript

n _{Rotational speed} p2 Pressure

Pm Root mean square acoustic pressure

P Pitch of blade q Local fluid speed Q Torque

R0 _{Radius of orbit circle} R Reynolds NUmbOZ

t Blade thickness T Thrust

tY Induced velocity component

L

Reference uniform stream speed VR Resultant speed,kiriematjc (x,y) Cartesian cOordinates

Z Number of blades

a Angle-of-attack

fi Blade angle from orbit circle y Angular orientation of transverse

component of mean onset flow

I' Circulation along blade span

i Various efficiency or loss factors

o Blade orbit angular position X Advance coefficient, ratio of

ad-vance to orbit speed c Cavitation number

r Thickness ratio of blade section; time scale

Blade pitch angle, function of 0

INTRODUCTION

The cyclojdal propulsor is a thr-usting device with an axis of rotation that is approximately perpendjcula _to the direction of travel. _{A number of} lifting blades (typically 4 to 7) equal-ly spaced at a fixed orbit diameter, _D0, rotate about this axis. Each blade is further controlled in its orbit such that a component of force in a desired direction is obtained, It is thus a special case of a controllable-pitch propeller with a unique means of thrust vectoring. A sketch of the overall configuration of a cycloidal propeller is shown in Figure 1, and the reference frames and nomenclature for otientatjon

No.2

2-I

Deift University of Technology

Ship flydromechanics Laboratory

Library

Mekelweg 2, 2628 CD Deift

The Netherlands

of the blades along am orbit path are shown in Figure 2. _{The blade-angle or} pitch control of the blades during the orbit is critical to acceptable cavita-tion and powering performance. Other names for this configuration are verti-cal-axis, trochoidal and vane propulsor. Its usage has included implementation both as an auxiliary component of the thrust and maneuvering system and as the sole component of combined thrust and maneuvering units. Courtesy: VC!1lI I Rotor casing 2 Blade 3 Beurir'g-laritern piece 4 Roller bearing 5 Praoller casing 8 Driving sleeve 7 Control rod 8 Servcsnotors B Gear oIlputnp

Fig. I Schematic of Cycloldal Propulsor

Different configurations of the cycloidal propulsor were independently developed by Kirsten (1928, 1950) and Schneider (1933) during or prior to the

l92Os. The blade-angle control of Kirsten is fixed by gears and has expe-rienced only limited development. The more complex but controllable blade-angle orientation using linicages pro-posed by Schneider has received consid-erable development. _{it is performance}

2.2

KLNEtAT1C

+ PITOI P01ST

Fig. 2 Nomenclature and Reference Frame for Cycloida2. Propulsor

of units with this variable-pitch con-trol that is addressed in this paper. There are two traditional regions of pitch control, one for low-pitch pro-pellers (associated with orbit rotation-al speeds, c.,D/2, where w is the angular velocity of the orbit rotation in radius per second, greater than the advance speed V0) and one for high-pitch propel-lers (associated with orbit rotational speeds less than the advance speed). Kirsten (1950) describes the philosophy of a universal pitch control mechanism. The flight path is a cycloid arid for low-pitch operation has a loop as will be discussed In subsequent paragraphs. Low-speed maneuvering requirements have led to ectensive development of the low-pitch configuration. Such propellers have top speeds usually less than 20 knots. High-pitch propellers have hy-drodynamic efficiency values greater than for low-pitch propellers and permit high-speed operation, but to date have received most attention at only the research level.

Because the cyc].oidal propulsor is complex mechanically, a considerable experience base is required of a

martu-facturer in order to produce reliable units. Worldwide, manufacturers have been located in Germany, Japan, USA and

Transom Stern

USSR. Perhaps the dominate manufacturer has been J.M. Vojth GmbH in Heidenhejm, Germany, with a consequence that another popular name for tbe unit is Voith-Schneider propulsor. It is a cutaway view of one of the Voith units that is shown in Figure 1. _{Sarchin and Tryner}

(1970) discuss the impressive reliabili-ty of those units which have had sus-tained development.

When fit to a ship, the rotating disk from which the blades project _is mounted flush to the hull. _{In some} ap-plications, this disk is near midohins so that it projects below the baseline; but equally typical is the situation with the propulsor near the stern (see Figure 3). _{In such a position, the best} propulsion efficiency (Sarchin, 1966 and Reed, et al, 1969) has been found with a transom submergence of no more than a

foot, and buttock slopes of no more than 5 degrees to a distance of about one orbit diameter forward of the rotor centerline. _{A broad flat transom is} desirable to produce large maneuvering forces. _{The rotor centerline is} recom-mended to be at least two orbit diam-eters forward of the stern. _Typically the blade span is limited to about _3/4 of the orbit diameter.

_With

thjgujd-ance, the tips of the blades are gerier-ally well above the baseline, _{even for} relatively shallow-draft vessels.

Water Tractor

Pig. 3 Hull Forms Employed with Cycloidal Propulsors

Because of the ability to rapidly change the direction while maintaining a great level of thrust (see Figure 4), there is no need for a rudder and of course the exposed shafts and struts seen on many transom-stern hulls are absent. Thus, the hull resistance is only the bare-body drag with perhaps a minor increase for bilge keels (if pres-ent). Because the blades oscillate to develop thrust, they are not as eff i-cient as fixed-geometry propellers but with the reduced hull resistance, the required propulsion horsepower may not be much different than for a

convention-2-3

Fact DtVftOPt fOCt MtDr F IWS

SSCII SCSW TIS SIT CYOIDAL TI

Fig. 4 _{Thrust Vector Plot}

al propulsor. Hence, comparison of propulsive efficiency of a cycloidaj. propulsor with conventional propulsors is often not productive. _Hull interac-tion coefficients (thrust deducinterac-tion, t, and wake fraction, w1, are typically in the range 0.1-0.15 (Clerc and

Goldsworthy, 1935, Sarchin, 1966 and Reed, et al, 1969).

The blade-angle or pitch-control mechanism is complex mechanically and may produce significant noise due to the

linkage oscillations as well as signifi-cant power losses associated with its increased mechanical-friction drag. One procedure to reduce the mechanical noise is to use worm-gear drive(Sturmhöef ci, 1978 and Almy, 1984) but additional power losses may accumulate to the point that such losses are as great as 8 per-cent of the input power (Voith Publica-tion ts 2750). _{The operating rotational} speed of the blades is generally reduced relative to other propulsors. The com-plex mechanism of the unit results in large starting torques that must be considered when selecting the shaft system (typically an elastic coupling is used).

The flat stern near the cycloidal propulsor may be of concern for

seakeep-ing since it may be imagined that there is a tendency to slam in a seaway. How-ever, flow studies indicate that the suction created by the operating unit will pull water with it and not result in slamming (Clerc and Goldsworthy, 1935, Foerster, 1936, Sarchin, 1966 and Reed, et al, 1969). One German

mine-sweeper of the R130 Class crossed the Atlantic after World War II and won praise for its seakindliness (Alay, 1984).

As a final attribute, it is to be noted that the narrow blades are the only projection from the hull and these blades are at differing local pitch angles. _{Hence, for a} horizontally-di-rected shock load, there

is

_{only a small} amount of surface area to be impacted for a cycloidal propulsor. _{Thus, one} may expect some interest in these units for mine countermeasure vessels for which both shock-absorbing capability

and high maneuverability are needed (see, e.g., Daniel, 1984).

In appi.ication, the signiricant hydrodynamic performance and geometric properties are the quick thrust vector-ing and shallow-draft capability. A special tug-boat configuration has been developed (called the water tractor, see Figure 3) based on the cycloidal propul-ser and other applications with preci-sion maneuvering/stationlceeping require-ments (such as buoy tenders, drilling platforms, ferries and mine countermea-sure vessels) are natural for utilizing such propulsors. Rickards (1970) pro-poses use of a cycloidal propulsor for a commercial submersible, noting especial-ly that the propulsor produces stability in a cross-flow current. Previously Rirsten (1950) has noted and proposed an explanation for the roll stabilization of a cycloidal propulsor. Some of the special capabilities of the cycloidal-propelled tugboats are discussed by Baer

(1954) and Sarchin and Tryner (1970). As with any complex piece of ma-chinery, the cycloidal propulsor manu-facturer requests to be involved in selecting a particular configuration for a given application. Because of this involvement of the manufacturer having a large performance database, what an independent trade-off analysis requires is a relatively simple tool that will give guidance as to the expected

hydro-dynamic performance in a number of areas. In particular, it is desirable to be able to predict performance in the following areas:

Hydrodynamic Powering Cavitation Inception

Acoustic Levels due to Cavitation near top speed

Thus, we present herein a methodol-ogy and implementation of an approximate analysis of a cycloidal propuisor that addresses these three areas. The objec-tive of this effort is to produce a pre-1 iminary-assessnent computational tool that is sufficiently accurate to define the performance in a short run-time so that alternatives can be evaluated and quantified. Other analysis procedures already exist, including two-dimensional unsteady approximations by Isay (1958), Sparenberg (1960), Mendenhal and

Spangler (1973), and James (1971), as veil as three-dimensional quasi-steady and unsteady analyses by Just (1939), Taniguchi (1962) and Isay (1968, 1970). Haberman and Caster (1961) indicate deficiencies in Isay's two-dimensional model while Haberman and Harley (1961) show promise for Taniguchi's quasi-steady three-dimensional method when applied to symmetrical blade sections with semi-elliptic planforms. Zhu

(1981) presents further analysis based on Taniguchi's Method and removes the restriction on blade-section symmetry. No implemented method is known to pre-dict (or even address) all three

perfor-2-4

inance varlaDjes aove a zuiiy satis-factory way for arbitrary blade sec-tions.

A survey of cycloidal propulsion has been given by Henry (1959). He presents a history and a description of some kinematics of the notion of a cy-cloidal propeller, as well as a critical review- of early attempts to model the flow field. Some of his suggestions for

improved modeling have yet to be imple-mented. Improvements in performance were addressed by Sparenberg (1960,

1969) and James (1971) who attempt to define blade-angle control specifica-tions for improved efficiency based on two-dimensional flow models. Systematic series data are presented by Nakonechny

(1961/1975), van Manen (1966), Ficken

(1966) (who includes Nakonechny's data

in his faired curves), Bjarne (1932, see also Kaistrom and Loid, 1984), and Bose and Lai (1980). The most extensive of these series are those of van Manen and Ficken. The main variable in these investigations is pitch control, but several blade shapes are also evaluated. Van Manen includes evaluation of the blade-angle motions derived by Sparen-berg (1960), which did produce improve-inents in efficiency over his convention-al cycloidconvention-al blade-angle variations. Van Manen's experiments (with low-pitch propellers) produced generally lower efficiency values than expected which has been partially explained by Ruys

(1966) as a viscous effect. Van Manen's data for high-pitch propellers had quite respectable efficiency values (j-0.7). The experimental evaluation of a eycloi-dal propeller is difficult as noted by Ficken (1966) who reports erratic torque readings at Reynolds number values less than where the Reynolds number is characterized with the quantity;

m

cm/V+(rnDo)2

(1)

where cm is the mean chord, V0 the axial speed of advance, n the rotational speed and v the kinematic viscosity.

Ibragi-nova and Rousetsky (1969) describe clearance features of their mechanism that must be carefully controlled to prevent anomalous thrust values at model scale. They were able to correlate their data with that of Ficken (1966)

for similar blade-angle variations. Hence, the technology base available in the literature is sparse relative to

both a design/analysis capability as well as standards of hydrodynamic

per-formance.

Even scarcer are data that reliably predict local details of the blade flow.

Isay (1958, 1969) does present some predictions of blade load.s as a function

of position and compares his overall loads with measured data for a high-pitch cycloidal propeller (for which the

interactions are reduced relative to those for the more common low-pitch prcpe].ler). Measurements of instanta-neous blade loads are presented by Panov

(196Ô) and Baer (1973) but sufficient information about the geometry of their test case is not given to permit under-taking calculations to compare with the data. Flow-visualization and instanta-neous blade-load results for operation of cycloidal propuisors are presented by McKillop (1965) and Jobst and Bandler

(1972). Although

their

results are for extremely low Reynolds number values, p. O.2.1O, the data are some of a very limited set of detailed flow information available and we use them herein to help confirm the adequacy of our- flow model of the cycloidal propulsor.

In the following sections of this presentation, we first construct a sim-ple model of the flow field that matches to a large extent the experimental data just described. This analysis Starts with kinematics

and

includes blade and point trajectories and arbitrary

blade-angle control specifications. _Following this geometric introduction, we present

a discussion of the previously-mentioned experimental data that will help guide the formulation of a mathematical model. A quasi-steady, approximate three-dimen-sional model is ultimately proposed that appears to reproduce the measured per-formance with reasonable accuracy as demonstrated by extensive comparisons with available experimental data. Cavi-tation. properties are assessed with

the

minimum pressure values of a typical

blade

section as a function of orbit

position.

_{Cavitation noise levels are}

-predicted based

on

functions for

the

non-dimensional spectra presented by

Blake

(1986) with additional

modifica-tions for

unsteady

flow.

The paper

closes with recommendations to improve

the performance of

this unique thrusting

device.

KI1EMATZCS OF FLOW AND OPERATION

In Figure 2, there are two coordi-nate systems: _{one global and inertial}

in nature, the other fixed to the moving blade as it translates and rotates

through space. _{The global system is} aligned with the axis of rotation at t =

0 with the reference blade at top-dead center but does not move with the pro-pulsor (as it rotates in

_the

counter-clockwise direction).

_{The inertial}

coordinates are denoted with the symbols

x and y.

The rotating and pitching

blade-fixed coordinate system

is denoted

with primes

(xr, y').

For a

uniform

speed of angular rotation,

,

we define

the angular position

of

the blade-angle

spindle-axis centerline point:

2.5 tURVATURE 3-2 -3--e .1

- 0.573

0 0.7 z. 4 = A - 1.114 e -1.4286

z-4

e . dr =,t (2)

where 0 = t = 0 is the starting position with a blade at top-dead center of Fig-ure 2. For a uniform speed of advance, V0, the inertial coordinates of the ref-erence -blade-axis (x' = 0, y' = 0) for an orbit diameter of D0 are:

- (X0 + sin 0); cos 0

where X = V0/(ITnD0) is the advance

coef-ficient and R0 = D0/2. _Flight-path

tra-ces are shown in Figure 5 for both high (X > 1) and low (X < 1) advance coeffi-cient values. _{Note that the blades} cross the path trace of

other

blades and that at low pitch operation the blade loops across its own flight path. The radius of curvature, RVAThRE, along this path 3.S given by:

D/2

_XcosO+i

(1. + 2 X cos 8 + X2-)312 (3)

(4)

4

4-2

-1 HIGH PITCH

Fig. 5 Blade Path Trajectories, for Cycloida]. Propulsors

We define a blade angle (0) that is the orientation of the nose-tail

line-(the

x' axis) relative to the tangent to the orbit circle (positive when the

-6.4

1 0 1

leading edge is 1splaceQ outboard of the orbit radius). The angle

=

(5)

is the blade-angle orientation relative to a line through the blade spindle axis and parallel to the x axis. _{The pitch} angle is 0 = r/2 - a (see Figure 2). An arbitray point (x',y') in the blade-fixed coordinate system then has a posi-tion vector in inertial space given by:

=

+ XcOS

a + Y'sin a

=

XS1fl

a + y'cos

The absolute velocity and accelera-tion of these points in space can be obtained by straightforward differentia-tion. After some calculation efforts, one can define a frame velocity,

expressed in terms of base vectors fixed to the foil reference system as:

VR(0) = (1 + 2 cog 8/k + 1/k2)112

a(8) = tan1 [ - sin 8 ]

Hence the geometric or kinematic angle-of-attack (see Figure 2) is given by

(8) = a a, (9)

The time-rate-of-change of this angle-of-attack about the orbit path is

readi-ly

found to be

=0

-

(X + cos 0)

(7)

i_ut w.

..j

is defined as an integral of the local increment of axial distance a blade advances along Ø(0) for an orbit-angle increment dO which produces a transverse movement of R0 sinG dO. Hence, we have the defining equation:

P 1. r2y

-

--I sin Otan0d6

lTD0

27rJ0

p

= 1 (21 sin 8 dO

21r Jo tan

which has an integrand that must be specially evaluated when tan q tends to infinity, which generally happens near B - n,, n 0,1,2.... In general, the integral exists only as a Cauchy Princi-pal Value unless both 0 and a tend to

zero with finite slope at the same point.

B].ade-Anale Contrc],

The key to proper operation of a cycloidal propulsor is the phasing of the blade nose-tail line relative to both the onset flow and desired thrust direction. Here we consider only the case that the thrust is developed in the onset flow direction but otper situa-tions are of interest also.

Perhaps the most common representa-tion of the blade angle, (0), is that

of pure cycloidal motion (see, e.g., Mueller (1955), Nakonechny (1961/1970), Ficken (1966), etc.). In such a motion, the normal to the blade nose-tail line at the pivot point passes through a

fixed point in space, called the pitch point. The computation of the blade angle for such a case is straightforward and one finds

(9;e) =tan1 e sin 8 1 + e cos 8

(12)

esinO

= sin

+ e2 + 2e cos B

where e = sin is called the eccen-tricity. Blade-angle variations are shown in Figure 6 for various values of e. The distance advanced in one revolution

(along without slip) defines the

pitch P. For this !9 distribution, the pitch is

= e = sin - _(13-)

ff00

As discussed by Mueller (1955), cycloidal blade-angle variation does not necessarily produce the best efficiency 1Maneuvering loads are desired for quantifying station-keeping and tracking and even at the nominal forward thrust condition,_ the cycloidal populsor_ may develop

a significant side force which can be

canceled by special pitch

control. 2.6 - =E-cOs0p_ V0

cos

C _y1

_{.Ej i'}

X

R0cXdO

-+ (-sinO sin$ - E..!..

_i] j'

p

X

R0CX d8

Relative to the blade-fixed coordinates, there is onset velocity given by

_-.

The onset velocity vector relative to the blade section consists of the vector sum of the steady rotational speed and steady advance speed. The blade-axis point (x' = y' = 0) experiences a re-sultant onset frame speed of V

(8)

at a kinematic angle a(8) (relative to the x axis, see Figure 2):

}

(6)

}

(8)

dt dO

₍₁₀₎

=0,

fXcos9+i.

l+2Xcos8+X2

dO J

Fig. 6 Blade-Angle Variations for

Pure

Cycloidal Motion

and has high accelerations near 6

= it.

He states that improved efficiency (and reduced accelerations) is obtained with amplified cycloidal motion for which the angle at Some reduced overall pitch setting, say corresponding to e*, for cycloidal motion is multiplied by an amplification factor A:

(8; A,e*) =

A taicl[e*sjno/(l+e*coso)J

Both

the

pure cyöloidaj. and

the

amplified cycloidal motions are mechani-cally complex to bring about and Mueller suggests that a simple sinusoidal blade-angle control fl(0)

B sin

6, even though it will produce reduced efficien-cy values, ay have payoffs in mechani-cal simplicity and robustness (see also Goldsworthy and Brown, 1957). Figure 7 reproduces Mueller's data for various pitch-angle distributions for the sane maximum value. In.

Figure

8, schematics of mechanisms for producing both sinu-soidal and amplified cycloidal blade motions

are shown.

The 5-bar linkage system for amplified cycloidal blade motion corresponds to the mechanism of Figure 1. A careful check of Figures 1 and 8 shows an initial angular offset of the blade lever

arm

and an initial of

f-set in

_$,

called angular

compensation.

This linkge-arin offset produces a dif-ferential in forward and aft angles because of the cosine effect (this is

simIlar to the Ackerman principle used

so successfully in automobile steering

systems; see Baumeister, 1967, p.

II-13). For known values of the linkage arm lengths, pivot positions and off-sets, one can compute $(6). We have done this for the set of linkage lengths of Figure 1 (and a steering or pitch-point offset sufficient to produce an average of the extreme values equal to

(14)

2-7

Angle of Rotation e- Deg.

SINUSOIDAL BLADE MOTION:

(6) = 53.13° sin8 CYCLOIDAL BLADE MOTION

-1 O.Bsin8

fl(6) = tan

0.scosO AMPLIFIED CYCLOIDAL BLADE NOTION

(6) = l.3B7tan1 0.62sinO 1+ 0.62 cos9

COMPENSATED AMPLIFIED CYCLOIDAL BLADE

MOTION

13 13

(8) =

Ebsinn9

-s a0 ₊

_EacosnO

i.1

,-;

Fig.

7 Various Blade-Angle Distributions 53.13 degrees,

_{the maximum}

of Figure 7) and include the results in

Figure

7. We

have also constructed a trigonometric

approximation of this computed

blade-angle variation to use in subsequent

calculations..

We. represent the

istri-bution as

N-i N

(0) =EbsinnO

+

Eacosn8

(15)

n-i

n.O

where the coefficients a and b can be

selected to exactly pass throug"h a given Set of 2N+1 cyclic data points at equal angular spacing (see Sokolnikoff and Redheffer, 1966) but the coefficients here tend to zero rapidly

and qnly a

limited

ntthbër of terms exist. Table 1 gives these harmonic coefficients for

the. selected set of linkage lengths.

In

addition to such harmonic distributions,

a constant can be included in the

ini-tial blade setting (called angular

com-pensation) as well as arbitrary varia-tions of can be specified

(particular-ly if individual active blade control is

rPitCh ritlo .lt'p y C,. Inde. b7 '

frsi,: Kirst (1950)

IU

NURUUN

2ViVA!4N12

iWdW$R

r.

r:

. '50 lAO0

TABLE 1

BLADE-ANGLE HARMONIC COEFFICIENTS FOR A SELECTED SET

OF UNKAGE LENGTHS AND STEERING CENTER STATIC ORIENTATION IS: ANGLE A 11.515° (ACKERMAN)

ANGLE A, = 12.474°

GEOMETRY FOR 5 BAR UNKAGE

ACTIVATING LEVER 750 (L5)

CONROD = .8500 1L51

CRANK .2300 1L4

DIST .2300 (L31

Al .2300 (INNER PART OF 11)

A2 800 (OUTER PART OF Li)

STEERING RADIUS .0930

HARMONIC ANALYSIS OF BLADE-ANGLE VARIATION FROM

LINKAGES )BETA IN RADIANS1

Fig. AN -.02624 .00482 .01856 -.0054c .00016 .00029 .00012 -.00022 .00011 -.00001 -.00002 .00002 .0000 öoôoo SINIIDAL BN .0)000 .83119 -.241 16 .04608 -00303 -00229 .00079 .00022 -.00034 .00017 -.00001 .00001 -.00001 1 I IIik 2 I..ef 'P1'!z' buiI 3 bsar1q cm crank S cc.mcng ruâ 6 cuaIing I.w.

8 Mechanisms for Producing Sinusoidal and Compensated

Amplified Cclçidal Blade

Motion

2-S

specified). in Figure 9, a blade-angle variation for improved performance as derived by van Manen (1966) from Sparen-berg's (1960) analysis is presented; it is van Manen's curve. II, with load-car-ryiñg specification for a constant cir-culation value on both the forward and aft orbit arcs.. To obtain a smooth representation of this curve using harmonic analysis,-we have found it necessary to round off the curve just prior to 1800 _{as indicated by the dashed}

line in Figure 9. _{Experimental data} presented by van Manen show an improve-ment in efficiency relative to his sea-surments with pure cycloidal blade motion.

Fig. 9 Blade-Amgle Variations for Improved Efficiency According to van Manen/Spareñberg

Coefficients for imprOved efficiency according to van Marten/Sparenberg Curve II - Least-Square Fit (v/some mods to

data) N AN SN 0 -. 020505 1 .054376 .667805 2 .100990 -. 150503 3 -.022977 .077729 4 .033500 -.018725 5 -.007562 .021846 6 013266 -.003900 7

jo

.007688

From these blade-angle variations, the geometric or kinematic angle-of-attack (Equation 9) can be derived, where the value is given by

a(9)=

aF $ - 0 (16)

Values of a. for an eccentricity and advance coeffidient less than 1 and for an eccentricity and advance coefficient greater than 1 are shown in Figure 10. These blade oscillations are of such an extent that at some orbital positions

0, ORBIT ANGLE

2Even more extreme curves are presented by Van Manen (1966) that would present a challenge to represent by a harmonic

series with only a few terms.

N 0 1 2 3 4 5 6 7 8 9 10 11 12 13

3

I

.1 - 1.8 A - 0.575

Rotation Angle 0

- Degrees

90 180 270

Rotation Angle 0 - Degrees

360

Fig. 10 Geometric 1ngle-of-Attack and Mean Flow Speed for Pure Cycloidal Blade Motion

near

0 =

ii, separation may occur and the suction peaks may be quite low. This separation is most likely to occur in

2.9

regions for which little thrust is pro-duced so it may not be too serious a problem in practice, but the suction peaks may produce detrimental cavitation and noise. The possible stall is some-thing known to be delayed in unsteady applications (as is the case herein) relative to steady-flow situations but as yet is not amenable to rational quantification.

The most common pitch control is that with total pitch less the

i.e., e<i, which we call low pitch val-ues herein. An upper limit of these low-pitch values is near a = 0.9 to restrict the accelerations experienced by the blade (see Mueller, 1955). To obtain pitch Values greater than e = 1, special mechanisms are required and even then the acceleration limits prevent a continuous change across e = 1. The special planetary gear mechanism used by Kirsten (1928) fixes the eccentricity at e 1, but in later work Kirsten (1955)

describes a pitch-control mechanism that covers both high and low pitch values.

To obtain some perspective on the circumferential load distributions, we compute the quasi-steady load variations for a purely two-dimensional flow at the spindle axis location (x' = y' = 0) for a chord length to radius ratio, _cmJRo, of 0.352 having a camber-to-chord ratio

/c = 0.044, for two cases. The first is at an advance coefficient X ₌ 0.573 with a distribution correspond-ing to e = 0.7 and the second at A = 1.114 with $ corresponding to e = 1/0.7 = 1.4286. The lift coefficient is given by CL =

2n_(aIN

+ 2 fm/C) for a reference speed of V0. and the circulation is qiv-en by the non-dimqiv-ensional quantity C = r/(D0 oR0) ₌ XC0/RØ CL/2. A local drag

coefficient is given by a constant fric-tional value plus a term depending on the square of the lift coefficient. The

local lift and drag can be resolved into thrust and torque directions. The

re-sults are shown in Figure 11 for a non-dimensionaljzatjon such that the two cases operate at the same rotational speed and geometry scale. Here the camber produces a differential in the loads in the forward and aft orbit arcs and the loads tend to be small in the region near 180 degrees where the great value of the lift coefficient suggests a tendency to have flow separation. The circulation exhibits a gradient that is greatest near the l80 position. Since it is the time variation of circulation that produces a shed-vortex element in two-dimensional flow, and since there is a region about 1800 that has the slowest relative speed, this region may be crit-ical to model properly to account for interaction with the shed-vortex sheet that remains near the blade. Hence, this region will be one to which we pay special attention as we examine experi-mental data.

3.5 S 2.5 2 1.5 Os 0 33 3 25 2 1.5 as C .251

4

-3 8- et, Red. 7 SINGLE BLADE IWO-OIme4uIonaJ Flow 1-0.573 o-0.7 DTMB BLADE SINGLE BLADE Two-memIonaI Flow I - 3.5 110.7 DIMB BLADE

0

C 2 3 4 6 6

Fig. ii. Illustration of Blade Load Variation along Flight Path for Two-Dimensional Flow

OBSERVED PLOW ABOUT CYCLOIDAL PROPULSORS Flow-field visualizations (McKil-lop, 1965, Reed, et al, 1969, and Jobst and Bandler, 1972) show an overall in-crease in fluid speed into the cycloida]. propulsor (Figure 12) caused by the lift of the blades. Measurements undertaken by Mcxillop (1965) and Jobst and Bandler

(1972) define a circumferential average total, speed, V at positions along the orbit path. The induced velocity compo-nent is greater along the aft side of the orbit relative to the forward side

and generally is in the axial direction (Figure 13). The angle y defines a transverse velocity component (V1 sin y) that is small for high-pitch values but greater for low-pitch operation. For a nearly constant-chord blade operating at a P/D0 value greater than ir (e > 1) the induced velocities are nearly the same along the span, but show spanwise vari-ability for tapered blades at reduced-pitch values for which the interaction effects are greater. In general, the

interaction is greater at low pitch than at high-pitch values. For a pure cy-clojdal blade motion,, the force on a single blade in the direction normal to

2-10

---

---fro:

Sarchin C196)

Fig. 12 Flow Direction iii Vicinity

- of Cycloidal Propulsor

the chord, FM, was also measured in those experiments and the results are

shown in Figure 14. The predicted loads in two-dimensional flow for the same eccentricity and advance coefficient are given in Figure 11. Note the reduction

in absolute magnitude of the force over the aft portion of the orbit (relative to the forward orbit arc) resulting from a combination of the greater values of

induced velocity there as well as the blade camber. The data show a smooth variation of loads as the blade transits the orbit circle, and hence the crossing of its own shed-vortex sheet'by the blade operating at a low pitch value in the orbit region near 0 = 270 degrees

(see Figure 5) does not significantly modify the load on the blade. (The time variation for the circulation (see Fig-ure 11) is expected to leave behind a region with concentrated vorticity (near

8 = 1800) and there will be vorticity shed because of spanwise variations in load.] A similar smooth variation is evident in the data presented by Panov

(1960) for the normal force component on a symmetrical blade and in data

present-ed by Baer (1973) for the root bending moment. (Baer also demonstrates that angular compensation can be used to

equalize the bending stress extremes along each arc of the orbit). In

gener-al, we conclude that the crossing of the shed-vortex sheet does not play as sig-nificant a role as does the mean flow at a point in space, although this

conclu-sion is tempered by an observed tendency to have an unusual oscillation in the loads on the aft orbit path that may be associated more with the vorticity shed due to spanwise circulation variations than that shed because of time varia-tions of the circulation. we conclude

this since XcKillop's (1965) data in Figure 14 at low-pitch values includes comparisons of a three-dimensional blade and the same blade with an end plate. Since the blade is tapered with a tip-to-root chord ratio of 0.57, there are still significant three-dimensional effects in the data with an end plate but they should be considerably reduced. The decrease in load along the orbit path from 180° to 270° is believed to be a consequence of the shed vorticity near the 180° position. ifowever, the data with end plate do indicate a more nearly uniform load distribution along the aft portion of the orbit cir1e (a is also

1 3

8- set.Red.

1.3 12 1.1 1.0 09 0.8 1.2 PREDICTED 1.0 02 0.8

- v,Iv,

MewredIi*wPropeiiIe.tVJ,1ND- .575, J-1.8 15 10 5 5 10 PREDICTED 10 S 0 -5 -10

predicted from our simple model). The predicted results in Figures 13 and 14 will be described later.

Cavitation patterns as a function of orbit position are presented by van Nanen (1966) and Bjarne (1984) (repeated

in Figure 15). _{In general, the region} of cavitation indicates considerable spanwise uniformity. Van !4anen's data show leading-edge cavitation starting toward the tip at 8 30° and spreading up the blade while the inside of the blade has bubble cavitation starting near the root and spreading out along the span at 330°. mjarne shows free-route operation with cavitation on only one side of the

blade.

These data de-fine trends as well as help evaluate the

location and value of minimum pressure peaks. _{Ta complement this data, we note} '.that both tip vortex and leading-edge

pressure-side cavitation on cambered sections are known to be Reynolds Number sensitive.

Measurements of propulsor effi.cien-cy as a function of Reynolds number indicates a significant scale effect since the model-scale propulsors typi-cally operate at Reynolds numbers of l0, see Figure 16 from Ruys (1966).

S 02 11 1.0 .03 - .02 - cj4 S C S 3oo o Vj/V0 PREDICTED

'

2 0.9 0.8 11 ¶0 0.9 0.8

Rolion AngIe 8 - DegTees.

0

Fig. 13 _{Average Induced Velocity Component about Orbit of} Cycloidal Propulsors with Pure Cycloidal Motion

-2 .4 0 -2

4

.4 2 0 -2 .4 MeasuredlnfIowPropertIeeatV/iND-I.114. J-&50

Fig. 14 Individual Blade Loads on Cycloidal Propulsors having Pure Cycloidal Pitch Motion

Other possible problems with model-scale measurements are described by Thragimova and Rousetoky (1969) and Ficken (1966).

2 02 -0.50

nil,

-

-!

0.RCO2O3

I 4 PEDWD :

0

C 12 1.1 1.0 0.9 0.8 1.1 1.0 0 -S .9 0 90 180 270 360

Rotation AngIe 8 Degiees

Fw.

...l P500 .. 514

E

with 1.4 01791115171719*51157

3The load coefficients KT and K3 are suggestive of two-dime3sional

performance. 1 014j4* If blodn 0.tsl. 01 11.Id If 25. 35

Cavitatiónpatt.ziis, cyceida1 blada Sotion

(0 - O.6 O

- .0

-- fro2: v.5 XanSn (1966)

IS_IISI TI..SI V.I7?V, ..v

' IS.w

Cavitatiolb pattatna. influanca of .hlp apeod f?0'a Bjarne (1953

Fig. 15 Cavitatio3 Patterns on Cycloidal Propulsors

2- 2

A tabulation of the geometry and operating parameters of several cyclol-dal propulsors at model scale is

ien

in Table 2 (this is an extension of a table given by Ruys, 1966). We present a comparison of load ooefficierttb3for thrust, T, torque, Q, and efficiency, values in Figure 17, for three of these propulsors in open water, where

TABLE2

SOME CYCLOIDAL PROPULSORS KS

=

>s c;

as a function of advance coefficient X V0/,rnD0 Or 3 = V0/nD for two pitch values of. the

same unit.

The series are-. those of icken (1966) for pure cycloi-da-1 motion with a 6-bladed propulsor, those of van Mahen (1966) fOr pure cy-cloidal motion with a 6-bladed propulsor and those of Biarn (1982)

with a

5-bladed Voith-supp].ied model with its own pitch control which we take

to

be

com-pensated amplified cycloidal (CAC) pitch Oontrol. These performance curves are similar in shape to those for conven-tional open propellers. The van Manen data (for a symmetica1 blade section) and Fiken data (for a cambered blade section) are. reasonably in agreement for-thrust Obefficient at the tsio pitch values but not in agreement for torque

P)am. A,.. Rob. Zo.. 2R. Z ..L A c_ Bbodsrm a at Mid Opsa Slod.-Mq Modan 81.d. PaliI. Ozbll OIaat.r lm} R. of T.ot R.919-SItlTAOI'lal SIIOPi c,., SpIdI.

Go-NSMD/von PSan.n 0.721108 4/6 1.20 0.36 Real. 1.0 0.50 0.743 CYC. + S,...,..u4..If 0.200 !.4.10

DT5'aJicken -0.35/0.53/1.06 I 2/316 L -. 1.0 -0.352 Rest. 1.0 0,47 . 0.1.1 CYC C*ni94md. 0.044 0.229 7.6.10°

ERAIM.KJltep 0 707 4 1 25 0 394 Teawed 057 0 47 0 15 CfCI?) Co,nbwsd

0.018

0.150 02 10

ERA/Jobat 0.71 4 1.0 0.352 Rest 1.0 0.47 0.11 CYC 0.160 0.2.10! SSPAJB).m. 1.11 5 1.50 0.4.44 Tops,wi 0.35

-.

0.33 0.1 .-. -CAC Cu,thoid.'f 0.0.3 0.200 71OiQ -CKT T =Rr/(L/D) = / (LID) (17) pn2 D0 L Pfl2 D' L T pn2 D:

I

U U IS IS

SPEED COEF1CIENT .J

Fig. 17 Performance of Cycloidal

Pro-pulsors with both pure and Modified Cycloidal Blade Motion

SPEED COEF9Crr .1 2-13

fr:

(2.8) .i.I SIN.t,lr_CI bnjarne (1982)

0

van Manen (1966) -5 .' a O.E

-11111111

_..

o. z-: 3 10 'I Fig. 17 Continued

values. _{Van Manen's data show reduced} values of efficiency, which has been examined by Ruys (1966). As can be seen, for the Ficken and Bjarne data, the efficiency values are about the same but a greater load is developed by the pitch control with amplified cycloidal blade motion. _{We have also plotted van} Manen's efficiency values for the Spar-enberg optimum blade-angle motion in Figure 17 and it can be seen that even these data for increased efficiency values are less than those measured for the other evaluations.

We have interpolated Bjarne's load coefficient data to intermediate pitch ratios and analyzed it to define a set of polynomial coefficients for perf or-mance of similar units that produce the large thrust values. These coefficients are given in Table 3. An additional efficiency gain of a few percent was applied to the torque curve according to the trend of Figure 16. _These polynoini-als are of use in preliminary design _as are similar polynomials for other pro-pulsors (as for example, the extensive series of polynomials derived by NSMB). They can be easily manipulated to find the optimum efficiency as a function of either thrust-loading coefficient

CThT/(_PVAAQ) =2

cK

2

A

or the special power coefficient, B, made up of dimensional quantities (N is

D is British horsepower and VA

speed in knots):

Rfl105

fr Ri.zys (1966)

Fig. 16 Reynolds-Number Effects on Cycloidal Propulsors 010 OcM,..d,,I 010 fi 0, 4,

TABLE 3

POLYNOMIALS FOR THRUST AND ToRauECOEFFICIENTS OF CYCLOIDAL PROPEIiERS AT PROTOTYPE

REYNOLDS NUMBERS

5 BLADES, dO. - 0122. LiD. -0.75,AMPLW CYCtO1OAI MOTION, WITH5.15 COMPENSATION FOR PP!DJPI - &5 TO 0.9

COEFFSFOR THRUST

CKT - (2223868+2.7755PODJPOD+(4.239779 POLO

(10.634861 +PO08.394C57sP004I.5852S3)DJA +3.819543P00(-185344+P00122140708 P009.810305)))JAJA

COEFFSFOR TOROUE

CKQ - (.818878.1.949753P0D)PODPDD (1.051328P00 J3128.POD916B79.POU1.9688flJA .-.420327+P0Dm253B25+POD1-1.493886+poo .534485}rJAJA.(0005958+P00.005052)'JAJA'JA =

N(P0/V)1'2

= 33.07 E 11,2. (19)

DJ

where A0 D0L and 3 = 3 (1-w1).

Com-parison of such coefficients for several

types

of propulsors for which similar polynomials are available is made in Figure 18. As can be seen, a moderate reduction in efficiency occurs with a cycloidal propulsor which, compensated with about a fifteen percent savings in hull resistance when the appendages

are

eliminated on a transom-stern hull, makes the cycloida]. propulsor quite competitive relative to more

convention-al units. *

In summary,

the data presented indicate that:

the loads

developed by a cycloidal

propeller are similar in form to those of conventional propellers

the interaction between the blade

and shed-vortex sheet doesn't play

an overwhelming role in the blade

loads,

the field

may be more similar to two-dimensional flow than is a conventional propeller,

the

time variation of the blade

loads is

at a

frequency

character-ized by the rotation rate

of the

propulsor and

the limited experimental data for

toad coefficients does not form an

entirely consistent trend.

The interaction with the shed-vortex sheet may be more significant relative

to that component of vorticity due to

the spanwise variation in circulation

(three dimensional effects) than

rela-tive to that from the time variation of

circulation.

2-14

uIII,J

iLIiIII_1

'1iL PiI

o,t.

----.7*'

--KdSmD

CRP -

F-I

- OPTN

iinuii.i

I!;

I!!I11i

li-IS,.,..,

'

COP..r.n s

I

IIIIIløIII

_ItiIIlfl

SI

03 0'000007 _'1;/z 2

am

3

3 1043

II 14 10110074 104 4. 344

Fig.

18

Comparison of Efficiency Values

for Several Propulsor Types

EYDRODYNAMIC ANALYSIS

For a

uniform

_{onset flew, the} ma-jority of the flow field will

_remain

potential as the blade experiences

the

kinematic motion

just

defined.

An exact

potential-f low model is straight-forward

to formulate in terms of time-varying

source and vortex/doublet distributions

over the surface and time-varying

vor-tex/doublet distributions over the

shed-vortex sheet trailing from each blade.

The singularity strengths are determined

by

the boundary

condition that

the

nor-mal

velocity component of the flow be the same as that for the solid surfaces and that there be no pressure difference

across the shed-vortex sheet as it is

convected

with

the

mean flow on the sheet. The kinematics previously described give information about the general nature of the flow and one can easily suppose a load variation due to this motion (see Figure 11). The flow field will be unsteady and periodic. The field is governed by the Laplace equation for potential flow as is

appro-El U I51 02 00 05 0' 3 03 02 II 0

priate for lifting-surface analysis for b2.ades operating in uniform onset flow (here we ignore the boundary layer de-veloping along the surface of the orbit

disk).

Drag forces can be added

sepa-rate].y as is conventional for lifting surface analysis. The boundary condi-tions give sufficient information to define the strength of time-varying singularity distributions over the blade surfaces and shed-vortex sheets. Such a model is discussed by Isay (1958/1970), Sparenberg (1960), etc., but as yet has not been fully implemented. To date its approximate application as a two diinen-sjonal field has not been successful in predicting the behavior of the flow (see

Haberman and Caster, 1962). I4enderthall

and Spangler (1973) present an approxi-mate two-dimensional model with variable vorticity shed from the blade but they were unable to correlate with some fea-tures of the flow field, while Ibragi-mova and Rousetsky (1969) report good correlation with an unsteady two-dimen-sional analysis for the spindle-axis moment. An adequate numerical implemen-tation with the boundary-element (or panel) method can start with an initial condition and time march to a fully periodic local flow about the blade (for which the starting vortex will be a few

orbit diameters downstream). Loads at this fully periodic state can be derived with an extended Bernoulli equation that

includes a term for the time-variation of the potential as well as velocity components. _{Unfortunately, such a} pro-cedure is lengthy to execute and is more appropriate to the final selection of a unit rather than an initial sorting out that is the focus of our analysis. Simule Math Model of Flow

The features we include in our simple quasi-steady analysis are predom-inately the two-dimensional flow at each blade section together with

modfica-tions for the unsteady pitch-variamodfica-tions in the onset flow (equation 10), which produce a boundary condition that is kinematica].ly equivalent to a change in the camber ratio (and possibly an incre-ment in the angle-of-attack if the blade spindle axis is not at mid-chord; simi-lar to the description iii Rickards, 1970, and MendenhaU. and Spangler, 1973) as well as a camber adjustment resulting from the curvature along the flight path

(equation 4). In addition, the local axial velocity component is increased at each orbit position by a contribution derived from an actuator-disk model. A further modification is included for the flow acceleration normal to the flight path using the added mass for a flat plate of the same chord. Then the flow about the modified blade section acting in a flow defined at the spindle axis location is computed based on a quasi-steady lifting-line planar wing model for which the kinematic flow angle is

2.5

reduced by an induced angle-of-attack. An empirical function is proposed to decompose the component of induced

ye-locity from the reference blade into two contributions, one that is local and one that has already contributed to the mean value from the actuator-disk model.

Specifics of this model are described in the following paragraphs and have been guided by Scherer's (1968) modelling of an oscillating three-dimensional wing.

Based on the measured data in Fig-ure 14, the frequency of load variation for the blade is the orbit rotational speed and hence the reduced frequency is characterized with the value;

c/R0

2VR _2X

_Il

₊

_1/x2

C2 0J

which is typically less than

_0.2

for the chord length values in Table 1 and an operating advance coefficient X of 0.5 or greater. Hence, at the outset we assume that such low values of reduced frequency provide further impetus to ignore the time variations per se and treat the field in a quasi-steady man-ner, including the pitching motion given by the angle-of-attack time variations

(equation 10) and added mass of a flat plate in heave. In particular, we choose to use the kinematic effects of the boundary condition rather than the dynamic solution known as the Theodorsen Function (which includes additional effects from a periodic shed vortex

sheet f or

both pitch and heave motions).

We take the time-varying velocity field in the vicinity of the blade (as seen by an inertial observer) to be composed of an average (or mean) and a local component that are functions of only position in the rotating frame:

(21)

where g,(0) is the sum of the onset flow plus an average induced velocity and

i(8) is another component of induced velocity that acts on only the blade of local interest (rotating with it). We approximate g, by adding a mean induced velocity, _{to the onset flow:}

;(0)

=V0+U+wR e

° where = - sin

Uj + cos 91

Vi = 4U V0 i/i + Th

- 1

(9r)

V0 2

where is the local increment of thrust-loading coefficient. The

expres-si.on for UA is applicable for positions

at the blade and the induced velocity is twice this value far downstream. To approximate this far-downstream effect

AtJ

we add (1 + sin 8)

"1F'.S

_{to the aft}

orbit values.

The local component of the velocity is that portion of the induced velocity from the blade's own shed-vortex sheet not included in the above mean value. Local and mean velocities can be

extracted from some configurations (such as cascades and propellers in uniform f low) but here we cannot easily do this and instead we make an approximation that the local effect is a fraction of what would occur if the blade were in local mean flow with only a change in direction according to the induced an-gle-of-attack from an elliptically-load-ed wing for which the inducelliptically-load-ed angle is

a1

_{= - S.}

A

(24)

where is the mean lift coefficient and A is the blade aspect ratio. This value is adjusted by a factor

F=l/Z

(25)

where Z is the number of blades, and added to the other components of the total effective angle-of-attack. Note that F produces the full effect of in-duced angle for one blade and that the local induced angle disappears for a large number of blades (corresponding to an actuator disk).

The flow is assumed to be indepen-dent of span and hence the lifting-line model of the flow field gives the over-all local lift as

L(0) =½pIJ2c01L Ctm(c1N -a -a0-Fa1) (26)

where C2u,1172, the lift-curve slope.

is a lift reduction due to thickness as described by Hoerner (1965): e (1- t/c) is a reduction in lift due to planform correction (- 0.9 for

rectangular planforms).

L0m is the span of the blade (from j the mean chord

the mounting disk or hull to the tip)

da c

a = __.!(l/2

-(27)

dO R0

is the change in angle due to pitch motion

2.16

a0 = - 2(_) K

Cm

is the angle of zero lift (K is viscous correction) and the equivalent camber ratio is

ft geo Cm 0m 1 - 1 -( (29) Cm 'c1iRVAtURE

1dac

-

-s

_{- /(XV1/V0)} B dO R0

In the above equations, a is due to time-varying pitch angles as is the last term of the modified camber ratio, and the middle term of the modified camber ratio is due to the radius of curvature along the flight path. The variable s is the spindle-axis location as a frac-tion of chord, measured from the leading edge.

The overall drag coefficient (non-dime.nsionalized by 1/2 pIg ICmL) is giv-en by (see Hoerner, 1965):

=

_{2Cf+!+1OOIrl(I+S2),}

(30) c

{cjj irA

where Cf is the friction coefficient on one side of a flat plat at the appropri-ate Reynolds number (derived from the ITTC friction line), and óCL = CL - _Cu is the change from the lift at the ideal angle-of-attack. The first term is the component of viscous and form drag act-ing on a blade section and the last term is the three-dimensional component of induced drag for an elliptic circulation distribution. When the lift coefficient exceeds the stall limit, a reduction in lift, linear in increasing effective angle-of-attack, is specified and then equations given by Hoerner (2.965) for separated flow are used. The stall

limit is here taken as a difference in lift coefficient of 1.5 from the value at the ideal angle-of-attack (which is

an arbitrary choice and subsequent eval-uations indicate a greater value may be

appropriate). A hysteresis effect (for angles-of-attack exiting separated re-gimes) is included by limiting the lift to 2/3 the LCL for the full separation value. This trend is suggested by the data reviewed by 11cCrosky (1982).

These lift and drag components are assumed to act normal and parallel to the resultant onset flow at aF (from equation 8) respectively and can be resolved into components in other

tions. The local thrust and torque elements are given by

CK

[SsinaF-CDc05aF)

.z1X2

V0 4

cK=[Ssin(aF-O)

-c.,COS(aF-9)]

2

..

ZLX.-V0 8 R0

and are summed over the orbit to find the net thrust and torque coefficejnt. Additional frictional drag and torque on the rotating plate can be included in the summed value if appropriate (most model-scale experimental data have it removed but it is present-full scale).

The mean value of induced velocity according to equation (23) and the nor-mal component of load resolved from _CL and C0 (equations 26 and 30) are includ-ed in Figures 13 and 14, respectively and show relatively good agreement ex-cept for a phase shift in time or 0. Other application of the analysis will be described in subsequent paragraphs.

To illustrate the various compo-nents of the loads we present several possible computations of the local

thrust value in the top half of Figure

19. _{This application is for the DTMB}

blades at e = 0.8 and X = 0.6336, a case for which some data are presented by Mendenha].l and Spangler (1973). First, it is possible to predict the performance of the foil as a two-dimen-sional section in the onset flow and angle-of-attack given by equations (8) and (9). _{The lift coefficient is given} by

lift

= CLO(t.,CIN - a0)

(32)

where a0 is the angle of zero lift

(approximately -2_K

with K is an empirical correctiox equarto about 0.8 for parabolic arc meanlings) and C is the lift curve slope equal to 2r,j1. The drag coefficient is drag

l2

- PVR Cm 2 = 2Cf [1 + + l00(.!)4](l

Another flow representation is that (31) of a lifting line in planar flow with

elliptic load (see van Dyke, 1972), for which the lift-curve slope is replaced

by:

2.17

These values can be resolved into the thrust and torque direction using equa-tions (31) with VR in place of

(This is the same procedure used for the data in Figure 11). Results for the present application are given in Figure

19.

CLa=

* _in +A,re918)

[

A

,T2A2

This correction for aspect ratio is shown by van Dyke (1972) to give quite accurate values for three-dimensional wings and captures both the induced angle-of-attack and flow curvature across the blade. A drag coefficient identical with that given by equation

(30), but with the lift coefficient just derived for three-dimensional flow is used.

The top part of Figure 19 shows these two approximations for the local thrust component from an individual blade as well as the final value using the lift and drag from equations (26)

and (30) and a computation without the induced velocities from an actuator disk. These data show an expected re-duction in load from two-dimensional

flow to a three-dimensional lifting-line model and then a further modification for the various other components includ-ed in equations (26) and (30), which in this case is mostly the induced velocity from the actuator disk.

In the lower part of Figure 19, we sum the components of both blades and compare the time-varying values with those predicted by Nendenhall and Spangler (1973) who explicitly consider the time-marching aspects of tracking a shed-vortex sheet. Here again, the data show no anomalous effects associated with crossing the vortex sheet (but Mendenhall and Spangler do report some irregularity in their calculated instan-taneous thrust coefficient at reduced

advance coefficients). We again con-clude there are minor effects associated with vortex-sheet crossing, either the

sheet from the reference blade itself or those from the other blades. For these cases, the stall limit has not been exceeded. Our total-load prediction is considerably less than the measured value for this advance coefficient as will be described further in subsequent paragraphs.

(34)

(33)

I

0

Fig. 19 Load Variations Computed for Several Field Representations There is another force component of interest and that is the side force. Our calculation of this value is not in acceptable correlation with experimental data, which in itself is not consistent as discussed by Menderthall and Spangler (1973). They present some investigation to document the sensitivity of this quantity and hence we do not further discuss this compoflent herein.

Cavitation Inceoticr

Cavitation is the macroscopic mani-festation of rupturing multiple weak spots in the fluid. Hence, both a re-duced pressure (typically a value close to the vapor pressure) and sufficient number of weak spots or nuclei must be present for this rupture to occur. In general the minimum pressure will be near a boundary (as required by the Laplace equation for potential flow) and consists of a mean plus turbulent fluc-tuating component (several real-flow possibilities are described b Huang and Peterson, 1976). The types of cavita-tion of concern here are blade surface (leading-edge sheet and bubble) as wel]. as trailing-vortex cavitation. For trailing-vortex cavitation, the minimum pressure may be in a vortex core that is some distance downstream from the foil tip. The cycloidal propeller operates in a time-varying manner that is intrin-sic to its ability to develop lift and hence the cavitation is unsteady.

Although there may be certain situ-ations that have a population of nuclei

2-IS

reduced sufficiently to cause a major difference between the local vapor pres-sure and the prespres-sure at inception, such cases are rare in engineering practice. A situation of greater concern is the determination of the local minimum pres-sure along or near the body of interest resulting from the complex viscous flow patterns that may occur on the foil surface (see Acosta, 1979 for a histori-cal perspective). _{For applications at a} Reynolds number greater than one mil-lion, the flow is generally fully turbu-lent and anomalous separation is not expected. In this case, when there is fully attached flow on the body, a cal-culation of the steady minimum pressure on the blade surface is easily computed from a potential-flow model and turbu-lent fluctuation are not of great con-cern. If the flow is not fully

attached, the minimum pressure is more difficult to predict. Since the cycloi-dal propeller blades can experience

large angles of attack (see Figure 10), we expect some separation on the blades. Our approach for blade-surface cavita-tion is as follows:

For those cases for which the angle-of-attack is less than the stall limit, a quasi-steady pressure distribution is determined on the blade section using the actual thickness and modified camber and local angle-of-attack at the spindle axis as previously described (equations 26 and

29). The minimum pressure will be used to assess the possibility of Cavitation. For conditions where separated flow over the blade is expect-ed, the minimum pressure is limited to the minimum pres-sure that is predicted for the orientation just prior to the occurrence of separation. The remaining form of cavitation is tip vortex. In this case, a vortex is formed at the tip as the fluid migrates from the pressure side to the suction side arid is convected downstream. _The structure of this flow is complex and has resisted development of a rational model for computing flow-field details. The inception of this type of cavitation

is predicted with an empirical equation derived by fitting an expression con-taining variables expected to be in-volved to available experimental data

for conventional propellers.

Blade-Surface Cavitation. An as-sessment of the cavitation performance of a cycloiclal-propu].sor blade can be made with a knowledge of the minimum pressure on the blade surface as a func-tion of orbit posifunc-tion. When this mini-mum pressure falls below the vapor

pres-sure of

the

liquid, cavitation is as-sumed to occur. For the minimum pres-sure coefficient at a fixed orbit posi-tion defined with the expression:

2 11 2

I i.1

(35)

V0

and operating cavitation number defined

by

o ₍₃₆₎

1/2 pV

where p0 is the static pressure at mid-span, one expects cavitation to be

pres-ent when

or

-

C ₍₃₇₎

Although

there may be significant scale effects associated with this inequality, a knowledge of the minimum pressure coefficient is believed sufficient to estimate the boundary between cavitatirig and non-eavitating flow at high values

of the Reynolds number.

The minimum pressure coefficient

can be computed for several (spanwise)

blade sections at the local

angle-of-attack and the least value defined.

Here we select only the mean thickness

and

mean

span point to characterize the

blade but could just as easily compute

the minimum pressure over several blade

sections.

Or

this representative blade

section there are three likely positions

of the minimum pressure coefficient: one

near mid

chord

on the upper, outside, or

suction side of the blade, a second near

the

leading edge on

the

same side, and

the third near

the

leading edge on the

other side of the blade (see Figure 20).

A fourth possible point is near

mid-chord on the pressure side.

_{If the}

section were an ellipse, the position of

the

maximum

speed is given by the

im-plicit equation (see Brockett, 1965):

r2/tan e -CpImth = I I. ½pv, _I IVR tan =

cos

(1 -

r2) - 1

where ₌ cos' (2x' - 1), r is the thi-ckness to chord ratio and a0 is the

ef-fective angle-of-attack. This position is hardly changed if the meanuine were one of the NACA a series meanlines with constant load up to at least mid chord. Once this equation is solved for the three possible maximum speed values

_(Ø,

I

= 1,2,3),

_{the three surface speed}

values are calculated

(38)

2-19

2

1, 4 pO 2, 3 4: UTS

Fig. 20 Points of Maximum Surface

Speed on Blade Section

=I1

= (1+1-)

5incosa0

_{(1-cos) sina0}

(39)

+

r2cos21

+ 4.09 ..!

c

where the last term

is the

velocity component for the NACA a series meanline

with a = 0.8 the subscript e denotes

the equivalent value,

_{; includes a}

correction for

the

ideal angle-of-attack

(a

=

_{1.54 (fm/Cui,)0/0.0679 in}

degrees

for

the NACA a

=

_{0.8 meanline) and the}

equa-tion is valid for the forward "a"

chord

positions.

For blade sections that do not have elliptic cross sections, the thickness is increased by 15 percent for position 1 and the leading-edge radius is used instead of r for positions 2 and

3 in

Figure 20

(the

equivalent

thickness

ratio is twice the leading-edge radius

non-dimensionalized by the chord). These

choices are in keeping with

_the

ideas proposed by Breslin and Laridweber

(1961).

The

three extreinum values of speed

are compared and the maximum value of

surface Speed selected at each blade-orbit position and the value and blade-orbit location of the minimum pressure coeff 1-cient determined. The minimum values generally are on the suction (or out-side) near 8 = 90° for the forward orbit arc and on the "pressure" (or inside)

near 8 = 270° for the aft orbit arc (see Figure 15).

Tb-Vortex Cavitation. For tip-vortex cavitation inception, an empir-ical fit to available propeller data

employing flow parameters believed to be

important to the cavitation process on

marne

_{propellers has}

been

defined by

Cox to be

4Private communicatjon with

Dr. Bruce _Cox

.8

.4

-7.0(G09) 2

_(c/R0)

0.4 (40) 1 _pvoz where (41) D0V0 V0 R0 4

is to be taken at the 0.9 span position which is 0.436 of the iaaxiinuin value for a siru1sgidal circulation variation, and V / v is the local Reynolds

num-ber. This equation is defined at each angular position of the evaluation and the greatest value used to define the initial inception. This approximate equation is believed "best used to com-pare various propulsor over a range of Reynolds Numbers near that of the under-lying data. Souders and Platzer (1981) show that local modifications at the tip can change the predicted value by a factor of 2.

HYDROACOUSTIC PERFORNA2CE

- The noise levels near the top speed

of a vessel are expected to be dominated by cavitation sources (Stuurman, 1974). This cavitation noise arises from the collapse of the ruptured voids. For cavitation, the noise is modelled with the collapse of a single isolated bubble

(see, e.g; ; Fitzpatrick -and- Strasberg, 1956). Several investigations have been undertaken to quantify the noise associ-ated with individual types of cavitation (e.g., Noordzij, et al, 1977, Bark and Berlekom, 1978, and Blake, 1986). The general trend is that the least noisy type of cavitation is trailing vortex and the most noisy is bubble or cloud cavitation. The noise-levels of iridi-vid'1al types of cavitation are not con-sistently defined in the various refer-ences and hence there is some uncertain-ty in their use in a prediction proce-dure. Blake (1986) presents nondimen-sionalized spectra in an appealing man-ner and here we use his spectra to pre-dict the cavitation noise for a cycloi-dal propeller. Alternatives are predic-tions based on more global parameters such as proposed by Brown (1976) and his coworkers (Abbott, et al, 1987). When cavitation occurs in a flow field that is unsteady, the noise has a tendency to increase, especially when sheet cavita-tion becomes unstable and transicavita-tions to cloud cavitation. This instability is believed highly likely to occur in the present case where cavitation changes from side to side during one revolution and, hence, at least at a condition with extensive cavitation, we expect that additional noise will occur over that predicted by the levels derived from Blake's data for steady flow. Here we taJce this additional level to be 20 dB, similar to the change in the

nondi-2.20

mersional noise levels produced when bubble cavitation is present as cited in the previous references. We expect, the sharp collapse of cavitation volume caused by the blade oscillations to also produce high-amplitude blade-rate compo-nents of the noise, even close to the inception condition.

""Noise levels' contain-such small power levels that its importance is not in any significant energy balance but rather in how it is perceived by a sen-sor, whether that sensor be the human ear or an objective transducer. As is traditional, noise is defined on a laga-rithntic scale based on a reference level in units called a decibel:

I

L = 10 log10 - (42)

'ref

where I is the intensity of the signal of interest and 'ref is a reference level in the same units. In addition to a level, the frequency spectrum is of great concern for noise since again the sensor perceives a frequency distribu-tion in different ways. Cavitation noise may occur in both discrete-fre-quency tones and a broad-band spectrum. We are here concerned with only the broad-band spectrum and we seek to de-fine levels in one-third octave bands for

which

the bandwidth is determined frcm the upper and -lower frequency lim-its related by

= 2f

and the cen-ter frequency is ₌ . In general,

the levels in the vicinity of the pro-pulsor(s) will be spatially variable but at some distance away, it is nearly as though it were created by a point source with an intensity that varies inversely with' the square of the distance from the hypothetical source. When this intensi-ty at some distance is converted to a level at a distance of one meter from the hypothetical source, it is called a source level, L. It is this source level that we seek to predict for the cycloidal propulsor. The specifics of our procedure are as follows. The as-sumption is made that for each type of cavitation (tip vortex, and sheet both on the suction and pressure side) there

is a unique far-field spectrum that shifts in level and

frequency

with in-creasing extent of cavitation associated with increased loads beyond the

incep'-tion point. L,ovik (1978) argues that the spectrum function for cavitation

should be of th& form' shown in Figure

21. For a

one-third

octave band analy-ses we take the region below c1, to be a constant, and for frequency values greater than this we take the levels to decrease as the inverse of the

frequen-cy. Blake (1986) has found both the level and frequency value at c based on hydrofoil and propeller experiments. Here we reassess the tip-vortex