A theory for the quasi steady and unsteady thrust and torque of a propeller in a ship wake

ARCHIEF

I.LissJ.

\\!5.14'

Experimental Towing Tank

Stevens Institute of Technology

Hoboken, New Jersey

lab. v. Scheepsbouwkunde

Technische Hogeschool

Delft

Z/212vi/

REPORT NO. 686 July 1958

A THEORY FOR THE QUASI-STEADY AND UNSTEADY THRUST AND TORQUE

OF A PROPELLER IN A SHIP WAKE

by

PAUL D. RITGER

and JOHN P. BRESLIN

EXPERIMENTAL TOWING TANK

STEVENS INSTITUTE OF TECHNOLOGY

HOBOKEN , NEW JERSEY

A THEORY. FOR THE QUASI-STEADY AND UNSTEADY .THRUST AND TORQUE OF A PROPELLER IN A SHIP WAKE.

by

Paul D. Ritger and

John P. Breslin

PREPARED UNDER

'SPONSORSHIP OF THE BUREAU OF SHIPS FUNDAMENTAL HYDROMECHANICS RESEARCH PROGRAM TECHNICALLY ADMINISTERED BY THE DAVID.W. TAYLOR MODEL BASIN

UNDER CONTRACT Nonr, 263(16) (E.T.T. PROJECT No. HZ 1863)

R-686 TABLE OF CONTENTS Page Abstract 11 Nomenclature Introduction .. 1

Statement of the Problem 3

Solution by the Strip Technique 5

Quasi-Steady Case ... . . .. .. -

. .... .

. S

Correction for Unsteadiness . . 9

Determination of Effective Angles of Attack 12

Total Thrust of a Blade Section 15

Computation of Lj(2) 15

Computation of 14j(3)

. ... .

. 17

Computation of Lj(4) 18

Expressions for Total Thrust and Torque 19

Fluctuating Thrust and Torque for a Single-Screw Ship

Calculation of Contribution T(k)(t) to Total Thrust . . . 23

Total Thrust and Torque

.... .

. . 24

Discussion of Results 26

Conclusions and Recommendations . 29

Acknowledgements . .

. ... .

...

30 References 31 Tables 33 Figures 43 Appendix I 49 Appendix II 52

ABSTRACT

This report provides the results of an analysis in which the unsteady aerodynamic theory is utilized to compute vibra-tory thrusts and torques developed because of the circumferen-tial inequalities in the wake flow produced by s. propeller.

A comparison is made with the quasi-steady theory now in use by some -naval architects by applying both the unsteady and quasi-.steady theories to a specific example of a single-screw ship for which the model wake distribution is available. The

peak-to-peak thrust. variation at blade frequency is found to be 34.pexcent of the small thrust when the quasi-steady theory is used, but the unsteady theory, as applied herein, predicts a variation of only 13 percent of the mean thrust. Three questions which may be raised are answered in order to give insight into the main effects predicted by the unsteady theory

and also to point out some of the expected shortcomings of

the analysis.. Recommendations are given for extensions of

this work and for experiments to secure data for correlation with the theoretical results stated herein.

an,bn A Altri,Bwn crodn CL,CD f(w) Fr,Fi G(w4) _{G( )= cFr(w)} I

see Equations (79) and (80)

, , ,Y

o

Eq Pqg

subscript designating j11-1 blade

KE,K0 Goldstein correction factors

Kgs,Ks,rb

three-dimensional flow corrections

Ko,K/

LI,L2

L(k)

rn number .designating. harmonic at multiple

of

blade frequency

M.(x torque of section of

j" blade

J '

number designating harmonics point of blade

P(x) B7rpcRU2.KgsKr.oKs cos

Qs quasi-steady torque, from unpublished

cal-zulations by A.J. Tachmindji

Qj(t).

torque of

NOMENCLATURE

axial inflow velocity correction

tangential inflow velocity correction Fourier coefficients of Wc/U

point in plane of propeller

harmonic coefficients of total thrust number of blades

length of chord

Fourier coefficients of Ws/U lift and drag coefficients

fici)EK0(iw) +

K1(ic4j1-1,

generalized Theo-dorsen function

real and imaginary parts of F(w)

modified Bessel functions of second kind

lift

quasi-steady lifts (Appendix I) contribution to lift

jth blade

8-686.

-Q(t) total torque

Qj(x,t) torque of j.!11- blade section

Q(x,t) torque at radial distance x

radial distance from center of propeller

1.0 hub radius.

one half the tip diameter of propeller Re,Im real and imaginary parts, respectively'

distance along chord of blade section h

f jt

sj distance along chord o blade section

Sn tq

T(t)

T*(k)(t) Ti(t) Tj(x,t) T.(k)(1c,t) T(k)(t) T(k)(x,'t) see Equation (65) time

(w/1809)(10°/n) q, time for propeller to ro-tate 10q degrees

quasi-steady thrust from unpublished calcula-tions by A.J. Tachmindji

quasi-steady thrust

contribution to quasi-steady thrust thrust of j.-t-A blade

thrust of Wl.. blade section contribution to thrust

contribution to total thrust, see Equation

(74)

contribution to thrust at radial distance x, see Equation (70)

total thrust

total thrust at radial position x

1

resultant velocity [(rn)2 + V2_111

U' resultant stream velocity (Figure 3)

U1 ,U2 resultant velocities (Appendix I)

V forward speed of ship

Vc velocity component perpendicular to airfo 1

Vs velocity component parallel to airfoil

cross_veloci.ty

Wc

mean value of Wo, Ws

Wo,Ws

WL longitudinal component of W

gust velocity factor

Wo

WR radial component of W

Ws "surge velocity'

WT _- tangential component of W

x r/R, dimensionless radial position.

xo r0/R, dimensionless hub radius

co-ordinate in direction of fore-aft lime of

Ship

a angle of attack

ai,a2 angles of attack (Appendix I)

a* - , angle of attack of U

a.(k) contribution to 'effective' angle of attack

8i 27T(j-1)/Bn

angular displacement of dth blade

7 tan-l(CD/CL)

X tan-1(V/rn) (Figure 3)

variation index _[(Tmaz - Tmio)/Tsvs]x 100

(in percent)

circular frequency of gust reduced frequency vc/.2U

reduced frequency nf2c/2u

wn

angular speed of propeller

0 angle between propeller plane and U' (Figure 3)

tan-1 [(V-wL)/(r _-WT)] (Figure 3) mass density of sea water

a solidity factor Bc/27TxR

0 geometrical pitch angle (Figure 3)

angle between propeller plane and zero lift 6o

line (Figure 3)

R-686

-R-6&6

- vi.

-angular displacement from center-line of ith blade

INTRODUCTION

AS a ship moves.it disturbs the surrounding water and pro-duces a wake. The water velocity in this wake varies

in

both magnitude and direction at different positions in the plane of the propeller, and, hence, the flow relative to the propeller

blades produces periodic variations in velocity and angle of

attack. As.a result of this circumferential inequality

in

the

flow, some highly-powered ships experience troublesome thrust variations with attending vibratory effects on the hull and

machinery. Circumferential non-uniformity generally is most

. pronounced on single-screw ships and, lately, has become more

important since ships are being built withever-increasing power. In recent years some naval architects have been calculating

the, vibratory thrust of single-screw ships on the assumption

that the lift and, hence, the thrust

and

torque are developed by the propeller blade sections as though the flow were steady. Such quasi-steady calculations (whichare.generallypessimistic, ,give an over-estimate of the Vibratory thrust) are used to determine. whether.or not objectionable thrust fluctuations are to be expected in the proposed design. Although no exact specification is known to the authors, it is generally believed that serious vibration may be expected for thrust fluctuations (double amplitude or peak-to-peak differences) in excess of ten' percent of the mean thrust..

Results of the study reported herein provide a method for computing thrust and torque which takes into account the

fact

that flows about the blade sections of

a-propeller are unsteady.

This is accomplished by utilizing (in strip technique) the two-dimensional theory of airfoil

sections mOving

rectilinearly .through sinusoidal gusts.: A general formula is derived and applied to a representative single-screw. ship,. Numerical re-sults are obtained for the thrust and torque variation

by

use

of both the quasisteady theory and unsteady. flow theory.

It is of interest to note here one simple but important

feature of the .analysis presented in.this.report.: . Since the

unsteady lift theory enables: one

to

compute: the section lift

in

a sinusoidally Varying flow, .and since the wake flow may be

R-686

-1-R-656

- 2

-simply regarded:as:a sumof.sinusOids of all frequencies, the response of each section-to:each:harmonic of the wake can be determined,

:How-ever, upon summing-up over Bblades, the ofilyterinswhioh.reinain art. those.which.contain_wakelarmonics of blade frequency and multiples

thereof. In other words, the .vibratory thrusts arise from only part Of the wake produced :angle,ofattack flOctuatiOns, :and, in particular, only from those .constituents at blade frequency and integral multiples

of that irequency. A simple proof of this fact is given in Appendix II.

Furthermore, the unsteady theory shows.that the unsteadiness

only modifies the angle of:attack by.a function of the blade frequency,_

thus ;accounting for fluctuations in flow.both.normal'to and. along the

chord: Ofthe,blade,section. Results for the. entire propeller are obtained;by numerical.integratioh.over.the_blade.span: Numerical re, SUlts:for a, particularwake show that the.inclusion.of:the effects .Of unsteadiness produces,a,vibratorY thrust of less than half: that found

by the quasi-steady theory:

-This work, conducted at the Experimental Towing Tank, -Stevens Institute of Technology (E.T.t,Project HZ 1863), has been .sponsored by the Bureau. of Ships Fundamental-Hydromechanics Research Program and technically:administered:by the Ship Division of the David Taylor Model

Basin under Contract Nonr 263(10,

*An interesting quasi-steady analysis which came to the authors' attention after completion of the final draft of this report-has .been'

presented by Schuster, S. and Yalinski, E.A.: 'Beitrag zur Analyse des Propellerkraftfeldes', Schiffstechnik HEFT 23, Band 4, Septem-ber 1957.

STATEMENT OF THE PROBLEM

In order to visualize the problem under consideration,

imagine a ship, without a propeller, coasting along at a con-stant speed V . At any point in the water the wake velocity,

, (measured with respect to the undisturbed water) can be

expressed as a vector. In particular, consider the wake at a point A in the plane of the propeller as shown in Figure 1. Let r be the radial distance of A from the center of the

pro-peller and the angular position of A , measured from the

ver-tical (positive counter-clockwise looking forward). At A , the wake velocity

(see Figure 2) has the com-ponents WL, WT, and WR where WL is the longitudinal compo-nent (positive forward), WT is

the tangential component

(pos-itive in the clockwise

direc-tion) and WR is the radial

component. Note that WT and

WR are in the plane of the

propeller and WL is perpendic-ular to this plane.

Now consider the point A

as a point on the propeller, rotating clockwise at an

angu-lar speed U . Thus, with re- WAKE VELOCITY COMFONENTS spect to the surrounding water,

A has a tangential speed 141 - W1 , a longitudinal speed V - WL

and a radial speed -WR.

Let the propeller have B-blades and a radius of R. Consider

a section of one of the blades at a distance r from the center. The effect of the radial wake, WR , is neglected and it is

as-sumed that the blade section then behaves like an airfoil in a two-dimensional flow. The lift of the section now can be

com-puted as a function of r and time, t . The approximate thrust

and torque of the blade then can be determined by integration over r .

FIGURE 1

DEFINITION OF CO-ORDINATES

FIGURE 2

Quasi-steady corrections for the three-dimensional effects such as induced inflow velocities, cascade effect, etc. must

be made.

R-686

-3-fi-686 - 4

Thus, the problem is to determine the lift of a blade

sec-tion as a function of r and t , and from this to deduce the

SOLUTION BY THE STRIP TECHNIQUE

As is customary in approximate wing theory, the thrust and torque of the propeller will be computed by use of the so-called strip method which providesa way of introducing the two-dimen-sional behavior of a section into a flow which is

three-dimen-sional. The total thrust and torque then may be considered to

be constructed in the following fashion. Let T(t) and Q(t) be the total thrust and torque, respectively, of the propeller at

any time t. If. T(t) and Qj(t) are the thrust and torque of

theiL/1 blade of a B-bladed propeller, then

T(t)

T(t)

j=1

and

Q(t) =

Q(t)

j=1

In order to obtain Tj(t) and Q(t) the thrust, Tj(x,t), and the torque, Qj(x,t) , for a cross section of the blade at

any radial distance, x = r/R, must be determined. The flow at this section is assumed to be equivalent to that about a two-dimensional section at the same hydrodynamic angle of attack.

Thus 1

T(t)

=

Jr

T.(x,t)dx xo and (2) 1

Q(t) =

jr

mx,t)dx

xo

Before determining T. Qj for the general case of un-steady motion the results for the quasi-steady case will be

reviewed.

QUASISTEADY CASE

On the assumption that a steady flow exists about a blade

section at any instant, the following expressions for Tj(x,t) and Q.(x,t) may be adapted from Burrilll:

Ti(x,t) = 47rxR2pK a(V - W )2(1 + K )/B

and (3)

Qi(x,t) = 47Tx2R3PKE a' (V - W

)(ail -

W1)(1 +aK )/B ,

(I)

R-686

-5-B-686

- 6

-where p is the density, KE and Ko are the "Goldstein correction factors", a is the axial inflow velocity correction, and a' is the tangential inflow velocity correction.

In terms of lift and drag coefficients, (CL and CD,

re-spectively), Burrilll gives

Tj(x,t) = fpcRU'2(CL cos 0 - CD sin

0)

and ( 4 )

c(x,t) = fpxR2U'2(CD cos 0 + CL sin 0)

where c is the chord length of the blade section and

0

is the

angle between the resultant stream velocity U' and the plane of

the propeller as shown in Figure 3. Figure 3 shows that

N's a (V -WL) -a' (xRct-W ) V -WL xRS2 -WT Zero-lift line FIGURE 3 V

U' = (V - WL)(1 + a) csc 0 , (5)

and, upon substituting Equation (5) in Equation (4)':

Ti(x,t) = fpcR(V -. WL)2 (1 + a)2 csc2 (Ci. cos 0 - CD sin

0)

and (6)

())(x, = fpcxR2(V - WL )2(1 + a)2 csc2 0 (CD cos 95 + CL sin 95)

On substituting Equation (7) in Equation (3) one obtains:

47rxR2pK6 aCL _{cos(0 + Y)} -and Ql However, (V - W) (l + a 4mi2R3pK6 aCL x,t)

-an.- w

- a') B _{21(6 2 sin2 0} cos 7

.(xin _

wT)2(1 - a' )2 tan2 0 sin( + 7) B 2K6 sin 20 cos 7

(V -

WL)(x110 - W )(1 - a')(1 + a) sin 0 sin 20 = tan 0

-cos 0

sin 20

= sec2 sin 295 = sec 0 sin 0 , (10)

and, hence, Equation (9) becomes

(9) Now, by combining Equations (3) and (6)

+aK4_ CLa

a cos -(0 + y)

1 + A 1 + ) 2K6 _{2sin20 cos 7}

and

a, 1

+ aK4_

Cca

sin (0 4. 7)

(7)

1 - a 1 + a ) 2K6 sin -20 cos y

a = Bc/27rxR .

where 7 =

_tan-"(CD/CL)

and, the "solidity", 0 , is given by

(8)

R-686

-7-R-686 - 8 -Tj(x,t) = 7rxR 1)(xRCI - WT)2(l - at.)240-C and T.J '(x t) sec2

0

cos

(0 + y)

----rrT;TT

1

Q j(-x, t) = 7Tx2R3P(xRn - W ) (1 - 1)2c-CLsec2 sin (0 +7) B cos y B cos Y (11). However,

ocRn

-woci -

a') = U' cos 0 (12)

and hence, Equation (11) can be written in the form cos (0 + y)

= wxR2paCLU'2

Qj(x!t) = xR tan (0 + y) Tj(x,t)

The lift coefficient CL is expressed by Burrilll in terms of the angle of attack,

a

,

as follows

CL = _27T0.Kg sKsKrb (14)

The correction coefficients Kg s , Ks , and

Krb

depend, in

general, on a , , and the shape of the blade section as shoWn

by Burrilll.

Substituting Equations (14) and (8) into (13) results

T cos (0 + y) . (x t) =

wpcRU12aKgsKs

j , _Krb cos y and (15) Qj(x,t) = xR tan (0 + y) Tj(x,t)

If

a is

small, this result can be interpreted as the thrust

and torque due to 'a parallel flow U' and crossflow

Usa.

Since y = t

n-l(cD/co

, y usually

will

be small (see Table

An unpublished method employed by Tachmindji at DTMB cor-responds to Equation (11) except that the term (an - Wi)2 is replaced by (xRn)2. This difference may account for the diS-crepancy between the result of the DTMB 'calculation and the

quasi-steady result of this paper (see Figure 12)- Another factor might be the DTMB method of smoothing off the peaks in the wake profile.

IV)*-- Consequently, thesilbsequentdiscussion can.be siMplified by assuming that cos 7 1,. In addition,'since.the angle -0 +

does not vary greatly.fromthe angle X..(see Table IV), cos -(0 4 7).

may be replaced bycos X in Equation (15). A significant sim-plification results since X is independent of the

blade'posi-tion. _{Computations were carried out} to check the effect of

this approximation nd it was found that the values of Ti(x,t) were not materially affected, However, an error of possibly

seven percent wasintroduced

into, the value of Qi(x,t) when

tan (0 +:7) was repladed by tan, X.'

With these simplifications, Equation (15) is replaced by

TI(xt) -7.1-rpc1W24KgsicsKrb

cos

X (16)

and

Qj(x,t) = !c2- Ti(x,t) (17)

CORRECTION FOR UNSTEADINESS

It 'is known that when a foil encounters a single cross-gust (or perturbation

in

the angle of the relative flow) the lift changes, and a free Vortex is shed simultaneously atthe

trailing. edge.

The circulation strength of

this Sled vortex

is opposite and equal to the circulation change which takes place about the foil. The primary effect of this

Vortex

is to

induce a crossflow against the foil- which acts to mitigate

the angle of attack produced by the gust alone- When a

foil

encounters acontinuously varying gust pattern, it Sheds a con-tinuously varying free-vortex sheet which produces a-varying correction to the instantaneous gust-angle of attack. This

problem has.been studied

thoroughly

by aerOdyhamiciSts, partic-ularly in the regime of speeds for which the air flows can be regarded

as

incompressible. It

is

therefore possibla to exploit this work in order to obtain a correction (to the quasi-steady analysis) which will account for the effects of .unsteady flow. Thus, in this report, all the effects of unsteadiness are assumed

to be generated by the blade elements which are taken to be

equivalent to foil sections moving rectilinearly through

4

flow

field providing harmonically varying flows perpendicular to and along the section.

All tables are numbered consecutively starting on page 3

Fls 686

-9-R-686 10

-It must be noted here that this approach to the action of the flow on the section differs from that taken by Timman and van Manen whose works were utilized by van Lammeren2. Timman and van Manen assumed that the flow about a section traversing a varying wake was equivalent to that generated by a section performing heaving and pitching motions while moving through a calm fluid. The two approaches give about the same results for low frequencies of encounter or for long "wave lengths" of the flow oscillation, but do not agree at high frequencies. The

difference stems from the fact that the boundary condition on a foil moving through a gust is basically different from that of a foil performing heaving and pitching oscillations. This

latter motion does not apply

to the rigid propeller motion

assumed here. If one were to include elastic motions of the

propeller section, then it would be necessary to construct a solution for a section which moves through a varying flow and

at the same time performs heaving and pitching oscillations. It appears reasonable at this time to consider the blade

sec-tions as being driven rigidly througlya spatially variable flow field and, therefore, to take the results of studies of foils

in sinusoidal gusts and apply them strip-wise along the span

of the propeller blade.

Sears3 has shown that the lift, L , on an airfoil in a

stream with a constant parallel velocity, V

= U

, and

var-iable cross-velocity,

Vo = U Re[Woeild"

s")]

is given by

= 77-pcR U2 Re[Woel tF(w)] (18)

where Wo is a real constant, w is the reduced frequency w = vc/2U

and the function F(w) is shown by Kemp4 to be

F(w) 1 (19)

iw[Ko(iw) + KI(iw)]

Ko and K1 being modified Bessel functions of the second kind.

for a _{"lifting line" or "chordless" foil representation as} currently employed in the quasi-steady calculations, a cross-velocity of the form

The incident angle of attack, a , in both the unsteady,

and the steady case is given by

a = tan71(V0/V ) .(21) R-686 Vc = U Re[!seivt] produces a lift L = rrpcRU2 given Re[Vicel't] by (20)

Comparison of Equation (20) with (18) shows that the lift ex-perienced by the foil of finite chord in a crossflow which varies along the chord as U - Re[Wsei'(t-s")] is equivalent

to the lift on a "lifting line" at s = 0, modified by the fre-quency-dependent function F(w) . Thus, F(w) represents the

correction factor arising from the effects of unsteadiness and the chord-wise distribution of the flow over the section.

In the linear approximation a = Vc/Vs = Re[w

a = Re(Wceilit).

err.v" - still] (unsteady)

(steady)

(22) (23)

where it is to be noted that the angle a is found only for the chord mid-point, s = 0,

in

the steady flow case.:

, Finally, .upon comparing:, Equations (23), (20) and (18),

F(W) can be interpreted as a-change in the effective

angle Of

attack due to 0-Steadiness,

In other words, the effective a used in computing the lift is

R-686

-12-Vs = U cos a* _-. Ws (30)

and

sin a* + Wc

(al)

DETERMINATION OF EFFECTIVE ANGLES. OF ATTACK.

In the case of the propeller, the resultant flow encount-ered by a blade section (after inflow corrections) consists of an incoming flow of velocity U' at an angle a with respect to

the zero-lift line of the section (see Figure 3 on page 6). That is, the velocity of the parallel flow is

Vs = U' cos a

(2)

and the velocity of the perpendicular flow is

Vc = U' sin a (26)

In order to utilize Sears result, it is necessary to

seperate the fixed

and variable parts of

the velocities in Equations (25) and (26). This can be accomplished as.follows:

=

U'

cos (00 - 0) (21)

or, upon expanding, and applying Equations (5) and (12),

Vs = [x110 cos 00 V sin ea] [{WL - a" - WO} sin 00

fWT as(xRn

w-01

cos 0] (28)

Similarly, the cross-velocity, Vc , can be written

Vc = [xRn sin 00 - V cos 00] [fwi - a(V - W ) cos 00

+ a.(xRn - WT)} sin 00] (29) + - WT) sin 00(33) where and Ws Wc = a(V - W WL - "V

-w

) sin cos 00 page

If one now introduces the angle a* = 13) and the perturbation velocities

00

Ws

X (see Figure and Wc , then

4,

Thus, the total surge velocity, Vs , is seperated into two

parts, U cos a* which is independent of time and -Ws which is the variation in the surge velocity produced by the variable

wake and inflow.Nelocities. Similarily, Wc contains that part

of the cross-velocity, Vc , which depends on time as shown in

Figure 4.

FIGURE 4

COMPONENTS OF CROSS AND SURGE VELOCITIES

In general, the angle a5 will be small enough to justify a linear approximation (see Table I) and, thus, Equations (30) and (31) can be simplified by the assumptions

R-686 13

R-686 14

-sin 80 = sin (k +

a*)

= sin k(1

+a

cot X + "-sin X , (35)* and

cos

80 = cos (X + a*) = cos X(1 -a*

t n +-..) "-cos X (3.6)* Thus, the parallel (or surge) velocity and the cross-vel-ocity are expressible in the forms:

and

Vs = U - Ws

Vc = U a* + Wc

respectively, where now

and

ws = {wL

{ wL

This approximation was made to simplify the mathematics but need not be made if one wishes to achieve greater accuracy.

cos k (39) sin X - (40) w-r + a'(xRC2

S'(xRn

-- a(V -- sin X+ -a(V -M

)1

cos

-TOTAL THRUST OF A BLADE SECTION

Wc/U = [an(x) cos n

+ b(x) sin n

] (43)

Consider a point, P , on the j01 blade at a distance sj

behindthemidpoint,(s.is measured along the

chord; sae

R-686 15

-Based on the work described shove, the total thrust, Tj(x,t),

of a blade section is the sum of .four component thrusts

pro-duce4 by the velocities Vs and Vc as given by Equations (37')

and (38). That is,

Tj(xt;) Tj(1) Ti(2)

.r.wi

4. t. (41

( 41 )

where

is due to U and Ua*

is due to U and Wc is due to -Ws and Ua*

and

T-(4) is due to -Ws and Wc .

Since U and a* are independent of time, .(1) may be

corn-puted by using Equation (16), i.e.,

T (1) = wpcBU2 a*KgsKsKrb cos. X

(42)

3

In order to compute the thrusts Tjc2),

T ()

, and

it is convenient to compute first the corresponding lifts

Li"),

Lj(3) and Li(4). For example, for T.(2) the lift L-(2) is

com-/ NI .

puted, fromwhicl the effective angle of attack ai2' is _obtained. This angle is then substituted into Equation (16) in order to

obtain the thrust Tj").

COMPUTATION OF Ljt21

In order to compute Lj(2), Sears'. result

will

be used to account for the unsteadiness of the flow.

At any point in the plane of the propeller, the cross-ve-locity, Wc, will depend on the radial position x and the angular position C (positive

in

the 'counter-clockwise direction; looking

forward). Since Wc is periodic in C , Wc/U may be written as

R-686 16

-Figure 5). The

angulai-posi-tion C now can be

deter-mined as a function of t as

follows:

sj

xRej sec 0,

(44)

where 0 is the geometric pitch angle and ej is then the angular co-ordinate of P in the plane of the

pro-peller. (See Figure 6)

In the linear approxima-tion 8 X so that Equation

(44) now can be replaced by

sj = Ufj/n . (45)

But, from Figure 6,

t = 71; + gj (46)

where nj is the angular

posi-tion of the centerline of

the j11: blade. If blade No.

1 is assumed to be vertical

when t = 0 , and if the

blades are numbered in

coun-ter-clockwise order, then

FIGURE 5

BLADE SECTION CO-ORDINATES

(43):

FIGURE 6

ANGULAR CO-ORDINATES

77j= 2-n(j - 1)/B - nt. (47)

Combining Equations (45), (46) and (47):

t = 27qj - 1)/B - n(t -

sj/u)

(48)

Substituting Equation (48) into CO

We/U=

an(x) cos ni./(t - si/U - 8j) - b( x) sin nfl(t - s j/U - 8j),

(49)

where

J = 2771j. -

1)/Tin

lows

Equation (49) can be written in exponential form as

fol-w./ (x)e-innsi einn(t - si/u))

COMPUTATION OF LI(3)

The computation of Lj(3)1 the surge effect, is complicated by the fact that thevelocity parallel to the chord varies with

time and position on the chord. However, the total surge effect is, in general, relatively small (see, e.g., Table VIII) and, hence, the authors feel that the unsteadiness effects in this case are small enough to be neglected.1 Therefore, only the quasi-steady value of Lj(3) is determined here.

It is proved in Appendix I that a parallel flow U - Ws and a crossflow Ua* produce the same quasi-steady lift in the linear approximation as a parallel flow U and a crossflow

(U - Ws)a*. Hence, the technique used in obtaining L.(2) can

be used again here, setting wn = 0 and hence F(w) E 1 to

ob-tain the quasi-steady part of Lj(3)

The surge velocity Ws can be expressed in the same form as Equation (43):

R-686.

17

-m4

..Imfbn(x)e-innai - sj/U))] (51)

Hence,' on applying Sears' method [Equations (18) and (24)]

the lift Lj(2) is given by

03

L.")

TrpcRU2 [Refan( )e-inn81 einntF w

)1

n=0

-Imp) ( )e-inn8j inntF(w ))]

where

(52)

con = rig-lc/2U (53)

The effective angle of attack .(2) becomes

03

ai(2) = [

(x)Refe"n"

8i)Nwno

n.o

11-686 -

18'.-CO

Ws/U =

riso

The uniform crossflow in this case is Ua

[co(x) cos nt d(x) sin 'IC ] (55)

snd, on applying

COMPUTATION OF

tj(4)

In order. to determine Lj(W) one is faced with variable

flows in both the parallel and the crossflows, i.e., -Ws and

Wc

However, since both these velocities are, in general,

relatively small (see Table III), it is possible to neglect all variable effects and simply determine the 'mean effect Of this part of the flow.

That is,

L.(4) . -1TpcRWi(Wo/W8) = .-77.PcRU2(Wo/U)(Wo/U) .C58), where Wc and Ws are the mean values, i.e.,

Wc/U a0.(x) and

Ws/U =c0(x)

Thus, Equation

(58)

becomes

LAO.

= -ITCRU2a (x)co(x)

and ai(4). becomes

qj

()=ao(x)-cv(x)

( 59 ) Sears' result and on setting

CO .(3) =_wpcRU2a* n=4) F(wn) = 1, one obtains fcn(x)einn(t - Si)} - Imfdn(x)einn(t - 8.0)] (56)

with the effective angle of attack, .o.i() , 'being given by

OD

a

(3)

= _a* y, _{[cn( )Re[einn(t} Si))

EXPRESSIONS FOR 'TOTAL THRUST AND TORQUE

The expressions_ Lj(2), Lk(3), Lj(4) give the lifts due to

various

parts of the flow.. That is, 1.,j(2) is the unsteady lift

due to the variation NC in the cross-velocity, 1,3(3) is the

quasi-steady lift due to the variation W. in the surge velocity,

and Li(4) is an approximation for the combined (second order)

effect. of Wc and Ws

In order to obtain the corresponding thrusts, T3(2), Tj(3) and3(41, the effacti.ve ,angles of attack

a(2)

, a.(3) and

-a-0" are substituted successively into Equation (16)', and thus the total thrust

of

an element

of

the j111 blade is given by

Ti(x,t) = (7TpcRU2KgsKsIcrb cos A.) aj (62)

where al

is

the sum Of the affective Angles of attack which were determined above. Thus

=

a.(1)

a.(2)

a.(3)

a.(4)

(63) where a.(1) = CO

a(2)

= [anRefeinn(t 8j)F(wn)1

CImfeinn"

80F(wn)1]

co a.(3) [tnRefeinfl(t -n=0 dn

Im{e"Ckt

8.0) a_I(4) = -a c_{o o} R-686 19

-i

(64)

Finally, in order to obtain the total thrust, T(t), an

integration with respect to x and a summation over j must be

carried out.; However, examination of Equations (62) and (64)

reveals that the 8j's are the only quantities which depend on

The exponential factors containing 8j are independent of

x. and hence, one can reverse the order of the integration and

R-686 20 -and

=o

m=o - bmB Im{e1mBf2t (comol] co

= - a*

[cmBRefeimBnt) - d_mB Im m=o .a(W) _{= -a c} o o

B

= )

eihn"

(W

appears.

It is easily shown (see Appendix II) that .

BfmB(x) eimBnt if n-= mB; m = 0,1,2,.... Sn =

0 if n mB

:

That isi the only. terms which will COlitri.bute to the final

result are the terms having frequencies 'equUl to 'a: multiple of

the blade frequency.

Hence, if T(x,t) is the total thrust at radial distance x,

then T(x,t) = (B7Tpc111J2K9sKi r cos A) a (67) where (66) That is 4 T(x,t). T(k)(x,t) where .1(k)(x,i) =1)(x)a( )(

,t)

4 a = a(k) kd a(i) = a* a) and P(x) = BviocROK _{s s r} cos X (71)

Therefore, the total thrust T(t) may be expressed either as an integral jr1 = T(t) T(x,t)dx (72) xo or as a sum 4. .T(t) T(k)(t) (73) k=1 where T(k)(t)

ElT(

)(x,t)dx (74)

The corresponding torque, Q(t), is given by

1

Q(t) = _I Q(x,t)dx (75)

where, from Equation (17),

V Q(x,t) = T(x,t) (76) That is , V Q(t) T(t) -(77) R-686 21

-R-686 22.

FLUCTUATING 'THRUST AND. TORQUE FOR A SINGLE SCREW SHIP

Thrust 'and torque are calculated in this section for la

four-bladed propeller operating in the wake of a vessel. The

ship, wake and propeller characteristics were provided by the David Taylor Model Basin. Wake tests were made at DTMB :dfl 'a

model of the ship under conditions corresponding to the follow-ing ship conditions:

Length 500 feet

Displacement 15,450 tons

Speed 17.0 knots

Trim 3 feet (by stern)

The pertinent propeller characteristics are Number of blades, B 4

Propeller radius, R 10.66 feet

Hub radius, rc = 2.13 feet

Design, rpm 84.4

Shaft horsepower (shp) 7300

Values of the propeller characteristics (c, 00, Kgs, KS)

which depend on the radial position are listed in Table I. These

figures were extrapolated from information s-oPPlied by DTMB in the form of tables for x = 0.4, 0.7 and 0.95. Also listed in

Table I are values of U, X, a*, sin a* and P(x) =47TpcRU2KgsKs cos k.

The measured wake components, 1NL and WT, are tabulated in Table II as functions of for x = 0.4, 0.7, 0.9, 0.95.

In order to determine the parallel and cross-wake veloc-ities, Ws and Wc, Equations (39) and (40) were used. These

computations were carried out forx = 0,4, 0.7, 0.95, and select-ed values of The results are given in Table III. The quantities a and a' were

computed by the method given

by Burrilil.

The values of Wc and Ws are plotted in Figure , and * .

smooth curves are drawn through the data points. The harmonic analysis of Wm/U and Ws/U was carried out by using a 24-point scheme which was given by Scarborough5. The points used in this analysis were taken from the curves in-Figure

9.

The wake coefficients

amB, bmB,

clog and

dsi,

which were obtained from the harmonic analysis, are tabulated in Table V for B = 4 and m = 0,1,2,3. These values are plotted in Figure 10 where smooth curves are drawn in order to obtain values for intermediate values of x.

Next, One requires thevalties of wills = mBnc/2U and F(w)=

Fr

+

Fi

. The .appropriate wmB are tabulated in Table VI.

The function F(w)

has been computed and tabulated by Kemp4

whose values were the basis for the curves in Figure 11. The

values of

Fr

and

Fi

in Table VI were taken from the curves in Figure 11.

Evaluation of the a(k)

in Equation (68) also requires

computation of cos mBnt and sin mBnt . These are listed in

Table VII for m.= 1 and 2 and for values of t =

tq

, correspond-ing to every ten degrees of propeller rotation.

CALCULATION OF CONTRIBUTION T(k)(t) TO TOTAL THRUST

The calculation of the various T( k ) (0 proceeds as follows":

The total thrust, T(1), due to the resultant free-stream velocity, U, is given by

1.1

T(1) = a*P(x)dx (78)

xo

Using the values of a* and P(x) from Table land integrating by Simpson's rule results in

T(1)' = 121,000 pounds

The total thrust, T(21(t), due to the cross-wake, at

time, t =

tq

is given by 1 T(2)(tcl)

Jr

T(2)(x,t0dx

Xo (79) 11-;-686 23

-R-686 24

-where

T(2)(x,te) = P(x) Yq(x)

Yq(x) = 110 + Hq1,0. KtiN10. +

H8Nq + Keg

= anFr(con) - b0F1.(wri)

-asFi (con) - b 11F r(o)n)

The values of T(t) are tabulated

in Table VIII for t = t q,

q = 0,1...8 .

The total thrust, T(3)(t), due to the surge component, of the wake is given by

T(t)

=

f

T(3)(x,t)dx where T(3)(x,t. ) = P(x)Ici(x) - a.* E q.Mq + c8Nq - deg

Eq = co + cLq

The results for T(3)(t) also are given in Table VIII.

Note:

unsteadiness effects were neglected in T(3)(t).

Finally, the approximate thrust, T"), due to the combined effect of W0 and Ws is given by

fxo

T(4) P(x) a_o(x) c (x)dx_o 1)

The integrated result is

T")

= -5,300 pounds.

TOTAL THRUST AND TORQUE

The values of the total thrust, T, are shown in Table VIII

for various blade positions. These values are plotted in Figure

12.

The harmonic analysis of the total thrust, T, yields the

following result': c0

T =

A0 + (A m cos 4m77 + B m sin 4m7)) m=o (80) ( 8 2 )

where A0 F 137;400 Pounds /44 6;800. pounds -A8 100 pounds AI2 ,400: pounds . A1.8 -100 pounds_ T. 0, 151,000:-.108,000 125;000 x 100%7-= 13% x 100% -= 41% x 100% .= 34%

For comparison T2(t) is computed without the unsteadiness effect, denoting this quasi-steady part of 14.2)(i) by ,T*(2)(t),. .T*(.2(t) is

obtained by setting. Fr 71, Fi 70. in the equation .for T(2)t). The

results are given in Table IX. T*(t) is then the quasi-steady Value of

T( t).

double amplitude variation can

be

computed as a percentage .of as follows:

. 148,000 - 130,000

137,000

= 174,000--.118,000

.136,000

(from complete theory including effects of unsteadiness) (83) (quasi-steady) (84) 8-686 25 -(unpublished quasi-steady computation by A.J. Tachmindji) (85)

Finally, the torque variation is obtained by using Equation (77), i.e.,

Q(t) = 'V T(t) = 3.25 T(t) (86)

The values of (N.t) are listed in Table X and plotted in Figure

134 _{-4200 Pounds} 136 = pounds BI2 = 200 pounds B16 0 pounds . The the Mean For T,/i. *

R-686

- :26

DISCUSSION OF RESULTS

In view of the fairly complicated structure of the theory presented herein, and particularly because of the fact that the numerical work obscures many facets of the analysis, it thought advisable to answer a few questions which may be asked regarding interpretation of the results. Three of these ques-tions and their answers are':

Is there not one term which provides the dominant

contribution to the vibratory thrust in the foregoing example? The answer is yes, since -- by inspection of the

calcula-tions -- the dominant vibratory-thrust term was found to hie

p(x)sw(x)Fr(dx, cos 4.t (Er,

where P(x) = 47riocU2RKg _sKs cos X, w4 = 4Qc/2U, and a4(x) is tlie magnitude of the fourth harmonic of those wake components which contributed to the flow normal to the blade section at tKe

radial station x = r/R.

Can one determine a criterion which would minimize the vibratory thrust?

The answer is a qualified yes.. Assuming the wake is given

and the ship speed and propeller angular velocity are fixed,

the only quantity in Expression (87) that 'can be' varied is the

chord distribution c = c(x). Since

c is in P(x) as well as in

Fr(w4),

it is necessary to minimize the quantity of cFr(20c/U).

In terms of wit, the quantity to be minimized is given by

The variation of w4 for the case considered is shown in Figure

7 on page 27. Examination of Figure 11 reveals that

Fr(w4) is

a decreasing function of w in the range of interest and can be

represented by a straight line in the range 0.7 w 2.0

Thus

G(04)

(44(0.66 - 13,29wo , 0.7 wq 2.0 .

But, unfortunately this function does not exhibit a minimum

(89)

G(cott

2C2

wFr(c4)

within the interval but only at

the end point, as shown in

Figure 8.

FIGURE 7

-VARIATION OF REDUCED FREQUENCY WA WITH RADIAL POSITION

G(w4) AT W4.= 1.14 G(w ) 4 MAX _{OR X "' 0.75} .1;

r /

1.0 2.0 ./ TIGURE 8

VARIATION

OF G(ro) WITH

REDUCED FRDOENCY W

FIGURE 8

VARIATION OF

G(c4)

WITH REDUCED FREQUENCY ci.)4

R-686

27

R-686

-28-However, it may be possible to adjust the ship lines and

propeller clearances so that the harmonic content of the wake at blade frequency is minimized. This would require harmonic analysis of model wakes to determine how the harmonic content is affected by the afterbody shape and the location of the plane

of the propeller. Then, the minimization of the vibratory thrust

would require adjusting c in such a way that Fr(w4) -4 0. This implies that to minimize vibrations, c must be such as to make w4 = 2nc/U 2.3 (see Figure 11). Examination of Figure 7 shows that this condition is nearly met near the blade root, but is widely departed from in the region where most of the thiust

arises. In order to keep w4 near 2.0 in the vicinity of

X =

it would be necessary to double the chord, but this is not

feasible when taking into account the propulsive efficiency. However, it may be possible to compromise and provide a chord

distribution such that w4 is as near to w4 =2.3 as is possible. This then would be one way of mitigating the vibratory thrust.

3. Will not the application of the strip method to account

for the unsteady effects lead to an over-estimate of the

in-fluence of the shed vortices?

Yes. It is noted that the total unsteady thrust variation is computed to be 13 percent of the mean thrust as compared:to 33 percent when determined by quasi-steady calculation. This large difference represents the effects of unsteadiness (as computed by two-dimensional theory) which arise from the ,dim-inuation of the angle of attack produced by the flow generated by the vortices cast off from each section of the blade.

It can be expected that the counteracting or mitigating effect produced by these vortices as the thrust changes will be smaller than that predicted by the two-dimensional theory, :and, hence, the unsteady thrust may be somewhat greater than 13

per-cent. Correction for the effect of aspect ratio does not appear

simple, but it is hoped that this matter will be studied

fur-ther. In the meantime, it would appear highly profitable to

find out how accurate the foregoing theory is by making moldel measurements of the vibratory thrust for the case evaluated: in this report.

CONCLUSIONS AND RECOMMENDATIONS

The foregoing analysis shows that:'

application of quasi-steady methods to the calculation of the vibratory thrust and torque developed by a

pro-peller in awake will give values of the order of twice

that predicted by the strip-wise application of the un-steady-flow theory,

the vibratory effects at blade frequency (and multiples

thereof) are dependent on the magnitude of the wake

harmonics at that frequency and on no other components

of the wake flow, and

trends for minimizing the vibratory thrust can be

ob-tained from the unsteady flow theory. It is recommended that:

ship model tests be made in order to compare experimen-tal results with the predictions given herein,

further work be undertaken to investigate the effects of the _aspect ratio on the unsteady theory,

further calculations be made for ships On which large thrmstand torque vibrations have been found, and

a _{theoretical investigation be undertaken to extend}

Sears unsteady crostflow theory to the case of an

un-steady incoming flow 'at an arbitrary angle of attack.

-R -686

30

-ACKNOWLEDGEMENT

The authors wish to express their gratitude to Messrs J. Hadler and A. Tachmindji of the David Taylor Model Basin fo their co-operation in supplying data

conducted at DTMB and for their overall interest ment during the project.

from ship model tests

encourage-REFERENCES.

Burrill, "Calculation of Marine Propeller Performance

Characteristics",

North-.Esst

Coast:Institute of Engineers

and Shipbuilders, March 1944, p.269.

2, van Lammeren, W.P,A, "Testing Screw Propellers in a Cavi-tation Tunnel with Controllable Velocity Distribution

over the Screw D i " , International 'Shipbuilding Progress,

Vol.. 2, No. 16-, 1955, Appendix I, pp. 588-594-.:

Sears., "Some Aspects.of Non-Stationary Airfoil Theory and Its PracticalAppli'cution",Journalof theAeronauticul Sciences, January 1941, Vol.. 8, NO. 3, T..104.

Kemp, N.H.'; "Onthel,ift and Circulation .of.AirfoiLs in Some Unsteady Tlow.:Problems", Journal of the Aeronautical Sciences, October 1952, Vol. 19, No, 10, p. 713..

Scarborough,

Wumerical.Matheuatical Analy",Second

Edition, p. 487, Oxford University Press, 195.0.,

R-686 31

-TABLE I

UANTITIES INDEPENDENT OF 'TIME

R-686 33 -x (ft) 00 deg) Kgs _K U X. (deg) a*

de)

sin a* P(x) 0.2 3.9 61.8 0.66 092 34.3 56.8 5.0 0.087 O.41 x 106 .3 4.5 49.9 .68 .92 40.3 45.4 4.5 .078 0.86 .4 4.9 41.3 .71 .93 47.4 37.3 4.0 .070 1.54 .5 5.2 34.8 .74 .93 55.2 31.3 3.5 .061 2.48 .6 5.3 29.9 .78 .94 63.4 26.9 3.0 .052 3.89 .7 5.2 26.1 .82 .94 71.9 23.6 2.5 .044 5.08 .8 4.8 22.9 .87 .95 80.7 20.9 2.0 .035 6.45 .9 4.0 20.2 .94 .95 89.6 18.7 1.5 .027 7.24 1.0 0.0 17.9 1.00 .95 98.6 16.9 1.0 .018 0.00 "

R- 686

34

-TABLE II

WAKE COMPONENTS (ft./sec.)

(From Model Wake Measurements at David Taylor Model Basin)

Degrees x = 0.4 x = 0.7 x = 0.9

x=095

WL WT _{WL WL} 0 18.5 -1.5 19.8 -1.1 21.2 0.9 21.6 1.8 15 17.7 0.8 13.7 2.0 11.1 3.5 30 14.9 0.6 10.2 2.9 7.9 4.2 7.5 4.4 60 12.6 0.9 4.7 4.7 3.1 4.6 3.2 4.2 90 9.3 2.1 2.4 4.1 1.7 3.6 2.1 3.1 120 8.5 1.2 2.1 3.0 1.6 2.8 2.1 2.5 150 12.7 1.4 2.4 1.8 1.7 1.8 2.2 1.5 165 16.5 -2.6 5.4 -0.2 2.1 1.4 2.0 -180 20.0 -2.3 21.7 -0.4 - 15.7 1.5 195 21.0 -1.4 7.5 2.8 2.1 1.2 1.8 210 17.1 1.2 2.7 0.2 1.7 -0.0 2.1 0 240 11.9 -0.2 1.8 -1.6 1.7 -1.5 2.4 -1.3 270 12.0 -1.8 3.9 -4.1 2.4 -3.1 2.5 -2.2 300 13.4 -1.8 4.8 -5.3 3.0 -4.2 3.0 -3.1 330 14.0 -0.8 10.3 -3.9 8.0 -5.0 7.5 -5.2 345 15.2 -0.9 13.1 -2.4 12.1 -3,9

TABLE III

COMPUTED WAKE COMPONENTS '('ft ../sec.)

R-686 35 -co WL WT a Wc Ws 0.4 0 18.5 -1.5 0.931 0.148 4.57 8.87 " 60 12.6 0.9 359 .110 2.42 8.07 120 8.5 1.2 .201 .085 0.90 6.11 " 165 16.5 -2.6 .749 .145 3.90 7.05 . 195 21.0 -1.4 1.397 .160 5.14 10.08 . _{240 11.9} -0.2 .356 .111 2.25 6.78 " 270 12.0 -1.8 .394 .116 2.60 5.49 " 330 14.0 -0.8 .482 .123 3.07 7.33 0.7 0 19.8 -1.1 1.530 0.079 4.00 6.31 " 15 13.7 2.0 .586 .058 2.22 7.20 . 90 2.4 4.1 .071 .017 -1.56 4.93 . ₁₅₀ 2.4 1.8 .097 .022 -1.42 2.88 180 21.7 -0.4 2.127 .083 4.20 7.41 " 210 2.7 0.2 .122 .024 -1.14 1.44 270 3.9 -4.1 .193 .033 -0.10 -1.99 330 10.3 -3.9 .440 .052 2.13 0.63 0.95 0 21.6 1.8 2.424 0.063 1.95 8.33 " 30 7.5 4.4 .257 .022 0.05 6.61 " 90 2.1 3.1 .088 .015 -1.57 4.13 " 150 2.2 1.5 .094 .011 -1.03 2.27 . ₁₈₀ 15.7 1.1 .962 .045 1.59 5.81 " 210 2.1 0.0 .105 .012 -0.99 0.81 240 2.4 -1.3 .121 .015 -0.76 -0.18 " 330 7.5 -5.2 .354 .028 0.78 -2.43

R-686

36

-TABLE IV

FLOW ANGLES FOR VARIOUS :POSITIONS OF THE PROPELLER ctz +7 cos (4) + Y)

cos X

.

0.4

0

37.3

30.4

1.0

31.4

0.854

0.796

n ₆₀

_33.8

_1.2

_35.0

_.819

. " 120

36.0

1.6

37.6

.793 " 195 "

29.3

0.8

30.1

.866 330

32.8

1.0

33.8

.832

0.7

15

23.6

21.6

1.2

22.8

0.923

0.917

" 90 n

24.9

4.6

29.5

.870 . _{180 "}

_19.7

_0.9

_20.6

_.937 II . ₂₇₀ .

_23.6

_2.1

_25.7

_.902 . ,, ₃₃₀ "

22.2

1.3

23.5

.918 "

095

0

17.8

16.5

1.4

17.9

0.952

0.953

,,, ₃₀ 90 "

'

17.7

18.7

2.8

11.3

20.5

30.0

.937 .866 " 'V " 240 330 . "

18.2

17.4

4.5

2.1

22.7

19.5

.923 .943 I,

TABLE V

HARMONIC COEFFICIENTS OF WAKE COMPONENTS

TABLE VI

REDUCED FREQUENCIES AND GENERALIZED THEODORSEN FUNCTION

R-686 37 -x so ail, a8 al2 bq b8 b12 0.4 .0633 .0063 .0000 .0000 .0063 .0000 .0000 0.7 .0014 .0125 .0056 .0000 .0000 .0000 .0000 0.95 -.0021 .0053 .0021 .0000 .0000 .0000 .0000 x c4 c8 _c12 di',

4

_dI2 0.4 .1561 .0021 .0000 .0000 .0063 .0000 .0000 0.7 .0473 .0056 .0042 .0000 .0084 .0000 .0000 0.95 .0256 .0128 .0053 .0000 .0053 .0000 .0000 X ait wEl Fr(N) F(w11) Fr(c08) F (w8) 0.2 2.01 4.02 0.08 0.27 -0.20 -0.03 .3 1.97 3.94 0.09 0.27 -0.20 -0.01 .4 1.82 3.64 0.13 0.26 -0.20 0.05 .5 1.66 3.32 0.18 0.25 -0.18 0.12 .6 1.48 2.96 0.23 0.23 -0.14 0.19 .7 1.28 2.56 0.29 0.19 -0.06 0.24 .8 1.05 2.10 0.35 0.14 0.06 0.27 .9 0.79 1.58 0.43 0.06 0.20 0.24 1.0 0.00 0.00 1.00 0.00 1.00 0.00

R-686

38

-TABLE VII

QUANTITIES. DEPENDENT ON TIME

L

_ cos 4 n

_tq

Mg = sin 4 n tq

Nq . cos 8

SI

t

q

Pq =

sin 8 n t,

tq = qt/

ti = 1.973 x

i02

sec -

7r

10°

cz

180°

L A 0

1.00

0.00

1.00

0.00

1

.77

.64

.17

.99

2

.17

.99

- 94

.34

3

-.50

.87

-.50

-.87

4

-.94

.34

.77

-.64

5

-.94

-.34

.77

.64

6

-.50

-.87

-.50

.87

7

.17

-.99

-.94

-.34

8

.77

-.64

.17

-.99

TABLE VIII.

RESULTS OF .THRUST CALCULATIONS

average values computed by Simpsoes rule

R-686

39

-Time

Blade

Position

Contributions to Thrust

Thrust

Total

T(lb.)

-77 1411 142)

T(3)

TM

0 0 121,000

39,500

-10,100

-5,300

144,000

ti

10° "

32,100

-9,100

ff

139,000

t2

20°

.

26,600

-8,000

. 134,000

t2

300 "

24,400

-7,800

. 132,000

t4

400 " -

23,100

-8,300

" 130,000

ts

500 "

23,800

-8,900

"

131,000

t6

60° .

29,600

-9,200

" 135,000

t7

70° "

38,300

-9,600

.

144,000

t6

80° "

42,900

-10,200

148,000 Average Values* 121,000

31,200

-9,000

-5;300

137,000 (T)

R-686 40

-TABLE IX

COMPARISON OF UNSTEADY _AND .QUASI-S.TEADY 'THRUST

as computed by Tachmindji:from quasi-steady-theory-.

J*** averageJvaluescomputed,by Simpsoes rule

t -77 T(2) T*(.2) T T T-T* -TB.** 0 0 . 39;:500

69;2-

00 144;000 174;000' . ,-30,000 J 151;000 , , ti 10° 32,100 53,100 '139,000 160,000. -21,000 138A00 t2 20° 26,600 24;500 134;000 132,000 2000 121;040 J t3 .30° 24,400- 10,000. 132,000 118;000 14,000 112;000 , J t4 40° 23,100 10,800 130,000 118,000 12,000 108;000. 1 t5 50° 23;800 12,500' 131,000 120,000 11,000 109,000 i Jt5 60° 29,600 14,400 135;000 120,000 15,000 115,000 j t7 70° 38;300 26;700 144,000 122;000 12,000 129,000' . I tii 80° 42;900' '56;400 148;000 162,000' -14,000. 144-,000' Average._ . J. Values

(f)

.*** 31,200 -31,100 137,00a 136,000 230' 125,000 . 1

TABLE X .

RESULTS OF TORQUE CALCULATIONS

R-686

41

-t

-77- Co(t) :0

ir

'468,000

458,000

t..1 100

451,000

430,000

t2

200

435,000

388.,000

t8

30°

428,000

372,000

400

422,000

368,000

t5

50°

425,000

370,000

t5

600

438,000

383,000

t7

70°'

468,000

408,000

te

80°

480',000

443,000

10

FIGURE 9

WAKE COMPONENTS

COMPUTED FROM MODEL WAKE MEASUREMENTS MADE

AT THE DAVID TAYLOR MODEL BASIN

N/

8

_\

\

/1

\

/ /...

_\

/1

/

1

\

\\

\\

\

/

I

\

\

/

_I I I

\

\

/

:0.7

\\

\ \

_Ir\

_\

\

'

If

/ li 1

\

li

0.4

\

I II

\

\

N

\

0.7

/

"SURGE" WAKE, Ws "CROSS u WAKE, Wc t

60

120 180

240

300

360

ANGULAR POSITION, (DEGREES)

0.95 R- 686

-43-/.

\

\

/

\

x =0.4

/ 1

/

/ /

0.95 RADIAL POSITIONS x

R-686

-44--FIGURE 10

HARMONIC COEFFICIENTS OF WAKE COMPONENTS

CO

WC "

'Ea

n cosnt +bn sin nt Ws/U

&les

cos nt+cls sin nt

0.2 0.4 0.6

CO, RANGE OF IMMEDIATE INTEREST

FIGURE

II

GENERALIZED THEODORSEN FUNCTION, Flu)

Jo(CIAICIUCIA 1J1 (0)) KDOW) F(W) WO IAIW [Ko0(41)

(1W)]K1

(Ia) + N0(I10) Fr (0)

REAL PART OF F (CU)

§ WI IMAGINARY PART OF F 4.5 5.0 5.5 6,0

R-686

-46-180 170 .160 150 140 130 120 110 100 FIGURE 12

VARIATION OF COMPUTED THRUSTS WITH BLADE POSITION

UNSTEADY

QUASI -STEADY

1

1 1 1 1 1 1 1 1 1 1

10 20 30 40 50 60 70 80

DEGREES OF PROPELLER ROTATION,

(-nj

1 t

III IL 440

zz

0

420 4 fa 400 380 a

0

FIGURE 13

VARIATION OF COMPUTED TORQUES WITH BLADE POSITION

TACHMINDJ1

i

10 20 30 40 80 80 70 80 90

ANGULAR POSITION OF PROPELLER BLADE FROM _{VERTICAL,(DEGREES)}

R-686

-47-APPENDIX I

QUASI-STEADY LIFT DUE TO SURGING

Sears2 gave a formula for the unsteady lift due to a var-iable velocity perpendicular to tfle chord of an airfoil where the uniform main stream velocity was taken to be parallel to the chord.

In this appendix it is shown that Sears' result can be applied to obtain the quasi-steady lift when a variation in the parallel flow takes place while the crossflow remains uniform.

If the flow pasta foil section

has a parallel component, V3

+Lsvs,

and perpendicular component, Vc, (see Figure A-1) where Vs and Vc are constant, but AVs varies with

time, then the resultant velocity,

U1, is given by

Ui = {(vs +Avs)2 +vc2]* _A-1)

In order to use Sears' result a variation AV in the cross-ve-locity has to be determined so

that this, together with a parallel

flow Vs, will produce the same

lift as in Equation (A-3) (see Figure A-2). In this case, the

lift is given by

L2 = wpcRU22 sin a2 (A-4)

FIGURE Ao..1

VARIABLE PARALLEL FLOW

at an angle of attack al. The quasi-steady lift on the foil is given by

LI = wpcRU12 sin al (A-2)

FIGURE A-2

EQUIVALENT VARIABLE CROSSFLOW

R-686 49

-Or

I

Vs

Hence, in order to have L2 = Ll, it must be true

AVc = Vc Vs Ua* U, Ua*

That is, the equivalent crossflow is

AVs (A-1Q)

In order to apply this result in the computation of Li (3) one ha'p

(A-11)

The above analysis shows that this is equivalent to a parallel flow of Vs E U and a crossflow of Vc + 4V, .where

(A. 12). R-686

- 50

-Or

L2 = wpcR[Vs2 + (Vc + In the linear approximation,

Vc <, 1 AV C and 1 AVc-)14(Vc + AVc) it is assumed V + AV C C that imply that (A-5) (A-0) ( A-7 ) (A-8) (A-9) 1 Vs Then L2 Vs Vs2 + (Vc + AVc)2 1 Or (Vs AV5)2 Vc2 -(Vc +

AVc)-vc.

AVc Ars ) 4-G:)2 (1 Equation

+c)

(A-6)

Then the approximations in AV

(1 + Vc

LI

'AV

Ua* - a*Ws

and this result is applied in Equation (56) of the text.

R-686 51

-R-686 52

-APPENDIX II

PROOF Of PREDOMINANCE Of BLADE FREQUENCY TERMS

It has been shown [see Equation (65) in the text] that the total thrust T(x,t) at any radial position depends on sums of the form = fn(x) einn(t - 8j) Or Sn = fn(x) einnt, Substitute x = e±277ni/B in order to get 8 ,if x = 1 =

)7,

0-1

_{1-3cB if}

1-x

j=1

It will be proved here that a sum of this form vanishes unless n = mB, where m = 0,1,2,.. That is, only terms having a frequency n = mB contribute to the total thrust. It is to be proved: eting18.

fB,

if n = mB, m

0,1,2.

f=1 0, if 11 mB, m = 0,1,2... where S. = 277.(j L 1)/Bn J. Proof: let et27rn i ( j -1)/B

But, if = m13, = et2wmi = 1 while if n mB, = et2wni/B and 1-xB 1 1-x _1-e±2 -07"1"B

Therefore, it has been shown that

BfmB(x)eimBnt if ii = mB; m = 0,1,2... Sn =

0 ,

ifmB

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