A new interpretation of the free-space pressure field near a ship propeller

ARCHIEF

I

A

NG

i'

Experimental Towing Tank

Stevens Institute of Technology

Hoboken, New Jersey

Labs

v

Scheepsbouwkunde

Technische Hogeschoo4

Deift

REPORT NO. 687

February 1958

A NEW INTERPRETATION OF THE FREE-SPACE PRESSURE FIELD

NEAR A SHIP PROPELLER

by

EXPERIMENTAL TOWING TANK STEVENS INSTITUTE OF TECHNOLOGY

HOBOKEN, NEW JERSEY

A NEW INTERPRETATION OF THE FREESPACE. PRESSURE FIELD

NEAR A SHIP PROPELLER

by

John P. Breslin

PREPARED UNDER CONTRACT Nonr263(16)

FOR THE SHIP.DIVISION OF THE DAVID W. TAYLOR BASIN (Err PROJECT NO. HZ 1863)

TABLE OF CONTENTS

Page

ABSTRACT ii

NOMENCLA TURE

INTRODUCTION 1

ThE VELOCITY POTENTIAL OF A SINGLE-BLADED POPET,TFR IN

FREESPACE

...e....e.ó ... .. ...óá...e...

PRESSU IN FREE SPACE 7

Basic Form of Linearized Presire ₇

'onnii1as for Uniform Distribution of Circu1tion ₉ General Characteristics of the PiêsSure Field 10

CONCLUSIONS ₁₃

REFERENCES

lii

R-687

-1-ABSTRACT

The pressure fluctuations at a point in the vicinity of a propeller operating in a boufldless incompressible fluid are found. from the velocity potential function. This approach, which yields results' in ageement with those obtained by d4fferent derivations, has the advantage of enabling one to interprete the character of the pressure field. It is shown that the linearized pressures are purely convectiv

in

nature and,, hence, the pres-sure field acts as though it is a ntatictt distribution which moves with each blade. _{This result provides some conceptional gain in the application} of the free-space. field tO the action expected on a. flat plate. Some broad characteristics of the field are butlined. Detailed evaluation in terms of elliptic integrals is deferred to a later report.

i,j,k J .zn PQ r R R 0 S y NOINCLATU propeller radius propeller diamete±'

distance from propeller tO a plane unit vectors in

X,y,Z

directions advance ratio U/nd

number of blades

doublet strength density

number of revolutions per second

unit vector normal to helicoid.ai sheet pressure change

pressure change due to thrust-producing distribution pressure change due to torque-producing 4istribution radial distance of any point from axis of rotation

-distance between a point on helicoida-1 sheet and point in spaQe alternate -form of R

distance between point on propeller blade axis and point in space radial distance from propeller axis to any point on blade

time

fre.e-stream speed

Cartesian co-ordinates as shown in Figin'e 1 relative blade position angle, a e -Y

Ux

angular co-ordinate as shown in Figure 1

-?a ?ao

vector operator i _{+ j} + k

a dw

variable R-687 x,y,z a

R-687

r

circulation distribution as a function of s F0 cohstànt or uniform F

A a dumnrvariable= x +

Ci) angular velocity of propeller velocity potential function

p mass density. of fluid

dummy time variable

INTRODUCTION

It is known that the velocity field about a propeller can be found in an approximate fashion from a representation of the propeller by a line vortex which has three parts:

a hub vortex which streams aft along an extension of the axis of rotation

a bound vortex which lies along the quarter-chord line of the blade

a tip vortex which extends into the wake in the form of a helix.

Orientation of this system with respect to the axes is shown in Figure 1 on page 2.

The velocity field of this array of vortices is computed from the Biot-Savart law as demonstrated by the author1 and others. Reference 2 shows that the pressure field found in this way is identical to that employed by Garrick and Watkins3 (for the sound-pressure field of air screws) when their result is specialized to the case of any incompressible fluid. However, these approaches do not permit an identification of the nature of the pressure field which is important for the discussion of the force on nearby boundries.

The derivation presented here consists in finding the pressure field from the velocity, potential function, an approach which hitherto has been avoided because of the supposed complexity of the potential function. The work reported herein was carried out at the Experimental Towing Tank, Stevens Institute of Technology, sponsored by the Bureau of Ships under

Contract Nonr. 263-16 and technically administered by the David Taylor Model Basin.

R-687

-R-6$7

-2-FIGURE I

ARRANGEMENT OF FIXED AND MOVING AXES

AND THE FIXED CYLINDRICAL COORDINATES x,r,Y

-BOUND VORTEX

/

I I

I

HUB VORTEX HELICAL VORTEX I

THE VELOCI POTENTIAL OF A S1NGLE-BLADED PROPELLER IN FREE SPACE

The velocity potential correspondingto the abOve dés'ibed _represent-. ation of a. propeller by means of a line vortex can be obtained, from the fact that the potential function for any vortex configuration of circulation

density,

1',

_{is equivalent to that of a distribution of doublets of strength,}

*

N, normal to the surface spanning the vortex array.. The strength-density of the doublet distribution M is taken equal to r. Since the hub, botmd and helical vortices form a helicoidal surface, it i necessary to construct a doublet distribution há.ving axes normal to this surface.

Any point on this surface can be expressed in the following parametric form:

x-tlr

y

= S

cos(e -wr) = s sin(e

-wT)

where

;

....

(1)

;.O<s<b

J

x,y,z are co-ordinates of any point on the heli.codial surface U

S

e

is the speed of the free stream

is a duzimiy time variable

is the radial distance of any point along the blade is the blade position angle as shown, in Figure. 1,

is the propeller angular velocity is the .radius of the blade

The 'normal to the helicoidal surface may he constructed by finding a.unit vector which is perpendicular to both the radial line in the surface,and also to the resultaift velocity relative to the blade formed by the imposed free stream velocity and the tangential velocity imposed _{by th propeller} blade. _{This vector, N, is then given by}

.

*

See, for example,.H. Lamb, tlHydrodynandcstt, Dover Publications, _N.Y. Sixth Edition,

_1932,

page 212.,

R-687.

-3-R-687

or

-=

-

U sin(e -wr) +

U cos(e -r)

]

/u2,2

+

where i,

j,

iE

are

the

unit

vectors in the x, y, z directions. The

potential of the doublet distribution over the helicoidal surface is then

1b

fs)N.

V()ds'ds

, 5=0

SO

where =

rA+

oz

R={(x+ur)2+[y_scos(e_c4.rr)12+fz_ssin(e_wr)]2J

,

and dst is an element of arc length along a helical line in the surface. For

fixed

radial co-ordinate s

ds'

= /u2 +

(us)adr

_{. (Li.)}

introduction of Eq.iations (2)

and

(Li.) into Equation

₍₃₎

then

gives

b b

0

-F(s)

I

drds

uf

F(s)i

r sin(e -

y -

&rr) drds

R

5=0 /z=o

so

ro

R3

ere r ₌ y + z and y = tan (5)

wh 2 2 2 -1I..-) y Now let = Ut dr bd U

I

-0 s cos(8

-ar)

s sin(e -)

-'U ws sin(e -&r) -(is cos(e -wr)

(6)

and replace all dimensions such as x,y,z, etc. by bx, by, bz, etc.,

where the new x,y,z, etc. are in multiples of the propeller radius.

Further-more, let be the first term of Equation (5)

and

the second term. Then

with

these changes incorporated one obtains:

0 _{= +}

(7)

1

5h2

()2

(a)

where R1 now has the form

1

[(x+)2+r2+s2-2rs

cos(e-Y+

_-u-)]

It is most advantageous to express Equations (8) and (9) in forms which involve x in the l:linits of integration since they must be subsequently

differentiated with respect to that quantity. To do this let A

x + and

then,

0T = +

s dAds

R

0

if'f

Fr sin [e -Y) +

(x -x)] dAds

OX

R13

where

=

{x

+ r2 + S2 -2rs. cos,. [(e -y) + (x - A)]) .

For convenience and to illuminate the way in which terms annul in the following development, it is advantageous to introduce two new variables. They are: 1.1 sf'' decls /

'in

l

R1 a

= e

-y

wb

= -U-Hence, a and

- =

ae âa

Now and can be expressed compactly by

8r

dAds 1

b2

=

()

,Jo

.'

$1

1

/

r

;in(e _y+!e) dds

.10 .10 _R13

(10)

I

(13) (]J4) (16) R-687

50Q =

-R-687

-6-1

0Q =

-where

now Is

wrIt%'n

as

[2

+

r2

+

s2

rrsin(a+p--x)dxds

3

2rs cos(a +

caJbX)1 3

(17)

(18)

'U J

PRESSURE IN FREE SPACE Basic Form of Linearized Pressure

It is possible to proceed with the calculation of the pressure field now that the velocity potential function has been obtained. The need f or

this function arises because it is required in order to calculate the pressure at a point which is stationary with respect to the angular rota-tion of the propeller. Heretofore, rotating axes were employed which en-abled one to find the pressures directly in terms of velocities computed from the Biot-Savart law, thereby side-stepping the need for the velocity potential function. The translatory motion of the propeller is brought to rest with respect to a set of axes fixed at the center of the propeller by superposing a free stream -U. Then, the linearized pressure change, p, can be written as

U

ua

cx

E0

or, utilizing a and ,

a

p

da0

,

where p is the fluid mass density and x is in.multiples of the pro-peller radius. It is important to draw a clear distinction between the nature of the first and second pressure term in Equation (19). _{Both terms} vary with time, or equivalently with the blade position angle 6. However, the first term, p , called the impulsive pressure, depends not only upon the position of the blade with respect to a fixed point in space, but also upon the time rate of change of the blade position. In distinction to this, the second term, - p , called the convective pressure, can be

said to be a "steady-state" pressure in the sense that it is developed sole-. ly by virtue of blade position. Hence, this convective pressure can be imagined to be produced by a stationary blade (and its vortex wake) which somehow has the same thrust distribution as developed by its motion. In a steady-state motion no impulsive pressure term arises and, hence, only the convective pressure exists.

It will now be shown that. the pressure field of the propeller

con-R-687

-7-R-687

-8-sists only of convective pressures. Hence, the pressure distribution in space may be regarded as a steady-state field which is carried along with each blade. _{This interpretation is vital in order to comprehend the} pro-peller effects on a nearby infinite plane parallel to the axis of rotation.

The pressure changes associated with is obtained by applying Equation (20) to (16). ThIs results in

p

or

w ,ib2

a .a ô

(Jb)2

()dXde}(22)

The last term of Equation (22) is clearly zero since involves a and

p in exactly the same fashion. Hence part of the convective pressure completely annuls the impulsive pressure leaving only aconvectivé remainder in . Thus, after carrying out the differentiation there is obtained:

,.1 I

xsrds

i 2 2 2

32

p

_{x +r + s}

- 2rs cos aj /

Similarly, the contribution of is

-I

r

wb

lrn

1 1

irsin

T'

I

JoJ

E

.i'sinds

tx + r +

32 - are cos a] 3/2 sin (a 4. p

-R3

1

As above, the last term of Equation (2L) vanishes leaving X)dXds

which also is a

convective

pressure, the contributions of having been annulled completely. Thus, the total pressure change

T contains only

conveOtive-type pressures so that the propeller field in free space may be regarded properly as the pressure produced by virtue of the relative posi-tion of the observaposi-tion point and the propeller blade and is independent of time-rate of dhange of blade position. Put in another way, it is as though the fluid motion were derivable from a time-independent potential which can only give rise to a steady-state pressure depending on the relative. blade position.

Formulas for Uniform Distribution of CirculatiOn

In order to bring the expressions for the pressure into final form, it is necessary to relate the circulation density to the propeller thrust. This will be done for the simplest case,

viz.,

that of a lightly loaded propelle±. In this way the complications of slf-indiced flows are

avoid-ed. Considering an element of the bound vortex as shown in the adjacent sketch, the thrust of the blade, according to the Kutta-Joukowsk law, is

Ô=w T1

fb

r(s)ds (26)

For constant

r'=

F (which is assumed for simplicity), then T1 =

pb F

(for a blade having zero hub radius). If one further assumeC that the blades of an m-bladed propeller act independ-ehtly arid eachproduce a T1, thenT1 may be replaced by where T is the total thrust of the propeller. The

circulation density then can be. written

-ro.= --

,

. ... (27) ()IT1 Figure 2 Flows Relative to 1

u

r

F(s) r sina ds.

'6

TIEI

12

2 2

13/2'

x + r

s - 2rs cosaj Bound Vortex (2S) R-.687

R-687

-:1.0-upon replacing r by Equation (27) in Equations (23) and (25),

PPTPQ '

x 8. T t/m 2 R wo ₀ and PQ T'/in I.].

Jr

I sinads

-0.

--0 where R = (x

+ r *

S 2rs 0 :i. cosa),2 advance ratio,

revolutions per second, diameter (2b).

one obtains

(30)

Equations (29) and (30)have been obtained from the vortex line

rpresenta-.2

. ..

.3

tion and also as a specialization of the results of Garrick and Watkins ..

Identification of these pressures as doublet distributions or accelera-tion potentials has ben made previously by the author2. By regaIing each. term as a "directed pressure", p1 and PQ were shown to arise from the thrust

component of the propeller loading and from the torqueproducing component, respectively. However, it did not appear possible to determine the char-acterbf these presures, i.e., .thether convective or impulsive, without resorting to the velocity potential as described above.

General Characteristics Of the Pressure Field.

The integrations required in Equations (29) and (30) are, elementary and will not be written down for sake of brevity. In practical

applica-tions, it iS the blade-frequency content of the total pressure. change, -p,

which is of interest. This requires harmonic analyses of Equations (29).. and (30) which, in application, has been done numerically. However, very

recent work at the Stevens Experimenta]. Towing Tank shows promise that the blade-frequency content of the pressure can be written in terms of

tabulated ellitc integrals If these forms are found to be useful, they will be given in a subsequent report.

J U/nd, the

n propeller

It is, nevertheless, possible to point out several facets of the pressure field without detailed calculations. It is clear that in the vicinity of the plane of the propeller, i.e., for x near zero, the dimensionless pressure p/ri/rn will depend linearly on the advance ratio J. _{On the other hand, for some distance fore and aft of the plane of the} propeller, in the plane of the axis of rotation, the term will pre-dominate so that the dimensionless pressure will be virtually independent of the advance ratio. A simple calculation shows that at large x and y (the order of only a few diameters) the pressure approaches

p

_-

+_(

_ine

_e'1

T'/m 2

2)3/22

lt'Y5

ZCOSij,

which gives the equation of the plane, viz.,

x ₊

J,

- y

sjn6 - .z cos8) 0 ,

to which the surface of zero-pressure change is asymptotic. _{Thus, for} values of x, y and z which make the sum on the left side of Equation (32) greater than zero, the pressure change is negative, whereas, for values of x, y and z which make this sum less than zero, the pressure change is positive. _{At the other extreme, for very small x and y, the surface on} which p 0 is given by

It

=

-.7 _J

_(2z-

X

at the instant the blade is vertical, e n/2.. Thus, the trace of the surface on which p = 0 in the plane z = f a constant (f > 1) is as shown in Figure 3 for the instant at which 0 it/2.

It is to be noted that for large distances Equation (31) shows

that the pressure has only single-blade frequencies. _{Hence, the region in} which the rn-tb harmonic of an m-bladed propeller is of sensible magnitude is restricted to much smaller values of x, y and z. For values closer to the propeller, it is necessary to carry out calculations, results _{of which} will be presented in a subsequent report.

(33)

R-687

-11-R-687 '-12-Region of p > 0 Locus of trace of surface of p = 0 in plane z = f (interpolated between asymptotes)

\

Region of p < 0 Everywhereto r.ght of trace, pO 11 f

y-

_(?f-1)

X

Figure 3 - Regions of Positive and Negative Pressure Change in a Plane Above a Single Blade at the Instant the Blade Is Vertièal.

CONCLUSIONS

It may be concluded from the foregoing analysis that the pressure field:produced at a pointwhich moves along with, but does not rotate with, the blades of a propeller is composed only of convective, or steady-state pressures. This finding is of considerable conceptual value ,in

under-standing the effects produced on near-by surfaces. The formulas for the time-varying pressure are found to agreeS with results obtained from other derivations which do not make use of the velocity potential for the flow produced by a vortex-line representation of a propeller. The advantage of the velocity potential approach taken here is that the nature of the

pressure field can be readily identified.

R-687

-13-R-687

CJENCES

1) Breslin, J.P. : "The. Unsteady Pressure Pield Near a Ship Propeller

and the Nature of the ibratory Forces Produced on an Adjacent Sur-face", Experinntal Towing Tank, Stevens Institute of Techno1o, Report No. 609, June

_1956.

2 Breslin, J.P. : "The Pressure ie1d Near a Ship Propeller", to appear

in the Journal of Ship Research during

_1958.

.3) Garrick, I.E. and Watkins, C.E. : "A Theoretical Study of the Effect

of Forward Speed on the Free-Space Sound-Pressure Field Around Propellers", NACA Report

1198, 1951L.

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