The axial velocity field of an optimum infinitely bladed propeller

(1)

PRNC-TMD-48a (Rev. 3-58)

lab.

y.

Scheepsbouwk'md.

Technische Hogeschool

Deift

..---- ___. -

-

_a - t

- s

t-THE AXIAL VELOCITY FILLE OF AN HYDROMECHANICS OPTINUFI IHEINITELY BLADED PP.UPLLLLL

bî

o

A. J. Tachrnindji

AERODYNAMIC o STRUCTURAL MECHAN ICS

F.ESEALCH AND DEVFLOPNENT FEPCLT

HYDEOMECRANICS LABOEATÛEY

o

APPLIED

.TJnury J..99

Leport No. 1291+

MATHEMATICS

(2)

EEEEAECH ANt DEVELOPMENT EEPOET HYDTOÌCBANICE LABOr.ATOIIy

January 195

EEpOrt No. 1291+

H.

2,

H O L L A N D.

THE AXIAL VELOCITY FIELD OF AN OPTIMU} INFINITELY BLADED PECJFELLEL

(3)

TABLE OF CONTENTS

.6

il

Pa ge ABSTPACT INTEODUCTJON

ANALYSIS.

2 FESULTS CONCLUSIONS "

(4)

E (K) Complete elliptic integral of the second kind

ga Distance factor

K(k) Complete elliptic integral of the first kind

k Modulus of elliptic function, Eouation

₍₉₎

P Padius vector

Unit vectors

P Propeller radius

Radius of ring vrtex r,e,z Cylindrical co-ordinates

Vector tangential to thea vortex ring

Induced velocity

Axial induced velocity x,y,z Rectangular coordinates

r

Circulation Advance ratio

NCTAT ION

(5)

AB STFLACT

This report gives the axial induced velocity ahead of an infinitely bladed propeller. The propeller is simulated

by a close succession of ring vortices whose strength vary along the propeller radius and which extend from the propel-1er plane to infinity. The results are compared with those

obtained for a propeller represented by a uniform sink disc.

INThODUCTI ON

The flow field in the vicinity of a propeller can be calculated by replacing the propellerby means of flDw singu-larities. The usual representation has been to simulate an

infinitely bladed propeller in a steady inviscid incompressible fluid by means of a distributed sink distribution over the

entire propeller disc. The problem was originally formulated

by Dickman1 for a. propeller having a uniform distribution of

loading along the radius and the induced velocities resulting for such a distribution has been given in the literature2. Furthermore, a propeller with a. radially varying distribution

has also been simulated by superimposing sink discs with a 3

common center

1,2,3

(6)

The approach of the present method has been to simulate an infinitely bladed propeller with radially varying distri-bution of loading, by means of a series of ring vortices

ex-tending from the propeller to infinity. euch a representation

is not expected to give radically different results from the sink representation for relatively large distances from the propeller plane. In the vicinity of the propeller plane,

however, the magnitude of the induced velocities will be sub-stantially affected by the mathematical representation _{of the} propeller.

The present report is confined to the formulation _and calculation of the Thduced axial velocities for _{a propeller}

having an ortimum distribution of loading. _Eimilar

formula-tions, however, can also he performed for both the tangential and radial induced velocities.

Analysis

The induced velocities in the vicinity of a propeller

are a result of the system of free vortice' which originate _at the propeller blades and constitute the propeller _slipstream.

A propeller with a very large number of blades (i.e., _an

infinite number of blades) can be considered _{as shedding a}

(7)

Figure la

Figure lb

(8)

system of closely spaced helical vortex surfaces which are

equivaient to a compact system of helical vortex sheets. These

helical surfaces can be resolved into two parts: (1) a close

succession of transverse vortex rings with center at the oropel-1er axis and (2) a system of vortex lines which are parallel to the propeller axis (Figures la and lb). The axial induced

velocity at any point in space is a result of only the vortex rings, as the parallel vortex lines do not induce an axial component Similarly, the tangential induced velocities are

induced only by the parallel vortex lines.

In order, therefore, to evaluate the axial velocities we will consider a vortex ring of radius r7 with the coordinate

system r,, z and x,y,z with z in the direction of the axis of the ring as shown in Figure 2.

By the law of Biot-Savart

(1)

giving the velocity vector c9 induced by an element of length

r'de at a point (r,e,z) or (x,y,z). ?ith the unit

vectors in the direction of x,y,z axes we can write

(9)

Figure 2..

(10)

a. nd

Hence,

ch

=

the radius vrctor from the vortex element to the

2 k

2.

2-p =

The z-component of the inducec velocity will 5e _{obtained by}

integrating the entire ring.

cs(-e') -so

Li

_ 'c0(e

This integration can be performed by means of complete elliptic integrale (Ref. +) to give

V(,z)

= (k) -

E (k)

where kz

_{zz± ()1}

4 Pf zs

_T-

[co(e-e)-E (2) 7

ZL+ (_f)2-j5

(Lf) and r point P.

(11)

and where K(k) and E(k) _{are the complete elliptic integrals}

of the first and second kind.

For a vortex ring placed at the axial position

_z0,

Ecuation (+) becomes L(k)

(6)

i1:h ere kz (z-z0m-

(,)L

For a propeller having _{an infinite number of blades,}

the

optimum circulation distribution _{along the radius is given by}

r

-- c

j/+

_IL

Where C is a constant, _{is the advance ratio}

and I is the propeller radius.

In order, therefore, _{to find the induced axial velocity} for the entire system _{of ring vortices,}

Enuation (6) can be written as:

(7)

(12)

and

The induced velocity at any point (i/E,d) is given by

Ecuations (8) and (n). _{This velocity can be norrnali7ed in} terms of the axial induced _{velocity at the propeller}

disc.

For points ahead of the _{propeller, we can write}

't

-* k2 = t

K(k)

4

/R

(

d)

_Ç/

(8) (9) (1G)

where _{is known as a distance factor for the axial}

velocity. Similar exìressions can easily he obtained for the _tangential induced velocity by _{considering the system of hound vortices} and line vortices.

8

Where =

= z/

(13)

1.0 2.0 Table I = 0.0 3.0 1-f-.0 _5.0 _6.0 0.02 0.032 0.036 _0.039

0.0i

0.0+

0.07

0.10 _0.156 0.172 0.186 _0.197 0.206Lf 0.2JJF 0.20 0.295 _0.319 0.339 0.355 _0.367 _0.377 0.30

0Jl2

0.0 _O.62

_0.78

_0.90

0.500

0.0

_0.509 _0.537 0.559 0.57Lf 0.585 Q.5'9Lf 0.50 0.588 _0.615' 0.635 Û.6+9 0.659 _0.667

0.60 _0.652

_0.678

0.696 _0.708

_0.717

_0.72k

0.70

0.705

0.728

0.7

0.755

0.763 _0.769

0.90 _0.783

0.801

0.818

0.822

0.829

0.833

1.10

0.833 _0.851

_0.861

_0.867

_0.872

_0.875

1.50

0.899

0.909

0.915

0.919

0.92 0.P2f

1.00

0.933

0.939

0.9 0.9-f6

0.9-f8

0.950 r,'i. = 0.2

0.02 _0.036 0.0+0 0.0--3

_0.05

_0.0+7 _0.0+7 0.10 0.168 o.i8+ 0.196 0.203 0.209 0.212 0.20 0.309 0.331 0.3)+9 _0.360 0.366 0.371 0.30

_0.25

0»+52

_0.69

_0.8o

_0.87

_0.92

0.0

0.521 0.5L6 _0.56k 0.57k _0.581 _0.586 0.50 0.598 0.622 0.638 0.6-f8

0.65

_0.658 0.60 0.660 0.683 _0.697 _0.706 _0.712 0.715 0.70 0.711 0.731

0.75

0.752 0.757 0.761 0.90 _0.787 0.803 _0.813 _0.819 _0.823 _0.826 1.10 _0.838 _0.851 0.859 0.86k _0.867 _0.869 1.50 0.899 0.908 0.913 _0.917 _0.919 _0.920 1.90 0.933 0.939

0.92

0.91f 0.9-6 0.9-f.7

(14)

j 10

0.02

0.01+0 0.01+2 0.01+3 0.01+2 0.31+1 0.01+2

0.10

0.181+

0.192

0.191+

0.193 _0.191

_0.191

0.20

0.330

0.31+1 0.31+3 _0.31+3 _0.31+2 _0.31+1

0.30

0.1+1+5 _3.1+57 _o.1+6o _0.1+60 _0.1+59 0.1+0

_0.537

_0.51+8

_0.552

_0.553

_0.552

0.552

0.50

0.610

0.621

0.621+

0.625

0.60

0.668

0.679

0.682

0.683 _0.683

_0.683

0.70

0.90

0.716 _0.787

0.726 _0.795

0.729 _0.78

0.730 _0.799

_0.7q

0.730

0.730 _0.800

1.10 _0.836

0.81+3 0.81+5

_0.8+

o.81+y _0.81+7

1.50

0.896

0.901

0.903 _0.903

0.901+ 0.901+

1.90

0.930

0.933

0.931+

0.935

1.0 Table I (Cont.)

r/F

0.3

2.0

3.0

1+.0

_5.0

_6.0

0.02

0.038 QQ1J

3QLf _0.01+1+ _o.o1+5 _0.01+1+

0.10 _0.176

0.100 _0.197

0.200

0.201

0.20 _0.320

_0.339

0.31+9

_0.35+

_0.356

_0.357

0.30

0.1+36 0.1+57 0.1+68 _0.1+73 _0.1+76 _0.1+77 0.1+0

_0.529

_0.550

_0.5b1

_0.566

_0.569

_0.571

0.50

0.605

0.621+ 0.631+

_0.639

_0.61+2 _0.61+3

0.60

0.665

0.683

0.693

0.697 _0.700

_0.702

0.70

0.715

0.731

0.739

0.71+1+ 0.71+6 0.71+8

0.90 _0.788

0.801

0.808

0.812

0.811+

_0.815

1.10

0.838

0.81+9

o.85+

_0.857

_0.859

_0.860

1.50

1.90

0.899 _0.932

0.906 _0.937

0.910 _0.939

0.912

_0.91+1

0.913

_0.91+1 _0.91+2

0.913

0.1+

(15)

i i i j

i

T8ble I (Cont.)

= 0.5

= 0.6

11

d \

1.0

2.0

3.0 .0

_5.0

_6.0

0.02

0.10

0.0Lf2

0.191

0.02

0.191

0.01

0.188

0.00 _0.185

0.00 _0.183

0.039 _0.181

0.20

0.30 0.LfSl

0.337

0.51 0.33k

L7

0.3

0.330

0.327 _0J1

0.325

0.38

0.0

0.50

0.539

0.609

0.50

0.611

0.537

0.608

0.605

0.533

0.531

0.603

0.529

0.601

0.60

0.70

0.90

1.10

0.662

0.712

0.782 o.8o

0.667 0.71L

0.783

0.832

0.665

0.712

0.782 o.8j.

0.663

0.710

0.781

0.830

0.661

0.708

0.780 0.82Q

0.659

0.707

0.779

0.828

1.50

0.891

0.892

0.892 _0.891

_0.891

_0.890

1.90 _0.925

_0.926

_0.925

0.02

0.10

0.03

0.19 0.CLfl

_0.187

0.00 _0.179

0.039 _0.175

0.037 _0.172

'0.037

'0.170

0.20

0.339

0.328

0.318 _0.311

_0.307

_.Q.3Q

0.30 0.O

0.8

0.36

0.25

0.18

0.11

0.50

0.60

0.532

0.600

0.521 O.58q

0.65

0.511

0.580

0.637

0.50

0.57 0.6,1

0.50O

0.57o

0.628

0.97

0.567 p0.625

0.70

0.90

1.10

0.699

0.768

0.817

0.691

0.762

0.812

0.68L

0.756

0.808

0.679

0.752

0.805

0.675

0.750

0.803 p0.673

'0.78

0.802

1.50

0.880

0.17

0.877

0.915 0.87L

0.913

0.873 _0.912

0.871 _0.911

0.870 _0.910

(16)

r

Table I (Cont.)

0.02

0.10

0.20

0.30

0.1+0

0.50

0.60

0.70

0.90

1.10

1.50

1.90

12

1.0 r/f. =

2.0

0.7

3.0

1+.o

5.0

6.0

0.01+3

0.039 _.0.036

0.031+ 0.031+

_0.033

0.191

0.171+

0.163 _0.157

_0.153

_iO.151

0.326

0.302

0.286 _0.277

_0.272

_0.268

0.1+27

0.1+oi

_0.381+

_0.373

_Q.367

_p0.363

o.oE

0.1+80 0.1+63 0.1+53 o.1+1+6 .0.1+1+2

o.570

.0.51+6

_0.530

_0.520

_0.511+

c.io

0.623

0.601

0.586 _0.577

.0.571

_0.567

0.668

0.61+8 0.631+

_0.626

_0.621

_p0.617

0.738

0.722

0.711 0.701+

0.700

0.697 '0.790

0.777

0.768 _0.763

_0.760

_tO.757

0.859

0.851

0.81+5

o.81+i

0.839 _0.837

0.901 _0.85

_0.891

_0.888

_0.887

_0.886

r/. = 0.8

0.02

0.039 _0.032

0.028

0.026

0.021+ 3.021+

0.10

0.162 _0.135

0.120

0.111

0.106 '0.103

0.20

_0.271+

0.232

0.209

0.196

0.188 o.l83

0.30 _0.358

_'0.312

_'0.285

_0.270

_.0.261

_0.256

0.1+0 0.1+29

_0.382

_0.355

_0.31+0

_0.331

_p0.325

0.50

0.60

0.1+900.51+1+ 0.1+1+6

0.502

o.1+190.1+77 0.1+01+ 0.1+63

0.395

_0.1+55

.0.389

.0.1+1+9

0.70 _0.592

_0.553

_0.5O

_0.517

_0.508

_a.5O3

0.90

1.10 _.0.730

0.670

0.637 _0.703

_0.687

0.618 '0.607

_0.E78

_0.672

0.600 .0.596

_0.669

1.50

0.811+

_0.795

_0.781+

_0.778

_0.771+

_0.772

1.90

0.866 _0.853

0.81+5

o.81+i

_0.838

_0.836

(17)

4

Table I (Cont.)

13 \ I/

1.0

2.0

0.9

3.0 .0

5.0

6.0

0.02

0.006 -0.016

-0.022

-0.026

-0.028

-0.029

0.10

0.011 -0.051

-0.079

-0.093

-0.101

_-0.105

0.20 _0.032

_-0.057

-0.099

-0.121

_-0.133

-0.1+l

0.30

0.080 -0.020

-0.069

-0.095

-0.111

-0.120

0.+0

0.Ji+2

0.039 -0.012

-0.0J.

-0.057

-0.068

0.50

0.208 _0.107

0.055

0.026

0.010

0.000

0.60 _0.272

_0.176

_0.126

_0.099

_0.083

_0.072

0.70 0.33)

0.214i

_0.197

_0.170

_0.155

_0.l--5

0.90

0.5 _0.367

_0.326

_0.30

_0.290

_0.281

1.10 _0.536

0.)-1-70 0.»+36

0.+l6

0.1+05

_0.397

1.50

0.671

0.621+

_0.599

_0.585

_0.577

_0.572

1.90 _0.753

_0.726

_0.707

_0.697

_0.691

_0.687

(18)

Ftesuits

The distance factor ga, has been computed, using Eouations (8), (9), and (io), on the IBM 70 Computer for values of r/? from O to 0.9, _{d from O to 1.9 and}

from 1.0 to 6.0. _{The results are given} _{in Table I, computed}

to an accuracy of 0.001.

The results can be compared with those obtained _from a sink disc representation

of a propeller of uniform loading. The comparison is made _{in Table II, and it}

is noticed that the difference is considerable, particularly in the _{vicinit.y of} the disc. _{This is primarily due to}

ielatively large change _of the axial velocity

component near the propeller plane, indi-cating that the approximation

of uniform loading along the

Table II

Comparison of _a _{with uniform sink disc}

r/R = 0.6 _r/F. ₌ _0.8

d = 1.0 = 6.0 Sink disc = 1.0 = 6.0 Sink disc

0.3 _0.448 _0.411 _0.373

0.358 0.256 0.477

0.5 0.600 0.567 0.536 _0.4.90 _0.389 _0.617

0.7 0.699 0.673 _0.646 _0.592 _0.503

(19)

propeller radius, leads to inaccuracies of he induced

velocity field. Examination of the results, shows that the

distance factor ga is larger than that obtaine from a

uni-form disc representation for propeller radii smaller than 0.6, however, for propeller radii of 0.8 or largr the

distance factor is smaller. This would indicate that although

the sink distribution representation may result in the correct total effect, the radial distribution of such velocity is a sensitive function of the radial distribution of loading.

Conclusions

Comparison of the axial induced velocity in the vicinity of a propeller indicates that the loading distribution is

important for points near the propeller plane. The velocities

have been calculated for an optimum loading distribution which in general is not too different from the radial distribution usually used. In specific instances, however, it may be

necessary to use the actual circulation distribution and this can most easily be performed by calculating the induced velocity resulting from the differences in loading between the actual and optimum distribution.

(20)

ACKNOEPGMENTS

The author is indebted to the members of the Applied Mathematics Laboratory and particularly to Nr. W. Mann for

the programming and computations.

REFEPENCE S

Dickmnan, H. E., "Giundlagen Zur Theorie _Mngformiger Tragflugel9, IngeniEur - Archiv, Vol. II, l9+0.

Korvin-Kr'oukovky, B. V., Stern Propeller Interaction

with a Streamline Body of Pevolut.iont', _{International}

Shipbuilding Progress, January lP6.

Fleisher, I. and Neyerhoff, L., _{The Field of a Uniform} Circular Source Disk, Eastern Eesearch _{Group Peport,} Contract NONE 2282 (00), Jan 1958.

1+. _{Jahnke, E. and Ee, F.,}

Tables of Functionst1, _Dover

Publications,

l95.

(21)

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(22)

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