-REPORT 69-3

INERMNATION OF THE DESIGN

PARAMETERS FOR OPTIMUM

HEAVILY LOADED DUCTED FANS

TERRY WRIGHT

Thlis work was done in partial

fulfillment of the requirements

qfor the degree of Doctor of Philosophy

NOVEMBER 1969

Prepared forU. S. Army Research Office, Durham

underProject THEMIS

Contract No. DAC04 68 0004

School of Aerospace EngineeringGeorgia Institute of Technology

Atlanta, Georgia 30332

ACINOWLEGENTS

I would like to express my appreciation to Dr. Robin B. Gray

for his suggestion of the thesis topic and for his guidance during

the course of this research. I would also like to thank Dr. A. Alvin

4 Pierce and Professor James E. Hubbartt for their careful examination

of the manuscript.

The mmerous discussions with Dr. Robert K. Signan, Mr. C. E.

Hammond, Mr. Larry Wright, and my wife Mary Anne iave provided me with

valuable insight into the various aspects of the problem from time to

time. Their advice is greatly appreciated. I thank Dr. Ke-iton D.

Whitehead and Mr. Howard L. Durham for suggestions concerning the

computational procedures of the thesis.

The financial support of the National Aeronautics and Space

Administration, the U. S. Army Research Office, Durham, and the

Georgia Institute of Technology are gratefully acknowledged.

Finally, I wish to thank .v parents, Mr. and Mrs. Roy J. Wright,

for their assistance and wholehearted support during the earlier years

of my education.

iW

-

iii

TABLE OF CONTEINS

Page

ACKNOWLEDGMENTS ........ ... ........................ . ii

LIST OF TABLES ........ .... ......................... v

LIST OF ILLUSTRATIONS...... . . ..................... .... vi

NOMENCLATURE ....... .. ........................... ... viii

SUMMARY ....... .... ............................. .... xiii

Chapter

I. INTRODUCTION ....... .... ...................... 1

II. DEVELOPMENT OF THE WAKE VORTEX MODEL ..... .......... 4

III. ANALYSIS AD SOLUTION OF THE WAKE MODEL .... ........ 24

IV. CALCULATION OF THE THRUST, POWER,AND-INDUCED EFFICIENCY. . ....... ................. 37

ThrustPowerEfficiency

V. NUMERICAL PROCEDURES .... .................. .... 54

Evaluation of the Velocity ComponentsSystem of Linear EquationsThrust and Power Integrations

VI. RESULTS ........ ........................ .... 74

Appendix

I. GEOMETRY AND MOTION OF THE INNER HELICAL SHEETS . . . . 86

II. VELOCITY FIELD OF THE UNIFORM BOUNDARY SHEET ......... 90

III. CHECK CASES FOR THE NUMERICAL PROCEDURES ........ . 98

IV. PERFORMANCE AND DESIGN TABLES .. .. .. .. .. . .. 105

TABLE OF CONTENTS (Continued)

Page

CITED LITERATRE ........... ......................... 139

,7

k

K N1

LIST OF TABLES

Table Page

1. Accuracy of the Wake Integrations ............. ..... 103

II

vi

LIST OF ILLUSTRATIONS

Figure Page

1. Conceptual Diagram of Vortex Shedding at the Duct

Trailing Edge .... ...................... 6

2. Helical Coordinate System ................ 7

3. Ultimate Wake System Showing Pathsof the Line Integrals ... .............. 9

4. Velocity Diagram at an Inner Helical Vortex Filamentof the Ultimate Wake ................ . . . . . 10

5. Ultimate Wake System Showing Pathsof the Line Integrals ................. 13

6. Concept of the Ultimate Wake Vortex System .......... 19

(a) Boundary Sheet of Uniform Strength

(b) Boundary Sheet of Varying Strength

(c) Result of Superposition

7. Velocity Diagram with respect to the Rotating Bladesof the Outermost Filament of the Inner Vortex Sheetand at the Adjacent Point of the Boundary Sheet . . .. 21

8. Control Vlume used in Determining the Thrust ..... 38

9. A Path of Line Integration in the Ultimate Wake .... 44

10. Schematic Diagram of the Arrangement of Vortex Filamentsand Control Points in the Ultimate Wake of theDucted Fan .... ...................... . 64

11. Network Used in Calculating the Velocity Integrals onthe i - Surface ...... .................... .... 71

12. Comparison of Two Methods of Determining the BladeBoun Vorticity for a Lightly Loaded Ducted Fan . . . . 75

13. Comparison of Two Methods for Determining the BladeBound Vorticity for a Lightly Loaded Ducted Fan . . . . 76

vii

LIST OF ILLUSTRATIONS (Continued)

Figure Page

14. Comparison of Two Methods of Calculating the BladeBound Vorticity for a Lightly Loaded Ducted Fan . . . . 77

15. Comparison of Two Methods of Calculating the BladeBound Vorticity for a Lightly Loaded Ducted Fan . . . . 78

16. Comparison of the Blade Bound Vorticity for a LightlyLoaded Ducted Fan for Successive Numbers of Blades(Digital Computer Solution) with the InTinite BladeSolution (Exact) ...... .................... .... 79

17. Comparison of Mass Coefficients for Increasing Numbersof Blades (Digital Computer Solution) with the MassCoefficient for the Infinite Blade Case (Exact) . . .. 80

18. Blade Bound Vorticity Distributions for a Family ofLightly Loaded Ducted Fans .... ................. . 82

19. Variation of the Load Scaling Factor . . . . . . . . . 83

20. Variation of Thrust Coefficient with Load for theFamily of Ducted Fans ................... .. 84

21. Variation of the Power Coefficient with Load for theFamily of Ducted Fans ........ ................. 85

22. Incremental Thrust Efficiency Diagram. ........... .... 88

23. Geometry of the Ring-Element Sheet ..... ........... 92

24. Geometry of the Straight-Line-Element Sheet ... ...... 95

25. Schematic Arrangement of Filaments and Control Pointin the Ultimate Wake of a Free Propeller .... ........ !00

26. Comparison of Two Methods for Calculating the BladeBound Vorticity for a Free Propeller ..... .......... 101

27. Convergence of the Wake Integrations to the ExactResults for the Infinite Blade Case ..... .......... 104

viii

NOMENCIATURE

an coefficients of the trigonometric series for the non-

uniform boundary sheet strength distribution

a, R, U dummy variables used in Appendix II

A1 defined as A,1 = sin (CP- rB)

A2 defined as A2 = cos ( -cPrB)A. elements of the coefficient matrix of a system of lineari,j

equations

b number of blades

B n coefficients of the trigonometric series for r(x)

C p power coefficient, C, Pl[(D)'7 2

C T thrust coefficient, C T =Ti[P (QR)% R2]

e non-dimensional energy loss, e = E/[p(MR) R0 ]

E energy loss in wake, E = -TV

f(t) function of time

f Cintegrand of the Biot-Savart equation for u

f integrand of the Biot-Savart equation for u

f integrand of the Biot-Savart equation for ur r

defined as gn = - ndT'gMn

G load scale factor G = 1 -u /w

h, h2 distances used in Appendix II

i integer subscript

I integer

I . integral of f,, equation (22)

i -

ix

NOMENCLATURE (Continued)

I r integral of fr' equation (21)

J integer subscript

J integer

k number of divisions of a turn of a helical filament

K(x) non-dimensional blade bound vorticity, K(x) = L~2

K(X) K(x) for R = 0 casel0integer subscript

L characteristic axial length in wake, L 2 BA/b

n integer subscript

p static pressure

p. static pressure in undisturbed flow

po total pressure

SP non-dimensional distance from a vortex element to a

control point

P induced or ideal power

P' P2 non-dimensional distances used in half-range integrations

Q torque

r, Y, z polar-cylindrical coordinates

r', Y', z' polar-cylindrical coordinates locating a vortex element

R radius of the ultimate wake and the fanIR inside surface of the wake cylindrical boundary

R outside surface of the wake cylindrical boundary

.sa length of a vortex element

S surface area

_i4

x

NOMENCLATURE (Continued)

t time

T total thrust of the ducted fan

u disturbance velocity component in direction of

subscript

u tdisturbance velocity along wake axis

Unon-dimensional disturbance velocity, U = u/v

UCB velocity describing motion of filaments of uniformboundary sheet

v total disturbance velocity

V total velocity

V velocity of the undisturbed flow

w apparent axial displacement velocity of blade trailing

vortex system

wB apparent axial displacement velocity of uniform boundary

sheet

rnon-dimensional apparent axial displacement velocity,

=w/nR

x non-dimensional blade radial station, x = r/R

x, y, z Cartesian coordinates

x', y, z Cartesian coordinates locating a vortex element

z 0 axial distance between the z = 0 plane and the point

where vortex filament intersects the xz-plane

z non-dimensional characteristic length on the wakec

boundary, z c = L/2rrR = X/b

angle defining direction of induced velocity in

Appendix Ii

xi

NOMENCIATURE (Continued)

angle between ds' and

y a vortex filament strength

non-dimensional vortex filament strength, y = /(4TRvXG)

y vortex sheet strength per unit length normal to the

filaments

Y(CB) vortex sheet strength of the uniform boundary sheet

r(x) blade bound vortex strength distribution

6,(Au/w) contribution of the th turn of a helical vortex

filament to Au/w

e defined as e=e r +e +6

e0 numerical factor for CT, equation (45)

e defined as er =f T2U2 F dr dz d/2rt

000

CT defined as e= ff lf F dF d d/2T000

e defined as e = J't Uz 2 dF dz dT/2rTZ Z1z 000

edefined as e j IfS U I- + ] 1; dF d~z dlY/2TR 000 R R

Chelical coordinate normal to the filaments

induced efficiency, W) =(K " 1 CT/CP

mass coefficient, X 2f K(x) x dx

0X 0 X for = 0 case

0K pitch of the inner helical sheets, x = (V~ +w)R

!

xii

NOMENCIATURE (Continued)

XB pitch of the uniform boundary sheet filaments,

X= (B + wB ) IOR

defined as p, = 2 2 dx0 X +

Po r Pfor = 0 case

p fluid density

CP pitch angle of an inner helical sheet at r

PR pitch angle of an inner helical sheet at r = R

cPB pitch angle of the uniform boundary sheet

pitch angle of a filament of an inner helical sheet

at r/

r dummy pitch angle defined by pr = tan (tan cJ/)B B

disturbance velocity potential

T 0 an arbitrarily fixed polar angle

W half-width of a strip element of a vortex sheet

0blade rotational speed (radians per second)

I,

ItI

i !i!ISUMMARY

Like the free propeller in axial flightha single-rotation ducted

fan of highest induced efficiency is characterized by an ultimate wake

vortex system shed from the blade trailing edges whose apparent motion

is that of rigid helical surfaces. In addition and concentric with this

inner sheet system there is a cylindrical surface of helical vortex

filaments shed from the duct trailing edge. For zero hub diameter and

neglecting compressibiLity, viscosity, and tip clearance, a consistent

mathematical model of the constant-diameter vortex wake is developed

and the compatibility relationships to be satisfied are presented.

Using the Biot-Savart equation, the vortex strength distribution in

the ultimate wake is determined and then related to the blade bound

vortex strength distribution. In addition, expressions are developed

for the thrust, power, and induced efficiency which depend on numerical

integrations of velocity profiles in the ultimate wake. 'It is shown

that the wake vorticity and the velocity distribution in the wake, for

all loadings from the lightly loaded limit to the heavily loaded static

thrust conditior, may be extracted from the lightly loaded result through

the use of a simple algebraic scale factor, The performance parameters,

thrust, power and blade bound vortex strength, are thus expressed in

terms of a lightly loaded solution for a given freestream velocity,

blade tip speed, blade number and the loading parameter. The lightly

loaded case is compared to existing theoretical and experimental

!4

----. .

xiv

results. Some sample results for heavily -.oaded performance parameters

and design parameters are presented and compared to the exact results

for a heavily loaded ducted fan with an infinite number of blades.

Results for the heavily loaded system over a broad range of conditicns

are presented.

}i

I.

1

CHAPTER I

INTRODUCTION

In recent years there has been an increasing interest in low

speed thrust and high-lift devices for application to aircraft with

short or vertical take-off and landing characteristics. One of the

devices under consideration for this role is the ducted fan or shrouded

propeller. Analytical and experimental work in this field have been

summarized in an excellent review of the state of the art through 1963

by Sacks and Burnell 1. As pointed out in the review, there have existed

for some time adequate means of sizing and designing a duct for a given

choice of fan blade design but there has been no method available for

optimizing the blade design for the finite bladed system which operates

at any but the lightest loadings. For the lightly loaded case the

problem has been solved by Tachmindji2 , Theodorsen3 , and Gray4' . In6 , 8

more recent years Morgan , Ordway, Sluyter, and Sonnerup', Chaplin ,

and others have published papers in the field of ducted fans, still

with primary emphasis on the shroud design. It is the purpose of this

research to provide the information needed to size and design the

optimum heavily loaded ducted fan.

The vasis for this work stems from the classical analysis of

the free propeller. It has been shown by Betz9 that an isolated

propeller having the highest possible induced efficiency, that is,

an optimum free propeller, will generate an ultimate wake vortex I

2 1!I

2

system which moves through the fluid medium as if the vortex sheets

of the wake formed a rigid helical structure of constant pitch.

Application of this constraint to the motion of the sheets provides

a straightforward means of determining the radial distribution of shed

vortex sheet strength distribution. This determination has been carried

out by Goldstein and Theodorsenll .

It has been shown by Gray4 that, for the ducted fan having the

highest possible induced efficiency, the same arguments and

considerations are valid concerning the geometry and motion of those

vortex sheets which are shed fron the blade trailing edges. These

arguments are reproduced in Appendix I for conpleteness.

In addition to this constraint, the system must satisfy the

Kutta condition at the duct trailing edge; that is, the flow at the

trailing edge must be tangent to the duct mean camber surface. As a

consequence of this condition a sheet of vorticity must be shed from

the trailing edge forming a boundary sheet enclosing the screw-like

sheets shed from the blade trailing edges. It is the determination of

the geometry, mcrcion, and strength distribution of this boundary sheet

and its mutual interaction with the inner sheets which constitute the

development of the ultimate wake vortex model for the ducted fan.

When the geometry and motion of the vortex sheets of the ultimate

wake have been defined, a straightforward application of the Biot-Savart

equation determines the strength distribution of the vortex sheets.

The blade bound circulation or vortex strength distribution is then

obtained by integration of the inner helical sheet strength. This

bound circulation is the basic informatinn frm1 . hich the fa-. biadcs

4

can be designed using any of the methods available in the literature

(e.g. Theodorsenll). In addition, the vortex sheet strength distribu-

tions enable a detailed calculation of the flow in the wake from which

the thrust and induced power may be calculated.

In the development and analysis which follows the effects of

hub diameter, blade tip clearance, compressibility and viscosity are

neglected. The results will therefore give an upper limit on the per-

formance of single-rotation fans without stator vanes for comparison

with other design methods.

i

I2-

! " ItI

CHAPTER II

DEVELOPMENT OF THE WAKE VORTEX MODEL

Earlier analyses by Gray (see Appendix I) have shown that the

optimum condition is obtained for the ducted fan when the helical

vortex sheets shed from the blades appear to move as rigid screw

surfaces in the ultimate wake. The principle difference between the

arguments used by Gray and those for the free propeller is that the

induced velocity in the wake, at the surface of a helical vortex sheet,

need not be normal to the sheet surface. Beyond the need for satisfying

the Kutta condition at the duct trailing edge no other information is

obtained and additional relationships must be developed in order to

assure a compatible vortex model.

Further information may be obtained through consideration of

the flow associated with the vortex systems. A blade trailing vortex

sheet has large radial velocity components associated with it and, with

{ respect to a coordinate system fixed in the duct, this flow is periodic.

Consequently, if the duct mean camber surface is to correspond to a

streamline, a distribution of vorticity must be placed along this surface

to cancel all normal velocity components. Part of this distribution

must be periodic in nature and must also be a function of the number of

blades in the fan and must rotate with the blades. A portion of this

periodic distribution is considered to be a continuation oi each blade

bound vortex which passes directly from the blade tip onto the duct

I

contour where it spreads out over the duct mean camber surface. Then,

in accordance with the Helmholtz theorems, it moves aft along the mean

camber surface and is shed at the duct trailing edge as free vortex

filaments of varying geometry and density wrapped on a cylindrical Isurface. While these filaments are within the duct contour it is

assumed that they make all of the necessary adjustments in phase

relationship and density with the blade trailing vortex system so

that the motion of this wake boundary system is along the tangent to

the mean camber surface at the trailing edge of the duct as shown in

Fig. 1. In addition to the blade bound vortex continuation, a portion

of the duct trailing vortex sheet is considered to be shed from bound

vortices of varying strength rotating in the duct with the blades.

It is also assumed that the duct is designed so that there is no

contraction or expansion of the wake downstream of the fan. This is

a feasible design problem and guarantees that the duct is compatible

with the wake geometry of the analysis.

Continuing with the examination of the velocities associated with

the wake vortex sheets, consider a helical coordinate system r, , C

defined in terms of the cylindrical coordinates r, T, z as shown in

Fig. 2. At a given instant an inner helical vortex sheet in the

ultimate wake coinciaes with the C = 0 surface. Then

r=r , O5r :

=r cos +zcos , -

s - +w) cosc( (V +w) COSMrY sine + z2b ((2/2rr) co 2b (0/2rr)

NM CAE SURFACE"--

W AKE BO N A Y

rL E I

Fiuc1 ocpulDarmo ore hdiga h utTalnEdge.

7

a.

r p.4x

N Co

C,.4.,

*r4

p.4

*r4'-4U,

C"

I,*r4

a

~ ~- U

t

8

where ir, measured along the helical vortex filaments and ( is measured

along a helical line normal to the vortex filaments. It should be noted

at the outset that w, the apparent velocity parameter, is not an actual

disturbance velocity associated with the vortex sheets of the wake.

Rather, it is the speed along the wake axis with which the inner helical

sheetsappear to be moving relative to a coordinate system fixed in space.

Similarly (V + w) is the axial speed with which the inner sheets appear

to move relative to the ducted fan.

If the apparent rigid structure of the inner helical sheets in

the ultimate wake is to be maintained, an observer fixed to any point

on a sheet must see the same distribution of vorticity in the wake

regardless of his angular position on the sheet. He must further see

the same distribution above as he sees below. These constraints on

the distribution of vorticity preclude the possibility of radial dis-

tortion of the inner sheets and amount to a requirement of helical

symmetry of the wake vortex system. Subject to this helical symetry

the disturbance velocity vector will be constant along the helical

lines r = constant and = constant ooth inside and outside the wake.

Now, consider a line integral of the velocity along the path

ABCDA within the ultimate wake as shown in Fig. 3. The velocity diagram

with respect to he rotating blades is shown in Fig. 4. Along BC

is constant since r and C Are constant. AB and CD are radial lines

at two values of T on the sheet surface and DA coincides with the wake

axis. Since no vorticity is enclosed by the path, the line integral

is zero so that

I

IgAT fi

I..

Fig. 3.Ultimate Wake System Showing Paths of Line Integrals.

10-7

01

E:71

N4

F II

A B C D

or

where u t is the value of It on the axis (r = 0).

Employing the helical coordinate relationships

/u rr v +w v +w

- - cos CP + s(Clr) Sin P) + Ur b(O/r) =0

Then

r (V + w)/ORut =ul U§ . V +w V

0 Cos T snR ORw sini

but tan pR = (V + w)/2R and for the coordinate

r tan cp = R tan 3o that

u = U9o {Cos + tany sin yJ

or

u, =u, sinp .

Consider next the line integral along the path EFGHE as shown

in Fig. 3. The portion EH is at r = R-. From the requirement of

helical symmetry, the filaments of the boundary vortex sheet must

cross the line of intersection with an inner helical sheet at every

iI

12

point on the intersection line at the same filament pitch angle

regardless of angular position. In particular, if the pitch angle

of the boundary sheet filaments is identical to that of the inner

sheet as the boundary is reached (r = R) then no vorticity is enclosed

by the path so that the line integral is zero. (The equal pitch

constraint is a lightly loaded condition which will be relaxed later).

Thus

It ( - 9F) + u ° sin PR (§ E -0

or

ut = uo sin

outside the wake as well as inside.

Further, the path ABCDA may be shifted axially so that the helical

portion of the path lies along an arbitrary helical line at the pitch

angle w anywhere inside the wake. The restrictions of symmetry still

require ug to be constant along this path so that ug = U sin p

throughout the wake.

These results are subject to the additional constraint that the

line integral of velocity taken along a path enclosing the wake must

be zero. The line integral may then be taken along the path ABCA in

Fig. 5 where u along ABC is constant and known so that

u z dz + 2'Ru tanw=R 0

C 0

j The integral is then taken along ACMPA Sbjict to hlbjca 1 mmetry

I

13

1A

0

%V, ~

Fig. 5.Ultimate Wake System Showing Paths of' Line Integrals.

° 1 4

the induced velocity is zero along the radial liaes 3 and V. If DE

is allowed to approach infinity then uz along DE is zero. Since no

vorticity is enclosed by the path

f u dz =0

C

Then, refering to the previous integration and subject to the condition

of identical filament pitch angle along the intersection line, this

result requires that u be zero. Thus, the induced velocity must be0

normal to the inner helical sheets. These two coexistent conditions

of equal filament pitch and normal induced velocity characterize the

lightly loaded conditions as discussed by Gray4

The more general condition of load requires only that the

filaments of the boundary sheet all cross the intersection line at

the same pitch angle so that a relative motion between the inner and

outer systems may exist. Thus, the path EFGHE in Fig. 3 will enclose

vorticity and no information will be obtained from the integral for

a heavily loaded ducted fan. However, the requirements on u still

apply inside the wake, and for the apparent rigid motion of the inner

helical sheets u must be proportional to cos cp. That is, from the

previous results and the velocity diagram of Fig. 4

u =u sincp

uC =w Cos Tp

15

A geometry anI motion must now be established for Lhe boundary

sheet which will maintain helical symmetry of vortex r.trength distri-

bution and which will permit the boundary sheet to have an axial motion

relative to the inner sheets. Further, the disturbance velocities must

be zero outside the wake in order that the flow be irrotational there.

This may be shown by considering the line integral about ACDEA as shown

in Fig. 5. EA and CD are radial lines along which u must be zero byr

helical symmetry. ED lies at r = so that uz is zero along this path.

Since the path encloses no vorticity the integral of uz blong AC must

zzbc zero. For this to occur either u z is identically zero or u Lis

part positive and part negative. From considerations of continuity,

for the latter condition to occur the flow must form closed streamlines

within the area defined by the path of integration. However, integration

of velocity along these closed streamlines would yield finite values of

circulation. Thus, a contradiction is arrived at in terms of the

irrotationality of the flow so that uz must be zero along AC. Another

line integral is taken along ABCDEA. For the helical portion of the

path the velocity u§ must be constant due to helical symmetry. Since

the velocities along the other branches of the path are zero and no

net vorticity is enclosed, u§ must be identically zero along ABC.

The path ABC may be shifted along the z-axis to any axial position

with no change in the results. The path is closed by joining the

helical part to the radial parts along the z-path AC. Thus with

u , uz and u all identically zero on the outer surface it may bez r

cotnclildior thatal indued 1 I - J.L- arc-zero on W=

of the wake boundary. This results may be extended to include the

pow

i{ 16

entire region outside the wake by allowing the paths of integration

to expand radially.

The strength of a vortex sheet is equal to the discontinuity

in the velocity components as the sheet is crossed and the motion of

the sheet along the line of discontinuity is equal to the mean value

of the velocity at R_ and R+. At r = R., the induced velocity at the

inner surface of the boundary sheet is given by the components

=~ Uo sincPR

wcosCP

while the induced velocity at R+ is zero. Thus at the line of

intersection with an inner helical sheet the boundary vortex sheet

strength is given b

2 .2 2 2(iY(CB) N sin (PR + w cos cR) (

The velocity of the filaments of the boundary sheet as they cross the

intersection line must be in the direction of the induced velocity

at R and normal to the filament direction; that is

U (u ° sin2 +2 2: CB 2TR + wcos 2 (2)

The filaments of the boundary vortex sheet must then all cross

the lines of intersection between the inner screw surface and the

ny].nnirinal hniindary v As nnnn.rn .n, en~ nB Pno F en a t R his

angle may be determined from the light speed (V,), the blade rotational

Ja

17

speed (OR) and the total disturbance velocity of equation (1). The

two vortex systems are related through equations (1) and (2) but only

along their lines of intersection. On the wake boundary between theso

helical intersection lines, the filament density or sheet strength and

the filament pitch angle will vary with the helical coordinate C.

The boundary sheet serves several basic purposes. First, it

must cancel the radial velocity field at the boundary associated with

the inner helical sheets. Second, it must accommodate the discontinuity

in velocity as the boundary sheet is crossed. Third, it must not induce

radial velocities, and hence radial distortions, at the inner sheet

surfaces. Its remaining function is to preserve, in conjunction with

the flow fields associated with the inner sheets, the apparent rigid

axial motion of the inner sheets and to cancel the sum of vorticity

of the inner sheets. The first and second requirements, along with

the rigid axial motion constraint and the constraint on net vorticity,

are satisfied by the strength distribution and the filament geometry

both as yet unknown. The third condition may be automatically satisfied

by a strength distribution and geometry that are symmetrical about the

lines of intersection. This is simply a restatement of the helical

symmetry requirements on the vortex systems.

Having to solve for both the strength distribution and the

filament geometry of the boundary vortex sheet presents considerable

difficulty and implies the need for some kind of iterative procedure

for locating the compatible strengths and positions of the boundary

sheet filaments. This difficulty may be eliminated through a

consideration of the implications of thp heinsoa syvmnetry requi4remnt.

II_

J-

F 18

It has been established that the boundary sheet strength and filament

pitch angle are constant along an intersection line. At an arbitrary

distance Az below this line, the strength and filament pitch angle

will be different. However, at any other angular position at the

same Az below the intersection line the pitch angle and strength

must differ from the values at the intersection line by the same

amounts due to the helical symmetry requirement. That is, at a

z-position between intersection lines it is possible to change the

sheet strength and filament pitch angle only through the addition of

an infinitesimal strength vortex filament at the pitch angle yR"

tThus, it is possible to replace the boundary sheet by two simpler

systems whose combined effects satisfy all of the conditions mentioned

above.

The first of these sheets is a uniform sheet of helical vortex

filaments having constant density y( B) and constant pitch angle pB

as shown in Fig. 6a. This sheet satisfies the required compatibility

conditions as previously discussed. The second is a cylindrical sheet

of helical filaments of varying and unknown strength but having a

constant and known pitch angle yR as shown in Fig. 6b. This sheet

must have zero strength along the lines of intersection in order to

preserve the compatibility established by the first sheet. It must

have a symmetrical strength distribution about these lines and the

lines midway between adjacent intersection lines and must cancel the

radial velocities at the cylindrical boundary associated with the

inner sheets. Superposition of the two sheets, as shown in Fig. 6c,

must maintain the apparent rigid axial motion of the inner sheets in

19

(a) Boundary Sheet of Uniform Strength.

(b) Boundary Sheet of Varying Strength.

(c) Result of Superposition.j Fig. 6. Concept of Ultimate Wake Vortex System.I

...... .- . . . . . .-..

20

conjunction with the flow fields associated with those sheets, and

must satisfy the net vorticity constraint. These are the conditions

which will be placed on the solution.

It should be noted that to an observer fixed on an inner helical

sheet the boundary vortex sheet (consisting of the superposition of the

uniform boundary sheet and the non-uniform boundary sheet) appears in

terms of the local strength distribution and,geometry to be fixed

relative to the inner sheet. Although the bounjary sheet is actually

slipping forward relative to the inner sheets, the observer sees the

same local vortex sheet strength at a given point on the boundary at

any instant while the identity of the filaments at the point is

constantly changing.

Now consider the geometry and motion of the uniform boundary

sheet relative to the last outboard filament of an adjacent inner

helical vortex sheet. The velocity diagram of Fig. 7 ilustrates

the relationship between the velocities associated with the two

systems according to the compatibility condition expressed in

equations (1) and (2). From Fig. 7

V +W

Xr tan VCB +

where

B 2(~ sin 2 2~ e pvB u si cp.,1+ vi cos cpR) ec90

ii

21

adr

0k

*r40

-P

a1) 012

4 -

94-0

04

8-4 'nG

0

4P

22

From the same figureu, sin tpR

0 = tan ((pR PB) (5)'w cos TR

Combining equation (4) and (5) to eliminate u and substituting for

the fuiictions of equations (3) yields

i+ X .2

X B X ffL 1 + + X L'"

where ii v=)R, 0 W - X. This result shows that, for any choice

of X and W, X B can be uniquely determined. It should be noted that

for vanishingly small values of w, XB approaches X which is in agreement

with the lightly loaded case. Farther, there would seem to be no simple

redefinition of X which would reduce the solutions to a single case for

all values of '. This is in agreement with the earlier analyses of

Gray 4

Solving the last expression for the pitch of the uniform

boundary sheet yields

XB=X - 9+ [(X _ X -w, ) + (6)

Since PB = tan'l XB' the value of Y(CB ) is known according to

Y(CB ) =vwcos YR sec (YR IB) "(7)

The ultimate wake vortex system of the heavily loaded ducted fan

is thus defined in terms of the inner helical sheets of Fig. 5 (one for

23

each blade) having unknown strength but known geometry, a vortex sheet

of uniform strength y(CB) and constant filament pitch angle TB lying

on the cylindrical wake boundary as shown in Fig. 6a, and a sheet of

varying strength but constant filament pitch angle R also lying on

the cylindrical wake boundary as shown in Fig. 6b. For a given blade

number b, pitch X, and loading parameter R, the motion and geometry

of the system are determined along with the strength of the uniform

boundary sheet. The solution of this model for the unknown strength

distributions of the inner sheets and the non-uniform boundary sheet

proceeds directly through the application of the Biot-Savart equation.

-.

1.i -

F-7

CHAPTER III

ANALYSIS AND SOLUTION OF THE WAE MCDEL

From the discussion of the preceding chapter, the geometry and

motion of the ultimate wake vortex system of infinite length are

known. The problem is now to find the distribution of vorticity in

the wake which will satisfy the velocity boundary conditions implied

by this geometry and motion. In this analysis, the Biot-Savart equation

supplies the required relationship between the geometry, motion, and

vortex sheet strengths. For an elemental length of an arbitrary vortex

filament

dv. cos j ds' (8)idi

The integral relations for the velocity components in Cartesian

coordinates are given by Lamb and are repeated here for a single

finite strength vortex filament.

z-z' dz'

Au ~ dz I x- x dx.z-zds' ds' "d

f dx x-x ds

Au di S

25

A vortex sheet of the wake system is divided into a number of

equal width strips. The,,e strips ar-i replaced by vortex filaments of

finite but unknown strengths lying on the centerlines of the strips.

The strength of the filament must be equal to the integral of the sheet

strength across the strip width. For purposes of numerical calculation,

an adequate representation of the vortex sheet is achieved by placing

a filament along the centerline of its corresponding strip, provided

the strip width is sufficiently small compared to a characteristic

sheet width.

The integrals for the velocity components associated with the

finite strength helical vortex filaments introduced in this manner

are more conveniently expressed in polar coordinates so that the

following transformation is made. (See Fig. 2.)

x/ =r/ cosT"

x = r cos T

y r/ sin/ '

y =r sin T

z =z + r' Y tang'0

Z =Z

s =r Y' secp

Enploying the transformation yields .

I! 'Q0

26

uf [r t (r sin T r sin Y)

-r cos i' (z - z- r'Y' tan cp')] p3

U y [r' tan q' (r cos T- r' cos ')

+ r' sin T ' (z - z' " r'"T' tan ') do0 -

U. = Jr' 2 rr' cos (Ti - )Y7

where

-2 r2 + r, 2 2 rr' cos (T' - T) + (z -z' - r'f' tan cp') 2

P

The boundary conditions are more conveniently expressed in

terms of the velocity components along the vortex sheets and

perpendicular to the sheets. Thus

u = u cos T + u sin Tr x y

= (uy cos T - ux sin T) cos y + uz sin

uC = uz cos cp- (Uy cos T - ux sinT) sin

The integrals for these velocities due to a single finite strength

filament become

I

27

AU f [r'~tn~ sin (T'

+ r - - r'T' tan cos (Y' -) dT

0~p3

A r' tan Er - r cos (T T)-0

+ r I[z I- r'Y' tan w'] sin (TY' . cos T0

+ {r' 2 - rr' cos (T'- i)}sin dTL

Au, =Y [{r2 - rr' cos ('- f)I cos Cp

-{r' tanp' [r - r' cos (Y' - T)]

+ r' [z - z' - r'T' tan cp'J sin (' -T)} sin cp] d

where the limits on the integrals refer to the infinite extent of the

ultimate wakte.

Now, non-dimensionalizing these equations with w and R, and

employing the helical relation

r tango = r' tanc' =R tancPR

the elemental velocities associated with one filament become

4I

28

Aur

-- = .R J' tan cD sin (Y' -)

I + - -' tfl(R) OB s?'~)Jd1' (9)+ - - 1" tan YR] cos (T' - 3) -

Y C (D tan p + 2cos (T T)) (1o)

[z Z' " ' ancos (' - ]

T rr

w 4rRw - rr co(T

-tan El [ -Cos (V" - t)

sin (I't - Y) tanR [ g " c0 " I tanoPe] 3

where

f2 p 2 + y,2 2 F-r cos ]2)+[g -z+(Y' -2) co Qv- Ir E -'' -' tan (PR]

cp = tan-l[ tan cp

-I

and F, F' i, z are non-dimensionalized by R.

The boundary conditions may be written by summing the contribu-

tions of every filament of the system. They are,A

29

on the inner sheet:

X (A = cos (p (12)

and on the cylindrical boundary:

Au

-- L 0(1)4)

The uniform boundary sheet induces no radial velocities anywhere

in the ultimate wake and the non-uniform boundary sheet induces no

radial velocities on an inner sheet due to its symmetry above and below

a line of intersection. Examination of the integrand of equation (9)

reveals that no radial velocities are induced at an inner helical sheet

by the evenly spaced inner sheets themselves. Thus, equation (13) is

satisfied identically. The condition on the cylindrical boundary,

equation (14) involves only the inner sheets and the non-uniform '

boundary sheet. The condition of equation (12) involves all three

systems. The remaining constraint requires that the sum of the

strengths of all of the vortex filaments comprising the wake be zero.

HThe boundary conditions may now be written on the inner sheet (from

equation (12))

Au Au AtuAu( + ~ - ( A + = coP ('5)

t-on the cylindrical boundary (from equation (14))

II

30

Au AU (6

and

(I~rrR-) + (IAR-w + (TA ) 0(17)

where refers to the inner sheets, refers to the uniform boundary

sheet, / refers to the non-uniform boundary sheet and

A 1 = cos (p -r) rB

A2 = sin ( PrB

tanBco tan coB and tanD =--tan P

r r *B

For a fixed choice of X and R, the contributions of the uniform

boundary sheet to equations (15) and (17) are fixed in terms of the

sheet strength Y(CB ) and the pitch angle cpB* Thus the system of

equations can be written more conveniently as

SAu Au A

AUr + Au

__)I

311h 3j

andI

(Iilv) WtR) kyy"-W)(20)

Equation OM~3 can be evaluated at a number of control points

on an inner sheet. These points are placed between the filaments

comprising the sheet and are equal in number to the number of filaments

on the sheet (the final point being placed at r = R). Equation (19)

can be evaluated at control points on the cylindrical boundary between

an intersection line and the ucint midway between intersection lines.

The points are again placed between filaments with one less point than

filaments. Equation (20) includes the filaments on one inner aheet,

the filaments on the non-uniform boundary sheet lying between an

intersection line and the adjacent intersection line, and those

filaments of the uniform boundary sheet passing through a line

connecting two adjacent intersection lines. These filaments comprise

a characteristic portion of the vortex wake, altbugh the velocities

must be calc"2lated using alU of the filaments of the wake system.

Now, the integrals of equations (9) and (l1) may be defined

respectively as

I (21)

Ii "l"C = cos p) (22)

and the system of equations may be written as follows:

32

for the control points on an inner sheet

4 ' ( -)+ IC V fj.)A - A)&Jicos CP (23)

for the control points on the cyli. -ical boundary

Ir (TRw + Ir (FRw) 0(24)

and

c R- + c cVW L U -

where

is over the filaments of one inner sheet,

Sis over the filaments of a characteristic

c portion of the non-uniform boundary sheet,

and

is over the filaments of a characteristic

C portion of the uniform boundary sheet.

With the equations in this form, the influence of i is confined

to the right hand sides of the equations through the velocity field of

the unifom boundary sheet, and the coefficients I r and IC depend only

on the choice for A (the pitch of the inner sheets) and b (the number

F of blades).

I

m

33

Owing to the simplicity of the uniform boundary sheet, the

velocity field inside the wake associated with this sheet has been

evaluated explicity in terms of X and V in Appendix II. The result

is

u z Y(CB)=- Cos (P

= 0w W

Using equation (7), can be written as

Sz= cos cPR sec - B) cos B

or

Uz- 1w 1 + XXB

Then, the right hand side of equation (23) becomes

1 -A s - C j P = 1 i + XxB (26)

Further, the right hand side of equation (25) becomes

- = (2rR XB COS CB/b)c

Since 2rrR XB/b is the length of the line defining the character:-.stic

portion of the uniform boundary sheet and y(61) is the sheet strength

Ii

[ ,a

34.

per unit length. Thus

A B (27)

Some rearrangement of the terms of equations (26) and (27) yields

± =__ 1_ V KX B (28)

X ~2II~- NX B~i(29)1 +B

From equation (5)

w 0Co C R tan (9) R cDB)

or

So that, de fining

G 1 w 0(30)

equations (28) and (29) may be rewritten as

B 1 + 2

735

I

Thus, the system of equations can be written

at the inner sheet control points

I v + Ig=GICITrR) IW -) 1 + 2'

at the cylindrical boundary control points

X Ir (Rw) + r 0

and

V 2)

c1 X

Since the right hand side of every equation is multiplied by G (or is

zero) a new vortex filament strength can be defined as

and the system written

at the inner sheet control points

SI + 3 iq 2 (3'

*~ X.

te

-r.--**- -- -*----

36

at the cylindrical boundary control points

Ir + L3Ir 0 = (32)

and

c C

The system of equations in this form does not contain the R parameter,

so that a solution may be obtained which may be scaled directly for

any value of i. That is, the equations are solved for G = 1 (V O)

and the wake vorticity distribution for any value of R is obtained by

multiplying the result by the appropriate value of G.

The blade bound vortex strength is found at any radial station

by summing the filaments of an inner sheet lying inboard of the radial

station in the ultimate wake. The remaining elements of the solution,

the strengths of the filaments of the non-uniform boundary sheet, are

required in the calculation of the power and thrust. The evaluation

of the coefficients I and I of the unknown filament strengths, ther

positioning of the filaments and control points on the sheets, and the

simultaneous solution of the system of linear equations for the unknown

strengths are all considered in some detail in Chapter V.

37

CHAPTER IV

CALCULATION OF THRUST, POWER, AND INDUCED EFFICIENCY]IThe solution for the distribution of vorticity in the ultimate

wake allows a detailed calculation of the induced velocities within

the wake. A knowledge of these velocities, and hence of the momentum

and energy in the wake, leads to a straightforward calculation of the

thrust, power and efficiency.

Thrust

Following the analysis of Theodorsen consider a control volume

enclosing the ducted fan and its ultimate wake as shown schematically

in Fig. 8. Using the momentum theorem the thrust of the ducted fan may

be found by considering the average pressure forces acting on the control

surface and the average flux of momentum through the surface. These

averages are taken over a time At = 2rr/bO and the integration is with

respect to time, dt = dz/(V + w). Thus, from Fig. 8

p + f dt dS= f p (V+ u)dt (V + uz)dS

p V dt V dS

which csa be written as

4Z

38

t

cx4.5

+E-"

I I

I I

|I I

I I!I

X

39

T+ bO f(p - P) dz'dS = f P )(V Uz + Uz2 dz dS (34)

In order to integrate equation (34) the pressure term must first be "

eliminated by employing the equation of motion for an unsteady,

i incompressible, potential flow.

LO + + 1 V2= f(t)

Since there can be no induced velocities at an infinite radial distance .

from the wake axis

.Ft' + pip ", V22 p Aol (35)

The unsteady term may be eliminated by considering the potential

field in a steady coordinate system such that

t(r, T', z) § (r, T orz

Then

Ft-T g - ut Or x N M

Nov u,, roy be written as

= osin , cos co - u. sin c

and

-A

7._ _ _

tt p

4o

I;a! az = sin 2 CD + u, Cos CP

Thus

V 1 flsin = tan Cta - 9sin " p tanep

u, becomes

= tan cp w(l -G) -1 tan Tp

V + wand, since tan cD = =x L inside the wake becomes

: (v + w)uz + W(v + w,) (1 - G).

Then inside the wake equation (35) is

p/p + V2/2- (V+ w)u + w(V + w) (I- G)= p /p (36)OR

The pressure must also be constrained by a static pressure

balance at the wake boundary. Since all disturbances vanish outside

the wake, the constraint becomes PR =p or

,p 1 R - .+P(vw+w)UzR - (V + w) w( -G)oR 2 R- 00

Then using equation (36) to eliminate p yields

41

(p. -p) .P(V 2 -V12 )-p(V~ + W) ( - u )(37)2 ~ZR_

where

2 U

, V = (v + u) 2 + N5kR 'IT

Substitution of equation (37) into equation (34) yields

2T= (V +w)J L~uz z 2\IVVolume

InsideWake

~B uZ 2 + 2 2 R

+ (V + w) (U' - U ] r dz d

The limits of integration are taken over a characteristic volume of

the wake such that

0 O z ! 2iprR/b

Then defining non-dimensional lengths as

42

zz/(2rRX/b) 0 o 9 1

= r/R , 0 ' 1

the thrust can be written as

b~~~p 2 2rr -2 12T R (2rrRX/b fIpfU+u [(V +uZ)2

+ rV ' JJ L zz20 0 0

-(V u + +(v + )] d12(V Zu _ r uy -- CO ( z - _R

Now, non-dimensionalizing the velocities according to i = u/w and

= w / R y i e l d s

S=2 (O l r' 1 z + U2 1 +

w ZRr z 2 R!O 2 -2-

%c/OR1+ (-:/. + l - )JF dF d! dY /aT

Noting that (V/oR) V( - i) and defining a thrust coefficieat az

CT T/[p (OR).R 2] (38)

the resu]lt may be reduced to

I

~43

C 2 -2 2[U2 + r + ]U2Z + (39)T0 0 0 .z z 21 2

]Y d-r d-z dfI/2r

Some of the terms may be integrated immediately as follows.

Consider a line integral within The wake as shown in Fig. 9. The

integral about the path ABCDA encloses those filaments shed by a blade

bound vortex between x = 0 and x = x (x =) so that the line integral

yield P(x) and

B?

ur dr + C"z dz + ur dr + uz dz r(x)A B C D

But, from the requirement on velocity discontinuity across the sheet

(for which ur of the filaments is zero)r?

ur dr =-J urdrA C

so that

Suz dz- u (2rrRkb) r(x)

D 0

or

dz 2rRwXbr(x)-

00

I 44

R CYLINDRICAL

r ~ HEIA

C DI

WAKEAXIS

I:N

* Pi;,nrc- 9. A Path of Line Integration in the Ultimate Wake.

~45• f

I

Rnploying the definition of Theodorsen for the non-dimensional blade

bound vortex strength as

K(x) = 2(. (40)

yields

Jo i d = K(x) + U

0

so that

fl flfr 7r d l 1 [K(x) + U ] x dx dT/2rr0 0 0 0 0

[ 2 fK(x) xax .y +0 0

But the integral term is defined as X, the propeller mass coefficient,

=2 f K(x) xcdx . (1i)

0

Thus

1 Il rJ' i d Uz d dY/2rr + Ut (42)0 0 0

If Y is taken as = x = 1 then uz becomes u so that the integration

of the last term of equation (39) becomes

" R

F'

f46

f

f 17 Fdr d! d,,,/2r [Kl + rt(3)0 C ' -

The remaining terms of the integral of equation (39), which

must be evaluated numerically, may be defined as

1 1 2iTz 2= F . df dZ dY/2n

0 0

r r dF di d'/2rT0 0 0

=~r -2 y~ dr- di dY/2rri0 0 0

1R ~ 1 l2rT >2 2~ ]d7ddY/-R "R

and e e + e + e Employing these definitions and equations (42)

and (43), equation (39) becomes

C,- K(l) + ( 1) + 2 C" + CR] (4)

It has been established that the flow field of the uniform

boundary sheet is known and that the vortex sheet strength distributions,j and hence the flow fields, of the sheets of varying strength need only

rW|I

be solved for the lightly loaded condition. It is possible, then, to Icalculate the thrust coefficient in terms of the 9 = 0 veake solution

and the scale factor G. Referring to equation (39) the integral terms

can be modified as follows. The velocities are separated into those

associated with the inner helical and non-uniform boundary sheets

(variable strength, subscript vsR) and those associated with the uniform

boundary sheet. The vs. velocities can be scaled according tow

%s = %So = Giivs. vs0 vs

Further K(x) = G K(x)o G Ko(x ) and G X From

equation (28) the velocity associated with the uniform boundary sheetG X 2

is simply U z = 1 + k2 for all Y. For simplicity in. the following

1~J 1 2rrdevelopment d(vol) will be taken to represent 7 d F dF d/2rT.•j

0 0 0

The integrals in equation (39) can then be written as

f Vd(vol) = (1 - + ]2 d(vol)vs l+

G2 U 2 d(vol) + 2G (1 .X. d(vol)

vs Piz vs

2 + 2

f d(vol) G? G 2 iv d(vol)

II

~J

D m

48

J d(vol) G2 2 d(vol)

2 1 1 +2

d(vol) d(vol) + 2

From equation (42)

,fiz do(vol) = [

vs 2 + 2

and from equation (43)d~vol 1 [Ko(1)

z vsR 2 +2

Substituting these results and the results of equations (42) and (43)

into equation (39) yields

C+ 2W 2[1 .2 G, , Ho + _), 1 ,+ ; RG

-l--

vs 1+ x2 2

4+9

12+ G- G )2 2 2 CJ'fi 2 d(vol) + JU s d(vol)l

vs +s S

:1. 2 G X2)( 1+ ' [G dvol) + G~2 K0l

+ + 2"

+ G X 2 )2 1 +X2

+ G2 VSR d(vol)J- (G K(1) + 1 - G)]

The remaining integrals, which must be evaluated numerically, may be

regrouped and defined as

2ri l -2 -2 _2 2e f f f[. z. - Ur - +u z r dr dz d'/2i (45)

0 00 vs vs vs vs R

Using the definition of equation (45), the thrust coefficient may be

written as

+22 G go (46)

G) 2 (+- 12)2]

+ 2," i+ X2

X -,-

1+X

50

where, for a given b and X, Xo0 Ko(1) and e are calculated from the

= 0 solution. Since G is calculated algebraically for a given X

and 9, equation (46) provides an entire family of values for CT whereas

equation (4) provides only a single value.

KPower

The ideal power required by the heavily loaded ducted fan may

be determined through a consideration of the induced energy loss in

the ultimate wake. Following Theodorsen the energy loss, E, is

given by the methods of classical mechanics as

E Q TbO E! Vv 2 (I P v2 pP)u)d (vol) ('47)2rt (V + ) 22

2where Q is the torque of the fan and v is the magnitude of the total2 2 2 2

induced Velocity, v = uz + u e y+ . Substituting from equation (37)

for (p - p), the energy loss becomes

SE= fw [I Vv + Wzu+U u gv -WUz d(vol)2rr (V + w)J 2 1 z 2 wu)]Rv

Then, defining a non-dimensional energy loss as

e = E/Ep (OR) 3 TrR 2] (48)

and non-dimensionalizing velocities and lengths as was done for the

thrust coefficient, e becomes

~= w j j j Lu ++gu vR -i )]f d-r d-d'/2r

0 0 0 " ZR

51

Again, dividing the induced velocities into those associated with the

uniform boundary sheet and those associated with the variable strength

sheets, the expression for e can be calculated in terms of G and the

w=0 solution. All manipulations and integrations are similar to those

performed in the thrust analysis. The result is given as

3F +l G X2 -G X 2

3. [G 2) (49) +X

MIR,

2 2 2 02

_+ 1)l+_X ) +X )

+X (2 X 2 2

2)2(++ 1)+Xol

vs

+ G2 (XlJ' + u- r d(vol)vs vs

+G2 (1 G X f[ 2 +u,-2 )d(vol)

-2 ~ z ~vl

l+X vsvvs

22 2 U ~v

vs vs vs R

52

Of the remaining integral terms the first three are evaluated in the

calculation of C for the thrust coefficient. The last two must be0

numerically integrated for the R = 0 case in a similar manner.

From equation (47) the power can be written as

so that defining a power coefficient as

Cp= P/[p (OR) 3 rR' ] (50)

yields

CP (X - ) CT + e (51)

The power requirement for a constant diameter wake can also be

calculated directly by the Kutta-Joukowski theorem. An increment of

torque is given by dQ = bp Vaxial r rdr where Vaxial is given by

Vaxial V + it sin cp +u, cos

v +w ( - GXx + x

Then

d= p (2rrRwX) K(x) V+ w (1 - 2,G X2 2 R2 x dxx + X

and

'P kiaimj ±~AL V + W)J K(x) xdx 2 GA" j K(x) 2 2 j0 0 x + X

I.

53

From equation (41)

1

=2 K(x) xdx0

Defining the remaining integral as~x

2 K(x) "2 x2 dx

0 x +

the power can be written as

p = ; p(TrR2 ) (nR)3 j. [.X - p]

Then, since X = c2o and p = Go' the power coefficient is given by00

C, G w2 [Xo " - i~o .10 (52)

Efficiency

The induced efficiency is defined as = TVJP. Thus from

equation (50)

(x - )T1i--( R- c T -e •(53)

Efficiency may also be written as

= (x .R)CT/Cp

54

CHAPTER V

IdMERICAL PROCEDURE

In the preceding chapters the model and its mathematical solution

have been outlined in terms of the basic procedures and developments

required. To obtain such a solution it is necessary to evaluate the

velocity contributions of a filament of unknown strength at an arbitrary

location in the flow field of that filament. This evaluation will be

carried out by numerically calculating the integrals of equations (9),

(10) and (11). Using these results a system of linear equations will

be developed in terms of the unknown filament strengths of the system

by equating the sums of velocities at control points in the ultimate

wake to the required normal velocities at these points. In order to

specify the parameters of these equations, decisions will be made about

the manner of subdividing the vortex sheets into finite strength fila-

ments and the placement of control points on the sheets.

After the system of equations is determined and solved, the

integrals of equations (45) and (46) will be evaluated in order to

calculate the thrust and power of the ducted fan. In addition the

values of Ko(x), X and p will be evaluated.

Evaluation of Velocity Components

The evaluation of the velocity components associated with a

single finite strength helical vortex filament at an arbitrary location

in the flow field of the filament depends (in this analysis) on a

1TO

55

numerical integration of an integrand which is solely a function of the

geometry of the filament and the point at which the velocity is to be

calculated. The expressions for the components, equations (9), (10),

and (11), are repeated here for convenience.

+r -' tan sin s (T) Y -

Au o 3

-- I etanR + -- - cos (T'

+ ( - o - " tan cos ) -(Y } 3d-.0 R ,

,i

Cosw { - cos y2 -

V iiV

tan2 -- cos ((1 -)

. sin (Y' - T) tanPR ( " tan R "

where

2 =.2 + r - 2 ' cos (T' -T) +[ -'' Y' tan- -2,

cp tan- tan D)

F- -. . . . . . . .2 . . . . .

MIPM

56

and the primel dimensions , T' refer to the location of an

elemental length of the vortex filament. The unprimed dimensions refer

to the location of the point at whicb the velocity is to be calculated.

The computations needed to numerically evaluate these integrals

may be simplified by converting their limits from - - 1 15 - to

0 : Y' 5 -. This is done in the usual manner by splitting the integral

at Y' =0, switching the limits on the - T 'S0 portion and

rredefining T' to - T' in the negative range. The resulting velocity

component relations are (using the substitution X = tan CR)

Au rr f f [Y' X sin ( t -

0

+ r' [ - - xY'] cos (y' - T)] --

- [Y' X sin (' + T)

- r' [- 2o + Xl ' cos (' + 'T)] }2

Au Cos f y r +V 4ff ;x 1-r -2 cos (Y ' ',)

0

+ [i- V- XY'] sin ('T' -'T)] ..1'31

57

+ ' x -,-+ 2 cos (T*' + 1)

- - o2 [--r

+----- ] sin (' - T)] d- '-A ]

2

"I -py 2 Fr' cos (T' + )0

0--3

2 F- ' (T +T

_--cos ' +

r

+-- X sin (T' + T) ' - + XT'-r]

r o p3

where

-2 -2 -/2 t.X,2P[ = r + r '2 ' cos ('' T) + [I

22 -0

- A £1 '- ( + + ,]

-2-2 + 2 r 'csi (' +) +z [- + vo~ a

whr

II

58

For convenience in the ensuing discussion the integrands of~ the velocity

components ar'e defined by f r f£ and f Csuch that

Aur

AugC

au trpzia rule inegrto suc that E )

VU 3.,w61 -i = ark) ~ 0~

of ncldedange 2T/k f 1 caclae at .2T1 = , 2n (2r/),..

k,(2r/k an thee vlue a r re as~-- the ar calulae.Te h

59

(Note that 81 is a coefficient of velocity subject to multiplication

by the non-dimensional filament strength y/4TTRw.) 8 measures the

contribution of the first turn of the helical filament (and its

reflection for - 2a!5 T' - 0) to the radial velocity associated with

the filament. This result is stored and the contribution of the turn

defined by 2rr T 4n (and its reflection) is calculated in the sameAumanner yield r Then &2 is compared to the sum of 61 and 82

A percent change is defined as 100 x 162(61 + 82)1. If this change

is not sufficiently small then 63 is calculated and the comparison is

made again according to 100 x The process is continued untilE i

i

Aurthe percent change in the coefficient of - is less than some specified

tolerance. The coefficient is then set equal to the sum of the contri-

butions of all terms for which 8i was calculated.

Clearly, the accuracy of this result depends on the choice for

k and the specified tolerance. The magnitudes of these parameters

depend in general on the geometrical values F, r/, YR and on the

number of blades in the fan. It would be possible to control these

values very closely by a process of repeated calculations and compari-

sons within a given integration. For example, 81 could be evaluated

with k = 90, then reevaluated with k = 120, again with k = 150, 180

and so on until two successive values were identical to an arbitrarily

specified number of significant digits. Then 82 wmild be handled in

the same manner. Similarly the 8 contributions would be summed until

the final contribution failed to change the sum in the chosen number

I "

60'

of significant digits. It is apparent that such a procedure could be

enormously time consuming, and the choice for the number of significant

digits to be carried must be made very conservatively. Choosing the

best alternative at hand, reliance is made on the specification of

k minimum requirements over a broad range of cases.

The values of k and the tolerances for the velocity componentsFwere sized somewhat subjectively on the basis of their effects onsolutions of the wake vorticity distribution. The method of sizing

a parameter consists of fixing all but one of these at very conservative

ft levels and systematically varying the remaining parameter over a wide

range. This procedure is repeated for various values of b and X and

the resulting wake vortex strength distributions are compared. For

example, k was varied over a range from 90 to 720. It was found that

for k greater than 180 the solutions for wake vorticity were essentially

identical for b and X taken over very wide ranges. That is, the value

of K 0(1) at b and X was changed by less than 1/2 of one percent by

increasing k from 180 to 720. On the basis of this result k was fixed

at 180 for all subsequent solution, obviating the need for any repetitive

calculations.

The permissable tolerances for summing the i contributions to

and Au were treated in a similar manner. The minimum values wereW w

found to vary with the choice of b and X but in general could be fairly

large due to a favorable compensation of the truncation errors. For

example, the value for L for ten turns of a filament might be only

95 percent of the value for 30 turns, but the errors for the w valuesiwfor the otner filaments would be nearly the same so that solutions for

61J6

wake vorticity (or for K(l)) would differ by perhaps 1/4 of one per-

cent. Thus it was decided that the tolerances on w and could be

set at two percent for all subsequent solutions without incurring

significant errors.

For the calculation of Ur the truncation tolerance was found

to be strongly dependent on the characteristic axial separation of two

adjacent inner helical sheets as measured at the wake boundary. This

characteristic length may be non-dimensionalized by the wake cir-

cumference to yield

~2 R X/b =Xb(5~ (55)

When z c is small problems arise in accurately calculating the radial

velocities. That is, when a calculation point lies on the cylindrical

boundary, the radial velocity due to a filament in a helical sheet

lying to one side of the point must be summed with the velocity

contribution of a filament at the same radial station of the helical

sheet lying to the other side of the point. The radial velocity

contributions of two such filaments will be nearly equal in magnitude

but large and opposite in sign. Thus an t.cceptable calculation of the

sum of these contributions requires accuracy to a large number of

significant digits.

The primary influence of these calculations on the wake vorticity

solution occurs in the sizing of the non-uniform boundary sheet strength.

If the percent change of 16 8 i is not held sufficiently small the

strength distribution of this sheet may become quite Irregular, changing

sign and generally degenerating. Fortunately, the magnitude of the

InA

62

non-uniform boundary sheet strength becomes very small (as compared to

K(x)) when 'c becomes small so that the effect of inaccuracies in this

sheet strength is of secondary importance to the solution for K(x) and

to the calculation of velocity profiles in the wake. (When ' becomesc

zero the model reduces to the infinite blade case for which the radial

velocities vanish along with the strength of the non-uniform boundary

sheet.) It was found that the solutions could be adequately controlled

if, for z < 1/4 the tolerance was held to 0.1 percent; for 1/4 <

< 1/2 the tolerance was 1/2 percent; and for 1/2 < < 1 the tolerancec

was held to 1.5 percent. Higher values of z are not considered to bec

of practical interest.

System of Linear Equations

As may be seen in the preceding section, the velocity component

contribution of a single filament at any point in its flow field is a

linear function of the vortex strength of the filament. If the

contribution of n filaments to a velocity component, say Au /w, at a

given point are summed then the result is of the form

g1 + g 2 2 +g 3 3 + " +gnn = /cs cP

where gi is the integrated functions of geometry, f related to the

set of filaments (one for each blade) of strength 7i and to the

calculation point, F, T, Z. If the filament strengths (7i) are

unknown and the geometry of the filaments is known, then specifying

a value for C at the calculation point yields a linear equation in

Yl' Y2 . . . n" If n such calculation points are chosen and the

63

velocity component is specified at each point, the result is a system

of n linear equations in n unknowns. Such a system of equations is

used in the solution for the distribution of vorticity in the ultimate

wake. The inner helical sheets and the non-uniform boundary sheets

must be divided into strips and the strips replaced by finite strength

vortex filaments. In order to arrive at some criterion for a minimum

number of finite strength vortex filaments to be used to represent a

sheet, the blade circulation for an optimum free propeller was calculated

using the methods of this chapter for comparison with the results of

10Goldstein. Solutions were generated using 4., 6, 8, 10, 12, and 18 .,

filaments per sheet. The results, which are considered in greater

detail in Appendix III, indicated that ten filaments constituted an

adequate approximation to the vortex sheet (i.e., a maximum value of A

K(x) of about 98 percent of Goldstein's value).

Initial calculations for the ducted fan wake model were performed

using ten filaments to replace an inner helical sheet and eight filaments

to replace a characteristic portion of the non-uniform boundary sheet.

Because of the symmetry of vortex strength in the boundary sheet the

eight filaments introduce only four unknowns. Control points were

placed between adjacent filaments using ten points on the inner sheet

at YT = 0 and three points on the cylindrical boundary. This arrangement

is illustrated schematically in Fig. 10. At the ten control points on

the inner sheet the contributions to :ycoscp = 1 are calculated and

summed to yield ten equations. At the three control points on the

cylindrical boundary the contributions to U_ = 0 are summed to yield

three equations. A final (14th) equation is written in terms of the

lo

WAKEAXIS

INNER HELICAL SHEET

X( CONTROL POINT

0 VORTEXC FILAMENT

CYLINDRICALBOUNDARYANDI NON-UNIFORMBOUNDARYSHEET

INNER HELICAL SHEET

Figure 10. Schematic Diagram of' the Arrangement of' Vortex Filamentsand Control Points in the Ultimiate Wake of the Ducted Fan.

65

-net vorticity. The forms of the right hand elements of the equations

are modified to include the effect of the uniform boundary sheet as

discussed in Chapter III so that the system of equations takes the

form

'A,1 ql + 1,2 q2 + + A,14 q14 X2 /1(l + X 2

A2,1 l + A2,2 V2 + + A2,14 Y14

= X2/(l + x )

A1o,1 ql + A10,2 q2 + * ' " + A1o"i4 q14 =X 2/(i + 2)

All,171 + Al,202 + " " " + A ll ,1 4 q l 4 = 0

A13,1 ql + A13,2 q2 + + A13,14 14 = 0

A1411 + A142'7 + + * 1,l.l bF_/ + X20

A,', through A0,10 are sums of b of the integrated function, f, with

the sum taken over the corresponding filaments of each inner helical

sheet. A,, through A10 14 are the sums of b integrations of f for

the corresponding filaments of the characteristic portions of the non-

uniform boundary sheet. Similarly, All', through A13,10 are integrations

of fr for the inner sheets and A1,11 through A13,14 are the integrated

coefficients for the non-uniform boundary shent. A1, 1 through A,10

are 1.0 and A.4 ,1 1 through A4,14 are 2.0. i through qIO are the

66

unknown strengths, y/(4TRvG) of the inner sheet filaments and Y

through q14 are the unknown strengths for the non-uniform boundary

sheet filaments.

Thorough investigations of the effect of the total numbers of

filaments and control point locations and their spacing on the inner

sheets indicate that the arrangement shown in Fig. 10 is satisfactory

and does not required modification with' (i.e., X and b). Thec

satisfactory numbers of filaments and control points on the cylindrical

boundary were found to be clearly dependent on z . That is, forc

z c < 1/4 6 filaments and 2 control points suffice; for i/4 < 'z < 1/2

8 filaments and 3 control points suffice; and for 1/2 <7 < 1, 12

filaments and 5 control points are adequate. The decisions for

adequate arrangements were made rather subjectively on the basis of

comparisons of a blade number family of solutions to the exact infinite

blade solution by Gray13 . A further criterion which involves the

smoothness of the vorticity distributions on the sheets was used as

a measure of the convergence of the sheet strength distributions.

In Chapter II it was noted that the strength of the non-iuiform

boundary sheet must be zero at the lines of intersection and must

maintain helical symmetry between lines of intersection. In order to

meet these requirements the sheet strength was defined as a series of

trigonometric functions such that

N r2n - 1y= Z a sin I- Tr

n=l n

67

where N + 1 is the number of control points on the sheet. The series

is integrated over a strip width of the sheet to yield the corresponding

finite filament strength. If 2 w represents the strip width such that

w = 1/4N then the filament strength is

I -W n-- 2 2i-w

The strength distribution of an inner sheet is also expressed in

series form. As F approaches 1.0 it is necessary that OW become zero

so that for points immediately above and below the sheet Ur may vanish. ISince the strength of the sheet is*' d r(x) this restriction requires ,

Ydx

that be zero at x = 1.0. Thus the bound vortex strength may be written

as

= BN- sin [ L - X • (55)2rRwX n=l 2

To calculate'the strength of a filament of the inner sheet the difference

in bound strength is taken at the radial stations corresponding to the

edges of the strip which the filament replaces. Thus

N-1 2n2 1 [r Co [2n -1 xS2Bn sin wl os 2

n=l

The system of linear equations is formed in the same manner asj

before by summing the velocity component contributions at a given point

due to all of the filaments of unknown strength. However, fhe unknowns

now become the coefficients of the two series for V. That is, at the Ipoint (F, T, 1) n is fixed and the contributions of all filaments for

......... i.

68

B n (or a n are summe to yield the coefficient Au.A For exuMleg set

n =1 in the B nseries so the coefficient for B 1 is

=- Z~ f ily [sn(f.w CB(f ~

The system formed in this manner is equivalent to the previous system

(with discrete filament strengths) vith the adeLitiolal const~raints on

the end values of the vorticity distributions.

Solution of the linear sys tem for either the filament strengths

or the series coefficients is carried out by the Gauss method of14f

successive eliminations as presented by Fa-deeva . Having solved for

the wake vorticity the blade bound vortex strength distribution may

9 be calcul~ated directly from the series of equation (55). X may be

calculated by integration of the series to yield

4N-1 /2n -1 r- [ \ f_)~ 2Ho Er '-1 n T B n

and

2b N1 [2n- 1K (x) E B sinL2--r x]n=l n

POmay be obtained by a simple strip integration based on filament

strength such that

PIb.i x 2 2 w}

0 (X 2 + X2

69

Thrust and Power Integrations

In Chapter IV the thrust and power coefficients were shown to

contain integrals of the velocity distribution through a characteristic

volume of the wake. These integrals, which must be evaluated numerically

are

J'2 [-u2 -2 -2 -2 -2 ]y dr-, < d '

0 0 0 vs vs vs vsR vsR 1

and the integrals occurring in the power calculation

2rrS k dd~~/r

0 0 0 vs vsR

and

i'1 ~~ S'S );~+ ] diF dY d'Y/2rr0 0 0 vs vsR vsR

When the strengths of all of the vortex filamLents comprising the

ultimate wake are known, the components of induced velocity can be

calculated at any point in the wake by evaluating the integrals for

each of the filaments and multiplying the result by the filament

strength. The procedures for integrating and summing the contributions

of the filaments are the same as those used in setting up the system

of linear equations. These results are then projected to yield the

components of induced velocity in the r -, z -, and T - directions.

Due to the helical symmetry of the vorticity and velocity distributions

in the ultimate wake, the volume integrations may be performed by

I' I '- ~"....

!A

70

obtaining a detailed knowledge of the flow field on a IF - surface.

This surface is bounded by the wake axis, the cylindrical boundary,

an inner helical sheet, and the radial line midway between two adjacent

inner helical sheets. The surface is divided into a network or grid.

At the intersection points of the grid the induced velocity components

are calculated, their squares are calculated, and all of this informa-

tion is formed into two-dimensional arrays of data. A typical

1. integration surface is shown schematically in Fig. U1.

Since the motion and vorticity of the wake are known, nume.rical

evaluation of the velocity components at the edges of the Z - surface

is not required. Specifically, at Y = 0 = Ur = 0 and Uz = o At 1= O,

22 2 2 2 X dr(x)U = - G/I(Y + X2 ), . /(F + X ), and = .I.z r b dx

2 2 2 OAt Y = 1 Ur = 0, Uz: 1 - GX2/(F +X)+ 2 cos pR' =;Y G F/

(12 + X2) + 2 sin (Here,w is the strength of the non-uniform

boundary sheet.) At ' = 1/2 Ur -- 0 but Uz and I must be evaluated

numerically. For the arrangement shown in Fig. 1l, all of the velocity

components must be evaluated at the internal grid points of which there

are 27, and the i and U values must be calculated at the 9 lower edgez

points.

The elements of these two-dimensional arrays of data are then

squared to form three new arrays denoted by U2z, ur and "

i~ ij i ,jiU i jTaking the U array as an example, the integration mayJ be performedzi~

by a simple strip method such that

71

WAXEAXIS

i, CYLINDRICALBOUNDARYINNER HELICAL SHEET BJD-

S3 4 5 6 78 9Jo

, 1 0

-L-2 2 "

' I

L LINE OF SYMvMY

INNER HELICAL SHEET

Figure 11. Network Used in Calculating the Velocity Integralson the F - surface.

L Ii I

72

2 r I Ji 1 ) [ ,U 2 + - 2Uz Ffr d d d/ = E E + uz -i=0i-0 412,ji+j0 0 0

+ T12 + U2zi,j+l Zi+l, J+1j

where (I + 1) is the number of grid points in the F - direction and

(J + 1) is the number of grid points in the Z - direction. For the

network illustrated in Fig. 9, I 1 10, J = 4 and 412j = 1600.

For the integrations involving the (vs) velocities, calculation

of the array members is restricted to the filaments associated with

the inner helical sheets and the non-uniform boundary sheet. The edge

values become: at F = 0 = U-r = 0, U - -/(l + X2); at= 2 - ...2 + x2) at -1 1 vs vs vs

'i=0UZvs +x 2 P Tv =-GX F/(r + x) and i .

X x at 1; Ui =0" ri =-G/(l + x2)+ 2~sn2b dx r vs vs Ysn(R

Uz =2 At 1/2 Ur =0 and U and z must be

zvs YcsPRvs zvs vs

evaluated numerically.

The accuracy of the integration technique is clearly dependent

on the choice for I and J and on the strip method employed. As a check

on the method the Uz velocity distribution was calculated for several

values of X and b and integrated using the method outlined above. From

equation (42) of Chapter IV

dvol = [X + i

73

Using values of I = 5 and J = 4 the numerical results agreed with the

more exact results of equation (42) to within about 1 percent in all

cases. (See Appendix III.) As a further check the values of e r'

and ez were calculated in the same manner for X = 1.356 and for

increasingly large blade number. Their convergence to the exact value

for the infinite blade case1 3 was satisfactorily observed and is

considered in more detail in Appendix III.

4:E

/p

N ': /

I . .,

i

_ . . . . .. . . - ~ - - ~ .I .

It

t

74

CHAPTER VI

RESULTS

Following the initial checkouts of the numerical procedures,

the methods for the generation and solution of the system of equations

for the wake vorticity distribution of the lightly loaded ducted fan

were programmed for the digital ccmputer. The system was solved first

for a value of X = 1.356 with b = 2 and 4. The results for the blade

bound circulation, K(x), were compared to the electro-potential analogy

results of Gray . These comparisons are shown in Fig. 12 and Fig. 13.

In both cases the agreement is considered to be good.

Solutions were also generated for comparison to the theoretical

results of Tacbmindji 2 who used the velocity potential approach. K(x)

is shown for the two methods in Fig. 14 and Fig. 15, with X = 1/3 and

2/3 and b = 4 in both cases. The agreement is again considered to be

good.

3For compari-son to the exact theory for a ducted fan having an

infinite number of blades, solutions were obtained for b = 2, 4, 6, 8,

12, 16, and 24 with X = 1.356. The results for the blade bound vortex

strength show excellent convergence to the exact solution as seen in

Fig. 16. Since the measure of convergence may be seen more easily in

terms of the mass coefficient, the values of 9o for increasing bladeInumber are shown in Fig. 17.

1 75

0.0X=1.356

b=2

0.1.5 --

00

0.05

-- DIGITAL COMM

0 POTE UAL TAZ~

10 0.2 0.4 0.6 0.8 1.0

Figure 12. Comparison of Two Methods for Determining the Blade1Bound Vorticity for a Lightly Loaded Ducted Fan.

76

I 0.25

X=1.356

b=4

0.20 -

,0I/

0.15

K(x)

0.10

/4/

-- - DIGITAL COM MU

/ 0POTE IAL TANK

M EMAS4

0 0.2 o.4 0.6 0.8 1.Ox

Figure 13. Comparison of Two Methods for Determining the Blade BoundVorticity for a Lightly Loaded Ducted Fan.

77

1.0

X=1 /3

o.8

o.6,

K(X)

o.14

jI2o.2 DIGITAL COMPUTER

0 0 0.* 2 0. o.6o 0. 1.07

Figure 14i. Comparison of Two Methods of Calculating the BladePA-And~ Vewtigiitv frsr aL gT~i~y-l TLAaA Nu'~ted Fin.

78

1.0X=2/3b=4i

0.8

0.6

K(x)

0

0.2 O44 76

Figure 15. Comparison of Two Methods of Calculating the BladeBound Vorticity for a Lightly Loaded Ducted Fan.

79

0.4-

IPI TE UM MRoF B:ADES30.35 -DIGIAL COMP M 24

16

0.308

0.25 - -

4i,/0.20

K(x)

0.15o ___ /I _

0.10/ -

O _lO / V

0.05

00 0.2 o.4 o.6 0.8 1.0

x

Figure 16. Comparison of the Blade Bound Vorticity fora Lightly Loaded Ducted Fan for SuccessiveNumbers of Blades (Digital Computer) withthe Infinite Blade Solution (Exact 3 ).

80

0.24

0.20 b= 00_

o.16X=1.356

0.12

0.08

D13xITAL CON UTER

oAo-- IN INITE NW BER

OF BLADES 3

0 - -1 -S-

0 4 8 12 16 20 24b

Figure 17. Comparison of Mass Coefficients for Increasing Numberof Blades (Digital Computer) with the Mass Coefficientfor the Infinite Blade Case (Exact3 ).

I

81

In order to illustrate the behavior of the design and performance

parameters with the the variation of the load parameter, a family of

solutions is presented in Figures 18 through 21. Fig. 18 shows the

basic lightly loaded cL'oculation curves, KoW, for X = 1/2 and

b = 2, 3, 4, 6, and 8. The variation of the scale factor G with R

is illustrated in Fig. 19. This result is plotted versus W/AX; since

R ranges from zero to X, ;/X yields the convenient range of zero to

1.0 with W/k = 1.0 corresponding to the static thrust condition. The

variations of 0T and Cp with W/X are shown in Fig. 20 and Fig. 21 for

the heavily loaded ducted fans corresponding to the lightly loaded fans

of Fig. 18. The exact results for the infinite blade case are included

in these figures for comparison and to illustrate the convergence of

the results, with increasing blade number, to the exact solution.

In Appendix IV a collection of results is presented in several

tables. The range taken for X was 0 < X 5 1, and for R/X from zero

to 1.0. For the values of X considered, the blade numbers were taken

as 2, 3, 4, 6, 8, 12 and 16. As may be seen in the tables, the higher

blade numbers were not considered in the calculations for the smaller

values of X. Rather, the blade number was increased until the results

had closely approached the cxact infinite blade solution.

"" Al!I!

82

X =1/2 8

0.73

0.6

0.5

K(x)

o.4

0.3 /0.2

0.1)A.- / - DIGITAL coMUM

0 -

0 0.2 o.4 o.6 0.8 1.0x

Figure 18. Blade Bound Vorticity for a Family of LightlyLoaded Ducted Fans.

83

1.0_____________ _

0.9

0.8

G

0.7

o.6

0.5-0 0.2 o.4 o.6 M. 1.0

Figure 19. Variation of the Load Scaling Factor.

0.16b

I86

o.14 I

0.12 ____

0.10

CT

0.08

m.6

m. 4

0.02 EAF

0 0.2 0.4 0.6 0. 1.0

Figure 20. Variation of Thrust Coefficient with Loadfor the Family of Dlucted Fans.

85

0.032 -

0.028 - - ______ /~--2

X=1/2

0.02 411

0.020

0.16

0.008 _____

EXACT 31

-.0 DIGIB COMPUTE

00 0.2 o.4 0.6 0. 1.0

Figure 21. Variation of the Power Coefficient with Loadfor the Family of Ducted Fans.

86

APPENDIX I

GEOMETRY AND MOTION OF THE INNER HELICAL SHEETS 4

The argument as to the geometry and motion of the wake vortex

pattern of an optimum ducted single-rotation fan is essentially the

same as that presented by Betz9 and Theodrosen . Following these

approaches, consider a non-cptimum ducted fan which is producing the

required thrust at the expenditure of the necessary amount of power.

At a distance behind the first ducted fan system such that the duct

interference velocities axe negligible, arrange a second ducted fan

having the same number of blades and rotational speed as the first

fan and so phased with the first fan that each blade intercepts one

of the sheet of discontinuity that are shed from the former fan's

blade trailing edges. The diameter of the duct of this second system

is set equal to the wake diameter so that it intercepts the sheet of

discontinuity that is shed from the first duct's trailing edge.

Assume that the second fan is mounted on an extension of the shaft of

the first fan and assume further that neither the second fan nor the

duct contribute to the motion of the wake nor disturb the flow in

any way. Similarly, place a third ducted fan, and so forth, until a

large number of ducted fans are arranged in tandem, all mounted on the

shaft of the first fan, all having the required phase relation, and

none contributing to the motion of the wake ncr to the thrust or

power required.

II

87

In general, certain of the blade elements of the first fan will

be operating at relatively high efficiencies while other elements will

be operating at relatively low efficiencies. This will be evident in

the wake, as will be seen later, by the pitch of the wake spiral with

the efficiency being higher where the pitch is lower and vice versa.

Suppose now that on the second ducted fan a positive increment of

thrust is added to a blade element where the pitch is low and an

equal increment of negative thrust is added on the third fan to an

element operating in a region where the pitch is high. The thrust of

the complete system remains unchanged but the third fan adds more power

to the shaft acting as a windmill than the second fan requires to produce

the thrust increment so that a net reduction of the power required by

the system is realized. (Of course skin friction is neglected and it

is assumed that the thrust increments are very small so that the power

recovery factor is 100%.) The efficiency of these added increments may A

be obtained by considering Fig. 22. Using the Kutta-Joukowski theorem,

the increment of thrust is

AT = pAr(Or - u )

The increment of torque is

AQ pAr(V + uz)r

This gives for the efficiency

AT Arrt - ISAQ pAIO(V rlow

1*. 88

IIvi , I

r

Figure 22. Incremental Thrust Efficiency Diagram.

I

89

I I tan (P

V/0

or 1. + i/V (56)CIDThe elemental efficiency is thus simply a function of the ratio of the

apparent velocity of tbh helical vortex sheet element to the free stream

velocity. The process of adding an increment of thrust on a blade

element of one fan and removing the same amount on the following fan

* with a net reduction in power required is continued until no further

reduction is realized. At this point the efficiency of the added

element of thrust will be the same regardless of the radius at which

it is added. From equation (56) this occurs when w/V0 is the same at

each blade station for the last fan in the array. The wake behind this

last fan represents the wake for the optimum case. Thus the optimum

condition is obtained when the ultimate wake vortex pattern appears to

move as a rigid body and the pitch of the inner helical -,-e spiral is

uniform along the radius. The problem, of course, is the determination

of the single ducted fan which will yield the same wake configuration

as the array.

IIi _ __ ___________

90

APPEIDIX II

VELOCITY FIELD OF THE UNIFORM BOUNDARY SHEET

The uniform boundary sheet of the ultimate wake vortex model

consists of equal strength helical vortex filaments of constant and

equal pitch wrapped on a right circular cylinder at the pitch angle PB"

The strength of all filaments is the same and is given by Y, the sheet

strength per unit length normal to the filaments.

This sheet can be divided into two sheets to be superposed.

One sheet consists of a system of infinitesimal strength ring vortex

elements whose axes are the axis of the cyll .. rical wake boundary

surface. The strength of this sheet is given by ~ cos B" The other

sheet is a sjstem of straight line filaments of infinitesimal strength

lying on the cylindrical boundary surface and parallel to the wake axis.

Again, the strength of all filaments is the same and the sheet strength

per unit length normal to the filaments is given by y sin cB

The velocity fields of these two sheets are calculated separately,

in terms of a Biot-Savart integration of the filaments of the sheets,

and then superposed to yield the velocity field of the entire uniform

boundary sheet.

Ring Element Sheet

From considerations of symmetry, the angular and radial velocities

associated with the ring element sheet are identically zero. The geo-

metrical relationsnips necessary to calculate the axial velocity field of

91

this sheet are shown in Fig. 23. From the figure

dyds' " Rddz

h2= R-r cos Yi

h2 = (R-r cos y)2 + z2

p2 =R 2 + r2 -2rR cos Y +z 2

Cos = h/

cos ct = h2/h

so that

z =2 J4OcP 2 cos a Rd~dz0 0

fp2 rT co co B (h/P) (h2/h)

0 0P

2~ f.cos PB (1 - Ycos ) d~d

02 j 0 (1 + F - 2F co + z3/

where P and 1 are nondimensionalized by R. Nov look at the ~

integration and define, a = (1. + Y 2- 2r cos T)so that

2j 2 3/2 z1/2 0 2 1 2 =2/a

0 (a + ) a~a+Z2 S+0a(_2

92

zd

FILA1=N

1:z

dY dY

h

/h I zhofof

Figure 23. Geometry of the Ring-Elemaent Sheet.

93

or 2 d d 2/(1 + - 2F cos T)

0 (a +)

This integral may be found in reference 15.

Then u becomesZ

u=2 cos 9B (1- Cos)uz 2T dT

0 (1 + F2 2r cos ,)

cosT Bff 2rr d, .F f cosT df

Y 2rr A + 5 cost r a + B cos 'J0 0

where = 1 + r2 and = -2F. But

CosT 1 B Cos, i{ 1 xR+' COST = If + 5cos V b -+-o

so that

Cos B f (F + [ T) a d-- -r ?_--d, + Fi+ST

0 0

Now

+ 5 cos = 15, which is of0 + 1Co + - cos T00 a

the form

2) aJ 1+ a os which yields from reference 150

94

2.rax cos = a < 1 or, 7 < 1 (inside the wake).f 1+ a cos-x 1/2

0o (l -a)

Then

2IT -f 2rT anf + 5 + c os T a 1/2 and

0 ( 1)

a~)2)

Cos(PB 2r.( -1 + (1 +41±)[/, 1 2 1/2uz = 212

or

Su = y cos TB "

Straight Line Element Sheet

From considerations of symmetry, the axial and radial velocities

associated with the straight line element sheet are identically zero.

A The geometrical relationships necessary to calculate the angular velocity

field of this sheet axe shown in Fig. 24.

The velocity at a point due to the filament can be evaluated

directly from reference 16 as

V= r sin cPB dS sin CB RdfP --- ,orv y vr=:, , =--=

95

FiuR R

Figure 24. Geometry of the Straight-Line-Element Sheet.

96

From the figure =r 2 + R2 2Rr cos

si , R2 - r 2 .p2sin = 2r2P

2rP

Then

2 _ R 2)

du, y sin RdY

so that

r' y 'T F R+ Cos0

where i = 1 + F2 and B = -2F. Again, the integral is of the form

_"_dx 2rr 2+ a cos x 21/2 , a < 1, or F< 1

0 (-a)I

so that

-r2 12rT dY2 1 12 1/2

r (i 2 =r "(2 rr/(( F2

(1+Y - 2? cos T) r +r

P2 F -j2rr/(1 272 + r = r -2

Thus

uY =Y- - - -r

i,*

97

Superposition

Fram the results for the two sheets the velocity field inside

the wake due to the uniform boundary sheet is given by

u =0 ,(57)

IY 0

98

APPENDIX III

CHECK CASES FOR THE BUMERICAL PROCEDURES

In developing the computer programs to handle the numericalcalculations required for the solution of the wake vorticity distribution

and the numerical calculation of thrust and power coefficients, a number

of check cases were developed in order to obtain accuracy criteria for

the various approximations required for a solution. The first of these

checks involves the replacement of helical vortex sheets by a number of

finite strength vortices.

The solution for the blade bound vorticity of an optimum free

propeller is well-established and has been solved for a number of

cases by Goldstein 0 . A numerical model similar to the one employed

for the ducted fan was developed in order to check the accuracy of the

integration of the velocity contributions of the finite strength helical

vortex filaments and to establish a criterion for a minimum number of

filaments required to adequately represent a helical vortex sheet.

The conditions placed on the solution were as follows. The velocity

normal to a helical sheet must be proportional to the local pitch of

the sheet along a radial line on the sheet, and the vorticity of a

single blade trailing sheet must sum to zero. The blade trailing

sheets were divided into J stry.ps, the strips replaced by filaments

lying along the centerlines of the strips. Control points at which

the normal velocity component was summed and specified were placed

99

at (J-1) points midway between the filaments representing the sheet

oriented at T = 0. The arrangement for J = 4 is shown schematically

in Fig. 25. The result is a set of 3 linear equations in 4 unknowns.

The 4th equation is supplied by the zero net vorticity constraint.

The system is illustrated as follows.

A , V + A1 ,2 Y + . • • + A l, = cos r

Aj-I, 1 i+ ....... + Aj-I,J J =cosPr2 J-1

+ ... .......... .+ j=0

Where, for example, A1,I is the sum of the geometrical integrations of

f for the inboard most filament of the b helical sheets. The system

was calculated and solved for X = 1/2 and b =2 for values of J = 4, 6,

8, 10, 12, and 18. The results are shown in Fig. 26. Based on these

results, it was decided that the ten filament system represents an

adequate approximation to the vortex sheets. Greater numbers of

filaments improve the solution very slowly and it was felt that the

correspondingly larger computation times were not justified.

In order to provide a check on the calculation and integration

of the functions of velocity in the ultimate wake, the integral for

was set up in the manner described in Chapter V and evaluated forZ

several cases of b at X = 1/2. The result was compared to the exact

value provided by equation (42) according to

!I

100

WAKEAXIS

SHEET 1OXOXOXO (BLADEl1)

SHEET 2(BLADE 2)

SHEET 1O 00 0 (2nd TUR)

X CONMOL POINT

0 FILANERT

Figure 25. Schematic Diagram of the Arrangement ofFilaments and Control Points in the UltimateWake of a Free Propeller.

I

101

0.35X =1/2b=2

0.30 16i

0.25A

0.20

K(x)

0.25

0.10

0.05 DIGITAL CO

0 1 _ _ _ _ _ __ _ _ _ _ _ _0 0.2 o.4 o.6 o.8 1.0x

F'igure 26. Comparison of Two Methods for Calculating the BladeBound Vorticity for a Free Propeller.

lipy

102

S1 1~ S r il aF ~r dF d! [xT +0 0 0

The comparisons are listed and reduced to a percent difference in

Table 1, where

Uz d vol -Uz d vol

percent difference = exact numerical

fz d volexact

As the table shows, the difference in all cases was of the order of

1 percent or less so that the calculation and integration methods were

considered to be sufficiently accurate.

As an additional check on the volume integrations, the values

of er, C T and ez were calculated for X = 1 and b = 2, 4, 8, and 12.

These results are compared to the values for an infinite blade fan.

As can be seen in Fig. 27 the results show satisfactory convergonce

for increasing blade number.

103

Table 1. Accuracy of the Wake Integrations

b d vol d vol Difference % DifferenceJuz Juznumerical exact

2 .520 .526 .006 1.14

3 .552 .557 .005 0.90

4 .569 .572 .003 0.53

8 .589 .590 .001 0.17

x=1/2, = 0

F74w4

.10 e b=:O

-, .o8/

.o6

z~~r' e¢z

.o4-

z

.02Z

rr

: 0121 r

b

Figure 27. Convergence of the Wake Integration to the ExactResults 13 for the Infinite Blade Case.

4.

105

'4APPEN~DIX IV

PERFORMANCE AND DESIGN PARAMETERS

A='/a s

x K, x) W/A cr cp Crp/C G

00 0 0 /0 Ao

0 _0 .0 5" .0007/. .0000677 .97,7 .175if

0./ .10 .00/435 .0001705 ,9532 -9507.1S .002/8 .oooz.iaf 9,24 .02 60

.20 .0oo28.85. .0003Z/9 *a X2 - Vo /20.2 .4920 .0.5 .0o.3567 , 390 .8809 .8765

.30 .0042 75" 00 "4 *e.563 .85/703 ,8208 ,3 .00+.9 2 .0003/7 .83/5 .8.26

4..8894 0 .5r-99 .0056s .996 8024.8 . 006394 .000,01,7 .7812 .777-

.2 0 .0O 07/05 .0oo 0- .115 .75s .7522.. .007815 ,Ooo7Zos9 .730i .72 72

,9-f .d/ -6 aS2 7 .OOo7-t27 .70417 .7023

.65 * o .00.'22 .000775/ ., 79O .6 773

.70 .O09961 .oo0 eo3 .4532 .4 523'7 . .75 ,a/ore ,ooo 8270 .6273 .4273

,8 .o/41 .oo 8404' .60/3 .60220.8 .9723 .85 .0/214 .00 m4 .5753 .772

0 .o /ooo 872 , 5 13 .3/S09 .0773 .S o ./3,g2 .0000786 .5.233 .. 270

/0 .0/437 ,ooo 808 .19572 .50/.9/0 .9735 - _ _ _ __ _ _ _

V.

S106

k8x6°c) / C,. - - __,/c__

0 0 0 .O _,o

.05 -0067Z3 .oov0oee .,9765

0.2 .10 '001443 .0oo014 .9270. .15 oo1,. .9Z _ _

0.2 7006 .2O .0028,, -OOO39.39 .A8_0

.30 .ooo3o .0oo-5f66 .853

o.Sg z .6005162 a309

-!0,1 .900/ .40 .00572f' .0005712 .8054___" .45 .o436 -ooG'2/7 ...900

~ 95 50 .430 -? 1 .oOt,77 .75fd_ _.55 .007896 .000 709Z .72P3

.60 .00&S1 .0007463 .7037~~~0.6, ..651.00:0300 .oot7af .6760

0.7 .9670 1 ./OO2 .OO08t I .6523

.75 ./075 .0008310 .626 _

0.8 .970 1.80 -,,ie .oc08505 .66O.851 0/2.21 -000656 .571f 7

0 9 .9 7r t , .0/2-95 .o c ca8o74 .5 -i8

_ _ .5 .0/36 0 OO BZ9 .S02ZQ l._

~* 9819/-0 Q& . 44sf -GOO 8850 ~ -

-- - -

T

107

A = 1/8 b= 4x_ k<°(X) C1 c, ¢ , % c, ,G

o 0 0 0 . O A.0o_ .05 .0c ro7lz , -00 1 _97_44

.10 1. 00 44 .0001-117 .1Z26, _0.1 4/30 . .5285 ,___.

.L. .20 .ooC eI .&)Osz4 o .9 2 46_-f_7063 .25 .03s 96 00011730 e7

.3 ,t'1 35 - O0.5a3 .0005871 83o0

.40 .005737 .0O0 S 723 .e s9__. 06 .. 645 1 O "-7 79Z

.. 50 .00-7166 .000 6689 -7j50

. , -7 .55 OCY7I13 .0007105 7Z89" ,. llt '6 ( O€VAGOC:I .0W 07476 .7033

.6-95 1 .!0.32 -CO0Z4a6 3 f(o 777

0.70 .0100.5 .ooo80a4 .6,5/9

751_ . -0/077 O83Z# .. 201

0a8 *l.0o//So . 085o as*z0o ._00_

,:. .8 " . O t z e ' 0 0 8 6 7 1 . 7 4 ,4.85 .01298 .ooor00rO .54d,__0.9 .aci .X .0o/,298 .Oco8T,0 .__

1O.,0 0 1.0 .o/1V7 .00088.45 .'V J

to .f- --

i,2

1' 108

X='A bo=2× a(X) O r Or ,, ¢-,/,

0 0 0 1.0 ,0-05 1 ..?_ 1 . 3 60 ,f ,902 .9 76,

,~ l~ 0 oq ,? 01b ,II :,,96.-527

.0 ,06%36 602-77 ,

0.2 .0119,9 00.61 .3 A 9-. 0$07

.30 Q 14,U o0305A Bog _

____.35 f0667 003445 4S 83.

.40 Oig 003863 8

05 762( .50 .0z370 004 25 .7_80 .75

.55 ,Oe(oO7 7O4 91 -2S .7. 37

.60 .-QZ84, o0927• 75 .7080

.65 03 05 13 f____ A 84 -839

0.71 .8soO 3333 005464

.05 .00 5 5 6 oo..O..3..339

0.8 .8747 . ,_____" -

_ .85" ..04 005662 .5771 .585,

Crr

0.9 .889,0,0 57 o s $o,95 -f Of (p27 005790 s zOfiI .-S329

/I 093.080 42 JO71.0 a 92

109

A/4 b=3x k°6c) 0 /) % C.. CT /C.1 , G

0 0 0 0 10 _40

0o 0O50 W00____ • .05 , ioQ.5Oo ,CX.)a(oO .9794 ____

.10 .0 7o't ,oo1ISz , o -_660

1 ozO S . o 746 . 007-(80.

.20 /.Q.8.i 0 *C .. I T "9133

_____ .25 .o22 . .ooz Gaz .9 oo -

0L3 .-30 .01:26S .00312 a (O(P I ___

.5839 .35. .0#705 .03509 •641

..O .C4 .0C038 74 -6(660/ .¢a989 # .c4o 7j ___

,t15 ,ol ?Z1S o 07 "7911

.50 O~Z426 . .7652

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.. 75 .*R *03r4h .630 5AZL. ____

.80 032 62

_8__.___ .005760 -_57_5

.8 .0%7.2 it .0570o .

1.0 .05002 .0059/5 .

OF I WI,-- ,-'

111~1.10

& A1/f4 b=4x k°(X) 1 C C, CT/C-

o 0 1.0 /.o -0 .05 ,Ob?_SI 00Q.0(aJ& .-9790

1183 .15 ."l& .01730 .9350_.20 .o)9927 -OOZ.33 .9Zl

Oz.-q084 .251 .01234 ,002 701 886 -03.30 -0 (4,'.9 -ooa=V . W,. ....

.5838 .35 .o721 ,oos..s .Sg9

0.4.40 .01963 .CO!903 .61490 745 .4 .ZoS -CA 2 .8 .724

0.5 .7856 .501 .>z-f9 .C ..7630555 (. _.737/

.60 .94 .o.10050se .Tn1s

0e402 .65- .C 191 "005271 .6a3(

0.7.70 _,C)3,43 .c995455 .0 sj_ 772_ .75 1, )3 69 .0.0 5( , 1

0.8 .80 . 6195e OJS 736 .Gol7_____ .85 .0422 "2Z 0. ,574_

0.9 9/85 .9 . .X 4190 .*0 901 Jf; -_(p

_. _ .S . 124,I .00524%6 -. 9/90

o zAOL .0,5044 j-90 1/

%1l4 b=

x KK(x) /,_ C _ c _ Or_0 o 0 0 0 o /.o

_ _ _.05 'c Ic0bA .2i.0 ~ ~ 0 O c 3f .60061A .978 7

0.1 .10 ,.oso4z .aoii 8 .9

I S49' .15 , .001"40 .9 _,

02 -go -O" 19t .00 Zz4 "9//o.4001 .25 *Q-OOZ7I1 -e87 __

.30 •0149o .o5s4 -663'2

-.51-5 * .0 1 7.34 -COB 556 ~ i. 3AIS0 .4 . .40 , 17?8 .. of j .e3Q

.70_9 .4 OZZZ3 G43ieLas .50 .OZj .0O45C9 .7 .Za L.- .B .55 6 .ooq4e .?,/.7890 .5 71 -# .Y,4 B44,73571

a .60 -0?64 .ooo089 "7091.8508 ,65 .n 5 -005303 .6623

0.7 .70 o34,* os.8 .6S _a.e5z1 1.7 .03727 .asel.f9 .428l

a8.90 .03968e .005771 .6008V02 .85 .0253 .c 76 .733

0.a g : .o 2 o~g, .54.l~9

.0, 5 . 0 ?_ 9A Z _5 _ .4/0 .05076 ____509 _____

?ki1 •

/

112

7 kax) CT cP cC7 /cT

01 0 0 0 ho /,

._0 _006_W .003_m 1.02 fSi . , 5 .15 ,6/329 00 46,2-0 _'2O/ .9-330

.2 a -00176 .964 e .7/30Z 3078 , . . oS 38/0

.____ ._,_ ., 8 _ •8 .887

43 .'37 A A• o-4- .657,0 .4102

.40 - 4a; ... .7 .4 3 .,/0.4 .5292 .103B57 .0QU .8073 .

*-( I _i_ __ .,_ __.74 .0o5o .O.?/p .606 .5766

0.7 .7075 "0. m" .0(005 .0141 .66,, .608

_,8 ___8 .85 .07506 0/50.9 . 7, . 6 -5

09 .75 75 -5,s,

' L --'~~~~ . 0 1 6 3 .7 , 9/ o o62• t / 3 3 .S , ,6 V

113

A31,e b 3x ka°e) W1X CT cp .P,/cr0 0 0 0

:!0 . 9. 00 03ZaI .f 43o /.2 .! .o , 0 o . - - -

____Of AA .0014 0 8 7 "a__

a'290 .01827 . Ob, ro .P_ _2775 .25 .O 74s9 .___

_____417 _ .5 0../,2 . 9 701 ± L-3. t4 . oio,,7 .-

i ,it 0.4 .530 0 Rr. AS 0106 '8312o,7 rs

0.5 . 2/ 0 C 44.7 , g123, "77O _

60.55 .040M .01306 ,7X/o'' .69 .o_.,,. .0,7 •42 ,z.,

.7 , .7 620i •.0M4 .,6657

______ 0 .07 74f ~ L~Li~ iL0.8 ,7770 .80 , 727t - • _____

o3.22 ._011 .0 88 .5450V9 .8oo

. WN /o .0948 .0/f62

0 o 0 00 __ _ _

_ _ _ .05 .0 477; 00 14 -

a, . 90 0(4?.L~Z~ eA(*zB& P2M -

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o_ _ _

z67

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Ls --SI il 0 78.;/ 6 6,!i2 03:0 .-0

I6 ,..92 0/w .

115

x K,(c) li/A~ Cr cp. C,1I/c?

o 0 oi h0 A.J_____ 05 .00,M.3. .00/770 !9& 16 ___

__ _ _.15 -,0040-y . M/ __ __

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(t3 ~~gj -.30 .0.2601 083 87j__

.4 .oini 7080.4 $0 j___

___.55 .00 -. /349 .70

0.8 e9 .60 C~. a .~01414 .7/0

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07- -0

,0-9

li6

0 0 0 0 0 4i O

OU ~ .10 oi~.. 2 - . 0011 -:969 .95921417Z .1$1 ot f .0085,41 192 0.32

all 2-7 .20 .02500 .01120O -933Z MA*7 .2s5 609 I i 43IL '91 S -8-90

aj3 .30 .036S .01545 -89/0 =9

. -3 ' - .5 1 .040 72 *667A &W

__5__ .066 or .OZZ77 -76O

0.C I.5332 -f .072A7 ._*02574 .7309 92

(P____.5 Zg3AL .6?456 .7001 -. 0Z

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~~~~~9~~6 Sao ./1 *Z68.~i 50

1.0/631 L . 133 0*68 4_____ 701

I3

117

x k06c e/p Or__ __ /C- C0 0 0 1.0 __0_

o 0_ _

025 .083s7 (a LA1 0 N-S .0857___

CLI.20 -n 6 . P 3 -0____

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0.7 0 A .j 3ZS *ol7 __ __ __615

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0.8 .6.3-01 o5 ~

69/2 -0023940 . 70Z& 0

S01 .h OU -OZ0 6

0.7 .0 ONS

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x k4'(c) ___x __ __ __ __ __ __ ____

0 0 0 0 0 1.3 A0______.05 . 8 _ _ _

U 0618 e0 ~ oro "o f~ __9_16 -=

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a-3.30 *csza *O0b2a .89-* ~ 35 -66PD ~ 081 -

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05.06, 022 .74037

0.9

1.0~~97 of.2 -768L.- ~g

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A=

l19

x Mx) "'/9 C, C¢ CTI,./C? -j0o 0 -0 /EO4o 0 o- o o 0

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120

K kOC) IA _ _ cp OrP/c_ _0o 0 0 4.0 Aoo o

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A='~b=e __

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124i

K k.6c) __ ___ __ ______ _ ____

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I. L _..4.4

125

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4

CITED ITIATUE

1. Sacks, A. H., Burnell, J. A., "Ducted Propellers -A Critical Review of the State of the Art," Progressin Aeronautical Sciences, Vol. III, pp. 87-135,Pergamon Press, Inc., Now York, N. Y., 1962.

2. Tachmindji, A. J., "The Potential Problem of the OptimumPropeller with Finite Number of Blades Operating in aCylindrical Duct," Report 1228, July 1958, David TaylorModel Basin, Washington, D. C.

3. Theodorsen, T., "Theoretical Investigation of DuctedPropeller Aerodynamics," Vol. III TREC TR 61-119, Sept.1961, U. S. Army Transportation Research Command, FortEustis, Va.

4. Gray, R. B., "An Investigation of an Approach to theProblem of Determining the Optimum Design of ShroudedPropellers," TREC 60-44, May 1960, U. S. Army Transpor-tation Research Command, Fort Eustis, Va.

5. Gray, R. B., "An Investigation of a Digital ComputerMethod of Determining the Optimum Design Parameters ofShrouded Propellers," TREC 61-124, Oct. 1961, U. S. ArmyTransportation Research Command, Fort Eustis, Va.

6. Morgan, W. B., "Theory of the Annular Airfoil and DuctedPropeller," Fourth Symposium on Naval Hydrodynamics,Office of-Naval Research, Department of the Navy,Washington, D.C., ACR-73, Vol. I, pp. 161-212.

7. Ordway, D. C., Sluyter, M. M., and Sonnerup, B. 0. U.,

"Three-Dimensional Theory of Ducted Propellers," TAR-TR

602, Aug. 1960, Therm Advanced Research Division of Therm

Corporation, Ithaca, New York.

8. Chaplin, H. R., "A Method for Numerical Calculation ofSlipstream Contraction of a Shrouded Impulse Disc in theStatic Case with Application to Other Axisymietric FlowProblems," Report 1857, June 1964, David Taylor ModelBasin, Washington, D. C.

9. Betz, A., "Screw Propellers with Minimum Energy Loss,"Technical Translation 736. National Research Council of

Canada, Ottawa, Canada, 1958.

14o

10. Goldstein, S., "On the Vortex Theory of Screw Propellers,"Proceedings of the Royal Society (London), Ser. A., Vol.112, pp. 440-465.

11. Theodorsen, T., Theory of Propellers, McGraw-Hill BookCompany, Inc., New York, N. Y. 1948.

12. Lamb, H., Hydrodynamic3, Dover Publications, New York,N. Y., 1945.

13. Gray, R. B., "Analysis of a Heavily Load Ductu Fan withan Infinite Number of Blades," to be publisheu.

14. Fadeeva, V. N., Computational Mcthods of Linear Algebra,pp. 67-72, Dover Publications, Inc., New York, N. Y. 1959.

15. Hodgman, C. D., Standard Mathematical Tables, The ChemicalRubber Publishing Company, Cleveland, Ohio, 1959.

16. Keuthe, A. M., Schetzer, J. C., Foundations of Aerodynamics,John Wiley and Sons, Inc., New York, N. Y., 1959.

4

UnclassifiedSecurity Classification

DOCUMENT CONTROL DATA - R & D(Security clasfication of fille, body of abstract and indexing onnotation must be entered when the overall reprtIsclalfifL,,.

1. FORIGINATING4 ACTIVITY (Corporae author) Iza.-RIEPORT SICURITY CLASSIFICATION

Georgia Institute of Technology Unlasiie

a. R9PO'PT TITLE

Determination of the Design ,Parameters for Optimum Heavily Loaded Ducted Fans

4. OESCRIPTIVE NOTUS (7)po of epart and Inclusive datse)Technical 'Report

a. Auiti4@iss (pitet mea., Zld~ro Init~l seat nat)

Terry Wright

1 . REPORT DATS 70. TOTAL NO. OF PAGES9 7b NO. or XKFSNovember 99lf 16

68. CONTRACT OR GRANT NO. IIe..ORIOINATOWS REPORT HUIDRtb)

DAHCO4i 68 c ooo145PROJECT No. Report 69-3r

S.OTHER REPORT MOIll) (Any 99tWtavw"~ Oat aay be assifind

T-2 .8-EIa. ISTRISUTION STATEMENT

This document has been approved for public release and sale; itsdistribution is unlimited.

II. SUPPLEMENTARY NOTES 12- SPONSORING MILITARY ACTIVITY

Ue S. Army Research Office-DurhamBox CM,, Duke Station

________________________________ Durham,# North Carolina 27706

Like the free propeller in axial flight, a single-rotation ducted fan of highestInduced efficiency is characterized by an ultimate wake vortex system shed from theblade trailing edaea whose apparent motion is that of rigid helical surfaces. In

addition and conce tic with-this inner sheet system there is a cylindrical surface

and neglecting compressibility. viscosity and tip clearance, a consistent mathemati-cal model of the constant-diameter vortex wake is developed and the compatibilityrelationships to be satisified are presented. Using the Biot-Savart equation, thevortex strength distribution in the ultimate wake is determined and then related tothe blade bound vortex strength distribution. In addition, expressions aredeveloped for the thrust. power and induced efficiency which depend on numericalintegrations of velocity profiles in the ultimate wake. It is shown that the wakevorticity and the veloc ty distribution in the wake for all loadings from the lightiloaded limit to the heavily loaded static thrust condition may be extracted from thelightly loaded result through the use of a simple algebraic scale factor. Theperformance para~mters, thrust power and blade bound vortex strength are thusexpressed in terms of a lightly loaded solution for a given freest ream velocity,blade tip speed blade number and the loading parameter. The lightly loaded case iscompared to existing theoretical and experimental results. Some sample results for

heavily loaded performance parameters and design parameters are presented and compareResla for the heavily load-d system over A broad range of conditions are presented.

D seoves5 RPLACES 00 FORM 11471. 1 JAN 64. WNICN IS

DD90OV91 73 ObS1OLeTE PIOR ARMY USE. Miclassified

Secriy Lasifcato

Secu~Ity ClassficationS4. LINK A LINK S LINK C

KECY WOROS

ROL% WTY n RLS WY ROLE WY

Ducted fans

Vortices

Wae

________dynamliicsm mL