Wake-Integral Determination of Aerodynami Drag Lift and Moment in 3d Flows

8/17/2019 Wake-Integral Determination of Aerodynami Drag Lift and Moment in 3d Flows

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(c)2002 American Institute of Aeronautics Astronautics or Published with Permission of Author(s) and/or Author(s) ' Sponsoring Organization.

AIAA

A02-14371

AIAA

2002

-

0555

Wake-Integral Determination

of

Aerodynamic Drag,

Lift and Moment in Three-Dimensional Flows

J.C. Wu

Applied Aero, L L C

Zephyr Cove, N V

C.M. Wang

Applied A ero,

L L C

Zephyr Cove, N V

K.W.

McAlister

Army Aeroflightdynamics Directorate

Moffett

Field, CA

40

th

AIAA

Aerospace

Sciences Meeting Exhibit

14-17 January 2002

Reno, Nevada

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( c )2 0 0 2 A m e r i c a n I n s t it u te o f A e r o n a u t i c s A s t r o n a u t i c s o r P u b l is h e d w i t h P e r m i s s i o n o f A u t h o r ( s ) a n d / o r A u t h o r ( s ) ' S p o n s o r i n g O r g a n i z a t i o n .

AIAA 2002-0555

WAKE-INTEGRAL

DETERMINATION

OF AERODYNAMIC DRAG, LIFT A ND

MOMENT

IN

THR EE-DIEMENSIONAL FLOWS

J. C. Wu*

Applied Aero, L L C , Zephyr Cove, Nevada

C.

M. Wangt

Applied Aero, L L C , Zephyr Cove,

Nevada

K.

W .

M cAlistert

Army

Aeroflightdynamics

D irectorate, Ames

Research Center,

Moffett Field,

California

Abstract

New wake-integral expressions for the

determination

of aerodynamic

load

on finite wings an d rotors are

established using

a vorticity-moment

theorem.

Com pared to previous

wake-integral expressions

based on the

momentum

theory,

the new

expressions

connect the wake

flow

properties

more

directly to the aerodynamic load.

They offer

enhanced physical understanding

of the flow

mechanisms

responsible for the

production

of

aerodynamic

force

and moment and are

simpler

an d more efficient to u s e .

Wind-tunnel

experiments are performed to

validate

the

wake-integral

expressions

for the thrust and the torque o n

rotors

in slow

climb.

A three-dime nsional particle-image

velocimetry system is used to obtain velocity values in the near-wake of a model

rotor.

Thrust and torque

values

determined

using

the wake data are presented and compared with balance-measured values.

1. INTRODUCTION

A

lifting body

in flight

always

leaves

behind

in the

fluid a footprint - the

wake.

Fo r

more than

a

century,

the

aerodynamicist

ha s

searched

for the connection

between

this footprint

and the aerodynamic load on the

body. L. Prandtl connected the down

wash

induced by

trailing

vortices - parts of the wake - to the induced

drag on the finite wing. The

profound

contribution of

the resulting

lifting-line

theory

to theoretical

aerodynamics cannot be overemph asized. The research

described in the present paper is centered on the wake-

integral

approach,

which

also connects

the wake to the

aerodynamic

load.

This

method, however, differs from

the lifting-line theory in that it focuses not on the

downwash induced by the wake, but on the

wake

itself.

Th e wake-integral method

does

not require the inviscid

fluid

idealization

and is useful in evaluating both the

inviscid

drag and the viscous

drag.

A

wake integral in a

general context

is an

integral

over a transverse surface

downstream

of a lifting solid

body. For the present work, the term

4

wake

integral' is

used

in a

more

restricted

context

to

designate

a

special

surface integral whose

integrand vanishes

outside the

vortical wake

region. A. Betz

2

pioneered

the

wake-

integral

concept

an d

successfully established

a wake-

integral expression for the

steady profile

drag

(also

*

President,

A ssociate Fellow

t Chief Aerodynamicist

t Research Sc ientist

Copyright ©

2002

by J. C. Wu. Published by the

American Institute of Aeronautics and Astronautics,

I n c . with permission

called

the

parasite dra g

3

)

in

1 9 2 5 .

E. C.

Maskel l

4

and

J. C. Wu et a l .

5

derived

a

w ake-integral expression

fo r

the induced

drag

in the

1970s. These

wake-integral

expressions allow

the

separate determination

of the

induced drag and the

profile

drag on the lifting body

through

wake surveys over a single wake plane. Since

measurements

ar e

required only

in small

wake regions

where the

vorticity

is

non-zero, both

the

profile drag

and the induced drag can be

determined

efficiently an d

accurately. The

advantages

offered by the method in

design diagnostics are obvious.

Efforts

have

been

in

progress

in

recent

years at

several

universities and

governmental

an d

industrial

laboratories at various points of the world to

further

develop the wake-integral method. Wind

tunnel

studies

of

many

aerodynamic shapes of practical importance,

including ca r shapes, have been performed

using

the

method.

In a

recent review article

6

on drag

prediction

and reduction,

I.

Kroo referred

to

many

recent

efforts,

noting that successes

have been

reported

along

with

several open

issues

that require further investigations.

Previous

studies

of the wake-integral method are

mostly concerned

with

steady aerodynamic

drag. The

present paper reports

selected

results of a research

program initiated in

1 9 9 6

and completed recently

7

.

The aim of the

program

is to

generalize

the

wake-

integral

method

for

unsteady

flow applications, in

particular

helicopter rotor

applications.

Under

this

program, new wake-integral expressions are derived for

the

finite wing. New wake-integral expressions a re

also

derived for the

thrust

and the torque on the rotor in

axial flight,

including hover.

Wind-tunnel experiments

are performed to validate the rotor

expressions.

1

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Institute

o f

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a nd

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2. VORTICITY

M OM E N T

A ND

VORTEX LOOP

Th e

conceptual

foundation of the present research

is

described

in

detail

in a

recent report

.

Several

long-

standing

issues

of

viscous aerodynamics

are examined

from

the vorticity-dynamics

viewpoint

in the report

8

.

In this

paper,

a vorticity-loop method fo r aerodynamic

analyses is

described. This

method is

based

on the

previously developed

vorticity-mom ent

theorem

9

.

New

wake-integral expressions are derived for the

finite

wing

and for the

rotor

in

axial flight using this method.

Th e vorticity-moment theorem

9

contains

several

mathematical statements involving integrals of vorticity

moments. These

statements

are derived

mathematically

rigorously from the Nav ier-Stokes equations. For the

aerodynamic force F on a solid

body,

the statement is:

=

-^-p—f r x c o d R +

p — f v d R

2

dt JR. dt

J R

S

(1)

where p is the

density

of the fluid; R«, is the infinite

unlimited

region composed of the solid region R

s

and

the fluid region R

f

; r is a position vector; v is the

velocity vector; and CO is the

vorticity

vector

defined

by co = V x v.

Th e

last

term in (1)

vanishes

if the

solid

motion is

rectilinear and does no t change with time. For

most

practical applications, the

contribution

of this term to

the

aerodynamic force

is negligibly

small

even if the

solid

is accelerating or

rotating.

In

such

applications,

the first

term

in (1)

determines

the

aerodynamic

force.

This term states that F is equal to -Vip times the

rate

of

change of the first moment of vorticity in R o o . This

region reduces to R

f

if the

solid

is not rotating.

A

Cartesian system of coordinates (x,y,z) with the

unit-vector se t

( i

j,k) is used in the following discussion

of the

vortex

loop method. If the freestream

velocity,

V

= Ui, is

aligned

to the

x-axis

and the span of the

solid

is in the

y-direction, then

the lift L and the drag D on

the solid are the z- and the x- components of F

respectively. The vectors r, v, and

C O

are stated as r =

xi + yj + zk, v = ui + vj + wk, and C O = £ i -f T j j +

£k.

The vorticity field, as the curl of a vector field

(specifically, the velocity

field),

is

solenoidal, i.e.,

divergence free.

It has

been show n

8

that

the regions R

f

an d

R

s

can be considered

together kinematically.

A

vorticity

field

in R

f

(or more generally in

RJ

can be

viewed

as being composed of closed tubes of vorticity

8

whose

walls a re vorticity

lines, i.e.,

lines

whose tangent

at each point is in the

direction

of the vorticity

vector

at

that point. The

strength

of

each

tube

(the integrated

vorticity

strength

co

over

the

tube's cross-section)

is the

circulation F around the

tube.

Since the

vorticity

is

solenoidal, F is the same at all

sections

of the tube.

Hence the vorticity strength co is inversely proportional

to the

cross-sectional area

of the vorticity tube.

If

on e views the vorticity

field

in R

f

(or R^) as

composed

of a system of vorticity tubes with small

cross-sectional

areas, then the vorticity in each tube can

be approximated by a vortex loop F = Ft, where t is the

unit

tangent vector of the loop's path C, as

shown

in

Figure la. The vector t

points

in the

direction

of the

vorticity

vector

in the

tube.

The term 'vortex loop' is

used in the

following

discussion for

convenience.

Th e

conclusions are obviously valid for the closed

tubes

of

vorticity that t he

vortex loops approximate.

Th e elemental

vorticity moment rXCOdR

of an

elemental

region dR is approximated by

rxFds,

or

F(rxtds), where ds is an elemental

segment

of the

loop.

If the vortex

loop lies

in the x-y plane z = z

h

then

tds =

id x

+ jdy and

rxtds

= Zi(-idy + jdx) + k(xdy -

ydx).

The integration of rxcodR over the vorticity tube R

t

is

frxcodR^-Fziif d y +

Fzjfdx+rkf

( x d y - y d x ) .

J R ,

Jc J c J c

Th e first two integrals in

this

expression are zero.

Using Green's theorem, it can be shown that the last

integral gives twice the area enclosed by C. Hence the

vorticity

moment A of the

vortex loop F

is

normal

to

the

plane

of the

loop

and its magnitude is

twice

the

loop's

circulation

F times the loop-enclosed area A:

A = 2FA

=

2FAn

(2)

where n is the unit vector

normal

to the

plane

of the

loop an d

points

in the

direction

of

advance

of a

right-

handed screw

as the loop is

traveled

in the

direction

t.

A change with time of the vorticity moment A

causes

a

force

F

r

on the solid

which

is, according to (1) an d (2):

F

r

=-p-(FA)

3)

The

force

F on the

solid

is the sum of

F

r

over

al l

loops of the system representing the vorticity field.

If the

path

C is

divided

into two

parts,

C \ and C

2

, as

shown

in Figure

Ib ,

and the two dividing

points

ar e

connected by a

line

C', then one has two closed paths: a

path Cy formed by joining C' to

C\

and another path

C

2

' formed

by

joining

C' to

C

2

.

Consider a

vortex

loop

FI on the path

Q'

an d

another

loop F

2

on the path

C

2

'.

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and A stronautics


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(c)2002

American

Institute of Aeronau tics Astronautics or Published with Permission of Author(s) and/or Author(s) ' Sponsoring Organization.

If F

= T

2

an d

t

2

= - t j on the

line

C ',

then

F, +

F

2

= 0

and

the combined

strength

of the

vortex line

on C' is

zero.

The two smaller vortex

loops

FI and

F

2

together

are thus

equivalent

to a single vortex F on the path C.

This

means

that

any vortex

loop F

is

divisible into

tw o

smaller loops.

Successive divisions give an arbitrary

number

of smaller

loops

that

are,

in

aggregate,

equivalent to the

loop

F. A non-planar vortex loop can

be divided into a

number

of small loops that are

approximately planar.

Any specific vorticity

distribution

can be approximated by various systems of

vortex loops configured with a great deal of flexibility.

The

vector area

A can be expressed in the

component form

A

= A

x

i + A

y

j +

A

z

k,

where A

x

, A

y

, A

z

are the

projected

areas of

A

in the

y-z,

z-x and x-y

planes

respectively. Also, (2) indicates that the

vorticity moment

A of a vortex loop

depends only

on

the strength of the loop and its

size

and

direction.

Hence

A is independent of the shape and location of the

loop. In other words, a planar

vortex loop with

a

fixed

enclosed

area

m ay deform in its own plane and undergo

rectilinear

motions

without altering its

vorticity

moment. These facts

greatly

facilitate the use of the

vortex-loop method

in

aerodynamic analyses.

3.

LIFTING

LINE THEORY

The lifting-line

theory models

the steady

flow

around a

wing

of

finite

span by a

horseshoe-shaped

vortex system

l

This

system is composed of a lifting

line

representing

the circulation F(y)

around

the

wing

and a trailing

vortex sheet representing

a thin

wake.

With this

flow

model, the Kutta-Joukowski

theorem

is

used to

derive

expressions for the

lift

L and the induced

drag Dj on the wing. Th e downwash, w, at the lifting-

line

location

is

viewed

as a

modifier

of the fresstream

velocity, hence also the ang le of attack, thus causing the

induced

drag.

The

expressions

for L and D j are then:

fb/2

=

pUj

r(y)dy

J—

b/2

4)

5)

Th e

vortex

loop method is used to re-derive (4) and

(5) as follows. Consider

first

the idealized case of a

wing with a

constant

circulation F. The vortex theorem

of Helmholtz -squires that this lifting line not to end in

the fluid. Th e

lifting-line flow

model is, in this case, a

vortex

line

composed

of the

lifting

line and two semi-

infinite vortex lines, called tip vortices, trailing from the

tips of the

lifting

line. O ne

thus

has an open-ended

horseshoe-shaped

vortex system. This system is

complete

if the presence of the starting vortex is

recognized

8

.

The complete

system

is a

rectangular

vortex loop. The starting vortex connects the tip

vortices

and

closes

the horseshoe-shaped system far

downstream.

As the

starting

vortex

moves away from

the

wing,

the tip

vortices

grow. The rectangular

closed

vortex loop

elongates and the

loop remains closed.

Let the

lifting line

lie on the

y-axis

an d

extend

between y = -b/2 an d

b/2,

b being the span of the

wing.

If the rectangular vortex loop lies in the z = 0 plane,

then t = j, i, - j, and -i respectively on the lifting line,

the tip vortex at y =

b/2,

the starting vortex, and the tip

vortex

at y = -b/2. Then n = -k and the area A enclosed

by the loop increases at the rate Ub. Hence, according

to (3), the growth of the

rectangular

vortex loop

causes

a

l i f t

on the

wing

in the amount pUbF.

With the wing circulation F(y), the

strength

y(y) of

the trailing

vortex

sheet in the lifting-line flow model is

required

by the Helmholtz

vortex

theorem to be

l

= -dF/dy

(6 )

The

complete

flow

model includes

the

starting

vortex 'closing' the trailing vortex sheet far

downstream of the

lifting

line. Consider a system of

rectangular vortex

loops placed side b y side in the z = 0

plane. The vortex loops a re

labeled sequentially

from 1

to J. The

vortex

loop j has a

lifting-line

segment on the

y-axis with the

strength

Fj =F(Vj) and the

length

8y =

b/J.

Let yj =

-(b/2)+j8y

and the jth lifting-line

segment

be in the range y

}

.\ < y < yj. There are two trailing

vortices belonging to the

vortex loop

j, one at y = y^

with

t = - i and the

other

at y = yj

with

t = i. Coexisting

at

y = ^

(except

the tip points j = 0 and j = J) are two

vortex

lines: the

vortex line Fj i belonging

to the vortex

loop

j and the

vortex

line -Fj+ii

belonging

to the

loop

j+1. The combined strength of the two vortices is [F(Vj)

- Fty,)].

As

8y-»0, [F(

yj

) - r(y

H

)]/8y ->

-dF/dy.

The tip

vortices

of the J

vortex

loops in the vortex

loop

system become

the trailing

vortex sheet

with the

strength given by (6). The set of J vortex loops is thus

an approximation of the lifting-line

vortex

system.

With the lifting-line flow model, the trailing

vortex

sheet lies in the z = 0 plane. The jth vortex

loop

has the

strength

Fj and its

area A j increases

at the

rate U8y.

According to (3), this vortex loop causes a lift pUFjSy.

Th e total

l i f t

caused by the system o f vortex loops is the

summation of this quantity over all the loops in the

vortex loop system.

In the limit 8y — •» 0, the

summation

becomes (4).

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With the vortex

sheet

lying in the z = 0 plane, the

lifting-line

flow

model predicts a zero induced drag.

Prandtl developed

the flow

model

by

assuming * th e

vortices move

away

from the

wing backwards with

th e

rectilinear velocity V .

To

re-derive

the induced

drag

expression

(5 )

using

the

vorticity-moment

theorem, this

assumption

needs

to be

modified

to

include

the velocity

component, w, in the

analysis. With

vortices moving

with

the

flow,

w

causes

the vortex loops to be inclined

to the z = 0

plane.

The

disturbance velocity caused

by

the wing is small

compared

to the freestream velocity.

Thus w«U and the angle of

inclination

of each vortex

loop to the z = 0 plane is very small. The z-component

of the

area

Aj ,

(Aj)

z

,

is =

A-

r

This component

area

grows at the rate U5y and causes, as shown, the

l i f t

pUFjSy. Th e

x-components

of the

area

Aj, (Aj)

x

, is =

(w/U)Aj . This component area grows at the

rate W j 5 y

and, according to (3),

causes

a drag in the

amount

-

W jp

Fj8y. Th e

total

drag

caused

by the

loop system

is

the

summation

of this

amount

over all

loops.

In the

limit

8y—»0,

one has

(5).

As discussed, vortex

loops

can be divided into

smaller

loops.

This

fact

leads to a simpler way to re-

derive

(4) and (5). At the time

level

t = T, introduce a

cut

at the

plane

x=xi >0 to

divide

the

system

of J

vortex

loops

into two systems each

containing

J smaller

loops:

a system

S

u

upstream of the cut (in the region x <

\\

containing the lifting

line

and a second system S

d

downstream of the cut (in the region x > x ̂ containing

the starting vortex, as shown in Figure 2a. At the

subsequent time level T + 8t, the system S

u

has

expanded and the system S

d

has moved downstream

with

the

flow.

If the

shape

and the inclination of the

vortex

loops in

S

d

collectively remain unaltered

during

the time

period

8t,

then, according

to the discussions in

the last paragraph of

Section

2, the vorticity moment of

the system of

loops

in

S

d

at the new time

level

T + 8t is

the

same

as

that

at the old time

level

T. The system S

d

therefore

does no t cause a force. At the new

time level

T + St, again introduce a cut at the plane x = X i to divide

the

system

S

u

into two new systems.

With

a steady

flow,

the new

upstream

system at the new

time level

is

identical

to the

system

S

u

at the old

time level I.

Therefore the change of vorticity moment that took

place

during 8t is

attributable entirely

to the

vorticity

moment of the new downstream

system,

shown in

shade in Figure 2b. This

system

occupies the region xi

< x 0, one

has (4) and

(5).

4.

W A K E

INTEGRALS FOR THE FINITE WING

Using (6), on e obtains F =

d(yF)/dy

+ yy. The

integration of

d(yF)/dy

over the

span

of the

wing

is

zero

since F=0

outside the wing

tips.

O ne thus

has,

from

(4),

pb/2

L =

pU

yydy

J-b/2

7)

Using

(6), one has wF = wd(yF)/dy

+ywy.

For a

symmetric wing, the

term

wd(yF)/dy is anti-symmetric

with respect

to y=0. The

integration

of this term

over

the span of the

wing

is therefore

zero

and (5) becomes

f

/2

yw(y)y (y )dy

b / 2

(8)

The strength

y

approximates th e integrated vorticity

value

across

the wake layer. With the

layer

inclined at

a very small angle to the plane

z=0, y

is the integration

of

£

respect

to z over the

wake

region. O ne

thus

re-

expresses

(7) and (8) in the

w ake-integral form:

(9)

(10)

puJ

=p f yw^dydz

where Wis th e

wake

cross-section.

Equation

(10) is a new

wake-integral

expression

for

the

induced drag.

An

wake-integral

expression

for

the induced drag, developed previously

5

on the basis the

momentum theory, is in the form

(11)

where \|/ is a stream function in the y-z plane.

It ha s been

shown

8

that

the new expression (10) is

equivalent

to the previous expression (11). With

measured wake

velocity

values,

corresponding

vorticity

values can be

computed

easily. The induced

drag

can

then

be ev aluated

using

(10).

Th e

numerical

procedure

required is simple and

efficient.

In

contrast,

the use of

(11)

requires the

computation

of the stream

function

\|/ by integrating the velocity values. The

procedure

for

the integration is relatively complex and

prone

to

error.

The use of the new wake-integral

expression

(10) is

therefore

preferred

over the previous expression (11).

The equivalence of (10) and (11)

endorses

the use of

the vortex-loop

method

in

aerodynamic analysis.

American Institute of Aeronautics and Astronautics


6/12

(c)2002

American

Institute of Aeronau tics Astronautics or Published with Permission of Author(s) and/or Author(s) ' Sponsoring Organization.

A

wake-integral

expression for the profile drag

8

o n

the

finite wing

can be

established

using the vorticity

moment theorem.

The wake-integral expressions link

the

footprints

of the

wing

to the

aerodynamic force

on

the wing. This linkage is discussed in Section 6 in

connection

with

wake-integral

results

for the rotor.

W A K E INTEGRALS FOR THE ROTOR

A cylindrical coordinate system (r,

0,

z) with the

unit-vector set (e

r

, e

e

, e

z

) is used in the present study o f

the hovering

rotor problem.

Th e vectors r, v, and

C O

are

stated

as r = e

r

r + e

z

z, v = e

r

v

r

+

e

e

v

e

+ e

z

v

z

, and (0 =

e

r

0 )r

+ ee (O e +

e

z

civ A stationary reference frame at

rest relative

to the fluid far

from

the rotor is used. Th e

rotor

disk

is placed in the z = 0 plane. The

rotor

rotates

about the z-axis with the angular velocity

Q.

Th e

circulation

F(r)

around

the blade depends on

the span location.

According

to the Helmholtz vortex

theorem,

the blade

must leave behind wake vorticity

as

it

advances azimuthally. For a

thin

wake, the vorticity

content

of the wake can be approximated by a vortex

sheet

with

the strength

y

=

dF/dr. (The negative

sign in

(6) is absent with the

ordering

of the unit-vector set e

r

,

e

e

,

e

z

.)

For a rotor in hover or in

climb,

the

velocity

v

z

transports

the wake

vorticity

continually in the

axial

direction.

A

helical wake

is

therefore

present

under

the

rotor

disk. The blade circulation is connected

through

this

helical

wake, which

is in

turn connected

to the

starting vorticity

at the far end of the

helical

wake.

Consider

a system of J

helical

vortex

loops.

Let

the j th

loop

contain a lifting-line segment of strength Fj

=

F(VJ).

Let

this segment

be on the

r-axis

in the z = 0

plane and

occupy

the radial range T J \ < r

61 ,

containing the

blade,

and a

second

system

S

d

in the

region

0<

61, containing the starting

vortex.

At the

subsequent time

l^^el

T + 8t, the

system S

u

ha s

expanded.

Introduce

a new cut at the plane 0 =0i+ QSt

to divide the system

S

u

into

two new systems each

containing

J

smaller loops.

Following the

discussions

of Section 4, the

newly emerged

vorticity moment in

the pie-shaped region

0

t

< 0


7/12


6. EXPERIMENTS AND

RESULTS

Rotor

tests

were

performed in the U.S. Army

Aeroflightdynamics

Directorate

(AFDD)

7' by 10'

Wind Tunnel

at the

NASA

Ames

Research Center.

A

two-bladed model

rotor

was

mounted

in the settling

chamber of the

wind

tunnel. The axis of the model

rotor

was aligned with the tunnel

flow

direction.

The

wake of the model rotor passed through the contraction

section

and the test section of the tunnel. In previous

rotor

tests in the

AFDD 7'

x

10'

wind tunnel, F.

Caradonna

et al.

M

demonstrated

the advantages of

simulating

rotor climb flows using this test

configuration. Descriptions

n

of the test configuration ,

the

physical layout, the rotor, and the instrumentation of

these previous tests are for the most part applicable to

the present tests. Modifications and

additions

were

made

to

obtain particle images

in the

wake

of the

model

rotor

and to

address

the issue of rotor-driven

flow

returning

to the

settling

chamber.

The

flow

circuit of the wind

tunnel

is shown in

Figure

3. A flow

seeder

was used to introduce particles

into the flow for particle imaging. A three-dimensional

particle-image velocimetry (PIV) system was used to

obtain particle images in the near wake of the model

rotor.

Major

components

of the PIV

system

are two

2,000

x 2,000 pixel

digital

cameras, lens

sets,

remote

focus

system,

high-speed interface and digital

links,

control

cables, computers for acquiring and storing

particle images,

and

laser light

source and

mirror

systems.

Figure 4 shows the camera and

light

sheet

configuration used.

The AFDD 7' x 10' wind tunnel is a closed-circuit

tunnel. The cross-section of the settling

chamber

is 30 '

x 31'.

The cross-section of the

test

section is 7 ' x 10'.

The

model rotor

has a

nominal

diameter of

7'

and a

true

diameter

of 6.283'. The wake of the model

rotor

was

expected to

flow

through the test section with

minimal

interaction

with

test-section walls.

With

the tunnel

drive-fan

o f f , the model rotor

acted

as a substitute drive

f a n

and created a flow through the tunnel's flow circuit.

Thus,

with the tunnel drive-fan o f f , a climb

condition

rather

than

a true hover condition was expected to exist

in the settling

chamber.

To evaluate the strength of the rotor-driven flow, a

curtain was installed at the air exchanger section of the

tunnel

to block the rotor-driven flow from returning to

the settling chamber. Fresh air was admitted to the

settling chamber through openings downstream of the

curtain, as

shown

in Figure 3. Prior to acquiring wake

data

using the PIV system, tests were run both with the

curtain in place and with it removed. During these

tests, the tunnel drive-fan was off and the

rotor

operated

at 870

rpm. Tests

were run

with

the collective pitch

angles of the

rotor

blade set at 1°, 3°, 5°, 7°, 9° and 11°.

Thrust and torque were measured using the balance

mounted on the

rotor's drive shaft.

The measurements

showed

that

the thrust and the torque on the model rotor

were not

significantly affected

by the blockage of the

rotor-driven flow.

In

these

test

runs,

flow velocities

were measured

in

the test

section

using vane- and

thermo-anemometers.

Total

volume

flow rates through

the test section were

determined from

the measured

test

section velocities. It

was

found that

the

flow

through the test section was

reduced between 14% and 21% by the blockage of the

rotor driven flow. Balance-measured thrust values were

used to estimate the

flow

through the rotor disk using

the axial

momentum theory.

It was

found

that

flow

through

the

test section

was

between

2.5 and 2.9

times

the

estimated flow through

the

rotor

disk,

indicating

that

a sizeable

portion

of the

flow through

the

test

section

did not go through the

rotor

disk.

Measured velocity contours in the test section

indicated

that

the rotor wake was diffused by the time it

entered the test

section.

It is postulated that the wake,

together with the fluid it entrained on its way to the

test

section, accelerated

slightly

in the

contraction

section.

The

acceleration lowered

the static

pressure

in the

test

section

slightly.

This

lowered

test

section pressure

created a

flow external

to the

'slipstream'

of the

wake.

The flow

through

the

test section

is

therefore composed

of the rotor

wake

and a

flow

external to the rotor wake.

The momentum

f l u x

at the

test section

was

estimated using measured average velocities and found

to be

greater than

the

thrust

on the rotor.

This excess

of

momentum

f l u x

supported

the

view that

a pressure

difference

existed

between

the settling chamber and the

test section.

The average settling chamber flow

velocity was very low.

(With

the curtain installed,

this

velocity was 1.55 fps for the 11° case, of which 0.54 fps

was due to the estimated flow of the rotor wake.) Only

a minuscule pressure difference would produce the

measured amount of flow through the test section.

Experimental

verification

of this

minute

pressure

difference

is therefore d i f f ic u l t .

Wake-integral expressions presented in Section 5

are applicable to

rotors

in

axial

flight, with hover as a

special

case.

The question as to what specific flow

rate

corresponds to the true hover state is not essential to the

present work. Future test

runs

with the test section

access doors open

to

equalize

the static

pressure

in the

test section with ambient air are desirable. With the

access

doors

open,

the measured flow rate

through

the

test section

can be used to

establish

the

true hover

state.

American

Institute of Aeronautics and

Astronautics


8/12


Th e

curtain

at

the

air exchanger section of the

tunnel was installed to block the return of the rotor-

driven flow during PIV test runs. The tunnel drive fan

was o f f . Two pulsed laser-light sheets were

introduced

in a plane

parallel

to the rotor's

axis,

as shown in Figure

4. As the

rotor rotated,

its

blades passed

through

this

light sheet

repeatedly.

Th e rotor was

operated

at 870

rpm to

match

the

maximum pulse

rate of the

laser.

Th e

blade-tip speed

wa s

about

286

f j p s

and

compressibility

effects

were

not

important.

Th e light sheet was aligned

to the trailing edge of the blade at the instant the

blade

advanced

passed

the sheet. This instant of time was

used as a reference time level. A set of 25 images was

acquired at the same blade azimuth during a series of

blade

revolutions. The images

were combined

to

produce time-averaged

velocity

fields

at specific

relative positions

between

the

blade

and the wake-

survey

plane.

Figure 5 shows the

geometry

of the rotor blade and

the position of the

light

sheet

relative to the

blade

at the

time levels particle images w ere acquired. The

light

sheet was stationary while the blade advanced during

the tests. Th e distance between the blade tip and the

light sheet designates the relative position of the

particle images (wake-survey

plane)

and the blade. Fo r

example,

a 2.0-c (two-chord)

wake-survey

plane

designates

particle images acquired

at the

instant

the

blade tip advanced two chords

from

th e

plane.

Particle

images were

acquired for two

collective

pitch

angles, 5° and

11°,

in three contiguous

rectangular

data patches along the blade

span.

Each patch covered

approximately 6 of span and 10 of

axial

distance.

The three patches together

covered

about

17.6"

of span

extending between

2 0 . 9 from the rotor

axis

to

0.8

outboard

of the

blade

t i p . In the

axial direction,

the

boundaries

of

each zone

were

about

3

upstream

and 7

downstream

of the rotor

disk.

Figures

6 and 7 show contours of the velocity

v

e

and the vorticity

C f l e

at the

wake survey

planes 2 . 0 - c for

the 5° and the 11° cases. The v

e

velocity

deficit layer

represents

a

layer

of

ov

This

layer

is the

footprint

of

the two boundary

layers

on the blade

surface.

This

layer is

composed

of two sub-layers, one

from

the

upper boundary layer and the

other

from

the

lower

boundary layer.

The

(O r contents

of the two

sub-layers

have different signs. The positive and negative

vorticity o\

in the two layers are connected by c i > z to

form closed vorticity loops

in the

9-plane.

The

presence of the vorticity co

z

, though not shown, can be

inferred from

the presence of the 0^ sub-layers. As

time

progresses,

vorticity

loops

emerge in successive 0-

planes.

With

downwash, a

helical

wake layer

composed

of

vorticity

loops

that lay in

planes normal

to

the helical layer is formed. In order to determine the

profile

drag, the sub-structure of the helical o\ layer

must

be

recognized.

If the

a^ layer

is

approximated

as

a

vortex sheet,

in other w ords, a layer of zero

thickness,

then

the

profile drag cannot

be detected.

This

is

because

the approximation makes vorticity

moment

zo\

zero and therefore (16)

gives

a zero profile

drag.

Th e

approximation

masks the

deficit

of

v

e

in the wake and

thus

(17)

gives a zero

profile drag.

The (O e

vorticity

in the

wake

is the footprint of the

circulation change

along the

span

of the blade.

This

footprint

is

linked

by

( 1 4 )

an d

( 1 5 )

to the thrust and the

induced torque on the rotor.

Figures

6 and 7 show

that

the c o e wake

associated with

each blade is composed of

a

strong

tip

vortex,

i . e . , a

helical

tube of

intense c o &

trailing the blade t i p , and a weak helical layer of

c o &

inboard of the tip vortex. The

blades

of the

present

tests

are

twisted.

The

observed

(0 &

distribution

indicates

that

the circulation around the

blade

changes

slowly

along the span an d

drops abruptly

to

zero

outside

th e

t i p .

The sign of

co ^

in the

inboard

layer is

opposite

to

that in the tip vortex. The (O e layer can be

approximated

by a

vortex sheet, without losing pivotal

information about

either the thrust or the induced torque

on the rotor.

This

is because

this inboard

co ^

layer,

unlike the

co ^

layer, is not composed of sub-layers

containing

vorticity of different signs. This

O G

layer is

a part of the vortex

loops lying

in the helical wake

sheet,

no t

normal

to the

sheet.

Th e

presence

of a hub

vortex and a starting vortex is inferred by the presence

of the helical layer of c o g . The hub vortex and the

starting

vortex, together with the circulation

around

the

blade, complete

the

vorticity

loops

containing

the

vorticity

0 0 9 .

The hub

vortex

and the

starting vortex

are

both outside the three data patches of the present tests.

Th e presence of tip vortices is

evident

in

Figures

6

and

7. Two

traces

of tip vortices

appear

in

Figure

7 for

the 11° case.

The one

very close

to the rotor

disk

is

associated with the blade that most recently passed

through the wake-survey plane. For convenience, this

blade is called the

first

blade. The

second

trace is

associated with the second blade, which is

about

1 8 0 °

from

the survey plane at the

instant

particle

images

are

taken.

A

third trace

of a tip

vortex

is observed in

Figure 6 for the 5°

case. This third trace

is the

footprint

of

the first

blade during

its previous

passage through

th e wake-survey plane.

The

layers

of

C 0 r

and

0 )9

leave the blade

together

and they are transported in the fluid by identical

American

Institute

of

A eronautics

a nd

Astronautics


9/12

(c)2002 American Institute

o f

Aeronautics

Astronautics

or

Published with Permission

o f

Author(s)

and/or

Author(s) '

Sponsoring Organization.

physical

processes

of

convection

an d diffusion. Th e

two

layers

therefore

occupy the same

physical

space.

The structures of the rotor

wake

described above

are

evident

at all

wake-survey planes.

Th e

thinness

of

the

inboard

vorticity layers in

Figures

6 and 7

indicates

that

for both the 5° and the 11°

cases,

any significant

flow separation, if present, is restricted to the root

portion

of the blade not covered by the data

patches.

As the tip

vortex moves axially,

it

also moves

inboard.

Figure 6 shows that, for the 5°

case,

the tip

vortices move axially at a speed substantially slower

than

that of the inboard

wake layer.

Th e movements of

the second blade's tip vortex bring it to the path of the

first

blade's inboard wake layer. A strong interaction

between the inboard wake layer of the

first

blade and

the

tip vortex of the second blade then occurs. For the

11°

case, the axial

speed

of the tip vortex is greater.

Th e

strong interaction

between the inboard vorticity

layer of the

first

blade

and the tip vortex of the

second

blade is not

observed

in

Figure

6 .

Spurious vorticity

along the

boundaries

connecting

the three data

patches

is observed in

Figures

6 and 7.

This spurious vorticity is attributable to an inexact

matching

of the three

data

patches in the

tests.

Fo r

wake-integral analyses, this spurious vorticity is filtered

and disregarded. Figure 7 also

shows

widespread

traces of

background noises.

The noises are

weak

and

do not have

significant effects

on wake-integral results.

Wake

data at the

0.5-c

wake-survey plane contain

excessive

spurious

values.

Th e

quality

of

these data

is

no t sufficiently high for meaningful

aerodynamic

analyses.

For the

11° case, wake data

for the

innermost

data

patch are either

missing

or not of

sufficiently high

quality at the 1.0-c,

4.0-c

and

5.0-c

wake-survey planes.

The qualities of all other

acquired

wake

data

are

comparable

to those

shown

in

Figures

6 and 7.

Because

of the

strong

interaction

between

the

vorticity layers

left

behind by the first blade and the tip

vortex

left behind by the

second blade,

the wake data

for the 5° case are not suitable for the evaluation of the

profile torque.

Profile

torque values

are

determined

using (17) and wake

data

for the 11° case. As

noted,

the three data patches cover only the outboard r >

20.9

portion

of the wake. In evaluating the

profile torque,

the contribution of the missing inboard wake

data

is

estimated by assuming the inboard o\ layer does no t

change with the span in the root portion of the wake.

Based

on

this

assumption, the missing

c\

layers in the

wake-survey planes

2.0-c

and

3.0-c

are estimated to

contribute

17% of the

total profile

torque. For the

wake-survey planes 1.0-c, 4.0-c

and

5.0-c,

the missing

(O r

layer

in the

root portion

of the

blade,

including those

in the innermost

data patch,

is estimated to contribute

36% of the total

profile

drag.

Wake data at the

2.0-c

and the 3.0-c wake-survey

planes

for the

11° case

show

that

the

o\

content in the

wake

layer

does

no t

change

rapidly in the two inboard

data patches.

Th e

estimated contributions

of the

missing inboard data do, however,

introduce

uncertainties in the evaluation of the profile

torque.

This uncertainty is due in part to the physical

presence

of

the

root structure

of the

model rotor Also, with

the

twisted blade, it is possible that

flow

separates over a

root portion of the blade , especially in the 11° case.

The vorticity

0 )9

in the

inboard

layer is

found

to be

very weak. For example, for the

11°

case, the

magnitude

of the

integrated c o & value

in the

inboard

layer is

determined

to be

1.4%

of

that

in the tip

vortex

at

the

2.0-c

wake-survey

plane.

As

(15)

and (14)

show,

the contributions of (Oeto the induced torque and the

thrust

a re

weighted

by the

factor r

2

.

Th e

missing

data in

the root

portion

of the blade

span

is therefore

unimportant in the evaluation of the induced torque a nd

the

thrust

using

wake-integrals.

Since the tip vortex is

located in the outermost data patch, the induced torque

an d the

thrust

on the

rotor

can be

accurately determined

using only

wake data in this outermost

data patch.

Induced torque

values, determined

using

(15), are

shown in Figure 8 for the 11° case. Total torque

values

are

obtained

by adding the

values

of profile

torque,

determined using

(17), to the induced

drag

values. The

very

good

agreement

between the balance-measured

value

and the

total torque

values

determined using wake

data

at survey planes

1.0-c

and

2.0-c

is

unforeseen

since, as discussed, the missing inboard wake-data

introduce uncertainties

in computing the profile torque.

Figure 9 shows the thrust on the rotor

determined

using

(14). Th e agreements

between

the wake-integral

results and the balance-measured thrust at all wake

survey planes

for both the 5° and the 11° cases are

reasonably good

and

encouraging.

Wake-integral expressions are derived in Section 5

by analyzing the

rate

of

emergence

of new vorticity

moment in the wake. It is

therefore

preferable to use

wake-survey planes close to the blade. As discussed,

the o> r layer is composed of two sub-layers containing

OT with opposite signs. As the wake ages, diffusion

disperses

the vorticity and partially

annihilates

the

positive

Or and the

negative 0 )r

in the two

sub-layers.

The

wake-integral

expression (16), or equivalently (17),

therefore provides more

accurate profile torque values

8

Am erican Institute of

Aeronautics

and Astronautics


10/12

(c)2002 American Institute of Aerona utics Astronau tics or Published with Permission of Author(s) and/or Author(s) ' Sp onsoring O rganization.

at wake-survey planes

closer

to the

blade.

Diffusion

effects are

less important

in the

determination

of the

thrust and the

induced torque since

(0 &

resides

nearly

wholly in tip

vortices.

7.

CONCLUSIONS

Th e

wake-integral

method connects the footprints

left

behind by a

solid

body in flight to the aerodynamic

force and moment on the

body.

Through this

connection,

the task of solving a

three-dimensional

aerodynamic

flow

problem is reduced to one of

evaluating the footprints in a

two-dimensional

planar

area. Information about

these footprints

can be

acquired

either

experimentally or comp utationally. By

reducing the dimensionality of the information required

to

determine

the

aerodynamic

load

from

three to two,

the method offers major

advantages

in all

three

branches of

aerodynamics

- theoretical, experimental

an d

computational.

Th e

method

is

efficient since

the

required

footprint information is

restricted

to the small

vortical

wake

region

of the

flow.

The central theoretical task of the

wake-integral

method is the establishment of wake-integral

expressions.

In the

present research,

a vorticity-loop

method was

developed

and used to derive new wake-

integral

expressions

for the

finite wing

problem.

Compared to previous

wake-integral

expressions for the

induced and the profile

drags,

the new wake-integral

expressions

are

remarkably simpler

a nd more

efficient.

New wake-integral

expressions are also derived,

using the vorticity-loop

method,

for the thrust, the

induced torque and the

profile

torque on the rotor.

These

expressions

connect the

footprints

of the rotor

blade to the aerodynamic load on the

rotor.

Th e

azimuthal component of the wake vorticity is

connected

to the thrust and the induced torque. The radial

component

of the

wake vorticity

is

connected

to the

profile

torque. The

axial component

of the wake

vorticity

does no t

need

to be known

explicitly.

Its

presence

in the wake and its contribution to the

aerodynamic load

are

inferred from those

of the

azimuthal

and radial components of the wake vorticity.

With the new wake-integral

expressions,

the use of

wake

data

very close to the

trailing

edge of the lifting

body

is

preferred. This

fact

offers

an

important

advantage to the use of

CFD

in wake-integral analyses.

Numerical

methods capable

of accurately

simulating

the near

wake

are useful,

even

i* ihe far

wake cannot

be

accurately simulated

because

of

numerical

diffusion.

Experiments performed

in the present research

have

validated

the p racticality and the major advantages

of

the

wake-integral

method. Th e power of three-

dimensional

particle-image velocimetry

in

experimental

aerodynamics has also been demonstrated. In addition

to providing quantitative wake

data, particle

imaging

has brought into focus wake

features

often

disregarded

in the past. These wake

features

are relatively

inconspicuous, but important to

viscous

an d

unsteady

aerodynamic analyses.

Efforts of the

present

program have

laid

the

foundation for continued efforts to

construct

a

practical

aerodynamic design tool

using the wake-integral

method.

Acknowledgements

The contribution of the wind-tunnel task-team fo r

the

present research

is

gratefully acknowledged.

Members

o f this

team include

Anita I.

Abrego,

Brian H .

Chan, Steven Chan, Lauura

Galvas, Joel T.

Gunter,

Elizabeth M. Hendley, Jon L. Lautenschlager an d

David

W . Pfluger.

Samuel

S.

Huang

served as the on-

site

engineer

of

Applied Aero

throughout the

planning

an d

execution

phases

of the wind

tunnel tests.

Dr. Luiz

Lourenco designed the particle image

velocimetry

system

an d provided related technical

support,

including the processing of particle images. Dr. Chee

Tung's support an d

timely

advice throughout

this

research program is also gratefully acknowledged.

References

1.

Prandtl, L.

Applications

of

Modern

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to Aerodynamics ,

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16 ,

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"Ein

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1925

3. Anderson, Jr., J. D. "AHistory of Aerodynamics",

Cambridge 1997

4.

Maskell,

E. C. "Progress Towards a Method of

Measurement of the

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Wu, J.

C., Hackett,

J.

E.,

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D. E. A

General

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Approach for Drag

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Kroo,

I. Drag due to Lift: Concepts for Prediction

an d Reduction", Annual Review of Fluid Mechani cs ,

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7.

W u,

J. C. and

Wang,

C. M.

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Rotors , USAAM CO M

T R-

01-D10, Aviation Applied

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Command, 2001

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Theoretical

Aerodynamics

based

on

Vorticity Dynamics , Applied Aero LLC, Report

2001-A1231,2001

9. Wu, J. C.

Theory

for Aerodynamic

Force

an d

Moment in

Viscous Flows", AIAA Journal,

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Caradonna,

F., Henley, E., Silva, M ., Huang, S.,

Komerath,

N. , Reddy, U. , Mahalingam, R., Funk ,

R.,

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O .,

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Gregory, J.,

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Aerodynamics, American Helicopter Society,

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Closed tube

of

varticitv

Figure la . Vortex

loop approximation

of

vorticity tube.

or

2

Figure Ib.

Division

of

vortex

loop into smaller loops.

Air

Exchanger Section

J

Settling Chamber

Figure 3.

Wind tunnel flow circuit.

r

Figure 4. Particle imaging system layout.

Figure

2a. Lifting-line

vortex-loop

systems a t time t = T.

-1

K U S t

New

upstream

system Transported Sd

Expanded S

u

Figure

2b. Lifting-line vortex-loop systems at time t = T + 5t.

75.40-

4.03

7.55 degree linear

twist

Figure 5.

Blade

geometry and wake-survey planes.

10

American

Institute

o f

Aeronautics

a nd

Astronautics


12/12

c)2002 American Institute of Aeronautics Astronautics or Published

w ith Permission

of Author(s)

and/or

Author(s) ' Sponsoring Organization.

velocity

-

:

/

, . i i .

_______©

1

1

1

1

u

S

-6.00

-7.36

-8.71

I 1007

-~

11.43

^ s w 12.79

te a

14

-

14

15̂ 0

B

1 6 > 8 6

S

18̂ 1

P19̂ 7

B

Ks>

20.93

22.29

23.64

25.00

- vorticity

•

: i

-C*=>

o

=>

0 o

-=>

°•*•

• , , , 1 , (

c o «

£K̂ OĈ C-K

<

-f)

, , , , ,

t

e

^-^~^~-^S^

~ r

,

1

1

. r

VORX

500.00

463.57

>W 427.14

M390.71

r*-n 354̂ 9

r-TJ 317.86

n

281

-

43

H

245

-°°

li

208

-

57

S172.14

l̂

135.71

id

99-29

S

62.86

26.43

-10.00

r ln)

Figure 6. Streamwise velocity and vorticity contours.

2-chord wake survey plane

5-degree

collective-pitch angle

_ _ _ _ Batence-MeasuredJotal

a

Wake-Integral,

Induced

•

Wake-Integral,

Total

2 3 4 5

Distance between Blade and

Wake

Plane chord)

Figure

8. Rotor torque, 11-degree

collective

pitch.

velocity

30

tin)

vorticity

5-degree collective

_ _ _ _

Balance-Measured

• Wake-Integral

VORX

500.00

463.57

427.14

390.71

H354.29

317.86

281.43

245.00

208.57

172.14

135.71

9929

62.86

26.43

| -10.00

Distance between Blade

and

Wake Plane chord)

t 11 -degree collective

- - - -Balance-Measured

•

Wake-Integral

rlane chord)

Figure 9. Rotor thrust, 5- and 11-degree

collective

pitch.

11

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