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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
American
Institute
o f
Aeronautics
a nd
Astronautics
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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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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.
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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
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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.
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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
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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
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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.
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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
-
8/17/2019 Wake-Integral Determination of Aerodynami Drag Lift and Moment in 3d Flows
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
American Institute of Aeronautics an d Astronautics