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7/25/2019 A Method for Measuring Skin Friction Drag on Flat Plate
1/2
JULY 1984
ENGINEERING NOTES
543
M
Y
= N
Z
Ny y
M
Z
=N
X
L
Y
-N
Y
L
YX
(14)
(15)
Next
an
equivalent Rmatrix using homogeneous coordinate
transformation
equationsis
developed
by
combining
the
three
rotational
matrices in the references As mentioned
previously, the order of combination is imp ortant If they are
combined as roll times pitch and then times
yaw,
the
followingmatrix
is
formed:
[cosa
cos/3]
[(shvysine*
cos/3)-
C O S T sin/3
]
[
C O S Tsinacos/3)
+
sin7sinjS
]
[cosasin/3]
[
(siiry
sina
sin/3)+ C O S Y
cos/3
]
[
C O S Y
sina
sin/3)
- sin7cos/3]
Equating terms to the matrix in Eq (1) gives nine equations
in
three unknow n variables From these we can determine that
a=sm-
1
-L
z
)
/3
=cos~
7
(L
x
/cosa) =sin~
7
( L
y
/cosct)
7 =
sin
7
( M
z
/cosof)
=cos~
7
(A^/cosa)
(17)
(18)
(19)
Either of the twopossible values of a.are correct /3 and 7
are uniquely determined onceais selected
The
required
X i
y
1
an d
z
t
local axis origin values
are the
coordinates of P
l9
which were subtracted
from
the coor
dinates
of
P
2
an d
P
3
to get apure
rotation
problem prior to
calculation
ofth eL M an dN
terms
There
is a
trivial case
to
consider
Ifcosa= 0 ,
there
is no
solution by this metho d This occurs when the local X axis
aligns withtheglobalZaxis This isindicated w hen \L
Z
\=1
As we have selectedrotation order of roll, then pitch, then
yaw
it
followsthat whenpitch
is
-Tr/2),
therolland yaw
must
sum to the
totalangle between
the
local Y axis
and the
global
Y
axis
If the
pitch
is +ir/2 the yaw has
negative sign,
thus
@-L
z
y= d (20)
where
-M
x
) = c o s ~
7
(M
y
)
(21)
Any combination of
/3
and 7 meeting the equations is
correct
Application of the Inverse Procedure
W ehaveno w identifiedaprocedurefor determining
X j
yj
Z j roll, pitch,
and yaw for a
local
axis system given three
points;
the
local axis system origin
P
7
a
point
P
2
on the
desired
local
X axis
and a point
P
3
on the
desired localX
Y plane
Applications of this procedure no w
will
be described The
most direct application is to allow the designer to
specify
that
a
component will
go
from here
to
there
by
input
of
three
points
The first
point
is here an d
becomes Pj
in the
preceding calculations
The
second point
is
there
an d
becomes P
2
The third point, P
3
, specifies only the com
ponent roll about
th eX
axis
by
defining
th e
X
Y
plane
This capability is shown in Figs 2 and 3
Figure 2 showsa typical design problem A w ing strut is
required extendingfrom point Pj to pointP
2
,and is oriented
such that the airfoil streamwise direction
(Y
axis) is ap
proxim ately parallel to the globalX axis
P
3
is coincident with
P
2
in rear viewand forw ard ofP
2
in top view thus orienting
th e X Y plane prop erly No te the initial strut location
whichis irrelevan t to our solution
Figure 3 show s the resulting strut location The calculated
values obtained in this example ar e A: /=70 35
> >
7
= 9 0 7 ,
j = -9 28, roll = 0 7
deg,
pitch = 18 8 deg and yaw= 87 9
deg
A
second
utilization of
this procedure
for
determining
the
local axis system values permits component rotations in the
global axis system regardless of the original orientation of the
local axis system
[-sina]
[ siirycosa]
1 6 )
This can be implemented using the preceding equations by
a simple trick W e take three arbitrary points in the local axis
system,convert them to global coordinates rotate them in the
global system, and then use the given procedure to
find
the
new
local axis system
x
t
y
}
Z j,
roll pitch, and yaw To
simplify
matters the three local axis system points selected
ar e
P,
= (0, 0 0),P
2
=( l ,0, 0), andP
3
=(0, 1 0)
Trunnion axis
rotations
can be implemented in a similar
fashion This and the related problem of trun nion axis
location are detailed in Ref 4
References
Computer
Graphics Prentice
HallGiloi W Interactive
Englewood C l i ff s N J
1978
2
Foley J and Van Dam A
Fundamenta l s
of
Interactive
C om
puter
Graphics Addison
Wesley Publishing Co
Reading Mass
1 9 8 2
3
Faux I D andPratt ,M J Computat ional
Geometry
for Design
and
M anufacture
John
Wiley Sons New
York
1979
4
Raymer
D P
Maier
R A and
Killian
M J Conceptual
Kinematic Design Using Homogeneous Coordinate Trans
formations AIAAPaper83 2460 Oct 1983
A M ethodforM easuring Skin Friction
Drag on a
Flat Plate
in
Contaminated G asFlows
R
B
Oetting*andG
KPattersont
Universi ty
of
M issour i -Ro l la ,Rol l a ,M issour i
Introduction
T
H E
most s t raightforward
way to
measure drag
on a
surfaceimmersedin afluid
flow
is bydirect measurement
of the force on the exposed surface
l
This usually involves
replacing
a
portion
of the
surface with
an
imbedded sensor
surface generally about 0.5 to 1 in in diameter (some
noncircular sensors have been used) Th e surface is directly
connected to a sm all force transducer beneath the plate It is
possible
to
achieve good results with these sensors
bu t
care
must betaken intheir installation
2
to avoid misalignmentof
th e
surfaces
an d
binding between
th e
sensor surface
and the
surroundingplate surface
An alternate method of determining surface drag is through
indirect methods based on similarity arguments
and/or
ce r
Received Feb 15
1984Copyright
American
Institute
of
Aeronautics and Astronautics, Inc
1984
All rights reserved
* Professor, D epartmentof MechanicalandAerospace Engineering
Associate
FellowAIAA
tProfessor
Department
of Chemical Engineering;
currently
at
University ofArizona
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7/25/2019 A Method for Measuring Skin Friction Drag on Flat Plate
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544
J
AIRCRAFT V O L 2 1 N O 7
L e a d i n g E d g e
Fa n
R o u g h e n e d
S u r f a c e
T r a i l i n g
d g e
F a i r i n g
Fi g
1 Drag
plate
configuration all
dimensions
in inches)
3 1
D N O P A R T I C L E S
O
P A R T I C L E S
Iv
D
f
l 2 p V C
D
A
p
Fig
2
Flat
plate
dragcom parisons
tain assumptions about th e
velocity profile
an d turbulence
levelin the surface
boundary
layer In
this broad category
of
drag determination,
th e
method
most easily
understood
in
volves
the measurement of the velocity
profile
of the boun
dary
layer, usually with a hot wire anemom eter Once the
velocity profile
is
established
it is possible to determine the
surface friction coefficient by
techniques
such as the
logarithmic law of the
wallpresented
by
Clauser
3
Neither the method
utilizing
the embedded sensor
surface,
nor an
indirect method
involving th e
measurement
of the
surface boundary
layer velocity
profile, isadequatewhen th e
fluid flowfield is
contaminated
with
particles Particles
as
small
as 50/xmdiameter
suspended
in the
flow
wouldinterfere
with
th e
function
of both the embedded sensor and the hot
wireprobe
The technique of
skin friction drag
measurement
presented here was developed to overcome the problem of
suspended
particlesin the fluid flow
This
activity was
part
of
a larger projec t
4
to investigate the
potential
for
drag
reduction
of
suspended particles in the developing
boundary
layer on a
flat
plate The key to the success of thepresent technique of
measuring the flat plate
surface drag
is the suspension of the
testplateon air bearings
Apparatus
The flat plate
arrangement
isshownin Fig 1 The 0 25 in
thickaluminum testplate(6 in widex72 in
long)
islocated
between tw o d u m m y plates of the same size an d material A
gap of about0 02 in separatesthe testplatefrom th ed u m m y
side plates and the leading and trailing edge fairings The
leading
edge fairing
5
provides some
flow
conditioning
an d
th e
rear fairing provides trailing edge streamlining
and a
housing for the Revere
model UMP1
0 005 A
load
cell
transducer Side
rails
and a
lower
plate
support
th e
d u m m y
plates,
test
plate
ai r
bearings
an d
force
transducer This
arrangement also
protects
the air bearings and
force
trans
ducer from
particle
contam inat ion from the underside of the
test arrangement
The air bearing arrangement for the test
plate suspension
includes six
horizontal bearings
fo r plate support, with four
mo re vertical bearings for
plate alignment
The
horizontal
a ir
bearings are located in
pairs equally spaced along
the length
of
the testplate V ertical alignment air bearings arelocatedin
pairs
equidistant
between
the
front
and
center,
and center and
rear horizontal
air bearings This
arrangement
allows for
good plate support
with
the six
horizontal
air bearings, and
the ability to controlthe gap
separating
the testplatefrom th e
d u mmyplatesusingthefour adjustable vertical air bearings
The experiments were
carried
out in the University of
Missouri Rolla Subsonic Wind Tunnel a closed circuit at
mospheric tunnel
with
a test section 32 in
highx
48 in
wide
x 11ft
long
Low turbulence (on theordero f 02 )an d
good
flowdirection
controlar eobtainedby acombinationof
features, including two screens in the stilling chamber 9:1
stilling chamber to test section contraction ratio,
feedback
command control of fan speed, and a 6 5 deg maximum
diffuser angle Testswererun at tun nel speeds up to 150ft /s
Particles were injected
through
a spread
nozzle
(6 in
wide
x
1/16 in deep) at a position in the inlet contraction
upstream
of the test plateto m inimize flowfield interference
Oxides of aluminum and iron particleswereused rangin g in
size
from
20 to 150
/m i
Particledensities
were
as
high
as 0 3
Ib
of
particle
pe r
pound
of air
Results and
Discussion
Compari son of the experimental drag measurements
with
theory for the 6 in wide x 72 in long (3ft
2
)test
plate
is shown
in
Fig 2 The
experimental
measurements are compared
with
th e
dragforce
calculatedusing theequation
6
=
0074/(Re
L
)
1/ 5
fo r aturbulentboundarylayerstarting at th e test plate leading
edge A roughened surface on the leading edge
fairing
(see
Fig
1) causes the
boundary
layer to trip and
become
a tur
bulent
boundary layer before
reaching
th e
test
plate
leading
edge This was confirmed through boundary layer velocity
profile
measurements
using a hot wire anemometer with no
particle injection into th e flowfield A turbulent
boundary
layer
exists over the
entire
test plate for tunnel
speeds
above
50
ft /s
The results of
this
study
indicate
that th e
measurement
of
drag on a flat
plate
i n a
particle
contaminatedairflow can be
readily
madeutilizing a system which
supports
the testplate
on air bearings
Adjustment
of the air bearing support
system
is
easily
made to
properly align
the
plate
to
prevent
in
terferencewiththe
surrounding support
system
Acknowledgments
This
work w as performed as
part
of a
research project
supported
through the NASA
Langley
Research
Center
NASA NSG 1452 The
authorswould
liketo
acknowledge
th e
contribution of Walter A
Lounsbery, aerospace
engineering
graduate
student who was instrumental in organizing and
performing
the experiments
References
^hawan
S D irect Measurement of Skin Friction NAC A
K e p t
1121 1952
2
Issa
R
K
and Lockwood F C Experimental
Study
of
Error
Sources in Skin Friction Balance Measurements Journa l of F lu id
M e c h a n i c s V ol
99 No
1
March
1977
pp 197 204
3
Clauser ,
F H
Turbulen t Boundary Layers
in
AdversePressure
Gradients
Journa l o f t he Aeronau t ica l
Sciences
Vol 21 No 2
Feb 1954 pp 91 108
4
Pat terson G K
Getting
R B Anderson R A and
James
W
J
Study
of Gas Solid Airflow Over a Flat Plate, Final Tech
Rept NASA
NSG 1452
Universi ty
of
Missouri Rolla Rolla
M o
Jan 1983
5
Davi s
M R
Design
of
Flat Plate Leading Edges
to Avoid
Separation
AIAA J o u rn a l Vol 18 May
1980 pp
598 600
6
Schlichting H B o u n dary L ay e r Theory 4th Ed McG raw H ill
BookC o New York 1960 pp 5 3 6 5 3 9