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

7/25/2019 A Method for Measuring Skin Friction Drag on Flat Plate

2/2

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