An Outline of the Techniques Available for the Measurement of Skin Friction in Turbulent Boundary...

P~og. A~rospuee Sci. 1977, Vol. 18, pp. I - 57. Pergamon Press. Printed in Great Britain.

AN OUTLINE OF THE TEChnIQUES AVAILABLE FOR THE MEASUREMENT OF SKIN FRICTION IN TURBULENT BOUNDARY LAYERS*

K. G. Winter

Royal Aircraft Establishment, Farnborough, Hants, U.K.

Summary-The techniques covered include force-mdasur~ment balances, the use of the velocity profile, pressure measurements by surface pitot tubes or about obstacles, and the use of the analogies of heat transfer, mass transfer or surface oil-flow. Hot-wire or laser techniques for determining the shear stress within the fluid are not included. The sources of error and ranges of application of the various techniaues are discussed.

i. INTRODUCTION

In most applications of fluid mechanics a knowledge of the drag created by fluid flowing over

a solid surface is essential to the understanding of the performance of a system whether it be

a ship or an aircraft or the flow through a pipe. Considerable effort has therefore been de-

voted to the measurement of skin friction. This brief review concerns itself only with exter-

nal flow and with measurements primarily related to the performance of aircraft.

It was, however, the need to estimate the performance of ships which led to the first measure-

ments at high Reynolds number. Probably the first systematic investigations were made over

iO0 years ago by Froude (1872) who measured the drag of a series of planks towed at various

speeds along a tank using the elegant apparatus shown in Fig. I. It is interesting to note

that at that time even the qualitative effect of Reynolds number on skin friction was not gen-

erally understood. Froude did apparently have a concept of a boundary layer and states:

The investigation of skin friction may be separated into three primary divisions: (i) the law of the variation of resistance with the velocity; (2) the differences in resistance due to differences in the quality of

surface; (3) the differences in the resistance per unit of surface due to dif-

ferences in the length of surface.

The necessity of investigating the latter of these conditions may not be at once apparent, it having been generally held that surface-friction varies directly with the area of the surface, and will be the sanm for a given area, whether the surface be long and narrow or short and broad. It has always seemed to me to be impossible that this should be the case, because the portion of the surface that goes first in the line of motion, in experiencing resistance from the water, must in turn co~mmnicate to the water motion in the direction in which it itself is travelling, and consequently the portion of the surface which succeeds the first will be rubbing not against stationary water, but against water partially moving in its own direction, and cannot therefore experience as much resistance from it. If this reasoning holds good, it is certain that doubling, for instance, the length of a surface, though it doubles the area, would not double the resistance for the resistance of the second half would not be as great as that of the first.

*Notes prepared for von Karmon Institute for Fluid Dynamics Lecture Series on "Compressible Turbulent Boundary Layers", March 1 - 5, 1976~ Copyright ~ HMSO (London) 1976.

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Skin friction in turbulent boundary layers 3

Later Ke~f (1929) ~de measure~nts of local skin friction at several stations along the bot-

tom of a pontoon 77 m long using fairly large panels (309 x I010 m) ~unted on balances (Fig.

2). These measure~nts achieved Re~olds numbers of up to 5 x 108. ~e direct ~asurement

of skin friction by force balance was an essential step in setting ~ the basic skin friction

laws and these measurements of Ke~f together with those of others, notably Schoenherr (1932),

formed the basis for the generally-accepted skin friction esti~tion for incompressible flow

(see, for exa~le; Goldstein 1938). Because of our limited understanding of turbulent flows

-! Platte I-

o.~J'~30° ~ [ t ,- IO00mm ~ -I

.

,], I ~Jfh~ngun~ -.-"- ~" . . . . . . . . . . F - - - ~

/ o ] ~ II ",Feder spannmol"or

Gml~ewicht ~J.O mX,,~ I 0 1 0 - 3 0 9 ~ .LI.O mm~ Uhr- [ Eichgewichte Bewegliche Versuchs~platte "x~e rk

--3chreibtrommel

H • N,

aor | u i i

F

m

V Fig. 2. Apparatus used by Kempfe for measurement of local skin-frlction

coefficient.

4 K.G. ~inter

there has been the need to extend direct skin-friction measurements to compressible flows but

the difficulty of applying the technique in many situations such as flows with pressure grad-

ients has led also to the development of indirect means of obtaining skin friction. Aerodyna-

mics is largely an empirical science and many experiments are made in which the pressure dis-

tribution over bodies is measured for comparison with calculation, or as an aid to understand-

ing the characteristics of the flow. It is the author's view that the value of many of these

experiments would be considerably enhanced if, in addition, the skin-friction distribution

were also to be measured, even if only approximately, by one of the simple techniques discus-

sed in the paper. A good review of the variety of techniques which have been devised to cover

the diversity of situations encountered in practice is given by Brown and Joubert (1967).

They present a relative classification of techniques on which the following chart has been

based.

Classification of techniques for measuring skin friction

Wall shear stress-- measurement

-- Wall similarity--F-Velocity profiles

--Liquid tracers ~Analogies

-- Momentum ~--Flow about balance obstacles

-- Direct measurement

Heat transfer

L Mass transfer

i Preston tubes

Stanton tubes

Razor blades

Steps and fences

Static pressure holes

The remainder of the paper discusses the various techniques as outlined in this classification

and omits any consideration of hot-wire or laser techniques by which the shear stresses with-

in a fluid may be measured and the wall shearing stress obtained by extrapolation. The use

of the momentum equation is also not discussed, since in principle this is straightforward,

but in practice is difficult because it is not easy to take account of the three-dimensional-

ity of the flow which generally occurs even in nominally two-dimensional situations, and also

because often the skin-friction term can only be derived as the difference of two large terms.

A,B

AI, A2

C

cf

C P

d

2. SYMBOLS

!

a2 _ y - I ~/TWe defined by - 2 M , also speed of sound

T a +-~-~- 1 constants in the law of the wall

area of pitot tube

constant in Spence's law of the wall, alson non-dimensional thickness of viscous

sub-layer u 6L/~ local skin-friction coefficient TW/½0U2

specific heat of air at constant pressure

diameter of Preston tube

diameter of static pressure hole

diameter of floating element of skin-friction balance

force on skin-friction balance


g

h

i

J

J

k

K

K m

m

M

M T

P

Pf' Pr

q

R

Re x, Re 8 ,

S

S

t

T

U

u

u T

u =

u =

P

v W

w

w(y/6)

x

Y

x*, y*

gap round element of skin-friction balance

height of step, height of razor blade, thickness of film of oil, enthalpy

current

molecular diffusion

function in equation (8-12)

thermal conductivity of air

thermal diffusivity of air k/OCp

mass transfer coefficient

length, length of heated element, length of mass transfer element, distance apart

of elements in Section 8.3

rate of mass transfer per unit area

Mach number

friction Mach number uT/a w

pressure

pressure-rise parameters for forward and rearward-facing steps (Ap/T W)

rate of heat flow per unit area

electrical resistance

Reynolds number based respectively on streamwise length, momentum thickness~

Preston tube diameter and length of heated element

radius of floating element of skin-friction balance, also temperature recovery

factor

developed total velocity, equation (4-16)

area of floating element of skin-friction balance

thickness of floating element of skin-friction balance, also time

temperature

velocity at edge of boundary layer

velocity in direction of U

friction velocity

(u 2 + w 2)

gj velocity of wall injection

cross flow velocity

wake component of velocity profile

s treamwise coordinate

coordinate normal to wall

Preston-tube coordinates

Apd 2 Twd2

x* = log 4pv2 , y* = log 40--~/~

6 K.G. Winter

Aph 2 Twh2

razor-blade coordinates x* = log 7 , y* = log--~-f~

f~

B

Bo B q

Y

(S

6 L

q

kinematic pressure gradient ~ dp P dx

crossflow angle in boundary layer

crossflow angle at wall

heat transfer parameter q/PxDpTwU T w

ratio of specific heats

boundary layer thickness

thickness of viscous sub-layer

thickness of thermal layer

A P

pressure gradient parameter ~--~ u 3

T

A 't

T1

shear-stress gradient parameter---- pu 3 ~y

Y

I O_ dy 0 0m

variation of g with displacement of skin-friction balance

von Karman constant

viscosity

kinematic viscosity

mass concentration

density

T

Subscripts

e

m

~C Prandtl number -~

shearing stress

, also o =

edge of boundary layer

intermediate-enthalpy conditions

adiabatic wall conditions

½ (~ - I)M e e

1 + ~(y - I)M 2 e

w wall conditions

Superscript

i refers to incompressible flow.


3. DIRECT MEASUREMENT

Apart from the experiments of Schutz-Grunow (1940) interest in the direct measuren~ent of skin

friction* lapsed until the increasing speed of aircraft called for precise measurements in

compressible flows. As a result there have been numerous skin-friction balances designed over

the past few years, and much ingenuity has been exercised in design to overcome the essential

problem of obtaining accurate measurements of the shear forces which are very small, three

orders less than the inertial forces.

The problems which have to be considered are listed below.

(i) Provision of a transducer for measuring small forces or deflections, and the compromise

between the requirement to measure local properties and the necessity of having an ele-

ment of sufficient size that the force on it can be measured accurately.

(2) The effect of the necessary gaps around the floating element.

(3) The effects of misalignment of the floating element.

(4) Forces arising from pressure gradients.

(5) The effects of gravity or of acceleration if the balance is to be used in a moving

vehicle.

(6) Effects of temperature changes.

(7) Effects of heat transfer.

(8) Use with boundary-layer injection or suction.

(9) Effects of leaks.

(i0) Protection of the measuring system against transient normal forces during starting and

stopping if the balance is to be used in a supersonic tunnel.

The basic choice to be made is the size of the floating element which dictates the sensitivity

required of the measuring system which may he passive (displacement) or active (force-feed-

back). To illustrate the sensitivity required the table below shows over a range of Maeh

number the force in milligrams to be measured by a balance with a head of IOnnn diameter in a

zero-pressure gradient flow at one atmosphere stagnation pressure and a Reynolds number of

IO million.

M 0.I 0.5 i 2 3

Force mg 16 290 680 540 210

For adverse pressure-gradient flows the forces will be even less. A very sensitive transducer

is therefore needed and a variety of transducers and sizes of floating element has been used.

The table on the following page lists some of the designs developed in the past few years.

The balance used by Schultz-Grunow (1940) is included for historical interest. In this

balance the floating element was rather large and was mounted on offset torsional pivots and

restrained by a torsion bar. With the exception of the balance of Ozarapoglu (1973) (Fig. 5)

in which the floating element is supported on air bearings, the remaining balances have either

a parallel-linkage supporting arrangement (Fig. 3) or effectively a pivot below the floating

element (Fig. 4). A similar arrangement to that of Schultz-Grunow was used in a small balance

by Kovalenko and Nesterovich (1973). In their balance the floating element was pivoted about

an axis normal to its surface with the axis offset to one side of the element. The most popu-

lar device for detecting the position of the floating element is a Linear Variable Differential

*It might be noted that the classical work of Wieghard (1942) on the drag of surface excrescences made use of direct force measurements.

Reference

Test conditions

Size of floating

Type of suspension/

Force

element (nmO

position/ force transducer

range

Schultz-Grunow

1940

Dhawan

1953

Coles

1953

Weiler & Hartwig 1952

Lyons

1957

U = 2Om/s

1.6 x 106 < Re

< 16 x 106

x

Low speed 6 x 104 < Re

< 60 x I0"

x

Subsonic 0.2 < M < 0.8, 0.3 x 106 < Re x

Supersonic 1.24 < M < 1.44

M = 1.97 0.4 x 106 < Re

< I0 x 106

x

M = 2.57 0.4 x 106 < Re

< 9 x 106

x

M = 3.70 0.5 x 106 < Re

< 8 x 106

x

M = 4.54 0.4 x i0

< Re

< 8 x 106

x

Supersonic wind tunnel

Supersonic flight

< 1.2

x 106

300

x 500*

11.5 × 63

2 x 20

6.2 x 37.9

25 dia

50 dia

Optical/manually-operated

offset torsion bar

Parallel interchangeable

LVDT

Parallel linkage

LVDT

Translation of element

support by micrometer

LVDT

Double parallel inter-

connected linkage to eliminate

sensitivity to linear and

rotational accelerations

LVDT

20 mg

to

800 mg

3 g

30 g

Smith &

Walker

1958

0,ii < M < 0.32

106 < Re

< 40 < 106

x

50 dia

Parallel linkage

LVDT

Kelvin current balance

14 g

MacArthur

1963

Moulic

1963

Shock tunnel

M = 6 Low density

6.4 dia

Parallel linkage

Piezoelectric beams

0.25 x 25

Side flexure pivot

LVDT

5 g

20 mg

Young

1965

Supersonic flow with heat transfer and surface

roughness

25 dia

Parallel linkage

LVDT

Dershin et al.

1966

*Estimated from sketch

Supersonic flow with mass transfer

"Pointed ellipse"

Parallel linkage

LVDT

Moore &

McVey

Brown &

Joubert

Fowke

Bruno,

Yanta &

Risher

Winter &

Gaudet

Hastings &

Sawyer

Paros

(Kistler)

Miller

Franklin

Morsy

van Kuren

Ozarapoglu

1966

1969

1969

1969

19

70

19

70

19

70

1971

1973

1974

1974

1975

High temperature hypersonic flows

Low-speed adverse pressure gradients

Supersonic speeds

Supersonic speeds including flows with heat transfer

0.2 <

M <

2.8

16 x

106 <

Re

< 200 ×

106

X

M=

4

iO x

106 <

Re

< 30 x

106

x

Used in a

wide range of conditions including

flight.

Cooling system available

Low-speed flow.

Favourable pressure gradient

Subsonic wind tunnel and water channel

Low speed flow past circular cylinder

High-temperature hypersonic flows with heat

transfer.

Floating element water cooled

Low-speed.

Adverse pressure gradients

19 dia

127 dia

20.3 dia

368 dia

7.9 dia

9.4 dia

25 dia

16 dia

50.1 x

3.2

12 x

12

127 dia

Flexure pivot

3 g

Pneumatic position sensor

High temperature motor

Parallel linkage

200 mg

LVDT

Flexure pivot

200 g

LVDT

Permanent magnet plus coil

Flexure pivot

2 g

LVDT

Var iab i e

by changing

Motor-driven spring

loading

spring

18OO g

Parallel linkage

Resistance strain gauges

Parallel linkage

500 mg

LVDT

Pivoted about crossed-spring

IOO mg

flexure.

Differential capaclty.to

Permanent magnet plus coil

IO g

Parallel linkage

200 mg

LVDT

Pivoted.

Variable geometry

i g

electronic valve.

Jewelled pivots.

130 mg

Clock springs

LVDT

Parallel linkage

5 g

LVDT

Air bearings

I g

LVDT

o"

= rt

o"

0 m m

10 K.G. Winter

Atrflow Dashpot ~ /Floating element "~ _ _ 25rim dia ,./ - t ~ /

I I

x/////////// j[~L/7~ v o T Cover

Fig. 3. Parallel-linkage balance (D.R.L.).

Transformer (LVDT) which is capable of a resolution of displacement as small as 0.05 ~m. In

many balances the force is determined from displacement of the element against a spring as

indicated by the signal from the LVDT. There are disadvantages in the use of a displacement

balance since the necessary gaps round the element vary with the load and this variabion may

produce spurious effects. In balances of the nulling type the position transducer is used to

provide the signal to a force system which maintains the floating element at a given position

The force system is usually either a Kelvin current balance or a coil and permanent magnet.

As an alternative to separate position and force transducers the Kelvin current balance may

be arranged to serve both purposes as shown by Franklin (1960).

Airflov _ ~ ~Floating element

_ 9ram dla/~

::;:~t:#¢¢ ~ H ~ ; ~ H'clt Sh''ld

I I ~ t L ~ _ . ~ _ _ ~ V , ~ I I ,-.t.,-,, . i . j " " ' . . ' ~ T I ~ / A I l K / / ~ I " ~ / / / ~ F o ~ . b,,oo,, • p,",',': """ I zllv//y/ z, IN ]--,,,to"o,-,,,

(; tossed - sprinq pivot

Fig. 4. Pivoted balance (Kistler).

The difficulty of measuring small forces in a displacement-type balance was overcome by

MacArthur (1963) by supporting his floating element on a piezo-electric crystal and Winter

and Gaudet (1970) were able to use resistance strain gauges in a large balance. Franklin

(1973) has obtained high sensitivity by using a variable geomatry electronic valve. Moore

and McVey (1966) have investigated a wide variety of position and force transducers for ap-

plication at high skin temperatures.

Skin-friction balances have been used in flight on rockets (Fenter and Lyons, 1957) and on

high-speed aircraft (Garringer and Saltzman, 1967; Fisher and Saltzmann, 1973). To eliminate

their sensitivity to linear accelerations of the flight vehicle it is necessary to arrange a

mass balance for the measuring system as is done in the Kistler gauge (Paros, 1970). Weilet

(~954) and Lyons (1957) produced designs which were also insensitive to rotational accelera-

Skin friction in turbulent boundary layers I!

Airflov Position ~ Floating adjuster 127 mm d i a / e l e m c n t

Force transdu ldjusti.g s©revr (spring-mounted f ~ for floating .lem¢,t

Fig. 5. Air-bearing balance (Ozarapoglu).

tions. In wind tunnels measurements have been made with heat transfer (McDill, 1962; Young,

1965; Young and Westkaemper, 1965; Bruno et a~., 1969) and roughness (Gaudet and Winter, 1973;

Young, 1965) and also with surface transpiration (Dershin et a~., 1966). Westkaemper (1963)

investigated the effect of a mismatch of the surface temperature of the floating element with

that of the surface in which it was mounted, and in his particular conditions found no corre-

lation between the force measured and the temperature difference in contrast to the large ef-

fect found on heat transfer.

All balances are subject to the effects of misalignment of the face of the element with the

surface in which it is mounted and these effects will be different for parallel-linkage or

pivoted systems. Fig. 6 illustrates the way the pressure forces due to misalign~ent will act.

For an element which protrudes the pressure rise and pressure drop produced by the forward-

facing step and rearward-facing step respectively, acting on the edges of the element will

result in a moment or a force increasing the balance reading. For a moment-measuring balance

there will be an additional increment produced by the suction in the local separation which

is likely over the upstream surface of the element. For a recessed element the effects of

the pressure changes may still be large since, in addition to the forces acting on the edges,

which will affect the readings of a parallel-linkage balance, separations will occur both at

the upstream and downstream edges of the element and will produce pressure forces acting on

the face of the element and contributing to the moment on a pivoted balance. That these re-

gions of influence may be of considerable extent may be judged from the measurements of Good

and Joubert (1968) who showed that for a fence of small height h in low speed flow the pressure

12 K.G. Winter

upstream was influenced for a distance of up to I00 h. A moment-measuring balance is also

clearly at a disadvantage in flows with pressure gradients.

Parallel-linkage balance

///Flff~fffFfFW

Pivoted balance

Normal force

f P--

Protruding floating element

II NormQI fore* Recessed floating

element

Fig. 6. Effects of misalignment of skin-friction balance.

Systematic investigations of the effects of misalignmant have been made by O'Donnell (1964),

O'Donnell and Westkaemper (1965) and by Allen (1976). The balance used by O'Donnell was of

the parallel-linkage type and so the errors incurred should derive only from the axial forces

experiences by the element. It should be possible by making use of available data on the drag

of excrescences to make an estimate of these forces. Gaudet and Winter (1973) showed that the

drag of shallow holes or of very short cylinders could be estimated by assuming that the pres-

sure variation on the vertical faces was that of forward or rearward-facing steps multiplied

by the square of the cosine of the local angle of sweep of the edge of the hole, and assuming

also that the skin friction on the flat surface was unchanged. For a protruding cylinder the

ratio of the axial force on the vertical faces to the skin friction on the flat surface be-

comes

AF 4 h ~- = --? (Pf + Pr ) 3~

where h is the height, r the radius and Pf and P r are the ratios Ap/T w

(3-1)

for the pressure rise

on forward and rearward-facing steps. A comparison has been made with the example given by

O'Donnell for a balance disc of 25 mm dlamter with an edge thickness of 0.25 mm at M = 2.67

and Re e = 10,O70. O'Donnell does not quote a length scale for his boundary layer, and so in

order to deduce a roughness Reynolds number u h/~ for the height of his protruding or recessed T


balance a value for boundary layer thickness has been taken from Stalmach's (1958) results for

M = 2.73. The values of u h/v are below the range covered by Gaudet and Winter and a plausible

extrapolation has been used to obtain Pf and Pr" Now equation (3-1) gives only the force on

the protruding faces and for a floating element surrounded by a gap there will also be forces

on the edges within the gap. It is usual to assume that pressure forces effectively act over

half the thickness of the edge of the element. Adding half the thickness of the submerged

edge (0.125 mm) in fact gives an overestimate of the force and an effective depth of penetra-

tion of the pressure of 0.02 nnn has been taken. For a recessed balance the only forces will

be within the gap and the same value of the effective penetration has been taken. The compari-

son given in Fig. 7 shows that though this estimate does not give precise agreement with

O'Donnell's measurements their general character is well reproduced, suggesting that the physi-

cal model assumed is correct.

30

I 2O

Error IS

I0

$

, 1 i

-o.o° -o.o,

/ / I /

I

Estlmste /III/ (Ah: O'OZ ~)/) ~//p

M o c ~ s u r e m e n . t

0.02 0"04 ~ 0 06 0"08 Protrusion •a ~ I

Fig. 7. Error in skin-friction measurement due to misalignment of floating element-parallel-linkage balance. O'Donnell: M ffi 2.67 Re 8 = 10070.

The balance investigated by Allen has been described by Fowke (1969) and is of the pivoted

type. Some of the measurements made by Allen at M - 2.19 for his floating element, both re-

cessed and protruding, have been plotted in Fig. 8. These measurements confirm the flow model

sketched in Fig. 6 in showing that for his geometry the effect of the moment due to the pres-

sures on the face of the recessed element is to cancel the moment of the thrust on the edges

and produce an apparent, and very large, increase in the indicated friction force so that the

indicated increment in force is positive for a recessed balance as well as for one which pro-

trudes. Allen also varied the size of the gap, g, round his floating element, and as can be

seen in Fig. 8, increasing the size of the gap reduces the effects of the balance misalignment.

An attempt has again been made to estimate the effects. The pressures on the vertical faces

are estimatedfrom the drag of forward and rearward-facing steps as was done for O'Donnell's

results. For the protruding balance the pressure forces on the flat face of the element have

been i g n o r e d . For t he r e c e s s e d ba l ance the Simple e x p e d i e n t has been adopted of d e t e r m i n i n g

the i n v i s c i d f low d e f l e c t i o n s a p p r o p r i a t e to the p r e s s u r e c o e f f i c i e n t s f o r two-d imens iona l

s t e p s , which can be done e a s i l y because the f low i s supersome, and hence f i n d i n g the a r e a of

the f l a t f a c e on which t h e s e p r e s s u r e s a c t . As in t h e ' c a s e of O ' D o n n e l l ' s r e s u l t s , t a k i n g

14 K.G. Winter

the forces only on the protruding parts of the faces underestimates the loads. The effective

depth of the edge on which the pressures act has been obtained by matching approximately the

measured loads for protruding balances. It was found that for g/r = 0.002 the increment in

depth was approximately Ah/r = 0.04 and for g/r = 0.02, Ah/r = 0.O1. The dotted lines in

Fig. 8 show estimates made with these empirical values of Ah/r. For g/r = 0.002 the estimate

matches the measurements remarkably well over the whole range of h/r, but for g/r = 0.02 the

effects for the recessed balance are overestimated badly. The implication of this result is

that the increase in gap size also reduces the pressure forces on the balance face. This re-

duction is perhaps not unexpected since as the size of the gap is increased so will the air

circulate more freely round the gap and ameliorate the pressure variations.

\

\ 4.0 \ \ \

\ \ \ \ 3-0

",, . \ ? \ \ , , +.o

-.

i

-0-020 -O-OIS

Fig. 8.

! r

O-OOZ Estlmote /

(.+=o.o+) " \ - / / " 0-012

/ /

J / / /

/ o o2o / / ' . d -

, / /' 47:i / / . / I ' ] '+ Estimot¢ ,,), .G-y (++.+0:o,)

i I i I I

-~010 - 0 " 0 ~ ~ &O~ 0-010 0-015 r

Error in skin-friction measurement due to misalignment of floating element--pivoted balance--Allen: M = 2.19.

0-020

The results of Allen illustrate the advantages of having a large gap round the element from

the point of view of reducing the effect of misalignment. On the other hand a large gap may

be expected to disturb the flow over the surface. Dhawan (1953) made a brief investigation

of the effect of a gap at low speed and could measure no change in the velocity profiles on

a flat plate downstream of a groove in the plate. His gap had a Reynolds number u g/v ~ 60.

However, Gaudet and Winter (1973), in an investigation of the drag of excrescences, found a

measurable drag increment for grooves normal to the flow when uTg/v exceeded IO. If a high

standard of accuracy is aimed for it is suggested that u g/v should not exceed iOO, when ac-

cording to Gaudet and Winter the shearing stress across a gap normal to the flow will be about

three times the undisturbed value of skin friction.

Ozarapoglu (1973) studied the effect of changing the geometry of the gap when the pressure in

the balance casing was varied. In zero pressure gradient he found only a smell effect on the

balance reading for casing pressures exceeding ambient pressure and suggested that small posi-

tive pressures should be used to avoid the large errors which can arise if a stagnation line

occurs on the upstream edge of the floating element because of a flow into the balance.

Moulic (1963) in a study of the strong interaction region near the leading edge of a flat

plate in a low-density flow at a Mach number of 6 found a significant effect of casing pres-

sure on his balance reading. He used a very small element of 0.25 × 25 rmn, and the effect he

measured was probably due to the asymmetrical cross-sectional shape of the element.


Brown and Joubert (1961) studied the secondary forces arising in adverse pressure gradients

with a view to applying their balance in three-dimensional flows. They argued on dimensional

grounds that the force F indicated by a balance should be given by

w = ~w , ~ , (3-2)

where T is the undisturbed shearing stress; w

S the floating element area;

D diameter;

kinematic pressure gradient i dp. p dx'

× variation of gap with displacement.

This expression (3-2) is for a given instrument with a given gap. A variable guT/~ or g/D

should be included in general, and the results will also depend upon the thickness of the

floating element. They compared the shearing stress indicated'by their balance with that de-

duced from Preston tubes. Though their instrument was not of the self-balancing type they

were able by tilting it to take all their readings with the floating element centred, so eli-

minating the variable X. Miller (1973) later extended the work of Brown and Joubert to favour

able pressure gradients. He ensured compatibility by choosing a value of g/D close to that of

Brown and Joubert. The combined results for the influence of secondary forces are shown in

Fig. 9. This figure shows that for the particular balance configurations used the secondary

forces do not exceed 15% of the friction force. The pressure gradients had values of u~/u 3 T

up to 0.03. A feature of the results is that even at zero pressure gradient the balance read-

ing depended upon Du /u. It might be surmised that this variation is really dependent upon

gu. r/~. 1'20 F

IResuKs of Miller * I = Results of 8torn I Joubert

1.15 I$00

I. iO ~ 00 ~ ~

,.o 5 f ,ooo

, , \ ,,1 , , , , , -zo -is - IO\ ~ - / ] v - o 5 IO. I'~ +CD 20 ~s 30

o.J Fig. 9. Effects of pressure gradient on reading of skln-frlctlon balance

in low-speed flow.

Everett (1958) calibrated a skin-friction balance with four different floating elements in

channel flow with three different values of favourable pressure gradient obtained by changing

the height of the channel. He found that the commonly accepted expression for the force aris-

ing from a pressure gradient

. td Fp - S ~ ~ , (3-3)

which assumes that the external pressure gradient decays linearly over the thickness.of the

]6 K.G. Winter

edge of an element, was satisfactory for a large edge thickness, t/D = 0.02, but progressively

underestimated the force for smiler values of t/D.

Ozarapoglu (1973) (and Vinh, 1973) studied the errors in the reading of their large balance at

two positions in a strong adverse pressure gradient where values of ~/u 3 were 0.02 and 0.07.

Because of the large size of the balance (127 ~mn) their values of ~D/u$, 90 and 170, were con-

siderably in excess of those reached by Brown and Joubert. As a consequence, the pressure

changes across the surface of the balance in a streamwise direction were very large, and they

were able to show, by traverses with a Preston tube over the surface of the balance, a large

reduction in skin friction over the forward part of the balance when the casing pressure was

set equal to the mean static pressure over the floating element. As the casing pressure was

reduced, reducing the flow out of the forward part of the gap round the element, the defect in

skin friction was also reduced. They showed that the errors in skin friction increased if the

gap was increased.

The choice of the size of the gap round an element will clearly he a compromise between con-

flicting requirements. If flows in zero pressure gradient only are to be studied a wide gap

will reduce the sensitivity of the balance reading to misalignment of the floating element

because of the reduction of the pressure changes, but the absolute level of the readings will

be uncertain because of the change in shear stress across the gap, and the resultant uncer-

tainty about the effective surface area of the element. For flows in pressure gradients the

minimum possible gap and edge thickness are desirable to reduce the flow through the gaps and

its effects on the pressure at the edges.

However, even if the greatest care is taken in the design and use of a skin-friction balance

it is difficult to establish confidence in the accuracy of the results from it. Mabey and

Gaudet (1975) compared the results from five different Kistler balances, and also compared the

results from one of these with the large balance used in the tests of Winter and Gaudet. All

the tests were made in supersonic flows with nominally zero pressure gradient. The Kistler

balances were used generally at ranges up to 25% of full-scale deflection. The results from

two of the balances were discarded because of lack of repeatability and two agreed to within

1% accuracy on skin friction. A comparison between the results from one of these two and the

balance of Winter and Gaudet is shown in Fig. i0. The Kistler balance reading showed consider-

2.2

Fig. iO.

2"1 ~ W

2"0

1"9 inter t Gnudet

,o'o, I I I"7

1.6

I'$

1.4

'~ I I I , . i . i 2 4 6 8 2 10 4 I0 5

Re 0

Comparison between results from Kistler balance and large balance of Winter and Gaudet.


able fluctuation and the mean reading varied by as much as 4% from that of the large balance.

Mabey and Gaudet tentatively ascribe the variation with Reynolds number of the difference in

reading to changes in the rate of variation of the tunnel temperature with time as the tunnel

total pressure was changed. The fifth balance was of a different design, with a vent hole

provided downstream of the floating element and coulnunicating with the balance chamber, to re-

duce the risk of damage from the pressure transients acting across the floating element which

occur in the starting and stopping processes of a supersonic tunnel. Fig. II compares the re-

suits of this balance with one of the two balances found previously to perform acceptably for

a range o f conditions in the RAE 3 ft x 4ft tunnel, and for one condition in the NOL Boundary-

Layer Channel. The results show that the difference in reading between the two balances is

reduced by sealing the vent hole, and the method of plotting (against a Reynolds number based

on wall conditions) implies that the difference in reading may be correlated with the differ-

ences in surface geometry between the two elements.

O.Z Acf.

ef

0"I

-0 .1

• / . ~ • Q ~ . s I

9 o

A . z ^ * * +~++~++"--+, ,o e e y~X ~ e 0 uv Im

1 o

VENT HOLES 14 OPEN CLOSED

:l 4.0 • RA[ 4.5 4.7 • NOL

20

I Fig. 11. Comparison of two Kistler balances showing effect of vent hole.

4. VELOCITY PROFILE

It has been accepted for many years that the velocity profiles of turbulent boundary layers,

at least in moderate pressure gradients, have inner layers for which the velocity scale is the

friction velocity u . Clauser (1954) was the first to point out that the resultant similarity T

of the viscous sub-layer, adjacent to the wall, a blending region and a logarithmic region.

be derived from measurements of the velocity profile. The inner part of the profile consists

f the viscous sub-layer, adjacent to the wall, a blending region and a logarithmic region.

With the assumption of constant shearing stress the velocity distribution in the viscous sub-

layer is given by

_u_u = YuT-. (4-I) u T

and the value of the friction velocity, and hence the skln-friction coefficient can, in prin-

cipl~be readily determined. However, in most aeronautical applications the laminar sublayer

is too thin for its velocity profile to be determined accurately. Instead the logarithmic or

"law-of-the-wall" region is used. In its original form the Clauser chart for determining skin

friction for incompressible flow was constructed in the following way. The velocity profile

is

u = A l o g - - yuT ÷ B u T

(4-2)

whence

u

uU uU = A l o g YUv + A l o g -~- + B ( 4 - 3 ) T

18 K.G. Winter

so that a series of lines expressing u/U as a function of yU/~ may be drawn with u U = (cf/2) 1

as a parameter. The chart in the form suggested by Clauser is given as Fig. 12. In use the

chart is superposed on a measured velocity profile and the value of the skin-friction coeffi-

cient obtained by matching. The value of the skin friction is of course dependent upon the

values adopted for A and B. Clauser chose the values A = 5.6 and B = 4.9. The values adopted

by the organizers of the Stanford Conference on Turbulent Boundary Layers (Khine et a~., 1968),

as giving the best fit for the data available, were A = 5.62 and B = 5.0, and these values

have been used in constructing Fig. 12. Other values have, however, been obtained ranging

from those of Ludwieg and Tillmann (1949) A = 5.0, B = 6.5, to those of Winter and Gaudet

(1970), A + 6.05, B = 4.05.

I-0

0.9

0.8

0.7

0'6 tJ

O 0.5

0.4

0..3

0.2

0.1

0

I0

Q

2 3 4 5 6 7 8 I0 z 2 I 4 $ 6 7 B 103 ~' yU3 ,4 5 67~ 104 2 .~ 4 5678105

Fig. 12. Clauser chart.

The velocity profile obtained from the usual method of using pitot tubes will contain errors

arising from the apparent displacement of a pitot tube in a shear flow, from the proximity of

the probe to the wall, from the effects of turbulence and from the effects of viscosity. De-

spite the many investigations into these effects, of which a good discussion is given by Chue

(1975), they are still not completely understood and their consideration is beyond the scope

of this paper. Suffice it to say that generally the experimental, points, if uncorrected, will

fall above the logarithmic line near the wall, but the errors will be sufficiently small

further away from the wall for a logarithmic region to be apparent.

Alternative and more convenient forms of the Clauser chart have been suggested; amongst them

are those of Bradshaw (1959) and Ozarapoglu (1973), Bradshaw suggests that one suitable refer-

ence point on the inner-law curve be taken and gives as an example the point yuT/~ = i00 for

which u/u ~ 16 (For the range of values of A and B quoted above u/uTat yu /~ = i00 lies be-

tween 16.1 and 16.5). By taking a range of values of U/u (that is effectively el) a curve

of u/U against y can be drawn for the particular Reynolds number of an experiment. The value

of u/U at the intersection of this curve with the measured velocity profile leads to cf since

c = 2 2 ÷ V

ref


The example shown in Fig. 13 is in fact plotted in axes u/U vs. yU/~. The experimental points

are for a high Reynolds number and so a reference value of yu /v ffi iOOO has been taken for

which (with A ffi 5.62, B = 5.0) (u/UT)re f = 21.86. A curve of cf is also included. As can be

seen the method has the advantage that the reference line crosses the curve of the velocity

profile nearly at right angles.

0 . 7 0

0.65

u

U 0.60

0.55

3.0

", o ~ I I . ~ . - "-'~

/ ' i - - t . . . . . . . . . . . . . . . . . . . .

?

I J I I ~ - J i I 2 5 6

2.5

Io%, 2.0

1.5

I.O

o.~o~ .~ 0 3 4 7

y u x I 0 - 4 u

Fig. 13. Skin-friction from velocity profile: Bradshaw's method.

Ozarapoglu obtains a direct relationship between u/u and yu/v by expressing the log-law (4-2)

as

u + A log ~ = A log y__~u + B. (4-4) u u ~) T T

Hence for each point on a velocity profile a value of u/u T can be obtained from the value of

yu/v using either a plot such as Fig. 14 or an interpolation process. A similar suggestion

has been made by Rajaratnam and Froelich (1967).

19-- /

U 16~ Wi7 -

,6-/ @ 1 4 - -

12 -

1 3 -

" / I I I I I J J [ I i 0 2 2 4 6 10 3 2 4 6 i0 4 4

yu

v

I i0 ~

Fig, 14. Skin-friction from velocity profile: Ozarapoglu's method.

A more c o m p l i c a t e d p r o c e d u r e has a l s o been proposed by D i c k i n s o n (1965) i n v h i c h a v e l o c i t y

p r o f i l e i s l o c a l l y f i t t e d to a pove~- law i n t h e form 1

n÷l u = F(n) yu E'- ~ (4-5) T

where n i s o b t a i n e d from n = d log u "

20 K.G. Winter

For flows in pressure gradients the extent of the logarithmic region is reduced, but unless

the flow is either in a strong adverse pressure gradient and close to separation, or in a

strong favourable gradient and close to relaminarization, a logarithmic region can usually be

identified. Since the velocity profile will be available, if it is to be used to determine

skin friction, it should be self-evident whether the determination for any particular flow is

satisfactory. The definition of limits is more important when the Preston-tube method is used

and will be discussed further under that heading. However, attention should be drawn to the

use by Coles (see Kline et al., 1968) of a complete outer velocity profile in the form

u = A log yu ~ Y ~-- v + B +--~ w--~ (4-6) %

where w = 2 sin 2 ~ ~ .

In the survey lecture for the Stanford Conference he pointed out that equation (4-6) could be

used to obtain the skin-friction coefficient by finding values of u and 6 such that the RMS

deviation of the data from (4-6) is minimized. The strength of the wake, 7, the third para-

meter in (4-6), is eliminated by evaluating the equation at y = 6. The whole of the profile

cannot be used since the viscous sub-layer is not represented in (4-6) and also a discontinuity

in slope at the edge of the boundary layer is indicated. Galbraith and Head (1975) show that

the complete velocity profile may be used if this is represented by Thompson's (1965) profile

family, This family includes a representation of the sub-layer and blending region, and by

using the concept of intermittency for the wake region avoids the discontinuity at the edge

of the boundary layer.

The discussion so far has been confined to incompressible flows but the method can also be

used for compressible turbulent boundary layers, though with somewhat less confidence. Allen

(1968), in connection with his investigations into the use of Preston tubes in compressible

flow, also examined some of the forms proposed for the law of the wall describing the logarith-

mic region. He showed that consistent results could be obtained with the Baronti-Libby (1966)

transformation of the velocity profiles, which is complicated in that the determination of the

transformed coordinate normal to the wall involves the running integration - ~Y ~ dy, but o ;

that equally good results could be obtained with the Fenter-Stalmach (1957) law ~ the wall

for adiabatic flow. With the constants for the law of the wall used previously for incompres-

sible flow the Fenter-Stalmach form is given by

U I sin -I ~ = 5.62 log ~ + 5.0 (4-7)

u 7 ~w T w

where u T w

y- IM2 2 e

and o = I + " ~ - ~ 2 1 M 2

e

This method has been used by Mabey et el, (1974) for Mach numbers between 2.5 and 4.5 in com-

paring the values of skin friction obtained from velocity profiles with measurements by skin-

friction balance. Figure 15 shows a sample of their results in which cf was deduced from each

point of the velocity profiles. The accuracy of the method can in part be inferred from the

variation of the derived cf over the region of the profiles where the log law may be e~pected


to hold. The variation of the skin friction in Fig. 15 may indicate a shortcoming in the

compressibility transformation used.

2.4

2.3

2.2

2.1

2 0

~3) 1.9

u ~

0

0 0 0 0 0

0 0 o 0

0 oo o

o o Reo =5.97x103 o

o

"°c"P'm°=~"~- n ° ~ ° ° o o Cf b a l a n c e oOCP°°~°o

° o ° ° o 1.8 -

1.7 o o °°° Reo=17"Sxl03

1.6 -- 1.5 _ ~ ~ v o v o ~ oo~ , ° ° -9 - -c f b a l a n c e

~ I I I I I I I I I I I I 0.3 0.4 0 .50.6 0 .8 I 2 3 4 5 6 8 I0

y m m (a) M = 2 . 5

20

1.7

1.6 o o ° o o o

o ° o 1.5 o %o

^ o ° R e 0 = 4 . 8 0 x 1 0 3

ba lance Cf 1.3 - o d"

1.2

I.I

1.0

0.9

0.8

0.3 0.40.50.6 0.8 I

0oO°°°°% J

_ ~o o ° ° l ~ e e = 2 8 . 6 x 1 0 o O O o o o o o o O O o o o ~ " c f b a l a n c e

I I I i I I I I I I I I I < 2 3 4 S 6 8 I0 20 60

ymm ( b ) M - 4 . 5

Fig. 15. Skin-friction from velocity profiles in supersonic flow: Mabey et al.

Other forms of the law of the wall have been used. For example Spence (1959) suggested the

form

(nuTo u A log + B

u Vm m

(4-8)

with A = 5.76, B = 5.5 and C = 2.3, and where the density p m used in the definition ~ = P u2 w mT m

and the kinematic viscosity v m are evaluated at "intermediate enthalpy" conditions, for which

Spence found Eckert's formula

h = O.5(h + h e ) + 0.22(h - h ) (4-9) m w r e

to give a satisfactory evaluation. The extra constant C enables a continuous profile to be

continued into the laminar sub-layer.

A particular simple form was shown by Winter and Gaudet (1970) to give an accurate representa-

tion of their velocity profiles in adiabatic flow at Mach numbers between 0.2 and 2.8. Their

expression is

i u yu --~ = 6.05 log +

1 v u e T

4.05 (4-10)

22 K.C. Winter

= + 0.2M \PeJ

For flows with heat transfer, relatively few velocity and temperature profiles have been

measured. Hopkins et a~. (1972), in one of a series of papers on skin friction at hypersonic

speeds, compared their measured velocity profiles on non-adiabatic flat plates with various

forms of the law of the wall. They concluded that the transformation due to van Driest (1951)

was the most satisfactory. In this transformation the law of the wall is represented as

where a 2

u 4a 2) 11

I U in -I 2a 2 ~- b h

a u w (b 2 + 4a2) ½ + sin-I (b ~ +

Y- 1 M 2 T 2 e b = a ~ ~ --e T ' +T - I" w w T e

yu w

= A l o g - - + B ( 4 - 1 1 ) v

w

: I + ~(~ - I)M~ , equa- For isoenergetic flow, with a recovery factor of unity so that T /T w e

tion (4-11) reduces to the expression (4-7) given by Fenter and Stalmach (1957) for adiabatic

flow.

It should be noted that all the expressions for the law of the wall given above are for smooth,

flat surfaces. For surfaces where the boundary-layer thickness is not small compared with the

radius of curvature of the surface the form of the law will change, and for rough surfaces the

constant B'will have a reduced value.

Surface injection or suction will also change the form of the law. It has been shown that for

flows with transpiration the so-called bilog law may be used

I< I u + B. (4-12) 2uT + - i = A log v w

Stevenson (1963) showed by comparison with experiment that A and B are independent of v w and

u . Jeromin (1968) showed that it is possible to find transformation parameters such that T

velocity profiles in compressible flows with injection can be transformed into Stevenson's

law of the wall, but the derivation of the parameters is too complex to be used for the deter-

mination of skin friction. Squire (1969) applied mixing-length theory to the problem and ob-

tained

u I p~du' YUTw

0 (PwVw u' ÷ ~w )~ = A log Vw + B

which, with the assumption of the quadratic temperature profile

T -T T-T (u) 2 __T = 1 + r________wu+ e r T w T U T

w w


becomes

U

~ d ~ = A log w

- - - - W

- + I + r + e r

T w U T w

+ B . ( 4 - 1 3 )

Squire compared (4-13) with experimental results and found that, although a logarithmic region

occurred with a constant value for A, the additive constant B varied both with M e and Vw/U T . W

Thus, though it may be possible to identify a logarithmic region and so to determine cf,

a simple form of Clauser plot cannot be derived from (4-13).

Because of our current inadequate understanding of the shear stress distribution in three-

dimensional turbulent boundary layers the determination of the skin-friction coefficient from

velocity profiles in three dimensions is somewhat speculative. East and Hoxey (1969) give a

good review of the possible forms of the law of the wall which might be used:

(a) The velocity profile is treated as being simply a twisted form of a two-dimensional pro-

file:

u yu = A log-- + B (4-14)

U V T

where u is the resultant velocity.

(b) The velocity profile is resolved in the direction of the external stream and the friction

v e l o c i t y r e s o l v e d i n t h e same d i r e c t i o n i s o b t a i n e d :

YUz (4-15) U X

= A log - - + B . u T X

(c) A "developed total velocity" is used in the form suggested by Perry and Joubert (1965):

= A log yu + B (4-16) U T

where s = IO 1 + \d--u~ / du'

in which the primes denote running variables, and w is the crossflow velocity.

(d) As suggested by Johnston (1960) an effective velocity in the direction of the wall shear-

ing stress is used. If the direction of the wall shearing stress relative to the external

streamlines is BO, then at the wall, and also in the inner region of the boundary layer as

shown by the linear region of the polar plot of velocity, (the Johnston triangular model)

+ u sec B O. Therefore at the wall

u sec B o /yu~ =f

U T

and it is assumed that this similarity can be extended into the logarithmic region so that

u sec 13 0 yu T u = A log--~ + B. T

(4-17)

24 K.G. Winter

A fifth possibility (e) is also considered by Pierce and Zimmerman (1972), based on a sugges-

tion by Coles (1956), in which the relevant velocity is taken to be that resolved along the

wall shearing stress direction. This assumption leads to

cos (8 - 8) yu = A Iog--+ B.

u v T

(4-18)

Figure 16 shows two sample comparisons from East and Hoxey of Clauser plots derived from

methods (a) to (d) (method (e) is omitted in Fig. 16b), together with the two-dimensional log

lines based on skin friction obtained from Preston tubes and razor-balde surface pitot tubes.

Also shown are the profiles of the cross-flow velocity w/U. The profiles were measured in the

boundary layer on a flat surface from which an obstacle protruded in the form of a circular

cylinder with a fairing. Figs. 16a and b are respectively for the flow at points upstream and

downstream of the inflexion point in the external streamlines. That is to say the crossflow

is respectively increasing and decreasing. In the latter case the outer parts of the velocity

profiles bear little resemblance to forms expected in two dimensions. On the basis of compari-

sons like Fig. 16 for a large number of profiles East and Hoxey conclude that method (d) is

the most satisfactory. This conclusion is in agreement with the outcome of the comparisons of

Pierce and Zimmerman in which method (d) is shown to give the largest apparent logarithmic re-

gion. In three dimensions the flow in the outer regions is strongly controlled by pressure

gradients and as Fig. 16 shows a simple interpretation like the two-dimensional law of the

wake is not possible.

o.9 ~ u u x , , / u ~Iode \ / v,,- I

o.8 v usecBd u v Razor I vusecBo/u /Preston,, + , , , u " + x b,o~,~ 0.8 /:~bx, x ,~ ',,

0.7 v 5

v t 0.7 So~O" ~ o°~°°° / 0.5

D 0.5

0.1 °° o o 0.4 0,3 0.3 ~L% d " ,5

0.2 dA~A,%AAd A ,% ~ 0.2 A,%~ Ad '%

I I I i I I I I I I 02.5 3.0 3.5 4.0 4.5 .5.0 .5.5 6.0 02.5 3.0 5.5 4.0 4.5 5.0 5,5 6.0

t O q i o , ;y U/v t O q l o , Y U/v

Fig. 16. Velocity profiles in three-dimensional flow: East and Hoxey.

5. THE PRESTON TUBE

A variant of utilizing a fit of measured velocity profiles to the law of the wall is to measure

one velocity only at a known distance from the wall. Preston (1954) realised that this could

be achieved with a circular pitot tube lying on the surface, and demonstrated by measurements

for a range of sizes of tube in fully established pipe flow that the unique relationship


F ~ , pv 2 \ Ov 2 ,J

which was to be expected from considerations of wall similarity, was in fact obtained.

(5-1)

In (5-1) Ap is the difference between the pitot pressure and wall static pressure;

d is the outside diameter of the pitot tube;

v is viscosity;

T is wall shearing stress. W

Despite the close relationship between (5-1) and the law of the wall it is not to be expected

that one can be derived simply from the other. The pitot tube reading will be affected by

the various sources of error mentioned previously and the static pressure, to which the pitot

pressure is referred, will also be subject to errors as first quantified by Shaw (1960). The

author is of the opinion that the calibration should be regarded as purely empirical within

the framework indicated by the similarity scaling.

Following Preston's work, others, notably (Hsu (1955), the Staff of Aerodynamics Division,

NPL (1958) and Walker (1959) checked his calibrations in various forms and doubt was cast on

its universality. By comparing the readings of a Preston tube and a sub-layer fence in both

the developing flow in a pipe and the fully developed flow, Head and Rechenberg (1962) were

able to show that similarity in the two situations existed. (Because of the small size of a

sub-layer fence (see Section 6.1) its reading should give a reliable measure of the skin fric-

tion in both situations) Using essentially the same appartus, Patel (1965) produced what is

currently regarded as a definitive calibration, covering a wide range of flow conditions and

sizes of Preston tube. His calibration is given in terms of

d 2 x* = log Apd2 and y* = log w

40v 2 40~ 2

a S ~

y* < 1.5 y* = Ix* + 0.037

1.5 < y* < 3.5 y* -- 0.8287 - 0.1381x* + 0.1437x .2 - 0"0060x'3 I (5-2)

3.5 < y* < 5.3 x* = y* + 2 log (1.95y* + 4.10)

These three ranges correspond roughly to the three regions in the velocity profile, the vis-

cous sub-layer yUT/V < 6, the transition region 6 < yu /~ < 60 and the logarithmic region

60 < yu /v < 500.

As was pointed out subsequently by Head and Vasanta Ram (1971) the expressions (5-2) do not

quite match at the changeover and also the expression for the outermost region is inconvenient

to use because of its implicit form. Furthermore Twd2/4pv2 varies by more than four orders

of magnitude over the full range of Patel's calibration. They therefore suggested the use of

two alternative forms of the calibration, the first

*The 4 in the denominator of the logarithms was originally adopted by Preston, since if the height of the centre-line of the pitot tube is taken as the relevant variable in the wall similarity parameter (y = d/2),

~ d2w ~ I ~ 2

4pv 2

26 K.G. Winter

Ap Apd 2 - - v s . - -

T w Pv 2

is tabulated in Table I, and the second is in effect a Clauser plot for a Preston tube.

the calibration is expressed as

If

j

then

w ~p ~p 1 Ud cf = = ÷ F

~ u 2 ½0u 2 ~ou 2 2 ~ •

u This expression is shown as a chart in Fig. 17 where U -~

variables with cf as a parameter.

I d__~__) ~ dU = and- are used as the ½PU2 "o

o8[

0.~ ~p

V

0.. ~

0.~

0.~

i ~ K i l l I l ~ n R i 3 ~ i s n ~ i K i l

0.2

0.,

2.0 2.5 3.0 3.5 4.0 4.5 5.0

Fig. 17. Preston tube calibration chart: Head and Vasanta Ram

Bradshaw and Unsworth (1973) give a further alternative, but implicit, expression,

du I duT~ AP=Tw 96 + 60 log s--~+ 23.7 log5-~-j;

du --< IOOO. valid over the range 50 < u

(5-3)

Patel also investigated the limiting pressure gradients, both favourable and adverse for which

his calibration might be expected to apply. His proposed limits are based on values of the

parameter


Table . ww as a function of Apd2/pu2 (Head and Vasanta Ram).

M AP AP@ AP -- -- pv’ TV

x IO-’ p9 Is9

x IO”

4.0 918 4.2 9.41 44 9.63 4.6 985 48 10.06 5.0 IO.27 5*2 IO.47 5.4 10.67 5.6 IQ87 5.8 llt% 6.0 II.25 6.2 II.44 6.4 11.62 6.6 11.80 6.8 Il.98 7.0 It.16 7.2 12.33 7.4 1250 76 12-67 7.8 12.83 8.0 12.99 8.2 13.15 8.4 13.31 8.6 13.47 8.8 13.63 9.0 13.78 92 13.93 94 14.08 96 14.23 98 14.38

A&’ AP T_ pfi r.,

x IO-’

I.0 14.53 I.02 14.67 1.06 14.95 1.10 15.23 l-14 15.51 I.18 15.78 1.22 16.04 I.26 16.30

1.30 16;56 I.34 16.81 1.38 17.06 I.42 17.31 1.46 17.55 I.50 1779 1.54 18.02 1.58 IS.25 1.62 IS.48 1.66 18.71 1.70 18.94 1.74 1916 I.78 19.38 1.82 19.59 1.86 19.80 1.90 20.01 I.95 20.27 2.00 20.53 2.05 20.79 2.10 21.05 2.15 21.30 2.20 21.54 2.25 21.78 2.30 22.02 2.40 22.49 2.50 22.96 2.6 23.41 2.7 23.86 2.8 24.30 2.9 24.73 3.0 25.08 3.1 25.43 3.2 25-78 3.3 26.13 I.4 26.48 1.5 26.82 1.6 2716 1.7 27.50 I.8 27.83 I.9 28.15 1.0 28.46 1.2 2907 1.4 29.66

ApdJ AP -- pd Tu

I( 10”

4.6 30.23 4.8 30.79 5.0 31.33 5.2 31.84 5.4 32.34 5.6 32.84 5.8 33.31 6.0 33.78. 6.2 34.23 6.4 34.68 6.6 35-l 1 6.8 35.52 7.0 35.94 7.2 36.34 7.4 36.72 7.6 37.11 7.8 37.50 8.0 37.87 8.5 33.74 9.0 39.58 9.5 40.40

I.00 41.18 I.05 41.93 I.10 42.65 I.15 43.34 I.20 44.00 1.25 44.54 I *30 4527 1.35 45.87 I.40 46.45 i-45 47.01 !*50 47.56 -55 48.09 -60 48.61 -65 49.12 -70 49.62 -8 50.56 *9 51.46

-

I APT’ AP AP# AP .- - m- p4 Tw pg 710

< 10-4 x10+

2.0 52.32 2.1 53.15 2.2 53.93 23 54.68 2.4 55.40 2.5 56.62 2.6 56.80 2.7 5745 i-8 58.07 2.9 58.68 3.0 59.28 3.2 60.40 3.4 61.46 3.6 62.47 3.8 63.43 4.0 64.34 4.2 65.20 4.4 66.01 4.6 66.80 4.8 67.57 5.0 68.32 5.2 6905 5.5 70.00 5.0 71.55 5.5 73.00 PO 74.35 7.5 75.60 6.0 76.80 5.5 77.95 a.0 78.95 P5 79.90

I -4 86.8 1.5 88.0 I.6 89.1 1.7 90.2 I.8 91.2 1.9 921 20 93.0 2.2 947 2.4 96.2 2.6 97.6 2#0 98.8 3.0 99.9 3.2 100.9 3.5 102.4 4-o 104.5 4.5 106.4 5.0, 108.0 5.5 10s 3 6.0 1 IO.4 6.5 III.4 7.0 112.2 7.5 113.0 8.0 113.7 9.0 114.9

IP~ AP

r-;“; -6

,*o 8@80 *05 81.70 *IO 82.55 -15 83.35 -20 84.1 -30 85.5

1.0 116.0 I.1 117.1

1.2 118.1 1.4 119.8 1.6 121.5 l-8 123.1

PO 124.6 1.2 126.0 1.5 127.8 I.0 1307 j-5 133.3 I.0 135.6 4.5 137.7

AP# AP

jg-‘ x

90 1395 60 142.7 7.0 145.4 8.0 147.8 90 1500

AP@ AP -- P9 TV

I< 10-l

zrzir 1.2 155.3 I.4 158.1 1.6 160.6 1.8 1628

2.0 164.9 2.2 166.8 2.5 169.3 3.0 172.8 3.5 175.8 4.0 178.5 4.5 180.9

5.0 183.1 6.0 186.8

7.0 1900 8.0 182.8 9.0 195.3

1.0 197.5 I.2 201.4

I.4 204.7 1.6 207.5 I.8 210.1 2.0 2125 L5 2174 1.0 221.6 3.5 225.0

28 K.G. Winter

de h = p Pu B ~ dx '

and are:

Adverse pressure gradients:

du

Maximum error 3% 0 < A < 0.01 T ~ 200; p v

Maximum error 6% 0 < A P

du

< 0.015 ---~ < 250.

Favourable pressure gradients:

du dA Maximum e r r o r 3% 0 > ~ > - 0 . 0 0 5 ~ < 200 d-~x < O; p

du dA > -0.007 ~ ~ 200 d~x < O. Maximum error 6% O > Ap v

As Patel points out these limitations are a rough guide only.

(1962) law o f t h e w a l l i n p r e s s u r e g r a d i e n t s i s a c c e p t e d ,

u _ f YUT, &

u ~ T T

He notes that if Townsend's

v ~T where & - --

pu~ ~y

and that in general AT, which is equal to &p only at the wall, might therefore be expected to

be the controlling parameter, l~s use a prior~ in a boundary layer investigation is, of

course, not possible. The limit on du /~ is perhaps also rather sweeping since the limiting T

value might be expected to depend on the Reynolds number of the boundary layer, tending t@

increase as a boundary-layer Reynolds number increases. The limitation for favourable gradi-

ents of dAp/dX < 0 was introduced to ensure that a boundary layer, if it is in a condition

where it may be subject to relaminarization, should only he approaching that state. The phy-

sical features of the flow which lead to the limitations have been discussed by Patel and

Head (1968) and are illustrated in Figs. 18 and 19. Fig. 18 presents profiles from Newman's

(1951) experiments (from the Stanford tabulations) and shows how the velocity profiles depart

from the law of the wall for zero pressure gradient as the adverse pressure gradient increase~

This departure is predicted by Townsend's velocity profile for a linear shear-stress gradient

away from the wall. On the other hand the recent reassessment by Galbraith and Head (1975)

of eddy - viscosity profiles implies that the mixing length increases in strong adverse pres-

sure gradients and that this delays the departure of velocity profiles from the conventional

law of the wall.

The profiles as plotted in Fig. 18 differ from those in Fig. 4 of Patel and Head and add

weight to the conclusion of Galbraith and Head. The difference is presumably due to a differ-

ent derivation of the skin-friction coefficient in the Stanford tabulations from that used by

Patel and Head. It is interesting to note, on the basis of Fig. 18, that the logarithmic re-

gion is entirely absent from the curve for the largest pressure gradient but the profile ap-

pears to follow a blending curve of the form suggested by van Driest (1956) without allowance

for the normal shear-stress gradient; hence Patel's calibration, for a sufficiently small

Preston tube might be applicable.


I00 --

90--

80 -

70 -

u 6 0 -

Uz

5 0 - -

4 0 - -

3 0 - -

20

o

o o o

o

o

o

o

o

o :r O

+ o +

o +

A~ o 2 . 0.0075 o + x 0 . 0 2 1 4 + + 0 . 0 9 0 5 o . ~,

o 0.2980 : . : /

o / 0 +

0 4- ,

0 4- ÷

o ..r.. xo. x o.~.-'- .~X + "4~'x

i O ÷

0 2 3 4 ~ 6 s l O 2 3 ~ , 6 102 YUr

v

I IIII lJ I 2 3 4 56 8103 2

Fig. 18. Velocity profiles in adverse pressure gradient: Newman.

Figure 19 shows a series of profiles in a strongly favourable pressure gradient. As the flow

progresses into the favourable gradine (x 0 - x B increasing) the velocity profiles, whilst re-

maining turbulent, first fall below the line of the law of the wall and then lie above the

line as the flow starts to revert to a laminar state. The departure from the law of the wall

= is evident at x 0 - x B = -2 in where Ap -0.01, in accord with Patel's limiting value for 5p

of -0.005.

Brown and Joubert (1967) also investigated the limitations on the use of Preston tubes in ad-

verse pressure gradients, basing the analysis of their experimental data on the dimensional

analysis of Perry et a~. (1966). This analysis indicates that the first departure from the

law of the wall due to a pressure gradient will be in the form of a half-power region, that is

U T

where a = I dp pdx'

and that the half-power region will start at a constant value of ay/u 2 = 1.41. They therefore

analysed their results for a range of Preston tubes in terms of ud/u 2 and obtained the follow- T

ing table of errors:

Preston tube error % I 2 3 ~5 7

ad - - 1.35 1.74 2.06 2.55 2.98 u 2 T

30 K.O. Winter

30

20

I0

22in. en'try 124in.entry A 0

cf U cf - A (f~/s) (ft/s) 8G.18 0 .00492 87.32 0 . 0 0 4 6 7 0.0178

77.93 0 .00537 78 .65 0.00521 0.0212

68.91 0 .00578 68.32 0 .00578 0 .0244

0 U Ur

59.65 0.00690 59.72 0.00597 0.0288

53.98 0 .00527 54.24 0.00516 0.0275

50.02 0 . 0 0 4 4 9 52.97 0.00:387 0 .0204

0 4 8 . 6 5 0 . 0 0 3 8 6 52.41 0.00318 0.0103

0 47.74 0.00351 51.87 0 .003055 0 .0038

0 2 I0 I0 ~ 10 3 10 4

yur v

Fig. 19. Velocity profiles in accelerating flow: Patel and Head.

which are broadly in agreement with their proposed limit of the log law as ~y/u 2 = 1.41. They

also suggest that, since the log law region will vanish when the outer and inner limits become

the same, that is when y = 30v/u = 1.41 u~/u or av/u 3 = A 0.05, Preston tubes should not T T p

be used for stronger pressure gradients than this. Since, however, Patel's calibration extends

to the viscous sub-layer this restriction may not be necessary for very small tubes. Criteria

based on the results of Brown and Joubert are shown in Fig. 20 in terms of du /v vs. A In T

terms of this mapping Patel's criteria for 3% accuracy are in good agreement at their inter-

section but are conservative as general criteria, but his criteria for 6% accuracy overestimate

the permissible limits. Ozarapoglu measured velocity profiles in strong adverse pressure grad-

ients and determined the outer limit of the logarithmic region. His critical values are also

shown on Fig. 20 and are roughly in agreement with Patel's 6% criterion for low values of A P

but indicate that Preston tubes may be used up to larger values of d than accepted previously. P

This is confirmed by recent results of Chu and Young (1975).

Patel's calibration expressions have an upper limit of y* = 5.3 which corresponds to a value

of duT/9 of approximately iOOO. This should be regarded as an upper limit for which the cali-

bration was obtained in Patel's experiments and not an upper limit for the application of the

technique, which will be the outer limit of the log-law region. This limit derived from the

experiments of Winter and Gaudet in zero pressure gradient is shown as a function of the Rey-

nolds number based on momentum thickness in Fig. 21.


-0"12

r-- -O' lO l

~ . . . . ~ Oza r s poglu

I - 0 , 0 8 I

,B-,,

Ap I

I - 0 " 0 6 ~., •

Pig. 20.

0 SO I00 150 l u u uT._.dd

I

Criteria for limits of Preston tube.

~v~

IO 40 i

6

+I

2

i0 3 6 6

4

2

SO 2 #

6

4

1 s

• , ,,p**| , * , i,,,i| * * , ,,,,,|

1(3 Z 4 6 8103 2 4 6 I1104 2 4 6 8 102 Rel) 105

Pig. 21.

' ~ ' ~'~'lO 6

Outer limit of logarithmic region of flat plate.

Several ways have been proposed for transforming the pressure and friction parameters used in

the callbration of a Preston tube to enable the calibration to be applied to compressible flow,

The first of these was probably that due to Fenter and Stalmaeh (1957); this was derived from

their compressible form of the law of the wall already referred to, that is

U ~ sin_ 1 ~ = A log , UT w w

+ B. (5-4)

32 K.G. Winter

Fenter and Stalmach assume that (5-4) may be applied in place of the incompressible form at

y = d/2* where d is the Preston tube diameter, and recast (5-4) as

or

dU i ( i ) r w 2 w o T sin-i o~ u ..... A log + B

~w

Ow ~e I Red sin-I <°~ U I p e ~w o ~

- - Re d log ~ _~e Re d + B ~w ~ ~w

~ee --~w Red(cf) ~ A log --~w Red(of) ~ +7-7B A log 23/2 I . (5-5)

The Preston tube calibration is thus expressed as

~w Redc~ vs. P e ~w o T

u where ~ is obtained from the usual expression for pitot tubes in compressible flow, together

with the assumption of constant total temperature through the boundary layer.

Sinalla (1965) obtained a simpler expression by evaluating the Preston tube parameters at the

"intermediate temperature" using Monoghan's (1955) expression for adiabatic flow

T = I + 0.35r(y - I)M 2 o

m e

From the data of Fenter and Stalmach he obtained the empirical equation

w /p2d2u2\O 873 Omd2"r 0.0290 m (5-6)

IJm m

in which the p r e s s u r e r i s e Ap used in i n c o m p r e s s i b l e f low i s r ep i aced by tOmU2 where u i s the

v e l o c i t y i n d i c a t e d by the P res ton tube.

Hopkins and Keener (1966) also used an intermediate temperature hypothesis but instead of re-

placing Ap by ~OmU2 they used Ap = ~ou 2 = (y/2)PeM2. Their calibration expression is

O d2T m w ~2 m

- 0.0228 OOmd2U2~ 0.883

(5-7)

Allen (1973) investigated the accuracy of these various calibrations for adiabatic flow using

a wide range of sizes of Preston tubes over the Math number range 2 to 4.6, comparing the re-

suits from the Preston tubes with skin friction measured by a balance.** He also made use of

*They also investigated the transformation to the calibration for incompressible flow that would be obtained if the average p~tot pressure over the opening of the Preston tube were taken as the relevant quantity. They conclude in accordance with Hsu's (1955) result for incompressible flow that the difference from simply taking values at d/2 could be neglected.

**Alien has recently discovered that the balance, against which he calibrates his Preston tubes, read erroneously. His revised equation (5-9) is log F 2 = 0.01239 (log Pl) 2 + O.7814 log F I - 0.4723.


the data of Fenter and Stalmach (M = 1.7 to 3.7), Hopkins and Keener (M = 2.4 to 3.4) for

adiabatic flows, and of Keener and Hopkins 41969) (M = 7) for non-adiabatlc flows. In addi-

tion to the calibration laws listed above he also devised a calibration for compressible flow

based on Patel's expression for incompressible flow for his highest Reynolds number range, by

using the intermediate temperature hypothesis with Ap = ½PmU2 to obtain

m - 2 1 . 9 5 l o g ÷ 4 . 1 ( 5 - 8 ) 2 2

The general conclusion of this experiment was that of these various calibrations those of

Fenter and Stalmach, and of Pateli which are based on logarithmic profiles gave the best re-

suits over the widest range of parameters. Since both of these calibrations are implicit,

Allen proposed an interpolation formula which gave a slightly improved rms deviation from the

d a t a . This formula is

log F 2 = 0.01659 (log FI) 2 + 0.7665 log F I - 0.4681 (5-9)

Pm d u Pm }ae u - Re d where F 1 ~m Pe Vm

F2 = 42Pm~ w)~ d = 0(~ / m , e ,

II e

The Sommer and Short (1955) value for the intermediate temperature is used, that is

= i + 0.035M 2 + 0.45 T e

and Sutherland's formula is used for the dependence of Viscosity on temperature.

The last word written so far on Preston tube calibrations in compressible flow appears to be

that of Bradshaw and Unsworth (1973, 1974). They express justifiable doubts about placing re-

liance either on the assumption that the pressure read by a Preston tube may be taken as that

registered by a small pitot tube placed at the position of its centre, or on the concept of

calculating fluid properties at an intermediate temperature. The propose an empirical cali-

bration law based entirely on dimensional grounds as

for compressible adiabatic flow. In 45-10) fi represents a calibration for incompressible

flow and fc a compressibility correction 4~ is the speed of sound at wall conditions). Equa-

tion 45-10) is, of course, an implicit form of the calibration but is used because the func-

tional relationships are clear in this form so that subsequent ~endment on the basis of new

e~eri~ntal data can easily be ~de. As noted earlier Bradshaw and Unsworth devised a fit to

Patel's results over the range 50 < du /~ < I~ as T

2

= 96 + 60 log ~ + 23.7 log ~ (5-3) T w

and expressed the compressibility correction based on Allen's results as

34 K.G. Winter

f C

u T W

where M = Y a

w

= IO~M~ - 2 (5-11)

They tentatively claim an accuracy of about 2% at low speed and small du ~v decreasing to TW/ W

about 10% for du w/~ w up to i000 and M up to 0.I. It is the author's view that the table of

Head and Vasanta Ram provides the most satisfactory calibration for incompressible flow; in

particular it extends to lower Reynolds numbers than the formula of Bradshaw and Unsworth.

For compressible flow the use of the Fenter-Stalmach functions appear to result in somewhat

less scatter than the Bradshaw-Unsworth formula, though it is admitted that only a small sam-

ple of data has been examined.

One aspect of the calibration of Preston tubes which has not been adequately explained is its

sensitivity to the Reynolds number of the static pressure hole to which the Preston-tube pres-

sure is referred.

Equation (5-10) is for adiabatic flow. For flow with heat transfer a further parameter

8q = q/0wCpTwU w where q is the heat flow per unit area, will enter into the equation. There

is a need for further systematic experiments to determine the effect of heat transfer. David-

son (1961) attempted this for M = 5 but found inconsistent results; Holmes and Luxton (1967)

in an experiment at low speed and Yanta et a~. (1969) at M = 4.8 found that their results were

best correlated by use of the intermediate temperature concept.

The measurements at supersonic speeds of Yanta et al. for favourable pressure gradients and of

Naleid (1958) and Hill (1963) for adverse pressure gradients give satisfactory results but

there do not appear to be any general criteria for the use of Preston tubes in pressure

gradients in compressible flow.

The Preston tube has also been used in flows with transpiration. Stevenson (1964) developed

a calibration equation using a power law approximation to the velocity profile (4-12) to find

a mean dynamic pressure over the opening of the tube, as had been done by Hsu for impermeable

surfaces. If the law of the wall is taken locally as

flu) I T + - I = C , v w

then an approximate expression for a Preston tube calibration can be derived as

(Apd2~ ~ = C 2 v (du~l l + 2 n ~ + cIdUTl l+n (5-12)

where C and n will depend upon the Reynolds number of the boundary layer but should have the

same values as for an impermeable wall. Typical values are C = 8.4, n = 1/7. Stevenson found

fair agreement between the skin friction deduced from Preston tube readings by means of (5-12)

and that from the law of the wall and from momentum traverses for the boundary layer growing

on a porous cylinder.


Simpson and Whitten (1968) suggest an alternative form of a Preston tube calibration for a

transpired boundary layer in which in addition to the usual shearing-stress and pressure-dif-

ference parameters they introduce a third parameter Vw(0w/Ap)l to take account of the tran-

spiration, but they do not give an explicit result.

For three-dimensional flows the Preston tube has been used, for example by East and Hoxey,

see Fig. 16. They set the tubes in the flow direction obtained from surveys of the boundary

layer by a yawmeter - in fact in the flow direction at a height ~ times the diameter of the

Preston tube. Their results, evaluated with Patel's calibration, appear to be satisfactory.

In two experiments on the flow past obstacles on a flat plate, one a circular cylinder and

the other an inclined fence, Prahlad (1968) showed that in the powerful favourable pressure

gradients which may develop in a three-dimensional flow the Preston tube should only be used

with caution. He used a vectorial pressure grad p and resolved this in the wall shear-stress

direction to show that at the points where the Preston tube failed, negative values of A as P

high as 0.04 were 0tbained, considerably in excess of Patel's criterion.

Prahlad (1972) also studied the yaw characteristics of Preston tubes, of surface tubes with

openings cut off obliquely at 4~ and of double tubes in the form of a 90 ° Conrad probe and

gives detailed charts of their characteristics in low-speed flow. Rajaratnamand Muralidhar

(1968) present calibration curves for a three-tube yawmeter used as a Preston tube.

Other variants of the Preston tube have been suggested, notably that of Rao et al. (1970) who

used two tubes of different diameter, though the tubes were not mounted in contact with the

wall. They argued that if it is assumed, as has been done so often, that the pitot tubes read

a mean dynamic pressure over their faces of area A], A 2 the difference between the reading of

two tubes in a two-dimensional flow is

Ap = 2 ~i (log y)2dA] - ~2 (log y)2dA2

AI A2

+ 0u~BA + 0u~A 2 log i log y dA 1 I At - ~2 log y dA 2 .

Thus if two tubes are chosen so that

(5-13)

i f i f A~I log y dA I ffi A-~ log y dA 2 ,

A1 A 2

a calibration may be obtained which is dependent of B and also of a static pressure measure-

ment. Rao et aZ. give an example for the flow on a flat plate in which the skin friction ob-

tained from the dual-pitot-tube calibration was within 5% of that obtained from either of the

tubes separately. The technique might be applied to rough wall flows in which B will be un-

known a priori.

An improvement to the dual pitot tube of Rao et al. has been explored by Gupta (1975) in which

two probes of different diameters rest on the wall slde-by-side and themouth of the smaller

one is chamfered at 45 ° away from the larger one, as shown in Fig. 22. In a limited investi-

gation a t low speeds Oupta showed t h a t dev i ce ( a ) , w l t k a d i ame t e r r a t i o o f 0 .78 , gave a p r e s -

su re d i f f e r e n c e of 71% of t h a t f o r t he l a r g e r tube u s e d a s a P r e s t o n t ube , and d ev i ce (b ) ,

36 K.G. Winter

with a diameter ratio of 0.64, gave 80%. Device (c) is suggested for use in three-dimensional

flows in which the pressure difference between the outer tubes is first used to align the de-

vice with the flow and, when the probe is aligned with the flow, the pressure difference be-

tween the centre tube and the outer tubes may be used to determine the skin friction. Bertelrud

(1976) has explored the use of a combined pitot-statie tube as a Preston tube.

0.7 mm

A PPreston ,, "

O.9mm Ca)

0 . 4 5 r a m 45"

-- ~ 0.80 - /, A PPrest°n I' "~

0.7mm

0.7mm

r

0.9mm

(b)

Fig. 22. Modified Preston tubes: Gupta.

(c)

6. OBSTACLES IN TWO-DIMENSIONAL FLOW

Because of the wall-similarity of the flow in a turbulent boundary layer the pressures around

almost any obstacle can be used to derive skin friction. Some of'the devices which have been

used are illustrated in Fig. 23 which shows their sensitivity Ap/~ where £ is a representative

height or diameter. The sensitivity of a Preston tube is also shown for comparison, and it

can be seen that the sensitivities are all of the same order as that of the Preston tube and

cover a range of about 2:1, at a given Reynolds number. The fence or the square ridge have

higher sensitivities than a Preston tube since the pressure difference they create is enhanced

by utilizing the suction at the downstream face as well as the pressure rise at the upstream

face. The devices are considered below in more detail.

6.1. Fence

The sub-layer fence was probably first suggested by Konstantinov (1955) and has been used by

Head and Rechenhert (1962), and by Vagt and Fernholz (1973). As well as giving a relatively

large pressure difference it has the advantage that it may be made sufficiently small to re-

main within the viscous sub-layer and hence to be used with confidence in flows with strong

pressure gradients. Because of its fore-and-aft symmetry it can also be used quantitatively


in flows which are separating. Its small size has the disadvantage that its geometry is dif-

ficult to define, and the recommended technique is to calibrate each particular fence against

a Preston tube in a well-behaved flow, thus using a Preston-tube calibration as the primary

standard giving the surface shearing stress. However, the calibration curve for the fence for

hu /~ up to ii shown in Fig. 23, and taken from Rechenberg (1962), can be joined by a specula- T

tlve fairing between hu /9 = ii and I00 to the line representing the results of Good and T

Joubert (1968) who investigated the characteristics of large fences. The calibration for the

square ridge in Fig. 23 is taken from the excrescence drag measurements of Gaudet and Winter

(1973). The fact that this is quite close to the results of Good and Joubert suggests that

the geometry of a large fence is not too critical, but the sensitivity of the drag of ridges

shown by Gaudet and Winter to the rounding of the upper corners indicates that the radius of

the corners should be made as small as possible. The sub-layer fence is also attractive as

a device for compressible flows since its small height will minimize the effect of variations

of density away from the wall.

4 x I0 z

2

8

6

4

¥ 2

tO 8 6

4

Z

I

Fence , i ~ '~

/~-Submerged . - - - step

i i | i i i | i

2 4 6 810 2 u~t 4 6 8102

Y __

$ M 'e

5...i"" " ~ l x k with ~ / cut-cut

Static -hole / pair / /

| d J i |

2 ,4 6 8 tO 3

Fig. 23. Sensitivities of various obstacles as skin-friction meters.

6.2 Razor blade

In order to explore the flow in a viscous sub-layer of very small thickness Stanton et al.

(1920) used pitot tubes with a rectangular opening of width much greater than height with the

test wall forming the inner surface of the tube. They determined the effective height of the

tubes which would make their readings consistent with a linear velocity profile at the wall

and thus produced a skin-friction meter which became known as a Stanton tube. Other workers

subsequently used larger tubes and showed that it was not necessary that the tube shouldbe

sufficiently small to be within the sub-layer for it to act as a skin-friction meter. A re-

view of the investigations up to 1954 covering both experimental and theoretical work was

given by Trilling and H~kkinen (1955). Hool (1955) suggested that a Stanton-type tube could

readily be formed by attaching a portion of a razor blade over a static-pressure hole and this

simple device has since become widely used.

A detailed saudy of the use of razor-blades at low speeds was made by East (1966) with a view

to their use in three-dimensional flows. He calibrated against a Preston tube with skin fric-

tion determined from Patel's calibration. Th~ ,,~e nf a pnTe~nn of a razor blade has obvious

38 K.G. Winter

attractions since the skin friction can readily be measured in any experiment in which static

pressure holes are provided to measure the pressure distribution.* East simplified the tech-

nique by forming his pressure holes in magnets so that the portion of the razor blade could

simply be placed over the hole, obviating the need for adhesive, and enabling the height of

the opening formed by the blade to be determined from previous measurements of its thickness.

The calibration obtained by East is shown in Fig. 24 and has the equation

y* = -0.23 + O.61x* + O.O165~ .2 (6-1)

ph2T where x* = log ph2Ap and y* = log w

u2 u2

4.0

3 . 0

f "" b l a d e

/ O.O0~in

* 2 .0 - f z ~ °

*-"0.23*0.~18~ % +0.0165x*

~, 1 I I 2.0 3.0 4)~phZ ~ 5.0 6.0

x•= L°q'° k - ~ - )

-Fig. 24. Calibration of Standard razor-blade surface-pitot tubes: East.

Equation (6-1) is for the "standard" position of the razor blades that is with the edge of the

razor blade over the leading edge of the static hole. The change in the pressure rise for

various fore-and-aft positions is shown in Fig. 25. An additional error can arise from the

dependence of the static pressure on the size of hole used, and East suggested as a standard

that d/h = 6, and that the breadth of the blade should exceed 30 times the height to avoid end

effects. Smith et al. (1962) had assumed in their calibrations at supersonic speeds that re-

sults would be unaffected by grinding away the upper portion of the blade to minimize the in-

teractions which might occur between blades in an array. East showed that the removal of the

1.2

, o " % 4 x .. x

0.8 - Pos i t ion ~ _ x + ,9 h A B ' T ~ k . z ~ ° x

, ~_ 0 .60 .O020 inx +

O . O 0 ~ O i n o • = ~ " ~ 0.4 -0"0153in ~' •

I I I I ~r -3 -2 -t 0 7

0.2

0

- d ( a p p r o x )

I I I I I I I 2 3 4 5 6

Ax/h

Fig. 25. The effect on Ap of varying the razor-blade longitudinal position relative to the static hole; East.

*The use of "plug-in" Preston tubes by Peake, et al. (1971) should however be noted.


upper portion of the blade in fact altered the calibration. Inspection of a typical blade of

thickness 0.2 uln indicates that this is to be expected since the edge is composed of two cham-

fers, the first being of total angle about 20 ° and length of 0.25 mm, and the second of angle

about 12 ° and length 0.85 mm, so that the flow over a blade is not likely to attach before the

first shoulder and perhaps not before the second shoulder. Gaudet (unpublished) reanalysed

East's results and other unpublished results and found an empirical expression for the effec-

tive height of a blade

hef f = h + Ah,

h - h Ah t

where h h ' (6-2) t

h being the height to the edge of the blade, and h t the total height. His expression for

East's calibration then becomes

y* = -0.459 + O.962x* - 0.O813x .2 + O.OO8x .3 (6-3)

where hef f is used in the expressions for x* and y*. Pal and Whir,law (1969) found a similar

effect from adhesive tape used to hold down razor blades. As a secondary experiment during

the course of the direct skin friction measurements of Winter and Gaudet, measurements were

made of the sensitivity of razor blades for a wide range of heights, achieved by inserting

packing under the blades. It was found that an empirically satisfactory, if conceptually un-

satisfying, fit to the results for compressible flow was obtained if ~ in (6-3) was replaced w

by FcT w , where F c = (I + O.2M~) ½ is the compressibility factor on skin friction empirically

determined by Winter and Gaudet. Equation (6-3) modified in this way was also found to fall

between the two sets of results of Hopkins and Keener (1966) for forward and rearward positions

of blades relative to the static pressure hole; these authors used large specially-made blades

.ith a single-sided chamfer only. Their static pressure holes were very much smaller relative

to the height of the blades than those of East and Smith et aZ., so that qualitatively the

mean of their results for the two positions of the blades corresponds to the results for the

standard position of East and Smith et al.

6.3. Forward-facing step

Nituch and Rainbird (1973) (see also Nituch, 1971), appreciating the difficulty of specifying

readily the geometry of chamfered blades, set out to determine an alternative form which could

be easily specified and which should give as large a value of pressure rise as possible. Their

starting point was a plain rectangular block mounted downstream of the static pressure hole

but they discovered that provision of a semi-cylindrical cut-out in the upstream face of a

block, fitting the downstream half of the static pressure hole, led to an increase of about

40% in the pressure rise. This effect is illustrated in Fig. 23 where their line for the

block with cut-out is shown and also results for a plain block derived from pressure measure-

ments made during the tests of Gaudet and Winter, over the range of hur/v < i00 and from those

of Nash-Webber and Oates*(1971) over the range 6 < huT/~ < 160. For a block with cut-out "

diameter 1/3 times the height, of width-to-height ratio 1.5, and length (in the stream direc-

tion)-to-height ratio of 3, Nituch and Rainbird give the calibcation equation

log Ap = 1.307 + O.117 log oh2Ap ~ (5-4)

w u 2

for 3.1 × 10 6 < ph2Ap < 8.7 x 10 9 . u 2

*Note that there are typographical errors in the paper of Nash-Wehher and Oates a power of 2 being omitted from both the terms Re d and Ms/M ~.

40 K.G. Winter

They found that for zero pressure gradient the calibration was valid within 1% for blocks of

height up to 6 times the momentum thickness, that is extending well beyond the wail-similarity

region of their boundary layers.

6.4. Submerged step

The submerged step shown in Fig. 23 was suggested by Bradshaw and used by Edwards and Sivase-

garam (1968). This device is very small, the length of the ramp being 0.5 r~n and the depth

0.04 mm. It was made relatively wide, 4 mm, so that a slot O.i m~ wide, could be used to

measure the pressure at the base of the step. A similar slot alongside the ramp was used to

measure static pressure. The advantage of the device is that its calibration is apparently

independent of compressibility effects. The line in Fig. 23 is derived from the calibration

of Edwards and Sivasegaram obtained both at low speed and at M = 2.2, the length scale being

the height of the step.

6.5. Static-hole pair

The other possibility shown in Fig. 23 is the use of a pair of static pressure holes. Shaw

(1960) showed that there is an error incurred in the measurement of static pressure because

of the shearing stresses acting across the face of the hole. Duffy and Norbury (1968) realized

that this effect could be used to measure skin friction if the pressure were measured simul-

taneously at two holes of different sizes. The pressure error varies nonlinearly with du /~ T

and thus, if the smaller hole is sufficiently small, say du /~ < 50, the calibration is very

nearly dependent only on the size of the large hole. The curve shown in Fig. 23 is derived

from the paper of Duffy and Norbury with the length scale taken as the diameter of the larger

hole of a pair. The pressure signal available is some two orders less than that from a Pres-

ton tube of comparable diameter. A variant on the static-hole pair was explored by Green and

Coleman (1973) which used the pressure difference between a pair of slots one inclined 45 °

forwards and the other 45 ° backwards. Their aim was to produce a device the reading of which

dependent entirely on viscous forces, so that the calibration could be universal for any flow

provided the temperature and density at the wall were known. In this they succeeded in prin-

ciple by showing that the pressure difference was directly proportional to • and independent w

of ~u /~, where ~ is the slot width, for ~u /~ < iOO. However, the effects of local pressure T T

gradients and inaccuracies in their device in which the slots were displaced streamwise, pre-

vented an absolute calibration being obtained. They suggest alternative designs which would

be worth studying if an instrument is required for their purpose in future.

7. OBSTACLES IN THREE-DIMENSIONAL FLOW

7.1. Fence

Vagt and Fernholz (1973) reconm~end the use of a surface fence in three-dimenslonal boundary

layers by determining its alignment with the surface flow by rotating it so that zero pressure

difference is obtained, and then setting it at 90 ° to this direction to find ~w" They also

give an expression for the change in pressure difference with angle of inclination so that the

calibration for two-dimensional flow may be used if the inclination of the flow is known.

7.2. Razor blade

For three-dimensional flows East (1966) suggested a very simp]e way of using razor blades. He

investigated the change in pressure with change in the angle of inclination of the flow to a

blade and found the unique calibration shown in Fig. 26. Using this unique dependence East

observed that if measurements were made for two different settings of a blade differing by a

known amount, say 30 ° , then both the magnitude and direction of the skin friction could be


deduced by means of a curve such as shown in Fig. 27 in combination with Fig. 26 and the cali-

bration for two-dimensional flow (Fig. 24). This technique could be used with other devices.

Fig. 26.

A,.

1.0 ~ll~el~ll:l~" \ h !. / h o+ 36 60

~ o , = O.O05in x + O.OISin o n

0.8 -- + ' ~ \

0.6 ~ \ \ ~Z~pp = C OS 2

0.4 -- "t't~\\x\ "

'¥ \ \

0.2 x ~ \ \ \ ,

l I I ~ 0 20 40 60

I/~1 ~e~ The effect of razor-blade yaw in a two-dimensional boundary layer: East.

40

30

20

I0

-I0

I a \ l i 0.5 hO "~.5 2.0

A PB/A PB':s°

Fig. 27. The relationship between 8 and ApS/Ap( 8 30) deduced from empirical curve of Fig. 26: East.

7.3. Shaped block

A brief investigation of the possibility of utilizing the principle of the cut-out block of

Rainbird and Nituch has been made by Dexter (1974). The triangular prism adopted is shown at

the top of Fig. 28, and the shape should readily be accommodated in a circular plug. The in-

tention was effectively to use pressure Pl, as equivalent to the pressure measured for a single

block, but replacing the static pressure by the mean of the two pressures P2 and P3 on the

42 K.G. Winter

rearward surfaces, and also to determine the direction of the shear stress indicated by the

difference of these pressures. Dexter found that blocks of the relatively low frontal aspect

ratio suggested by Rainbird and Nituch were unsatisfactory because the pressure difference for

the two rear faces did not vary unambiguously with the angular setting, presumably as a result

of the nature of the separation from the front corners. However, a satisfactory calibration

was found, as shown in Fig. 28, for a block having an aspect ratio of 9 for its faces. This

is not necessarily a lower limit since the next smaller aspect ratio tested was 3. The tests

were made only over a very limited range of Reynolds number and further work should be done

before the calibration is accepted. However, Fig. 28a shows how the angle of the flow rela-

tive to the block can be obtained, and Fig. 28b that the angle can be used to determine the

pressure difference between the front face and the mean of that at the rear faces for zero

angle. A calibration for zero angle should then give the skin friction. Dexter did not pro-

duce a calibration in this form but his results showed that the pressure difference was in-

creased by about 30% compared with using a reference static pressure. It should be noted that

for a block with an equilateral triangular planform a calibration is required only over 60 °

angular setting to deal with a full 36~ variation of flow direction. As with any device of

finite area, the calibration will be sensitive to pressure gradients but it is clearly worth

further investigation of the potentialities of the instrument.

4 =0.3 h PL

h

1.0

0-8

0,2

D dqroes

0 I0 20 .30 40 50 60 a Determination of sheor-stress direction

o IO 20 30 40 50 60

O . O - t

0.6~

0.4

O.Z Determination of datum pressure rise

0"6 PJ" P3 P,'Pz

0,4

pz.p3, ~

Fig. 28. Surface block for three-dimensional flows: Dexter.


8. ANALOGIES

8.1. Heated film for two-dimensional Flow

Though Fage and Falkner had investigated the relatlonship between local skin friction coeffi-

cient and heat transfer from platinum strips embedded in a surface as long ago as 1931, the

first practical instrument using the analogy between skin friction and heat transfer was de-

signed by Ludwieg in 1949, and subsequently used in the classical experiments of Ludwieg and

Tillmann (1949) on the variation of turbulent skin friction in pressure gradients. Ludwieg

showed that, if a heated surface could be made of sufficiently small stremnwise length that

its thermal layer lay within the linear velocity region near the wall, then the heat flow was

proportional to the wall-shearing stress to the power of 1/3. He deduced the following equa-

tion

1

q£ = 0.807 f ' ~ (8-1) kAT

with the condition

i I

1.86 / c f_~6(_Re £_h ~ 6_~q < 1 (8-2)

where q is the heat flow per unit area;

k is the thermal conductivity of air;

AT temperature difference between the heated element of length £ and the unheated surface;

o Prandtl number;

uT~ L C - the non-dimensional thickness of the sub-layer;

6 the thickness of the thermal layer. q

Ludwieg's analysis was repeated by Diaconis (1954) for compressible flows and he showed that

equation (8-1) could be used provided the quantities appearing were evaluated at wall condi-

tions.

The instrument designed by Ludwieg consisted of a block of copper 2 ~m long in the stream dir-

ection, by 9 mm wide, by 6 mm deep, and heated by an electrical current. The block was mount-

ed beneath a thin sheet of celluloid which formed the airswept surface, and which in turn was

attached to a cylindrical housing. The celluloid and an air gap round the remaining sides of

the block provided thermal insulation. The temperature difference between the heated block

and the unheated wall was measured by thermocouples. The calibration performed in the turbu-

lent boundary layer of a low-speed blower wind tunnel confirmed the relationship (8-1) when

the substantial heat losses from the block other than to the airstream were taken into account.

Liepmann and Skinner (1954) devised an alternative form of Ludwieg's instrument with a much

reduced effective streamwise length. Their instrument consisted of a platinum wire of 13 um

diameter cemented into a groove in the surface of a piece of ebonite. They reassessed the de-

rivation of the form of the calibration of the instrument including the effect of pressure

gradients. They also drew attention to the existence of a limitation on the minimum length

of an element. For the boundary-layer type analysis used to be valid the thickness of the

thermal layer should be small compared with the length of the element. This was expressed

as

q~ > 1 (8-3) kAT

44 K.G. Winter

Bellhouse and Schultz (1966) realised that the thin-film techniques, which had been used to

make resistance thermometers for surface temperature measurements in hypersonic flows, could

be applied to measure skin friction in place of the hot wire of Liepmann and Skinner. Their

first investigations at low speeds revealed however that different calibrations were obtained

in laminar and turbulent flow.

In a subsequent paper (1965), which is concerned with compressible flows, they recast equation

(8-1) in the form

q--~£ = T3C o 3 (p~)3 (8-4) AT w p

They noted, following Liepmann and Skinner, that over the thickness of the thermal layer of

the element, C and o could be taken as constant and that the factor (p~) is almost independent P

of temperature so that the calibration should be independent of compressibility effects. They

confirmed this experimentally for Mach numbers up to 3 using platinum films about 2 mm × 0.2 nun

fired onto a pyrex glass substrate. The resistance of the heated film was used to measure its

own temperature and that of a similar unheated film to measure the reference temperature. For

a film heated to a given temperature, so that C , ~ and a are independent of the external P

flow, they showed that the calibration took the form

i2R i AT = A(OTW) 3 + B (8-5)

where i and R are the heating current and resistance of the element and p is evaluated at the

film temperature.

As well as being simple the thin-film gauge has a high frequency response and hence can be

used to measure fluctuating quantities (Bellhouse and Schultz, 1968) and in facilities with

short running times.

Brown and Davey (1971) describe the use of a simple apparatus for calibrating hot-film skin

friction gauges in which the gauge is mounted in a stationary plate separated from a rotating

plate by a small air gap.

In a general theoretical investigation of heat transfer in shear flows Spence and Brown (1968)

determined series expansions relating the heat convected into a stream from a heated element,

with a "top-hat" temperature distribution, to the skin friction and pressure gradient and ap-

proximated these by

i9 ~2~ 5 dp kAT (8-6) T = ~ - - '

w I0 0o 18 dx q

under the conditions

Lu i T > 6.6

(i) V 7 to ensure that the T3-1aw apply;

~u (2) T < 640 for a unique calibration in laminar and turbulent flows to be obtained

Brown (1967) describes experiments in which he showed that equation (8-6) gave accurate re-

sults for skin friction in a laminar boundary layer approaching separation in which circum-

stances a Stanton tube gave erroneously high readings. He also showed that the same calibra-

tion could be applied in laminar and turbulent flow provided condition (2) above was met. In


order, however, to ensure this he made a special gauge in which the glass slide of thickness

0.15 mm, on which the platinum film was mounted, was embedded in a block of copper by an epoxy

resin with a small groove left clear round the slide on the airswept surface. The purpose of

this arrangement was to limit the effective length of the heated element. Later, however,

Pope (1972) investigated theoretically the consequences of heat conduction from the film to

the adjacent surfaces and concluded that this was not responsible for the differences in cali-

bration obtained in laminar and turbulent flows. He suggested instead that the source of the

discrepancy is a variation of effective viscosity through the viscous sub-layer so that the

velocity profile is of quadratic rather than linear form, and showed that velocity profiles

imply

ueff = u ( 1 + 0.042 ? . W

)

(8-7)

Using (8-7) he was able to obtain identical calibrations for laminar and turbulent flow. The

condition which results is much more stringent than the second’one of Spence and Brown above,

and is that

3

Il”r AT

?Y < 0.51 X 106 $

( ) W

where AT is the error in r W W

consequent upon applying a calibration obtained in a laminar flow

to a turbulent flow so that for aur/ov = 64 an error of 5% in rw is predicted.

Rubesin et al. (1975) have recently re-examined the errors arising from thermal conduction into

the substrate which effectively increases the length of the element. Guitton (1969) also drew

attention to the importance of the heat losses into the substrate. He used a thin film set on

the end of a glass rod fitted into a brass plug. He noted that a calibration of his instru-

ment obtained with it mounted in a brass pipe required correction when applied to the same in-

strument mounted in a perspex plate. They point out that the glass substrate used by Bellhouse

and Schultz has a relatively high thermal conductivity, some 14 times that of the ebonite used

by Liepmann and Skinner. They investigated the use of a plastics substrate of low thermal

conductivity. Because of the difficulty of depositing a narrow, thin-film element onto the

material they reverted to the arrangement of Liepmann and Skinner and used a platinunrrhodium

wire of 25 urn diameter, which was laid on the surface of the substrate in a layer of epoxy

resin. The surface was then handworked to expose the wire< Gauges produced by this method

gave reduced heat losses at zero flow and small effective streamwise lengths. Rubesin et al.

also showed that the heat convected from the heated wire could readily be detected by unheated

wires installed upstream and downstream so that a triple-wire unit made a sensitive separation

indicator. An alternative means of manufacture is described by Singh and Railly (1970) (but

see also Railly, 1972) in which a Sum diameter tungsten wire is pressed into the surface of a

small block of perspex.

8.2. Heated films for three-dimensional flow

The single film has directional properties and may be used for finding the surface flow direc-

tion by rotating it and finding a minimum in the heat transfer, which indicates that the film

is aligned along the flow. (The maximum when the film is normal to the flow is very flat.)

The determination of flow direction by this means, of course, requires an adjustable insert.

Drinkuth and Pierce (1966) used this technique with an instrument in which the element was a

heated wire (50 urn diameter platinum - rhodium) mounted beneath a 25 urn thick Mylar film.

46 K.G. Winter

McCroskey and Durbin (1971) investigated the possibilities of producing a dual-film gauge

which would give both the surface shearing stress and its direction, with the aim of studying

the boundary layers on helicopter blades. They pointed out that the calibration of films is

extremely sensitive to the properties of the material and the physical dimensions of the films

so that matching a pair of gauges would be very difficult with laboratory methods of manufac-

ture. They therefore obtained films made commercially using the techniques developed for the

production of thin-film strain gauges with two elements set mutually perpendicular, of nickel

5 ~m thick, 0.05 mm wide and 5.5 mm long. They also developed a special twin-bridge constant-

temperature system for operating the gauges, and found that a linear relationship existed over

±40 ° between the flow angle and the heat transfer from the two elements expressed as (QI - Q2)/ 1/3

(Q1 + Q2). They confirmed the T relationship for the heat transfer but noted that the abso-

lute calibration depended upon the nature of the suhstrate material.

8.3. Pulsed heated film

A variant on the heated-film friction gauge, analogous to the pulsed hot-wire technique, has

been studied by Ginder and Bradbury (1973). They use three parallel films set normal to the

flow. The films are about 12 mm long with the centre film I mmwide separated by gaps 0.5 mm

wide from the outer films, which are O.I mm wide. The mode of operation is to heat the centre

film by a short duration electrical current and to measure the time interval between the heat

pulse and the occurrence of a change in resistance of one of the outer elements, two outer

elements being provided in order that the gauge may be used in separating or attaching flows.

They use a very simple model to make an estimate of the time interval t. The model assumes

that the process takes place entirely within the linear velocity region at the wall, and that

heat diffuses outwards a distance y, is convected along the stream a distance Z (the separa-

tion of the heated and receiver elements) and then diffuses back to the wall. The time is then

t =y2 + 2K ~u

where K is the thermal diffusivity, K = k/0Cp.

The first detection will occur for the value of y which gives a minimum for t, which leads to

Kt 3 K~g_.~ui ~2 2 (8-8)

The conditions on the use of the expression (8-8) are that the time must be less than the

direct diffusion time between the two films, but the distance must also be sufficiently small

that the height, y, to which the heat diffuses must be such as to give yum/~ < 5 say. For

Prandtl number near unity this leads to

~u

4 < < 125 .

Ginder and Bradbury calibrated gauges in a laminar-flow channel and obtained an expression

2

Kt - 2.4 + 0.82 (8-9)

which agrees in form with (8-8). The signals to he measured were extremely small, the tempera-

ture rise of the passive element being of the order of IO -~ times that of the active element,

and the time interval being of the order of a millisecond. Ginder and Bradbury applied their


gauges in a separated flow behind an aft-facing step and obtained the result shown in Fig. 29,

which demonstrates the potential of their instrument for investigating the unsteady properties

of separated flows. However, they found that in attached turbulent boundary layers their basic

calibration gave values for skin friction some 20% higher than indicated by a Preston tube,

and further study is needed before the instrument can be considered for general use.

.I .0 o

0 .6

0 . 4

0.2

I - 4 3

-0.8

o

- e> Gauge span o 3 . 3 m m

"~ ~ 7.6mm

I I -3 - 2 -I 0 I 2

I0 -4du s" dy '

8.4.

Fig. 29.

Mass transfer

Measurements in separated flow behind a step with "time-of- flight gauge": Ginder and Bradbury.

The use qualitatively of a surface coating which will subl~e is a common technique for

determining transition from laminar to turbulent flow and is based on the increased rate of

s~limation due to the increased skin friction in the turbulent region. Owen and Ormerod

(1951) made a quantitative investigation of the s~limation technique and obtained a relation-

ship of form similar to that for heated films, for the mass transfer from a small finite re-

gion, which can be e~ressed as i 3

mR (~2Tw 1 ojA~ = 0.54 \-~/ , (8-I0)

where m is the rate of mass transfer per unit area;

j is the molecular diffusion coefficient, and

A~ is the difference in concentration between the surface and the airstream.

~nerally the concentration at the surface may be taken to correspond to the saturation vapour

pressure. Konstantinov (1955) also derives this equation but with the constant having a value

0.807 as in the corresponding equation (8-1) for heat transfer, which can be e~ressed in the

same form by replacing k the thermal conductivity by the thermal diffusivity K = k/pCp to give 1

0.807 -F~-/ (8-11)

Equation (8-i0) holds only when the diffusion process remains wi~in the s~-layer. When the

diffusion layer has a thickness of the same order as the turbulent bounda~ layer the relation-

ship becomes the same as the yon Karman e~ression for heat transfer, that is

48 K.G. Winter

m

where K m = oUA$ and J is an analytic function which is zero for j/~ = I.

Konstantinov tested diffusion gauges in a low speed wind tunnel. His gauges used water as the

working fluid which was evaporated from the surface of a small porous ceramic block of exposed

area 2 mm × 18 ~ml. A capillary tube was used to measure the rate of flow of the water to the

block. From the results given it is difficult to assess the accuracy.

Murphy and Smith (1956) showed that the mass-transfer analogy could be applied to determine

the skin friction on a flat plate. They used a film of silicone fluid and measured the time

history of the variation of the film thickness by an interferometric technique on a flat plate

in a wind tunnel at low speed. It is not clear how they distinguished between changes in the

thickness of the film due to evaporation and due to streamwise movement under the shear

stresses. Wazzan et a~. (1965) extended the investigation to supersonic speed, and limited

their measurement to the film front where the film was very thin so that convection could be

neglected. They concluded on the basis of a comparison of their results with calculations

for a laminar boundary layer that mass transfer was proportional to skin friction.

A more detailed study of the possibilities of the technique was made by Sherwood and Tr~ss

(1960) using a subliming material, naphthalene. Their experiments were made on a flat plate

at subsonic and supersonic speeds. A smooth surface on a layer of up to I n~n thickness was

obtained by applying naphthalene in molten form. The thickness was measured before and after

a wind-tunnel run by mechanical means. They showed that their results for K were in good m

agreement with the analysis of Deissler and Loeffler (1958), extended to mass transfer, which

indicates that the analogy factor for mass transfer becomes increasingly smaller than that

for heat transfer as Mach number increases.

9. LIQUID TRACERS

9.1. Oil flow

Squire (1960) investigated theoretically the movement of an oil film under a boundary layer

and showed that the oil follows the surface streamlines except near separation. However, he

did not attempt to produce a quantitative method for determining skin friction. Meyer (1966)

described a variant of the oil-flow technique, the "oil-dot" technique, in which rows of spots

of a suitable oil and pigment are placed on a model in a wind tunnel. When the air flow is

started the dots run into streaks defining the surface streamline direction. Meyer discovered

that the length of the streaks depended little on the dot size, as shown in Fig. 30, and that

the length of the streak was proportional to the surface shear or heat transfer as shown in

Fig. 31 for the flow over a half-cone delta-wing model at M = 12.6. Though the technique is

used regularly for qualitative measurements it has not been fully exploited quantitatively.

Tanner and Blows (1976) suggest a quantitative method of utilizing the temporal variation of

the thickness of a liquid film on a surface due only to convection to find the skin friction

of air flowing over the surface. They show that the variation with time of the thickness h

of a liquid film in a two-dimensional flow is given by

~t~h 3~I ~x~ ( h3 ~ 2~I ~x~ (h2Tw) . (9-1)

If oil is deposited on a surface in a line normal to the flow then, except near separation,

the second term in equation (9-1) may be expected to be dominant and the oil will flow down-


Fig. 30.

A o R u n 6 9 7 / I \&~ x Run 698 z~ Run699

\

,(E

o"

C3

015

010

0 05

x

x

I

Ao

__ I I I I I 0 0.1 0.2 0.3 0 .4 0 .5 0 .6

Streak length, in

The dependence of streak length on dot size for a fixed condition of surface flow: Meyer.

4 0

• ~ 3 o c o

e~

" - 2 0

i---

~r

- &

A ~&

O

O /

B

A Heat transfer measurements

o FI0w indicator measurements

Curves matched at A and B

Fig. 31.

0 0 .5 1,0

S / S T

A comparison of streak length distribution and heat transfer distribution for an incidence of 15°: Meyer.

50 K° G. Winter

stream. Once the film has spread over the surface to be investigated the thickness h w[l] de-

crease with time. Since the pressure-gradient term depends upon h 3 and the friction term h 2

the latter will become increasingly dominant with increase of time. It is therefore togical

to look for a solution initially ignoring the pressure-gradient term and then to determine a

correction. By evaluating the development of the oil film following a surface particle Tanner

and Blows show that ht = constant at given x. Hence from equation (9-1)

x x

T w - 2U I hdx - 2~ I (ht)dx . t h 2 (ht) 2 (9-2)

O 0

Tanner and Blows show that, if in a pressure gradient a correction term ~ is introduced so that

h = ho(l + c) then

= . . . . l ho dp + 2U I dp (9-3) 3 T w dx 3t T2

w

They measured the time history of flows using an interferometric technique on a glass surface

and confirmed the relationship ht = const for flows in zero pressure gradient. They also exa-

mined a flow approaching separation caused by a spoiler on a flat plate and deduced the skin-

friction distribution. Figure 32 shows the film thickness distribution at various times in

terms of the number of fringes. The results are replotted in Fig. 33 as the product ht, which

is not independent of t because of the effect of the pressure gradient. The curve labelled

t = ~ is obtained by applying the pressure-gradient correction. From this curve the skin

friction can be calculated by use of equation (9-2).

350

300

2.50

200

c L. h 150

I00

5O

0

- 5 0

J~ /+, / ~ +---+ =T / , Y--

dp / ~ 6 ~ / / '~'

l / / " / / ÷\ t : 5 m i y / ~

_ // '5 / ,/"'+

, , / . . _ ~ . - - - - . - - ~ v _ . ~ ~ ---'--" bU

~ --2.0 40 x m m 60 80

09.

J IOO

~ Glass plate No oil

Fig. 32. Distribution of oil-film thickness under flow approaching separation: Tanner and Blows.

The technique needs a special environment to make the interferometric measurements possible

but clearly could be valuable in specialized cases, and has the great advantage over many

methods that no knowledge of the properties of the test medium is required. It can be extended

to three-dimensional flows as shown in a subsequent paper by Tanner and Kulkarni (1976). As

Tanner and Blows point out care must be taken in flows near separation that the accumulation

of oil does not modify the flow. There may, however, be limitations on the combinations of

shear stress and film thicknesses for which the film will remain stable. Figure 34 shows a

Skin friction in turbulent boundary layers 5]

photograph taken by the present author of a very thick oil film on a flat plate at a speed of

35 m/s. The photograph shows the flow in the region of the transition front near the centre

of the plate where the wavelength of the pattern in the oil is clearly decreasing through the

transition region. In the lower right of the photograph the turbulent wedge spreading from

the intersection of the plate and the sidewall is indicated by the shorter wavelength of the

disturbances compared with those in laminar flow.

7 0 - -

8 0 - -

50 - -

4 0 - -

E

g -~ 3 0 - -

2O

I0

o

Fig. 33.

• 5 min + I O m i n o 2 0 min z~ 4 0 m i n {~ 6 0 m i n

_ T - 5 m i n - - ~ . ~ ' '

20 40 60 x mrn

IT-oo i

J

6u--J/

/

80

Limiting form of oil-film thickness distribution: Blows.

I ~oo

Tanner and

Surface of + polished wood /

Surface 1 painted black

!

3S role

Camera Flat plate

Tunne window

Laminar

TurNlent wedge from ~ ~ -

junction of plate leading edge end tunnel sidewall

I I I I m from leading edge 1.7 1.6 I.S 1.4

Trop-~ition region us indicated by surface pitot tube

F i g . 34 . S u r f a c e w a v e s i n t h i c k f i l m o f o i l .

52 K.G. Winter

These difficulties can be avoided by the use of a simplified version of the technique as de-

scribed by Tanner (1976), which may be applied to polished metal surfaces. In this version

only the leading edge of an oil film is used where the thickness varies linearly with down-

stream distance so that

~x = -- (9-4)

w ut

Hence by finding the gradient with distance of the film thickness the shearing stress can be

obtained. This is done by using two laser beams with a known separation, x, one to define

the leading edge of the film and the other to record the evolution of the thickness by count-

ing fringes as indicated by zero crossings of the output of a photocell on which the reflected

light is focused.

9.2. Liquid crystals

Another suggestion for a surface-coating technique is that of Klein and Margozzi (1969, 1970)

who have made an exploratory investigation of the use of liquid crystals. Liquid crystals ap-

pear to be viscous liquids and yet show many of the properties of solid crystals. One of

these properties is selective light scattering so that when illuminated with unpolarized white

light incident at a given angle liquid crystals reflect strongly only one light wave length

at each viewing angle. Small changes in conditions can cause a shift in the wavelength of the

reflected light in a reversible way. Klein and Margozzi showed that a mixture of liquid cry-

stals could be produced, the properties of which were primarily dependent upon the shearing

stress imposed on the mixture, although the mixture also exhibited sensitivity to temperature

and to the angle between the specimen and the direction of illumination and scattering.

Figure 35 shows a calibration obtained in the shear flow between a fixed annulus and a rotat-

ing annulus. The calibration shows a reversal of colour change for ~w greater than about 300

N/m 2 (3.06 x 10 -2 g/mm2). In practice this reversal would not often be of significance since

shearing stresses would rarely exceed the critical value. (For example in a wind tunnel at a

Mach number of unity and a stagnation pressure of one atmosphere a skin-friction coefficient

of 0.002 gives a shearing stress of about 75 N/m2.) However, it was found on making experi-

ments in a pipe that a film of sufficient thickness to exhibit the light-scattering properties

Fig. 35.

E

% 4

2 I 550

? I

l

o

IHIII IIIIII

560 570 580 590 600

Woveleng ' l 'h , nm

Wavelength of light scattered from liquid crystals under shear stress: Klein and Margozzi.

O \ \

o o~ \

\ \ \

\


produced marked ridges normal to the flow, and that the scattered light signal fluctuated con-

siderably making it extremely difficult to measure the wavelength. A great deal of further

development is therefore required before the technique can be considered for routine applica-

tion.

IO. CONCLUDING REMARKS

Of all the techniques reviewed it is apparent that none can be considered an absolute and re-

liable standard. The obvious technique of directly measuring the surface shearing stress by

a force balance is beset by many pitfalls which may be overcome in particular cases but the

possibility of specifying a priori the requirements to be met in general seems remote. An

analysis is given of the errors arising from various causes and this may provide some guidance

in design. The most reliable device at present seems to be the Preston tube because of its

simple geometry and because it has been investigated the most thoroughly. However, there is

still room for further work on the effects of pressure gradient, of flow unsteadiness and of

heat transfer and in three-dimensional flow. Potentially, sub-layer fences hold most promise

for devices of the pressure-measuring type if a design can be found with a geometry easy to

manufacture repeatably, which is very difficult because of their smell size. There is an

opportunity for the exercise of some ingenuity in devising instruments for use in three-dimen-

sional flows. In the long run the heated-element instrument is likely to prove the most re-

liable. For general application the discovery of a simple shear-sensitive surface-coating

agent would be most welcome.

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An Outline of the Techniques Available for the Measurement of Skin Friction in Turbulent Boundary...

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