EXTENSEON OF FRE$YQN'S SHEAR
MEASUREMENT TECHNEQUE TO
EQUGH BQUE‘QDARE‘ES‘

Thesis {or ”19 Degree of M. 5.
MICHIGAN STATE UNWERSITY

Li- San Hwang
1962

2n
U)

THE

This is to certify that the

thesis entitled

Emmanuel: of bestow a Shem:
Mutant Technique to
W Boundaries”

presented by

Li-San Huang

has been accepted towards fulfillment
of the requirements for

__!1_~S.__degree mwneering

Major professor

 

Date February 23. 1962

 

0-169

—___

LIBRARY

Michigan State
University

 

EXTENSION OF PRESTON'S SHEAR MEASUREMENT TECHNIQUE

TO ROUGH BOUNDARIES

By

Li - San Hwang

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Civil Engineering

1962

ii

ACKNOW LEDGEMENTS

The author is indebted to Dr. E. M. Laursen, without whose
guidance and encouragement, this thesis would not have been possible;
he also wishes to express his gratitude to Dr. H. R. Henry for his
review of the manuscript, to Dr. C. P. Wells for his review of the
analytical result, and to his colleague, Mr. J. R. Adams for his
assistance. The research upon which this thesis is based was part
of a larger study on "Pressure and Shear Distribution on Dune-shaped
Boundaries, " conducted under a grant from the National Science

Foundation.

iii

TABLE OF CONTENTS

Page
NOTATION ............................................. iv
I INTRODUCTION .............................. 1
II. ANALYSIS ................................... 4
III. EXPERIMENTS .............................. 10
IV. RESULTS AND DISCUSSION ................... 13
V. CONCLUSIONS ............................... l7

BIBLIOGRAPHY ....................................... . Z7

NOTATION

3o h/k
5
Inner radius of the stagnation tube
30 a/k
5
Outer radius of the stagnation tube
Coefficient in Karman-Prandtl velocity equation
Coefficient in Karmén-Prandtl velocity equation
Resistance coefficient
Height of center of stagnation tube from zero datum
Sand roughness
Dummy integration parameter
Stagnation pressure
Wall static pressure
Reynolds number
Inner radius of the tubing
Velocity at distance y from wall
Shear velocity = To p
Distance from zero datum
Height of bottom of stagnation tube from zero datum
Gamma function

Thickness of laminar sublayer

Integration parameter

iv

U Kinematic viscosity

 

 

p Density of fluid
0' Area of stagnation tube opening
To Wall shear stress
P—P
( O )a Analytical pressure-shear ratio, in completely rough regime
o
P-P
( o

) Experimental pressure-shear ratio, in completely rough regime
e
o

ABSTRACT

EXTENSION OF PRESTON'S SHEAR MEASUREMENT

TECHNIQUE TO ROUGH BOUNDARIES
by Li-San Hwang

Preston's shear measurement technique consists of placing a
pitot tube in contact with a wall and interpreting the dynamic pressure
reading obtained as a measure of the local shear on the wall. The
rationale of the technique is that the velocity distribution near the
wall is a function only of conditions at the wall, and the dynamic pres-
sure reading in the tube is determined by the velocity distribution and
the tube size. Assuming the Karman-Prandtl velocity distribution,
an analysis was performed for the fully rough flow regime which gives
the ratio of the dynamic pressure reading to the wall shear as a
function of the tube size and? of the groughness element. _ Experimental
measurements provided correction factors for the particular roughness
used and for the transition regime offlow. Satisfactory results were
obtained indicating that Preston's technique can be used for rough

boundaries.

I. INTRODUCTION

Detailed knowledge of the local boundary shear is needed in
many problems of fluid mechanics. Since analysis can seldom provide
the desired answer if the flow is turbulent, one must resort to experi-
mental measurement techniques. The empirical data from measure-
ments may be immediately helpful in practical problems, or may be
useful in the building of theory.

Direct measurement of boundary shear has been attempted, but
has not been entirely successful. To obtain a direct measurement of
the shear on the boundary, one must measure the shear force on a
small isolated floating element of the surface.1 Maintaining a small
gap around the floating element, keeping it in the correct position,
and recording the small shear force acting on the element are all
difficult problems. An inherent disadvantage to this type of instrument
is that it is restricted to a predetermined fixed point, thus inhibiting
the free exploration of the shear distribution on the boundary surfaces.

A standard indirect technique in boundary layer‘studies has been
the measurement of the velocity and pressure profile normal to the
boundary at successive sections and solving for the shear by the
momentum principle. 2 If the shear is small or varies in a manner
which is not simple, the results are less satisfactory, since the shear

is obtained as the small difference between large values.

Ludweig developed an indirect method which relates shear force
to the heat loss from a hot-spot on the boundary to the flow medium.
The insulation between the hot-spot and its surroundings must be
excellent or considerable error may result. This method is not
adapted to the evaluation of unknown shear distributions as it is
restricted to measurements at pre-chosen fixed points. Moreover,
the method can only be used when there is a laminar sublayer.

In 1954 J. A. Preston successfully developed a simple technique
for measuring the local shear on smooth boundaries using a pitot tube
in contact with the surfaces. 4 This method is based upon the
assumption of an inner law relating the local shear to the velocity
distribution near the wall. Using the pressure drop in a pipe to cali-
brate the instrument, Preston obtained equations relating the shear to
the pitot tube reading both for the case of the laminar sublayer
enveloping the tube, and for the case of the tube in the turbulent
boundary layer on the smooth surface.

E. Y. Hsu used the velocity distribution equations, u/u"< = k y1/7
(turbulent boundary layer) and u/u* = u"< y/V (laminar sublayer), to
establish analytical relationships between the dynamic pressure read-
ing'and the local boundary shear. 5 Hsu's analysis agrees well with
Preston's experimental results and also appears to give good results
for the boundary layer on a flat plate with ambient pressure gradients.

In the present investigation an experiment was performed with a pipe

to check equipment and technique. The results agreed, as expected,
with Preston's and Hsu's.

In practice the boundaries of the flow are more likely to be
rough than smooth. Thus, an extension of Preston's relatively simple
technique to rough boundaries would be very useful. An analytical
relationship has been developed between the dynamic pressures acting
on the pitot tube in contact with the rough surface and the local
boundary shear. The relation is a function of the inside diameter of
the tube, the size of the roughness, and the position of the tube in
relation to the zero datum for flow in the completely rough regime.
For the transition and hydraulically smooth regimes one more parameter,
the ratio of the roughness to the laminar sublayer is needed. The

effect of this parameter has been assessed experimentally.

II. ANALYSIS

The concept of the inner law is that the velocity distribution near
a rough boundary depends only upon the viscosity and the density of
the fluid, the roughness of the boundary, and the shear stress at the
wall. The Karman-Prandtl velocity distribution equation which is
generally considered to be a satisfactory approximation for pipes and
channels and is often used in other situations, can be expressed as

follow 5

u/u,< = C log y/kS + D ................ (l)

The classic systematic experiments of J. Nikuradse utilized pipes
covered tightly on the inside with ordinary building sand. 6 For
better adherence of the sand grains, the pipe was filled and emptied
a second time with Japanese lacquer after the inside had been coated
with sand grains. The values usually quoted for C and D are based
on Nikuradse's data, C = 5. 75 and D = 8. 5, except in the transition
regime where D is a function of u* kS/U or kS/o'. Although the
adequacy of this logarithmic equation can be debated, especially near
the boundary, and the indeterminancy of the zero datum presents
difficulties, Eq. (1) has been adopted for this analysis. Further
assumptions which have been used in the analysis are:

(l) The disturbance of the flow caused by the presence of the tube

on the rough boundary may be neglected.

(Z) The dynamic pressure acting on surface pitot tube, and therefore
the reading obtained, is the average of puZ/Z over the open area.

With the above assumptions, the dynamic pressure acting on
the tube, as shown in Fig. 1, can be correlated to the local boundary

shear by
- 2 _ l f 2
(P—Poy-n'a, — 2 p 0' L1 dO' ........... (2)

01'

 

 

P - 1Do 1 u 2
= (—) d U
‘r 2 a u*

_ o ZTra
In the completely rough regime, i. e. , where u* kS/V > 70, Eq. (1)

can be written in the form

u/uak = 5. 75 log 30 y/ks .............. (3)

 

I do = “/ a2--(y-h)Z dy

‘1 Li t | JV
‘7?

Y

 

 

. ”h
/ ,.--2°

G} 6 6366””;

line of zero datum

 

 

 

sand grains

Fig. l. A stagnation‘tube resting on a rough boundary

Substituting Eq. (3) into Eq. (2) results in

 

 

h+a
P-P
o .1 3O 2 2 Z
Z = E [5.75 u* 10g —k—Y] ° 2 ° \/a - (y-h) dy (4)
Tra 3
h-a

This integral relation can be evaluated in the following way: Let
y = h + a sin 9,

A: 30 h/k ,
s
B = 30 a/ks- (where A > B)

Making these substitutions in Eq.‘ (4), and changing integration limits,

one obtains,

("

Tr
P-P 2 '—
. 2
___i :(_5_7_5_)__ 2 [log(A+B sin 9] coszede
T TT
0 E
Z
2 2n
. Z 2
: (2—2???— [log(A+B sin 9)] cos 9 d9 ...... (5)
0
Zn
m B . m 2
Now let f(m) = A (1+; sm 6) cos 6 d6 ----------- (6)
0

Upon expanding the integrand, there is obtained,

217 2
B Z
f(m)=Arn (1+m— sine + (111) -B—- sin 9 + (r3)
. A 2 A2 3
O

K.

sinad + . . .)cosZGd8

bulbs

The odd powers of sin 9 disappear and the even powers can be

integrated through the relation between the gamma and trigonometric

functions,

1 1
5-1 1_ r<3s1r<§n1
' 2 1 1
O F(ZS+En)
then

3 3

2 I‘(—)1“(-)

_ m B} E}. _.__Z _2__
f<m)-4A [ +<21 22 N3) +<

Md

5 3
m) Ff. r(3)r<31+ ]
4 4 n4)

tvi -

:>
;>

The gamma functions can be evaluated by the relation

I‘(s+l) = sF(s),

and rm =1 and 11—5) 2W.

 

Then
ml m B2 1 m 13-1A"
f(m):ZTrA [2+(2): 2.4 +(4)2 4-63" I (7)
2 4
1 B ml 3-1 B
Let¢(m)=—+(m)— H) . — +
2 2 4A2 4 2 4 6 A4
Then
f(m) = 21TAm ¢(m) .................... (8)

Differentiating Eqs. (6) and (8) with respect to m twice and putting

m = 0, one obtains

2n 2 2
f" (0) =f [log (A+B sin 8)] cos 9 d9
0

— 2w log A¢(O) +4Tl' log A¢'(O) + 2n¢"(0)

Substituting Eq. (9) and the values of A and B into Eq. (5) there results

 

 

 

P-P
o _ 30h Z
—16.53l<:[log k ]
O S
30h a 2 a4 a6
-log ks [0. 25 (E) +0.0833(B-) +0.0704(h) +. . .]
a 2 a4 a 6
+[o.25(-}—1-) +0.1146(-fi) +0.0586(-£) +...]} (10)

This series converges quite rapidly, especially when the ratio a to h
is small.

Given an inside tube diameter and a roughness size a/ks, the
pressure-shear ratio (P-Po)/'ro, varies with the relative position of
the tube as shown in Fig. 4. The relative position of the tube (h - a)/k.S
can also be written as zO/kS + t/ks, where 20 is the distance from the
zero datum to the outside bottom of the tube and t is the thickness of
the tube wall. If the tube is large compared to the sand particle, 20
equals the distance from the datum to the top of the particles, or
something less than 0. 5 ks. If the tube is small it may rest below the
top of the particles but zo/ks would probably be greater than zero.
Intuitively, one might estimate that zo/ks would fall between 0.1 and 0.4.

Examination of Fig. 4 shows that the larger the tube opening, the
greater the pressure-shear ratio and the less the effect of unknowable

zo/ks values. Therefore, the tube should be chosen as large as

possible, but within the region in which the velocity distribution is
primarily determined by the boundary shear and in which the general
overall boundary configuration has little influence.

The above analysis assumes that the flow is in the completely
rough regime. In the hydraulically smooth and transitional regimes,
D is no longer constant, but a function of u>z<ks/V' The pressure-
shear ratio then would be a function of three parameters a/ks,
(h-a)/k , and ufik /1/.

s - 5

Given values of C and D for various values of u*kS/U one could
obtain relationships similar to Eq. (10), or a correction factor.
Because the values of C and D are open to question (in fact the
applicability of the logarithmic velocity distribution itself so close
to the boundary is debatable), this analysis has not been performed.
Instead, the correction factor is presented as an empirical formula-

tion based on the experimental data.

10
III. EXPERIMENTS

The general arrangement of the apparatus is shown in Fig. 2.
The tubing consisted of four sections each 10 ft. in length, of 6-in.
outside diameter and l/8-in. wall thickness. These were split
lengthwise for access when covering theiinterior surface of the tube
with sandpaper. A pair of steel bars, welded on each side of the
tubing before splitting, were used to bolt the split tubing together.
The width of the saw 'cuts was compensated for by gaskets which
maintained the circular cross-section and formed air-tight seals.

Two pairs of peizometers were used on each of the first three
sections of the tubing. In each section, the first pair of peizometers
is located 4 ft. downstream from the joint and the other pair 1 ft.
upstream from the' next joint.

Five series of tests were performed: one with the compara-
tively smooth surface of the steel pipe and others with rough surfaces
of four different grades of sandpaper. The sandpaper, with grit
numbers 36, 80, 150 and 220 corresponding to nominal grain sizes
of O. 75, O. 35, 0.17 and O. 06 mm. respectively, was obtained from
the Minnesota Mining and Manufacturing Company. The grades were
chosen for a ratio of approximately 1/2 between the different grades.
All were of the open-coat type of production paper with sharp grains.

Microphotographs of the sandpapers are shown in Fig. 5. The

ll

sandpaper was glued to the tubing with "Feathering Disc Adhesive, "
also a product of the 3M Company.

For the smooth boundary case, the peizometers were l/lé-in.
diameter holes drilled through the tubing with l/4-in. diameter and
3/4-in. long nipples soldered to the outside of the tubing. For the
rough boundary cases, the design of the peizometers is shown in
Fig. 2c. The smooth plate of l-in. diameter and 0. 008-in. thick
brass was used to avoid the possibility of the wakes from discrete
sand particles influencing the pressure measurements. The set screw
was used to orient the curved plate along the tube axis, and to permit
adjustment of the height of the plate for the various sandpapers. The
peizometer was connected to an alcohol manometer, which had a
direct vernier reading of 1/1000 ft. Fig. 6 is a typical plot of pressure
drop along the tubing. The measurements lie nearly on a straight
line. _Each pair of peizometers gave consistent results, with only
slight differences between the top and bottom taps.

The. dynamic pressure was obtained by subtracting the static
pressure from the total pressure measured by a stagnation tube. The
several stagnation tubes used were made from hypodermic needles
with the inside diameters and the ratios of inside to outside diameter
given in Table I. The measurement of the total pressures was taken

1. 5 in. from the exit. To aid in precise positioning of the tube on

the rough boundary, a lamp was placed so that the light visible

12

 

 

Table I
Stagnation tube number 1 2 3 4
Internal dia. 10-2 in. 8. 55 6. 25 4. 65 3.10
Ratio 1. D. /o. D. 0.78 0.75 0.72 0. 64

 

between the stagnation tube and the boundary could be observed, and
the stagnation tube adjusted to equalize the amount of light visible
along the length of the tube. The static pressure was measured by
a static tube which had the same diameter as stagnation tube number 3,
with four small holes of 0. 020 in. in diameter, located eight outside
diameters from the tip of the hemispherical nose.

For each run the total pressure was measured by recording
values at eight positions around the periphery of the tubing except in
the vicinity of the junction of the two halves of the tubing. The average

of these eight readings was used in the calculations.

13
IV. RESULTS AND DISCUSSION

The experimental results for all four tubes on the smooth steel
surface are plotted in Fig. 3. They agree with Preston's experimental
results very well. Preston's empirical and Hsu's analytical equations
are plotted in the figure. On the basis of this agreement it was
assumed that the equipment and techniques used in this investigation
were satisfactory and could also be used on rough boundaries.

The dynamic pressure readings taken with the stagnation tube
resting on the sand-roughened boundary are quite consistent. However,
the readings taken at the eight peripheral positions were not exactly
alike. The differences can probably be attributed to the variation in
the roughness texture and, therefore, the position of the tube relative
to the zero datum. A variation in the velocity distribution is also
possible, but would be small and have less influence. The possible
positioning errors in the pressure readings with the different sand
roughnesses and different stagnation tubes are shown in Table II.

The errors are expressed in percent error at a 0. 90 degree level.
of confidence using the. t-test. 8 The table shows that the error
increases as sand roughness increases and decreases as the size of
the tube increases. The larger, and erratic, errors of the low
velocity runs reflect the effect of the least reading of the manometer.
Although an individual reading may be several percent in error, the

average should be reasonably accurate.

14

Table II. Typical Error in % in Pressure Reading

 

 

 

 

sandpaper no. 36 80 150 220
Velocity » H L H L H L H L
Tube no.
' 1 f2.o ' f2.8 f1.5 f4.9 lo.9 f2.9 ll.o f2.9
2 i2 1 f3 2 +1 7 +4.0 i0.9 +2 1 +1 8 +3 0
3 f3.3 f4.2 l2.4 f2.7 l1.3 f1.o t1.o l4.7
4 l4.5 f5.9 f2.l l5.3 t1.o lo.4 l1.8 f4.4

 

H: Highest velocity; L: Lowest velocity

Figure 7 shows that the value of f is somewhat larger than the
value obtained by Nikuradse for the same roughness ks. The following
reasons might serve to explain this:

(a) The nominal size of the particles on the sandpaper may not be
equivalent to Nikuradse's.

(b) The sandpaper used in this testing is of an open coat type, i. e.
there is no glue or adhesive covering the sand grains. The effective
roughness may be larger than that of a closed coat as used in
Nikuradse's test.

(c) There are spaces between the particles on the sandpaper, whereas
the description of Nikuradse's roughness would indicate the particles

were quite tightly packed.

15

(d) The sand used in this investigation is a crushed product of
aluminum oxide having sharper corners and more irregular shapes
than ordinary sand.

In Fig. 8, the ratio of the pressure to the shear as measured
is compared to the pressure-shear ratio obtained analytically for
fully rough flow as a function of the particle shear-velocity Reynolds
number. The plot of

P-P P-P
°/< °)

1' T a
O O

 

 

against log u*kS/v is the familiar Nikuradse harp and represents the
effect of a systematic variation in the velocity distribution in the
transition region. The analytical pressure-shear ratio is obtained
from Fig. 4 knowing a, ks, and t and assuming 7.0/1eS = 0.15, 0.20,

0. 25 and 0. 30 for sandpaper 36, 80, 150 and 220 respectively. Only
for the roughest sandpaper does the zo/ks value present a problem.
For the thinnest walled tube with the No. 80 sandpaper the pressure-
shear ratio would only change 10% for extreme values of zO/ks of 0.1
and 0. 5. The experimental data appear to be about 12% below the
analytic line for the fully rough flow. Perhaps the reasons mentioned
for the discrepancy in f values could also explain this discrepancy.
Another reason could be the assumption of the velocity distribution
since the pressure-shear ratio is quite sensitive to the velocity profile.

J. M. Robertson analyzed Nikuradse's data and suggested the Karman-

l6

Prandtl velocity distribution should be written as u/u* = 5.6 log y/ks+8.39
instead of u/u* = 5. 75 log y/kS + 8. 5. If Robertson's values of C and
D are used in the analysis, the discrepancy between measurement and
analysis would be reduced from 12% to 7%. The data from the experi-

ments with the roughest sandpaper are plotted as

P-PO P-PO
1' /( ‘r )e
o o

 

 

against log u="ks/U in Fig. 9, where the denominator is the experimental
pressure shear ratio in the rough region. For this series of measure-

ments the tubes were kept in a fixed position and the rate of flow varied.
This could only be done for the experiments with the roughest sandpaper,
for with the other sandpapers the data did not extend into the fully rough
regime. This curve shows that the data obtained with the different sizes

of stagnation tubes do not show any systematic scatter.

17
V. CONC LU SIONS

Although certain difficulties remain, it would appear that
Preston's shear measurement technique can be used with rough
boundaries. The difficulties are due to our inadequate knowledge of
flow near rough boundaries, and may be simply stated as questions
of what is the correct form of the velocity distribution and where is
the zero datum.

The analysis which has been carried out using the commonly
used Karmén-Prandtl velocity distribution provides a relationship
which should give excellent qualitative values of shear, especially
when used in conjunction with the empirical correction values for
flow in the transition range. It would have been possible to achieve
an excellent correlation between analysis and experiment by assuming
other values for the coefficients C and D in the logarithmic velocity
distribution (or other forms of the velocity distribution equation).
This was not done because it would merely disguise the practical dif-
ficulty, which should be explicitly faced by anyone using this technique
for measuring local shearswhat is the "roughness" of any particular
surface, and what is the velocity distribution near this particular
surface ?

For many problems a qualitative, or relative, measure of the

local shear should be sufficient. For example, if one were to measure

18

the distribution of shear in uniform flow in a trapezoidal channel,
qualitative values could be corrected to agree with the total shear
obtained from slope measurements. For such problems the pressure-
shear ratio relations for fully rough flow presented in Fig. 4 and the
correction factors presented in Fig. 8, should be directly applicable.
For problems in which a quantitative measure of shear is

required, experiments such as performed in this investigation would
permit the development of correction factors similar to Fig. 8 for the

particular roughne s 5 involved.

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27

BIBLIOGRAPHY

SMITH, D. W. , and WALKER, J. H., "Skin-Friction Meas-
urement in Incompressible Flow, " NACA Technical Note 4231,
1958.

 

ABARBANEL, S. S., HAKKINEN, R. J., and TRILLING, L.,
"Use of a Stanton Tube for Skin-Friction Measurements, "
NASA Memorandum 2-17-59W, 1959.

 

LUDWEIG, H. , “Instrument for Measuring the Wall Shearing
Stress of Turbulent Boundary Layer, " NACA Technical Memoran-
dum 1284, 1950.

 

PRESTON, J. H. , "The Determination of Turbulent Skin Friction
by Means of Pitot Tubes, " Journal of the Royal Aeronautical
Society, Vol. 54, February, 1954.

 

HSU, E.’ Y.‘, "'The Measurement of Local Skin Friction by Means
of Surface Pitot Tubes, " David W. Taylor Model Basin Research
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