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NATL INST. OF STAND & TECH cations

A111DS T7MDAM

z

"^au of * NBS TECHNICAL NOTE 1151

A111D1 ^MSEST

U.S. DEPARTMENT OF COMMERCE/ National Bureau of Standards

l-QC

100

.U5753

No. 1151

1982
NATIONAL BUREAU OF STANDARDS

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NATIONAL BUMAt'

OF 8TAKDABDS

Surface Roughness Measurements of janTTw

Circular Disks and Their Correlation

with Hydrodynamic Drag

M6s

T. V. Vorburger

F. E. Scire

E. C. Teague

Center for Manufacturing Engineering

National Engineering Laboratory

National Bureau of Standards

Washington, DC 20234

Sponsored in part by: David 'laylor Naval Ship Research and Development Center

Annapolis. MD 21402

*4u

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Issued January 1982
National Bureau of Standards Technical Note 1151

Nat. Bur. Stand. (U.S.), Tech. Note 1151, 63 pages (Jan. 1982) CODEN:NBTNAE

U.S. GOVERNMENT PRINTING OFFICE

WASHINGTON: 1982
Abstract

The problem of relating hull roughness to the drag of ships

is a complex and important one in ship research. One of the

complications is that there are three fairly distinct roughness

regimes (microroughness, macroroughness , and structural roughness)

which make up a ship's surface and their relative importance is

not yet well understood. The present report focuses on stylus

measurements of the microroughness of rotating disks and their

significant correlation with drag measurements. In particular,

the roughest disks had drag coefficients Cm

that were 'V 30 percent

greater than those of the smoothest disks. The following empirical

formula was derived to relate Cm

with the roughness average Ra and the peak-count wavelength X pc at a Reynold's number of

1.5 x 106 .

C = b R /(X J 1 / 2

m

+ C , a' v pc 7 o»

where b = 3.35 ± 0.22 x 10"3 ym"1 / 2 and CQ

= 6.kQ ± 0.07 x 10" 3

.

The formula was observed to hold for both painted and bare metal

disks.

Key words: disks; drag; flow; friction disk; hulls; hydrodynamic drag;

rotating disk; roughness; ships; stylus; surface roughness; surface

topography.

iii
TABLE OF CONTENTS

PAGE

Executive Summary v

1. Introduction 1

1.1 Basic Ideas in Hydrodynamics and Ship Research .... 1

1.2 Scope of the Present Report 2

2. Surface Roughness Measurement 2

2.1 Roughness Regimes 2

2.2 Stylus Instruments 4

2.3 Instrument Bandwidth 6

2.4 Roughness Parameters 7

2.4.1 Amplitude Parameters 8

2.4.2 Wavelength Parameters 10

2.4.3 Functions 12

3. Experimental Details 15

3.1 Disks 15

3.2 Stylus Instrumentation 18

3.3 Stylus Measurement Procedures 18

3.4 Waviness Measurements 20

4. Results of the Tangential Surface Texture Measurements. . . 21

4.1 Roughness Average -Ra

21

4.2 Peak-to-Valley Roughness - Rmax 27

4.3 Peak-Count Wavelength - i\ pc 29

4.4 Other Parameters 30

4.5 Statistical Functions 30

5. Structural Roughness (Waviness) Results 34

6. Directional Measurements 36

7. Relating Drag and Roughness 40

7.1 Drag Measurements 40

7.2 An Empirical Model for Drag vs. Roughness 43

7.3 Final Working Model 50

8. Conclusions 51

9» Acknowledgments 52

10. References 53

IV
Executive Summary

The problem of relating hull roughness to the drag of ships is a complex and important one in ship research. One of the complications is

that there are three fairly distinct roughness regimes (microroughness,

macroroughness, and structural roughness) which make up a ship's surface, and their relative contributions to ship drag is not yet well

understood. The microroughness regime is caused primarily by machining

marks and paint texture; macroroughness is mainly due to inhomogeneities in the paint application, to corrosion, and to paint flaking;

and structural roughness consists of large-scale perturbations such as plate warping or metal joining imperfections. The present report

focuses on stylus measurements of the microroughness of test rotating

disks performed at the NBS and the correlation of these data with

drag measurements performed by the David Taylor Naval Ship Research

and Development Center (DTNSRDC). The report discusses the effects of

roughness wavelength as well as roughness amplitude on drag. The

key findings demonstrate that under the present experimental conditions

the roughness average Ra and the peak-count wavelength Xp C are related to the drag coefficient Cm

of the rotating disks as shown in

the figure. Thus, the roughest disks were found to have drag coefficients that are about 30 percent greater than those of the smoothest

8.5 xi<r 3

0.5 um v
disks. Based on a statistical analysis of the data, the following

empirical relationship between Cm , Ra , and ApC for a Reynold's

number of 1.5 x 10° was obtained:

Cm

= bR /(A,) 1 / 2 + c , m a pc o'

where b = 3.85 + 0.22 x 10"3 ym"1 / 2 and CQ

= 6.U8 + 0.0T x 10~3

.

The disks that were used for this study were all of uniform size and

shape, but their surfaces were produced by two different manufacturing

processes. One important result was that the above formula could be

used to predict Cm

for both sets of disks. The first was a set of nine

bare titanium alloy disks. Three of these were finished by a lapping

process and were quite smooth with Ra

' s ranging from 0.l4 to 0.17 ym.

The other six were finished by end milling with a V-shaped tool. They

exhibited roughness patterns that were grid like and highly periodic

with Ra

's ranging from ~ 5 to 16 ym. The second set consisted of

eight painted disks, all of which had random, gritty surface finishes

characteristic of the paints themselves. Four different combinations

of paint systems were used on these disks and they produced four

different surface finishes.

In addition to Ra and XpC , the other surface parameters and

functions computed from stylus measurements were the peak-to-valley

roughness R^x, the peak-to-valley waviness Wmax , the average slope

Sa , the root mean square roughness R

q

, the average wavelength X a , the amplitude density function, the autocorrelation function, and

the power spectral density. All of the results for the above parameters and typical graphs for the functions are given in the

accompanying report.

During the course of this work, we arrived at a number of conclusions as outlined below:

• At present , it seems that knowledge of one amplitude-sensitive

parameter and one wavelength-sensitive parameter is adequate for

characterizing increases in the drag of rotating disks due to

surface roughness.

• Rais the preferred amplitude -sensitive parameter because it is the

most widely used. We propose X pc as the wavelength parameter

because it is sensitive to the larger profile features rather than

to the fine structure.

• There is a need for all workers in the field to standardize on measuring the same parameters so that the results of one group can be more easily related to the work of other groups.

• The three dimensional structure of a surface is not so important in

the drag problem as the two dimensional profile along the direction

of flow.

vi
• The overall waviness (warp) of the disks does not seem to affect their

drag characteristics.

• Soaking the disks in salt water over the course of 2 weeks did verylittle to affect their Ra values.

• Both the painting and machining processes were fairly uniform since

the differences between the measured Ra of the front and back surfaces

or between pairs of the same set were not significant.

• The ratio between Rmax and Ra depended on whether the surfaces were

lapped, milled, or painted.

vn

1. Introduction

Surface roughness increases the hydrodynamic drag on ships.

Hence, a rough hull surface causes both a loss in ship speed for a given input power and a waste of fuel. This report discusses recent

state-of-the-art measurements of surface roughness parameters and their

correlation with results of drag experiments. It is part of a larger

investigation by the Navy to understand better the effects of roughness

upon drag and to improve the speed and efficiency of high-speed ships.

1.1 Basic Ideas in Hydrodynamics and Ship Research

The drag of a body moving through a fluid is the force required to

keep the body moving at constant speed. For a ship in water, the drag

consists of four components: skin -friction, form drag, wave making,

and air resistance [l,2]. The last three do not depend significantly

upon hull roughness. The skin-friction, on the other hand, is caused

by the tangential shear stresses exerted by the water on the moving

hull surface, and its magnitude depends strongly on the hull roughness

as well as several other factors including the speed of the ship, its

length and surface area, and the density and viscosity of the water.

As water flows past a ship there is a thin sublayer adjacent to the

hull in which the flow is laminar. As long as the surface roughness

heights are smaller than the thickness 6 Q of the laminar sublayer,

the ship is considered to be hydrodynamically smooth and the flow is

not affected by the roughness. However, for roughness heights greater

than 6 Q , the skin-friction increases rapidly with roughness. For

a large ship moving at a speed of 15 m/sec (30 knots), <$ is approximately T urn [3]. Typical ships, however, have peak-to-valley roughnesses an order of magnitude larger than this, resulting in perhaps a

kO percent or more loss of power with respect to that for a ship with

a smooth hull [l].

Hull roughness is increased by the corrosion and fouling processes

in the marine environment. To reduce these effects, paint coatings are applied to the hulls of both commercial ships and warships. In general,

anticorrosive (AC) coatings are applied first, followed by antifouling

(AF) top coats. Several factors, therefore, contribute to the roughness

of ships. First of all there is the basic texture of the coating itself.

Then there are factors in the coating process including poor surface

preparation and uneven application. Finally, the original coatings are

slowly degraded by corrosion and fouling. Indeed the marine environment

is so severe that the peak-to-valley roughness of commercial ships can reach 500 um or more [l]. However, even new coatings can have peak-tovalley roughnesses of U5 ^m or more, several times the thickness of a hydrodynamically smooth surface [k] . Therefore, the Navy's program to

improve the speed of ships seeks to develop coatings that will have

(l) smoother texture so that the condition of hydrodynamic smoothness

is approached, and (2) better AC and AF properties to reduce the

degradation of the surfaces of ships in service.
1.2 Scope of the Present Report

In this report we discuss the roughness of fresh coatings. One

question that needs to he answered is whether this part of the problem

is significant in view of the severe degradation prohlem associated

with corrosion and fouling. Unfortunately, because of the extreme com- plexity of the problem, it is next to impossible to predict the skin

friction of a ship simply from a knowledge of hull shape and surface

roughness parameters. On the other hand, experiments to measure the

drag of full-scale ships with various coating systems are prohibitively

expensive. Therefore, the Navy has taken recourse to small-scale

drag experiments using rotating disks with various painted and bare

metal surfaces. The experiments seek to correlate roughness parameters

of the disks (measured by NBS) with drag measurements (by DTNSRDC) in

a rotating disk instrument. It may then be possible to scale up the

results to predict the drag of real ships [ 5]

The focus of the NBS work is threefold:

1) to measure the roughness parameters of a series of circular disks

with various painted and bare metal surfaces,

2) to correlate these results with measurements of drag and, thereby,

to determine those roughness parameters important to hull drag, and

3) to develop adequate statistical procedures for the measurement and

analysis of roughness data.

Section 2 contains a general discussion of roughness measurement

and its application to hull surfaces. Sections 3-6 deal with the

NBS roughness and waviness measurements of the disks, and section 7

deals with the data analysis of these measurements and the correlation

with Navy drag measurements. Section 8 contains our conclusions and

recommendations.

2. Surface Roughness Measurement

The goal of surface roughness measurement in ship research is to

understand the complex relationship between the surface topography and

the skin friction well enough so that one could measure a few surface

topographic parameters on a ship and predict the effects on drag. In

the ideal situation, one would know which surface parameters to measure

and would have an instrument for making high-speed measurements over a relatively large area of the ship's surface, in situ , either in drydock

or underwater. There are, however, great limitations in our knowledge

of hull surface topography and its effects on skin friction and in our ability to make economical, thorough measurements. A short review of

the history and limitations of surface roughness measurement in ship

research is given in the following subsections.

2.1 Roughness Regimes

The roughness of ships may be divided into three regimes: microroughness, macroroughness, and structural roughness [6]. In general,

2
<g 100

3

- R max1 = 74"m R max2 = 66"m R max3 = 59"m

-100-

Black-Painted Disk P-11 Profile #1

3 4 5

Distance (mm)

Figure 1. Typical surface profile for a circular disk fabricated at

the Naval Ship Research and Development Center. The profile

has been arbitrarily broken up into 2-mm sampling lengths

and the maximum peak-to-valley-roughness Rmax has been calculated for each. The mean line is represented by the dotted

line.

each of these roughness regimes is caused by different processes and

each merits study independently of the others. The microroughness is

caused primarily by machining scratches and marks due to tool vibration

and feed, paint texture, or, in the case of replicas, the texture of the

material itself. The macroroughness is mainly due to inhomogeneities

in the paint application, corrosion, and paint flaking. Structural

roughness consists of large scale perturbations on the ideal form of

the object. For a ship hull, it includes plate warping, rivet heads,

large weld beads, and the edges of openings. For a rotating disk, the

structural roughness is primarily the warp of the disk, but it may

involve other long-scale errors of form as well.

These regimes may be roughly classified in terms of the characteristic distances between peaks and valleys along the surface. This

procedure is begun by Fourier decomposition of the surface profile

(e.g., fig. l) into sinusoidal component wavelengths. When this is

done for an object such as a large flat plate being tested for drag,

the resulting power spectral density (PSD) looks like that in figure 2

[?]• In general, the microroughness regime spans surface wavelengths

on the order of a few mm or less, the macroroughness extends from

several mm to several tens of mm, and the structural roughness extends

from there up to approximately several meters. In figure 2, the value

of the PSD at a particular wavelength is proportional to the square of

the amplitude of each Fourier component in the profile. It is important

3
Figure 2.

Power Spectral Density of a

20 m length of primer-coated

ship's plate resting in the

bottom of a flow channel. The

approximate wavelength ranges

of the microroughness, macroroughness, and structural roughness regimes are also shown.

The curve (taken from Ref. 7) was pieced together from measurements made with three

instruments: stylus instrument,

dial gauge and straight-edge,

and electronic level.

103 104 105 106 107 Wavelength (^m)

to note that the long-wavelength surface features are generally higher

in amplitude than the short-wavelength features [7]. It seems then

that a key problem in ship research is whether the smaller, more closely

spaced surface features contribute more or less to the drag than the

larger, more widely spaced features, i.e., whether one of these regimes

dominates over the others in affecting drag.

2.2 Stylus Instruments

The standard technique for surface topography measurement has been

the stylus technique [8]. This instrument (see fig. 3) uses a fine

diamond stylus, which is traversed over the surface, and a transducer

which produces a time-varying, electrical signal that, under the proper

conditions, accurately represents the undulations of the surface profile.

The resulting data may then be digitized, stored in a computer [8-10],

4
and conveniently analyzed to yield a wide variety of statistical parameters and functions [7-15] for characterizing the surface. Stylus

data are therefore accurate, quantitative, and analyzable.

Resolution and range are also strong features of stylus instruments.

The ultimate vertical resolution (given "by the rms noise of the instrument) can he as small as 0.1 nm [l6] and the range can be as high as 100 ym. The horizontal resolution is limited by the stylus tip width

[lT-l8]. It is typically several micrometers, but can be as small as 0.1 ym, more than acceptable for most topographic measurements.

However, the conventional stylus instrument has a number of shortcomings. The transducer and the stylus tips are fragile, so the instrument must be used in a fairly quiet, clean environment. A surface

profile measurement takes several seconds, a bit on the slow side for

some applications. Finally, the instrument generates two-dimensional

surface profiles rather than three-dimensional surface contour maps.

Therefore, the fraction of the surface area examined by the technique

is vanishingly small. Briefly then, the stylus is fragile and slow,

and under normal usage it produces only two-dimensional data. The

foregoing limit its usefulness for quality inspection of ship surfaces

in drydock and for measurements of ship hulls underwater.

Electrical

Signal -'WWW A/D

Conversion

Surface

.*.

• • • •

• • • • • • • •

• • • .

• • •

• • • Computer

Digitized Points

Figure 3. Schematic diagram of a stylus instrument for surface

roughness measurements
£ 100

0)

V)c

o

a(0

a> cc

CO

>» (0

5 10 50 100 500 103 5 104

Wavelength t^m)

Figure h. Typical response function for a stylus instrument. The

response is limited at lower wavelengths by the stylus

radius and at higher wavelengths by an electronic filter or by the trace length itself. Response curves for filters

with 0.8 and 8 mm cutoffs are shown.

The British Ship Research Association (BSRA) manufactures a portable, and more robust, stylus instrument called the electronic hull

roughness analyzer, which features a large vertical range for measuring

the extremely rough surfaces typical of ship hulls. A similar instrument, called the monotester, is manufactured in Norway. The drawback

of these instruments is that they output only one parameter, the peakto-valley roughness. Otherwise these instruments are every bit as slow as conventional stylus instruments. Since at least 50 profiles

are required for an adequate check of hull roughness [l.9]» the measurement process can be quite expensive.

There is then a need for an instrument capable of rapid, jLn situ

measurements. Optical, microwave, and ultrasonic techniques offer perhaps the greatest potential along these lines, but a great deal of

research and development must be done to establish the connections

between the parameters measured by these techniques and hull roughness

parameters.

2.3 Instrument Bandwidth

In all forms of surface roughness measurement, however, one must

take into account the approximate bandwidth of the probe, i.e., the

approximate range of surface spacings or wavelengths which it senses. Stylus instruments are typically sensitive to the range of wavelengths

extending from about 1 ym to about 10 mm. This range is limited at

the lower end by the finite size of the stylus tip and at the upper

6
end "by the sampling length, which in turn is usually determined by an electronic filter with specified cutoff length. A schematic plot of a typical stylus instrument's response vs. wavelength is shown in figure

k [15] • The curves show the characteristics of two filters available

with many stylus instruments. These have nominal cutoff lengths of 0.8

and 8 mm. In the absence of an electronic filter, the upper end of

the sensitive hand is limited ultimately by the traverse length of the

instrument, but that limit may be reduced by graphical division of the

profile into sections. For example, figure 1 shows a roughness profile

broken into 2-mm sampling lengths with the maximum peak-to-valley

roughness calculated for each.

The choice of sampling length greatly influences the measured

values of roughness parameters. This is particularly true because

roughness generally increases with surface wavelength [t! » as shown in

figure 2 and elsewhere [12], and the longest wavelengths within the

sampling length tend to dominate the roughness measurement. For example, the peak-to-valley parameter calculated in figure 1 is sensitive

neither to the larger features of the surface profile with wavelength

greater than 2 mm nor to the very fine features, but rather to surface

wavelengths on the order of a millimeter. The choice of sampling

length therefore implies a choice of which regime is most significant

to the function of the surface. In many industrial applications the

standard sampling length is 0.8 mm. However, for hull roughness

measurement a longer scale is required. The BSRA has traditionally

chosen 50 mm as the standard sampling length [l Q ]> but the present

report is primarily concerned with the finer surface texture. Therefore, since a standard electronic cutoff of 8 mm lies close to the

upper end of the microroughness regime and near the lower end of the

macroroughness regime, it becomes convenient to redefine these regimes

in terms of the sampling length. Henceforward in this report , the

microroughness will be defined as those surface structures which can

be measured by an instrument with a sampling length of ~8 mm, and the

macroroughness will be those structures with longer wavelengths which

can be measured using a 50-mm sampling length. This definition makes

the macroroughness regime equivalent to the regime traditionally

measured by the BSRA methods. Structural roughness will consist of

surface wavelengths greater than 50 mm, plus any imperfections caused

by metal joining, such as protruding fasteners or weld beads.

2.U Roughness Parameters

Once the surface profile has been measured, the question arises as to which of the many parameters most directly affect surface drag.

Although many different parameters have been defined to characterize

surfaces, we will introduce only those considered most relevant in the

present discussion. Additional information on surface parameters may

be found elsewhere [7-151 • There are basically three kinds of roughness parameters: amplitude parameters, wavelength parameters, and

hybrid parameters which are sensitive to both amplitude and wavelength.
2.U.1 Amplitude Parameters

In all forms of roughness measurement, amplitude parameters are considered to be most important. This is true in ship research as veil, since

there is a definite correlation between roughness amplitude and drag

[l]. Indeed, an often-used rule of thumb is that ships consume approximately 2.5 percent more power to maintain a constant speed for every

25 Vm increase in the roughness mean apparent amplitude (M.A.A. ) [l9l»

We discuss M.A.A. as well as some of the other more important roughness

parameters below.

a) Roughness Average-P^. This parameter is also known as the center line

average (CLA) or the arithmetic average (AA), but Ra

is now the internationally standardized nomenclature [15]. Ra represents the average

absolute deviation of the profile about its mean line, and it is defined

as follows

/

Ra

= (1/L) / |y(x)|dx, (1)

where x is the distance along the surface, y(x) defines the height of the

surface profile about the mean line (fig. l), and L is the sampling

length. For a digital instrument, like the one at NBS, which represents a surface profile by N equally spaced digitized points y^, the definition

becomes

N

Ra

= (1/N)^|yi |. (2)

i=l

Ra

is the most widely used surface parameter in the world, partly

because it is an easy one to calculate. All commercial stylus instruments that we know of, except the two manufactured specifically for

hull measurements, have direct readouts of the Ra value.

b) RMS Roughness - R

q

. This parameter, also known as o in the optics

and statistics communities, is defined by
L

R = |(1/L) / y

2 (x) dx

1 hi- (3)

or in the case of a digitized profile by

N

1/2

I^E'i2

]

R

q

=I(1/N) > y/| . (U)

i=l

Since R

q is equal to one standard deviation of the profile about

the mean line, a number of theoretical calculations of surface properties yield results which are naturally expressed in terms of Rq rather

than Ra

. It is therefore a parameter more widely used than Ra in

theoretical work. R

q is always greater than or equal to Rfl . c

Maximum Peak-to-valley Roughness - R^x* This parameter is defined as the height difference between the highest peak and the lowest valley

on the profile within the sampling length. When the surface profile

contains more than one sampling length (fig. l), the Rmax value is

generally calculated as the average of the results obtained for the

several sampling lengths. Discussion of this parameter poses the

important question of whether Raor R^ax is more significant to the

surface drag problem, i.e. whether the flow over the surface is more

sensitive to the highest peaks and lowest valleys or to some average

property of all the peaks and valleys. The BSRA and Norwegian instruments yield values for this parameter and none other. A schematic

diagram of a surface profile and the parameters Ra , Rq, and Rmax is

shown in figure 5. d) Mean Apparent Amplitude - M.A.A. This parameter was traditionally

used by the BSRA when manual methods were used to calculate roughness.

It was obtained by dividing the profile into 50-mm sampling lengths,

drawing envelope curves through the peaks and valleys of the surface

profile, and calculating the enclosed area per 50-mm sampling length.

With the advent of the BSRA' s own electronic hull roughness analyzer,

which yields R^x instead of M.A.A. , this parameter is probably obsolete.

e) Equivalent Sand Grain - Ks

. In the 1930' s, Nikuradse [20] performed

important experiments on the flow resistance of pipes which had been

roughened by cementing sand grains of various diameters to the walls.

Since his results for roughness were expressed in terms of grain sizes

Ks

, there has been some interest in expressing other forms of roughness

measurements in terms of the equivalent sand grain roughness [20-22] to

take advantage of his results for flow resistance. However, we do not

9
Figure 5. Schematic surface profile y(x) showing several roughness

height parameters and the amplitude density function (ADF).

pursue that approach in the present work.

f ) The Need for Standardization. From the above it seems apparent

that expedience has ruled the choices of roughness parameters in the

past. Theoreticians and statisticians tend to use R„. Nikuradse, on the other hand, probably chose Ks because that was the easiest parameter

to control and measure. The BSRA used M.A.A. when their calculations

were done manually, but now with the electronic instrument their parameter of choice seems to be R^x instead. Because the conversions

from one parameter to another vary from one type of surface to another,

it is difficult to compare the findings of different groups when they

have measured different parameters. There is a clear need for workers

in the field of surface drag to standardize on measuring at least one height parameter that is the same for all, and functionally it may not

matter a great deal which one. Ra

is currently the most popular with

engineers, and therefore, would enable workers in the field of ship

research to communicate their needs more easily with engineers and

scientists outside the field. There is, however, the possibility that

other parameters, particularly R^x* ma,y ^e more significant in

drag. In this paper we consider both Ra and Rmax as alternative

roughness amplitude parameters to be correlated with the results of

drag experiments on friction disks.

2.U.2 Wavelength Parameters

Surface wavelength effects on drag have not been studied as much

as roughness amplitudes. However, Schlichting [20] has shown experimentally that the spacing between asperities on the inner walls of

pipes affects the flow resistance of the pipes. There should be similar

effects due to asperity spacing on ship hulls as well. Three of the

wavelength parameters that can be used to characterize spacings are discussed below.

a) Average Wavelength - X a . Spragg and Whitehouse [23] first proposed

10
this parameter to answer the need in the manufacturing industry to

characterize spacings. It is defined by

X a = arVS^ (5)

where Sais the average slope of the surface profile (to be discussed

"below). The wavelength of a perfect sine wave is exactly equal to X a

as obtained from the above formula. For random surfaces,

X

a tends to be more sensitive to the shorter wavelength components within each sampling

length, whereas it is felt that surface drag should depend on the larger,

longer wavelength surface features.

The average slope Sa

is a hybrid parameter since it is sensitive

to both roughness and wavelength. We define Sa for a digitized profile

by

l+Pk

r) 5j l yj+k-y,il

j=l,l+k,l+2k,..

/, lyj*-yjl»

where t is the horizontal point spacing of the digitized profile and

kx is the horizontal point spacing that determines the resolution of

the calculation. The quantity kx is equal to an integral number of

point spacings of the profile itself and P is the total number of these

kx spacings in the profile. The numerical value of Sa depends on the

resolution of the instrument and on the spacing kx [8]. In the present

work, kx is chosen to be J. 6 ym, approximately equal to the stylus

radius.

b) Peak-Count Wavelength - XpC

. To address the need to measure

the longer wavelengths, we have defined the quantity XpC which is similar

to the peakcount parameter occasionally used in industry [2U]. As shown

in figure 6, X pc is defined as twice the trace length divided by the

number of times that the profile crosses completely through a mean band

centered about the mean line. The height of the mean band is rather

arbitrary. For the present measurements we have chosen a band height

equal to the measured Ra of the profile.

c) Autocorrelation Length - a. This parameter is a measure of the mean wavelength content of the profile. It can be obtained from the

autocorrelation function to be discussed in section 2.U.3.

Since XpC is felt to be a more relevant parameter to the surface

drag problem than Xa and since it is easier to calculate than a, we

11
Red-Painted Disk P-1

Profile #1

No. of Crossings = 18 2 (Profile Length)

DC — " " K No. of Crossings

I

"VjL_ J^J<1a_.A* TLA ii 1h /

=

_ 2x7.6 mm rt „,, = 0.844 mm

18 —\r is~~_JTJWJ_UL_

^"^pVU/V-- -V -^r/^X /-<-* \ .»* A . ji

i i i i i i i

-100

1 2 3 4 5 6 7

Distance (mm)

Figure 6. Typical profile trace shoving a sample calculation of A pc

obtained "by counting the number of times that the profile

is crossed by the mean band.

have used only X vc to characterize surface wavelengths in the present

report.

2.1+.3 Functions

In addition to the parameters discussed above, there are a number

of fundamental statistical functions which characterize a surface

profile in a more complete, though more complex fashion. These functions

may be used as a basis for calculating a number of the parameters

already discussed.

a) Amplitude Density Function-ADF. The ADF is a probability density

of surface heights. It is found by plotting a histogram of the profile

points in the vertical direction as shown schematically in figure 5« Ra , Rq, and Rmax can all be calculated from the ADF [9,15]

b) Autocorrelation Function-ACF. The ACF is one way of characterizing

both the wavelength and amplitude properties of a surface. It is a quantitative measure of the similarity between a laterally shifted and

an unshifted version of the profile. For a digitized profile it is

given by

12
N-k

ACF (fcr ) = -2_

N-k L—J Vy± yi+k. (T)

i=l

where x is again the point-to-point spacing of the profile and kr is

the shift distance. Typical autocorrelation functions for two painted

disks fabricated by DTNSRDC and autocorrelation lengths are shown in

figure 7. The ACFs were all normalized to have a value of unity at a shift distance of zero. This suppresses any amplitude information in

the ACF but allows a better comparison of the wavelength information in

the various profiles. The autocorrelation length a is derived from

the ACF and is yet another parameter for estimating spacings. It is

defined as the shift distance at which the value of the ACF (or its

envelope) drops to a certain fraction of the zero shift value, 0.1 and

O

<

o

a>

E

0.8 1.0 1.2

Shift Distance (mm)

Figure 7. Normalized autocorrelation functions for 2 painted disks.

P-T (new), P-T (used), -•-•-• P-15 (new),

P-15 (used). The autocorrelation lengths corresponding to the 0.1 point and the e~l point are also shown for

P-T (new).

13
e~l being two fractions that are commonly used (see fig. 7). Points on the profile that are separated by more than an autocorrelation length

may be considered as uncorrelated, i.e., portions of the surface represented by these points were produced by separate surface forming

processes. Autocorrelation lengths may range from the infinite correlation length of a perfectly periodic waveform to zero for a completely

random waveform [9l»

c) The Power Spectral Density-PSD. The PSD is another function for

characterizing both the wavelength and amplitude properties of a surface.

As discussed in section 2.1, it represents a breakdown of the surface

profile y(x) into its Fourier components. It is formally given by

PSD(f) = (l/L) /y(x) e -iarfx dx (8)

where f is the surface spatial frequency. The PSD is also the Fourier

transform of the ACF to which it is related by

L

PSD(f) = 4 lACF(s) cos2rrfsTrds. (9)

Several surface parameters may be calculated from the PSD by taking

moments of the form

00 /

M

n

fn PSD(f) df. (10)

1 /? 1/2

In particular^ is equal to M ' , and the rms slope is equal to Mp ' •

All of the above parameters are defined only in terms of twodimensional profiles. One could define a similar set of functions in

terms of three-dimensional surface maps. The question of studying three

14
dimensional topography is explored briefly in the present work. However,

it is expected that only the direction of flow is important in the drag

problem so that profiles, rather than maps, are adequate to characterize

the surface for drag. A study of weld beads on ships by Townsin, et al.

,

tends to support this viewpoint [k] .

Although we report the results of a large number of parameters and

functions for the surfaces of rotating disks, we will confine the

analysis to a few questions that we consider most important: Is Ra or

Rmax more significant in affecting drag? If neither has a predominating

effect, is there an approximate multiplicative factor that one can use

as a rule of thumb to relate Rmax and Ra for ship hull surfaces? What

are the relative effects of roughness and wavelength on drag, and can we

express these in terms of a simple functional relationship? These questions are addressed in section 7 after a discussion of the measurement

conditions and the results.

3. Experimental Details

This section describes the various rotating disks that were studied,

the stylus instrument, and the surface measurement procedures.

3.1 Disks

The. circular disks (fig. 8) were fabricated by DTNSRDC. They were

all the same size with a diameter of about 230 mm and a thickness of

about 3.2 mm, but they had varying surface textures produced by two different kinds of processes. There were eight painted steel disks (not

shown) whose roughnesses were dominated by the various paint textures and

nine bare, titanium alloy disks whose roughnesses were varied by varying

the metal removal conditions. The disks had five holes positioned near

the center for mounting in the NSRDC rotating disk apparatus. For all of

the disks, the front surfaces received the same surface preparation as the back surfaces.

The eight painted disks are described in table 1. There were four

different surface preparations. Disks P-l and P-9 received a vinyl

anticorrosive (AC) undercoat ing followed by a red antifouling (AF)

overcoat; their surfaces are designated as red/vinyl. Disks P-3 and

P-ll are identified as black/vinyl, disks P-5 and P-13 as red/epoxy,

and disks P-T and P-15 as black/epoxy. For each odd-numbered disk, there

was an identical even-numbered disk, i.e., disks P-2 and P-10 were red/

vinyl, disks P-U and P-12 were black/vinyl and so on. The odd-numbered

disks were measured for their surface finish in our laboratory, and the

even-numbered disks were measured for drag at DTNSRDC.

The outer edges of the disks were also painted, but in a few spots

the paint was chipped or nicked from the edges. This was not thought to

have a significant effect on the drag measurements, however.

15
Figure 8. Photographs of the Ti alloy disks [31]. a - smooth

lapped surface. Also shown in white are the tangential and directional patterns of stylus trace locations,

The directional pattern shows only 5 of 11 traces.

b - close Crosshatch pattern produced by end milling,

c - wide Crosshatch pattern produced by end milling.

The nine titanium alloy disks are also described in table 1. Three

of them are depicted in figure 8. The three smooth disks, T-l, 2, 3, were polished on a lapping machine to achieve a smooth surface, whereas

disks T-U , 5, 6, 7, 8, 9 were end-milled in two directions using a V-shaped tool to produce artifically rough surfaces. The depths of cut

determined the final roughnesses of the surfaces, and the resulting

surfaces had highly periodic, Crosshatch roughness patterns. The tool

16
Table 1. Descriptions of Disk Surfaces

Disk I.D.

Painted Disks

Coating Code Numbers and Descriptions

P-l

P-3

P-5

P-T

P-9

P-ll

P-13

P-l 5

117, 119, 121

Vinyl anticorrosive (AC ) undercoat followed

by Red antifouling (AF) overcoat (Red/Vinyl)

117, 119, 129

Vinyl AC followed by Black AF (Black/Vinyl)

150, 151, 15^, 121

Epoxy AC - Red AF (Red/Epoxy)

150, 151, 15^, 129

Epoxy AC - Black AF (Black/Epoxy

)

117, 119, 121

(Red/Vinyl)

117, 119, 129

(Black/Vinyl)

150, 151, 15^, 121

(Red/Epoxy)

150, 151, 15^, 129

(Black/Epoxy)

Disk I.D,

Ti Alloy Disks

Surface Finishing Process Tool Feed (mm)

T-l

T-2

T-3

T-U

T-5

T-6

T-7

T-8

T-9

Lapped

ii

End Milled with V-shaped cutter

iin

it

ii

0.6 (Close Crosshatch)

0.6 (Close Crosshatch)

1. h (Wide Crosshatch)

l.k (Wide Crosshatch)

1.3 (Wide Crosshatch)

0.7 (Close Crosshatch)

17
feeds shown in table 1 were determined by measuring the periodic grid

spacing with a ruler. The 3.2-mm wide outer edges of the titanium alloydisks were smooth.

3.2 Stylus Instrumentation

Surface topography measurements were taken with a Talysurf U

1 stylus

instrument interfaced to an Interdata 7/32 minicomputer 1 [9»10]. Using

an interferometrically measured step, the system was calibrated on each

value of magnification employed during a measurement and a calibration

constant was derived. Surface profiles were then taken with a stylus

having a tip radius of ~7«5 Mm and a stylus force of ~ 5x10"^ W.

The stylus speed was 1.52 mm/sec. Contrary to the normal procedure, the

electrical signal from the stylus transducer was not filtered. Rather,

the long-wavelength cutoff and the sampling length were limited by the

total trace length of 1,6 mm. It should be noted that having the cutoff

limited by the trace length is not a typical mode of operation.

Ordinarily the stylus instrument should have a trace length approximately

equal to or greater than three cutoff lengths. In the present experiment

it was important to include surface wavelengths up to several mm in the

output profile rather than to filter them out.

Surface profiles of the calibrating step and the roughness area under test were stored in the minicomputer memory as ^000 digitized

points after 12-bit analog-to-digital conversion. The sampling rate for

the roughness profiles was 1 point /l. 9 Mm over the 7*6 mm traversing

length. After a certain amount of data processing, the digitized numbers

were written onto magnetic disks for permanent storage.

In the early stages of this work, the first measurements on the

titanium alloy disks were made with an Interdata 3 computer rather than

the Interdata 7/32. The surface profiles taken at that time contained

U096 points rather than U000. In addition, the Interdata 3 did not have

the capability for permanent data storage on magnetic disks.

3.3 Stylus Measurement Procedures

Two modes of measurement were used to characterize the disk surfaces.

In the principal mode, profiles were taken along the tangential direction

at 10 equally spaced positions around the periphery of the disk,

approximately 11. 5 mm from the edge as shown in figure 8. The disks were

mounted on a rotary table which was positioned on the base table of the

stylus instrument (fig. 9) so that the disks could be rotated to a specified angle. As a rule, 10 measurements were taken on each side of

Certain commercial equipment, instruments, or materials are identified

in this report in order to specify adequately the experimental procedure.

In no case does such identification imply recommendation or endorsement

by the National Bureau of Standards, nor does it imply that the material

or equipment identified is necessarily the best available for the

purpose.

18
a disk. Each of the disks was measured in this way at least once. The

tangential measurements were felt to he the key ones "because they sample

the outer surfaces of the disks, which spin the fastest in a rotating

disk apparatus and have the greatest influence on the flow. In addition,

the tangential direction of the roughness profiles was close to the

direction of flow of the water over the surface in the spinning disk

experiment [25]. The direction of flow obviously figures most importantly in the interaction between the flow and the surface. The results

of these measurements are discussed in section k.

Tangential measurements were also used to test for surface

degradation of some of the disks. The Ti alloy disks were measured for

roughness soon after manufacture. They were then returned to DTNSRDC

for drag testing, including various experiments relating to marine growth

on the surfaces. After these tests were completed, the disks were cleaned

and their surfaces were remeasured in our laboratory.

The painted disks were handled differently. The new disks were measured for roughness and were then submerged in a saline solution for

extended periods in our laboratory. They were then dried and their

surfaces were remeasured for roughness. The results of the surface

degradation tests are discussed in section k along with the other

tangential measurements.

Figure 9» Photograph of a disk positioned on the

below the stylus instrument.

19

rotary table and
LVDT Probe

Rotary

Table

Figure 10. Schematic diagram of the waviness measuring instrument,

The positions of two of the hall hearing supports are

shown by the dotted parallelograms.

In the second mode, measurements were taken in 11 equally spaced

directions at one position on each of seven disks as shown in figure 8.

This was done to assess the possible directional properties of the surfaces. The center of the stylus trace was positioned by eye over the

axis of rotation of the rotary table, and the disk was mounted on the

table with its center located about 72.5 mm from the rotation axis. The

results of these measurements are discussed in section 6.

For both the tangential and directional modes, stylus profiles were taken three successive times at each position. The Ra was calculated for

each, and the average of the three Ra readings was recorded. Then the

digitized profile data for the third traverse were written onto magnetic

disk for permanent storage. The other parameters and statistical

functions were calculated from the stored profiles.

3.^ Waviness Measurements

During the course of the stylus measurements, it was noted that some

of the disks were obviously warped into a potato-chip shape. Since it was speculated that this warp could affect the drag properties, the

structural roughness (or waviness as we shall call it) of each of the

disks was measured around its entire periphery. The apparatus used for

these measurements consisted of an Ultradex rotary table, a Bendix height

20
gauge with an LVDT transducer, and a Hewlett Packard strip chart

recorder. A schematic picture of the setup is shown in figure 10". The

sensing element of the height gauge was cemented to a 2 cm x 1 cm x 1 mm

piece of flat copper sheet which served to filter out the roughness

structure from the waviness measurement. The system was calibrated on each value of magnification by traversing a gauge-block step with a height of 203 ym under the probe. Each disk was then placed on three

ball-bearings located 120° apart on the table and centered by a spindle.

The probe was positioned on the disk near the outer edge, and the table was set to rotating. The resulting 360° profile was output from the

height gauge and recorded. The results of these measurements for peakto-valley waviness are discussed in section 5«

h. Results of the Tangential Surface Texture Measurements

k.l Roughness Average Ra

The Ra results for the Ti alloy disks are shown in table 2, and those

for the painted disks are shown in table 3. Values for the average Ra from 10 measurements on both the front and back surfaces of the new disks

are given in column 2 and the averages of each pair of readings are given

in column h. These average values represent an important roughness

parameter for the disks, and they were used to characterize the disks in

subsequent calculations. After immersion, sliming, and cleaning, most

of the Ti alloy disks were measured again, and these results are given in

columns 6 - 9 of table 2. A scan of the numbers suggests that the new and used results are quite similar. This is not surprising, since the

processes of sliming and cleaning were not expected to remove any metal

from the surfaces. The Ti alloy disks revealed two regimes of roughness,

the lapped surfaces, T-l, 2, 3, being much smoother than the machined

surfaces, T-U...9.

Similar results for the painted disks are given in table 3. It

shows clearly that the pair of black/epoxy surfaces was the smoothest of

the set. The black/vinyl pair was rougher, the red/epoxy pair rougher

still, and the red/vinyl pair was the roughest. The red overcoat was clearly rougher than the black, but the surprise in these data was the

effect of the vinyl undercoat, which gave rougher surfaces than the

epoxy undercoat. Comparing the results in columns k and 8 suggests that

soaking in saline solution did little to change the roughness values of

the surfaces. The exception was the front surface of disk P-3 which,

for unknown reasons, became heavily corroded with soaking. The corrosion

consisted of green and rust-colored raised spots in large enough abundance that it was not feasible to make roughness measurements on the

surface afterwards. The roughness of the back of this same disk was measurable, however, and its R a value changed by only 9 percent with

soaking.

The uncertainties quoted in columns 3 and 7 of both tables represent

the sum of the estimated calibration uncertainty and one standard deviation of the mean for the series of readings. For a set of N numbers yj_

21
Table 2. Results of the Ra Measurements for Ti Alloy Disks. (The sources of the quoted uncertainty (u) are discussed in the text.

)

Disk

I.D.

T-5a

New

Front

Back

Ra U

ym ym

Average

R,Aaym

U

ym

Used

(After Immersion,

Sliming and Cleaning)

Front Average

Back

Ro U 1^ U

ym

vaym ym ym

T-l 0.176

0.168

+

+

0.0U7

0.027

0.172 + 0.0U7

0.155 1

o.i4o +

0.029

0.025

0.148 + 0.029

T-2a 0.1vr

0.132

0.026

0.022

0.139 0.026

0.117

0.139

0.124

0.145

0.020

0.037

0.024

0.030

0.131 0.037

T-3 0.176

0.154

0.039

0.028

0.165 0.039

0.133

0.139

0.025

0.024

0.136 0.025

T-4 9.5

10.5

1.1

0.8

10.0 1.1 — — — —

9.5 1.4

9.3

8.2

T7>T

1.4

8.8 1.4 8.8 1.4

8.2 1.4 9.5

8.4

1.4

1.4

T-6 17.3

15.0

1.7

2.2

16.2 2.2

15.7

16.9

2.5

1.9

16.3 2.5

T-7 13.6

12.7

1.1

1.7

13.1 1.7

13.5

12.8

1.5

1.7

13.1 1.7

T-8 9.5

8.5

0.9

1.3

9.0 1.3

9.3

10.4

1.1

1.1

9.9 1.1

4.87 0.64

T-9 5.20 0.64

4.53 0.53

aDisks T-2 and T-5 were measured twice after being slimed and cleaned.

22
Table 3. Results of the Ra Measurements for the Painted Disks. (The sources of the quoted uncertainties are discussed in the text.)

Disk I.D. New Used (After Soaking)

Front Average Front Average

Back Back

Ra U Ra U Ra u Ra

u ym ym ym ym ym ym ym ym

P-l ik.k + 1.0 ik.k + 1.1

(Red/Vinyl) 15.1+1.1 lU.6 + 1.1

15.9 1.1 1U.7 1.1

P-3 9.0 0.9

(Black/Vinyl) 9.0 0.9

9.0 0.8 9.8 0.T

9.8 0.7

P-5

(Red/Epoxy)

lU.5 1.1

12.5 0.. 13.5 1.1

lU.8 l.i

12.5 0.9

13.6 1.1

P-7 5.15 0.6k

(Black/Epoxy) 6.28 O.98

l.kO O.98

U.60 0.60

7.98 0.75

6.29 0.75

P-9

(Red/Vinyl)

Ik. 6 l.l

18.2 1.9

16. k 1.9

12.7 1.2

17.0 1.2

lU.8 1.2

P-ll 12.1 1.2

(Black/Vinyl) 10.5 1.2

8.8 0.8

12. k 1.1

8.8 0.9

10.6 l.l

P-13 12.8 1.0

(Red/Epoxy) 13.7 1.0

1U.5 1.0

13.2 1.1

13.8 1.0

13.5 1.1

P-l 5 ^.79 0.U6

(Black/Epoxy) 5.66 O.lk

6.52 0.1k

U.92 O.kf

6.33 0.6l

5.63 0.6l

23
with an average y, one standard deviation of the mean (l S.D.M.) was calculated by the following formula.

N

i s.d.m. = (i/K) f i E (y,--y)

2 1/2 [i £ . (11) (yi-7)2

]

1 ' [(nTTnH) i=i J

where the factor K is a slowly varying function of the sample size N.

The value of K varies between 0.797 and unity [26] ; for N=10, K is equal

to 0.973; for N=20, K is equal to O.987. For most of the surfaces, the

standard deviation of the mean was the larger of the two uncertainty

components.

The calibration uncertainty had a systematic part and a random part

and was attributed to six sources of uncertainty. The systematic part

was an estimate of the differences between a value for Ra obtained with

the use of the NBS stylus /computer system and a value for Ra obtained

with an ideal system. These uncertainties resulted from those properties

of the measurement process which are fixed prior to and during the procedure of obtaining data. The random part described the variations in the

results of a measurement process during repetitions of the procedure to

obtain data.

The random uncertainty was attributed to four sources: (l) the

variations in the calibration constant due to the surface finish of the

calibration step; (2) the variations in the calibration constant due to

sampling and digitizing processes, software computations, and non- linearities in the stylus instrument transducer and in the interface

hardware; (3) the variations in the measured Ra values due to the processes mentioned in (2) for a fixed calibration constant; and (h) the

variation in Ra due to variations in the average slope of the unfiltered

profile. Sources of the systematic uncertainty were: (5) uncertainty

in the height of the calibrating step obtained from interferometric

measurements and (6) uncertainty in the stylus tip radius.

The uncertainty of each average result in tables 2 and 3 was

simply taken as the larger of the pair of uncertainties for the front and

back surfaces.

A number of comparison tests was made with the data of tables 2 and

3 to test for systematic effects of several factors on the surface roughness. First, the differences between the Ra values for new and used

disks were calculated to determine whether the surface changed significantly with use. Second, the differences between Ra values for the

front and back surfaces of the new disks were calculated as indicators

of how consistent the machining or painting processes were. Finally,

the differences in Ra between the members of the various pairs of painted

disks were calculated as further indicators of the consistency of

24
Table U. Comparison Tests: Average Absolute

Ra differences between the Various

Sets of Disks

Front-Back

ym

New - Used

ym

Differences

Within Pairs

Of Similar

Surfaces

ym

Ti Alloy

1-3

0.015 + 0.0^5 0.020 + 0.0U9

Ti Alloy

k-9

1.2 + 1.8 0.25 + 2.4

Painted 2.0 + 1.1+ 0.U + 1. 5 0.9 + 1.6

the painting process. The calculations were done by taking absolute

differences between corresponding members of a pair and then calculating

average values for all of the pairs in a set. The results are shown in

table 4, and an example of one calculation is as follows. The absolute

differences between the front and back Ra measurements for the new

lapped, Ti alloy disks (T-l, 2, 3) were 0.008, 0.015, and 0.022 ym.

The average absolute difference of 0.015 ym is shown in table h. The

uncertainty of this value was estimated by taking the quadratic sum of

the uncertainties in each pair and averaging over all the pairs in a group. For example, the estimated uncertainty in taking the T-l frontback difference was [(O.OU7) 2 + (0.027) 2

] 1 / 2 = 0.05^ ym, and the

average uncertainty for disks T-l, 2, 3 was (.054 + 0.034 + 0.048)/3 = 0.045 ym.

Since almost all of the average differences are smaller than the

estimated uncertainties, it seems likely that the machining and lapping

processes were fairly consistent from front to back, the painting

processes were fairly consistent from one disk to the next, and the disk

surfaces did not change significantly with use. The above conclusions were further supported by standard analysis-of-variance tests [27]. The

average difference of 2.0 + 1.4 ym between Ra values for the front

and back surfaces of the painted disks suggests that there may have been

some inconsistencies in the paint application between front and back

surfaces, but the difference does not seem to be very large.

25
Table 5« Results for the peak-to-valley roughness Rmax for all disks.

(The sources of uncertainty are discussed in the text. )

Disk I.D. New Used Average

urn urn ym

T-l - 1.83 + 0.23

T-2a - 1 -?6 + O.lUf 1#51 + 0#1T

1.65 + 0.19) ~

T-3 - 1.U7 + 0.16

T-U ^0.2 + 2.8

T-5a - 37.6 + 3.M 37.1+'+ 3.1+ 37.2 + 3-3

J T-6 - 83.7 +8.5

T-7 - 68.2 + 5.5

T-8 - 56.U + U.l

T-9 23. h + 1.8

P-l 103.3 + 6.7 96.5 + 6.5

P-3 60.5 + U.6 60.8 + U.5

P-5 90.6+6.9 96.7+6.^

P-7 i+3.^ + U.8 U0.7 + U.3

P-9 107.3+8.0 100.7+8.1

P-ll 69.0+7.5 70.2+6.7

P-13 93.7 + 6.2 93.1 + 6.0

P-l^O.U + 3.9 ^0.2 + 3.0

aDisks T-2 and T-5 were measured on two occasions after they were

slimed and cleaned.

26
U.2 Peak-to-Valley Roughness - R'max

The results for the peak-to-valley roughness R^x are given in

table 5» Once again, the differences between the new and used painted

disks seem insignificant. The new Ti-alloy disks were measured before

the data storage system was in operation and before the system possessed

the capability of calculating Rmax . Moreover, disks T-U and T-9 were not

available for surface measurement after they were slimed and cleaned.

Therefore, in order to obtain at least one full set of Rmax results, we

calculated values for Rmax of T-h and T-9 by hand using the original

profile traces obtained when the disks were new. From the Ra consistency

checks of table h t we did not expect significant differences in the Rmax

values between the new and used disks, so it was not necessary to do the

hand calculations for all the new disks.

If the ratio Rmax /Ra were fairly constant, it would be possible

to propose an approximate rule-of-thumb for these surfaces to convert

Rmax results to Ra and vice versa. Values of Rmax/Ra are shown

in table 6. The average values can be used as approximate conversion

factors for each of the three types of surfaces. A truly random surface

Table 6. RmaX /Ra Ratios. (The uncertainty of each average value

is the standard deviation of the set of measurements.

)

Disk I. D. R,max' -"a /R, Average

Lapped

Ti Alloy

T-l

T-2

T-3

10. 6U

10.86

8.91

10 + 1

Machined

Ti Alloy

T-U

T-5

T-6

T-T

T-8

T-9

U.02

5. IT

5.21

6. 27

U.80

5.0 + 0.8

Painted P-l

P-3

P-5

P-T

P-9

P-ll

P-13

P-l 5

6. 8U

6.72

6.71

6.91

6.5^

6.57

6.8U

7.1^

6.8 + 0.2

27
with a Gaussian ADF would be expected to have a high Rjnax/^a ra/t io °f

~10 ; whereas, a truly periodic surface would be expected to have a Rmax /Ra ratio of only 3-*+ or so. These considerations agree with the

results of table 6. The surfaces produced by lapping, a fairly random

process, have large Rmax/^a ratios, whereas the surfaces produced by

the deterministic milling process have the smallest Rmax/^a ra

"t i°»

These differences are also revealed by the surface profiles shown in

figures 1, 6, and 11. There is clearly a difference between the lapped

surface profile of disk T-2 with its jagged peaks occurring at different heights and the machined profile of disk T-8 with its periodic

flat peaks.

The estimates of uncertainty in Rmax are derived from the same sources as those for Ra

. Since we had already tested the differences

between the front and back surfaces with the Ra results, no attempt was made to compare the Rmax measurements of front and back surfaces.

Consequently, the quoted values of Rmax are simply the averages of all

20 measurements on each disk surface. The random component of the

uncertainty was the estimated standard deviation of the mean (eq. 11 ) of

each set of 20 measurements.

Lapped Ti Disk T-2

Profile #1

1 i*m

i^yWH^W^

X

Milled Ti Disk T-8

Profile #1

1 2 3 4 5 6

Distance (mm)

Figure 11. Typical surface profiles of two Ti alloy disks.

28
k.3 Peak-Count Wavelength - X pc

The results for X_c are given in table 7. The lapped Ti alloy

surfaces have the finest spacings (X-p C ~ 0.1 mm) whereas the

machined surfaces have the widest spacings, resulting from the tool feed

of ~1 mm . The uncertainties of these values represent the sum of the

Table 7. A pc Results for All Disks.

Disk I. D. New Used

mm mm

T-l - 0.139 + 0.027

T-2 - 0.088 + 0.027) 0.102+0.027

0.115 + 0.025/ ~

T-3 - 0.090 + 0.012

T-U 0.92 + O.lU

T-5 - 1.16+0.18) 1.17+0.19

1.18 + 0.19/ ~~

T-6 - 1.88+0.27

T-7 - 1.61 + 0.15

T-8 - 2.10 + 0.30

T-9 0.899 + 0.080

P-l 0.88U+ 0.082 0.825+0.075

P-3 1.01 + O.lli 1.00 + 0.16

P-5 0.633+0.061 0.716+0.068

P-7 1.22 + 0.U0 0.87 + 0.11

P-9 O.96 +0.10 0.796 + 0.070

p-11 1.1U + 0.15 0.96 + 0.10

P-13 0.651+0.080 O.709+O.O85

P-15 0.7^5 + 0.091 O.769 + 0.083

29
estimated calibration uncertainty and the estimated 1 S.D.M. The

calibration uncertainty is simply equal to 3.5 percent of the X-p C value,

and it arises from the uncertainties in measuring the wavelength of the

periodic roughness artifact used to calibrate the horizontal travel on the stylus. The sample random uncertainty is the larger of the two

components. In the case of disks T-6 and P-7 , the large uncertainty was

due almost entirely to the sample random uncertainty.

h.h Other Parameters

The average results for the rms roughness R

q

, average slope Sa , and

average wavelength X a are shown in table 8. These parameters were not

used in any subsequent calculations to characterize the disk properties.

The Rq parameter measures essentially the same surface property as Ra and

yields very little new information. Sa and X a are sensitive to the

finer surface texture, whereas we expect that the surface drag of the

disks should depend more on the larger surface features with wavelengths

of ~1 mm.

k. 5 Statistical Functions

Some of the statistical functions calculated from the profile

measurements are shown in figures 7 and 12-lU. Each function is an averTable 8. Results for Additional Surface Texture Parameters

R

q

s a ^a

ym mm

0.020 0.0U7

0.020 O.OUl

0.021 0.0U1

0.10 0.58

0.1U 0.78

O.lU 0.60

0.12 0.55

0.28 0.31* 0.19 0.30

0.31 0.28

0.16 0.2U

0.28 0.36

0.22 0.31

0.31 0.27

0.17 0.21

30

T-l 0.21

T-2 0.17

T-3 0.17

T-H —

T-5 10.1

T-6 19.9

T-T 15.7

T-8 12.3

T-9 —

p-1 18.9

P-3 11. h

P-5 16.8

P-7 8.0

P-9 20.6

P-ll 13.0

P-13 17.2

P-1

5

7.1
0.6 0.8 1.0 1.2

Shift Distance (mm)

Figure 12. Normalized ACFs for Ti alloy surfaces. The curve for T-2

has been displaced downward from the other two for clarity.

31
J I I I I

' '

i

' I

I I I L

0.005 0.010 0.015

Spatial Frequency-f (^m~1

)

Figure 13. PSDs for three disks.

32
4

2

0.06

0.05

T 0.04

E

u_Q

< 0.03

0.02

0.01

-40 -30 -20 -10 10 20 30 40

Height (^m)

Figure ik. ADFs for the same three disks as in figure 13.

33
age of the functions calculated from the 20 surface profiles taken on the disk.

The ACFs for disk P-7 and P-15 (fig. 7) show a decay that is fairly

close to exponential, an observation which suggests that these painted

disks have a fairly random structure. Disks T-5,7 (fig. 12) have ACFs

with a very strong periodic structure, as expected. The curve for T-2

exhibits two decay rates, thus indicating two orders of surface structure. The fine microroughness structure yields the sharply decaying peak

with a width of ~50 ym,but there is also a longer wavelength waviness

structure in the surface profiles which yield the gently sloping part of

the ACF. Superimposed on both of these features is a fine periodicity

with a wavelength of ~12 ym. This latter feature is not due to any

real surface structure, but rather to a component of noise in the instrument which crops up at the very high magnifications required to measure

smooth surfaces like T-2. This noise component does not contribute significantly to errors in the Ra or Rmax measurements.

In figure 7, disks P-7 and P-15 are twins, and as expected, their

ACFs are quite similar. In addition, it is seen that their ACFs changed

very little with use.

The PSDs of figure 13 reveal both amplitude and wavelength information. The figure is plotted on a semilog scale spanning five orders of

magnitude to compare the smooth and rough surfaces. The periodic nature

of T-5 is once again revealed by the set of discrete peaks in its PSD.

The ADFs of figure lU are also enlightening. P-15 is a fairly random surface so its ADF is bell-shaped and resembles a Gaussian function.

T-5, however, is periodic so its ADF has two peaks at heights of

~+l8 urn in addition to the one at zero. T-2 is so smooth that its surface structures span only a very narrow range of heights; therefore, its

ADF looks like a spike when plotted on the same scale as the other two.

5. Structural Roughness (Waviness) Results

Subsequent to the drag and the roughness measurements, the structural

roughness of the disks was studied to measure the long-wavelength surface

structure, not accounted for in the roughness results, and, in particular, to test whether the obvious waviness of some of the disks affected

drag. Since the waviness could vary significantly from one disk to the

next, it was necessary to measure the very disks that had been studied in

the drag experiments. Consequently, waviness profiles were made on six of

the even-numbered, painted disks, in addition to their odd-numbered

twins, and the titanium disks as well.

The measurement procedure was described in section 3.U. Some of the more interesting waviness profiles are shown in figure 15, and the results

for the peak-to-valley waviness Wmax are given in table 9» As expected,

Wmax is many times greater than the peak-to-valley roughness Rmax #

34
It should be noted that the profiles, and hence Wmax , depend somewhat

on the leveling procedure described in section 3.*+. When a disk is highly warped, one can rotate the disk with respect to the ball bearings on which it lies and get different profiles, and hence slightly different

results for Wmax . Disk P-9 was lying level on the ball bearings. Hence

the profile in figure 15 shows two peaks, which correspond to the two

flaps of the potato-chip shape. These are separated by about l80°, or

~ 360 mm around the circumference, and they have approximately equal

height. On the other hand, the profile for disk T-l suggests that T-l

was probably laid upside down with one flap of the potato chip pointing

down and the other flap lying level. Hence, the peaks are not of equal

height. Although it is possible to develop a suitable procedure to

define the leveling component in the waviness profile and correct for

it, this was not considered necessary in the present study. The results

are meant to be only an approximate indicator of the waviness of the

disks, say to within a factor of two. As we shall see in section 7, the

results are accurate enough to suggest that the waviness did not affect

drag. In this respect, the profiles for disks T-l and T-2 (fig. 15) and

the corresponding waviness values (table 9) proved particularly interesting. These disks had highly polished surfaces with similar roughness

values, but the waviness differed by almost an order of magnitude.

"3

X

1

90° 180°

Angular Position

270<

Figure 15. Waviness profiles for three disks. A complete rotation of

360° corresponds to Tl8 mm around the periphery of the

disks.

35
Table 9.

Disk I. D,

Results for the Peak-to-Valley Waviness Wmax

Disks.

Front Back

mm mm

0.1+7 0.42

0.05 0.07

O.3I+ 0.36

0.1+5 0.1+1

0.1+6 0.50

0.20 0.20

0.11 0.12

0.33 0.26

0.2U 0.3*+

0.88 0.80

0.82 0.99

0.9*+ 0.89

0.86 0.91

0.59 0.1+8

0.71 0.93

0.69 0.66

0.91 0.9^

of the Test

Front Back mm mm

T-l

T-2

T-3

T-U

T-5

T-6

T-7a T-8

T-9

P-l

P-3

P-5

P-7

P-9

P-ll

P-13

P-l 5

P-2

P-1+ P-6

P-8

P-10

P-12

P-lU

p-16

1.1 >1.0

1.0 1.0

0.93 0.87

1.0 1.0

1.1 1.1

1.1 1.1

0.99 1.0

aThis disk was not leveled by the three ball approach discussed in the

text. It was measured while lying flat on the rotary table.

The waviness measurements were made several months after the drag

experiments. The warp of the disks could have changed in the interim,

depending on how they were cared for. Thus, a more definitive experiment

on waviness would involve waviness measurements taken before and after

the corresponding drag measurements. This consideration does not seem

important, however, since no correlation between the waviness and drag

results was observed.

6. Directional Measurements

The original reason for taking the directional measurements was to assess whether the overall three-dimensional surface structure affects

the hydrodynamic flow over surfaces. After the project was begun, we realized that the direction of flow was the predominant one in the drag

problem. This view is evident in the work of Townsin, et al . , [ 1+ ] in

their studies of the effects of weld beads and plate edges on drag and

in the work of Allan and Cutland [28]. A second factor that reduced the

importance of the directional measurements was the difficulty of study

36
ing directional effects on drag in a spinning disk apparatus. Such

studies would involve fabricating spiral roughness patterns on the disks,

certainly an expensive operation, and it is not clear that any important

new information would be gained in the effort.

Therefore, the measurements described in this section represent a preliminary attempt to assess the sensitivity of our methods to the

directional properties of the surface topography, in case it becomes

necessary to measure these properties later on. Previous work along

these lines was done by Kubo and Peklenik [29] and Tanimura, _et al. [30].

As described in section 3 and shown in figure 8, measurements were taken

at one position in 11 directions ranging from 0° to l80° ? and therefore spaced by 18°. The lapped disk T-3, the machined disks T-5 and T-8,

and the painted disks P-l, P-3, P-9, and P-ll were measured in this way.

From the data, we plotted polar maps showing the variation of Ra and

the e~l autocorrelation length with direction. Since the results should

be insensitive to a reversal of the stylus tracing direction, the values

at each angle were also plotted at 9+l80°. In addition, the angles

0° and l80° represent the same direction, so the results at these two

angles were averaged and the average value plotted for both the 0° and

9<r 90'

0.2 /im

180"

20fim

180'

270" 270"

90« T-8

20 Mm

Figure l6.

Ra maps for three Ti alloy disks.

These maps are based on measurements taken between 0° and l80°

as indicated by the heavy dots.

The Ra result at each angle 9 was also plotted at 9+l80°. The curves are therefore symmetric

with respect to rotation of l80°.

37

180'

270*
180* 180'

270< 270*

180<

90(

180'

270*

P-11

(Black)

270(

Figure IT: Ra maps for four painted disks. The results are based on measurements taken between 0° and 180° and the curves are symmetric with respect to a rotation of l80°.

l80° directions. As shown in the past [29-30] , the autocorrelation map

seems to be more sensitive than the Ra map to directional effects in the

surface structure.

The Ra maps are shown in figures l6 and IT. The Ra map for the

lapped specimen T-3 (fig. l6) seems to show the anisotropics of the

surface. As shown in figure 8, the lapping process left a faint laypattern on the surface. Apparently on T-3, the lay direction was close

to the 0° - 180°

greatest at 6=90'

direction, since R a was smallest in that direction and

The Ra maps for the machined surfaces, T-5 and T-8, (fig. 16) reveal

striking anisotropics, as expected for these highly deterministic surfaces,

The Ra values for 9=90° are much smaller than those for the other

38
90*

180*

270*

180

270*

90<

180(

P-3

(Black)

180° ^

P-11

(Black)

270*

Figure 18: e~l Autocorrelation length maps for the same four painted

disks as figure IT. The curves are symmetric with respect

to a rotation of 180°.

directions, because in that direction the stylus was tracing along one of

the grooves and completely missed the surface peaks. Indeed, the character of the maps should depend on the position of the center of the stylus

traces relative to the peaks of the surface. If the center were positioned on one of the peaks, then the Ra map should be fairly isotropic.

If the center were positioned at the intersection of two grooves, the map

would show four clefts, separated by 90°, rather than the two shown here.

By contast the painted disks appear to be more isotropic. The Ra maps

(fig. IT) show much smaller variations from one direction to the next.

39
The e --'- autocorrelation maps are plotted in figure 18. These are similar to autocorrelation maps discussed by Kubo and Peklenik [29] and

Tanimura, et al. [30]. The lag distance at which the autocorrelation

function is equal to e~l (0.3769) i- s plotted versus direction. Results

are shown only for the painted disks. The ACF results for the Ti alloy

disks were obtained by the older computer system. On that system, the

ACF calculation did not include a calculation of the zero line, thereby

making it impossible to calculate the e~l length for the titanium disk

directional data. It is clear from figure 18, however, that the e~^- autocorrelation length plots are much more sensitive to anisotropies

than the R^ plots. Slight changes in the surface topography in certain

directions cause striking lobes in the e--*- maps. It should be noted

that we are only discussing the local surface topography near one point

of the surface. The painted surfaces seem to be quite isotropic on the

large scale, and averaging data over several positions should reduce the

anisotropies considerably. However, if real anisotropies exist on surfaces, the e~l autocorrelation maps should be much more sensitive to

them than the Ra maps.

In conclusion, then, the overall three-dimensional structure should

not be as important in the drag problem as the two-dimensional structure

measured along the direction of flow. If three-dimensional structure

were important, the e~l autocorrelation plot would be a sensitive way to

search for directional properties.

7. Relating Drag and Roughness

In this section we show that a significant correlation exists between

different measures of roughness and the drag of the disks, and we discuss

empirical models to predict the drag behavior from roughness measurements.

7.1 Drag Measurements

The drag measurements were made by Belt and Smith [31,32] , using

the DTNSRDC rotating disk apparatus shown in figure 19« This instrument measures the torque (2t) required to spin a fully immersed disk as a function of the angular rotational speed w (expressed in radians/sec).

The symbol t itself is used to represent the torque on a spinning disk

that is wetted on only one side. Each disk was mounted on the end of a vertical shaft in the housing shown, which was then filled with water.

Power was supplied to the shaft by the motor whose rotational speed

could be varied between 800 and 2200 rpm, and both the torque and the

rotational speed were measured simultaneously [3l]» At these speeds,

the flow over the outer edges of the disks was in the turbulent regime.

The torque measurements had a repeatability of about 0.8 percent. After

undergoing suitable experimental corrections, the torque data were con- verted to dimensionless drag coefficients Cm , and the rotational speeds 00 were converted to Reynold's numbers. The Reynold's number Re, also a

40
1-1/2 hp motor

RPM magnetic

• pick-up sensor

Figure 19* Photograph of the rotating disk apparatus [3l].

dimensionless parameter, is the most fundamental parameter for characterizing fluid flow. For rotating disks, Re is given by [33]

Re = ur^/v , (12)

where r is the disk radius (~llU mm) and v is the kinematic viscosity,

defined as

v = n/p (13)

where n is the viscosity and p the density of the fluid (water in this

case). The drag coefficients Cmwere expressed in terms of the measured

torque ( 2x ) by

41
Table 10. Results for the Drag Coefficient Cm interpolated

to a Reynold's Number of 1. 5X10-6. a

Disk C^

T-l 6.70X10-3

T-2 6.65

T-3 6.60

T-U T.82

T-5 7.3^

T-6 7.9^

T-7 l.kQ

T-8 7.16

T-9 7.08

P-2 8.65

?-h 7.55

P-6 8.55

P-8 7.20

P-10 8.75

P-12 7.30

P-lU 8.50

P-16 7.30

aData supplied by G. Belt, N. A. Smith, and D. R. Laster [31,32]

Cm

= 2x , (Ik)

l/2PW2

r

5

The results for the drag coefficient Cm were then interpolated to

the same Reynold's number of 1.5 x 10°, which corresponds to a rotational

rate of ~1000 rpm, to facilitate the comparison with roughness. In this

connection it is important to restate that, while all of the titanium

alloy disks underwent both drag and roughness measurements, the painted

disks required the comparison of drag results measured for the even- numbered disks with the roughness results measured for their odd-numbered

twins.

The results for Cm are shown in table 10 and are plotted as a function

of average roughness Ra

in figure 20. In spite of the deviations, the data

of figure 20 show a strong correlation between the drag coefficient Cm

and

the average roughness Ra

. The drag coefficient of the roughest disk is

30 percent greater than those of the smoothest disks, and this dependence

of Cm

on Ra appears to be fairly linear over the narrow range of the data.

42
T»2 An Empirical Model for Drag vs. Roughness

We now attempt to develop a mathematical model for the drag

coefficients as a function of roughness, and along this line three key

questions now present themselves:

l) What is the suitable form for such a model? That is, are roughness

height parameters the only important quantities affecting drag or are wavelength parameters important as well? The waviness results show that

8.5x10-3 -

E

o

i co

«^

*«— a>oo

O)

CO

15 ^m

Figure 20.

Roughness Average—

R

a °m

in

vs. Ra showing fitted straight line.

Ra are derived from Tables 2 and 3, column 5. The uncertainties

43
roughness wavelength must have some importance because the long wavelength, structural waviness properties seem to affect the drag very little, whereas the short-wavelength microroughness had a significant

effect. For example, the peak-to-valley waviness of disk T-l is quite

large. If we assume that the waviness profile is approximately sinusoidal, then we can estimate a value for the Ra of the waviness profile

based on peak-to-valley waviness measurements. For a sinusoidal profile,

Rais equal to the peak-to-valley height divided by it . Since the average

peak-to-valley waviness height is equal to ~U^5 pm (table 9), "we es- timate that Ra

is equal to 1^2 ym, an order of magnitude larger than

the Ra values measured by the stylus. Yet the drag coefficient of T-l is

almost identical to those of the other lapped disks; therefore, it seems that any effects of waviness height are mitigated by the fact that the

characteristic waviness wavelength ( ~360 mm) is much longer than

roughness wavelengths. We would like for the model to account for this

insensitivity of the drag coefficient to waviness.

2) Is Ra or Rmax a more important parameter for drag? We leave R

q out

of this discussion because it is evident from equations 1 and 3 that Ra and Rq are very similar parameters and that there would be little to

choose between them.

3) Can the same model be used to characterize both the Ti alloy disks

and the painted disks?

The above questions are handled by using the method of least

squares, a statistical technique for measuring the goodness of fit of a mathematical model to a set of data. Given a set of N data points yi(xj_)

that depict the relationship between a dependent variable y and an independent variable x and a model y(x) involving p estimable parameters,

a]_,...,ap, the best model is the one that minimizes the residual standard deviation RSD of the data points about the fitted curve. RSD is

given by

RSD =

N 2

' V1 (y(x) -yi(xi))

L

A

f — (

1/2

(15)

where D is equal to the degrees of freedom (N-p) of the fit. The smaller

the RSD, the more credible the fit. The statistical calculations for

fitting the data to the models were done using the computer software

package, DATAPLOT, developed by Filliben and described elsewhere [3^,35]

<

In order to answer the first question, we consider three empirical

models for the dependence of Cm on roughness and wavelength. These are

Model 1: Cm

= aRa

+ C ,

44
Model 2: Cm

= W*

a /U pc ) 1/2 + CQ

, and (l6)

Model 3: Cm

= cRa

/X pc + C .

These models assume a linear dependence of Cmupon Ra but account for the

effect of the peak-count wavelength XpC in different ways. In all of

them, however, the constant C Q represents the drag coefficient expected

for a hydrodynamically smooth disk where the roughness effects are so small as to be negligible.

Model 1 does not admit any wavelength dependence at all. It assumes

that the drag coefficients depend only on the heights of the surface as- perities without regard to their separation. This might be a good model

under the limited range of roughness and C mshown in figure 20, but it

clearly breaks down for the long-wavelength asperities of the waviness

regime. Model 3 assumes a linear dependence of Cm

on the quantity Ra

/X-p C

< Since this quantity can be thought of as a measure of the surface slopes

over the sampling length of 1,6 mm, the model assumes that Cm

is primarily a function of the slope of the surface with respect to the average

flow direction. A plot of the data for Cm

vs. Ra

/X-pC is shown in figure

21, and indeed, the data seem to fit a straight line slightly better than

do the Cm

vs. Ra data of figure 20. Model 2 is a compromise between the

other two. It postulates a dependence of Cm

on both roughness and wavelength, but the dependence on roughness is the more significant. The Cm

data are plotted vs. R„/(X,) ' in figure 22. Here again, the data seem to approximate a straight line better than do those of figure 20.

The data were fitted to the three models in turn using the DATAPLOT

program previously mentioned. Each of the IT data points was weighted

equally in the fitting procedure, and, with 2 adjustable parameters for

each model, there are 15 degrees of freedom. Results obtained for a, b,

c, o and the residual standard deviations (RSD) are shown in table 11,

Table 11. Results of Fitting Three Models to the Data for Cm

vs. R . a

Model C Q a,b,c RSD

C_= xlO" 3 xlO -4 m

aR + C 6.53 + 0.16 1.15 + 0.15xl0

_i| ym" 1 3.28

£1 O — —

bRQ /(X J

1/2 + C 6.U8 + 0.07 3.85 + 0.22xl0~ 3 ym~1/2 1.60

cRa

/X pc + C Q 6.60+0.10 0.105+0.009 2.19

^he uncertainties are standard deviations obtained from the least

squares fitting routine [3^,35].

45
8.5x10- 3 -

E

o

C

Q) 'o

0)oO

(0

1.0 2.0x10-2

Rg/Apc

Figure 21: G^ vs. Ra Mpc with fitted least squares straight line.

The horizontal bar shows a typical uncertainty for Ra ApC

» The point W was obtained from the waviness data of disk T-l.

46
P10

8.5x10-3

E

o

c

0) 'o

a>oO

u>

CO

8.0 -

7.5

7.0 -

0.5 ^m

^a^^pc)

Figure 22. R /(X J 1 / 2 with fitted least squares straight

lines. The point W was obtained from the waviness data

of disk T-l.

m

47
and the best lines are the ones plotted as solid lines in figures 20-22.

Model 2 fits the data best since it has the smallest RSD. To get an idea

of how significant this result is, we calculated the F ratio [36], a statistic that can be used for comparing the fits of various models to

data. If, for example, we wish to compare Model 1 with Model 2, the

ratio F-^2 is given by

F

12 = (RSD1

) 2 /(RSD2

) 2 = k.2 (17)

Then with a value for F]_2 , we can use statistical tables [37] or computer

software statistics packages such as DATAPLOT [3^,351 to obtain the

probability Pp for F^2 to exceed 4.2, given the assumption that both

models are equally correct. Pp is a function not only of F but of the

degrees of freedom of the two models, in this case 15 for each. The

corresponding entry in the statistical tables is Pp(F, 15, 15)» For

Model 1 vs. Model 2, Pp is equal to 0.4 percent, a result which suggests

that Model 1 is an unlikely choice. To state it more specifically, if

Model 1 and Model 2 were equally good choices, there would be only a 0.4

percent probability that (RSD-^ 2 would exceed U.2(RSD2

) 2

. Model 2,

therefore, looks significantly better than Model 1.

There is little to choose, however, between Models 2 and 3. The

ratio F32 is 1»9« The corresponding Pp value of 11 percent suggests that

Model 2 is the likelier choice, but this conclusion get modified when one takes the waviness data of disk T-l into account. As before, we assume

that disk T-l has an effective waviness average Ra of 1^2 um and an effective XpC of 360 mm and that waviness affects drag according to Models

2 or 3. When we plot the C value of 6.7x10"^ for disk T-l versus

Ro/(^TM) an d RQ Ar^» we g^ the points labeled W on figures 21

and 22. Then, if these points W are included in the least squares fits,

the RSDs are almost equal, 2.3X10 "^ for Model 2 and 2.1xlO_I+ for Model 3«

Now, we turn to the question of whether Raor Rmax is the more

significant roughness height parameter for disk drag. We therefore

consider the models

Model 2a

Model 2b

Model 3a

Model 3b

Cm= bW)l/2 +CC

Cm

= b

l WtV^2 + Co>

(l8)

cm = c Ra/ Xpc + C o» and

^m = c l ^max' pc + ^o>

48
and we fit these to the data with the W points either included or excluded in the fit. For all of these cases, the RSD values are not

significantly changed by the use of ^max instead of Ra

; therefore, it

appears that Ra and Rmax are equally good parameters. Since Ra

is the more widely used of the two parameters, it seem that Models 2a and 3a are more convenient than 2b and 3b.

Lastly, we face the question of whether the painted disks and the

titanium alloy disks require different models or whether it is reasonable

to use a single model as we have been doing. This type of question has

been addressed directly by Neter and Wasserman (38], whose approach we

follow here. Using Model 2 for the moment, straight lines are fitted in

turn to the Ti alloy disk data only, to the painted disk data only, and

finally to all of the data. The results for the parameters are shown in

table 12. Then^an F statistic is calculated from the residual sums of

squares (RSS) of the three fits, where

RSS = RSD2 x D. (19)

The formula for F and the result of 1.68 are shown in table 12. To find

out whether this value of F is reasonable^we obtain the probability Pp

for obtaining a value of F larger than l.o8 assuming that the single

model with 15 degrees of freedom is the correct description. This probability corresponds to the entry Pp (1.68,2,13) [38] in the statistical

Table 12. Testing the Equivalence of the Painted and Ti Alloy Disk Data

for the Model:

C = b R /(X_J 1/2 + C, m a' ' pc

C Q

D RSD

xlO-3 x!0-U

6.57 + 0.06 T l.lU

6.32 + 0.19 6 1.89

6.U8 + 0.07 15 1.60

F = |RSS(all)-[RSS(P)+RSS(T)] }/{p(all)-[p(P)+D(T)lt = lm £Q

lRSS(P)+RSS(T)]/lD(P)+D(T)l

^he uncertainties are standard deviations obtained from the DATAPLOT least

squares fitting routine [3^,35]

49

Data Set b

Um-1/2

xlO-3

Ti Alloy (T) 3.33 + 0.28a

Painted (P) U.28 + 0.U6

All 3.85 + 0.22
tables. The result of 22 percent indicates that the use of a single

model for all disks is reasonable. The same hypothesis has been tested

using the Ra ApC model and the probability Pp is 39 percent.

7.3 Final Working Model

We are left with two reasonable models for describing the drag of

disks as a function of roughness. At present, there does not seem to be

any physical reason why one should be chosen over the other. Since the

R /(X )

' model fits the data better when the odd W point is

excluded, it perhaps should be chosen as the more reasonable one in the

microroughness regime we have been studying. It may be that subsequent

experiments will determine that the exponent of ApC is actually some fractional value other than 1/2. We have done some preliminary calculations along these lines, i.e. we have fit the data to a model of the

form Rg/U-nn) , where k is an adjustable exponent. The best values for

k vary between O.U and 1.0, depending on whether the Rmax or Ra data

are used in the model and on whether the painted or Ti disk data sets

are studied. Therefore, for the present, we relate the disk drag coefficients to roughness by the simpler model:

Cm

= [3 ' 85 R

a

/(X pc

)l/2 + 6.U8] xlO-3

. (20)

The above numerical results were checked by the inverse fitting

procedure, that is, the quantity R

a

/(^ r)C) was taken as the depen- / dent variable and Cm

was taken as the independent variable. Although

the relationship, R/ (A )' = a-,C + a , is awkward to interpret

physically, this procedure is more correct statistically because the

scatter in the R

a

/(X_ C

) data was significantly larger than the

scatter of 0.8 percent in the Cmdata, and the dependent variable should

be the one with the larger associated uncertainties. The best line

which results from this inverse fitting procedure is shown as the dashed

line in figure 22, and the fitted parameters are C Q

= 6.^3 x 10"" ^ and

b = U . 0U x 10~3 ym-1/2. Both of these are within one standard deviation of the original values given in equation 20. We could use the

later result as the final model, but it does not seem as physically realistic as the model given in equation 20 for the following reason. Disks

T-l, T-2, and T-3 were expected to be hydrodynamically smooth and the

close agreement in the measured drag results from these disks suggests

that the model should not dip too far below these points, and correspondingly, that the intercept should be close to 6.6 x 10~3. Hence, we conclude that equation 20 is the better model for disk drag vs. roughness

under the experimental conditions that were used.

The functional form represented by equation 20 is only one of many

that could be chosen to describe disk drag, and it is strictly an empirical

one. A more physically meaningful form might look like the following

50
Cm

= a(R

a-R )e-*pc/A +CQ

, Ra>RQ

, (21)

^m = ^o > ^alfto

»

where RQis the maximum admissible Ra for a hydraulically smooth surface,

CQis its corresponding drag coefficient, A is a horizontal length scale

factor, a is an adjustable parameter, and all of the aforementioned

quantities would depend on the disk Reynold's number. Such a model,

however, is probably too elaborate for the limited range of data taken

during this experiment.

The present model was derived for only a single rotational speed

with Reynold's number 1.5 x 10°, and at this rotational speed the outer

edges of the disk have a speed of approximately l8m/s (comparable to a ship speed of ~36 knots). Under these conditions, the maximum peakto-valley roughness for which the surface is considered to be a hydraulically smooth surface is estimated to be 5.6 ym according to a formula

discussed by Schlichting [39] • If this peak-to-valley roughness of 5«6 Mm

corresponds to Rmax , the Rmax/^a ra"tios from table 6 can be used to

estimate the maximum admissible Ra

for a hydraulically smooth disk under

these conditions. The value falls between 0.6 and 1.2 ym, a result

which suggests that at a Reynold's number of 1. 5 x 10" the smooth disks,

T-l,T-2,T-3, were hydraulically smooth, but the others were not.

8. Conclusions

Stylus measurements have been performed on a number of painted and

machined rotating disks, and from these a number of statistical parameters

have been derived for describing the microroughness surface topography.

We have tried to simplify the difficult theoretical problem of relating

skin-friction fluid drag to surface topography by postulating that the

drag coefficient Cm

is a simple function of the average roughness Ra and

a wavelength parameter which we call the peak-count wavelength X pc . The model,

Cm =bW)l/2 +Co> (22)

fits the data quite well and can be used to make quantitative predictions

of Cm for disks from a knowledge of these two roughness parameters.

In this experiment, a specific correlation has been drawn between

the roughness and the hydrodynamic drag of rotating disks. In addition,

the present experiment has modelled the effect of roughness wavelength,

as well as roughness amplitude, on drag and has formulated the parameter

X pc to characterize the wavelengths of random surfaces. Previous

51
studies of drag vs. roughness include those "by Townsin, et al » [ko] , Karlsson [Ul], and Musker and Lewkowicz [U2] on flat plates, and

that by Lackenby [l] on full scale ships, and the work of Nikuradse,

Schlichting and others [39l on rough pipes.

The empirical and theoretical limitations of the mathematical

model are considerable, however. First, the model was derived for only

one angular speed. Belt and Smith [3l] measured the drag of these

same disks as a function of speed, however, and their straightforward

data suggest that the model can be extended to a range of speeds fairly

simply with the addition of one or two parameters. Second, the study

was limited to the microroughness caused by machining marks and paint

texture. Even in this regime, the dependence of drag upon roughness

is quite significant. Although some measurements of waviness were

done to characterize the structural roughness of the disks, the relative

importance of the macroroughness regime should be studied by fabricating

and testing disks with significant macroroughness and comparing the

results with those of the present study. Finally, similar experiments

need to be performed on larger objects, such as plates and ships, to

determine how significant the effects of the microroughness are to the

operation of naval and commercial vessels in the light of the other

surface degradation effects on the macroroughness.

The theoretical problem is so complex that the proposed models are based on intuitive ideas about what the most pertinent quantities might

be, rather than on a fundamental knowledge of fluid dynamical processes.

Theoretical efforts probably should be concerned with relating the

characteristic roughness height and wavelength parameters to characteristic scale parameters of the fluid flow, such as the mixing length and

the scale of turbulence. As to surface topography, one can learn a great deal by characterizing the surfaces with two simple height and

wavelength parameters, like Ra and Xp C. At the present state of

knowledge, the key to the problem is not to find the optimum surface

parameters affecting drag, but rather to relate straightforward height

and wavelength parameters to drag by a valid physical model. Ra

is a convenient choice because it is the most widely used surface height

parameter. The quantity Xpc can be related to the peak count parameter used by engineers and surface metrologists. By contrast, the

traditional use of the parameters MAA and Ks is fairly restricted to

hydrodynamicists.

In future experiments, some analysis of the power spectral density

will be needed to sort out the relative effects of the microroughness

and macrogroughness regimes on drag, and eventually if the physical

models become more complete, it is possible that other statistical

functions may be needed to characterize the surface adequately.

9. Acknowledgments

The authors are grateful to H. S. Preiser and D. R. Laster for

their original suggestion of this work and for their support and encour52
agement along the way; to J. J. Filliben, S. Leigh, and K. R. Eberhardt

for their guidance with the statistical calculations; to B. Borchardt,

D. R. Flynn W. Liggett, and P. Purtell for their careful reading of the

manuscript; and to S. Morris for assistance in its preparation. We

would also like to acknowledge enlightening discussions with G. Belt,

A. Ticker, E. Neal, and N. Smith. This work was supported mainly

by the David Taylor Naval Ship Research and Development Center under

contracts N0016T-T8-WR8-208 and N00l68-79-WR7-0l83.

10. References

1. H. Lackenby, Proc. Instn. Mech. Eng. If

, 981 (1962).

2. D. R. Laster, H. S. Preiser, and E. Neal, private communication (Apr.,

1979).

3. H. Schlichting, Boundary-Layer Theory , 7th edition, trans. J. Kestin

(McGraw-Hill, New York, 1979).

k. R. L. Townsin, J. B. Wynne, A. Milne, and G. Hails, in Uth Int. Cong,

on Marine Corrosion and Fouling (Centre de Recherches et d' Etudes

Oceanographiques , Boulogne, 1976) p. 23.

5. P. S. Granville, Similarity-Law Characterization Methods for Arbitrary

Hydrodynamic Roughness , DTNSRDC Report 78-SPD-815-OI (Feb., 1978).

6. J. A. Malone, M. M. Murtagh, M. J. Ailman, K. D. Baetsen, and D. E.

Little, Ship Roughness Technology Program , Vol. I, Dept. of Commerce

Report MA-RD-930-8101U (Oct., 1980). Although we use the same terminology as those authors, we have redefined the three roughness

regimes slightly.

7. T. R. Thomas and R. S. Sayles , Mecanique Materiaux Electricite No.

337, p. 7 (Jan. 1978).

8. D. G. Chetwynd, Wear _57, 137 (1979).

9. E. C. Teague, Evaluation, Revision, and Application of the NBS Stylus /

Computer System for the Measurement of Surface Roughness , NBS Tech.

Note 902 (U.S. Dept. of Commerce, Wash., D.C., 1976).

10. T. V. Vorburger, E. C. Teague, and F. E. Scire, Dimensions /NBS _62,

No. 11, 18 (Nov. 1978).

11. D. J. Whitehouse, in Characterization of Solid Surfaces , ed. P. E.

Kane and G. R. Larrabee (Plenum, New York, 197*0 chap. 3.

12. J. Peklenik, Proc. Instn. Mech. Engrs. 182 Pt 3K, 108 (1967-68).

53
13. R. S. Sayles and T. R. Thomas, Trans. Am. Soc. Mech. Engrs. F: J.

Lub. Tech. 101, U09 (1979).

lU. D. J. Whitehouse, Pro. Instn. Mech. Engrs. 192 , 179 (1978).

15. American National Standards Institute BU6. 1-1978, Surface Texture

(American Society of Mechanical Engineers, New York, 1978).

16. J. M. Elson and J. M. Bennett, J. Opt. Soc. Am. _6£, 31 (1979).

17. J. Jungles and D. J. Whitehouse, J. Phys. E. _3, U37 (1970).

18. T. V. Vorburger, E. C. Teague, F. E. Scire, and F. W. Rosberry,

Wear _57, 39 (1979).

19* H. Hacking, Int. Ship Painting and Corrosion Conf. (197*0 p. 53.

20. Ref. 3, p. 6l5ff.

21. P. T. Manning, Wear _57, 365 (1979).

22. D. K. Brown, Proc. 5th Inter. Naval Corrosion Conf. (Defence

Scientific Establishment, Auckland, New Zealand, 1976) p. 7.1.

23. R. C. Spragg and D. J. Whitehouse, Proc. Inst. Mech. Engrs. 185, 697

(1970-1).

2k. Ref. 15, p. 27

25. P. S. Granville, J. Ship Res. 17, l8l (1973).

26. M. G. Nat re 11a, Experimental Statistics , NBS Handbook 91 (Dept. of

Commerce , Wash. , 1963) p. 1-10.

27. S. Leigh and J. J. Filliben, private communication (Nov., 1980).

28. J. F. Allan and R. S. Cutland, Trans. N.E.C.I.E.S. 72, 257 (1955).

29. M. Kubo and J. Peklenik, CIRP Annals l6, 235 (1968).

30. Y. Tanimura, E. C. Teague, F. E. Scire, R. D. Young, and T. V.

Vorburger, Bulletin of the Nat. Res. Lab. of Metrology (Japan)

30, 37 (1981).

31. G. S. Belt and N. A. Smith, Drag of Slimes and Smooth Surfaces as Measured by a Rotating Disk , DTNSRDC Report SPD-O865-OI (DTNSRDC,

Bethesda, MD, July 1979).

32. G. S. Belt and D. R. Laster, private communication (Nov., 1978).

33. Ref. 3, p. 105f.

54
3U. J. J. Filliben, in Proceedings of the Statistical Computing Section

of the American Statistical Association (San Diego, 1978) p. 3^»

35* J» J« Filliben, in Proceedings of the Twelfth Interface Symposium

on Computer Science and Statistics (Univ. Waterloo, Waterloo, Canada,

1979) p. W.

36. See, for example, P. R. Bevington, Data Reduction and Error Analysis

for the Physical Sciences , (McGraw-Hill, New York, 1969) p. 196.

37* See, for example, B. Ostle and R. W. Mensing, Statistics in Research , (Iowa State University Press, Ames, Iowa, 1975) P» 5^5ff.

38. J. Neter and W. Wasserman, Applied Linear Statistical Models , (Richard D. Irwin, Inc., Homewood, Illinois, 197^) V> l60ff.

39. Ref. 3, p. 659.

i+0. R. L. Townsin, D. Byrne, A. Milne and T. Svenson, Naval Architect

Sept. (1980) p. U59-

Ul. R. I. Karlsson, in Proc. Int. Symp. on Ship Viscous Resistance (SSPA,

Gothenborg, 1978) p. 9:1.

k2. A. J. Musker and A. K. Lewkowicz, in Proc. Int. Symp. on Ship Viscous

Resistance (SSPA, Gothenborg, 1978) p. 11:1.

55
NBS-114A (REV. 2-8C) U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

PUBLICATION OR

REPORT NO. NBS TN 1151

2. Performing Organ. Report No 3. Publication Date

January 1982

4. TITLE AND SUBTITLE

Surface Roughness Measurements of Circular Disks and Their Correlation with

Hydrodynamic Drag

5. AUTHOR(S)

T.V. Vorburger, F.E. Scire, and E.C. Teague

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

7. Contract/Grant No.

8. Type of Report & Period Covered

Final

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State, ZIP)

Sponsored in part by:

David Taylor Naval Ship Research and Development Center

Annapolis, MD 21402

10. SUPPLEMENTARY NOTES

J Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

The problem of relating hull roughness to the drag of ships is a complex and

important one in ship research. One of the complications is that there are three

fairly distinct roughness regimes (microroughness , macroroughness , and structural

roughness) which make up a ship's surface and their relative importance is not

yet well understood. The present report focuses on stylus measurements of the

microroughness of rotating disks and their significant correlation with drag

measurements. In particular, the roughest disks had drag coefficients C that were

*>-> 30 percent greater than those of the smoothest disks. The following empirical

formula was derived to relate C with the roughness average R and the peak-count

wavelength X % at a Reynold's number of 1.5 x 10 . pc

C = bR /(A ) 1/2 + C , m a pc o _} _i /? -3

where b = 3.85 + 0.22 x 10 urn i/z and C = 6.48 + 0.07 x 10 , was observed to hold for both painted and bare metal disks.

The formula

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

disks; drag; flow; friction disk; hulls; hydrodynamic drag; rotating disk; roughness;

ships; stylus; surface roughness; surface topography

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