Optically-Tailored Test Sections

for Pipe Flow Investigations

Ronald J. Hugo,∗ Benjamin J. de Witt†, and Haydee Coronado‡

Department of Mechanical and Manufacturing Engineering

University of Calgary

Calgary, AB, T2N 1N4, Canada

In this work, an optically-tailored contour shape was designed to correct for the opticaldistortions caused by a curved surface. This method can be applied to non-invasive imagingtechniques for fluid flow experiments. Software tools were developed to aid in the designof an optically tailored test section for pipe-flow experiments. Numerical models werecomputed for a standard acrylic pipe, inner diameter 57.15mm, with water enclosed. Acorrective optic prototype was machined on a 5-axis CNC machine, and polished with 3µm- 15µm diamond paste, alleviating any surface imperfections without significantly alteringthe surface. Experiments were performed to measure the emerging optical wavefront forthree test cases: (1) acrylic pipe with no enclosed liquid; (2) acrylic pipe filled with water;and (3) acrylic pipe/corrective optic assembly filled with water. The emerging opticalwavefront was slightly corrected when utilizing an optically-tailored test section.

I. Nomenclature

δ Off-axis displacement [mm]

G Gladstone-Dale Constant[

m3

kg

]

LED Lateral Effect Detectorλ Wavelength [nm]n Index of RefractionOCC Optical Contour CalculatorOPL Optical Path Length [mm]OPD Optical Path Difference [mm]OPLave Average Optical Path Length across entire [mm]

viewing aperturePIV Particle Image Velocimetryθ(x) Off-axis beam angle [radians]ID Inner Diameter of acrylic pipe [mm]OD Outer Diameter of acrylic pipe [mm]

II. Introduction

Ivestigations of industrial flow systems such as piping networks, heat exchangers, or reactor vessels of-ten rely upon non-invasive flow sensing methods for measuring velocity fields. Non-invasive flow sensing

methods are comprised of modern diagnostic tools such as particle image velocimetry (PIV), laser Doppleranemometry, optical tomography, or imaging with high-speed digital-video camera equipment. Curved sur-faces, such as transparent pipe walls, cause significant optical abberations that degrade imaging capabilities.

∗Associate Professor, Senior AIAA Member†Graduate Student‡Graduate Student

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36th AIAA Plasmadynamics and Lasers Conference6 - 9 June 2005, Toronto, Ontario Canada

AIAA 2005-4773

Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Figure 1(a) illustrates the degree of refraction caused by a pipe wall, where ray-tracing techniques1 havebeen used to compute light ray paths travelling through air into a transparent acrylic pipe filled with water.If non-invasive imaging techniques are to be used, modifications need to be made to correct for the opticaldistortions caused by curved surfaces.

A number of correction procedures currently exist that alleviate optical aberrations. One correctionmethod involves placing a pipe within a transparent box, filled with a fluid of the same index of refractionas the fluid within the pipe. Considerable improvements are made, as shown by the computed beam pathsin Figure 1(b). This method was used in a study done by Zhang,2 where Particle Image Velocimetry wasused to investigate an in-line vortex pipe flow.

Researchers at Physikalisch-Technische Bundesanstalt (PTB)3 crafted a pipe test section with outer glasswindows and a smooth inner wall contour using thin glass films 100µm thick to investigate pipe flows andinstallation effects using laser Doppler anemometry. A third correction procedure involves matching the indexof refraction of the fluid within a pipe to the index of refraction of the pipe-wall material. This approachwas used in investigations done by Uzol et al.4 and Agrawal et al.5 The latter investigation focused onthe problem of flow development in curved pipes, where glycerine and water were mixed to closely matchthe refractive index of Plexiglas. Once more, a ray trace shown in Figure 1(c) reveals the optical distortionfor this situation. Further enhancements can be made by applying digital image correction algorithms inpost-processing of image data,6 giving rise to yet another correction method.

1 1.5 2 2.5 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(a) Curved Pipe Walls Lead to Opti-cal Distortions (external fluid index ofrefraction = 1.0; pipe index = 1.49; in-ternal fluid index =1.33)

1 1.5 2 2.5 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(b) Fluid-in-Box Correction (externalfluid index of refraction = 1.33; pipe in-dex = 1.49; internal fluid index =1.33)

1 1.5 2 2.5 3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

(c) Index-Matching Correction (exter-nal fluid index of refraction = 1.0;pipe index = 1.49; internal fluid index=1.49)

Figure 1. Ray-Trace through a Pipe

An unconventional correction method involves custom machining an external shape to remedy the opticaldistortions that occur in pipe-flow investigations. Durbin et al.7 used this so-called “optical contouring”method for PIV experiments on a turbomachinery cascade where wall curvature effects were present, butnot as severe as for pipe-flow experiments. Optical contouring is an alternative correction method, with thepotential for application in non-invasive flow sensing experiments such as particle image velocimetry (PIV),laser Doppler anemometry, optical tomography, and imaging with high speed digital-video cameras.

The current paper will examine the performance of optical contouring to correct for refraction encounteredin pipe-flow experiments. Several objectives will be met in this study. First, software tools that utilize ray-tracing techniques1 will be developed that determine an external wall contour, correcting refraction due tothe curvature of a pipe wall. Second, a corrective optic contour shape generated from the software will bemanufactured and polished. Finally, verification of the design will be performed using a small diameter laserbeam and a lateral-effect detector to measure the laser beam’s deflection angle after it emerges from the pipe,with and without the corrective optic. The emerging optical wavefront will then be analyzed to determinethe effect of the optical contouring method.

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III. Theory and Background

There are four models that may be considered when analyzing the propagation of an optical signal throughan optical system: quantum optics; electromagnetic wave theory; scalar wave theory; and geometric

optics.8 Scalar wave theory considers the optical signal as an electromagnetic wave, whereas the geometricoptics model regards light as bundles of rays that travel in straight lines. These two models may be linkedthrough Huygens’ Principle.

A. Huygens’ Principle

Huygens’ Principle states that every point on a wavefront can be considered as a source of tiny wavelets thatspread out in the forward direction at the speed of the wave itself; the new wavefront is the envelope of allthe wavelets.9 This is illustrated in Figure 2, showing a wavefront travelling away from a source S. Thewavefront at a later time is shown to be comprised of a series of secondary wavelets defining the new positionof the wavefront. The notion of light rays follows from this concept, as being directed perpendicular to thesurface occupied by the wavefront at a given time.1 In other words, Huygens’ Principle maintains that lightrays are always perpendicular to the optical wavefront.

t

t+∆t

S

Source

Figure 2. Huygens’ Principle used to determine wavefront

Through a geometry exercise, the deflection angle of a light ray is equal to the local spatial derivativeof the optical wavefront, as shown in Figure 3. Hence, if a light ray is sent through an optical system, thedeflection of the beam would provide knowledge of the slope of the wavefront at that spatial location wherethe light ray exits the aberrated media. This is useful for the current investigation because knowing theoff-axis angle of a beam transmitted through an optical system, it is possible to quantify the distortion ofthe emerging optical wavefront.10

Original Planar Wavefront

Emerging Distor ted Wavefront

ρ

y

x

θ

θ

Figure 3. Linking Wavefront Distortion with Geometric & Scalar Optics Models

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B. Quantifying Wavefront Distortion

The spatial distortion of an optical wavefront may be described in two ways after it is has transmittedthrough an optical system: Optical Path Length (OPL) and Optical Path Difference (OPD). A wavefrontis the locus of all points at which the phase of vibration of a physical quantity is the same. Equation 1describes the OPL if the index-of-refraction field is known:

OPL =

∫ y2

y1

n(y) dy (1)

where n(y) is the index-of-refraction field along the path that the light travels. The index-of-refraction fieldcan be found by knowing the thickness and index of the material that the light is propagating through.With this field known, the OPL can be evaluated at a number of points in the viewing aperture of theoptical system. These points can either be along one-dimension, in which case the OPL is a one-dimensionalfunction OPL(x), or in two-dimensions, in which case the OPL is a two-dimensional function, OPL(x, z).

The OPD can be calculated by removing the spatial average of the OPL across the entire viewingaperture from the OPL itself. This is given as:

OPD(x, z) = OPL(x, z) − OPLave (2)

where OPLave denotes the spatial average of the OPL performed over the entire viewing aperture. AlthoughOPD is used to quantify the spatial distortion of an optical wavefront, the OPD is not equal to the opticalwavefront. Rather, the OPD is the conjugate of the optical wavefront. In order to minimize abberations theOPD must be zero across the viewing aperture. In effect, the wavefront is then planar. Knowing that theOPD of the emerging wavefront must be zero, the contour shape of the corrective optic may be solved foranalytically.

IV. Design Methodology of Corrective Optic

The outer wall shape required to correct for optical aberrations due to the curved surface of a pipe andan enclosed liquid may be computed from Snell’s Law of Refraction and ensuring that the OPD is zero

from the emerging wavefront. In this study, software tools were developed to predict and verify a correctiveouter wall optic.

A. Optical Contour Calculator (OCC)

The first stage in this study concerns the development of software to calculate the corrective optic. Afunctional breakdown of this algorithm is outlined in Figure 4 on the following page. Before proceeding, itshould be mentioned that re-examination of this analysis at a late stage of the investigation revealed thatlight ray deflections in the vertical direction were neglected; an incorrect assumption for the current problem.Hence, the computed contour shape from the following algorithm is only valid if a small angle approximationholds. Although the shape of the corrective optic is incorrect for transmission through a water filled pipe,the manufacturing and testing methods remain valid.

1. Details of OCC

The OCC performs a wavefront calculation, and computes the time it takes for an entire wavefront totransmit through the pipe. The computed time is then used to calculate the shape of the entire wavefrontacross the aperture. In the following derivation, the spatial coordinates x & y will represent the traditionalCartesian orientation as shown in Figure 5 on the next page.The following two equations describe all points that lay on either the inner or outer pipe wall:

x2 + y2 = R2i (3)

x2 + y2 = R2o (4)

and thus the thickness of the pipe wall in the horizontal direction at any y location is given by:

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Store optical

properties for pipe

Store geometric

parameters of pipe

Store info on

light source(s)

Define pipe in

cartesian

coordinates

Pipe

Geometry Optical

Properties

Output file

for CNC

machine

Define light source

in cartesian

coordinate system

Snell’s Law &

OPD to compute

contour shape

Define

index-of-refraction

fields

Light Source

Figure 4. Functional Decomposition of OCC

y

xR

i

Ro

Air - n1

Pipe - ni

Fluid- n2

Optical

Correction

Material - n3

(x,y)

∆

ε

δ

Figure 5. Geometry and Nomenclature for OCC

δ(y) = xo − xi =√

R2o − y2 −

√

R2i − y2 (5)

The thickness of fluid inside the pipe can be shown to be:

△(y) = 2xi = 2√

R2i − y2 (6)

Thus, the time taken for light to travel through a pipe of refractive index ni, filled with a fluid of index n2,can be computed to be:

τ =2δcni

+△c

n2

=2

c

[

ni

(

√

R2o − y2 −

√

R2i − y2

)

+ n2

√

R2i − y2

]

(7)

The longest time for the light to transmit is at the center of the pipe, and this time can be computed bysetting y = 0:

τmax =2

c[ni (Ro − Ri) + n2Ri] (8)

In time τmax the light at locations y 6= 0 travels further. Computing how far light travels in time τmax

results in the following, where the first term accounts for transmission through the media surrounding thepipe while the second and third terms account for transmission through the pipe:

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Γ(y) =c

n1

(τmax − τ) + 2δ + △

=2

n1

[

ni (Ro − Ri) + n2Ri − ni

(

√

R2o − y2 −

√

R2i − y2

)

− n2

√

R2i − y2

]

+2

(

√

R2o − y2 −

√

R2i − y2

)

+ 2√

R2i − y2

=2

n1

[

ni (Ro − Ri) + n2Ri − ni

(

√

R2o − y2 −

√

R2i − y2

)

− n2

√

R2i − y2

]

+2√

R2o − y2 (9)

where Γ(y) describes the shape of the optical wavefront after emerging from the pipe. The wavefront can be“zeroed” by removing the minimum wavefront across the aperture:

Γzero(y) = Γ(y) − Γmin (10)

The peak-to-peak wavefront distortion is computed by:

Γp−p(y) = Γmax − Γmin (11)

The wavefront will take a certain amount of time to transmit while it is “corrected.” This time will bereferred to as τcorr. During this time, the portion of the wavefront that is lagging the most (Γmin) will needto transmit through the thickest portion of high-index correcting material. At the same time, the portion ofthe wavefront that is leading the most (Γmax) will transmit through both surrounding media and high-indexmaterial. Therefore, τcorr can be computed using the following equations:

τcorr

c

n1

= Γ + τcorr

c

n3

Γ = τcorr

(

c

n1

−c

n3

)

τcorr =Γ

(

cn1

− cn3

) (12)

Relating this back to the optical contouring problem, the thickness of the material required to correct foroptical aberrations can be computed by:

ε(y) = τcorr

c

n3

Γ =Γ(y)

(

cn1

− cn3

) =Γ(y)

n3

n1

− 1(13)

The correction shape can then be described in terms of its inner profile:

xinner(y) =√

R2o − y2 (14)

and its outer profile:

xouter(y) = ε(y) +√

R2o − y2 + εoffset =

Γ(y)n3

n1

− 1+

√

R2o − y2 + εoffset (15)

where εoffset is a pre-determined machining tolerance. A number of contour points (337) were generatedfrom this algorithm, and the profile is shown in Figure 6 on the following page. These contour points wereimported into G-Code, a programming language that controls CNC machine tools and instructs the machinetool what type of action to perform.

B. Ray-Tracing Tool

Numerical tests were performed on this OCC computed contour shape by developing a Ray-Tracing Tool.The Ray-Tracing Tool is a program written to perform a ray trace analysis through an optical system calcu-lated by the OCC. This program is based on ray-tracing techniques1 detailed below. The program observeshow light refracts as it comes into contact with different materials.

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Corrective

Optic

ContourPipeID = 57.15mm

Figure 6. Profile of Corrective Optic Contour

1. Index of Refraction

The index-of-refraction of a fluid is a function of temperature. For instance, the index of refraction of air isdetermined using

n′

= n − n = G [ρ − ρ] (16)

where G is the Gladstone-Dale constant, a function of the wavelength of the optical signal:

G = 0.223 × 10−3

[

1 +7.52 × 10−15

λ2

]

(17)

If a HeNe laser, λ = 632.8nm, is used to generate the optical signal, then the refractive index of air at 1 atmis given as:

n = 1.0003 + 8.0208 × 10−2

[

1

T+

1

293.15

]

(18)

where T is the temperature in Kelvin.

2. Ray-Tracing Equations

Ray-tracing techniques are used to determine the path that a ray of light takes as it travels through mediawith index-of-refraction variations. The ray-trace equation1 is given by:

−→τ2 =n(−→r1)

n(−→r2)−→τ1 +

△s

n(−→r2)∇n (19)

where the gradient operator is defined as:

∇n =∂n

∂xi +

∂n

∂yj +

∂n

∂zk (20)

The terms used in Equation 19 and Equation 20 are graphically illustrated in Figure 7 on the next page. Anonzero gradient of the refractive index causes the ray direction −→τ2 to be nonparallel to −→τ1 . Knowing the raydirection at −→r2 allows one to approximate −→r3 = −→r2 + −→τ2△s, and so on.

3. Ray-Tracing Algorithm

Having formulated an optical system on paper, one can mathematically project beams of light through thesystem to evaluate its performance. A flowchart of this algorithm is given in Figure 8 on the followingpage. Testing was done for different optical systems including prisms, rectangles, circles, pipe geometry, andfinally for a pipe fitted with a corrective optic (calculated from the OCC). Predictions from the Ray-TracingTool were compared to Snell’s Law of Refraction and were found to be within +2%. The Ray-Tracing Toolsuccessfully predicts the path that a beam of light will take when travelling through an optical assembly,and was useful in verifying the final design of the corrective optic.

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O

y

x

r3

∇nr

2

r1

τ1

τ2

Figure 7. Symbol Conventions in Ray-Tracing

Begin Ray Trace

(Loop 1)

Begin Main

Compute position with reference to surfaces in optical system

Call function that assigns index of refraction

Calculate new ray direction

Output the position to file

Trace until certain

horizontal distance has

been reached

(Loop 2)

Declare variables

Open files for writing

Initialize variables & arrays

If horizontal

distance has not

been reached,

return to Loop 2

End of Loop 2

Close files

End Main

End of Ray Trace

(Loop 1)

Figure 8. Functional Decomposition of Ray Tracing Algorithm

C. Manufacturing

The functional specifications of the corrective optic must provide high transmissitivy/clarity ratings andshow little hazing. Table 1 shows the optical properties of some transparent materials. Standard acrylic wasdeemed the most appropriate choice of material for the corrective optic considering its optical properties,strength, and performance properties such as chemical and heat resistance, and ease of fabrication.

Material Transmittance Haze Refractive Index

Glass 92% 0 - 0.17% 1.52

Polycarbonate 86 - 89% 1-3% 1.586

Standard Acrylic 92% 2% 1.49

TYRIL SAN 87% 3% 1.57

Table 1. Optical Properties of various materials11

Contour points (337) of the corrective optic shape, generated from the OCC, were imported into G-code for simulation and machining on a 5-axis computer numerically controlled (CNC) machine, shown inFigure 9(a) on the following page. Manufacturing took place under a 0.79375 mm

(

132

”)

ball nose end mill,

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with A-9 lubrication oil [shown in Figure 9(b)]. The manufactured face then had to be polished to removesurface imperfections and opaqueness. Using diamond paste solution, the polishing stages were reducedincrementally from 15µm to 3µm. This significantly improved the quality of the finish, while minimizingalterations made to the contoured surface.

(a) 5-axis CNC Machine

Ball Nose

End Mill

(b) Close-up

Figure 9. Calibration Curves for Lateral-Effect Detector

A 3-axis coordinate measuring machine (CMM), accurate to +4µm, was used to assess the precise geom-etry of the corrective optic. A minute variation between the manufactured contour shape and the originalcalculated contour was discovered. The largest discrepancy between the two profiles was ≈ 0.127mm, shownin Figure 10. When the profiles were superimposed, it was revealed that they matched exactly, thus provingthat the discrepancy of the two profiles must have originated from the manufacturing set-up procedure. Thispossibly occurred when aligning the center of the prototype material in the CNC machine.

1.6 1.605 1.61 1.615 1.62 1.625 1.63 1.635 1.64−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Vertical (Y-axis) inches

Horizontal (X-axis) inches

Computed Contour

Manufactured

Contour

(CMM data points)

Figure 10. Computed Corrective Optic Contour compared with manufactured Corrective Optic (CMM datapoints)

V. Evaluation of Corrective Optic Design

The effectiveness of the corrective optic was determined by analyzing the distortion of the emergingoptical wavefront. As discussed previously, a light ray, after transmitting through an optically active

medium, will emerge with an off-axis angle θ. This off-axis angle will be equal to the local slope of theemerging wavefront. This measurement was performed using a small-diameter laser beam and a lateral-effect detector to measure the beam’s off-axis angle upon emerging from the pipe. The objective was todetermine the accuracy of the software in predicting the shape of the corrective optic and to quantify thewavefront distortion for three test cases:

1. acrylic pipe with no enclosed liquid, no corrective optic

2. acrylic pipe filled with water, no corrective optic

3. acrylic pipe/corrective optic assembly filled with water

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A. Testing Apparatus

A schematic of the optical bench layout is given in Figure 12 on the next page. Measurements of the off-axis angle were made with optical components mounted onto a single optical bench. The use of a singlebench reduced the possibility of unwanted vibrations between components that occur if the light source anddetector are on separate benches. The 63.5mm OD (57.15mm ID) acrylic pipe was placed on the translationstage and was capable of controlling one-dimensional movements of the pipe into and out of the plane of thepage, as shown in Figure 12 on the following page.

1. Lateral Effect Detector (LED)

The lateral-effect detector (LED) used in the experiment was a Graseby Optronics Model 1233(SC-10D)that has a 10mm × 10mm active sensing area. A LED is a large-area photodiode that can measure thecentroidal location of an incident laser beam. Since the LED is a photodiode, it will induce a current whenphotons come into contact with its surface. The resulting current will be proportional to the intensity of theincident light, and will also indicate the centroidal location of the incident laser beam. A transimpedance am-plifier converts the current into a voltage. The position of the laser beam can then be quantified and recorded.

The method of calibration for the LED consisted of placing the lateral-effect detector onto a x-z transla-tional stage (0.0254 mm resolution) and traversing the detector in both the vertical and horizontal directionwith the beam position fixed spatially. This resulted in a plot of the output voltage as a function of theposition of the translational stage. When performing an x-axis translation, the voltage corresponding to thez-axis position was nulled and vice versa.

−5 −4 −3 −2 −1 0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

6

8

10

Position (mm)

Ou

tpu

t V

olt

ag

e

X-axis Calibration

(a) Horizontal (X-Axis)

−5 −4 −3 −2 −1 0 1 2 3 4 5−10

−8

−6

−4

−2

0

2

4

6

8

10

Position (mm)

Ou

tpu

t V

olt

ag

e

Z-axis Calibration

(b) Vertical (Z-Axis)

Figure 11. Calibration Curves for Lateral-Effect Detector

The calibration curves are given in Figure 11. The calibration curves are seen to become non-linear nearthe largest displacements, and thus the data was fit to a cubic function. Equations 21 and 22 are equationsfor position of the laser beam on the LED surface as a function of voltage.

Horizontal (X − Axis) Position = −0.0367 · V 3 + 0.00184 · V 2 + 2.81 · V − 0.0679 (21)

V ertical (Z − Axis) Position = −0.0459 · V 3 + 0.00122 · V 2 + 3.08 · V + 0.0723 (22)

2. Light Source

The light source used was a 1mW Helium-Neon laser with λ = 633nm. It was not possible to experimentallyreproduce bona fide light rays, however the diameter of the laser beam used in this experiment (≪ 1mm)is a close estimate, and was used to characterize light rays. In connection with the experimental procedure,

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Laser

Source

Distance

Reference

Distance

Fluid Filled Pipe

Corrective Optic

Optical Bench

Lateral Effect

Detector

Translation Stage

PC

transimpedance

amplier

Figure 12. Experimental Set-up to verify corrective optic

the exact center position of the pipe was located by translating the pipe so that the laser beam transmittedthrough the central region of the pipe. Due to reflection of the laser beam off of the LED, it was possible toalign the position of the pipe such that the retro-reflection exactly coincided with the incident beam. Thisensured that the laser beam was passing precisely through the center of the pipe. Once the pipe was centered,the position of the emerging laser beam was recorded on a PC using a LabVIEW-based data acquisitionsystem. From this reference, the pipe was translated at 0.127mm increments while recording the position ofthe emerging laser beam on the LED. Since the LED was placed a known distance L away from the centerlineof the pipe, the off-axis angle of the beam can be calculated knowing the off-axis displacement, δ, of thebeam:

θ(x) = tan

(

δ

L

)

(23)

As mentioned earlier, θ was calculated for three test cases: (1) an acrylic pipe with no enclosed liquid; (2)an acrylic pipe filled with water; and, (3) an acrylic pipe/corrective optic assembly filled with water.

B. Wavefront Distortion

The geometry of the problem is defined in Figure 13. From Huygens’ Principle it is known that the off-axisangle of the beam is equal to the local spatial derivative of the optical wavefront.

Surface of

LED

L

θδ

Pipe

Figure 13. Wavefront Distortion Calculation

It can be shown that for θ(x) ≪ 1, the OPL is given as10 :

dOPL(x)

dx= −θ(x) (24)

The negative sign appears in front of θ(x) because the OPD and optical wavefront are conjugates. Integratingthis equation with respect to x yields:

OPL(x) =

∫ x2

x1

θ(x) dx (25)

This is an equation for the Optical Path Length at the location of the laser beam. The integral may beapproximated numerically with great accuracy, seeing that θ(x) was interrogated every 0.127mm. Using

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Equation 2 on page 4, the OPD can be calculated from the OPL, and hence, the emerging optical wavefrontcan be estimated. Figure 14 shows the shape of the emerging optical wavefronts across the entire ID of thepipe for the three test cases previously mentioned.

30 20 10 0 10 20 30

0.05

0

0.05

Central Axis of PipeOpt

ical

Wav

efro

nt (

mm

)

Position of Laser Beam (mm)Traversing across ID of Pipe

(a) Pipe (no water)

30 20 10 0 10 20 300.08

0.06

0.04

0.02

0

0.02

0.04

0.06

Opt

ical

Wav

efro

nt (

mm

)

Position of Laser Beam (mm)Traversing across ID of Pipe

Central Axis of Pipe

(b) Pipe (with water)

30 20 10 0 10 20 300.08

0.06

0.04

0.02

0

0.02

0.04

0.06

Opt

ical

Wav

efro

nt (

mm

)

Position of Laser Beam (mm)Traversing across ID of Pipe

Central Axis of Pipe

(c) Pipe & Corrective Optic (with water)

30 20 10 0 10 20 300.08

0.06

0.04

0.02

0

0.02

0.04

0.06O

ptic

al W

avef

ront

(m

m)

Position of Laser Beam (mm)Traversing across ID of Pipe

Central Axis of Pipe

Pipe and COPipe

(d) Superposition of Optical Wavefronts: Pipe with water,Pipe & Corrective Optic with water

Figure 14. Emerging Spatial Distortion

VI. Discussion

The emerging optical wavefront across the entire inner diameter (≈ 57mm) of the acrylic pipe are shownin Figures 14(a), 14(b), & 14(c). The trend in Figure 14(a) suggests the wavefront is diverging upon

exiting the pipe; in Figures 14(b) & 14(c) the wavefront is converging upon exiting the pipe, in agreementwith the ray-trace results shown in Figure 1(a) on page 2. This implies that the contour shape is alteringthe character of the wavefront. This is encouraging, however, the emerging wavefront is still far from beingplanar despite the small range of the vertical scale.

If errors had occurred in the machining of the corrective optic, they would have appeared as caustics(focused bundles of light rays). The smooth, clean quality of the data indicates that caustics are not an issue.It can be concluded that the machining process is not liable for the optical wavefront emerging distorted.

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The emerging optical wavefronts possess a similar fundamental shape, as shown in Figure 14(d) onthe page before, when Figures 14(b) on the preceding page & 14(c) on the page before are superimposed.However minute, there is a definite trend that the optically contoured correction method has had a positiveeffect. The large residual wavefront distortion can be ascribed to the previously described error in the OCC.A comprehensive inspection of all computations indicated that the small-angle approximation used in theanalytic solution for the contour shape contained the most significant error.

VII. Future Work

Most of the distortion occurs when light transmits from acrylic to fluid. A future direction for this workcould involve experimenting with combinations of contoured acrylic, and matching the refractive index ofthe fluid to the refractive index of the material of the pipe. For instance, a ray trace shown in Figure 15illustrates a ray-trace done for an acrylic pipe enclosed in a cube portion of acrylic, with the refractive indexof the fluid enclosed in the pipe matched to the refractive index of the acrylic5(n = 1.49). A second andmore immediate direction is to correct for the error in the OCC and manufacture and test a new correctiveoptic.

Acrylic

Water and Glycerinerefractive inde x

mat ched to acr ylic

Figure 15. Index-Matching Correction

VIII. Concluding Remarks

The focus of this work was to custom machine a corrective optic that corrects for optical aberrations.Experiments were done to measure the OPD, and hence the shape of an emerging optical wavefront in

three test cases. The data collected shows that for an optical assembly consisting of a pipe filled with waterand a corrective optic that the emerging wavefront is not planar. However, when compared to the emergingwavefront of a pipe filled with water, minor improvements were shown. As a result, optical contouring hasthe potential to be an effective method in reducing optical distortions due to a curved surface, provided thealgorithm used to compute its shape is correct. Although the final correction failed in the current problem,this paper has demonstrated the steps in the design process for manufacturing and designing a correctiveoptic for use in pipe flow investigations.

IX. Acknowledgements

This research has been made possible by funding through the support of a NSERC Discovery Grant.Thanks to T. Leung, J. Au-Yeung, S. Setiawan, E. Kang, & N. Danard for their hard work and dedicationwhile working on this project. Also, thanks to A. Vicharelli of Stanford University for her timely advice onmanufacturing & polishing.

References

1Klein, M., Useful Optics, The Chicago Lectures in Physics: The University of Chicago Press, 1970.2Zhang, Z., Particle Image Velocimetry (PIV) Measurements of an Inline Pipe Vortex , Master’s thesis, University of

Calgary, 2004.3Wendt, C., Mickan, B., Kramer, R., and Dopheide, D., “Systematic investigation of pipe flows and installation effects

using laser Doppler anemometry- Part I. Profile measurements downstream of several pipe configurations and flow conditioners,”Flows Meas. Insmrm., Vol. 7, No. 3 and 4, 1996, pp. 141–149.

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4Uzol, O., Chow, Y., Katz, J., and Meneveau, C., “Unbstructed particle image velocimetry measurements within an axialturbo-pump using liquid and blades with matched refractive indeces,” Experiments in Fluids, Vol. 33, 2002, pp. 909–919.

5Agrawal, Y., Talbot, L., and Gong, K., “Laser anemometer study of flow development in curved circular pipes,” Journalof Fluid Mechanics, Vol. 85, No. 3, 1978, pp. 497–518.

6Reuss, D., Megerle, M., and Sick, V., “Particle-image velocimetry measurement errors when imaging through a transpar-ent engine cylinder,” Meas. Sci. and Technol., Vol. 13, 2002, pp. 1029–1035.

7Durbin, P., Eaton, J., Laskowski, G., and Vicharelli, A., “Transonic Cascade Measurements to Support TurbulenceModeling,” Air Force Office of Scientific Research, Contractors Meeting in Turbulence and Rotating Flows, 2004, pp. 57–62.

8Welford, W., Useful Optics, The Chicago Lectures in Physics: The University of Chicago Press, 1991.9Giancola, D., General Physics, Prentice Hall Inc., 1984.

10Jumper, E. and Fitzgerald, E., “Recent advances in aero-optics,” Progress in Aerospace Sciences, Vol. 37, 2001, pp. 299–339.

11Weast, R., editor, CRC Handbook of Chemistry and Physics, The Chemical Rubber Company, 52nd ed., 1971.

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