14th int symp on applications of laser ecthniques to fluid...

14th Int Symp on Applications of Laser Techniques to Fluid Mechanics

Lisbon, Portugal, 07-10 July, 2008

Proposals for PIV of Near-Wall Flow over Curved Boundaries

Chuong V. Nguyen1, Thien D. Nguyen2

John C. Wells2, Akihiko Nakayama3

1: Dept. of Mechanical Engineering, Monash University, VIC 3800, Australia,

[email protected]

2: Dept. of Civil & Environmental Engineering, Ritsumeikan University, Shiga Japan,

[email protected]

3: Dept. of Science for Regional & Built Environment, Kobe University, Hyogo Japan,

[email protected]

Abstract: It is widely recognized that PIV measurements near a wall are dicult; problems includelow seeding density, high velocity gradient and the presence of strong reections and/or the appear-ance of rigid boundary that will create the anomalies in correlation map, hence biasing the detecteddisplacements and diminishing the reliability of image processing analysis. Implementing conventionalPIV techniques in this region require much care to increase the accuracy of detected velocity and wallshear gradient when encountering curved boundaries. This paper presents a near-wall measurementtechnique, named Interfacial PIV, which is a further improvement of our recent works on PIV, called In-terface Gradiometry (Nguyen & Wells, Journal of Visualization, V45, 2006) and its extension (Nguyen& Wells, FEDSM2006-98568, Miami, USA, 2006). In interfacial PIV, we stretch the PIV curved imageinto rectangle by means of conformal transformation. Then, one performs 1D correlation on each hor-izontal line of pixels within the template to produce a stack of 1D correlation curves. The wall sheargradient is measured by tting a straight line to this correlation stack. Analogously, the near-walltangential velocity prole is obtained by integrating the velocity gradient at each height up from thewall. In order to evaluate the capability of interfacial PIV and various PIV processing, synthetic imagepairs have been generated from a single DNS velocity of turbulent, recirculating ow over a sinusoidalbed. The velocity prole from application of interfacial PIV and Particle Image Distortion (Huang etal., Experiments in Fluids, Vol. 15, 1992) to synthetic images have been compared with DNS datain some circumstances. According results suggest that our method could attain about 0.18 sub-pixelaccuracy of velocity measurement. In addition, we introduce a new proposal for near-wall PIV, hereinthe conformally transformed images are rectangular, whence can be processed by any suitable PIVtechnique. The accuracy increase in wall shear measurement by our image transformation algorithmover the centroid shifting technique (Hochareon et al., Journal of Biomechanical Engineering, Vol. 126,2004) and by interfacial PIV versus PID from application to sets of synthetic images is demonstratedin this paper.

1 Introduction

The important roles of wall shear gradients and/or ow immediately adjacent to the wall invorticity generation, and in heat and mass transfer have attracted many researchers' interests.However, performing standard PIV measurements in the very near wall ows requires care toreduce the erroneous detected vectors. Hochareon et al [1] proposed a technique to re-locate thedisplacement vector to the centroid of uid region instead of the center of interrogation windowif it partly includes the non-uid region or wall boundary. Lowering the threshold value to searchlarge particles in cross-correlation map is a solution suggested by Huang et al [2]. However, this

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treatment raises other issues, such as augmenting the probability of inappropriately detectedvectors and diminishing the accuracy of peak detection. To overcome such diculties andimprove the accuracy of conventional PIV for high shear gradient regions, Huang et al [2]proposed Particle Image Distortion (PID) which distorts the template according to the velocityelds initially measured by standard cross-correlation to iteratively compensate the estimateddeformation. An increase in velocity resolution could be implemented by Theunissen et al [3]concept in which the image interrogation is rotated parallel to the boundary and stretched bya factor for excluding the interface from correlation process and taking into account a sucientnumber of tracers. The expensive cost of computing cross-correlation especially when one wouldlike to obtain high resolution velocity vectors, and the numbers of particles in template yieldingthe true displacement, limit standard PIV to the near wall measurements. In previous work, weproposed a technique [4], called PIV/ Interface Gradiometry (PIV/IG), to directly measurethe velocity gradients by shearing the PIV image templates parallel to a no-slip wall. Thisaords higher SNR and accuracy if considering the template height and the tracer concentrationis sucient to yield peaks in correlation map. In this paper, we reports an extension of PIV/IG,denoted herein by interfacial PIV, to generalize the technique to curved boundaries and tomitigate the ancestor' restrictions. In interfacial PIV, the wall identication is followed by aconformal transformation from curved templates to rectangles whose lower edge correspondsto the wall boundary. Then, one performs line correlation on each horizontal line of pixelswithin the template to produce a stack of 1D correlation curves. The wall shear gradient ismeasured by tting a straight line to this correlation stack. The near-wall velocity prole isthen determined by upward integration of the velocity gradient at each height within the stack,where this gradient is determined analogously to that at the wall. Independently, we introducea new proposal for near wall PIV to handle the complex boundaries in which the curved imageis conformally transformed into rectangle, then can be processed by any suitable PIV processingtechnique. Validation and investigation with synthetic PIV images of turbulent ow over a wavybed demonstrate the benet of conformal image transformation and capabilities of interfacialPIV to measure wall shear gradient and near-wall velocity proles.

2 Velocity Interpolation near Curved Walls

Building on our previous work [5], the new method called interfacial PIV consists of 5 steps:

a. Identifying the position of the wall boundary;

b. Transforming the near-wall image region to rectangular shape by conformal transfor-mation to handle the curved wall boundary;

c. Calculating line correlations between the 1st and 2nd exposures to obtain a stack ofcorrelation curves;

d. Measuring the wall shear gradient and deriving velocity prole from the correlationstack;

e. Reverse transforming of velocity prole and wall shear gradient to obtain physicalvalues.

Within this recipe, steps c and d are a more exible version of our PIV/IG technique [4]to generate an accurate prole of tangential velocity. Steps b and e introduce a new proposalfor near-wall PIV, and can be applied independently of c and d, i.e. can be followed by anysuitable PIV processing. Let us now describe each step in detail.

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2.1 Identifying Wall Boundary

The accuracy of wall identication strongly aects the accuracy of interfacial PIV techniquewhen the boundary condition is assumed to be no-slip [6]. Based on our experience so far,the following image processing, named Laplacian of Gaussian, seems to detect the wall mostaccurately in our experimental images. The raw images, rst smoothed by convolution witha Gaussian kernel, will be applied to a second-order method for detecting wall position. Inidentifying the wall, the tracers act as noise, therefore, the time-averaged sample of images isrst taken to suppress noise before image convolution. The time-averaged intensity distributionf(x, y) is then convolved with a Gaussian lter g(x, y) to yield a smoothed image:

s(x, y) = f(x, y) ∗ g(x, y) (1)

Next, to locate the boundary we evaluate the Laplacian of s(x, y). The raw wall positionsconsidered as local minima in the resulting image are then smoothed by low-pass ltering bya Fourier kernel. Fig. 1 gives an example of the detected boundary for experimental imagesof a sinusoidal wall in a water channel. As a matter of computational eciency, the equationbelow shows that it is equivalent to convolve the image f(x, y) with the Laplacian of Gaussianoperator instead of applying the Laplacian to the smoothed image s(x, y).

∇2s(x, y) = ∇2(f(x, y) ∗ g(x, y)) = f(x, y) ∗ ∇2g(x, y) (2)

Figure 1: Wall detection method. Detected boundary by Laplacian of Gaussian method(dashed), with closeup of white rectangle, overplotted on time-averaged experimentalimage.

2.2 Conformally Transforming Near-Wall Region To Rectangle

Next, we generate an orthogonal curvilinear grid [7] whose lower boundary lies on the curvedwall, and three other sides are chosen to enclose a region of xed height (100 pixels in theexample of g. 2a and b). Conformal transformation is applied to preserve the local angles,i.e. the orthogonality of the grids is maintained. In g. 2c, the intensities in the grid regionare then interpolated to produce the rectangular image. Such images can be processed by anystandard PIV technique. In the next section, a comparison of wall shear obtained by ParticleImage Distortion [2] on the original and transformed images, benchmarked by DNS data, showsthe benets of transformation.

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(a) Orthogonal grid generation

(b) Conformal transformation

(c) Rectangular template

Figure 2: Image transformation method. Orthogonal curvilinear grid overplotted on experimen-tal image and the near-wall region transformed into rectangular image by conformaltransformation. Given curved boundary (a) of near-wall region in (x, y) space, orthog-onal curvilinear grids (b) are generated; transformed near-wall image (c) is resampledby 2D interpolation.

2.3 Calculating Stack Of Line Correlations

Next, we apply a 1D form of PIV that assumes quasi-tangential ow, i.e. one where wall-normal displacement of tracers is less than the particle diameter, so that a purely tangentialsearch can produce correlation peaks. Denoting height and width of the transformed templateby (N,M), and pixel coordinates by (n,m), the line correlation is performed as follows; theintensity distribution I(m) along each line n on the rst template (of the 1st exposure) isnormalized, then cross-correlated with the line at the same height in the second image (2ndexposure), yielding the correlation coecient CU,n calculated by:

CU,n =

M∑m=1

[(Im+U,n − IU,n)(I′m,n − I

′n)]√

M∑m=1

(Im+U,n − IU,n)2

√M∑

m=1

(I ′m,n − I′n)2

(3)

where IU,n and I ′n are the mean intensities on each line of the rst and second template. Thecorrelation stack CU,n, an example of which is shown in gure 3b as a color contour map, isa function of horizontal interframe shift U and the wall normal position (vertical axis). Thepeaks in the correlation stack should normally lie at the horizontal position corresponding tothe tracer displacements.

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Figure 3: Velocity interpolation method. Color maps of instantaneous correlation stack formedby line correlation (equation 3) applied to a synthetic image pair, as overplottedby: (a) For n = 50 pixels, Gaussian-weight with σ = 10 pixels (white curve), andstraight line t to correlation stack (blue) that measures velocity gradient there, (b)velocity prole derived by interfacial PIV (white curve) and displacement vectors byPID with template height of 11 pixels (black vectors) compared to DNS prole (blackdash curve).

2.4 Integrating Velocity Prole

While Nguyen &Wells [5] identied strong peaks in a correlation stack and interpolated velocityproles, the present study introduces a new method of extracting velocity proles from thecorrelation stack by integrating the measured velocity gradient grad(n) upward from the wallone pixel at a time, i.e.:

u(n) = u(n− 1) + grad(n) (4)

where: n = 0, 1...N and u(0) = 0. grad(n) is chosen to be the slope of the straight linealong which the Gaussian-weighted sum of correlation values is maximal, i.e. which maximizesthe objective function of equation (5). Near the wall, the weighted sum includes values of CU,n

from negative values of n, specied by reecting the near-wall correlation stack around the pointy = 0, u = 0, i.e. CU,n = C−U,−n (cf. g. 3a). U(m) applied in CU,m depends on searching slopegrad via equation (6). The Gaussian-weighting function Ω is shown in equation (7). In g. 3a,the velocity gradient measurement at n = 50 pixels and Gaussian weighting distribution withσ = 10 pixels are overplotted on the correlation stack (top half) combined with its reectedimage (bottom half) to allow the no-slip condition at zero wall-normal position. Correlationvalues along a straight line (blue) rotating around the point A are accumulated with Gaussianweighting function Ω. The slope of line corresponding to the maximum correlation summation

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Figure 4: Reducing the eect of Gaussian weighting function on measurement results for thesynthetic image pair of g. 3. The detected velocity prole of interfacial PIV (whitecurve) is overplotted on the correlation stack and compared with DNS prole (blackdashed curve) (a) σ = 5 pixels (b) σ = 20 pixels.

is the measured velocity gradient. Compared to [5], the shear velocity gradient at the wall,which we refer to as wall shear or WS, is no longer separated from determination of thevelocity prole; rather it is simply the rst step in our integration process with point O xedto enforce no-slip at the wall.

F (grad) ≡

N∑

m=−N

CU,mΩ(m,n)

N∑m=−N

Ω(m,n)

(5)

U(m) = u(n) + (m− n)grad (6)

Ω(m,n) = exp

[−(m− n)2

σ2

](7)

A somewhat new aspect is the Gaussian weighting scheme, in contrast to the top-hat weighting-scheme in [5]. In the present limiting case of nearly-pure shear, our algorithm reduces to aone-parameter search over grad(n). Note that the step-by-step directional processing buildsin a reliable validation automatically. We have checked to conrm that the dierences due todownward integration are negligible. Another advantage of interfacial PIV over its ancestor,PIV/IG [4], is that various types of post-processing of correlation stacks can be tried withoutthe expensive re-computation of correlation tables.It is remarkable that the accuracy of interfacial PIV can be substantially improved by using

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the normalized version instead of the covariance stack. As seen from g. 3, the Gaussian kernelof width σ = 10 pixels yields a plausible prole that agrees well with the input snapshot fromDNS. The eect of Gaussian kernel width σ which may specify the seeding density and desiredresolution at measured point, however, has been diminished. A prole computed with σ = 5pixels, g. 4a, does not much suer in regions with low SNR from insucient particle imagesnear the wall in the correlation stack. On the other hand, referring to g. 4b, the prole ofσ = 20 pixels seems to be more reasonable that of σ = 5 excepting the region where owsalmost reach the freestream velocity.

2.5 Reverse Transformation to Obtain Physical Values

Horizontal position (pixels) with velocity profiles x 10 times

Ver

tica

l pos

itio

n (p

ixel

s)

10009008007006005004003002001000

500

600

700

800

900

0 4 .5 9 13.5 18 22.5 27 31.5 36 40.5 45

40.5

36

31.5

27

22.5

Horizontal position (mm) with velocity profiles x 10 times

Vertical position (m

m)

Figure 5: Detected proles of tangential displacement x 10 times by interfacial PIV processingfrom experimental image data on the background.

In the present interfacial PIV implementation for solid walls, only the wall-parallel componentis considered. The relationship between wall parallel component of the physical velocity up inlocal coordinate (x, y) with axis x aligned to the local wall position and horizontal velocity

component obtained in the previous step U can be described as up = U∂η∂y

. The local wall

shear gradient in the physical domain in obtained by∂up

∂y= ∂U∂η

∂η∂y

where ∂U∂η

is the wall shear

gradient in the transformed domain. Fig. 5 shows ten times of detected proles of tangentialdisplacement by interfacial PIV overplotted on experimental image of turbulent ow over awavy bed.

3 Testing with Synthetic Images

To evaluate the accuracy of various PIV processing, synthetic images have been generated froma single DNS velocity eld of turbulent, recirculating ow over a sinusoidal bed (Nakayama et al.[8]; DNS program by Yokojima [9]). Eight bit intensity distributions, with a maximum intensitylevel of 255, are computed following Lecordier et al. [10]: the tracers are randomly scattered intothe virtual 3-dimensional volume of the laser sheet over the wavy bed, and the rst simulatedexposure records the projection of the initial tracers' positions. After a delay ∆t, the secondexposure again captures the projected tracers as displaced according to spatial interpolation ofthe DNS velocity elds. In this algorithm, the integration of particle intensity distributed over

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the sensing area of CCD element, as parametrized by particle size (particle diameter), particleperpendicular position in laser sheet, and CCD sensor characteristics are used to compute thegrayscale intensity of tracer in synthetic images. By considering the background noise intensity,laser sheet thickness and the eects of out-of-plane particle movement, the resulted images havebecome more realistic than standard synthetic images.Two sets of images have been created. In the rst set, all the tracers are randomized in thecentral plane of the laser sheet, which yield the maximum grayscale intensities; and with theignorance of tracers' velocities in spanwise direction the number of out of plane tracers is zero,therefore the signal to noise ratio in the correlation stack is maximal. On the other hand, inthe latter set, by located in the virtual 3-dimensional volume of ow, the tracer intensity isdetermined by its position within the laser sheet.

3.1 Particle Image Distortion (PID) On Raw vs. ConformallyTransformed Synthetic Images

Applying the conformal transformation described in steps b and e of section (2) to the rstset of images, which corresponds to a delay ∆t = 5ms yielding the maximum displacementof particle about 12 pixels, the transformed templates have well-dened rectangular shape,therefore the PIV interrogation window can be located accurately along the wall boundary,hence eliminating the need for centroid shifting or similar techniques. However, the advantageof conformal transformation is not just in the simplicity of PIV interrogation.

Figure 6: Accuracy improvement by image transformation technique versus centroid shiftingtechnique. Wall shears from DNS data (square) comparing to those averaged fromforty synthetic images pairs by applying PID to raw images with centroid shiftingtechnique (circle) and to transformed rectangular images (triangle). Half-height oferror bar corresponds to standard deviation of measured values at each position.

A comparison has been implemented between the wall shears derived from applying PIDwith minor modication to untransformed images at curved-wall-embracing-grid-points, whichare extracted from the orthogonal curvilinear grid (cf. g. 2b), and those by applying PID totransformed templates at corresponding stretched grid points. Following Hochareon et al. [1],when we apply PID to untransformed images, the velocity vectors are shifted to the centroid of

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uid region if the interrogation window partly includes the non-uid region or rigid boundary.The inset in g.6 shows a close-up of wall shear measurement applied to untransformed imagesat the corresponding PID grid (red bullet) located near the wall in case the interrogationwindow invades the wall boundary. Wall shear stress is computed by dividing by the wall-normal distance from these centroid coordinates. Spacing of PID on transformed templatesare 5 pixels in vertical and 25 pixels in horizontal directions. The PID analysis programcross-correlates two 50Ö11 pixels interrogation windows with 50% overlap between horizontallyand vertically adjacent windows. The convergence of PID can be achieved by adapting theinterrogation window size suitably, i.e. the template height is decreased to 8 pixels close to thewall to handle the smaller displacements there; and a Gaussian-smoothing scheme analogousto step d is applied along streamwise and wall-normal directions to suppress the instability ofPID iterations.Fig. 6 shows wall shear averaged from 40 synthetic image pairs and its standard deviation at 38streamwise positions along the sinusoidal wall. Wall shears measured by PID with conformaltransformation (triangle) provides measurement results closer to the true value from DNS data(square) with smaller standard deviation than the results of PID with centroid shifting techniqueto untransformed images (circle). This shows the advantages of conformal transformation innear-wall ow measurement.

3.2 Interfacial PIV and Particle Image Distortion on TransformedSynthetic Images

In comparing wall shears by interfacial PIV and by PID on transformed images, forty syntheticimage pairs of the rst set used in g. 6 have been re-generated but with delay ∆t = 1ms,i.e. the maximum tracer displacement is 5 times smaller. The wall shears averaged from theseimages by PID (triangle) as in section 3.1, i.e. computed from velocity at the rst point dividedby its own wall-normal distance, and interfacial PIV (diamond) to transformed images are com-pared to those by DNS data (square) and overplotted on g. 7. An overall better agreementto DNS wall shear could be observed indicating the advantages of interfacial PIV in near-wallmeasurements.Forty synthetic image pairs of the second set have been created from the DNS velocity eldused in the former, ∆t = 1ms, to compare the measured PID and interfacial PIV particle dis-placement. In g. 3, the horizontal displacements from PID applied to transformed rectangularimage are measured at 5 pixels intervals, hence interpolation is normally required in case onewould like to achieve the displacement at all horizontal pixel lines. Computationally, it is muchmore tractable to derive the integrated velocity proles by interfacial PIV. Other interrogationwindow sizes have been tested to conrm that the PID displacements with a template height of11 pixels agree best with DNS proles, especially where the velocity is close to zero, as in thenear-wall region. The same template width of 50 pixels and horizontal search size of 8 pixelsused in PID have been conducted to interfacial PIV.The displacements uj

i obtained by interfacial PIV and PID from transformed images (cf. g.3) at 38 streamwise sections are ensemble-averaged from forty synthetic image pairs. Indexinglocation by j and image pair by i, each point in the g. 8 represents a local sample averagedened by:

uj =1

P

P∑i=1

uji (8)

In g. 8, the number of calculated points by PID with 5 pixel vertical spacing is about ve

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0 100 200 300 400 500 600 700 800 900 1000−0.06

−0.03

0

0.03

0.06

0.09

0.12

0.15

0.18

0.21

Horizontal position (pixel)

Wal

l she

ar (

1/fr

ame)

Wall shear by DNSWall shear by PID on transformed templatedWall shear by Interfacial PIV

Figure 7: Advanced wall shear measurement by interfacial PIV versus PID. Wall shears fromDNS data (square) comparing to those averaged from processing forty synthetic im-ages pairs by PID to transformed rectangular images (triangle) and by interfacial PIV(diamond). Half-height of error bar corresponds to standard deviation of measuredvalues at each position

times fewer than the those obtained over the height of the transformed images in our method.Velocity data from our present algorithm with a Gaussian kernel width σ of 10 pixels agree wellwith those of DNS although the 3D eects as out-of-plane tracer motion appeared in second setof images, which is practically impossible to correct with one camera view, cause the scatter ininterfacial PIV but less than the corresponding results from PID.

−1 −0.5 0 0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

1.5

2

2.5

3

Wall−parallel displacement by interfacial PIV (pixels)

Wal

l−pa

ralle

l dis

plac

emen

t by

DN

S (

pixe

ls)

Gaussian width σ = 10 pixels

(a)

−1 −0.5 0 0.5 1 1.5 2 2.5 3−1

−0.5

0

0.5

1

1.5

2

2.5

3

Wall−parallel displacement by PID (pixels)

Wal

l−pa

ralle

l dis

plac

emen

t by

DN

S (

pixe

ls)

Template height = 11 pixels

(b)

Figure 8: Point-to-point comparisons of velocity measurements averaged from forty syntheticimage pairs by interfacial PIV (left; σ = 10 pixels) and PID (right; template height11 pixels) to transformed images against those by DNS in case of tracers randomizedin the virtual 3D volume of the ow, spanwise velocity component included. Thestraight lines indicate zero error.

All velocity data by interfacial PIV shown in g. 8a correspond to a width of Gaussianweighting scheme σ = 10. The same process of extracting velocity prole from correlation

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stack has been carried out with various values of σ, and the resulting root mean square ofrandom error, ε, and of total error, Σ, is plotted on g. 9. The sample variance betweenindividual measurement uj

i and DNS data U j is spatially averaged over the forty image pairsyielding the mean square of total error Σ, and similarly for the random error ε, as follows:

Σ =

√√√√ 1

PQ

Q∑j=1

P∑i=1

(uji − U j)2 (9)

ε =

√√√√ 1

PQ

Q∑j=1

P∑i=1

(uji − uj)2 (10)

In equations (9) and (10), P , Q respectively correspond to the number of image pairs andspatial points from one pair. The total error Σ obtained for σ = 10 pixels is about 0.16 pixelsuggests the velocity prole adjacent to the wall by interfacial PIV could be attained in sub-pixelaccuracy.

0 5 10 15 20 250

0.05

0.1

0.15

0.2

0.25

0.3

Width σ of Gaussian weighting scheme (pixel)

r.m

.s o

f ra

ndom

err

or ε

and

tota

l err

or Σ

(pi

xel)

Random error ε, case of 2D laser sheetTotal error Σ, case of 2D laser sheet

Random error ε, case of 3D laser sheetTotal error Σ, case of 3D laser sheet

Figure 9: Root mean square errors of velocity measurements by interfacial PIV versus width σof Gaussian weighting scheme

4 Conclusions

This paper has presented an approach for extracting an accurate near-wall prole of tangentialvelocity from the correlation stack that is formed by 1D cross-correlation of pixel lines at thesame height of the rst and second exposures, then integrating the velocity gradient upwardfrom the wall. By this means the shear velocity gradient now is simply obtained from therst step in our integration process. The capability of our new method has been successfullytested with synthetic images from DNS of the turbulent, recirculating ow over a sinusoidalbed. The velocity data from application of the present algorithm to the synthetic images agreewell with those from the DNS data. Accordingly, these results suggest that by using correlationstack, the accuracy of velocity data by our proposed algorithm can be clearly improved overprior state-of-the-art for near-wall PIV. Additionally and independently, we introduce a newproposal for near-wall PIV that applies conformal transformation to stretch the curved imageinto rectangles so they can be processed by any PIV processing. This paper has not dealt withthe measurement sensitivity to the wall detection, which substantially aects the accuracy ofour PIV algorithm [6]. Future work will include extension to wall-normal velocity, combining

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the present innovations with the PIV/IG implementation of Phan et al [11] to obtain the 2Cvelocity prole near the wall.

5 Acknowledgments

The rst author would like to express appreciation to Dr. Frederic Plourde (Laboratoired'Etudes Thermiques, Poitiers, France) for commenting on PIV/IG presentation when he wasin France. We also express our special thanks to Nicolas Buchmann, Department of MechanicalEngineering, University of Canterbury, New Zealand by his advice on improving the accuracyof our proposal.

References

[1] Hochareon, Manning B., Fontaine A., Wall shear-rate estimation within the 50cc PennState articial heart using Particle Image Velocimetry, Journal of Biomechanical Engi-neering, August 2004, Vol. 126, p.430-p.43.

[2] Huang, Fiedler, H. E, Wang, J. J, Limitation and Improvement of PIV, Experiments inFluids, Vol. 15 Springer-Verlag 1993, p.263-p.273.

[3] Theunissen, R., Scarano, F., Riethmuller, M. L, On improvement of PIV image interroga-tion near stationary interfaces, Experiments in Fluids, Springer-Verlag 2008.

[4] Nguyen, C. V., Wells, J.C., Direct measurement of uid velocity gradients at a wall byPIV image processing with Stereo Reconstruction, Journal of Visualization, V45, p.5-p.27;adapted from Nguyen et al., Proc. Int. Conf. on Advanced Optical Diagnostics in Fluids,Solids and Combustion, Tokyo, Japan 2004.

[5] Nguyen, C. V., Wells, J.C., Development of PIV/Interface Gradiometry to handle lowtracer density and curved walls, Proceedings of FEDSM2006 European Fluids EngineeringSummer Meeting, FEDSM2006-98568, Miami, USA 2006.

[6] Nguyen, C. V., Nguyen, T. D., Wells, J. C., Sensitivity of PIV/ Interface Gradiometryto estimated wall position, Journal of the Visualization Society of Japan , Vol. 26, No. 2,p.203-p.206, Japan 2006.

[7] Denham, Charles R. (2000), SeaGrid Orthogonal Grid Maker for Matlab,http://woodshole.er.usgs.gov/stapages/cdenham/public_html/seagrid/seagrid.html

[8] Nakayama, A., Sakio, K., Simulation of ows over wavy rough boundaries, Center forTurbulent Research, Annual Research Briefs 2002, p.313-p.324.

[9] Yokojima, S., Modeling and Simulation of Turbulent Open- channel Flows EmphasizingFree-surface Eects, PhD thesis, Kobe University, Kobe, Japan 2002.

[10] Lecordier, B., Westerweel, J., The EUROPIV Synthetic Image Generator (S.I.G), EU-ROPIV II Project, Zaragoza, Spain.

[11] Phan, N.M.T, Nguyen, C. V., Wells, J. C., A P.I.V. Processing technique for directly mea-suring surface-normal velocity gradient at a free surface, FEDSM2005-77493 Proceedingsof 2005 ASME Fluids Engineering Division Summer Meeting and Exhibition, Houston,USA 2005.

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