Benchmarking PIV with LDV for Rotor

Wake Vortex Flows

Manikandan Ramasamy∗ and J. Gordon Leishman†

University of Maryland, College Park, MD, 20742

An experiment was conducted to identify and measure the sources of uncertainty thatare associated with the application of particle image velocimtery (PIV) to the measure-ment of the vortical wakes trailing from helicopter rotor blades. Phase-resolved 3-D PIVmeasurements were performed in the wake of a small-scale rotor operating in hover, andwere compared with high-resolution 3-D laser Doppler velocimetry (LDV) measurementsobtained with the same rotor under identical operating conditions. This helped formulatethe essential experimental conditions that need to be satisfied for PIV to accurately resolvethe high velocity gradient, streamline curvature flows that are present inside the rotor wakeand the blade tip vortices. Uncertainties associated with the calibration, acquisition, andprocessing PIV images were analyzed in detail. It was shown that the optimization of laserpulse separation time needed to reduce the errors associated with the acceleration and cur-vature effects, as well as the interrogation window size for velocity gradient measurements,are critical in accurately mapping out the wake flow using PIV. The correlation betweenLDV and PIV measurements of the tip vortex characteristics, such as core radius and peakswirl velocity, were found to be excellent.

Nomenclature

A Acceleration, m/s2

c Chord of the rotor blade, mCT Rotor thrust coefficientdi Image distance, mdo Object distance, mdp Seed particle size, µmdτ Particle image size, µmL Length scale of the coherent flow structureLm Measurement volumeM MagnificationN Number of points used in the estimation of σr Radial coordinaterc Core radius of the tip vortex, mV Absolute 3-D velocity, m/sVθ, Vr, Vaxial Swirl, radial, and axial velocity components of tip vortex, m/sVtip Tip velocity of the rotor blade, m/sX, Y Locations in the camera coordinate systemXP , YP Projected length of the calibration target as seen by the camerax, y, z Locations in the object / fluid coordinate systemα, β Inclination of the calibration target with respect to x and y axis∆R Absolute displacement, m∆t Pulse separation time, µs

∗Assistant Research Scientist. mani@umd.edu. AIAA Member.†Minta Martin Professor of Engineering. leishman@eng.umd.edu. AIAA Associate Fellow.Copyright c© 2006 by Manikandan Ramasamy and J. Gordon Leishman. Published by the American Institute of Aeronautics and

Astronautics, Inc. or the American Society of Mechanical Engineers with permission.

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∆X, ∆Y Displacement in the camera coordinate system, pixels∆S Orginal distance travelled by the seed particle, m∆x,∆y, ∆z Displacement in the fluid coordinate system, mδx, δy, δz Uncertainty in the estimation of x, y, and z, mε Absolute errorεb Bias errorσe Rotor solidityσ Standard deviationλ Wavelength of the laser light, nmΩ Rotor rotational frequency, Hz

Introduction

Developing a comprehensive knowledge of the evolutional characteristics of the wakes trailing from he-licopter rotor blades is essential for developing strategies aimed at reducing or alleviating wake inducedadverse effects, such as blade–vortex interaction noise, rotor vibration levels, unsteady blade airloads, androtor wake–airframe interactions. A considerable amount of experimental research has been conductedover the last several decades towards this objective.1–6 These measurements have been made at differentfacilities, with different rotors at different rotor operating conditions, and using different measurement tech-niques.1,2, 5, 7–11 Therefore, it becomes imperative to understand the measurement uncertainties associatedwith each type of experiment if the goal is to understand accurately the rotor flow physics. The analysis ofmeasurement uncertainty is also essential when the various measurements are used for validating computa-tional predictions. Furthermore, the application of semi-empirically developed rotor wake and vortex modelsin comprehensive rotor predictive analyses means the measurement uncertainty directly propagates into therotor loads and performance predictions.12

Different measurement techniques have been employed to understand the formation and evolution of theblade tip vortices and the associated wake structures trailing from helicopter rotor blades. For example,intrusive techniques such as hot wire anemometry (HWA) has been used for measuring the flow fields ofboth full-scale1,10 and sub-scale helicopter rotors.13,14 The biggest problem encountered in using HWA forrotor wake measurements, however, is spatial proximity concerns to the rotor blades, which prevents themeasurement of flows associated with young, newly created tip vortices. HWA also suffers from directionalflow ambiguity. Furthermore, the spatial resolution of HWA in now known to be insufficient to resolve thehigh velocity gradients inside tip vortices.12

Application of optical flow measurement techniques such as laser Doppler velocimetry (LDV) and particleimage velocimetry (PIV) has gained popularity for rotor wake studies because of their non-intrusive nature(except for seeding the flow) in measuring a given flow field. 1-D, 2-D and 3-D LDV measurements havebeen successfully made.3,15,16 The biggest advantage of LDV is its high spatial resolution when operated in3-D coincidence mode, which allows the high velocity gradients inside the vortex cores to be measured veryaccurately. Higher spatial resolution also means that the minimum measurable flow structure length scaleis smaller. However, LDV is a point-by-point measurement technique that requires substantial time to mapout a given flow field, especially for a rotor. This shortcoming is, however, mostly overcome by the use ofPIV, which measures the velocity in a plane at a given instant of time. Several studies have been performedwith PIV to understand the process of making accurate flow field measurements,17–20 but not previously forthe rotor wake environment. It is to this end that the present paper is focused.

PIV is a non-intrusive experimental technique, which allows the instantaneous 2-D or 3-D measurementof a planar flow field by imaging the reflected light from small particles in the flow that are illuminated by alaser sheet. The acquired images are divided into small interrogation regions (called interrogation windows),which are statistically analyzed to produce a correlation peak that represents the magnitude of the particledisplacement within the interrogation region. Similar displacements that are estimated through all of theinterrogated regions enables the development of a velocity vector map over the entire region of interest.Even though (in principle) PIV consumes a much shorter testing time than LDV to resolve a given flowfield, it has previously suffered from relatively poor spatial resolution compared to LDV. This has limited itsapplication to rotor wake flows in general, and for vortex flow measurements in particular.21 The importanceof spatial resolution was demonstrated by Martin et al.;12 it was shown that a measurement technique thathad poor spatial resolution22 yielded a non-physical result while that of a good spatial resolution yielded

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accurate measurements.3 Thanks to the recent improvements in optics and electronics, higher resolutionCCD cameras can give spatial resolutions in the flow comparable to those of 3-D LDV, and can now beemployed for successfully measuring the wake of a rotor.

However, understanding the uncertainties associated with any measurement technique is critical for in-terpreting the measurements accurately. Even though LDV and PIV are non-intrusive to the flow field(again, except for the seeding particle dynamics), both techniques contain various similar and different typesof measurement uncertainties, which can have substantial effects on the resulting flow measurements. Adetailed analysis of the uncertainties in LDV for measuring vortex flows was made by Martin et al.12 Nosimilar analysis has yet been undertaken for the application of PIV to the same types of flows. Most ofthe existing PIV literature examines the uncertainty associated with the spatial aperiodicity of tip vortices,and in developing procedures to correct for it (for example, conditional averaging by McAlister4 and Vander Wall et al.23). This uncertainty is significant while estimating the mean velocity from the a sequence ofphase-locked instantaneous velocity flow fields. However, no uncertainty analysis has been previously madefor measuring the instantaneous velocity vector field. Such an analysis is essential, especially in the highvelocity gradient streamline curvature flows produced in rotor wakes and, in particular, by the rotor bladetip vortices.12

In the present study, an experiment was conducted to identify and measure the sources of uncertaintyassociated with the application of 3-D PIV for measuring the flow field of a helicopter rotor. Such an analysishelps to formulate criteria that need to be satisfied for the successful application of PIV to rotor wake flows.The PIV measurements were compared with high-resolution 3-D LDV measurements of the same flow.

Experimental Setup

Rotor Facility

A single bladed rotor operated in the hovering state was used for all of the flow measurements. The advantagesof a single bladed rotor have been previously addressed.24,25 This includes the ability to create and studyin detail a representative helicoidal vortex filament without interference from any other of the vortical wakestructures generated by the other blades.24 Also, a single helicoidal vortex is much more spatially andtemporally stable than with multiple vortices,25 thereby allowing the wake and tip vortex structure to bestudied to much older wake ages. This also means that the measurements are relatively free of the highaperiodicity issues typical of multi-bladed rotor experiments, which can make detailed flow measurementsextremely difficult or even impossible.

The rotor blade used in the experiment was of rectangular planform, untwisted, with a radius of 406 mm(16 inches) and chord of 44.5 mm (1.752 inches), and was balanced with a counterweight. The blade airfoilsection was the NACA 2415 throughout. The rotor tip speed was 89.28 m/s (292.91 ft/s), giving a blade tipMach number and tip chord Reynolds number of 0.26 and 272,000, respectively. All the tests were made atan effective blade loading of CT /σe ≈ 0.064 using a collective pitch of 4.5 (measured from the chord line);the zero-lift angle of the NACA 2415 airfoil is approximately −2 at the tip Reynolds number. During thesetests, the rotor rotational frequency was set to 35 Hz (Ω = 70π rad/s).

The rotor was tested in the hovering state in a specially designed flow conditioned test cell. The volumeof the test cell was approximately 362 m3 (12,800 ft3) and was surrounded by honeycomb flow conditioningscreens. This cell was located inside a large 14,000 m3 (500,000 ft3) high-bay laboratory. The rotor wakewas exhausted out of the test cell into the bay.

The flow at the rotor was seeded with a thermally produced mineral oil fog. The average size of the seedparticles were measured through calibration to be between 0.2 to 0.22 microns in diameter, which was smallenough to minimize the particle tracking errors for the vortex strengths found in these experiments (Ref. 26).The fundamental requirement of seed particles to track the flow and their influence on the PIV imageprocessing will be discussed later. The entire test area was uniformly seeded by a fog generator before eachsequence of measurements, and the fog density was replenished as needed during the experiments.

For flow visualization, a laser light sheet from Nd:YAG pulsed laser source with a wavelength of 532 nmand capable of firing up to 15 Hz was used in synchronization with the rotor frequency to illuminate planesin the flow field. Laser light sheet was produced using a cylindrical lens that was placed at the end of anarticulated optical arm, and which transmits the laser light from its source to the region of focus (ROF). Aconvex lens was placed at the end of the optical arm to reduce the waist of the laser light sheet at the regionof focus and also to increase its intensity.

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Laser light sheet

Lenses

Image sensors(ccd camera)

Region of focus

Figure 1. PIV system schematic.

Cameras

Laser

Region offocus

Rotor

Figure 2. Schematic of the current experimental set-up.

The PIV system included dual pulsed Nd:YAG lasers that operated in phase synchronization with therotor, a pair of digital CCD cameras with 2 Mega-pixel resolution placed on the same side of the light sheet, ahigh-speed digital frame grabber, and analysis software. The cameras were placed at 20 from the mid-pointof the ROF, and satisfy the Scheimpflug criteria to have uniform focus across the entire region of interest, asshown in Fig. 1. The laser could be fired at any rotor blade phase angle, and so enabling the measurementsto be made at any required wake age (i.e., time after origination in the flow in terms of angular blade rotationand normally measured in units of degrees).

A schematic of the experimental set up is shown in Fig. 2. The two lasers were fired with a default pulseseparation time of 1µs. This corresponds to less than 0.01 of blade motion. Over 100 images were capturedat each wake age and were phase-averaged to estimate the mean velocity across the flow field.

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Principle of Operation

The PIV technique measures the instantaneous velocity in the flow field based on the expression

V =M ∆R

∆t(1)

where ∆R is the displacement of the seed particles over a given pulse separation time ∆t, and M is theimage magnification given by

M =h

H=

di

do(2)

Here h and H are the image and the object size, respectively. The magnification can also be defined as theratio of the image distance (di) to the object distance (do), as shown in Fig. 3. In the case of a digital CCDcamera, this is the ratio of the sensor size to the object size. Equation 1 clearly shows the fundamentalsteps involved in performing PIV measurements in any given flow field: (1) calibration for estimating themagnification involved in the process, (2) image acquisition for estimating the displacement of seed particlesin the region of focus of the flow, and (3) image processing to calculate the flow velocity.

Horizontal angleof view

Vertical angleof view

Image

Object

camera lens

di = f

doH

h

Figure 3. Schematic explaining the field of view of the camera optics.

Stereo Calibration

In the present study, the stereo calibration for making 3-D flow field measurements was performed using aDual Plane Double Sided (DPDS) calibration target that is black in the background with white markers – seeFig. 4(a). The target has two planes of marker points such that the alternate markers are in different planes.The planes are separated by 1 mm. The target is placed in such a way that the two planes of markers areon either side of the laser light sheet. A fiducial point in the center acts as the reference or origin of the PIVcoordinate system. The location of this point is an input, and the other coordinates are computed relativeto this fiducial mark. The two plane target set up allows the computation of the calibration coefficientswithout traversing the target in the out-of-plane direction. However, as a result, the measured out-of-planevelocity component is only first-order accurate.

The side of the target is fitted with a plate that has a mirror aligned with the centerline between the twoplanes of the target, as shown in Fig. 4(b). The laser sheet was aligned with the target using this mirrorsuch that the sheet illuminates a measurement plane between the two planes of calibration. The laser sheet

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Figure 4. Calibration target used in the experiment, with side mirror.

strikes the mirror slot and reflects it back. The optical path of the reflected sheet is made co-planar withthe incident sheet to ensure alignment of the laser sheet with the calibration target. Sufficient illuminationis provided such that the cameras can clearly focus and image the target. The calibration image analysisidentifies the calibration markers in the image and locates their centroid.

The top left of Fig. 5 shows the identified calibration markers with a box (or circle) around them. Thefiducial mark is highlighted with a diamond. The region of focus in the calibration target seen by the left andright cameras is shown in Fig. 6. The identified calibration markers as seen by the left and right cameras arealso shown in Fig. 5 on the top right and bottom left, respectively. The overlapping region of focus from thetwo cameras can be seen at the bottom right corner in Fig. 5. The smaller box embedded within representsthe common field of view of the two cameras in which the three components of velocity can be reconstructed.The calibration markers are sorted into a grid, with each marker point having a row and colum grid index.

The grid structure, with the fiducial mark being used as the base point, is used to match the calibrationpoints recorded on the image to target locations in the object plane (which is the plane of the laser lightsheet). The result of this image analysis is a calibration points file with a record of the (X, Y ) image pixellocations and the target markers (x, y, z) locations in the object plane for each calibration marker. This datafile is used to develop the mapping functions, which are given by

Xleft = f1(x, y, z)Yleft = f2(x, y, z)

Xright = f3(x, y, z)Yright = f4(x, y, z) (3)

where f is the mapping function that is also used to find the magnification for each 3-D vector in the leftand the right images.

The local magnification can be computed using the derivatives of Eq. 3 with respect to the spatialdirections. The derivative of the mapping functions represent the gradient of the particle displacement inthe image plane (in pixels) to the particle displacement in the object plane (in mm). These gradients give

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Figure 5. Calibration image processing results.

Left Right

Figure 6. Region of focus seen by the left and right CCD cameras.

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the pixel displacements in the X or Y direction in the image plane that are caused by particle motion inthe x, y or z directions in the object plane. Each point in the object plane has a unique set of displacementcalibration factors. To locate the velocity vectors in the object plane (the plane of laser sheet), a 3-D vectorgrid is created to define the locations where the 3-D velocity is desired. After calibration, the images areacquired and the inplane cross-correlation algorithm computes the relevant pixel displacements for eachcamera. The following set of equations are used to obtain the displacements:

∆Xleft

∆Yleft

∆Xright

∆Yright

Image

= ∆xfluid

[A

]+ ∆yfluid

[B

]+ ∆zfluid

[C

](4)

where

[A

]=

(dXleft/dxfluid)(dYleft/dxfluid)

(dXright/dxfluid)(dYright/dxfluid)

,[

B]

=

(dXleft/dyfluid)(dYleft/dyfluid)

(dXright/dyfluid)(dYright/dyfluid)

,[

C]

=

(dXleft/dzfluid)(dYleft/dzfluid)

(dXright/dzfluid)(dYright/dzfluid)

(5)

The particle image displacements (∆Xleft,∆Yleft,∆Xright,∆Yright) are the computed pixel displacementsfrom a cross-correlation based on each camera view. The three displacements in the fluid (i.e., ∆xfluid,∆yfluid, and ∆zfluid) are the three unknowns that need to be solved for. The system of three unknowns andfour equations are solved using an iterative least-squares method.

Processing Procedure

The acquired image pairs are sub-divided into smaller regions of analysis called interrogation windows.Each interrogation window produces one independent 3-D displacement vector based on Eq. 4. Determiningthe size of these interrogation windows is a separate task by itself that involves numerous factors to accuratelyresolve the flow field. These interrogation windows between the image pairs are cross-correlated using adefined algorithm, such as direct correlation, FFT correlation, the Hart correlation, or the Rohalay–Hartcorrelation. Once the correlation is complete, the correlation peak is estimated (for sub-pixel displacements)based on curve fitting the correlation signal. Various methods such as bilinear curve fit, three-point Gaussian,or parabolic fit and centroid estimation may be used for estimating the correlation peak.

A simple correlation algorithm involves the application of interrogation windows in both the referenceand displaced images to be of same size with no offset, i.e., both the windows are placed at the same placein the fluid. For a typical displacement, as shown in Fig. 7(a), it can be seen that this procedure (singleiteration) results in loss of image pairs; only four particle images are used for correlation peak estimation.Such a low number of contributing pairs may result in incorrect estimates of particle displacement and hencethe flow velocity. This can, however, be improved by offsetting the second window (dotted line) based onthe displacement estimated from the first iteration. This window offsetting process improves the number ofimages that contribute for the estimation of the correlation peak – see Fig. 7(b). Another alternate means toimprove the number of contributing image pairs is to have the second window larger than the first window,as shown in Fig. 7(c).

An improved technique than the one shown in Fig. 7 can be used to obtain better correlation. Thistechnique is recursive in nature and is similar to the one shown in Fig. 7(b). However, the size of theinterrogation window is reduced for successive iterations. This provides the benefit of being able to produceaccurate results without losing any resolution. However, there is a disadvantage associated with this process.For a flow with very high velocity gradients, the average displacement estimated by the largest window size(which are used to move the smaller windows) may not be optimum for the smaller windows. This meansthat the smaller windows may have moved by a larger or smaller distance than that is required for optimumcorrelation. This can result in a loss of image pairs. A judicious choice of the initial window size, however,can solve this problem.

For flow fields with very high velocity gradients such as a helicopter rotor wake, a new and improvedrecursive technique called the deformation grid correlation has been developed.27 Here, the procedure is

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Particle position at t = ΔΔtNew entraining particles

Particle position at t = 0

First windowSecond window

(a) (b)

(C)

Figure 7. Schematic of the correlation choices.

First window

Second window

R1/2

- R1/2

32 X 32

R2

Shear

R0First window

Second window

R0

Translation

64 X 64

R132 X 32

64 X 64

64 X 6464 X 64

Figure 8. Outline of the recursive correlation method.

similar to that of a simple recursive technique, however, the second window is deformed (i.e., both shearedand translated instead of just the simple translation) – see Fig. 8. The procedure starts with the correlationof an interrogation window of a defined size (say, 64-by-64), which is the first iteration. Once the meandisplacement of that region is estimated, the interrogation window of the displaced image is moved byinteger pixel values for better correlation in the second iteration. This third iteration starts by moving theinterrogation window of the displaced image by sub-pixel values based on the displacement estimated fromsecond iteration. Following this, the interrogation window is sheared twice (for integer and sub-pixel values)based on the velocity magnitudes from the neighboring nodes before performing fourth and fifth iteration,respectively.

Once the velocity is estimated after these five iterations, the window is split into four equal windows(of size 32× 32). These windows are moved by the average displacement estimated from the final iteration

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(using a window size of 64× 64) before starting the first iteration at this resolution. This procedure can becontinued until the resolution required to resolve the flow field is reached. The second interrogation windowis deformed until the particles remain at the same location after the correlation. This technique, especiallythe introduction of shear, has been shown in the present work to be very efficient for measuring the highvelocity gradients inside rotor wake flows.

Simple Validation

Validating any PIV processing algorithm requires experimental measurements with known variables. Gen-erally a Monte–Carlo type of simulation is used to identify the the influence of various parameters on theestimation of velocity, and the effectiveness of correlation and peak finding algorithms. Also, experimentswith known measurement variables have been used in the past for validation. In this case, an artificial flowfield is created using particles of known size distributed homogeneously on a painted board. After acquiringthe first image, the board is moved in the necessary direction with a known magnitude to acquire the secondimage. These two images are then cross-correlated to estimate the actual displacements. The differencebetween the actual and the estimated displacement will give a measure of the error associated with theprocessing technique.

There are advantages as well as disadvantages in using these techniques. Even though it is possibleto control the density or the fill-ratio within the interrogation window, as well as the displacement of theparticles in the acquired images, this does not represent the actual flow condition, especially the particle sizeand its random distribution.

In the present study, the first image is acquired from a real vortical flow field. The second imageis manipulated from the first image using various image processing techniques. The biggest advantageof acquiring the image from real flow conditions (same magnification, particle size, etc., as in the actualexperiment) is that it will enable a better understanding of the various PIV measurement accuracy issues(such as pixel-locking). Even though it is easier to have integer pixel value displacements, having sub-pixeldisplacement is useful although it becomes complicated.

To achieve the sub-pixel displacements, the images were first enlarged using bi-cubic interpolation. Thisstep was followed by moving the enlarged images by integer pixel displacements before reducing it back toits original size. For example, to achieve a 0.5 pixel displacement, the image was enlarged to twice its size,followed by moving it by one magnified pixel and then reducing the image back to its original size beforethe cross-correlation. Any error associated because of this enlarging/reducing process must be estimated. Itshould, however, be noted that the image manipulation can be performed only in two-dimensions becausethe information on third dimension is lost in the acquired images.

Figure 9 shows the displacement estimated by the two PIV correlation algorithms used in the presentstudy for a given 1-D in-plane displacement that is obtained through the aforementioned procedure. In thecase of a single iteration correlation technique, the window size was 16×16 pixels. In the recursive technique,

Figure 9. Correlation error with various window choices.

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the size of the first window was 128 × 128 pixels. This was gradually reduced to 16 × 16 pixels. It can beseen that the single iteration procedure results in significant errors for all displacements despite the flowbeing 1-D. Also, this procedure completely failed when the displacement was more than 4 pixels. On theother hand, the recursive technique provided a much better estimate of the original displacement of the seedparticles.

Generally, it is suggested that the maximum displacement of the particles should be less than one-fourthof the interrogation window size for a given ∆t. For example, this would mean that an interrogation windowof size 16×16 pixels cannot handle more than 4 pixel displacement of the particles images. This would meanthat for a required resolution of 16 pixels, the value of ∆t should be small enough to have a displacement thatis less than 4 pixels. At the same time, the value of ∆t should be large enough for the correlation algorithmto identify the movement of particle images. Typically, a minimum displacement of twice the particle sizehas been suggested for accurate estimation of particle displacements.28

These requirements generally limit the application of PIV to the measurement of flow fields that havelow dynamic range. However, such a limitation can be overcome by the recursive correlation algorithm thatis used in the present study. While the single iteration correlation technique failed to produce any resultwhen the displacement was more than one-fourth of the interrogation window size, the recursive correlationprocedure was able to estimate a displacement of 60 pixels. The only care that needs to be taken in the caseof the recursive correlation technique is that the first window should be large enough to handle the pixeldisplacement. This property of the recursive technique is very useful for rotor blade tip vortex measurementsthat have very high dynamic range within a very small spatial dimensions. It should, however, be notedthat both the techniques showed ≈ 0.03 pixel displacement even when there was no such motion in the ydirection. This was found to be consistent with the results from the uncertainty analysis, as discussed laterin this paper.

PIV Parameters

In the current study, the improved recursive correlation (deformation grid correlation technique) explainedpreviously has been used to estimate the flow velocities inside the tip vortex cores, which, as previouslynoted, generally have very high velocity gradients. Various pre-conditioning steps may be used to improvethe correlation. The average background noise, estimated from all the acquired images, was removed beforecorrelation. FFT correlation with a three-point Gaussian fit has been used in this case for the correlationand peak estimation, respectively.

The signal-to-noise ratio (the ratio of the primary peak to the secondary peak of the correlation) wasset to 1.5. Even although this is a stringent requirement, such a high value is essential for the accurateestimation of particle displacements. Once the correlation for the entire image was performed, spuriousvectors were removed based on the 3σ values and a median tolerance of 2 pixels based on a neighborhoodof 5× 5 points surrounding each vector. Any spurious vectors were replaced by interpolated vectors from aneighborhood of 3× 3 nodes. A 50% overlap has been consistently applied through out the entire processingarea.

Measurement Results

The 100 acquired PIV images were processed and simple phase-averaged to estimate the mean velocity inthe region of focus – see Fig. 10. It should be noted here that every eighth vector has been plotted to preventimage congestion. Because these measurements were made at a young 3 of wake age, the aperiodicity of thetip vortex locations were not significant to require formal correction procedures.3,29 The figure has vorticityas its background, which clearly shows the presence of a concentrated blade tip vortex. The presence of aturbulent vortex sheet that is trailed from the inner part of the blade can also be identified.

Figure 11 shows the measured tangential, radial, and axial velocity components by making a horizontalcut across the vortex core. By convention, the velocities and the radial distances were normalized by the tipspeed of the rotor, and the core radius of the tip vortex, respectively. The core radius was assumed to behalf the distance between the two swirl peaks across the center of the vortex. It can be seen that the peakswirl and axial velocity components are 30% and 24% of the blade tip velocity, respectively.

Notice that the peak swirl velocities are not symmetrical on either side of the vortex, mainly because ofits non-zero convection velocity through the flow. The difference between the two peaks on each side of the

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Figure 10. Flow field surrounding a tip vortex trailing from a rotor blade (every eighth vector on either sideis plotted to prevent image congestion).

core can be assumed to be its average convection velocity, which is also a measure of the induced velocityinside the slipstream boundary of the rotor wake. It can be observed that the axial velocity in the vortex hasits peak not at the vortex center, but slightly away from its axis; this may be because of the inclination ofthe helical vortex filaments with respect to the measurement plane. It should be noted that there are morethan 20 points present inside the vortex core boundary, which is a result of the combination of the reducedregion of focus and the higher resolution cameras that were used in the present study.

Figure 12 shows the measured tangential and axial velocities across the tip vortex, along with the LDVmeasurements.3 Good correlation can be seen between both measurement techniques. In the case of PIV,every other point is plotted to prevent image congestion. It can be seen that the peak swirl velocity measuredby PIV matches the LDV measurements. Very good agreement can be seen inside the vortex core wherethe velocity gradients are high. However, the velocities measured by the two techniques are slightly differentoutside the vortex core boundary.

Similarly, in the case of the axial velocity, the PIV measurements show higher velocities when comparedwith the LDV measurements. This may be because of the lower spatial resolution used in this particularcase of LDV measurements. For the axial velocity measurements with LDV, on-axis data acquisition wasused instead of the better off-axis mode; this reduces the resolution by an order of magnitude (from 3%of the probe volume to 30% relative to the vortex core size). Even in the case of PIV, the probe volume(interrogation window size) is dictated by the thickness of the laser light sheet.

In the present study, the laser light sheet thickness at the region of focus has been reduced significantlyby the use of a convex lens. Because both the measurements used exactly the same conditions (rotorblade, operating speed, seed particles, etc.), the differences in the measurements may be attributed to theuncertainties in the measurement technique.

The measurement of the standard deviation of the normalized swirl velocity across the tip vortex producedby phase-averaging is shown in Fig. 13. It is apparent the standard deviation reaches it maximum value nearthe vortex core axis. Even though aperiodicity has been previously assumed to be negligible in this case,such variations of standard deviation still suggests the presence of some aperiodicity in the flow field that

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Figure 11. Example of measured velocities across the tip vortex.

(a) Normalized swirl velocity. (b) Normalized axial velocity.

Figure 12. Example showing the comparison of the normalized velocity profiles acquired by LDV and PIV.

Figure 13. Standard deviation of the normalized tangential velocity across the vortex produced by phaseaveraging.

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spatially alters the phase-resolved tip vortex locations between every PIV acquisition image.

Error Analysis

The difference between the true value of a parameter and its measured value is defined conventionally asthe total measurement error, i.e.,

ε = xtrue − xmeasured (6)

There are two sources for the total measurement error: systematic error and random error. Systematic erroris defined as the part of a measurement error that remains constant in repeated measurements of the sametrue value. Multiple elemental systematic errors, such as the calibration and data acquisition errors, cancontribute towards the total systematic error of the flow measurement.

One particular property of this error is that it follows a consistent pattern that can be predicted, whichallows for methods to be devised to reduce or, even, remove them. Random error, on the other hand, willvary with repeated measurements of the same true value. The uncertainty associated with the random errorcan be estimated by inspection of the measurement scatter. The standard deviation, σ, is generally used tomeasure the scatter produced by the random error.

Calibration Error

Calibration error has contributions from two sources: 1. The error associated with improper placementof the calibration target, and 2. The error associated with the development of mapping functions. Eventhough the error associated with improper placement of the calibration target can be removed in stereoscopicPIV through various techniques,30 the error from the latter source is common to both 2-D and 3-D PIVtechniques.

Calibration Target

The largest source of calibration error occurs when the calibration target is not placed perfectly in the laserlight sheet plane. Figure 14(a) shows the coordinate system of the laser light sheet (x, y, z) along with animproperly placed calibration target of dimension (X, Y, 1) and its projection on the laser light sheet plane.The angles have been exaggerated to give a better understanding of the misalignment. An inclination ofα about the x-axis – see Fig. 14(b) – alters the vertical dimensions of the calibration target viewed by thecamera. Similarly, an inclination of β in the y-axis – see Fig. 14(c) – alters the horizontal dimensions of thecalibration target viewed by the camera. A small displacement along the z-axis can alter both the horizontaland vertical dimensions.

Here, it should be noted that the horizontal and vertical angle of view of the lens are different (see Fig. 3,which would have a different influence on the magnification along x- and y-axes (even though their aspectratio is maintained constant). Therefore, the uncertainty associated with the estimation of the horizontal

z

y

x

α

y

z

x

β

z

y

x= +

z

y

x+

h

(a) (b) (c) (d)

Figure 14. Schematic showing the possible sources of error in placing the calibration target on the laser lightsheet plane.

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and vertical dimensions of the target or their influence on the magnification are given by

δMx =

√(∂M

∂β

)2

(δβ)2 +(

∂M

∂z

)2

x

(δz)2 (7)

and

δMy =

√(∂M

∂α

)2

(δα)2 +(

∂M

∂z

)2

y

(δz)2 (8)

Because any vector in the flow field, r, can be represented by x and y coordinates using

r =√

x2 + y2 (9)

their associated uncertainties can be represented as

δM =√

2δMx + 2δMy (10)

In the present study, the calibration target was provided with a mirror in one of its edges. This mirroris utilized to reflect the laser light light sheet back to its source. This significantly reduces the misalingmentof the calibration target, however, it does not remove the error completely.30 This is because the laser lightsheet has a finite thickness (≈ 0.8mm at its waist in the present study) and it is difficult to detect thealignment of the Gaussian peak of the laser light sheet (source) and the reflected light from the calibrationtarget mirror.

The vertical dimension of the calibration target viewed by the camera from its inclination about x-axis(α) is given by

Yp = Y cos α (11)

where Yp is the projected length on the laser light sheet as seen by the camera. This would mean that

1M

∂M

∂α= sec α tanα (12)

This reduction in the actual distance alters the magnification (based on Eq. 2) and can, therefore, directlyinfluence the estimation of flow velocity. Similarly, an inclination in the y-axis by β results in

Xp = X cos β (13)

where Xp is the projected dimension of the calibration target seen by the camera. As a result, then

1M

∂M

∂β= sec β tanβ (14)

Here, β can be estimated from from the thickness of the laser light sheet, the distance between the laserlight sheet source, and the mirror on the calibration target, as shown in Fig. 15. The maximum value of βoccurs when the Gaussian peak of the reflected light coincides with the edge of the incident laser light sheetwhen it exits the laser optics.

A similar technique can be used to estimate the maximum value of α. The incident laser light sheet fromthe laser contracts in width (area). This is because of the convex lens that is used in the current experiment,so that the waist of the laser light sheet coincides with the region of focus to reduce the uncertainty associatedwith the thickness of the laser sheet. This point will be discussed further later in the paper.

As previously mentioned, compared to the rotation about x-axis (or y-axis), a displacement along thez-axis will alter the dimensions viewed by the camera in both the x- and y-axes. In the present study, a100mm micro lens with a f# of 2.8 was used; this has 4.36 and 3.55 angles of view along the x-axis andthe y-axis, respectively. As a result, the following relationships hold:

1M

(∂M

∂z

)x

=2 tan(4.36)

X − 2z tan(4.36)(15)

and1M

(∂M

∂z

)y

=2 tan(3.55)

Y − 2z tan(3.55)(16)

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α

Calibrationtarget

Laser optics

Incident lightsheet

RReefflleecctteedd lliigghhttsshheeeett

Reflectingmirror

Figure 15. Schematic explaining the alignment technique used in the current experiment.

As mentioned previously, because the calibration target mirror is used for alignment, the maximum possiblevalue of z will be half the laser light sheet thickness. Because the estimation of α, β and z is not possible(other than by knowing their maximum possible values), this would result in δα = δβ = δz = 1. Uponsubstituting the values of α, β and z, then

(δM

M

)x

=

√sec2 β tan2 β(δβ)2 +

(2 tan(4.36)

X − 2z tan(4.36)

)2

(δz)2

(δM

M

)y

=

√sec2 α tan2 α(δα)2 +

(2 tan(3.55)

Y − 2z tan(3.55)

)2

(δz)2

Substituting the corresponding values, the total uncertainty in magnification can be estimated using Eq. 10to be (

δM

M

)≈ 0.3%

This calibration target uncertainty can be reduced significantly with the stereoscopic PIV technique.This is done by measuring the calibration target location in the laser light sheet coordinate system using adewarping process, and cross-correlating the flow images. The process is similar to PIV processing, however,the correlating flow images correspond to the the same instant in time in this case. If the calibration wereto be perfect and the laser light sheet had zero thickness, then the particle images from the same laser pulse(from the two camera images) would be at the same pixel location in both of the dewarped images. Acorrection based on this principle is given in detail in Ref. 30, and has been used in the present study.

Mapping Functions

Another source of error that occurs during the calibration is when the maping functions, as given in Eq. 3,are created. Marker points from the object plane are related to the calibration marker points in the imageplane through curve fitting using polynomial equations. This curve fitting introduces residual errors thatneed to be accounted for. In the present study, it was found that the bias error and the standard deviation inidentifying a marker point in the image plane to that of the object plane through the developed polynomial

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equation were 0.0249 and 0.0491 pixels, respectively. A total of 19 points were used for developing thepolynomial equation. The total uncertainty would then be

δX =√

X2b + P 2

I (17)

where Xb is the bias error and PI is the precision error that is given by

PI =t95 σ√

N(18)

A student factor of 95% would result in δX as 0.032. This would mean that the uncertainty in identifying adistance in the image plane is 2δX. From Eq. 4, a distance in the image plane (along X, in the left camera)is given by

∆Xleft =dXleft

dxfluid∆xfluid +

dXleft

dyfluid∆yfluid +

dXleft

dzfluid∆zfluid (19)

Assuming that there is no uncertainty in measuring the distance in the calibration target (δxfluid = δyfluid =δzfluid = 0), the uncertainty is

2δXleft = 2(

δX

dxfluid

)∆xfluid +

(∆X

∆xfluid

)δ∆xfluid + 2

(δX

dyfluid

)∆yfluid

+(

∆X

∆yfluid

)δ∆yfluid + 2

(δX

dzfluid

)∆zfluid +

(∆X

∆zfluid

)δ∆zfluid (20)

Four such equations must be solved simultaneously using a least-squares technique to estimate the netuncertainty of the displacements in the x, y and z directions, respectively. However, a specific assumptionthat there is no contribution from y and z on the uncertainty in x will reveal some interesting outcomes.Based on this assumption, then Eq. 20 can be rewritten as

δ∆xfluid

∆xfluid=

1M

(2δXleft

∆xfluid− 2δX

dxfluid

)(21)

It is apparent from this expression that a larger displacement of a particle (∆xfluid) reduces the uncertaintyin the estimation of the in-plane particle displacement. This would mean that the uncertainty in the swirlvelocity near a vortex core boundary will be smaller because of larger particle displacements for a given pulseseparation time than would be found near the vortex core axis.

The out-of-plane displacement is the difference in displacement observed in the image plane of right andleft CCD cameras. This would mean that the uncertainty associated with the out-of-plane component istwice that of the in-plane components, at least theoretically. The velocity is calculated using the expression

V =∆xfluid

∆t(22)

Because the instrument precision for measuring t is 0.05, this would mean that the corresponding uncertaintyassociated with the estimation of the in-plane velocity is

δV

V=

δ∆xfluid

∆xfluid− δ∆t

∆t(23)

This clearly suggests that for the regions where larger particle displacements are observed for a given valueof pulse separation time, the biggest source of error will be from the instrument accuracy in precisely settingthe pulse separation time.

Figure 16 shows the representative uncertainty associated with the estimation of velocity across the vortexcore. Near the vortex axis, it is known that the absolute velocity is substantially smaller than that at thecore boundary. As a result, the uncertainty from the estimation of displacement (of particles) is dominant.However, as the velocity increases away from the vortex axis, the displacement uncertainty reduces andasymptotes to the uncertainty resulting from the instrument precision of setting the pulse separation time.

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Figure 16. Normalized measurement uncertainty for varying velocities across the vortex.

In the present experiments, the measurement bias error across the vortex resulting from the mappingfunctions is shown in Fig. 17(a). Combining the bias error and standard deviation measured across thevortex, the total error is given by

ε =

√(ε2b +

t95 σ√N

)(24)

where εb is the bias error, N is the number of samples used to determine the standard deviation σ, andt95 is the 95% confidence level in the measurements. The total errors from processing the images for theswirl, radial, and axial velocities are shown in Figs. 17(b), (c), and (d), respectively. In the case of the swirl(swirl) velocity, the total error is a maximum near the vortex core axis and then gradually decreases withan increase in radial distance across the vortex. This is consistent with Eq. 21, which suggests that thelarger displacement of seed particles (near the core boundary) reduces the uncertainty associated with themeasurements.

It can be observed that the axial component exhibits a larger measurement uncertainty than for thetangential velocity. It is known that the axial velocity suffers the most because of aperiodicity issues. Thisresults in a larger scatter in the axial velocity measurements and a larger standard deviation that, in turn,contributes to a higher measurement uncertainty. In the case of the radial velocity, both the bias andprecision uncertainties are much larger relative to the magnitude of the normalized radial velocity, eventhough their absolute value is comparable to that of the tangential and axial velocities.

Acquisition Uncertainties

The elemental uncertainty associated with data acquisition has multiple sources. These include thebias and random error associated with the size of the seed particles used in the experiment, random errorsthat result from the noise in the image recording and its subsequent processing, acceleration errors causedby the basic principle of PIV that approximates the Eulerian velocity components based on finite particledisplacements, velocity gradient errors that arise because of the difference in velocity across the interrogationwindow (both random and bias error),31 and registration errors32 that result from the imperfect matchingof two stereoscopic velocity fields.

Another unique error associated with the PIV technique when measuring vortical rotor wakes and tipvortices is the loss of information about the flow curvature. Among these errors, acceleration error andcurvature error are all dictated by the laser pulse separation time, and the velocity gradient error is causedby the choice of the interrogation window size (spatial resolution).

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(a) (b)

(c) (d)

Figure 17. Variation of different errors across the tip vortex: (a) Measurement bias error across the tip vortexfrom the calibrating mapping functions, (b), (c) and (d) Calibration and processing error across the tip vortexfor swirl, radial and axial velocity distribution, respectively.

Random Error

The acquired images were first divided in to small interrogation windows (generally, 32×32 or 64×64). Thesewindows have multiple particle images of size dτ , which upon correlation yields a signal peak whose diameteris of the order of dτ . This clearly suggests the need for smaller seed particles. Sub-pixel displacements can beestimated by various methods that include, but are not limited to, bi-linear fit, curve fits such as Gaussian,parabola, etc., or by calculating the centroid. However, this sub-pixel estimation is influenced by variousparameters.

Adrian33 found that the irregular particle images, imperfections arising from the film grain (in the case offilm-based PIV), pixel read out noise, non-uniform particle size distribution, and the variation of illuminationintensity across the interrogation spot, all contribute towards random error. These errors also scale with theparticle image diameter, dτ . Furthermore, the error associated with random correlation between particlesthat do not belong to the same image pair, as well as the error associated with the loss of in-plane andout-of-plane image pairs, will contribute to the random error. This random error is given34,35 by

σrand = c dτ (25)

where c = 0.05 to 0.07. It is clear from the above expression that the random error reduces with reduction

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in the particle image size. Also, a reduction in particle image size increases the intensity of the reflectedlight, which can result in improved correlation. However, this may also introduce pixel locking errors, whichwill be discussed in the next section.

Particle Size

The size of the seeding particle plays a substantial role in the experimental errors associated with themeasurement of velocity vector fields using the various types of PIV processing algorithms. Seed particlesare required to meet certain required criteria irrespective of the employed measurement technique. Seedparticles should be large enough to reflect the incident laser light, which is the primary source of themeasurement. However, the particles should be small enough to have low inertial and aerodynamic responsetime and so will follow the fluid motion without any lag or slip. These contradicting requirements need tobe met with a reasonable compromise.26 From a PIV perspective, there are further requirements to be metfor choosing the most suitable size of the seed particles.

The ratio of the particle image diameter and the pixel size of the camera have significant influence onthe velocity estimates. The particle image diameter is defined36,37 as

dτ =√

(Mdp)2 + d2diff (26)

where M is the magnification, dp is the particle size, and ddiff is the diffraction limited minimum imagediameter. Irrespective of the actual particle size, the image size of the particle seen by the camera is basedon diffraction principles. A detailed explanation on this topic is given in Ref. 37. This diffraction limitedminimum image diameter is given by

ddiff = 2.44f#(M + 1)λ (27)

where λ is the wavelength of the incident laser light. In the present study, f# = 2.8 and the laser has awavelength of 532 nm. An increasing f# increases the value of ddiff . However, this will also reduce theamount of light entering into the imaging system.

When the particle image size is too small compared with the pixel size (i.e., it is under-resolved), thiswill result in significant bias error because of the finite numerical resolution of the correlation function.35

The displacement of the particles resulting from this effect will appear to be biased towards an integral pixelvalue. This effect is called “pixel locking” or “peak locking.” The pixel locking effect increases with thereduction in the particle image diameter.38 A larger value of this ratio increases the random errors thatresult from imperfections in pixel read out noise, etc.33 This suggests that optimizing the particle image sizeis essential. An optimum value of 2.1 has been suggested by Prasad et al.35 based on the random and biaserror minimization for a cross-corrlation procedure.

In the present study, the ratio of particle image diameter and the pixel diameter was measured to be 0.66.Despite having such a relatively low value for this ratio, the results did not exhibit any peak locking effects– see the histogram shown in Fig. 18. In this case, several factors contribute to the successful elimination ofthe peak locking effect. These include: (i) the increased fill-ratio or higher sampling rate, i.e., the presence ofmore particles inside the correlation window (higher density), which result in multiple particles contributingfor the peak correlation,30 and (ii) improper focussing of the seed particles, which increases the particle imagesize seen by the camera. Even though pre-conditioning can also be applied to optimize the particle imagediameter with respect to the peak estimator,37 no such pre-conditioning has been applied in the presentstudy. Also, peak locking directly depends on the choice of the correlation peak detection algorithm. Whilea 3-point curve fit such as a Gaussian fit is susceptible to peak locking errors,37 other techniques such asbilinear fits or centroid based fits may not exhibit peak locking errors.

Pulse Separation Time

The laser pulse separation time plays a significant role in accurately measuring the instantaneous velocity ofa flow field.28 This is especially true in the case of rotor blade tip vortices, where significant flow accelerationsand flow curvature exists.

The importance of optimizing the pulse separation time can be understood from Fig. 19, which showsthe measured tangential (swirl) velocity distribution in a tip vortex for two different pulse separation times,while keeping other variables constant. It is apparent that the swirl velocity profile, peak swirl velocity,and the core radius (the three defining parameters of a tip vortex) all change with the selected value of

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Figure 18. Particle image displacement histogram showing no signs of pixel-locking effects.

∆t. Furthermore, in the case of tip vortices, another unique condition should be satisfied to properly makephase-resolved flow measurements, which is a point previously discussed by Martin et al.21 Because the tipvortices in a rotor wake convect at approximately half the average inflow velocity at the rotor disk, careshould be made that the vortex core remains the same position in both the PIV frames. Martin et al.suggested the condition that

∆t

(ΩRλ

c

)≤ 0.1% (28)

be met so that the vortex core position lies within 1% of its core size between successive frames. Here, λ isthe nondimensional inflow velocity. In the present study, care was taken to ensure this condition is satisfied.A ∆t value of 1 µs corresponds to less than 0.01%.

Figure 19. Swirl velocity distribution in a tip vortex estimated for various laser pulse separation times.

Acceleration Error

PIV operates on the principle of correlating images that are acquired within a time interval of ∆t. Thismeans that the location of particles at two instants in time X(t) and X(t+∆t) is the only stored informationused for analysis. Because real rotor wake flows can be unsteady (aperiodic) and the flow will always besubjected to accelerations, there will be an error associated with the estimation of velocity based on theparticle displacements. For particles that track the fluid without any lag or slip, the trajectory can be

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approximated34 as

X(t + ∆t) ≈ X(t) +dX

dt(∆t) +

d2X

dt2

(∆t2

2

)(29)

This means that the relative error resulting from the acceleration of the flow is given by

εaccel =∣∣∣∣dv

dt

∣∣∣∣ |∆X|2Mv2

(30)

where v is the velocity in terms of pixels per second and M is the magnification. It is clear from Eq. 30that a reduction in pulse separation time, which reduces the particle image displacement, always reduces theerror associated with flow acceleration effects.

In the case of tip vortices, centripetal and Coriolis accelerations act on the particles that are distributedacross the vortex core.26 To estimate the acceleration of the particles, it is essential to have 3-D measurementsin at least two different planes. This is because of the 3-D nature of the tip vortex that makes it essential tomeasure the axial velocity gradient across the measurement plane to compute the acceleration.

The variation of centripetal acceleration across the vortex core (in a 2-D plane) is shown in Fig. 20. Forillustration, this acceleration has been estimated based on the expression A = V 2

θ /r. Because the accelerationincreases with radial distance, the uncertainty associated with the acceleration increases with radial distanceuntil 75% of the vortex core boundary at r = rc is reached, at which point it starts to reduce quickly.

Figure 20. Centrifugal acceleration across the vortex.

Flow Curvature in the Tip Vortices

Another source of error that is significant, and which is lost in PIV measurements, is the flow curvatureeffects. Because the location of particles at two instants of time X(t) and X(t + ∆t) are the only storedpieces of information, the correlation algorithm assumes the smallest distance between the two correlatedparticle images to be the distance traveled by the particle. Obviously, a significant error can be introducedas a result of this assumption. This is especially true near the vortex core axis, where the flow curvature issubstantial. This effect can be better understood using the schematic given in Fig. 21. It is apparent thatthis error always reduces with a reduction in the pulse separation time.

This form of bias error can be reduced by introducing a correction factor obtained for axisymmetricvortices. Based on the estimation of the displacement of particle images measured by PIV and the radialposition, it is possible to retrieve this lost curvature effect information. From the schematic shown in Fig. 21,it is apparent the original distance travelled by the particle is ∆S, i.e.,

∆S = θπ

180r (31)

However, the distance traveled by the particle image as seen by the PIV images is given by

∆X = 2r sin(

θ

2

)(32)

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Figure 21. Displacement estimation error from curvature effects.

Substituting the value of θ derived from Eq. 32 into Eq. 31 results in the corrected displacement, with thecorrection factor being given by

∆S = ∆Xcorrected = 2r sin−1

(∆Xmeasured

2r

)(33)

Correcting the measured velocity by this factor retrieves some of the lost curvature information, as shownin Fig. 22.

Figure 22. Normalized error from curvature effects in a tip vortex.

The foregoing discussion helps in estimating the upper and lower bounds on the pulse separation time.It is apparent that the pulse separation time should be as small as possible (i.e., a very small particledisplacement) to reduce the errors associated with flow acceleration and curvature effects. However, thedisplacement of seed particles should be larger than the minimum displacement that can be accuratelyestimated by the correlation technique.

Typically, recent correlation techniques are capable of estimating the particle displacement within 0.1pixels. In the present study, the chosen pulse separation time of 1µs results in an approximately 1 pixeldisplacement at the vortex core boundary, which is where the peak swirl velocity is observed. For a magni-fication of 35µm per pixel and a pulse separation time of 1µs, this would mean that any estimated velocity

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value that is smaller than 3.5m/s is not reliable.However, the upper bound for the pulse separation time is typically dictated by the correlation analysis

software used. For the correlation software used in the present study, the maximum window size cannotbe larger than 128 pixels. To have relatively good correlation in the first iteration (before shifting thewindows in successive iterations), the displacement of particles should be less than one-fourth of the windowsize (i.e., 32 pixels or 1,120 m/s). An increase in the pulse separation time (that eventually increases theparticle displacement) reduces the minimum velocity that can be accurately measured, while also reducingthe accurately measurable maximum velocity.

Velocity Gradient Bias & Spatial Resolution

The spatial resolution of any measurement technique is dictated by the measurement size (volume) of theprobe. In 2-D PIV, the interrogation window size limits the minimum measurable length scale of any coherentstructure present in the flow field. Generally, based on Nyquist theorem, this is

∆X ≤ 14L (34)

where L is the length scale of the coherent structure. It is clear from the above equation that any accurateinformation on the coherent structure of length L requires an interrogation window size to be smaller thanone-fourth of its length scale. This suggests that for accurately measuring a smaller length scale such as atip vortex, requires window sizes that are very small indeed.

However, it should be understood that the accuracy in estimating the correlation peak increases with theincrease in the interrogation window size. This is because of the presence of more image pairs that contributetowards the estimation of the correlation peak. These two contradictory requirements suggest the need fora larger window size (in terms of pixels), that actually represent a smaller region in the flow field. This canbe either achieved by increasing the resolution of the camera or by reducing the region of focus and, thereby,increasing the magnification, i.e., by using the equation

∆X ≤ L

4R

ROF(35)

where R is the camera resolution and ROF is the region of focus. Here, the ratio R/ROF is the magnification(in pixels/mm).

However, it should be understood that reducing the region of focus negates the advantage of PIV becausethe entire set-up needs to be moved to resolve flow details (such as the convecting tip vortices) for every fewwake ages. Even though, Eq. 35 represents the condition that is to be satisfied for measuring any coherentstructure present in the flow field, it may not be suitable for high velocity gradient flows, such as near tipvortices.

Chue39 discussed the possible sources of error associated with measuring high-gradient flows using anymeasurement technique. The measurement error in the mean velocity was found to be a function of the ratioof the spatial resolution of the instrument and the length scale of the shear layer. This ratio is given by theequation

α =Lm

L(36)

where Lm is the spatial resolution of the instrument and L is the length scale of the shear layer. Kreid40

showed that the uncertainty in the flow measurement is directly proportional to the square of this ratio (α2).Grant41 showed that the dynamic range within the measurement volume should be less than 20%. Martin etal.21 applied the theoretical method given by Kreid40 to a self-similar vortex, and estimated the minimumresolution required to resolve a high gradient flow field accurately (with uncertainty less than 0.1%). It wasfound that a probe volume length of 0.1rc is required, where rc is the core radius (length scale) in the caseof a tip vortex. This suggests that a minimum of 20 measurement points inside the vortex core to properlyresolve a normalized tip vortex. Using two measurement volume sizes (0.03% and 0.3%) obtained with LDV,Martin et al.12 showed that the resulting differences in the measured peak swirl velocity and the measuredcore radius is significant.

Table 1 gives an overview of the spatial resolution obtained in the present study when compared withother PIV measurements of rotor wakes and tip vortex flows. The table also includes one set of LDV

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measurements3 for comparison. It should be noted that if PIV measurements were unavailable for youngvortices, 5% of blade chord can be assumed as the core size of the vortex. All the measurements were madein hover, and the scale of the rotor can be identified from the chord and tip speed of the rotor.

Reference CT /σ Vθmax/Vtip Vtip Core Chord dVθ/drc Lm/rc ∆t

(m/s) rc/c (m) (/s) (µs)Martin & Leishman21 0.087 0.17 89.28 0.15 0.045 2546 1.31 50Van der Wall et al.23 0.057 0.25 218 0.05 0.121 3940 0.51 17Richard & Raffel42 0.064 0.32 210 0.07 0.270 7526 0.47 10McAlister4 0.095 0.40 87.2 0.051 0.104 6735 0.36 5Ramasamy & Leishman11 0.083 0.305 27.1 0.165 0.019 2631 0.14 2Present experiments 0.064 0.33 89.28 0.032 0.045 21580 0.08 1Martin & Leishman3(LDV) 0.064 0.329 89.28 0.029 0.045 22553 0.03 -

Table 1. Pulse separation time and spatial resolution for various PIV measurements in rotor wakes.

In the case of PIV, which is based on statistical measurements of particle displacements, the displacementgradient across the measurement volume (interrogation window) will result in biased data because all particlespresent in the first window may not be present in the second window, even if the second window is displacedto account for the mean (average) particle displacement. As a result, the estimated mean displacementwithin the interrogation window will be biased towards lower pixel displacements. This is because of thehigher number of low velocity image pairs that contribute to the estimation of the correlation peak.

By plotting the uncertainty associated with the velocity gradient for two different window sizes (usingthe same normalized particle intensity), Raffel et al.37 found that the smaller windows tolerate the velocitygradients better than the larger windows. The uncertainty was higher for the larger windows. This is becauseof the wider correlation peak that occurs when using larger windows, which is directly proportional on thelocal dynamic range. It was concluded that smaller windows are best suited for high velocity gradient flows.

However, the interrogation window should be chosen such that the displacement gradient within theinterrogation window (or its dynamic range) is not more than the size of the particle image. Any largerdynamic range will result in a larger correlation signal that introduces errors into the displacement estimates.This clearly suggests that the interrogation window should be chosen by taking into account the size of theparticle images as well as the velocity gradient in the flow field. For a uniform grid (with constant distancebetween adjacent nodes through out the region of focus) and a constant particle image size, the requirementsbecome clear for high velocity gradient flows, i.e.,

∆Vi,i+1∆t ≤ dτ (37)

where ∆Vi,i+1 is the difference in velocity between adjacent nodes, dτ is the particle image size, and ∆tis the pulse separation time. This means that for a dynamic range of 35 m/s, as found in the rotor wakeflow examined in the present study (0 m/s at the vortex centers and 35 m/s at their core boundary), themaximum difference in velocity between adjacent nodes for a particle size of 4µm at 1µs is 4 m/s. Thisclearly suggests that there should be at least 9 points from the center of the vortex (its axis) to its coreboundary to enable accurate velocity gradient estimations. The present measurements clearly have morethan sufficient points for estimating accurately the flow velocities. The absolute value of the velocity ordisplacement (mean value across the window) may not have any influence on the velocity estimation becausethe window shifting nature of the algorithm takes care of the mean flow value.

The importance of optimizing the interrogation window size in measuring accurately the velocity gradientis apparent from the last three sets of measurements in Table 1. It can be seen that the PIV measurementsmade in the present study, with the spatial resolution similar to that of LDV, correlates well with the LDVmeasurements when determining the essential properties of a tip vortex. This includes core size, peak swirlvelocity, and velocity gradient. In the work of Ref. 11, measurements were made on a very small rotor(suitable for a rotating-wing type of micro-air vehicle) that was operating at very low tip Reynolds numbers.As a result, the vortex core size was reported to be larger than that expected based on previously reportedsub-scale measurements. This, in turn, resulted in lower measured velocity gradients, which is a result ofthe physics of the flow at this operating condition and not the shortcomings of the measurement technique.

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(a) Swirl velocity. (b) Axial velocity.

Figure 23. Measured velocity distribution at various resolutions.

(a) Swirl and axial velocity. (b) Core size.

Figure 24. Measured properties of the tip vortices at various resolutions.

Figures 23(a) and (b) show the swirl and axial velocity distribution estimated through interrogationwindow of various sizes. Here, except for the window width, all other factors remain the same. An increasein window size increases the chances of having more number of correlation image pairs. However, for highvelocity gradient tip vortex flows, the correlation will be biased towards the lower velocity, as mentionedearlier. As a result, the estimated flow velocity values throughout the vortex are much lower when comparedwith their higher resolution counterparts.

Figures 24(a) and (b) show the measured peak swirl and axial velocities and the radius of the vortexcore that are estimated using various interrogation windows of different sizes. It is apparent that both thevelocities, as well as the normalized core radius, asymptote to a constant value as the interrogation windowreduces. An interrogation window size of 8 pixels with 50% overlap (Lm/rc ≈ 0.08) seems to correlate wellwith the LDV measurements, suggesting that the current measurements of the rotor wake flow made withPIV are quantitatively reliable.

Conclusions

An experiment has been conducted to identify and measure the sources of uncertainty associated withthe application of PIV to measuring the wake flow of a rotor. Various critical factors have been found to havea significant influence on the estimation of flow velocities in the high velocity gradient, streamline curvature

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flow typical of those found in the rotor wake and in the blade tip vortices. The biggest source of uncertaintylies in establishing the optimum laser pulse separation time and the imaged interrogation window size. Aninappropriate value of either of these can completely change the magnitude and length scale of the measuredflow structures.

The following are some specific conclusions that were derived from this study:

1. PIV can be successfully employed for measuring the vortical flows and tip vortices trailing from ahelicopter rotor blade. Comparisons of the measured flow velocities between PIV and LDV were foundin good agreement. The application of an iterative deformation grid increased the spatial resolutionof PIV (120µm) to that comparable to LDV (80µm). As a result, the interrogation window size (≥ 25measurement velocity vectors inside the vortex core) is more than sufficient to resolve the steep velocitygradients present inside the tip vortex cores.

2. Calibration uncertainty reduces with an increase in the displacement of the seed particles. On onehand, this would mean that the swirl velocity measurements are more reliable near the vortex coreboundary than near the vortex axis (where the flow velocities are small). On the other hand, the axialvelocity has its maximum value at the vortex core axis. However, the calibration procedure followedin this study was shown to be first-order accurate in the axial flow direction.

3. Optimizing the laser pulse separation time is essential to reduce the errors associated with flow acceler-ation and curvature effects in PIV measurements of rotor wakes. Measurement errors in displacementestimation increases with increase in pulse separation time for highly rotational flows such as tip vor-tices. The pulse separation time should be small as possible, however, it should be chosen such thatthe particle displacement is sufficiently larger than the minimum measurable displacement.

4. The spatial resolution of the high velocity gradient tip vortex flows that can be measured stronglydepend on the particle image size (seed particle size). The local dynamic range between adjacentnodes should not be more than the particle image size, especially within the vortex core.

Acknowledgments

This research was supported, in part, by ARO under the Multi-University Research Initiative (MURI)Grant ARMY W911NF0410176.

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