RESEARCH ARTICLE

Effect of tip vortex aperiodicity on measurement uncertainty

Mahendra J. Bhagwat • Manikandan Ramasamy

Received: 22 December 2011 / Revised: 4 June 2012 / Accepted: 25 June 2012 / Published online: 24 July 2012

� Springer-Verlag (outside the USA) 2012

Abstract Vortex aperiodicity introduces random uncer-

tainty in the measured vortex center location. Unless cor-

rected, this may lead to systematic uncertainty in the vortex

properties derived from the measured velocity field. For

example, the vortex core size derived from averaged or mean

flow field appears larger because of aperiodicity. Several

methodologies for aperiodicity correction have been devel-

oped over the past two decades to alleviate this systematic

uncertainty. However, these do not always reduce the

accompanying random uncertainty. The current work shows

that the analysis methods used to derive the vortex properties

from the measured velocity field play an important role in

the resultant random uncertainty in these properties; perhaps,

even more important role than the aperiodicity correction

methodology itself. It is hypothesized that a class of methods

called global methods, which use a large extent of measured

data, yield a smaller measurement uncertainty compared to

local methods. This hypothesis is verified using a newly

proposed global method based on a planar least-squares fit.

The general applicability of the method is demonstrated

using previous particle image velocimetry measurements of

rotor tip vortices. The results clearly demonstrate a reduced

random uncertainty in the vortex core properties, even in the

presence of secondary vortical structures. Furthermore, the

results are independent of the choice of aperiodicity cor-

rection methodology.

1 Introduction

Over the last decade, particle image velocimetry (PIV) has

found increasing use in rotor flow field and tip vortex mea-

surements, and with great success—see, e.g., Raffel et al.

(2007). The applications have spanned a wide range from

very large-field (e.g., Wadcock et al. 2011) to microscopic

(e.g., Ramasamy et al. 2010). As the PIV measurement res-

olution increases, attention is again focused on understanding

the associated measurement uncertainties. The two types of

uncertainties are systematic uncertainty, which refers to

accuracy, bias, offsets, etc., and random uncertainty, which

refers to variability, repeatability, precision, etc. Systematic

uncertainty is typically quantified using calibration and spe-

cially designed measurements.

A unique contributor to uncertainty in tip vortex mea-

surements is their wandering or meandering motion; this is

a seemingly random movement of the vortex center loca-

tion normal to the vortex axis. For rotor tip vortices, this is

often referred to as aperiodicity because the wandering

motion appears as variation in vortex center location from

one rotor revolution (period) to another. While tip vortices

exhibit some inherent wandering motion, the blade aperi-

odicity, i.e., the variation in blade location from one period

to another (see, e.g., van der Wall and Schneider 2007),

also contributes to the perceived wandering motion of the

tip vortex. This vortex aperiodicity appears as the random

uncertainty in the measured tip vortex location. However, it

also results in additional systematic uncertainty called

‘‘Type II bias,’’ which is a result of transformation of the

measurement variable(s) through a nonlinear model. For

example, the vortex aperiodicity leads to a larger perceived

vortex core size. Several methods have been developed to

correct this uncertainty due to vortex aperiodicity. How-

ever, these methods do not always ensure that the random

M. J. Bhagwat (&)

US Army Aeroflightdynamics Directorate (AMRDEC)

Ames Research Center, M/S 215-1,

Moffett Field, CA 94035, USA

e-mail: mahendra.bhagwat@us.army.mil

M. Ramasamy

UARC/AFDD, NASA Ames Research Center,

M/S 215-1, Moffett Field, CA 94035, USA

123

Exp Fluids (2012) 53:1191–1202

DOI 10.1007/s00348-012-1348-7

uncertainty does not grow in the process of removing or

minimizing the systematic uncertainty. In this paper, the

aperiodicity correction methodologies are reviewed in the

context of their effect on the overall resulting measurement

uncertainty in the vortex properties.

Vortex wandering has been a particularly challenging

problem for point measurement techniques like hot-wire

anemometry (HWA) or laser Doppler velocimetry (LDV),

where an averaged flow field can only be constructed by

independently averaging measurements at each point. One

of the earliest attempts to assess the effect of wandering on

tip vortex core properties is reported by Baker et al. (1974).

Using this aptly called ‘‘simple average,’’ the wandering

motion also contributes to velocity fluctuations at a fixed

location, which may be mistaken for turbulence. This effect

can be eliminated by conditional sampling (Takahashi and

McAlister 1987) or must be corrected using some measure

of the vortex wandering. Devenport et al. (1996) present an

analytic correction procedure based on measured wander-

ing magnitude using an assumed velocity profile. Leishman

(1998) presents a similar numerical correction procedure

using the measured vortex velocity profiles. Planar mea-

surement techniques like PIV present other alternatives to

help mitigate the effect of vortex aperiodicity.

One of the first PIV measurements of rotor tip vortices

by Raffel et al. (1998) showed that averaging several PIV

planar measurements (images) gives close agreement with

the corresponding LDV measurements, which inherently

include vortex aperiodicity effects in the measured aver-

aged flow field. Instantaneous planar PIV measurements

provide the ability to identify and correct for the aperio-

dicity effects. Yamauchi et al. (1999) present tip vortex

PIV measurements for the TRAM rotor and describe the

aperiodicity correction methodology in detail. The proce-

dure is similar in spirit to conditional sampling, where only

a small set of samples with the vortex centers close toge-

ther are chosen. The individual samples are aligned using

their respective vortex centers prior to averaging, i.e., the

measurements are ‘‘corrected’’ for the movement of vortex

center location before averaging—hence the name ‘‘cor-

rected average’’. Unlike point measurement techniques,

PIV allows one to obtain the vortex core properties from

each individual planar measurement (image) as well. The

average of these individual (instantaneous) measurements

would also be representative of mean vortex properties.

To better understand these aperiodicity correction

methodologies, examples of simple average and corrected

average are shown in Fig. 1. Measurements are shown as

velocity vector field colored with velocity magnitude, with

the red being the highest and the blue being the lowest. For

visualization purposes, the vortex center is shown with a

filled circle, and the approximate circular vortex core

boundary, corresponding to the local maxima in swirl

velocity, is shown with a dotted line. The vortex core

shows a data void because of the difficulties in seeding the

flow inside the core. Large variations in the vortex center

location are readily apparent even in these four instances.

When a number of such individual images are averaged to

obtain mean flow properties, the resulting flow field is

smeared because of the variation in the vortex center

location. This is clearly seen in the simple average flow

field, where the vortex appears larger in size, and with

lower swirl velocity, than any of the instantaneous mea-

surements (core boundaries shown by dotted lines).

An intuitively better methodology for obtaining the

mean flow characteristics is to first align the instantaneous

measurements using their respective vortex centers, as

shown in Fig. 1. This is the central idea of the corrected

average or conditional average for aperiodicity correction.

In this case, the averaging does not produce the smearing

effect seen with simple average. In the corrected average

flow field, the vortex looks nearly the same size as the

instantaneous measurements.

The third methodology is the individual average, which

does not ‘‘correct’’ the individual measurements for ape-

riodicity. Instead, the vortex properties are derived from

each instantaneous measurement and then averaged to give

mean vortex properties. This also readily provides an

estimate of measurement uncertainty in terms of mean and

standard deviation of the desired vortex property.

Ideally, any of these methodologies would be equally

effective. However, Yamauchi et al. (1999) found that the

average of individual vortex measurements was signifi-

cantly different from the ‘‘corrected’’ average obtained by

aligning the individual measurements. This was attributed

to large scatter in the individual vortex properties, i.e., to

measurement uncertainty. Later experiments on a model-

scale rotor (Ramasamy et al. 2007) also showed such dis-

crepancy between averaged individual measurements and

corrected average profiles, while for the HART II mea-

surements, the conditional average is considered manda-

tory (Lim and van der Wall 2010). One motivation for the

current work is to better understand and, eventually, to

eliminate these inconsistencies.

The underlying experimental uncertainty in the velocity

measurements itself is outside the scope of the current

work. However, the choice of analysis method is important

in controlling the propagation and amplification of this

uncertainty into derived quantities of interest like the

vortex core size and circulation. Determining the vortex

center from the velocity field is an important first step for

the corrected average process. van der Wall and Richard

(2006) suggest that a convolution-based method improved

the accuracy of vortex center identification and resulted in

reduced scatter in the derived vortex properties of interest.

This claim is explored further in this paper, and the

1192 Exp Fluids (2012) 53:1191–1202

123

improvement in the final measurement uncertainty in vor-

tex properties resulting from the vortex center identifica-

tion method is examined. Note that while vortex

aperiodicity can be a big contributor to the uncertainty in

the derived vortex properties, the methods used to derive

those properties can be even bigger contributors.

The current work briefly reviews various methods for

determining the vortex center and other properties of

interest, and how these methods can potentially affect the

final uncertainty. The aperiodicity correction removes a

systematic uncertainty in the vortex core radius (and peak

swirl velocity). However, the analysis method determines

how the random uncertainty propagates to the derived

quantity. It is important to ensure that the method does not

add/amplify random uncertainty which may mask the

benefits of aperiodicity correction. A new method based on

a planar least-squares fit is proposed for obtaining both the

vortex center and other vortex properties of interest. This

method is shown to give reduced uncertainty in derived

quantities like vortex core radius, circulation, etc. Fur-

thermore, with the planar fit method, the three aperiodicity

correction methodologies give self-consistent results. The

method is first demonstrated using the AFDD model-rotor

tip vortex measurements (Ramasamy et al. 2010) and then

extended to other measured flow fields including the

TRAM rotor measurements (Yamauchi et al. 1999) and the

HART II measurements (Lim and van der Wall 2010).

2 Methodology

This section gives an overview of tip vortex aperiodicity

correction, followed by methods to determine the vortex

properties. These latter methods are categorized into local

and global methods based on the spatial extent of the data

used relative to the vortex core. The newly proposed

method, a planar least-squares fit, is then described in detail.

2.1 Aperiodicity correction

As described earlier in the introduction, there are three

categories of aperiodicity correction methodologies: the

simple average (SA), the corrected average or conditional

average (CA), and the individual average (IA). The SA can

Fig. 1 Schematic showing simple average and corrected average using a few sample PIV vector fields. The simple average gives a larger core

size than the original samples, whereas the corrected average gives nearly the same core size

Exp Fluids (2012) 53:1191–1202 1193

123

be traced back to fixed-point measurement techniques like

LDV/HWA and is often combined with some empirical or

semi-analytic aperiodicity correction—see Devenport et al.

(1996) and Leishman (1998). In the current work, the

analytic expression given by Devenport et al. (1996) is

used to correct the core radius obtained using SA. The

corrected average and individual average are possible only

with a field measurement technique like PIV, and can be

computationally intensive. However, these are applicable

even with arbitrarily large aperiodicity magnitude, unlike

the aperiodicity correction applied to the SA. The third

method, IA, has its advantages in simplicity; no flow field

averaging is required, and only the vortex properties

derived from each individual (instantaneous) measurement

are averaged. This also provides an estimate of measure-

ment uncertainty in the form of mean, l, and standard

deviation, r, of each vortex property of interest, i.e., the

measured property has a 68 % confidence level in

the l ± r interval.

All aperiodicity correction methodologies belong to one

of these three categories and differ only in their imple-

mentation details like the method for determining the

vortex center and/or other vortex properties. These meth-

ods are divided into two categories: ‘‘local’’ methods and

‘‘global’’ methods. As the names would suggest, the local

methods are mainly based on the flow field in close vicinity

of the vortex center while the global methods are based on

a much larger flow field surrounding the vortex. In other

words, the local flow field near the vortex center is the most

important for the local methods. It should be noted that the

global/local categorization is not very rigid; it is simply a

means to explain the observed differences in these meth-

ods. It seems intuitively obvious that a global method that

uses a large amount of measured data would be better than

local methods that use a very small subset confined to the

vicinity of a local maxima (often including measurements

with lower signal-to-noise ratio and excluding those with

possibly higher signal-to-noise ratio). However, this

hypothesis remains to be carefully examined.

2.2 Methods to determine vortex center

The most common methods to calculate the vortex center

from measured flow field are based on some flow invariant

property like vorticity, k2, the Q-criterion, etc. These flow

quantities exhibit a local extremum at the vortex center—

hence the name ‘‘local methods.’’ Commonly used local

methods include the measurement grid node that carries the

maximum value of vorticity (e.g., Vogt et al. 1996; Hei-

neck et al. 2000; Yamauchi et al. 1999; Ramasamy et al.

2007a), helicity, axial velocity (e.g., Ramasamy et al.

2009), or Q-criterion (e.g., Ramasamy et al. 2009; van der

Wall and Richard 2006). Centroid or area center of

vorticity (e.g., Ramasamy et al. 2009; Lim and van der

Wall 2010) has also been used to determine the vortex

center. Ideally, the vortex center defined by any of these

criteria is the same. However, the vortex center estimates

do not always agree (e.g., Lim and van der Wall 2010;

Ramasamy et al. 2007a). Cucitore et al. (1999) present a

summary of the effectiveness and limitations of such local

vortex identification criteria.

One short-coming of all of these methods is that they

rely mostly on measurements near the vortex core. The

measurements near the vortex core have a lower signal-to-

noise ratio and correspondingly larger statistical errors

because of the challenges in seeding the flow around the

vortex core. This often results in a seed void near the

vortex center and a corresponding, but often smaller, data

void. In fact, the seed void is a fairly reliable indicator of

vortex center location (e.g., Leishman 1996; McAlister

2004) and has even been used for corrected averaging

(McAlister 2004). For the AFDD model-rotor vortex

measurements, data near the vortex core with low signal-

to-noise ratio were eliminated and were not replaced using

interpolated vectors, rendering local methods inapplicable.

A global method is clearly necessary. One global method

found in the rotorcraft literature is the convolution-based

method used by van der Wall and Richard (2006); this

showed improved accuracy in determining the vortex

center. Although the measurements reported by van der

Wall and Richard (2006) did not show a data void near the

center, the results definitely demonstrated the benefit of

using a larger measured flow field rather than only a few

measurement points inside the vortex core.

2.3 Methods to determine vortex properties

Having determined the vortex center, the next step is to

determine key vortex properties like vortex core radius,

peak swirl velocity, circulation, etc. The choice of methods

for determination of vortex core properties from either the

averaged or the instantaneous flow field is also independent

of the aperiodicity correction. For the CA method, the

vortex center for each individual measurement must be

determined prior to determining the vortex core properties

from the averaged flow field. For SA, the subsequent cor-

rection, like the Devenport correction, also needs the vor-

tex center locations (aperiodicity magnitude). In fact, the

accurate estimation of the vortex center is a necessary

prerequisite for the accurate estimation of the vortex core

properties (van der Wall and Richard 2006).

The methods to determine vortex properties can also be

divided into local and global categories. In this case, the

local methods are usually based on measurements along

straight-line cuts through the vortex center, perhaps using

the analysis techniques developed for LDV/HWA

1194 Exp Fluids (2012) 53:1191–1202

123

measurements. The peak swirl velocity along these cuts

defined the vortex core radius and asymptotic value of

circulation defined the vortex strength (Yamauchi et al.

1999). These methods are considered local methods

because the core radius and peak swirl velocity are deter-

mined from a local maximum in velocity. A variation of

this uses a linear curve fit (based on an assumed vortex

model) to the data along a straight line (e.g., Lim and van

der Wall 2010; Bhagwat and Leishman 2000). A global

method based on a fit to the entire measurement plane (or a

suitably large enough subset of that) was used by van der

Wall and Richard (2006) with good success.

2.4 New method: planar least-squares fit

The newly proposed method is based on a planar least-

squares fit of the measured velocities to those given by an

assumed vortex model. The center of the vortex (xc, yc) is

defined in the measurement coordinates, x; y; zð Þ. The

vortex convection velocity or background velocity is rep-

resented by uc; vcð Þ; also in the measurement coordinates.

In general, the measurement plane is not normal to

the vortex axis. Two rotation angles are required for the

transformation from the measurement coordinates to the

vortex coordinates, xv; yv; zvð Þ. This is schematically shown

in Fig. 2 where the PIV measurement window is shown

along with the measurement and vortex coordinates. A first

rotation of hy around the y-axis results in an intermediate

coordinates x1; y1; z1ð Þ and a subsequent rotation hx around

the x1-axis gives the vortex coordinates. The transforma-

tion from the measurement coordinates to the vortex

coordinates is then given in terms of the transformation

matrices Rx and Ry as

Rx ¼1 0 0

0 cos hx sin hx

0 � sin hx cos hx

264

375 Ry ¼

cos hy 0 � sin hy

0 1 0

sin hy 0 cos hy

264

375

xv

yv

zv

8><>:

9>=>;¼ Rx½ � Ry

� � x� xc

y� yc

z

8><>:

9>=>;

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2

v þ y2v

qð1Þ

where r is the vortex radial coordinate. The vortex induced

radial, axial and swirl velocities Vr;Vz;Vhð Þ are

transformed to the Cartesian vortex coordinates as

uv; vv;wvð Þ through the transformation matrix Rc as

uv

vv

wv

8<:

9=; ¼ RC½ �

Vr

Vz

Vh

8<:

9=; RC½ � ¼

xv

r 0 � yv

ryv

r 0 xv

r0 1 0

24

35 ð2Þ

These velocities are then transformed back to the

measurement coordinates using the inverse transform.

The results shown in current work use the Lamb-Oseen

model for vortex induced swirl velocity, i.e.,

Vh ¼Cv

2pr1� exp �a r=rcð Þ2

� �� �ð3Þ

where Cv is the vortex circulation, rc the core radius and

a = 1.25643 is a constant parameter. The resulting

function for the vortex velocity field is given by

f rc;Cv; xc; yc; uc; vc; hx; hy

� �¼ Ry

� �TRx½ �T RC½ �

Vh

0

0

8><>:

9>=>;

þuc

vc

0

8><>:

9>=>;

ð4Þ

The planar least-squares fit uses the above function with

the eight independent variables, viz., rc;Cv; xc; yc; uc; vc; hx;

hy; to obtain the best match with measured velocities.

The laminar Lamb-Oseen model is perhaps the most

commonly used viscous vortex core model, and has also

been employed to identify vortex structures in separated

flows (e.g., Schram et al. 2004; Cierpka et al. 2008). The

proposed method, however, is not restricted to the laminar

Lamb-Oseen model and can readily incorporate other

vortex models. It was shown by Ramasamy et al. (2010)

that the choice of vortex models did not affect the vortex

center location—each model gave a slightly different core

radius corresponding to different values of the core to total

circulation ratio. Therefore, the vortex model should not be

predetermined, but should rather be chosen based on the

actual measured flow field. All the results shown in the

current work correspond to relatively low vortex Rey-

nolds numbers Cv=mð Þ; where m is the kinematic viscos-

ity. Therefore, the Lamb-Oseen model was found to be

x1

z1

y1=y

xy

z

y

x

Measurement window

Vortex coordinates

Measurement coordinates

Vortex axis

Vortex center xc, yc

xv

yv

zv

xv =x1

yv

zv

Fig. 2 Inclination of vortex axis relative to measurement plane

Exp Fluids (2012) 53:1191–1202 1195

123

satisfactory. Furthermore, the methodology can easily

include all three vortex induced velocity components,

although only the vortex induced swirl velocity is used

here.

3 Results and discussions

In this section, the results obtained using the planar fit are

shown for three different rotor tip vortex measurements.

The first set of measurements is the model-scale rotor tip

vortex measurements made by Ramasamy et al. (2010) at

AFDD using microscopic-PIV using highly twisted tilt-

rotor blades. The second set of measurements are the

TRAM rotor measurements originally reported by Yama-

uchi et al. (1999), and the final set of measurements is the

HART II measurements at position 17, studied in detail by

all the HART II partners, and summarized by Lim and van

der Wall (2010). The HART II measurements in hover

reported by van der Wall and Richard (2006), and a small-

scale rotor measurements from University of Maryland

(UMD) by Ramasamy et al. (2007b) are also shown for

comparison.

3.1 AFDD model-rotor tip vortices

The measurements presented here were performed on a

three-bladed model-scale rotor system with a highly twisted

rotor blade that was similar to the XV-15 blade. The blade

had a radius R = 656 mm and chord c = 49 mm and was

operated at a rotational speed of X ¼ 800 rpm, i.e., tip speed

of XR ¼ 55 m/s. Results at a rotor thrust coefficient

CT = 0.0076 are shown at two early vortex ages of 4� and

15�. The thrust coefficient is defined as CT ¼ T=qpR2ðXRÞ2;where T is the rotor thrust and q is the air density. The vortex

age, wv; is the azimuth behind the blade; 4� and 15� corre-

spond to a downstream distance of 0.9 chords and 3.5 chords

or a non-dimensional time based on tip speed and chord of

0.0008 and 0.003, respectively.

Figure 3 shows a summary of all the vortex properties

for CT = 0.0076 and wv ¼ 15�. A map of vortex center

locations and the representative core size are shown in

Fig. 3a. The grid lines correspond to the measurement grid

and show the high resolution achieved using a microscope.

Recall that the corrected average procedure used in the

current work does not re-mesh and interpolatemeasure-

ments after aligning the vortex centers. As a result, the

corrected average does not completely eliminate the ape-

riodicity, but only reduces it to a value less than one grid

cell. The residual aperiodicity can be seen in Fig. 3a from

the corrected vortex center locations. The distribution of

vortex center locations xc and yc shown in Fig. 3b is sug-

gestive of a Gaussian distribution. The Devenport

correction is based on the deconvolution of a Gaussian

wandering motion; the observed distribution suggests that

such an analytic correction would indeed be applicable.

Figure 3c and d show the results for the vortex axis

orientations, hx and hy, and the mean vortex convection

velocities, uc and vc, respectively. The vortex core radius

and circulation are shown in Fig. 3e and f, respectively. All

three aperiodicity correction methodologies give results

consistent with each other for all vortex properties. For the

core radius, in Fig. 3e, the SA method gives a larger core

radius as expected; the subsequent Devenport correction

gives a core radius that closely agrees with the CA and IA

results. Notice that in all cases, the vortex properties show

a roughly Gaussian distribution. The normalized standard

deviation in circulation and core radius is about 5 %, which

is indicative of fractional random uncertainty. The differ-

ences in vortex properties obtained using the three aperi-

odicity correction methodologies are smaller than one

standard deviation.

The measurements are compared with three previously

reported rotor PIV measurements: the UMD measurements

from Ramasamy et al. (2007b) of small-scale rotor tip

vortices; the TRAM measurements from Yamauchi et al.

(1999) for the high thrust case; and the HART II mea-

surements from Fig. 12 of van der Wall and Richard

(2006), for the wv ¼ 44� hover case. The corresponding

vortex aperiodicity magnitudes (standard deviation of

vortex location normalized by core radius) are also shown

for comparison. Interestingly, the two measurements with

higher aperiodicity magnitudes showed lower resulting

measurement uncertainties. It appears then that uncertainty

must result from other factors like the measurement

resolution.

For tip vortex measurements, resolution is not compared

as an absolute length scale, but relative to the vortex core

being measured. The ratio of the measurement length,

lm, or the PIV measurement window size, to the core

radius, i.e., lm/rc, has been used in the past as a measure of

relative PIV resolution. This is schematically shown at the

top of Fig. 4 with the squares representing the PIV pro-

cessing window relative to the vortex core size shown by

the circle. The AFDD model-rotor measurements have a

significantly lower lm/rc ratio owing to the use of a

microscopic lens. However, this appears to have played

only a small part in the resulting measurement uncertainty.

The UMD measurements have a small lm/rc ratio but a

larger uncertainty, while the HART II measurements have

a larger lm/rc ratio but a smaller uncertainty.

It must be reiterated that the uncertainty in derived

quantities such as the vortex core radius, or circulation, is

not the same as measurement uncertainty. Instead, it is the

compound result of how the underlying measurement

uncertainty in the velocity propagates into the derived

1196 Exp Fluids (2012) 53:1191–1202

123

(a) (b)

(d)(c)

(e) (f)

0

5

10

15

20

0.03 0.04 0.05 0.06 0.07 0.08

No.

of s

ampl

es

Vortex core radius, rc /c

SA=0.0746308SA+D=0.05717CA=0.0548237IA=0.056778±0.00333122

Fig. 3 Example of vortex properties obtained using the planar fit for the

AFDD model-rotor tip vortex measurements, CT ¼ 0:0076;wv ¼ 15�.Distribution of individual samples is shown as histograms along with the

simple average, corrected average, and individual values (mean and

standard deviation) are shown. a Vortex center locations before and after

center alignment, b Vortex convection velocities, uc, vc, c Vortex center

locations, xc, yc, d Vortex axis orientation angles, hx, hy, e Vortex core

radius, rc, and f Vortex circulation, Cv

Exp Fluids (2012) 53:1191–1202 1197

123

vortex properties (like core radius/circulation), along with

any additional uncertainty associated with the derivation

method itself. Therefore, it is important to examine the

analysis methods and to ensure that their contribution to

the resulting uncertainty, over and above the underlying

measurement uncertainty, is minimal. For example, gradi-

ent-based criteria can be severely contaminated by mea-

surement noise and lead to incorrect vortex center

identification (Kindler et al. 2011), and larger uncertainty

in the derived vortex properties. One advantage of the

current planar least-squares fit is that it directly uses the

measured flow field and does not require numerical com-

putation of flow derivatives.

One key difference between the four results shown in

Fig. 4 lies in the analysis methods—clearly, they have a

strong effect on the final perceived measurement uncer-

tainty. The UMD and TRAM experiments used local

methods for obtaining the vortex center and other vortex

properties. The HART II results, in particular, the results

presented in this figure from van der Wall and Richard

(2006), used global methods. The current work also uses a

global method as described earlier. Figure 4 strongly sug-

gests that using global methods for data analysis result in

significantly lower uncertainty in derived vortex properties.

This working hypothesis is tested here in two ways: first by

reanalyzing the AFDD measurements using a local method,

and then by reanalyzing previous measurements using a

global method (the proposed planar fit).

3.2 Local versus global method

The AFDD model-rotor measurements were analyzed

using a local method are shown in Fig. 5 for the case of

CT ¼ 0:0076;wv ¼ 4�. In this case, the vortex center was

first identified using the planar least-squares fit. The vortex

properties were, however, not obtained from the fit.

Instead, horizontal and vertical cuts were made through the

vortex center, and the vortex core was identified using the

peak swirl velocity along these cuts, the distance of the

peak swirl location from the vortex center giving the core

radius. These results are shown as a scatter plot in Fig. 5a

with the core radius for each measurement sample along

with the averaged values. The peak swirl velocity location

is not interpolated to a sub-grid level, but is restricted by

the grid resolution. This introduces some additional

uncertainty in the vortex core size. The results show some

inconsistencies—the SA core radius is much smaller than

0 5

10 15 20 25 30 35 40

UMDRamasamy et al

(2007b)

TRAMYamauchi et al

(1999)

HART-IIvan der Wall &

Richards (2006)

Currentwork

Unc

erta

inty

, σ/

μ (%

)

Core radiusCirculation

0

10

20

30

40

50

60A

perio

dici

ty,

σ/r c

(%

)

Fig. 4 Measurement uncertainty in the vortex core radius and

circulation compared for various experiments including the AFDD

model-rotor measurements. The vortex aperiodicity magnitude (stan-

dard deviation normalized by vortex core radius) is also shown for all

cases along with a schematic representation of the vortex core and the

relative PIV window size

(a) (b)

Fig. 5 Vortex core radius for the AFDD model-rotor derived from the measured velocity field using a a local method (based on peak swirl

velocity along horizontal/vertical cuts through the center), and b a global method (the planar fit). CT ¼ 0:0076;wv ¼ 4�

1198 Exp Fluids (2012) 53:1191–1202

123

the CA result and the IA result. The IA results show a

significantly large scatter of about 23 %, compared to those

obtained using the planar least-squares fit shown in Fig. 5b.

The local method clearly introduces a much larger uncer-

tainty in the vortex core radius as compared to the global

method.

This observation reinforces the intuition that the analysis

method used to derive the vortex properties adds to the

perceived measurement uncertainty, with the local methods

giving significantly larger uncertainty than the global

methods. Some possible contributing factors are asym-

metric data void and presence of spurious data points near

the vortex core—see Ramasamy et al. (2010). Having

confirmed the hypothesis that global methods for deter-

mining vortex center and vortex properties are better, the

planar fit is applied to two of the previous measurements

shown in Fig. 4 for further reinforcement.

3.3 TRAM measurements

The planar least-squares fit was applied to the TRAM

results from Yamauchi et al. (1999). Two cases were

considered: a high thrust case and a low thrust case that has

two counter-rotating vortices. These are forward flight

measurements, and the PIV flow field shows large mean

vortex convection velocities and larger vortex axis orien-

tation angles. Another complication was that the PIV

measurement window often contained two vortices in close

proximity. Yamauchi et al. (1999) overcame this by simply

making smaller analysis windows surrounding each indi-

vidual vortex. In the current work, however, the planar

least-squares fit can be easily modified to include two

vortices.

The vortex core radii derived using this two-vortex

planar fit are summarized in Fig. 6. The SA, CA and IA

results are compared with those reported by Yamauchi

et al. (1999). Note that only one vortex for the high thrust

case was reported in that work, whereas the current work

includes results for both the vortices. As expected, the SA

results are similar for the two methodologies but the CA

results show a consistent behavior with the planar fit. The

IA results for the planar fit show a much smaller standard

deviation than the previous results. The resulting fractional

uncertainty is about 8 % using the proposed new method,

compared to about 25 % as reported by Yamauchi et al.

(1999). This again confirms the hypothesis that a global

method results in a much smaller measurement uncertainty

than a local method.

3.4 HART II measurements

The planar fit was also applied to the HART II measure-

ments at position 17. This position has been studied in

great detail by the HART II partners, and a comparative

summary of analyses performed by US Army AFDD, DLR

and ONERA are presented by Lim and van der Wall

(2010). In this case, the flow field was characterized by the

presence of strong vortical flow structures in close vicinity

of the vortex core. An example flow field in Fig. 7 shows a

relatively strong vortex sheet in close vicinity of the vortex

core being examined. In this case, the contours represent

the velocity magnitude, and at least a part of the vortex

core boundary can be clearly identified by the large swirl

velocity region near the top-right corner. The derived core

0.02

0.04

0.06

0.08

0.1

0.12

0.14

CCW

low

CWCT

Yamauchi et al (1999)

CCWhigh CT

CCW

low

CWCT

current

CCW-1

high

work

CCW-2CT

Vor

tex

core

rad

ius,

rc

/c

IA ± std.dev.SASA+DevenportCA

Fig. 6 Core radius calculated from the flow field using the planar fit

compared with results presented by Yamauchi et al. (1999), where

only one vortex was reported for the high thrust case. The results are

shown as IA with standard deviation together with CA and SA values.

Current work include the Devenport correction with SA

localglobal

Vortex core given by different methods

Vortex sheet coordinates

xs

ys

Fig. 7 Simple average flow field for the HART II position 17 shown

as velocity vectors along with contours of velocity magnitude, blue is

the lowest and red is the highest. The vortex center location and size

identified using four different methods are also shown

Exp Fluids (2012) 53:1191–1202 1199

123

centers and sizes using four different methods are also

shown in the figure with different colors.

The planar fit can be easily modified to account for the

presence of a vortex sheet. The sheet appears like one half

of a vortex with the other half stretched along the sheet

length. A quick representation of such vortex sheet was

devised using the Lamb-Oseen vortex profile at the sheet

edge (semicircular region), combined with the tangential

velocity along the length of the sheet also given by the

Lamb-Oseen profile as a function of normal distance from

the sheet. In the vortex sheet coordinates, xs; ysð Þ; as shown

in Fig. 7, with the vortex sheet extending along the nega-

tive xs-axis, the sheet induced velocity field given by

xs� 0 Lamb-Oseen vortex (Eq. 3)

xs� 0 us ¼ Cs

2py 1� expð�a y2

r2s

� �; vs ¼ 0

ð5Þ

The planar least-squares fit now includes the sheet location,

circulation strength (Cs), core radius (rs) and orientation

axes as additional parameters. Note that the sheet repre-

sentation requires one extra orientation angle to account for

in-plane rotation of the sheet (orientation of the sheet axes

xs, ys). Such a simple representation is, perhaps, sufficient

to eliminate the influence of the vortex sheet on the tip

vortex core properties. However, a more appropriate rep-

resentation must be chosen to model the vortex sheet if the

vortex sheet properties are also of interest.

Figure 7 shows that the vortex center location and core

radius obtained using the proposed planar fit are in good

agreement with prior calculations. The global methods

show a similar behavior and are, perhaps, least affected by

the presence of the vortex sheet in the flow field. The

vortex core radius and peak swirl velocity are shown as

scatter plots in Fig. 8a and b, respectively. At this position,

the tip vortex exhibited a moderate aperiodicity magnitude

and, therefore, the Devenport correction applied to SA

results gave a smaller core radius and larger peak swirl

velocity. For the core radius, the corrected SA and the IA

results were close to the previous calculations by AFDD,

ONERA and DLR reported by Lim and van der Wall

(2010). There was a small variation in peak swirl velocity

among these different calculations. Contrary to the com-

monly held belief, the simple average, albeit with the

Devenport correction, gave as good results as the other

methods. The CA and IA results are also in overall good

agreement and show that the uncertainty in core radius or

peak swirl velocity is about 20 %.

Another aspect hidden in the HART II results is the

uncertainty in vortex center location. As mentioned earlier,

the AFDD calculations shown by Lim and van der Wall

(2010) used a local method for vortex center based on

centroid of vorticity. The ONERA calculations also used a

local method based on the Q-criterion, while the DLR

calculations used a global method based on convolution of

k2. The uncertainty in vortex center location (that is, the

standard deviation in the vortex center location) is shown

in Fig. 9 for these different methods along with the planar

fit. The distinction between local and global methods is

again apparent with the global methods (convolution of k2

and the planar least-squares fit) resulting in significantly

lower uncertainty.

4 Concluding remarks

Vortex aperiodicity, or the random variation in vortex

center location, results in a larger perceived core radius in

the averaged flow field measurements. This is often recti-

fied using methodologies that fall in three general types:

(a)

(b)

Fig. 8 Scatter plots for the vortex properties for HART II position 17

shown with the SA, CA and IA results using the planar fit to include a

vortex sheet. Results from the HART II partners reported by Lim and

van der Wall (2010) are also shown for comparison. a Vortex core

radius, rc, b Peak swirl velocity, Vhmax=XR

1200 Exp Fluids (2012) 53:1191–1202

123

the simple average (SA) followed by an analytical/empir-

ical correction, the corrected average or conditional aver-

age (CA), and the individual average (IA). However, the

analysis methods used to derive the vortex properties of

interest also play a crucial role in how the underlying

measurement uncertainty is transformed into the final

uncertainty in the derived vortex properties. Global meth-

ods that use a larger area of measured data were shown to

give less uncertainty in the vortex properties compared to

local methods that use only a small subset of data in close

proximity to the vortex core. The newly proposed planar fit

method is one such global method, and it was shown to

give lower resulting random uncertainty in vortex proper-

ties. In fact, all three types aperiodicity correction meth-

odologies gave essentially the same result when used with

the proposed planar fit. The method was also applied to

previous tip vortex measurements for the TRAM and the

HART II rotors. In all cases, the results showed the same

behavior, thus confirming the wide applicability and ver-

satility of this method. Specific conclusions from this study

are summarized below.

1. The analysis method used to derive vortex properties,

like the core radius, plays a significant role in the

resulting uncertainty, perhaps a more significant role

than the aperiodicity correction or measurement res-

olution. It was hypothesized that global methods,

which use all measured data, result in lower uncer-

tainty in the derived vortex properties than local

methods, which use only a small sub-set of data close

to the vortex core. The hypothesis was confirmed using

the AFDD model-rotor measurements as well as the

TRAM and HART II measurements.

2. The HART II measurements at position 17 analyzed

using the proposed new method showed that the global

methods indeed resulted in smaller uncertainty in

vortex center location as compared to the local

methods.

3. Using a global method for deriving the vortex center

and core properties, the results were insensitive to the

choice of aperiodicity correction methodology. The

mean vortex core properties estimated using all three

aperiodicity correction methodologies (SA, CA and

IA) were self-consistent. The same behavior was

observed in previous measurements reanalyzed using

the proposed new method.

4. Contrary to commonly held belief, the simple average,

which is both the simplest and the fastest, gave a good

estimation of vortex core properties, provided the core

radius and peak swirl velocities were later corrected

using the inverse convolution method developed by

Devenport et al. The TRAM measurements showed

that the simple average results were not sensitive to the

choice of local or global derivation method. This is

suggestive that the simple average minimizes the

random uncertainty in the transformation from velocity

field to vortex properties.

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