2003-ahs straining a vortex filament - experiment

8/14/2019 2003-AHS Straining a vortex filament - Experiment

1/15

THE INTERDEPENDENCE OF STRAINING AND VISCOUS

DIFFUSION EFFECTS ON VORTICITY IN ROTOR FLOW FIELDS

Manikandan Ramasamy J. Gordon Leishman

Alfred Gessow Rotorcraft Center,

Department of Aerospace Engineering,

Glenn L. Martin Institute of Technology,

University of Maryland, College Park, Maryland 20742

Abstract

An experiment was performed in an attempt to quantify

the interdependence of straining and viscous diffusion ef-

fects on the development of rotor tip vortices. The prop-erties of the blade tip vortices were measured in the wake

of a hovering rotor, and compared to the results for the

case when the wake approached a solid boundary. The

presence of the boundary forced the tip vortex filaments

to strain or stretch, allowing the effects of this process

on the tip vortex developments to be studied and mea-

sured relative to the baseline case. It is shown that vortex

stretching begins to decrease the viscous core size and,

when the strain rates become large, this can balance the

normal growth in the vortex core resulting from viscous

diffusion. A parsimonious model for the effects of vor-

tex stretching based on the conservation laws was used to

correct the measurements, which brought the results intoagreement with what would be expected on the basis of

viscous diffusion alone. A byproduct of the process of

progressively stretching the vortex filaments allowed their

properties to be measured to older wake ages than would

otherwise have been possible. The measured results were

used to compare with other vortex measurements, and to

augment previous correlations in an attempt to develop a

more general vortex model that would be suitable for use

in a variety of aeroacoustic applications. A new model

is suggested that combines the effects of diffusion and

strain on the vortex core growth. The empirical coeffi-

cients of this model have been derived based on the best

available results from many sources of both fixed-wing

and rotating-wing tip vortex measurements.

Graduate Research Assistant. [email protected] Professor. [email protected]

Presented at the 59th Annual Forum and Technology Display

of the American Helicopter Society International, Phoenix, AZ,

May 68, 2003. c2003 by the authors. Published by the AHSInternational with permission.

Nomenclature

a1 Squires coefficient

c Rotor blade chord, m

d Non-dimensional downstream distanceCT Rotor thrust coefficient, T/(A2R2)e Correlation coefficient

k Empirical constant

l Filament length, m

p Probablility density function

r Radial distance, m

rc Viscous core radius, m

r0 Initial core radius, m

r Non-dimensional radial distance, r/rcrl Position vector of the filament, m

rp, zp LDV measurement location w.r.t. rotor axes

rv, zv Vortex core location w.r.t. rotor axes

R Rotor radius, mRev Vortex Reynolds number, v/t Time, s

T Rotor thrust, N

V Swirl Velocity, ms1

V Free stream velocity, ms1

V Filament strain rate, ms1

z Downstream distance, m

Oseen constant = 1.25643v Tip vortex strength (circulation), m

2s1r, z Standard deviation of vortex location in

radial and axial directions, respectively, m

Effective diffusion constant

Filament strain Wake (vortex) age, deg Kinematic viscosity, m2s1 Azimuth angle, deg Flow density, kg m3 Vorticity, m2s1 Rotor rotational speed, rad s1


2/15

Introduction

During the past two decades, considerable research has

been conducted into the problem of measuring the de-

velopment of blade tip vortices trailed into the wakes of

helicopter rotors. The structure of the tip vortices define

the majority of the induced velocity field at the rotor, as

well as being responsible for a number of adverse prob-lems. These problems include large unsteady airloads

and high noise levels associated with blade vortex inter-

actions (BVI), and significant vibration levels associated

with rotor wake/airframe interactions. Therefore, better

models of the tip vortex structure should, at least in prin-

ciple, translate directly into improved predictions of over-

all rotor performance, the unsteady airloads on the blades,

rotor and airframe vibration levels, and also rotor noise.

The reduction of rotor noise has become an extremely

important goal from both military and civil perspectives.

Small changes in the structure of the tip vortices and their

positions relative to the rotor blades can have substantial

effects on BVI noise (e.g., Ref. 1). The blades may also in-teract with vortex filaments that are relatively old in terms

of rotor revolutions. During this time, the vortex filaments

will have undergone some amount of viscous diffusion,

as well as encountering steep velocity gradients that can

also affect their viscous evolution. This suggests that vor-

tex models need to be developed with very high levels of

fidelity to ensure sufficient confidence in the induced air-

loads predictions. Only then can effective strategies aimed

at the control or alleviation of adverse vortex induced phe-

nomena be considered.

There has been much experimental work done to mea-

sure and otherwise characterize the structures of blade tip

vortices. A good summary of the various experiments,

as well as the relative capabilities, limitations, and pre-

cision of different measurement techniques, is given by

Martin et al. (Ref. 2). Typical experimental goals include

the measurement of the induced velocity field close to the

vortex, the viscous core size, and the rate of growth of

the viscous core as the vortex ages in the flow. However,

the velocity field and viscous core size by themselves are

insufficient quantities to develop comprehensive and fully

satisfactory models of the tip vortices.

Making vortex flow measurements from rotating blades

is very challenging, e.g., see Ref. 3. For example, despite

the instrumentation issues in making rotor flow measure-

ments, one concern is that the measurements must usually

be made on sub-scale rotors. This is a situation of neces-

sity because of the enormous (and perhaps impractical)

difficulties in making measurements with the necessary fi-

delity on full-scale rotors. However, amongst other issues,

this raises questions about flow scaling effects, which at

model scale may have vortex Reynolds numbers that may

be up to two orders of magnitude less that those asso-

ciated with full-scale rotors. The vortex Reynolds num-

ber is known to affect both the structure and the viscous

growth rate of the tip vortices (Refs. 47), although the

dearth of quality measurements over a wide range of vor-

tex Reynolds number means that the dependency has not

been fully quantified.

Unlike fixed-wings, which trail rectilinear vortices, he-

licopter tip vortices are curved, and also lie in proxim-

ity to other vortices. This means that the rotating-wing

vortices are subjected to self and mutually-induced ef-fects from other vortices as they are convected in the rotor

wake. The inherent difficulties in performing detailed vor-

tex flow measurements inside rotor wakes and isolating

the effects of the individual tip vortices has forced rotor-

craft analysts to augment mathematical models by using

results from fixed-wing vortex flow measurements. This

is, in part, because rotor wake measurements have been

too sparse, too incomplete, or simply unavailable at older

wake ages to enable models to be developed with high

confidence levels. Also, the measurements themselves

may be inconsistent between similar rotor configurations,

making it difficult to rely on some experimental data.

The tip vortices also experience strain effects as they

are convected in the velocity gradients produced inside the

rotor wake. This means that filaments must undergo con-

tinuous changes in their vorticity field, even in the absence

of ongoing diffusion (Ref. 8). Ananthan et al. (Ref. 9)

showed that filament stretching may have significant ef-

fects on the local velocity field induced by the tip vortices,

which in turn, will have an effect on the development of

the overall wake as adjacent vortex filaments move rel-

ative to each other. This can be a particular modeling

issue in forward flight, where individual vortices tend to

bundle and roll up together from the lateral edges of the

rotor disk. These straining effects can make the mea-sured vortex properties different to those that might be

obtained with equivalent rectilinear vortices in a uniform

flow. Therefore, isolating any contributing effects is es-

sential for the development of better vortex models.

The present work has two primary goals. First, to per-

form an experiment in an attempt to examine any inter-

dependence of straining and viscous diffusion in a rotor

wake flow. The properties of the blade tip vortices were

measured in the wake of a hovering rotor, and were com-

pared to the results for the case when the wake was con-

vected near to a solid boundary. The presence of the

boundary forced the tip vortices to stretch in a known

(measured) strain field, allowing the effects of this pro-

cess on the tip vortex development to be measured and, to

some extent, isolated. Second, to further examine the is-

sues of vortex modeling, from the practical perspective of

including such models in rotor aeroacoustic simulations,

and with overall the goal of improving the overall fidelity

of the model based on the best available empirical infor-

mation. Problems considered include the modeling of vis-

cous diffusion, the interdependent modeling of strain ef-

fects, and the issues of vortex Reynolds number scaling.


3/15

z

x

(a) Baseline configuration rotor operates in free-air

Blade

Tip vortexWakeboundary

(b) Rotor operates next to ground plane

Blade

Ground plane

Measurementregion

Measurementregion

Wakeboundary

Filamentsare stretched

Flowdiverter

Description of the Experiment

Rotor Facility

A single bladed rotor operated in the hovering state was

used for the measurements. The advantages of the sin-

gle blade rotor have been addressed before (Refs. 10, 11).

This includes the ability to create and study a helicoidalvortex filament without interference from other vortices

generated by other blades (Ref. 10). Also, a single heli-

coidal vortex is much more spatially and temporally sta-

ble than with multiple vortices (Ref. 11), thereby allowing

the vortex structure to be studied to much older wake ages

and also relatively free of the high aperiodicity issues that

usually plague multi-bladed rotor experiments.

The blade 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 airfoil section was the NACA 2415 throughout.

The rotor tip speed was 89.28 m/s (292.91 ft/s), giving a

tip Mach number and chord Reynolds number of 0.26 and272,000, respectively. The zero-lift angle of the NACA

2415 airfoil is approximately -2 at the tip Reynolds num-ber. All the tests were made at an effective blade loading

of CT/ 0.064 using a collective pitch of 4.5 (mea-

sured from the chord line). During these tests, the rotor

rotational frequency was set to 35 Hz ( = 70 rad/s).The rotor was tested in the hovering state in a specially

designed flow conditioned test cell. The volume of the test

cell was approximately 362 m3 (12,800 ft3) and was sur-

rounded by honeycomb flow conditioning screens. This

cell was located inside a large 14,000 m3 (500,000 ft3)

high-bay laboratory. In the baseline case, the wake was

allowed to exhaust approximately 18 rotor radii before en-countering flow diverters. Aperiodicity levels in the rotor

wake have been measured, and were used to correct all of

the velocity field measurements - see appendix.

Ground Plane

To examine the effects of superimposed velocity gradients

on the tip vortex developments, an artificial means was de-

veloped using a solid boundary or ground plane. Based

on the experiments of Light & Norman (Ref. 12), calcula-

tions showed that significant strain rates could be expected

well before the vortex filaments reached the ground. A

schematic of the experimental setup is shown in Fig. 1.

In the current experiment, the ground plane was twice the

diameter of the rotor diameter. The ground plane could

be adjusted to give different distances between the rotor

plane, although for the present tests it was set to 0.5R. As

the rotor wake approached this ground plane, the vortex

filaments were rapidly strained in the radial direction.

The possibilities of aperiodicity and recirculation of the

wake was reduced by the use of a flow diverter placed

along the circumference of the ground plane. This di-

Figure 1: Schematic showing the experimental setup. (a)In the baseline case, the rotor operates in free air. (b) In

the other case, the rotor operates close to a solid ground

plane.

rected the flow away from and behind the ground plane.

A further series offlow diverters behind the ground plane

was used to control the flow quality, which was veri-

fied using flow visualization. The inherent difficulties

in measuring vortex velocities close to surfaces, and also

in measuring the velocity profiles of tip vortices at older

wake ages, however, imposed restrictions on how close

the ground plane could actually be placed to the rotor.

Measurements were made at up to 0.05R from the groundplane.

Flow Visualization

Flow visualization images were acquired by seeding the

flow using a mineral oil fog strobed with a laser sheet. The

light sheet was produced by a dual Nd:YAG laser, which

generated a light pulse on the order of nanoseconds in du-

ration at a frequency of up to 15 Hz. The light sheet was

located in the flow using an optical arm. Images were ac-

quired using a CCD camera, and were later digitized using

a calibration grid. The laser and the camera were synchro-

nized using customized electronics, which converted the

rotor one-per-rev frequency into a TTL signal that pulsed

every third rotor revolution. A phase delay was introduced

so that the laser could be fired at any rotor phase angle.

The seed was produced by vaporizing oil into a dense

fog. A mineral oil based fluid was broken down into a

fine mist by adding nitrogen. The mist was then forced

into a pressurized heater block and heated to its boiling

point where it became vaporized. As the vapor escaped

from the heat exchanger nozzle, it was mixed with ambi-


4/15

tx

z

Tip Vortex

tyx

BlueBeams

Blade

VioletBeams

GreenBeams

VioletBeams

Traverse

Fiber OpticProbe

GreenBeams Motor V

Vr

zV

Scale

0.5m

Tip VortexVelocity

Components

Enlarged Viewof

MeasurementVolume

BeamExpander

WakeBoundary

Figure 2: Setup of 3-component LDV system.

ent air, rapidly cooled, and condensed into a fog. From

a calibration, 95% of the particles were between 0.2 m

and 0.22 m in diameter. This mean seed particle size

was small enough to minimize particle tracking errors for

the vortex strengths found in these experiments (Ref. 13).The fog/air mixture was passed through a series of ducts

and introduced into the rotor flow field at various strategic

locations.

LDV System

A fiber-optic based LDV system was used to make three-

component velocity measurements. A beam splitter sepa-

rated a single 6 W multi-line argon-ion laser beam into

three pairs of beams (green, blue and violet), each of

which measured a single component of velocity. A Bragg

cell, set to a frequency shift of 40 MHz, produced the sec-

ond shifted beam of each beam pair. The laser beams

were passed to the transmitting optics by a set of fiber-

optic couplers with single mode polarization preserving

fiber optic cables. The transmitting optics were located

adjacent to the rotor (see Fig. 2) and consisted of a pair

of fiber optic probes with integral receiving optics, one

probe for the green and blue pairs, and the other probe for

the violet pair. Beam expanders with focusing lenses of

750 mm were used to increase the beam crossing angles,

and so to decrease the effective measurement volume.

To further reduce the effective size of the probe volume

visible to the receiving optics, the off-axis backscatter

technique was used, as described in Martin et al. (Ref. 2)

and Barrett & Swales (Ref. 14). This technique spatially

filters the effective length of the LDV probe volume on

all three channels. Spatial coincidence of the three probe

volumes (six beams) and two receiving fibers was en-

sured to within a 15m radius using an alignment tech-

nique (Ref. 2) based on a laser beam profiler. Alignment

is critical for 3-component LDV systems because it is ge-

ometric coincidence that determines the spatial resolution

of the LDV probe volume. In the present case, the result-

ing LDV probe volume was measured to be an ellipsoid

of dimensions 80 m by 150 m, which for reference was

about 3% of the maximum blade thickness or 0.5% of the

blade chord.

The high capacity of the seeder allowed the entire test

cell to be uniformly seeded in approximately 30 seconds.

Signal bursts from seed particles passing through the mea-

surement volume were received by the optics, and trans-

mitted to a set of photo multiplier tubes where they wereconverted to analog signals. This analog signal was low

band pass filtered to remove the signal pedestal and any

high frequency noise. The large range of the low band

pass filter was required to allow measurement of the flow

reversal associated with the convection of a vortex core

across the measurement grids. The analog signal was dig-

itized and sampled using a digital burst correlator. The

flow velocities were then converted into three orthogonal

components based on measurements of the beam cross-

ing angles. Each measurement was phase-resolved with

respect to the rotating blade by using a rotary encoder,

which tagged each data point with a time stamp. The tem-

poral phase-resolution of the encoder was 0.1, but themeasurements were averaged into one-degree bins. The

uncertainty in this process has been discussed by Martin

et al. (Ref. 2).

Experimental Results

Wake Displacements and Strains

In the current experiment, the vortex strain rates were de-

termined based on measurements of the spatial locations

of the tip vortices at various wake ages. Therefore, much

care has to be taken in locating the vortices to avoid er-rors. To do this, flow visualization images were acquired

at several wake ages using the strobed laser sheet tech-

nique. To make these images, the volume and distribution

of seeding were judiciously adjusted so that the core of

the vortices appeared as distinct voids of seed. These

seed voids were then used to find the centers of the vortex

cores. At each wake age, a minimum of 300 separate im-

ages were taken, from which the spatial average locations

of the vortex relative to the rotor were quantified by using

a calibration grid. Later, these locations were also cross-

checked when making the LDV measurements. The flow

visualization results were used to acquire statistics of the

small aperiodic deviations of the vortex positions from the

mean, and were used to correct the LDV measurements for

aperiodicity effects (see appendix).

From the results shown in Figs. 3 and 4, it is apparent

that the tip vortices move inboard radially from the blade

tip and axially downward at the early wake ages. In both

cases, the axial convection velocities are nominally con-

stant until the first blade passage at = 360. As a resultof the ground plane, the tip vortices then start moving ra-

dially outward, and become almost parallel to the ground


5/15

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 180 360 540 720 900

With ground plane

Baseline

Non-dim

.axiallocation,z/R

Best fit (Landgrebe model)

Best fit

(a)

0.7

0.8

0.9

1

1.1

1.2

0 180 360 540 720 900

With ground plane

Baseline

Non-dim.

radiallocation,x/R

Wake age, (deg.)

(b)

Best fit (Landgrebe model)

Best fit

-0.001

-0.0005

0

0.0005

0.001

0 180 360 540 720 900

With ground plane

Baseline

Convectionvelocitiesoftipvortex

Wake age, (deg.)

d(z/R)/d

d(x/R)/d

-0.001

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 180 360 540 720 900

With ground plane

Baseline

Filamentstrain,

Wake age, (deg.)

Strain rate increases rapidlyas filament approaches the ground

Figure 3: Results showing the axial and radial locations

of the tip vortices relative to the rotor tip-path-plane. (a)

Axial displacements. (b) Radial displacements.

Figure 4: Estimated components of the tip vortex convec-

tion velocities compared to the baseline condition.

plane at older wake ages. It can be deduced from Fig. 4

that the vortices move through the flow at a modest rate at

early wake ages, but encounter much higher overall veloc-

ities when in close proximity to the ground plane. Clearly,

the axial (slipstream) velocity component of the tip vor-

tices must asymptote to zero as they approach the ground

plane.

After the spatial locations of the tip vortices were mea-

sured, the average strain rates acting on the vortices was

calculated as they convected in the flow. This was done by

Figure 5: Estimated strain experienced by the vortex fila-

ments as they approach the ground plane compared to the

baseline condition.

fitting a series of curves to the displacements, and differ-

entiating the curve numerically by means of finite differ-

ences. From the locations of the vortices at various wake

ages, the length of the vortex filament was determined byits location in space defined by the position vectors of two

adjacent locations atrl and rl1. If the filament is assumedto be a small straight-line segment, the length of the fila-

ment is given by |l| = |rl rl1|. Therefore, the rate ofchange of the length of the filament as it convects through

the velocity field is given by

dl

dt=

d(rl rl1)dt

(1)

or

d

dt

=dl

dt1

l (2)

which implies that the strain, , is

=

d

dt

t =

d

d

(3)

where = l/l is the strain imposed on the filament overthe time interval, t.

Results documenting the estimated strains are shown

in Fig. 5. Notice that the amount of strain or stretch-

ing produced on the tip vortex as it develops near the

ground plane is much larger than that obtained in free air

conditions. For the baseline case, the filament strain is

negligible at older wake ages. This is expected, because

the results in Fig. 3 have shown that the radial locations

of the tip vortices in the baseline case begin to asymp-

tote to a constant value and the radial velocities approach

zero after the initial wake contraction. When the tip vor-

tices approach the ground plane, the strain on the tip vor-

tex is initially slightly negative at early wake ages similar

to that found in the baseline case. Thereafter, the strain

becomes quickly positive as the vortex filaments stretch

radially outward.


6/15

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

= 16

0

= 350

= 590

Non-dimen

sionalswirlvelocityV

/R

Non-dimensional distance from core center, r / c

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

= 91

0

= 1770

= 2020

Non-dimen

sionalswirlvelocityV

/R


-0.3

-0.2

-0.1

0

0.1

0.2

0.3

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

= 3540

= 438

0

= 4820

= 5210

Non-dimensionalswirlvelocityV

/R


-0.2

-0.15-0.1

-0.05

0

0.05

0.1

0.15

0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

= 727o

= 824o

= 899 o

= 972o

Non-dimensionalswirlvelocityV

/R


Figure 6: Swirl velocity profiles in the tip vortex at different wake ages, showing the vortex diffuses under the action of

viscosity.

Velocity Field Measurements

Phase-resolved LDV measurements of the vortex proper-

ties were acquired by making a radial traverse across the

vortex core at various planes in the wake between the rotor

and the ground plane. By estimating the blade azimuth at

which the vortex core was centered on the grid, the in-

stantaneous velocity field could be measured. Through

post-processing of the data, the vortex properties could be

studied as a function of wake age. Wherever possible, the

results measured with the ground plane were compared

to the baseline measurements at the same wake age. The

ability to measure results at older wake ages ( > 360)was considered a novel feature of the preset work relative

to what has been previously possible, in part because of

the precise spatial alignment of the LDV system, which

gives high quality data with good data rates.

Swirl Velocities

Figure 6 shows a series of tangential (swirl) velocity pro-

files measured across each radial grid as the convecting

vortices intersected the measurement grid in a cross flow

plane (see Fig. 1). These profiles are presented in terms

of the non-dimensional distance with respect to a coordi-

nate system centered at the vortex axis, and the velocity

is non-dimensionalized with respect to the rotor tip speed,

R. In other words, all the data is placed in a frame ofreference moving along with the vortex. Also, all the data

was corrected for the measured effects of aperiodicity (see

appendix).

Notice that the results in Fig. 6 show that peak swirl ve-

locity decreases as the vortex ages, which is symptomaticof the effects of viscous diffusion. The distance between

the peaks in each swirl velocity profile can be considered

as indicative of (but not equal to) the viscous core diame-

ter (see later). The peak swirl velocity at the earliest wake

age ( = 16) was about 35% of the tip speed, which istypical of the values measured on helicopter rotors. The

initial core radius was only 3.2% of blade chord (dimen-

sionally this is only 1.4 mm), which gives some idea as

the spatial resolution necessary to resolve the vortex core

dimension.

For the initial 600 degrees of wake age, the strain rates

are small, and vortex develops (diffuses) in a manner sim-

ilar to the baseline case (Ref. 15). However, the last four

measurements at = 727,824,899 and 972 show adecrease in the core size, with an increase in the peak ve-

locity. This is a reversal in the earlier trends, which sug-

gests that the imposed strain rates have affected the char-

acteristics of the vortex. Measurements at even later wake

ages were not possible because the vortex comes too close

to the ground plane to be able to exclude the consideration

of external viscous effects.


7/15

0

0.05

0.1

0.15

0.2

0.25

0 180 360 540 720 900

Squire / Lamb model

Baseline

With ground plane

Non-dimen

sionalcoreradius,rc/c

Wake age, (deg.)

= 1 (laminar)

= 2

= 8

= 16

0

0.05

0.1

0.15

0.2

0.25

0 180 360 540 720 900 1080

Net circulation

Core circulation

Non-dimens

ionalcirculation

/

R

c

Wake age, (deg.)

Figure 7: Measured growth of the vortex core radius as a

function of wake age.

Core Dimension

The vortex core radius was determined from the LDV

measurements based on a measurement of half the dis-

tance between two velocity peaks. This determination wasbased on a spline curve fit to the measured velocity pro-

files, which was then used to find the distance between

the two peaks. This technique helps remove the otherwise

subjective nature of this determination process.

The deduced vortex core radius is plotted in Fig. 7 with

respect to wake age. This plot, when compared along with

Fig. 5, throws some light on the physics involved as the tip

vortex is strained as it approaches the ground plane. At

early wake ages, the two sets of results in Fig. 7 seem to

agree well, but there are some differences. With the pres-

ence of the ground plane, the core growth was found to be

initially larger, although the differences were small. More

importantly, however, at later wake ages, the growth trendwas reversed as the strain rates became positive. These re-

sults suggest that the effects associated with straining be-

gins to balance viscous diffusion. At the older wake ages

(near the ground), the filament starts to stretch at a much

faster rate, as shown in Fig. 5, and this distinctly arrests

the core growth.

Results from the modified Squire/Lamb-Oseen core

growth model (Refs. 4, 7, 16, 17) are also shown in Fig. 7,

which are taken as a reference to represent the effects of

viscous diffusion on the core growth (see next). It is ap-

parent that at earlier wake ages, the measured results fol-

low this model quite well.

Circulation

The vortex circulation can be estimated from the measured

swirl velocity distributions shown in Fig. 6, the results be-

ing shown in Fig. 8. The net circulation was determined at

a distance of 0.25c from the vortex axis, and by assuming

flow axisymmetry in the reference system moving with the

vortex core. The core circulation is that value contained

within the dimension of estimated core radius. Notice

Figure 8: Measured circulation as a function of wake age.

that the values of net circulation decrease only relatively

slowly with wake age, and the core circulation stays essen-

tially constant. This confirms that disipation of the vortex

energy is small, and that the competing mechanisms in the

dynamics of the vortex evolution are diffusion and stretch-

ing, respectively.

Analysis

Treatment of Viscous Diffusion

When presented in an axis system moving with the vor-

tex core, the swirl velocity field induced by the trailing

vortex resembles that of a potential vortex at a large dis-

tance from the vortex center, a near solid body like ro-

tation in the viscous core of the vortex, and zero veloc-

ity at the center of the vortex see results in Fig. 6 and

also the schematic in Fig. 9. While a variety of math-ematical models have been suggested for the diffusion

of tip vortices, one of the simplest is the classic Lamb-

Oseen model (Ref. 16). However, the spin down of the

swirl velocity and core growth given by the Lamb-Oseen

model is found to be unrealistically slow when com-

pared to measurements. In light of experimental evi-dence (Ref. 7), empirically modified Lamb-Oseen growth

models are found to give better representations of the ve-

locity fields surrounding rotor tip vortices.

Bhagwat & Leishman (Refs. 6, 7) have modified the

Squire model (Ref. 4) with the inclusion of an average

apparent viscosity parameter to account for turbulence

mixing on the net rate of viscous diffusion, effectively in-creasing the viscous core growth rates to values that are

more consistent with experimental observations. Further-

more, at t= 0, the swirl velocity given by the Lamb-Oseenmodel is singular at the origin of the tip vortex, and so un-

realistically high velocities are always obtained at young

wake ages compared to measurements. Therefore, an ef-

fective origin offset can be used to give the tip vortex a

This is because of the laminar flow assumptions invoked inthe model; that is, molecular diffusion only is allowed.


8/15

Swirl velocity

Swirl velocity

Filament undergoesviscous diffusion Filament is strainedor "stretched"

Swirl velocity

Swirl velocity

Sl

S

l + l

Figure 9: Schematic showing the spin down of the vortex

and core growth resulting from viscous diffusion.

finite core size and finite induced velocity at its origin.

In light of the foregoing issues, Bhagwat and

Leishman (Refs. 6, 7) suggest that the viscous core ra-dius, rc, of the tip vortices can be effectively modeled as a

function of age, , using the equation

rc() =

4

0

r20 +4

(4)

with = 1.25643. The ordinate-shift, 0, is responsiblefor the non-zero effective core radius, r0, at the tip vortex

origin where = 0, to give a more physically correct (fi-nite) induced velocity there compared to the Lamb-Oseen

result.

Results from this viscous diffusion model have beenshown previously in Fig. 7. Notice that the growth of the

vortex core is relatively quick at young wake ages, but

grows less slowly at older wake ages, which is generally

consistent with experimental observations. The proper de-

termination of is clearly key to the success of the model,and the selection of this parameter is considered later.

Treatment of Strain or Stretching

To understand the interdependent consequences of strain

effects acting on a viscous vortex filament as it diffuses in

the rotor wake, consider a section of an axisymmetric vor-

tex filament of arbitrary length, l, and with the vorticityconcentrated over a cross-sectional area, S. In the absence

of external viscous effects, Helmholtzs laws require con-

servation of circulation, , of the filament, which can bemathematically stated as

=

S

dS = constant (5)

It has been shown previously in Fig. 8 that the measured

circulation in the tip vortex decays only very slowly with

wake age.

Figure 10: Schematic showing the positive straining or

stretching of a vortex filament.

Suppose after a time t+ t, the filament convects to anew position under the influence of the local velocity field

and it becomes strained, as shown in Fig. 10. Conserva-

tion of mass (constant density assumption) implies thatthe change in filament length is accompanied by a cor-

responding change in the cross-sectional area over which

the vorticity is distributed. Therefore, a change in the fil-

ament length is accompanied by a proportional change in

its vorticity. In cases where the strains are large, this can

have a pronounced effect on the induced velocity field in

the immediate vicinity of the vortex filament.

It can be assumed that the bulk of the vorticity is con-

tained with the vortex core, although this depends on the

assumption of a particular velocity profile. The change in

the core radius resulting from the imposed strain is calcu-

lated using the conservation of mass (see development in

Ref. 9), which gives

rc = rc

1

1 +

1(6)

This result in Eq. 6 also satisfies momentum conservation

implicitly.

It should be noted that the above argument is strictly

valid only in incompressible or constant density flow

fields. In compressible flow, the stretching of the fil-

ament need not necessarily be accompanied by an in-

crease in vorticity because the density of the fluid will also

change with filament stretching. The formation of rotor tip

vortices clearly involve compressibility effects (Ref. 18).However, it is reasonable to assume that to a first level of

approximation that any changes in flow density are small

enough so that an increase in vorticity can be considered

to be the primary effect offilament stretching.

Correction for Strain Effects

In light of the foregoing, it is apparent the mechanisms of

viscous diffusion andthe effects of strain fields can act to

change the size and growth rate of the viscous core. In


9/15

0

0.05

0.1

0.15

0.2

0.25

0 180 360 540 720 900

Squire/ Lamb Model

Baseline

With ground plane

Non-dimen

sionalcoreradius,rc/c

Wake age, (deg.)

= 1 (laminar)

= 2

= 8

= 16

All measurements correct for the effects of strain

0

0.05

0.1

0.15

0.2

0.25V

= 0

V= -0.5

V

= -0.25

V= -0.1

V= 0.1

V

= 0.25

V= 0.5

0 360 720 1080

Non-dime

nsionalcoreradius,rc/c

Wake age, (deg.)

= 10

Contracting

Stretching

Figure 11: Trends in the core growth when correcting the

measurements for the effects of filament strain.

Fig. 11, the core growth for the two cases are plotted as a

function of wake age. In this case, the results also include

the development of the core radius, which has now been

corrected to account for the effects of strain. This was

obtained by following a inverse procedure, whereby the

core radius was corrected assuming the validity of Eq. 6

and using the measured values of strain as given in Fig. 5.

Notice that while the correction only makes a difference at

older wake ages where the strain rates are large, the results

fall into much better agreement with the trends resulting

only from pure viscous diffusion (without the effects of

strain). The results suggest that an average value of = 8,in this case, gives good agreement with the experimental

results.

Contributions to Vortex Modeling

Combined Growth Model

The forgoing results suggest that a vortex model can be

developed that accounts for both the effects of viscous dif-

fusion and the effects offlow field strain. Using Eqs. 4 and

6, the two equations can be combined to give an equation

for the core growth as

rc(,) =

r20 +

4

1

1 +

1(7)

The strain rate, , can be determined based on a knowl-edge of the velocity field in which the vortex develops.

This would normally be calculated as part of the rotor

wake model, e.g., in a free-vortex scheme. The two em-

pirical parameters in this model that are used to describe

the viscous development, are the initial core radius, r0 (or

virtual time, 0), and the average turbulent diffusion pa-rameter, , which is known to depend on vortex Reynoldsnumber (see later).

The significance of strain or stretching effects using

Eq. 7 can first be illustrated with reference to the vis-

cous development of a rectilinear vortex filament. The

Figure 12: Representative growth of the viscous core ra-

dius of a rectilinear vortex filament as a function of time

(wake age) for uniformly imposed strain rates.

effects on the filament were examined as a function of a

prescribed strain rate, as defined by V = d/dt. Figure 12shows the growth of the viscous core radius of a rectilin-

ear vortex filament as a function of time (wake age) fordifferent uniformly applied strain rates using an imposed

strain rate of the form

V() = constant (8)

The results are compared in Fig. 12 to the baseline case

where viscous diffusion alone (with = 10) is allowed ina zero strain field.

Notice from Fig. 12 that the effect of strain on the core

development differs depending on whether the filament

length increases (stretches) or decreases (contracts) with

time. For negative strain rates the core radius is found to

increase relatively rapidly or bulge compared to that ob-tained with positive strain. This is because as the filament

approaches zero length, the core radius must increase to

infinity to conserve volume (conservation of mass, con-

stant density flow). Furthermore, in this case the effects

of viscous diffusion and strain on the viscous core devel-

opment are complementary. With positive strain rates, the

core radius becomes nominally constant for later values

of time. In this case, after an initial development, the in-

crease in core radius resulting from viscous diffusion can

be balanced by the decrease in the core radius as a result

offilament stretching.

Figure 13 shows the computed core growth for different

linearly imposed strain rates. The imposed strain rate isdefined by

V() = A +

dV

d

(9)

with the assumption ofA =0.5 in this case. Typically, ina practical rotor calculation A would take values less than

this. However, these conditions are representative of the

strain rate conditions found in the wake of a rotor operat-

ing in hovering flight, where the wake initially contracts


10/15

0

0.02

0.04

0.06

0.08

0.1

Baseline, dV/d =0

dV/d=0.1

dV/d=0.2

dV/d=0.3

0 360 720 1080

Non-dimensionalcoreradius,rc/c

Wake age, (deg.)

A = 0.5, = 10

0

0.05

0.1

0.15

0.2

0.25

0 180 360 540 720 900

Squire / Lamb model

Baseline

With ground plane

Diffusion / Strain model

Non-dimensio

nalcoreradius,rc/c

Wake age, (deg.)

= 1 (laminar)

= 2

= 8

= 16

Figure 13: Representative growth of the viscous core ra-

dius of a rectilinear vortex filament as a function of time

(wake age) for linearly imposed strain rates.

below the rotor (negative strain rate) and then becomes

constant or slowly starts to expand radially outwards (pos-

itive strain rate) as the wake gets older and is convected

into the downstream region (see Fig. 5).

The results in Fig. 13 illustrate an interesting conse-

quence of imposing varying strain rates along the length of

a vortex filament. In the baseline viscous diffusion model

(without strain rate effects), the cores of the filament at

later wake ages are significantly larger than found with

the applied strain rate. In other words, the segments of the

wake undergoing positive strain or stretching may have a

much smaller core radius, even though they have existed

in the flow for a longer time. This means that the peak

induced velocity surrounding those segments is larger and

the vorticity is more concentrated than that found at the

earlier wake ages.To further show the validity of the model in Eq. 7, it has

been used to predict the core growth with the assumptions

of = 8 (see Fig. 11) and using the strain rates definedby the results plotted in Fig. 5. The results are shown

in Fig. 14, where it is apparent that the model faithfully

predicts the measured core growth. Such levels of corre-

lation give considerable confidence in the ability of this

type of relatively parsimonious model to predict the vis-

cous core growth for a tip vortex encountering an arbitrar-

ily imposed strain rate.

A Model for As previously mentioned, the determination of the diffu-

sion parameter is key to the success of the model. Apurely laminar flow, i.e., where viscous diffusion of vor-

ticity takes place on the molecular level alone, then = 1.In such a case, with = 0 and r0 = 0, then Eq. 7 reducesto the classical Lamb-Oseen core growth model. In most

practical cases of lift generated tip vortex flows, however,

experimental measurements suggest that turbulent flow ef-

fects increase the average rate of diffusion of vorticity so

Figure 14: Predictions of core growth under the assump-

tions of viscous diffusion with = 8 in a known strainfield.

that > 1, therefore increasing the core growth rates. Thedetails of this core growth process, however, are not fully

understood or documented with rotors, and existing ex-

perimental results are not entirely conclusive. There isevidence that the inner core growth is laminar and there

is no turbulent mixing effects to enhance the diffusion of

vorticity in this region (Refs. 15, 19). There is also evi-

dence that turbulent flows surrounding the vortex core can

be re-laminarized. Other measurements suggest that there

is turbulence at the edges of the laminar core (Ref. 20),

which acts to enhance the net diffusive growth character-

istics of the tip vortex.

While complete understanding the details of tip vortex

flows still requires much further research, more readily

derived vortex properties such as the peak swirl velocity

and effective core size can be used to better understand the

overall vortex modeling requirements. While the value of

can be estimated from the present measurements (whichsuggests 8), in the general case will be a function ofvortex Reynolds number, Rev

v/ . For the presentmeasurements, Rev is of the order of 10

5. For a full-scale

rotor, however, the values of Rev may be of the order of

107 or greater. Therefore, the difficulty in constructing a

more general vortex model that has a wide range of appli-

cability is to establish how will vary with Rev.

Functional Representation for

Squire (Ref. 4) hypothesized that should be proportionalto the vortex circulation strength. The value of was thenformulated in terms of the vortex Reynolds number as

= 1 + a1Rev (10)

where a1 is a parameter that must be determined empiri-

cally from tip vortex measurements.

Existing vortex models assume that the velocity profiles

are self-similar, indicating that the vortex can be repre-

sented using a single shape function by appropriately scal-

ing length scales and velocities (Refs. 7, 15). Even when


11/15

0.01

0.1

1

10

0.01 0.1 1 10 100 1000

All fixed-wing data

Iversen's correlation

Lamb's model

Equivalentpe

akvelocity,

V

max

c/

v

Equivalent downstream distance, z v

/ V

c2

Correlation ~ ( - 0)

-0.5

~ -0.5

0.01

0.1

1

10

0.01 0.1 1 10 100 1000

Present dataMartin & LeishmanAll fixed-wing dataIversen's correlation

Lamb's modelCookMahalingam et al.Bhagwat & LeishmanMcAlister

Equivalentpe

akvelocity,

V

max

c/

v

Equivalent downstream distance, v

/ c2

Correlation ~ ( - 0)

-0.5

~ -0.5

Figure 15: Correlation of peak swirl velocity with equiv-

alent downstream distance for fixed-wing tip vortex mea-

surements.

the profiles may not be self-similar, it is generally eas-

ier to measure the peak swirl velocity in the vortex flow

than derived quantities such as its core dimension. Fol-

lowing an approach similar to Iversen (Ref. 5), Bhagwat &Leishman (Ref. 6) have shown a correlation between the

non-dimensional peak swirl velocity and the wake age, ,or equivalent downstream distance, d, of the vortex. This

correlation takes the form

Vmax

d+ d0 1

2 = k (11)

where the constants d0 and k can be determined empiri-

cally.

In the case of a fixed-wing, the non-dimensional veloc-

ity, Vmax , is defined as

Vmax =

VmaxV

Vcv

(12)

and the equivalent non-dimensional downstream distance,

d, is defined as

d=

z

c

vVc

(13)

where z is the distance downstream of the wing. In the

case of a rotating-wing, the non-dimensional velocity is

defined as

Vmax =VmaxR

Rcv

(14)and the equivalent non-dimensional downstream distance

is expressed in terms of wake age as

d=

v

c2

(15)

Examples of the Iversen-like correlation curves are

shown for fixed-wing tip vortex measurements in Fig. 15,

and for rotating-wings in Fig. 16. In the case of the fixed-

wing, measurements have been taken from Refs. 2127

Figure 16: Correlation of peak swirl velocity with equiv-

alent downstream distance for rotor tip vortex measure-

ments.

and 28. For the rotating-wings, measurements have been

taken from Refs. 2931 and 32. The measurements ob-

tained in the present work were obtained at relatively old

wake ages compared to that shown in other measurements,

and so can be used to further augment the correlation

curves. In both cases (Figs. 15 and 16), the measurements

show a definitive trend as given by Eq. 11. With the trans-

formation that t = z/V or t = /, the correlation givenby Eq. 11 shows that

Vmax

vt

(16)

The maximum swirl velocity as given by the Lamb-Oseen

model is

Vmax v

2rc v

t(17)

Therefore,

Vmax

vt

1

v

(18)

Comparing Eqs. 16 & 18, it follows that

1

v

=

Rev

= constant (19)

which means that the average apparent viscosity coeffi-

cient, , is proportional to the vortex Reynolds number,v/. Therefore, this analysis supports the initial hypoth-esis that

= 1 + a1Rev (20)

Determination of and a1

Figure 17 shows an assemblage of tip vortex measure-

ments (from the many sources cited previously) as the


12/15

0.1

1

10

100

1000

104

1000 104

105

106

107

Fully laminarCorsiglia et al., 1973Cliffone & Orloff, 1975Rose & Dee, 1963McCormick et al., 1963Kraft, 1955Jacob et al., 1995Jacob et al., 1996Govindaraju & Saffman, 1971Bhagwat & Leishman, 1998Mahalingam & Komerath, 1998Cook, 1972McAlister, 1996Baker et al., 1974Dosanjh et al., 1964Martin & Leishman, 2000Present data

Effectiveviscositycoefficient,

Vortex Reynolds number, Rev

a1

= 0.0002

a1

= 0.00005

Model scale Full scale10

-6

10-5

0.0001

0.001

0.01

1000 104

105

106

107

Laminar trendCorsiglia et al., 1973Cliffone & Orloff, 1975Rose & Dee, 1963McCormick et al., 1963Kraft, 1955Jacob et al., 1995Jacob et al., 1996Govindaraju & Saffman, 1971Bhagwat & Leishman, 1998Mahalingam & Komerath, 1998Cook, 1972

McAlister, 1996Baker et al., 1974Dosanjh et al., 1964Martin & Leishman, 2000Present data

Coefficient,a1

Vortex Reynolds number, Rev

a1

= 0.0002

a1

= 0.00005

Model scale Full scale

Figure 17: Effective diffusion parameter, , as a functionof vortex Reynolds number, Rev.

estimated value of from the measured core growth re-sults which is then plotted versus the corresponding vor-

tex Reynolds number. The data include results from fixed-

wing as well as rotating-wing trailing vortices. Lines are

shown for the predominantly laminar trend, along with the

trends obtained on the basis of Squires hypothesis.

For low Reynolds numbers, the measurements show

small and nominally constant values of, suggesting thatthe core is mostly laminar for these Reynolds numbers.

However, it is apparent that increases with increasingReynolds number, with an almost linearly increasing trend

at higher Reynolds numbers. Notice that any experimen-

tal values of < 1 are physically impossible, and the vari-ous challenges and uncertainties in experimental measure-

ments may account for such inconsistencies. The over-

all experimental evidence, however, strongly suggests the

validity of Squires hypothesis that turbulent diffusion of

vorticity from within the vortex core is directly propor-

tional to the vortex Reynolds number.

Figure 18 shows the same experimental data in the form

of Squires parameter, a1. The experimental data sug-

gests that a1 falls into the range O103 to O104.However, it appears that the rotating-wing results show

a slightly higher effective viscous diffusion rate corre-

sponding to an average value of a1 = O

104

, while the

fixed-wing results show a lower value of a1 = O

105

.

It must be recognized, however, compared to fixed-wing

vortex measurements most rotating-wing results will have

the implicit effects of vortex straining resulting from fila-

ment curvature and other wake distortion effects included

implicitly in the measurements, which may account for

part of these differences.

Figure 18: Effective viscosity parameter, a1, as a function

of vortex Reynolds number, Rev.

In light of the previous results and discussion, isolat-

ing the viscous effects associated with diffusion of vortic-

ity from those associated with strain or vortex stretching

is clearly a problem for further consideration. Further-

more, isolating and correcting for the effects of wander-

ing and aperiodicity in some of these measurements must

be accomplished. This may be difficult or impossible in

some cases because the necessary wandering (or aperiod-

icity) statistics have not been measured. (See appendix

for a method for correcting the vortex measurements forwandering or aperiodicity.) Clearly, however, the average

value ofa1 is of the order of 103 to 105 for all the data

shown here.

Therefore, on the basis of the foregoing results, it can

be concluded that Eq. 7 provides a best available model

for the growth of the viscous core of a trailing tip vortex,

with the value of the empirical parameter a1 being deter-

mined from various vortex experiments, as described pre-

viously. For full-scale helicopter rotors, which will have

much larger values of Rev, the correlation curve would

suggest values of 1000. This suggests that the tip vor-tex may exhibit diffusive characteristics that are orders of

magnitude larger than compared to those expected on the

basis of laminar diffusion alone.

Finally, some estimate for r0 is in order. The initial

core radius of trailing vortices has been measured to be

typically 510% of chord, i.e., of the order of the airfoilthickness at the blade tip where the vortex originated. The

effective origin offset, z0 or 0, can then be establishedfrom the initial core radius, rc0 , by using Eq. 4 or Eq. 7.

It would seem that from the data shown, 0 is typicallybetween 20 and 25 degrees.


13/15

Conclusions

An experiment was performed in an attempt to examine

and quantify the interdependence of straining and viscous

diffusion on the tip vortices in a rotor flow. The properties

of the blade tip vortices were measured in the wake of a

hovering rotor, and compared to the results for the case

when the tip vortex was strained. The results have beenused, in conjunction with other vortex measurements, to

construct a model for the vortex core growth that accounts

for both the effects of viscous diffusion and straining.

The following conclusions have been drawn from the

study:

1. The use of a ground plane was found to be success-

ful in imposing a strain field on the developing rotor

wake. This boundary forced the vortex filaments to

stretch, allowing the effects on the tip vortex de-

velopment to be examined. From the measured wake

displacements, the strain field could be estimated.

2. It was shown that vortex stretching slows the normal

viscous growth of the vortex core. When the strain

rates become large, the effects can balance or counter

the growth in the vortex core resulting from viscous

diffusion. A model for the effects of stretching was

used to correct the measurements, which brought the

results into agreement with what would be expected

on the basis of viscous diffusion alone.

3. As a byproduct of the process of stretching the vor-

tex filaments, it allowed their properties to be mea-

sured to much older wake ages than would other-

wise have been possible. Corrected results for zero

strain field were used to compare with other measure-

ments, and to augment previous correlations where

measurements could not be made to old wake ages.

4. The new results were used to help support a more

general vortex model valid for older wake ages that is

suitable use in a variety of aeroacoustic applications.

A model was suggested that combines the diffusion

and strain on the vortex core growth. The empirical

coefficients of this model have been derived based on

an average of available measurements that have doc-

umented the viscous core growth of trailing vortices.

Acknowledgments

The authors wish to thank Mahendra Bhagwat and

Shreyas Ananthan for their contributions to this work.

This research was supported, in part, by the National

Rotorcraft Technology Center under Grant NCC 2944.

Drs. Yung Yu and Thomas Doligalski were the technical

monitors.

References

1Schmitz, F. H., Rotor Noise, Chapter 2, Aeroacous-

tics of Flight Vehicles: Theory and Practice, Vol. 1,

NASA Reference Publication 1258, Aug. 1991.

2Martin, P. B., Pugliese, G. J., and Leishman, J. G.,

Laser Doppler Velocimetry Uncertainty Analysis For

Rotor Blade Tip Vortex Measurements, AIAA CP 2000-

0263, 38th Aerospace Sciences Meeting and Exhibit,

Reno, NV, 2000.

3Leishman, J. G., and Bagai, A., Challenges in Un-

derstanding the Vortex Dynamics of Helicopter Rotor

Wakes, AIAA Journal, Vol. 36, No. 7, July 1998,

pp. 11301140.

4Squire, H. B., The Growth of a Vortex In Turbu-

lent Flow, Aeronautical Quarterly, Vol. 16, August 1965,

pp. 302306.

5Iversen, J. D., Correlation of Turbulent Trailing Vor-

tex Decay Data, Journal of Aircraft, Vol. 13, No. 5, May

1976, pp. 338342.

6Bhagwat, M. J., and Leishman, J. G., Correlation

of Helicopter Tip Vortex Measurements, AIAA Journal,

Vol. 38, No. 2, February 2000, pp. 301308.

7Bhagwat, M. J., and Leishman, J. G., Viscous Vor-

tex Core Models for Free-Vortex Wake Calculations, Pro-

ceedings of the 58th Annual Forum of the American Heli-

copter Society International, Montreal Canada, June 11

13, 2002.

8Saffman, P. G., Vortex Dynamics, Cambridge Univer-

sity Press, Cambridge, U.K., 1992, Chapter 1.

9Ananthan, S., Leishman, J. G., and Ramasamy, M.,

The Role of Filament Stretching in the Free-Vortex Mod-

eling of Rotor Wakes, American Helicopter Society 58th

Annual National Forum, Montreal, Canada, June 1113

2002.

10Martin, P. B., Bhagwat, M. J., and Leishman, J. G.,

Strobed Laser-Sheet Visualization of a Helicopter Rotor

Wake, 2nd Pacific Symposium on Flow Visualization and

Image Processing, Honolulu, Hawaii, 1999.

11Bhagwat, M. J., and Leishman, J. G., Stability Analy-

sis of Rotor Wakes in Axial Flight, Journal of the Ameri-

can Helicopter Society, Vol. 45, No. 3, 2000, pp. 165178.

12Light, J. S., and Norman, T., Tip Vortex Geometry of a

Hovering Helicopter Rotor in Ground Effect, 45th Annual

Forum of the American Helicopter Society, Boston, MA,

May 2224, 1989.

13Leishman, J. G., Seed Particle Dynamics in Tip Vor-

tex Flows, Journal of Aircraft, Vol. 33, No. 4, 1996,

pp. 823825.


14/15

14Barrett, R. V., and Swales, C., Realisation of the Full

Potential of the Laser Doppler Anemometer in the Anal-

ysis of Complex Flows, Aeronautical Journal, Vol. 102,

No. 1016, 1998, pp. 313320.

15Martin, P. B., Pugliese, G., and Leishman, J. G., High

Resolution Trailing Vortex Measurements in the Wake of

a Hovering Rotor, American Helicopter Society 57th

An-nual National Forum, Washington, DC, May 911 2001.

16Lamb, H., Hydrodynamics, 6 th ed., Cambridge Univer-

sity Press, Cambridge, UK, 1932.

17Oseen, C. W., Uber Wirbelbewegung in Einer Reiben-

den Flussigkeit, Ark. J. Mat. Astrom. Fys., Vol. 7, 1912,

pp. 1421.

18Bagai, A., and Leishman, J. G., Flow Visualization

of Compressible Vortex Structures Using Density Gradi-

ent Techniques, Experiments in Fluids, Vol. 15, 1993,

pp. 431442.

19Cotel, A. J., and Breidenthal, R. E., Turbulence In-

side a Vortex, Physics of Fluids, Vol. 11, No. 10, October

1999, pp. 30263029.

20Han, Y. O., Leishman, J. G., and Coyne, A. J., On the

Turbulent Structure of a Tip Vortex Generated by a Rotor,

AIAA Journal, Vol. 35, No. 3, March 1997, pp. 477485.

21Govindraju, S. P., and Saffman, P. G., Flow in a Tur-

bulent Trailing Vortex, Physics of Fluids, Vol. 14, No. 10,

October 1971, pp. 20742080.

22Jacob, J., Savas, O., and Liepmann, D., Trailing Vor-

tex Wake Growth Characteristics of a High Aspect Ratio

Rectangular Airfoil, AIAA Journal, Vol. 35, 1995, p. 275.

23Jacob, J., Savas, O., and Liepmann, D., Experimental

Investigation of Forced Wake Vortices on a Rectangular

Wing, AIAA Paper 96-2497, 14th AIAA Applied Aero-

dynamics Conference, New Orleans, LA, June 1996.

24Ciffone, D. L., and Orloff, K. L., Far-Field Wake-

Vortex Characteristics of Wings, Journal of Aircraft,

Vol. 12, No. 5, May 1975, pp. 464470.

25Corsiglia, V. R., Schwind, R. G., and Chigier,

N. A., Rapid Scanning, Three Dimensional Hot WireAnemometer Surveys of Wing-Tip Vortices, NASA CR-

2180, 1973.

26McCormick, B. W., Tangler, J. L., and Sherrieb,

H. E., Structure of Trailing Vortices, Journal of Aircraft,

Vol. 5, No. 3, July 1968, pp. 260267.

27Kraft, C. C., Flight Measurements of The Velocity

Distribution and Persistence of the Trailing Vortices of an

Airplane, NACA TN 3377, 1955.

28Rose, R., and Dee, W. F., Airfract Vortex Wake and

Their Effects on Aircraft, Aeronautical Research Council

Report No. CP-795, 1965.

29Bhagwat, M. J., and Leishman, J. G., Measurements

of Bound and Wake Circulation on a Helicopter Rotor,

Journal of Aircraft, Vol. 37, No. 2, Feb. 2000, pp. 227

234.30Mahalingam, R., and Komerath, N. M., Measure-

ments of the Near Wake of a Rotor in Forward Flight,

AIAA Paper 98-0692, 36th Aerospace Sciences Meeting

& Exhibit, Reno, NV, January 1215, 1998.

31Cook, C. V., The Structure of the Rotor Blade Tip Vor-

tex, Paper 3, Aerodynamics of Rotary Wings, AGARD

CP-111, September 1315, 1972.

32McAlister, K. W., Measurements in the Near Wake

of a Hovering Rotor, AIAA Paper 96-1958, 27th AIAA

Fluid Dynamic Conference, New Orleans, June 1820

1996.

33Heineck, J. T., Yamauchi, G. K., Wadcock, A. J., and

Lourenco, L., Application of Three-Component PIV to a

Hovering Rotor Wake, 56th Forum of the American He-

licopter Society, Virginia Beach, VA, May, 2000.

34Devenport, W. J., Rife, M. C., Liapis, S. I., and Follin,

G. J., The Structure and Development of a Wing-tip Vor-

tex, Journal of Fluid Mechanics, Vol. 312, 1996, pp. 67-

106.

35Leishman, J. G., Measurements of the Aperiodic

Wake of a Hovering Rotor, Experiments in Fluids,Vol. 25, 1998, pp. 352-361.

36Gursul, I., and Xie, W., Origin of Vortex Wandering

Over Delta Wings, Journal of Aircraft, Vol. 37, No. 2,

1999, pp. 348350.

Appendix Aperiodicity Correction

Aperiodicity is the inherent random movement of the

phase-resolved spatial locations of the vortex cores in-

side the rotor wake. Measurements of aperiodicity were

made using laser light-sheet illumination of the seeded

flow. A laser pulse duration on the order of nanoseconds

was achieved using an Nd:YAG laser. The laser was syn-

chronized to the rotor so that the aperiodicity of the core

position could be measured at a fixed wake age. A CCD

camera with a micro lens acquired the images, which were

digitized and the vortex positions quantified with respect

to a calibration grid.

Various methods have been proposed for correcting

measurements for aperiodicity or wandering (Refs. 33

36). The method of Leishman (Ref. 35) is used here,


15/15

(rv, zv)

Measurement grid

-0.12

-0.08

-0.04

0

0.04

0.08

0.12

-0.5 0 0.5

Measured data

Corrected data

Non-dimensiona

lswirlvelocityV

/R

Non-dimensional radial distance, r / c

Figure A 1: Flow visualization image of the vortex core,

showing the measurement grid and the coordinate system

relative to the vortex axis.

and accounts for an arbitrary velocity distribution and

anisotropic variations in aperiodicity.

Consider the two-dimensional aperiodic motion of a tipvortex at a given wake age, . Define the LDV measure-ment location, which is fixed with respect to the rotor axes

system, as (rp,zp). The current location of the vortex coreaxis relative to a rotor based axis system is assumed to

be (rv,zv) - see Fig. A1. The velocity field measured at(rp,zp) at a wake age will be functions of r and z andthe position of the measurement point relative to an axis

at the center of the vortex, i.e.,

V(r,z,) = V(rp rv,zpzv,) (A1)Over a sufficiently large number of rotor revolutions,

the aperiodicity of the vortex location relative to the mea-

surement point can be described by using a probabilitydensity function (p.d.f.), say p = p(rv,zv,). FollowingDevenport et al. (Ref. 34), it may be initially assumed that

the aperiodicity is normal (Gaussian) so that a joint nor-

mal p.d.f. can be defined as

p(rp,zp,) =1

2rz

1 e2 exp 1

2(1 e2) r2v2r

+z2v2z

2 e rvzvrz

(A2)

where r = r() and z = z() are the measured r.m.s.aperiodicity amplitudes in the radial and axial directions at

each wake age, respectively, and e = e() is the correlationcoefficient. Using Eq. A2, the actual or measured velocity

V(rp,zp,) can then be determined by convolution where

V(rp,zp,) =

V(rprv,zpzv,) p(rv,zv) drv dzv(A3)

The discrete equivalent of Eq. A3 is

V(rp,zp,) = V(rp rv,zpzv,) p(rv,zv) rv zv(A4)

Figure A 2: Example results of applying aperiodicity cor-

rection at a wake age = 521.

This latter equation is solved by re-expressing V in a

Cartesian coordinate system, and the summations are

taken over length scales that are at least one order of mag-

nitude larger than . An advantage of the numerical so-lution using Eq. A4 is that very general velocity profiles

such as the non-axisymmetric tangential profiles gener-

ally found in rotor wakes can be solved to establish actual

quantitative effects of aperiodicity on the results.

Starting from an initial (assumed) tangential velocity

profile without any aperiodicity, a profile with the ef-

fects of aperiodicity can be obtained numerically by using

Eq. A4. By comparing in point by point sense this new

profile with a specified amplitude of aperiodicity to the

actually measured velocity profile, then a correction can

be applied and a new guess made at the true tangential

velocity. The process can be repeated using Eq. A4 until

convergence is obtained, which is typically within a few

iterations. This technique, therefore, yields an estimate of

the true velocity field based on the measured velocity field

and measurements of the aperiodicity of the tip vortex lo-

cations.

An example is shown in Fig. A2, which shows LDV

measurements of the tangential velocity in the tip vor-

tex at wake age of 521

in terms of the distance in core

radii from the vortex axis. The results for the corrected

(true) velocity profile in the absence of aperiodicity are

also shown in Fig. A2, where it will be apparent that the

true peak tangential velocities are about 30% higher and

the core radius is about 20% smaller than those actually

measured.

2003-ahs straining a vortex filament - experiment

Documents