2008-ahs-dual-plane piv paper -1

8/14/2019 2008-AHS-Dual-plane PIV paper -1

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Turbulent Tip Vortex Measurements Using

Dual-Plane Digital Particle Image Velocimetery

Manikandan Ramasamy Bradley Johnson J. Gordon Leishman

Department of Aerospace EngineeringGlenn L. Martin Institute of Technology

University of Maryland, College Park, MD 20742

Abstract

The formation and evolutionary characteristics of the tip

vortices trailing from a model-scale, hovering helicopter

rotor was analyzed by performing comprehensive flow

field measurements inside the rotor wake. The use of a

dual-plane stereoscopic digital particle image velocimetry

(DPS-DPIV) technique allowed measuring all the termsinvolved in Reynolds-averaged stress transport equations.

These include, not only all the three components of ve-

locity, but also all nine velocity gradient tensors simulta-

neously, a capability not possible with conventional PIV

technique. High-resolution imaging of the vortex sheet

trailing behind the rotor blade, and detailed turbulence

measurements on the process of the roll up of the tip vor-

tices revealed the presence of several micro-scale flow fea-

tures that play a critical role in the overall evolution of the

tip vortices. These measurements help provide a bench-

mark case for calibrating turbulence models, as well as

for validating CFD predictions. Also, the measurements

clearly confirmed that an isotropic assumption of turbu-

lence properties is not valid, and that stress should not

be represented as a linear function of strain for the de-

velopment of future turbulence models. DPIV measure-

ments were complemented with three-component, laser

Doppler velocimeter (LDV) measurements made on same

rotor. Good correlation was found between the two mea-

surement techniques.

Nomenclature

A rotor disk area

c blade chordCT rotor thrust coefficient, = T/A

2R2

Assistant Research Scientist. [email protected] Graduate Research Assistant. [email protected] Minta Martin Professor. [email protected]

Presented at the 64th Annual National Forum of the American

Helicopter Society International. Inc., April 29May 1, 2008,

Montreal, Canada. c2008 M. Ramasamy et al. Published bythe AHS International with permission.

i, j, k direction vectorsNb number of blades

p static pressure

R radius of blade

r, , z polar coordinate systemr0 initial core radius of the tip vortex

rc core radius of the tip vortex

Rev vortex Reynolds number, = v/u, v, w velocities in Cartesian coordinatesu, v, w normalized RMS velocities in DPIV coordinatesuv normalized Reynolds shear stress in X,Y plane

vw normalized Reynolds shear stress in Y,Z plane

uw normalized Reynolds shear stress in X,Z planeVax axial velocity of the tip vortex

Vr radial velocity of the tip vortex

V swirl velocity of the tip vortex

Vr, V, V

z normalized RMS velocities in polar coordinates

Vtip tip speed of blade

x, y, z tip vortex coordinate system

xT, yT, zT rotor coordinate systemX, Y, Z DPIV coordinate system Lambs constant, = 1.25643v total vortex circulation, =2rV ratio of apparent to actual kinematic viscosityi j Kronecker delta kinematic viscosity wake age air density azimuthal position of blade relative blade position rotational speed of the rotor2-C two-component

3-C three-componentBSPCs beam splitting polarizing cubes

Introduction

The thorough understanding of a helicopter rotor wake is

a challenging one to say the least, even in hover where

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the flow state relatively axisymmetric. Decades of re-

search have been focused towards a general understand-

ing of the complex vortical wake generated by a rotor, and

its effect on vehicle performance, vibration, and noise lev-

els (Refs. 111). Much of the research has been right-

fully focused on the understanding of the blade tip vor-

tices, which are the most important features of the rotor

wake (Refs. 1218). These vortices can remain in theproximity of the rotor disk for several rotor revolutions,

and can lead to rotor vibration and unsteady airloads. It

is, therefore, important to better understand and predict

the physics that determine the formation, strength, and tra-

jectories of these tip vortices to develop more consistent

and reliable mathematical models to predict rotor aerody-

namics. To this end, no model can be completely suc-

cessful unless it is able to accurately capture the three-

dimensional, turbulent flow distribution present inside the

vortex cores, and their changes as a function of wake age

and other functions.

Mathematical models of tip vortex evolution have been

developed by making several assumptions to the NSequations (e.g., the LambOseen model). These assump-

tions, such as incompressible, one-dimensional, inviscid

flow do not completely represent the real flow field com-

plexities, and can result in inaccurate predictions of the

growth properties of real tip vortices. Nevertheless, these

models have formed the basis for several empirical mod-

ifications, which are used today in many comprehensive

rotor analyses (Refs. 7, 19, 20).

Efforts towards resolving rotor wakes using Navier

Stokes (NS) methods have been carried out, and are

steadily on the rise. Because Direct Numerical Solution

(DNS) of the NS equations is presently unrealistic for the

rotor wake problem because of the enormous computa-

tional expense more effort has to be focussed toward solv-

ing the Reynolds-Averaged NavierStokes (RANS) equa-

tions. RANS simplifies the process by time-averaging the

NS equations by representing the flow velocity at a point

(ui) as a combination of the mean component, ui, and a

fluctuating component, ui, as given by

ui = ui +ui (1)

Applying Eq. 1 to the incompressible NS equations re-

sults in the RANS equation that is given by.

Dui

Dt=

xj

pi j

+

uixj

+uj

xi

uiu

j

(2)

where, D/Dt is the substantial derivative, p is the staticpressure, i j is the Kronecker delta function, is the den-sity of the fluid medium, ui is the instantaneous velocity,

and uiuj is the correlation (or shear stress) term. The over-

bar represents the time averaged mean values.

Time-averaging bypasses the need to computationally

represent the high-frequency, small-scale flow fluctua-

tions (i.e., ui) caused by the turbulent eddies. However,

this advantage is countered by the presence of an ad-

ditional unknown term (i.e., the Reynolds stress uiuj),

which makes the RANS equations unsolvable unless a

closure model to rebalance the number of equations and

unknowns. This stress term (or the correlation term) ba-sically accounts for the effect of velocity fluctuations cre-

ated by the presence of eddies of different length scales.

It is vital, therefore, that any turbulence closure model

adopted be consistent with the flow physics to correctly

model the contributions of turbulent flow, and also consid-

ering any numerical stability limits inherent to the specific

discretization scheme used.

The basis for turbulence models is empirical evidence

and measurements. Therefore, the objective of the present

work was to explore the complex wake (both the vortex

sheet and the tip vortex) of a rotating-wing, using high

spatial and temporal resolution DPIV and LDV to mea-

sure the mean and turbulent flow quantities in all three

flow directions. These objectives were accomplished by

employing: (1) A conventional DPIV with an extremely

high-resolution (11 mega pixel) camera that provided a

great deal of quantitative information on the formation

of the tip vortices, and the turbulence transport between

the vortex sheet and the tip vortex, and (2) A dual plane

DPIV (DPS-DPIV) technique, which can simultaneously

measure the velocity across two adjacent, parallel planes.

An advantage of DPS-DPIV is that it can determine not

only the 6 in-plane velocity gradients, but also the 3 out

of plane velocity gradients, which are needed for under-

standing the turbulence transport.

Separate LDV measurements on the same rotor were

also used to compare with the DPIV data. While typical

experimental goals of measuring the tip vortex core size,

peak swirl velocity, and their evolution with time were ac-

complished. The focus here was on understanding the tur-

bulent production, transport, and diffusion present in the

near- and far-wake of the rotor. This information can be

used to validate and improve vortex models, as well as to

validate turbulence models for RANS solvers.

Turbulence Modeling

Generally, the turbulence characteristics of any given flow

field depends on the length scales of turbulent eddies

present in the flow, the time history effects, and the in-

fluence non-local effects, such as the presence of a solid

boundary. Developing a turbulence model to address all

these effects for general flow conditions is not possible.

This can be appreciated from the fact that the turbulence

fluctuations observed in a free shear flow, for example, are

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Figure 1: Laser light sheet flow visualization of a rotor

tip vortex, displaying 3 distinct flow regions

completely different from those found in bounded flow.

As a result, numerous flow-dependent turbulence models

have been tailor-made for a given set of flow conditions.

The JohnsonKing model, for example, is often used forseparated flows (Ref. 21). Similarly, the V2F model is

more suitable for low Reynolds number flows (Ref. 22).

Because different turbulent models must be developed

for different problems and flows, they will understand-

ably vary in complexity, and in the number of equations

and coefficients used in the model. Regardless, every

model requires a certain number of closure coefficients (or

constants), and damping functions. The simple Prandtls

mixing length model, for example, requires just one con-

stant, while the modified BaldwinLomax model requires

six constants (Ref. 23). The frequently used Spalart

Allmaras (Ref. 24) model has 8 constants and 3 damp-

ing functions, while BaldwinBarth model requires 7 con-

stants, 2 damping functions, and 1 additional function for

the length scale (Ref. 25). These closure coefficients and

damping functions are derived from free shear flows or

homogenous flows, and are based on experimental mea-

surements.

A problem arises, however, when these models, whose

coefficients are derived from simple measurements of a

particular flowfield, are applied to a more complicated

flow that may involve shock waves or streamline curva-

tures, (such as those present in tip vortices at the blade

tip). It is known, for example, that tip vortices can produce

relaminarization or rotational stratification of turbulencenear the vortex core axis, which will not be accounted for

if turbulence models derived from non-rotational flows are

used without modification (Refs. 26, 27).

Applications to the Rotor Wake

There are many challenges in developing a turbulence

model for rotor wakes. Figure 1 shows a stroboscopic

Figure 2: Variation of Richardson number across the

vortex.

laser light sheet visualization performed in the flow field

of a hovering model-scale rotor. It is apparent that the

vortex comprises of three regions; (1) an inner laminarregion, where the vortex behaves similar to a solid-body

with little to no interaction between adjacent layers of

fluid, (2) a transitional region with eddies of different

scales, and (3) an outer potential region. This is simi-

lar to the make-up of a turbulent boundary layer, which

is composed of (1) a viscous sub-layer, (2) a logarithmic

layer, and (3) an outer free-stream flow. While Van Dri-

est (Refs. 28, 29) used a damping function to model the

logarithmic layer, Klebanoff (Ref. 30) used an intermit-

tency function to represent the turbulent fluctuations from

near the wall to the outer free stream flow. Based on a

similar idea, Ramasamy & Leishman developed an inter-

mittency function to represent the eddy viscosity variationacross the tip vortex, and derived a semi-empirical so-

lution for the evolutionary characteristics of tip vortices.

This eddy viscosity model is given by

T =+VIF 2

RL|| (3)

where T is the total viscosity, is the kinematic viscos-ity, VIF a vortex intermittency function, RL is an em-pirical constant, and is the strain. The model is basedon Prandtls mixing length hypothesis, and addresses the

effects of rotational stratification through a Richardson

number (Ref. 31).

The Richardson number can be defined as the ratio ofturbulence produced or consumed inside the vortex to the

turbulence produced by shear. Figure2 shows the distri-

bution of Richardson number measured inside a tip vortex

along with predictions from two vortex models. It can

be seen that the Ri number decreases quickly from infin-

ity at the center of the vortex, and approaches a thresh-

old value. It has been argued (Ref. 32) that turbulence

cannot develop until the value of Ri number falls below

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this threshold. The RL eddy viscosity model was basedon this idea, and the closure coefficients associated with

the model were derived by first finding a mathematical

solution to the one-dimensional NS equations, followed

by comparing the Reynolds number dependent, similar-

ity solution to the experimental measurements of the evo-

lutionary characteristics of tip vortices (such as the swirl

velocity distribution) obtained from various sources.Understanding the turbulence activity inside the tip vor-

tices also requires an understanding of the turbulence in

the inboard vortex sheet, especially at early wake ages,

where the sheet becomes partly entrained into the vor-

tex. However, understanding the roll-up process is diffi-

cult because the flow is turbulent, three-dimensional, and

involves high pressure and velocity gradients. The pres-

sure difference between the lower and upper surfaces of

the blade tip accelerates the flow from the lower surface,

which when combined with the free-stream flow results in

the formation of a tip vortex. However, this is only an in-

viscid description, and it masks the intricate details of the

flow physics. Secondary and tertiary vortices are known toform when the viscous nature of the flow couples with the

local pressure gradients, which results in flow separations

near the tip surfaces. These structures continue to evolve

on the upper surface, and ultimately merge into a coher-

ent vortex downstream of the trailing edge of the blade.

Furthermore, part of the shear layer from the trailing edge

is also entrained into the tip vortex flow. Although the

tip vortex is fully rolled-up and largely axisymmetric in

the far-field (Ref. 33), modeling the turbulence inside the

vortex cores (and their evolution) depend directly upon the

initial strength and its roll-up behavior.

Description of Experiment

The present study involved the application of both LDV

and DPIV techniques. In the case of DPIV, a simple 2-

component configuration was first used to analyze the vor-

tex sheet using an ultra-high resolution (11 mega pixel)

camera. This was followed by a DPS-DPIV technique,

where a pair of 4 megapixel cameras and a single 2 mega

pixel camera were used. Because a 2-component DPIV

configuration is relatively simple to set up, the discussion

in this paper is focussed on the DPS-DPIV technique. All

of the measurement techniques used the same seed parti-cles, whose average size was approximately 0.25 m in

diameter. This mean seed particle size was small enough

to minimize particle tracking errors (Ref. 34).

Rotor System

A single bladed rotor operated in the hovering state was

used for the measurements. The advantages of the sin-

Figure 3: Schematic and photograph of DPS-DPIV ad-

vanced optical setup

gle blade rotor have been addressed before (Refs. 35, 36).

This includes the ability to create and study a helicoidal

vortex filament without interference from other vortices

generated by other blades (Ref. 35). Also, a single heli-

coidal vortex is much more spatially and temporally sta-

ble than with multiple vortices (Ref. 36), thereby allowing

the vortex structure to be studied to much older wake ages

and also relatively free of the high aperiodicity typical of

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 and

272,000, respectively. The zero-lift angle of the NACA

2415 airfoil is approximately -2circ at the tip Reynolds

number. All the tests were made at an effective blade load-ing ofCT/ 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).

DPS-DPIV

DPS-DPIV differs from conventional DPIV because it

can measure all nine components of the velocity gradi-

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ent tensor. For example, a conventional, stereoscopic (3-

component) DPIV system is capable of measuring all the

three components of velocity in a given plane (Refs. 37

40). These instantaneous velocities can then allow for ve-

locity gradient calculations to be estimated in the in-plane

x- and y- directions. However, estimating the velocity

gradient in the out-of-plane direction (i.e., /z) requires

the measurement of all three components of velocity in atleast two planes that are parallel to each other, and are sep-

arated by a small distance in the z direction. The optical

set up of the DPS-DPIV technique is shown in Fig. 3.

For dual-plane measurements, two separate DPIV sys-

tems are required to simultaneously measure the flow ve-

locities in both the planes, independently. It should be

noted that DPS-DPIV can be arranged as a combination

of two stereoscopic PIV systems, or as a combination of a

stereoscopic PIV system and a 2-component PIV system

(Refs. 41, 42). While the former combination provides all

the three components of velocity in both of the two par-

allel planes, the latter provides all the three components

of velocity in one plane and only the in-plane velocities(i.e., two components) in the other plane. The out-of-

plane velocity is usually calculated using continuity as-

sumptions. This latter method provides some advantages

over the stereoscopic setups, mainly because of its simpler

configuration. The present set up is shown in Fig. 3. A

conventional DPIV setup (the 2 mega-pixel camera is la-

beled as C2 in Fig. 3) is used to measure two components

of velocity in one plane, while a stereo setup (a pair of 4

mega-pixel cameras labeled C1 and C3 in Fig. 3) is used

to measure three velocity components in the second plane.

Continuity in the form of Eq. 4 is then applied to estimate

the third component of velocity in the 2-C measurement

plane (shown in green), i.e.,

w1 =(u1x

+v1y

)z+w2 (4)

The resulting velocity fields in the two planes can then

be compared to determine all nine components of velocity

gradient tensor. To ensure the correctness of these gradi-

ent calculations, however, several precautions have to be

taken. In terms of the setup procedures, the two laser light

sheet planes must be both parallel, and adjacent (ideally,

just a small distance apart) to each other. Additionally,

the lasers must be synchronized not only with each other,

but also with both sets of cameras and to the rotor rota-tional frequency. For the present experiment, each laser

pair (i.e., lasers 1 & 2, and lasers 3 & 4 in Fig. 1) deliv-

ers two pulses of laser light with a pulse separation time

of 2 s. The first laser pulse from the green pair (laser 1)

must be synchronized with the first laser pulse from the

blue pair (laser 3), and the same for the second laser pulse

from each laser pair (lasers 2 & 4).

Each of the three cameras must then be synchronized

Figure 4: Schematic of DPS-DPIV timing.

with the lasers (i.e., the first image in each plane is cap-

tured upon the firing of lasers 1 & 3 and the second im-

age in each plane is captured during the firing of lasers

2 & 4). There are several challenges with simultaneous

measurement in adjacent, parallel, laser planes, however,

mainly resulting from crosstalk between cameras. This

occurs despite each camera having a finite depth of field,

and would record seed particle reflections from both illu-

minated laser planes (because the laser planes are apart by

only a few millimeters). If any camera captures reflection,

from seed particles in both the planes, not only will its

planar velocity map be erroneous after DPIV processing,

but also the comparison between the velocity map in the

first plane with that of the second plane (which is needed

to calculate velocity gradients in the z direction) would be

meaningless. This problem is heightened by the need to

have the intensity of each laser each laser set to high levels

so that the individual seed reflections can be captured by

the cameras. To guarantee that each respective set of cam-

eras is only seeing the flow in its designated laser plane, aspecial optical setup was used, as shown in Fig 3.

The purpose was to split the polarizations of the two

respective laser pairs (lasers 1 & 2 are s-polarized, and

lasers 3 &4 are p-polarized), and then to use appropriate

filters and beam-splitting optical cubes placed in front of

each camera to guarantee that they only see one type of

polarizedlight. In the present setup, the middle 2-C cam-

era (C2) was tuned to the s-polarization of lasers 1 & 2,

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and the stereo cameras (C1 & C3) were tuned to the p-

polarized light lasers from 3 &4. Figure 3 shows a dia-

gram explaining the setup, where the blue (p-polarized)

and green (s-polarized) light rays trace the paths of the

light reflections from each respective laser sheet. One

beam splitting cube in front of the 2-C camera initially

sees both sets of laser reflections, and allows the p-

polarized blue light to pass directly through and re-directsthe s-polarized green light to a second beam splitting cube.

The second cube simply acts as 45 mirror by re-directing

the s-polarized light to the camera. A linear filter is placed

over the lens to act as a final buffer against any p-polarized

light. Each stereo camera also has one beam splitting cube

in front of it, which re-directs the s-polarized light into

separate light dumps adjacent to the cubes, and allows the

p-polarized blue light to pass through to the camera lens.

Each stereo camera also has a linear filter over the lens

(oriented at a different angle than that over the 2-C cam-

era) to act as a final buffer against any s-polarized light.

Final verification of the working condition of the optical

set up was made before measurements were started. It wasensured that the cameras see only their designated lasers

and not the reflections from the other lasers, i.e., there was

no crosstalk.

The second challenge in the experiment involves the

need for simultaneous measurement in each plane. Even

after optically separating the two DPIV systems, care has

to be taken to ensure that both systems are synchronized

with each other so that the flow is measured in each plane

at exactly the same time. This will guarantee that the tur-

bulence estimates will be derived from measurements of

the same exact flow features. Figure. 4 shows the tim-

ing setup of the experiment, which takes a 1/revolution

TTL signal from the rotor, and uses it to synchronize both

ND:Yag laser pairs, with each other, and their respective

cameras.

The processing of the acquired images from the DPS-

DPIV technique used a deformation grid correlation algo-

rithm (see Ref. 43), which is optimized for the high ve-

locity gradient flows found in rotor tip vortices. The inter-

rogation window size was chosen in such a way that the

images from both the cameras were resolved to approxi-

mately the same spatial resolution to allow measurements

to be made of the velocity gradients in the out-of-plane

direction. The steps involved in this correlation algorithm

are shown in Fig. 5.The procedure begins with the correlation of an inter-

rogation window of a defined size (say, 64-by-64), which

is the first iteration. Once the mean displacement of that

region is estimated, the interrogation window of the dis-

placed image is moved by integer pixel values for bet-

ter correlation in the second iteration. This third itera-

tion starts by moving the interrogation window of the dis-

placed image by sub-pixel values based on the displace-

Figure 5: Schematic of the steps involved in the defor-

mation grid correlation..

ment estimated from second iteration. Following this,

the interrogation window is sheared twice (for integer and

sub-pixel values) based on the velocity magnitudes from

the neighboring nodes, before performing the fourth and

fifth iteration, respectively. Once the velocity is estimatedafter these five iterations, the window is split into four

equal windows (of size 32 32). These windows aremoved by the average displacement estimated from the fi-

nal iteration (using a window size of 6464) before start-ing the first iteration at this resolution. This procedure can

be continued until the resolution required to resolve the

flow field is reached. The second interrogation window is

deformed until the particles remain at the same location

after the correlation.

LDV System

A fiber-optic based 3-C LDV system was used to make

three-component velocity measurements. To reduce the

effective size of the probe volume visible to the receiv-

ing optics, the off-axis backscatter technique was used, as

described in Martin et al. (Ref. 44). This technique spa-

tially 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

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

nique (Ref. 44) 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 re-sulting LDV probe volume was measured to be an ellip-

soid 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. A coincident window of 80s was

used to ensure that the same set of particles provide all the

three components of velocity. The flow velocities were

then converted into three orthogonal components based

on measurements of the beam crossing angles. Each mea-

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Figure 6: Experimental set up for 2-D DPIV.

Figure 7: Presence of TaylorGortler vortices in the

vortex sheet trailing behind a rotor blade.

Figure 8: Close up view of the trailing tip vortex.

surement was phase-resolved with respect to the rotating

blade by using a rotary encoder, which tagged each data

point with a time stamp. The temporal phase-resolution

of the encoder was 0.1deg, but the measurements wereaveraged into one-degree bins (Ref. 44).

Figure 9: Close-up view of the tip vortex and various

segments of the shed vortex sheet

ResultsThe observed results have been classified into three cat-

egories, in which they will be analyzed: (1) High-

resolution imaging of the vortex sheet and the formation

of tip vortices, (2) The mean characteristics of the rotor tip

vortices, and (3) The turbulent characteristics of the rotor

tip vortices. It should be noted that the measurements of

the vortex sheet were carried out using 11 mega pixel cam-

era, while the remainder of the results were obtained using

the DPS-DPIV and 3-C LDV techniques.

Vortex Sheet and Tip Vortex Formation

Figure 6 shows the schematic of the experimental set up

used for the high resolution 2-C DPIV measurements. The

laser was fired along the span of the rotor blade and the

11 mega pixel CCD camera was placed orthogonal to the

laser light sheet.

An instantaneous DPIV velocity vector map obtained at

2 wake age is shown in Fig. 7, where the color contour

is stream wise vorticity (only every 4th vector is shown to

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Figure 10: Signature of the Taylor-Gotler vortex pairs

seen in the instantaneous flow field vector maps.

avoid image congestion). This image reveals the complex

turbulent flow pattern in the near wake, and shows the in-

terplay of the tip vortex with the turbulent inboard vortex

sheet.

Several observations can be made. First, the instan-

taneous vector map (Fig. 7) shows not only a clear

tip vortex forming behind the blade, but also a chain of

counter-rotating vortices clearly intertwined with one an-

other, and with blade tip vortex (note the interchange be-

tween red and blue vorticity contours which represent

clockwise and counter-clockwise rotation respectively).

Commonly known as TaylorGortler vortices (Ref. 45),

these counter-rotating structures are present along the en-

tire span of the shed vortex sheet. Various sections of the

flow field (along the span of rotor blade) are shown in

a close-up view in Figs. 8 and 9, which evidently showthe tip vortex and the presence of these T-G vortices, re-

spectively. These are highly unsteady, making the vortical

sheet behind the blade, and its roll-up into the tip vortex at

early wake ages a highly aperiodic and turbulent process.

The presence of these vortical pairs is directly attributed to

the streamline curvature of the boundary layer (Ref. 46).

This aperiodicity is seen using Fig. 10, which is ob-

tained by making a horizontal cut through the center of

the tip vortex along the entire span of vortex sheet. The

results include measurements from both the instantaneous

vector map, as well as from a phase averaged vector map,

which was obtained by simple averaging of 1,000 such in-

stantaneous velocity vector maps. It can be seen that theinstantaneous swirl velocity profile shows velocity fluctu-

ations that arise from the presence of the TaylorGortler

vortex pairs, while the phase average vector field essen-

tially eliminates many of the intrinsic turbulent structures

inboard. This is understandably a result of the aperiodic

nature of these vortex pairs that do not maintain the same

spatial location from one measurement to the next. Con-

sequently, the signature of these coherent structures are

smeared through the averaging process. Nevertheless, it is

these type of small scale (or even smaller), high frequency

aerodynamic structures that contribute primarily to the ini-

tial turbulence fluctuations in the wake flow. These fluc-

tuations, when combined with the high velocity gradients

found in these tip vortices, play a substantial role in the

momentum transfer process in the boundary layer.

Tip Vortex Measurements

While the small-scale turbulent vortices in the inboard

vortex sheet must play a significant role in defining the

turbulence-scale in the near-wake of the blade, the most

dominant coherent structure in both the near- and far-wake

of the rotor blade is still the tip vortex. Therefore, it is im-

portant to study and measure the mean and turbulent quan-

tities inside the tip vortices, and to do so as a function of

wake age.

Aperiodicity Correction

This distinction between mean and turbulent velocities in

the tip vortex, however, is complicated by the fact that the

wake is aperiodic. Even in hover, it is well known that the

convecting vortex filaments develop various types of self-

and mutually-induced instabilities and modes (Refs. 36,

47) from their interaction with each other. This causes

their spatial location to change slightly from one revolu-

tion to the next. This leads to the vortex center wandering

about a mean position in each instantaneous measurement

plane, and so this effect poses a problem in finding the

correct mean flow measurements. Unless corrected for,

this manifests in inaccurate estimation of turbulent flow

components based on Eq. 1.To achieve accurate mean flow velocities, the vortices

first have to be co-located for the averaging process, such

that the center of each vortex is aligned with one another.

In essence, this guarantees that individual mean velocities

at a point in the flow are calculated based on that points

location with respect to a defined tip vortex center, not

based on its unadjusted location with respect to the image

boundaries. The conditional aperiodicity bias correction

procedure used in the present study (helicity-based cen-

tering) was successfully analyzed in detail in (Ref. 37).

Mean and turbulence measurements made from 1,000 in-

stantaneous velocity vector maps, and co-locating them

such that the point of maximum helicity (zw) in eachof the instantaneous vector maps coincided (before phase-

averaging) were found to result in accurate estimates of

the core properties.

Mean Flow Characteristics of Tip Vortices

Only after applying the conditional helicity phase-

averaging technique can accurate mean flow properties to

Page 8


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Figure 11: Growth characteristics of tip vortices trail-

ing a helicopter rotor blade

Figure 12: Normalized swirl and axial velocity distri-

bution at various wake ages

be estimated. One important derived parameter is the coreradius of the vortex, which is usually assumed as the dis-

tance between its center (maximum point of helicity) and

the radial location at which the maximum value of swirl

velocity occurs.

In the present study, the core size was determined using

a two step process. First, a horizontal cut (along the span

of the blade), and a vertical cut (normal to the span) were

made across the tip vortex. The radial distances at which

Figure 13: Comparison of LDV and DPIV to axial ve-

locity

the swirl velocity reached a maximum value on these cuts

were then averaged to estimate the core radius. This pro-

cedure has been proven to result in accurate estimates of

the size of the tip vortex cores (Ref. 37).

The measured core sizes obtained at various wake ages

are shown in Fig. 11, along with measurements made on

the same rotor using LDV. All of the measurements were

normalized using the rotor blade chord. The figure also

includes core growth estimated from Squires model, as

modified by Bhagwat and Leishman (Ref. 48) given by

rc =

r20 +4

(5)

When = 1, the model reduces to the laminar LambOseen model. Increasing the value of basically means

that the average turbulence inside the tip vortex is in-creased, which can be expected to result in a higher core

growth rate. It can be seen that the present measurements

follow the = 8 curve, suggesting that the momentumtransfer occurs eight times faster than for laminar flow.

This was the case for both the present measurements, as

well as for previous measurements made on the same ro-

tor.

The swirl and axial velocity distributions measured us-

ing PIV (by making a horizontal cut across the vortex) are

shown in Fig. 12. The classical signature of the swirl ve-

locity distribution can be seen, with the peak swirl veloc-

ity continuously decreasing with an increase in wake age.

This, combined with the increase in vortex core size forincreasing wake age, suggest that viscous and turbulent

diffusion are important flow mechanisms.

For the mean axial velocity, it can be seen that the ear-

liest wake age (2) exhibited an axial velocity deficit of

75% of the tip speed of the rotor blade. This immediately

reduces to about 45% of tip speed within 2 of wake age.

However, any further reduction in the mean axial velocity

occurred relatively slowly, as it remained near 30%, even

Page 9


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Figure 14: Schematic showing the coordinates fol-

lowed in the present experiment.

after 60 of wake age. Such high values of axial velocity

deficits at the centerline of the tip vortices from trailing

from micro-rotor blades were reported in Ref. 49.

LDV measurements made on the same rotor at 45

wake age, as reported in Ref. 50, are compared to the cur-

rent results in Fig. 13. It can be seen that the axial ve-

locities estimated from LDV are lower in magnitude com-

pared to those estimated using DPIV. The difference can

be directly attributed to the lack of any correction proce-

dure for aperiodicity bias for the LDV measurements. The

DPIV measurements are corrected using helicity based

aperiodicity correction technique. It was shown in Ref. 49

that axial velocity is one of the most sensitive parameters

to the aperiodicity in the rotor wake. The result in this fig-

ure clearly shows the need to correct the velocity measure-

ments for aperiodicity bias, otherwise the estimated peak

values will be flawed. It should be noted that estimating

accurately the peak value of axial velocity and its gradient

is critical to understand the evolutionary characteristics of

tip vortices.

Velocity Gradients

Corrected mean measurements using DPS-DPIV allow for

accurate estimations of all 9 velocity gradients in 3 flow

directions. Figure 15 shows the nine gradients for a wakeage of 12 wake age. The solid circles represent the aver-

age core size of the tip vortex. As mentioned previously,

w/z is obtained using the continuity equation. The co-ordinates (and the sign convention) used in the present ex-

periment can be understood from Fig. 14.

It should be noted here that not only do all these gra-

dients have different orders of magnitude, but their dis-

tributions throughout the vortex cores are also different.

The presence of the lobed-patterns in Fig. 15 are a re-

sult of analyzing a rotational coherent structure in terms of

Cartesian coordinates. Comparing the gradients, both the

u/y and v/x components were to be at a maximumnear the vortex center, albeit with opposite signs. Their

peak values were also significantly higher than all other

velocity gradients, followed by w/x and w/y whose

higher magnitude can be explained by the steep rise in theaxial velocity deficit within the viscous vortex cores. The

w/y and w/x gradients predictably form a two lobepattern of opposite sign about the vortex center because

the axial velocity deficit should increase going radially in-

wards towards the vortex center, and decrease going radi-

ally outwards from the vortex center.

The other in-plane gradients, i.e., u/x and v/y, de-veloped a four-lobed pattern, that were approximately 45

to the x-y coordinate axes. Specifically, the u/x com-ponent displays negative lobes at 45 and 225, and posi-

tive lobes at 135 and 315. The pattern developed in the

v/y gradient is offset from that in u/x by 90. As a

result, their sum (which is w/z, based on Eq. 4) will berelatively small. Notice, the positive lobes in u/x willbe added to the negative lobes in u/x, and vice-versa.

For the other streamwise gradients of in-plane veloci-

ties (u/z and v/z), a two-lobed pattern should be ex-pected. Based on the coordinate system followed in this

work, u/z is negative on the lobe aligned with the posi-tive y-axis, and negative along the y-axis. In turn, the swirl

velocity gradient, V/z, will be negative (z is positivestreamwise) at all points inside the vortex core, indicating

a reduction in the swirl strength of the tip vortices. This

gradient can be expected to be positive on top of the blade

when the tip vortex is still undergoing its roll-up process.

Turbulence Characteristics

A detailed analysis was performed on the measured tur-

bulence characteristics to help in understanding the evo-

lutionary behavior exhibited by the tip vortices. 1,000 ve-

locity vector maps were used to estimate the fluctuating

velocity components. All the first and second order veloc-

ity fluctuations were normalized by Vtip, and Vtip2, respec-

tively, and the length scale was normalized by the rotor

blade radius.

Turbulence Intensities

Figure 16 shows the distribution of normalized turbulence

intensities u, and v from = 4 (z = 0.66 c) to 4

wake age (z = 0.33 c). The solid black circle representsthe blade. It can be seen from Figs. 16(d) and (e) that u

and v are biased along x and y axes, respectively. This

bias also occurs in the initial stages of the roll up (i.e., =2), and correlates with previous measurements made in

Page 10


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Distance from the vortex center, Y/R

Distancefrom

thevortexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dudx: -6.556 -4.635 -2.714 -0.793 1.128 3.050 4.971

(a) u/x


Distancefrom

thevortexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dudy: -2.000 1.000 4. 000 7. 000 10.000 13.000 16.000

(b) u/y


Distancefrom

thevortexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01

d ud z: -0 .3 6 9 -0 .22 8 -0 .0 87 0 .05 3 0 .19 4 0 .33 4 0 .47 5

(c) u/z


Distancefrom

thevo

rtexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01d vd x: -1 5.0 00 -1 2.0 00 -9 .0 0 0 -6 .0 0 0 -3 .0 0 0 0 .00 0

(d) v/x

Distance from the vortex center, Y /R

Distancefrom

thevortexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dvdy: -6.500 -4.500 -2.500 -0.500 1.500 3.500 5.500

(e) v/y


Distancefrom

thevo

rtexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dvdz: -1.20 -0.80 -0.40 0.00 0.40 0.80 1.20

(f) v/z

Distance from the vortex center, X/R

Distancefrom

thevortexcenter,Y/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dwdx: -7.000 -4.000 -1.000 2.000 5.000 8.000

(g) w/x


Distancefrom

thevortexce

nter,Y/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dwdy: -9.000 -6.000 -3.000 0.000 3.000 6.000

(h) w/y


Distancefrom

thevortexcenter,X/R

-0.01 -0.005 0 0.005 0.01-0.01

-0.005

0

0.005

0.01dwdz: -2.500 -0.500 1.500 3.500

(i) w/z

Figure 15: DPIV measurements of the nine velocity gradient tensors inside the tip vortex core

the flow field of a micro-rotor using DPIV (Ref. 49), and

those made behind a fixed-wing (Refs. 51, 52). While

the bias pattern remains the same at all wake ages, the

magnitude of the bias varies with wake age.

Figure 17 shows the distribution of turbulence intensi-

ties obtained by making a horizontal cut across the center-

line of the tip vortex at 2 wake age. Along the horizontal

cut, the Cartesian components of the velocity u and v will

be equivalent to Vr and V, respectively, in polar coordi-

nates. Because of the bias of u (i.e., Vr

in this case) along

the horizontal cut, the Vr distribution is noticeably wider

than the V distribution. Also, both turbulence intensities

reach a maximum value at the center of the vortex. This

can be seen from the contour shown in Fig. 18. Further-

more, it can be seen that Vr > V. This observation is

of particular significance, and has been used by Chow et

al. (Ref. 52) to explain the turbulence intensity bias ob-

served in Fig. 16.

Page 11


12/21

(a) =4

(b) =2

(c) = 0

Figure 16: In-plane measurements of turbulence intensities during the tip vortex roll up (Every third vector has

been plotted to prevent image congestion)

Page 12


13/21

(d) = 2

(e) = 4

Figure 16: In-plane measurements of turbulence intensities during the tip vortex roll up (Cont.)

Figure 17: Turbulence intensity distribution across the

rotor tip vortex at = 2 wake age.

Examining the turbulence production terms for Vr and

V transport shows that

Vr(prod.) =2

V2r

Vrr

+Vz V

r

Vrz

V

rVr V

(6)

and

V(prod.) =2

V2

Vrr

+Vz V

Vz

+Vr

Vr V

(7)

Comparing the two equations, the second term is the

streamwise gradient of the in-plane velocities. This termis relatively small, and becomes even smaller when multi-

plied by shear stress. The velocity gradient in the first term

is very small, mainly because the radial velocity is small,

so the gradient of radial velocity becomes even smaller.

However, the presence of a normal stress term (which is

usually significantly larger than the shear stresses) does

tend to compensate for the small gradient in radial ve-

locity. The last term in both equations involves the shear

Page 13


14/21

(a) V r (b) V

Figure 18: Turbulence intensity distribution (in polar coordinates) across the rotor tip vortex at = 4.

(a) V r (b) V

Figure 19: DPIV measurements of the nine velocity gradient tensors inside the tip vortex core

stress and swirl velocity gradient. Inside the vortex cores,

the components V/r, and V/r are very similar. Thedifference here, however, is the sign of the last term.

While, this term is negative for Vr, it is positive for V.

While VrV is usually assumed to be zero for solid body

rotation, the present results show it to be predominantly

negative. A non-zero VrV will therefore increase the pro-

duction ofVr and reduce V, resulting in V

r > V

. This, in

turn, explains the reason behind the turbulence intensity

bias observed in u and v, as shown in Fig. 16.

Figures 19(a) and (b) compare the turbulence intensities

measured across the tip vortex at 30 obtained using DPIV

and LDV. Good correlation can be seen between these two

measurement techniques, even though the LDV measure-

ments are not corrected for aperiodicity bias. Although

procedures are available to correct the core size and peak

swirl velocity in LDV, no such procedure is available for

turbulence measurements. However, at early wake ages,

the magnitude of aperiodicity is relatively low and allows

such a comparison to be readily made.

Reynolds Stresses

Figure 20 shows the distribution of Reynolds shear stress

(uv) and its associated strain (u/y+ v/x) from =4 (on the blade) to = 4. The reason to plot shear

stress along with strain directly stems from the basic as-sumption in linear eddy viscosity-based turbulence mod-

els, where stress is represented as a linear function of

strain. However, examination of the contours in Fig. 20

clearly suggests that this assumption is not valid, as has al-

ready been shown for curved stream lines (Refs. 32, 53).

This is true regardless of the wake age. The magnitude

of both the shear stress and strain continue to change with

wake age, and eventually form a clear four-lobbed pat-

Page 14


15/21

Non-dimensional distance, X/R

Non-d

imensionaldis

tance,

Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02UV: -0.006 0.001 0.008uv

___

Non-dimensional distance, X/ R

Non-d

imensionaldis

tance,

Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02

S tra in ra te : - 15 00 0 - 75 00 0 7 50 0 1 50 00

(a) =4

Non-dimensional distance, X /R

Non-d

imensionaldistance,

Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02UV: -0.006 0.001 0.008uv___


Non-d

imensionaldistance,

Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02

S tra in ra te: - 15 00 0 - 75 00 0 7 50 0 1 50 00

(b) =2


Non-dimensionaldistance,

Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02

UV: -0.017 -0.009 -0.002 0.006 0.013uv___

Non-dimensional distance, X/ R

Non-dimensionaldistance,

Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02

S train rate: -15000 -8182 -1364 5455 12273

(c) = 0

Figure 20: Reynolds shear stress and strain at various wake ages.

Page 15


16/21


Distancefrom

thevortexcenter,Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02UV: -0.006 0.0005 0.008uv___


Distancefrom

thevo

rtexcenter,Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02

Strain-rate: -15601.7 -4156 7936.85 26767.7

(d) = 2


Distancefrom

thevortexcenter,Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02U V: -0.006 0.001 0.008uv__


Distancefrom

thevortexcenter,Y/R

-0.01 0 0.01 0.02 0.03

-0.02

-0.01

0

0.01

0.02

Strain rate: -19819 -1301 12588

(e) = 4

Figure 20: Reynolds shear stress and strain at various wake ages (Cont.).

tern, as early as = 2. These lobes, whose magnitudesalternate in sign, are aligned along the Cartesian coordi-

nate axes for strain, and 45 with respect to the coordinate

axes for shear stress. The contours also suggest the pres-

ence of significantly high levels of shear stress inside the

vortex sheet at early wake ages, which can be expected

based on the instantaneous turbulent activity seen previ-

ously in Fig. 7(b).

Figures 21 and 22 show the Reynolds shear stress com-

ponent (uv) and its associated strain (u/y+v/x) fora fully developed tip vortex at = 60 wake age. The fig-ure also includes stress and strain contours from two other

experiments. While Figs 21(a) and 22(a) are from a fixed-

wing experiment performed in a wind tunnel (Ref. 52),

Figs 21(b) and 22b show DPIV measurements on a micro-

rotor. Qualitatively at least, it can be seen that the correla-

tion between all the three experiments is excellent.

Figure 23 shows a similar plot for another component

of Reynolds shear stress (vw), and its associated strain.

Here, the current measurements again correlate well with

the fixed-wing measurements, qualitatively. Unlike uv,

the vw term has only two lobes. However, the alignment

of the lobes are still 45 offset from the coordinate axes.

The associated strain also shows only two lobes, which

are aligned with the y axis. This suggests that the orienta-

tion of all the shear stress distributions are 45 offset from

the shear strain distribution, regardless of the the vortex

Reynolds number or the type of lifting surface used.

The importance of vw (the correlation term between

the streamwise and the cross flow directions) can be un-

derstood from the momentum equation in the z direction

given by,

UU

z+V

V

z+W

W

z=

1

p

z+2W

uw

x

Page 16


17/21

vw

yw2

z(8)

The pressure gradient, which is positive during the tip vor-

tex roll up (resulting in increased axial velocity deficit as

the wake age increases) can be assumed negligible at older

wake ages. Similarly, the effects of molecular viscosity

() can be considered negligible compared with the ef-fects of eddy viscosity. This leaves the gradients of the

stress terms uw, vw, and w2 to play a significant role

in defining the axial momentum at any wake age. This is

especially true at the centerline of the vortex, where the

axial velocity deficit is maximum. Figure 23(b) evidently

suggests that the gradient vw/y is very high inside thevortex core. Such high gradients directly transfer momen-

tum from the streamwise direction to the cross flow direc-

tion, resulting in the reduction of the peak axial velocity,

as shown in Fig. 12(b).

ConclusionsComprehensive measurements in the flowfield of a sub-

scale rotor operating in hover were performed using dual-

plane digital particle image velocimetry. This method al-

lowed for the successful, simultaneous measurement of

all nine velocity gradient tensors. The measurements con-

centrated on the shed vortex system behind the blade, and

studied the tip vortex evolution from as early as =4

(on top of the blade) to 270 wake age. These dual plane

measurements were complemented by high resolution, 2-

D DPIV measurements of the near-wake, and LDV wake

measurement on the same rotor. The present measure-

ments can be used to help validating CFD predictions aswell as to calibrate new turbulence models.

The following are the specific conclusions derived from

this study:

1. High-resolution imaging of the vortex sheet trail-

ing behind the rotor blade revealed the presence of

several micro-scale, high frequency, counter-rotating

Taylor-Gortler vortex pairs, which produce substan-

tial fluctuations in the flow velocity. These fluctu-

ations, combined with the high velocity gradients

found in these vortices play a significant role in the

momentum transfer properties of the boundary layer.

Consequently, this affects the roll-up process of the

tip vortices.

2. Turbulence intensity measurements clearly showed

anisotropy. Specifically, Vr was greater in magni-

tude than V in both the near- and far-wake. This

is consistent with previous observations made on the

tip vortices trailing a micro-rotor at very low vor-

tex Reynolds numbers. The very presence of VrV

(which is typically assumed to be zero for solid body

rotation) can be concluded to be the primary reason

for the anisotropy.

3. Qualitatively, good correlation was found in the

Reynolds shear stress distribution and strain rate be-

tween the current measurement and other measure-

ments made on tip vortices trailing behind a micro-rotor, and a fixed- wing. This suggests that the tur-

bulence pattern remains the same for all tip vortices,

independent of the operating Reynolds number or the

type of lifting surface on which they are measured.

The results conclude that shear stresses cannot be

written as a linear function of strain, as assumed in

most of the existing linear eddy viscosity based tur-

bulence models.

4. Excellent quantitative correlation found between

LDV and DPIV measurements of turbulent intensi-

ties clearly suggest that DPIV can be confidently ap-

plied for turbulence measurements. Additionally, theuse of 1000 samples in the case of DPIV was found

to be sufficient, based on a comparison with 200,000

samples used in the case of LDV. This establishes the

sample size requirement for statistical convergence

of turbulent properties of tip vortices using DPIV.

5. The measured mean characteristics of the tip vor-

tices, such as their core size, peak swirl velocity,

and their variation with time were found to correlate

well with previous measurements made using LDV.

The turbulent diffusion, especially, was found to be

eight times that of molecular diffusion. The mea-

sured peak axial velocity deficit (corrected for apre-riodicity bias effects) was found to be about 75%

of the tip speed at the earliest wake age behind the

blade. This reduced to 40% at 60 wake age. Such

high values were not found in any previous literature

because of the unavailability of any method to cor-

rect for the effects of aperiodicity. These high deficit

values clearly suggest that aperiodicity plays a sub-

stantial role in axial velocity estimation.

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VW: -0.004 -0.002 0.000 0.002vw___

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