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. mani@umd.edu Graduate Research Assistant. bjo212@umd.edu Minta Martin Professor. leishman@eng.umd.edu
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
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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
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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
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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
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
0.01dudy: -2.000 1.000 4. 000 7. 000 10.000 13.000 16.000
(b) u/y
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
0.01
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
Distance from the vortex center, Y/R
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
Distance from the vortex center, Y/R
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
Distance from the vortex center, X/R
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
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
0.01dwdz: -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.
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(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)
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(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
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(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-
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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-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
0.02
S tra in ra te: - 15 00 0 - 75 00 0 7 50 0 1 50 00
(b) =2
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
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.
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Distance from the vortex center, X/R
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___
Distance from the vortex center, X/R
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
Distance from the vortex center, X/R
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__
Distance from the vortex center, X/R
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
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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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(a) Bradshaw
Distance from the vortex center, X/R
Distancefrom
thevor
texcenter,Y/R
-0.00500.005
-0.005
0
0.005
VW: -0.004 -0.002 0.000 0.002vw___
(b) Current
(c) Bradshaw (d) Current
Figure 23: Reynolds stress (vw) and strain rate vz +wy at 60
wake age showing anisotropy.
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