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Three-dimensional reconstruction of the oscillatory
free-surfaces of a flow overantidunes: stereoscopic and
velocimetric techniques
D. Douxchamps1, D. Devriendt2, H. Capart2;3, C.Craeye1;4 , B.
Macq1 and Y. Zech21UCL Telecom. Lab., Place du Levant, 2, B-1348
Louvain-la-Neuve, Belgium2UCL Civ. Eng. Dept., Place du Levant, 3,
B-1348 Louvain-la-Neuve, Belgium
3Fond National de la Recherche Scientifique,
Belgium4TUE/Electromagn. Lab., PO 513, MB5600 Eindhoven, The
Netherlands
AbstractWe present imaging methods developed to characterize
the oscillatory free-surface of rapid flows and apply them
totorrential currents propagating over sediment antidunes. Theaim
is to obtain high-resolution relief maps of the
free-surfacetopography, in order to highlight the regular spatial
patternsassociated with the bedforms. Two measurement principlesare
outlined and tested, both based on the imaging of floatingtracers
dispersed on the rapidly flowing surface. The firstrelies on direct
stereoscopic measurements obtained using twocameras, while the
second exploits an original velocimetricprinciple allowing to
derive elevation from the velocity fieldacquired using a single
camera. The measurement proceduresand image analysis algorithms are
introduced for the twomethods, along with the physical assumptions
underlyingthe velocimetric principle. The results of the two
techniquesare compared for different free-surface patterns and
goodcorrespondence is obtained. The obtained relief maps
vividlydepict the variety of motifs that can evolve as a result
ofinteraction between shallow flows and loose sediment beds.
I. INTRODUCTIONOf interest to hydraulic engineers and
geomorphologists,
bedforms of various types are observed in a wide range ofmodern
flow environments and ancient sediment deposits. Inthe present
paper, we focus on one of the most evanescentfamily of bedforms:
antidunes. This class of fluvial bedforms[1],[2] appears when
rapid, shallow currents propagate overcoarse granular material, and
is characterised by in-phasecoupling between the oscillatory
sediment bed and the flowfree-surface. They can be observed both in
the laboratory[4] and in the field [3]. Among other features,
antidunesare notable for the fact that they vanish rapidly once
theflow wanes. As a result, they leave few lasting traces asidefrom
bedding and grain-sorting effects. Their geometry, inparticular,
has to be studied when the flow is active.
Long- or short-crested, arranged in regular arrays or inisolated
trains of peaks and troughs, antidunes come in avariety of
patterns. In the present study, a characterisation ofsuch patterns
is sought through measurements of the waterfree-surface topography.
Visual access to the underlyingsediment bed surface is impaired by
the rough and shallowcharacter of the flowing water layer, hence no
directmeasurement of the bottom topography is possible. Since
thetwo surfaces are locked in phase with each other, however,
the
water surface topography provides an indirect image of
thebedform pattern.
Measurements of water surface topography most ofteninvolve point
sensors, placed in multisensor arrays at fixedlocations or scanned
across the surface. Sensors used tomeasure water elevation include
resistance gauges, pressuretransducers and acoustic beams.
Resistance gauges areinapplicable in the present case because of
the high sensitivityof antidune flows to intruding objects. More
generally, thespatial and temporal resolution of point sensors is
limitedby the number of available devices or the time required
toscan a sensor across the field of interest. This makes
themunsuitable for the transient (on a time scale of a few
seconds),spatially varied surfaces of flows over evolving
bedforms.Imaging techniques, by contrast, permit non-intrusive,
fastwhole-field measurements (a review of velocimetric methodsis
given in [5]). In the present work, two distinct methodshave been
developed for the rapid profiling of a highly variedwater surface.
Both are based on the imaging of floatingtracers dispersed on the
flowing water, but the two rely ondifferent reconstruction
principles. The first is a stereoscopictechnique, based on the
matching of particles on image pairsacquired from different
viewpoints using two cameras. Thesecond is a monoscopic technique,
based on particle trackingmeasurement of the velocity field in the
horizontal plane. ABernoulli-type principle derived from the fluid
mechanicsof the water free-surface is then exploited to reconstruct
thevertical elevation map. As the two techniques are based
ondifferent principles and complete separation was
maintainedbetween the two data elaboration processes,
intercomparison ofresults makes it possible to estimate the
measurement quality.
The paper is organised as follows. The experimental setupused
for the measurements is first presented. Each of the twomethods is
then detailed. Finally, results of the two methods areshown and
compared.
II. EXPERIMENTAL SETUPExperiments are conducted in a hydraulic
flume having
the following dimensions: length = 6 m; width = 50 cm;sidewall
height = 50 cm. The flume is tilted to obtain a bottomslope of 1 %.
A 5 cm deep layer of loose sediment coversthe flume bottom and is
replenished during flow by means ofa silo. A coarse sand of nearly
uniform size distribution isused as sediment material. For the
preliminary experimentspresented herafter, the water inflow was not
tightly controlled
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X
Y
Z
Figure 1: A snapshot from one of the stereo cameras, with
thethree-dimensional frame of reference. The white particles on
theimage are the floating tracers. Symbols (+) mark the detected
particles.Flow is from left to right.
but rather freely operated in such a way as to produce a
varietyof patterns. Water discharges in the flume ranged from 15
to35 l/s, yielding flow depths of 4 to 6 cm and Froude numbersin
the range Fr = 1.4 to 2. For such high Froude numbers,the water
free-surface responds strongly to the underlyingoscillatory
topography of the bedforms, up to the point ofbreaking at the wave
crests when antidunes are fully developed.Starting from a plane
bed, the antidunes emerge as longitudinaltrains of crests and
throughs initiating from downstream butstationary in phase. The
antidunes are observed to respond totransient changes in the flow
rate (both increase and decrease)by temporarily growing in
amplitude. Amplitude responsesand gradual shifts in pattern occur
on a time scale of tens ofseconds. By contrast, on the shorter
timescale corresponding tothe image acquisition (of the order of 2
s), the hydrodynamicscan be assumed to be quasi-steady and this is
exploitedhereafter to derive a single surface from each
measurementsequence. Slight unsteady pulsations of the flow are
howeverobserved, and this physical source of noise will affect
theresults concurrently with measurement errors.
The measurement section is placed some 2 m upstream ofthe flume
outlet. Moments before image acquisition, a uniformdispersion of
floats is dropped onto the mean surface by meansof staggered metal
meshes. The tracer particles are whitewooden pearls 9mm in
diameter. Image sequences are obtainedusing digital cameras placed
above the flow. The velocimetricand stereoscopic methods require
rather different acquisitionsystems, hence some care is necessary
in order to operate themsimultaneously to image the same scene. Two
commercialdigital cameras (miniDV, PAL, 25 frames per second
(fps))are used for the stereoscopic measurements. These
camerasoffer good image resolution (576 by 768 pixels) but cannot
besynchronised with each other during the acquisition. To
avoidmotion-stereo ambiguity, it is thus necessary to
synchronisethem a posteriori using an interpolation procedure (see
below).
The stereo cameras are placed above the flow with obliqueoptical
angles contained in a vertical plane parallel to thedirection of
flow (see Fig. 1 for a sample image). For thevelocimetric
measurements, a high frame rate camera (250 fps)is placed directly
over the flow with a nearly vertical opticalaxis. Due to the high
frame rate, this camera requires stronglighting, obtained with four
2 KW light sources. Such powerfullighting saturates the commercial
cameras even at maximumshutter speed, and these have to be fitted
with dimming filters.After positioning, the viewpoints of all three
cameras aredetermined by placing a calibration target in their
commonviewing volume. This is essential for stereo reconstruction
andallows the results of the two methods to be obtained in thesame
three-dimensional referential.
III. STEREOSCOPIC TECHNIQUEIn the first measurement technique, a
stereoscopic procedure
is applied to the dispersion of floating tracers imaged by
thetwo commercial cameras. These recognisable feature-pointsare
detected on the two different views and matched in orderto derive
their three dimensional coordinates. This requiresprecise
synchronisation of the two views, achieved here by aninterpolation
procedure performed a posteriori. Finally, thewater elevation map
is obtained by fitting a surface throughthe collection of points
positioned in space. Presented inmore detail in [6], these various
steps are outlined in the nextparagraphs.
A. Particles detectionThe detection process is based on a
transformation of
the original image with a variation of the wavelet transform[7],
[8] presented in [9] and modified for our purposes.Strickland
developed in this paper an optimal filter for breasttumor
detection: a case where the texture of the images aresurprisingly
close to our images of floating particles overblurred
sediments.
Since the particles have a unique known size, we canuse a single
set of two adapted wavelet filters instead of acomplete recursive
filter bank. Filtering the image along its twodimensions with the
two different filters provides four filteredimages: dHH , dLL, dHL
and dLH where the indices “H”and “L” refer to the filter used
(High- or Low-pass). Whilethe dHH image could be used alone for
detection, we preferto recombine it with the dLH :dHL image to
lower the effectof high-frequency noise. The combination of the
differenttransforms is then
T =1
2
�pdHL:dLH + dHH
�; (1)
where all operators are applied element by element. Themaxima of
T represent the particles positions and are detectedusing a classic
neighborhood zeroing technique with anadditional low-derivative
constrain. The set of detectedparticles is shown on Fig. 1.
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B. Camera viewpoint calibration and matching ofstereo pairs
The information provided by the particles localizationbeing
expressed in pixels, a transformation is required totranslate world
coordinates x � (x; y; z) to image coordinates(Xk; Yk) for a given
camera viewpoint k. This transformationis usually modeled as a
central projection followed by an affinetransformation in the image
plane [10], [11]. It is then possibleto define for each viewpoint k
a matrix Ak and a vector bksuch that 0
@ XkYk1
1A = Ak
0@ xy
z
1A+ bk: (2)
The 9 coefficients of matrix Ak and 3 coefficients of vectorbk
can be calibrated by least squares using at least 6 pointsfor which
we known both world and image coordinates. Inpractice, it is
important to use a larger number of these points,with positions
well distributed in the viewing volume of interest.
A point having image coordinates (Xk; Yk) under viewpointk is
then known to belong to a ray defined by parametricequation
x = x0k + fkvk (3)
where x is taken as a column vector, fk is an arbitrary
scalarand vectors x0k and vk are given by
x0k = �A�1k bk; vk = A�1k
0@ XkYk
1
1A (4)
Consider two candidates for a stereoscopic match, having
imagecoordinates (X1; Y1) as seen by camera 1 and (X2; Y2) as
seenby camera 2. The corresponding rays are defined by
x = x01 + f1v1; x = x02 + f2v2: (5)
The locations x�1
and x�1
on rays 1 and 2 corresponding tothe least distance between the
two rays are then specified byparameters f �
1and f�
2which are solutions to the system�
v0
1v1 v
0
1v2
v0
2v1 v
0
2v2
��f�1
f�2
�=
�(x02 � x01)0v1(x02 � x01)0v2
�(6)
where the prime denotes the transpose of a vector. It is then
easyto find the midpoint x�
12and the distance between rays d12:
x�
12=
1
2(x
�
1+ x
�
2); d12 =
q(x
�
2� x�
1)0(x
�
2� x�
1): (7)
Due to errors in image plane measurements and cameracalibration,
rays corresponding to the same physical point willnever perfectly
intersect. A small distance d12 thus indicatesan encounter that is
close to an intersection at point x�
12.
This distance is exploited to establish which points on eachof
the two images correspond to one and the same physicalparticle:
they are likely to be the ones for which the distancedij is
minimum. To obtain a global matching subject to theconstraint that
each particle can only participate in one pair, anapproximate
optimisation algorithm (the Vogel method) is usedin the present
work.
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cale
d cr
iterii
Figure 2: Variation of the discrepancy indicator (average
distancebetween matched rays) with the trial phase lag between
theasynchronous cameras. The minimum discrepancy at 16.3ms
providesan estimate of the actual phase lag.
C. A posteriori camera synchronizationIf the particles are
moving, their images under two different
viewpoints have to be precisely synchronised in order forthe 3D
stereo reconstruction to be successful. As the imagesobtained from
our two commercial cameras are asynchronous(a situation also
encountered in other contexts such as airbornestereo acquisition),
a special procedure had to be devised tosynchronise the particle
positions a posteriori.
First, particles on one of the images are tracked in timeusing
the Voronoı̈ algorithm also employed for the velocimetricmethod
[12]. Positions of the particles at any given instantcan then be
obtained by interpolation along the interframedisplacements. If
this interpolation time is chosen to offsetthe phase lag between
the two asynchronous cameras, thenthe interpolated positions on the
first view can be preciselymatched with the positions on the
other.
As the phase lag between the two cameras is unknown, aniterative
procedure is used. A trial value is chosen for the phaselag, the
interpolation is performed, and a tentative matchingof the
particles is obtained. A measure of the quality of thematching is
then provided by a discrepancy indicator such asthe average of the
distances dij between the matched rays. Bytesting various values of
the phase lag, a value can be foundwhich minimises the discrepancy
indicator (Fig. 2). As the twoasynchronous cameras operate at
precisely the same frequency,this optimal phase lag correction has
to be determined only oncefor each run.
D. Free-surface reconstructionThe stereoscopic matching
procedure (after
synchronisation) yields a set of points positioned
inthree-dimensional space. If the process were perfectly
accurate,all points would belong to a common surface and
interpolationwould suffice to approximate the surface elevation
�(x; y) at
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Figure 3: Sample surface slice (centerline of run 3) showing
the3D stereo points and the surface estimates obtained with
differentaveraging methods. Dashed line: mean; continuous line:
median;thick line: Bayesian estimate.
arbitrary locations. In practice, dispersion arises because
ofphysical fluctuations, mismatches, and limited accuracy of
theparticle positioning and ray intersection.
Mismatching of stereo pairs constitute a particularlyproblematic
source of error because it tends to scatter theresulting spurious
points uniformly in the viewing volumerather than in a
neighbourhood of the actual surface. Theprocedure adopted consists
in binning the data into rectangularcells in the (x; y) plane, then
averaging the data of eachbin to obtain a value of the surface
elevation. To deal withmismatch error, Bayesian averaging [13] is
performed for eachbin. It consists in a recursive procedure which
assigns to eachpoint a probability that it constitutes a correct
match, thenobtains the free-surface position as an average weighted
by thisprobability. In Fig. 3, it can be seen that this special
averageyields better results than both the mean and median
estimatesin the present case.
Another issue concerns the size of the cells to be used inthe
binning procedure. If the cells are too large, attenuation ofthe
surface amplitude variations occurs, while if the cells aretoo
small, the noise is insufficiently filtered. Both these effectscan
be approximately estimated from the data, and we utilise abin size
of 50 to 100 mm which minimises the sum of the twoerrors. For the
stereo procedure, this results in a combined errorwhich is
estimated to be of the order of 1 mm (but which canbe considerably
larger in zones of the flow were few particlesare present). This is
reasonably small compared to the elevationrange, which is of the
order of 30 mm.
IV. VELOCIMETRIC TECHNIQUEThe second technique involves the
reconstruction of the
surface elevation map from the horizontal velocity field,by
means of a relation derived from the dynamics of thefluid
free-surface. A raw velocity field is first extractedfrom
monoscopic images using a pattern-based particle
tracking velocimetry (PTV) algorithm. The elevation field isthen
obtained after suitable along-streamline and transverseaveraging of
the individual velocity vectors. These steps aresketched in the
next paragraphs. For more detail, the reader isreferred to [14] and
[12].
A. Free-surface dynamicsTo describe the local dynamics of the
fluid free-surface,
we make the following two basic assumptions: (i) the
timeevolution of the loose sediment bed is sufficiently slow that
itappears stationary to the rapidly flowing fluid, hence the
flowcan be taken as quasi- steady; (ii) the flow can be
consideredinviscid (but it does not have to be assumed
irrotational).
Under these assumptions, and if the slope of the free-surfaceis
everywhere moderate, then the following equations holdalong a
surface streamline:
d
d�
�u � u2
+ g��= 0; (8)
w =pu2 + v2
d�
d�; (9)
where x � (x; y; z) is the position given by its
longitudinal,transverse, and vertical coordinates, u � (u; v; w) is
the fluidvelocity given by its three components, �(x; y) is the
elevationof the free-surface, and � is the curvilinear position
coordinatealong the horizontal plane projection of the
streamline�x(�); y(�)
�defined by
@x
@�=
upu2 + v2
;@y
@�=
vpu2 + v2
: (10)
Dynamic equation (8) is the Bernoulli equation, describingthe
conservation of energy for a steady flow with vorticityand written
for a surface streamline, while kinematic equation(9) expresses the
constraint by which an infinitesimal fluidparcel belonging to the
free-surface remains at the free-surface[15], [16].
From equations (8) and (9), it is at once apparent that ifthe
horizontal velocity field fu; vg(x; y) is known, then anordinary
differential equation can be solved for the free-surfaceelevation �
along each streamline. This constitutes the basis ofthe
velocimetric technique proposed here for the measurementof
free-surface topography. On a streamline-by-streamlinebasis, the
above principle can be applied to a wide class offlows.
An especially simple situation arises if, as in the presentcase
of antidune flow, the free-surface can be approximated as
asuperposition of small amplitude oscillatory perturbations upona
rapid mean flow in the longitudinal x direction (with only aweak
mean velocity gradient along the transverse y direction).In that
case, up to first order in a
uwhere a is a typical amplitude
of the velocity fluctuations and u is the mean surface
velocity,equations (8) to (10) reduce to the following simple
expressionfor the elevation:
� = � � uu0g
(11)
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Superposed upon a mean surface level �, the fluctuationsin
free-surface elevation scale linearly with the fluctuationsin
longitudinal velocity, by a factor equal to the meanvelocity
divided by the gravitational constant g. Qualitatively,conservation
of energy head along streamlines results inparticle velocities
which are slowest at the crests and fastestat the troughs of the
oscillatory flow, an observation whichwas already consigned be
Lonardo Da Vinci in his notebookssome centuries ago [17]. In its
quantitative version (11), theobservation provides a way to extract
from the horizontalvelocity field an estimate of the local
free-surface elevationrelative to its average level. By pursuing
the expansion tosecond order, an estimate of the error associated
with the firstorder approximation can be made, and amounts to �1 mm
forthe conditions of the present experiments.
B. Velocity field acquisition and averagingBefore applying the
above principle, velocity measurements
are extracted from the high-frame rate image sequences bymeans
of a pattern-based PTV algorithm. Particles are firstdetected on
the images by applying a method similar to theone sketched above in
section III.A. The Voronoı̈ algorithmof Capart et al. is then used
to track particles from one frameto the next by comparing the
patterns associated with a localneighbourhood of the Vorono ï
diagram built upon particlecentroids. This makes it possible to
match particles overdisplacements which can be of the same order as
the meanparticle interdistance (whereas the simple nearest
neighbourtracking methods of PTV require the displacement to be
smallwith respect to this interdistance). In acquiring the
surfacevelocity field, it is implicitly assumed that the floating
tracersclosely espouse the motion of the fluid at the
free-surface.Due to the relatively large size of the tracers (9
mm), somedeviations are however to be expected. These effects
aredifficult to estimate and are not accounted for in the
method.
To regularise the raw velocity field and project it ontoa
regular grid, averages are performed first along
particletrajectories (oriented in a predominantly
longitudinaldirection), then along the transverse direction. Top
hat filtersare used, choosing filter widths which are again
calibratedto minimise the combined effects of attenuation and
noise.Error levels of the order of 1 mm on the resulting elevation
areestimated from both the trajectory and the transverse
filtering.On the basis of the regularised velocity field, equation
(10) isfinally used to obtain the oscillatory elevation field
component.
V. COMPARISON OF RESULTS AND DISCUSSIONExamples of final
surfaces obtained by both methods are
presented on Fig. 5. Since the velocimetric method
cannotdetermine the mean water height, the latter has been
substractedfrom the stereo results for ease of comparison. Because
theyresult from a 2D binning which assigns an elevation even
toareas where few particles were identified, the stereo
resultsextend over the whole domain. The velocimetric results, on
theother hand, are only given on the restricted domain covered
bythe retrieved trajectories of the particles.
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Figure 4: Comparison of surfaces cross-sections for run number
3.From top to bottom: y=100, y=200, y=300 and y=400.
The two methods were designed in such way as to becompletely
independent. It is thus reasonable to judge theirglobal accuracy by
comparing the results of both methodswhile keeping their respective
accuracy in mind. Fig. 4 showsthe strong correlation between the
two methods: amplitude,wavelength and patterns of the antidunes are
similar. However,some differences still exist between the two
surfaces, mostly inthe phase and in the local elevation of the
surface.
The cause of the phase lag between the velocimetric
andstereoscopic profiles could not be identified with certainty.As
the velocimetric elevation lags behind the stereoscopicelevation,
it is possible that this effect is associated witha delayed
response of the floating tracers with respect toelevation changes.
Wavebreaking at the crests could be anothercause of discrepancy,
while an inadevertent spatial bias of oneof the steps of the
algorithms cannot be completely excluded.
Some local discrepancies in amplitude can be ascribedto the
surface reconstruction process used for stereovision.In zones of
lower particle densities, Bayesian averaging isperformed over a
cell which can be larger than the relevantphysical feature. A zone
subject to this problem can beseen on surface 6 around position (x;
y) = (125; 400)[mm](Fig. 5g). In this zone, the synchronisation
interpolation andstereo matching appears to have broken down due to
the localturbulence associated with wavebreaking. Consequently,
thesharp crest was badly reconstructed as a crater.
The surfaces also exhibit textures which reflect
thepeculiarities of each of the two methods. Longitudinal stripesin
the velocimetric results are the memories of the streamlinesalong
which velocities are measured and averaged. TheBayesian averaging
of the stereo method, on the other hand,results in staircase-like
profile gradients. These are fine-grainedeffects, however, which do
not overly affect the surfaceelevation maps.
Overall, the quality of the comparison is
encouraging.Discrepancies in amplitude are of the order of the
anticipated
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(g) (h)Figure 5: Results of the reconstruction for several
surfaces. Left column present stereo reconstruction, right column
velocimetric reconstruction.(a), (b): surface 3, depth map. (c),
(d): surface 3 rendered. (e), (f): surface 5. (g), (h): surface
6.
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errors. The discrepancies highlighted in distorted z-axis
plots,moreover, appear much less severe when the surfaces areshown
at their true aspect ratios (Fig. 5c). Most importantly,the spatial
patterns are well-captured by both methods. Themeasurements vividly
depict a variety of motifs, includingnarrow trains of peaks and
throughs, broad crested rolls, andzig-zag patterns. All the
surfaces also show some asymmetryin their patterns. Due to their
continuous evolution and to thecomplex way in which light interacts
with the rough watersurface, such organised patterns cannot easily
be observed bypure visual inspection of laboratory flows.
VI. CONCLUSIONSTwo distinct imaging techniques were developed
for the
measurement of the oscillatory free-surface of flows
overantidunes. Both measurement principles succeed in
capturingantidune patterns, and the results compare favourably
witheach other. Beyond the particular application at hand,
theresults illustrate how stereoscopic and velocimetric
matchingalgorithms can be exploited for the
three-dimensionalcharacterisation of flowing fluid surfaces. Future
prospectsinclude the use of synchronised high-resolution cameras
andsmaller particles in order to obtain more accurate,
denserinformation. Also, application of the techniques to
evolvingantidune fields at a larger scale is contemplated.
VII. REFERENCES[1] J.F.Kennedy, ”The mechanics of dunes and
antidunes in
erodible-bed channels,” Journal of Fluid Mechanics, vol.16, 1963
pp. 521-544
[2] J.R.L.Allen, ”Sedimentary structures-their character
andphysical basis,” Amsterdam: Elsevier, 1984
[3] J.Alexander, C.Fielding, ”Gravel antidunes in the
tropicalBurdekin River, Queensland, Australia,” Sedimentology,vol.
44, 1997, pp. 327-337
[4] G.V.Middleton, ”Antidune cross-bedding in a large flume,”J.
Sediment. Petrology, vol. 35, 1965, pp. 922-927
[5] R.J.Adrian, ”Particle-imaging techniques for
experimentalfluid mechanics,” Annu. Rev. Fluid Mechanics, vol.
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