气泡减阻

Click here for quick links to Annual Reviews content online, including:

? Other articles in this volume ? Top cited articles

? Top downloaded articles ? Our comprehensive search

Further

ANNUAL REVIEWS

PIV:particle image velocimetry DNS:direct

numerical simulation HWA:hot-wire anemometry LDV:laser-Doppler velocimetry

1.INTRODUCTION

In the past three decades,particle image velocimetry (PIV)has become a standard tool in exper-imental ?uid mechanics.The principal characteristic that has made it so useful is its ability to measure the instantaneous velocity ?eld simultaneously at many points,typically of the order of 103–105,with spatial resolution suf?cient to permit the computation of the instantaneous ?uid vorticity and rate of strain.To date,PIV is the only experimental method that provides such information in rapidly evolving ?ows.PIV measurements are most commonly snapshots of the two-or three-component velocity vector ?eld on a planar cross section of the ?ow,but in recent years new developments have made it possible to measure the velocity over volumetric domains and to measure sequences of velocity in time at rates suf?cient to resolve the temporal evolution.Undoubtedly,PIV has signi?cantly advanced experimental ?uid mechanics,especially the study of ?ows in complex geometries and turbulent ?ows,providing resolution and detail that can compete with modern numerical methods,such as direct numerical simulation (DNS)(Moin &Mahesh 1998).Applications of PIV range from creeping ?ows (Santiago et al.1998)to detonations last-ing only a few tens of microseconds (Murphy &Adrian 2011),from nanoscale ?ow phenomena (Stone et al.2002,Zettner &Yoda 2003)to motion in the atmosphere of Jupiter (Tokumaru &Dimotakis 1995),and from the motion in the beating heart of vertebrate embryos (Hove et al.2003,Vennemann et al.2006)to the accidental release of oil at the bottom of the Gulf of Mexico (McNutt et al.2011,2012).The evolution of PIV into the currently dominant method for mea-suring velocity is illustrated in Figure 1.Since its invention,it has largely superseded the two most important methods of measuring point-wise velocity,hot-wire anemometry (HWA)and laser-Doppler velocimetry (LDV).These methods have strengths that PIV has not been able to duplicate thus far.HWA has a superb signal-to-noise ratio,which makes it ideally suited to study low-intensity turbulent ?ows and their spectra,whereas LDV is well suited to high-intensity ?uc-tuations with respect to the mean and accurate measurements of long-time average,single-point statistics.But neither provides the spatial derivatives,?ow visualization,and capability for the spatial correlation offered by PIV,and Figure 1is perhaps best interpreted as an indicator of the importance of those capabilities in modern experimental ?uid mechanics.

19601970198019902000

1

2

3

4

R e l a t i v e o c c u r r e n c e (a r b i t r a r y u n i t s )

HWA LDV PIV

Figure 1

The occurrence of the trigrams hot wire anemometry (HWA),laser Doppler velocimetry (LDV),and

particle image velocimetry (PIV)in Google Books (https://www.360docs.net/doc/be184906.html, )between 1952and 2008.We note that a previous review on PIV in this journal (Adrian 1991)appeared when there was no obvious

prevalence for any of the three main measurement methods.In the two decades since,PIV has become the dominant approach in experimental ?uid mechanics.Data taken from Google Ngrams.

410

Westerweel

·

Elsinga

·

Adrian

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

PTV:particle

tracking velocimetry Image density:average number of particle images per interrogation window Interrogation domain:small subpart of an image pair containing a pattern of particle images for which the displacement is determined by

evaluating the spatial cross-correlation Image shifting:approach to avoid directional ambiguity in multiple-exposure single-frame recording

PIV has been reviewed in the literature several times (Adrian 1991,2005;Tropea et al.2007;Wereley &Meinhart 2010;Katz &Sheng 2010)and is also the subject of two books (Raffel et al.1998,Adrian &Westerweel 2011).The most recent book presents the current state of the art for PIV in its broad sense,i.e.,including approaches such as particle tracking velocimetry (PTV),microscopic PIV,and holographic PIV.In view of the material covered in these sources,this article brie?y reviews only the major developments in PIV over the past 20years,and it concentrates in more detail on recent major developments and their logical extensions into the foreseeable future.These are time-resolved PIV (Section 1.3),statistical PIV (Section 1.4),and tomographic PIV (Section 2).In Section 3,we consider the current limitations of PIV,and in Section 4we consider some pathways that might be taken by the next generation of PIV instruments.

1.1.Synopsis of a Basic PIV System

Our review begins where a previous Annual Reviews article on imaging methods in ?uid me-chanics left off (Adrian 1991).That article considered many different methods—particle streak,laser speckle,optical ?ow analysis,particle tracking—and argued that double-pulsed,high-image-density images analyzed by digital spatial correlation analysis offered the best chance of achieving a large number of vector measurements per unit area and hence the best spatial resolution for studies of complex ?ows such as turbulence.At that time,images were necessarily recorded on photographic ?lms that required chemical processing.Subsequently,?ve developments combined to crystallize the form of the modern single-camera PIV system.(a )Digital computing power in-creased enough to permit the computation of two-dimensional (2D)spatial correlation functions using laboratory computers.(b )The feasibility of digital PIV was established (Willert &Gharib 1991)and theory was developed that enabled optimal implementation and established the basis for further re?nements (Keane &Adrian 1990,1991,1992;Westerweel 1993,1997,2000,2008;Westerweel et al.1997).(c )Digital cameras became available that were well suited to PIV by virtue of having more than 1million square pixels.(d )Special PIV cameras were developed that could separately store two images in close (<1μs)succession (Lai et al.1998,Lourenc ?o et al.1994).(e )Double-oscillator solid-state Nd:YAG lasers became available with pulse energies suf?cient to expose images of micrometer-sized scattering particles.Keane &Adrian (1990,1991,1992)de?ned a set of four simple “design rules”that provide guidelines to the optimal concentration of particles and exposure time delay t between the two light pulses with respect to the size of the interrogation domain,the light-sheet thickness,and the local variations in velocity.

This combination of a double-pulsed laser forming a light sheet,a single digital camera imaging light scattered from particles at 90?to the light sheet,and spatial correlation analysis of the images to determine their displacement in the image plane is a standard that has remained remarkably constant over the past two decades.A complete PIV system is shown schematically in Figure 2.The laser pulses are separated by an easily adjustable time delay t ,and the pulse durations are short enough,～10ns,to effectively freeze the images of submicrometer particles moving at all but the hypersonic velocities.Pulse energies range from 5to 200mJ,depending on the size of the illuminated region.The scattering particles are selected to follow the accelerations within the ?ow ?eld with high accuracy (i.e.,small velocity slip),to scatter light of intensity suf?cient to form clear images,and to have relatively uniform size.Typically,～1-μm aerosol particles,liquid or solid,are used in gas ?ows,and ～10-μm solid particles are dispersed into water ?ows.Images from the ?rst laser pulse are stored in the ?rst camera frame,and images from the second pulse go into a separate,second frame.This two-frame recording determines unambiguously the direction of the displacement between any pair of particle images,and it allows the analysis of overlapping images,thereby eliminating the need to employ image shifting (Adrian 1986).The frames are divided into

https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

411

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

?eld measured by stereoscopic PIV in turbulent pipe?ow at Re

plane is normal to the pipe axis.The arrows indicate the in-plane

indicates the axial velocity component.Data provided by Sebastian

University of Technology.(Animation available as Supplemental

Material link from the Annual Reviews home page at https://www.360docs.net/doc/be184906.html, Supplemental Material

of coherent ?ow structures in turbulent ?ows.Moreover,PIV enables the turbulence production terms,and other turbulence properties,to be evaluated even in highly complex ?ow geometries,such as in turbomachinery (Soranna et al.2010)and stirred reactors (Sharp &Adrian 2001).Positioning the light sheet normal to the main ?ow direction,in combination with stereo-scopic PIV,provides a means to determine the quasi-instantaneous ?ow structures as they advect through the measurement plane (see Figure 3).This approach revealed the ?rst experimental observation of new solutions,in the form of traveling waves,for the Navier-Stokes equations in a pipe geometry (Hof et al.2004).

Multiple-plane PIV has made it possible to measure derivatives perpendicular to the planes.Two parallel planes generated by two separate dual-laser systems in combination with a dual stereoscopic PIV system (i.e.,with four cameras in total)can be utilized in various ways (K¨a hler &Kompenhans 2000).Besides setting up arrangements that reduce the measurement uncertainty due to large out-of-plane motion,one can record all three velocity components at the same instant in time at two nearby measurement planes.This requires the appropriate arrangement of the laser-beam polarization,polarization ?lters,and cameras.However,it is possible to measure all nine components of the instantaneous velocity gradient tensor in a planar cross section of the ?ow.This was utilized by Ganapathisubramani et al.(2005)to characterize the ?ow structure in the log region of a turbulent boundary layer.

The use of multiple (three to four)cameras makes it possible to determine the motion of individual tracer particles in a volumetric domain (Maas et al.1993,Virant &Dracos 1997,Ouellette et al.2006).This provides unique experimental data on Lagrangian ?ow statistics and was used to determine the pair dispersion function in highly turbulent ?ows (Bourgoin et al.2006,Xu et al.2007).

Single-camera and multiple-camera PIV systems with the appropriate optics and ?lters and ad-vanced image processing make it possible to separate various optical data,based on (tracer/particle)size or the separation of light wavelength.This can be used to investigate particle-laden turbulent ?ows (Kiger &Pan 2000,2002;Khalitov &Longmire 2002;Hwang &Eaton 2006;Poelma et al.2006,2007;Tanaka &Eaton 2010)or mixing in a turbulent ?ow by combining laser-induced ?uo-rescence and PIV (Webster et al.2001,Fukushima et al.2002,Westerweel et al.2002,Holzner et al.2006).The latter approach was also used to investigate the turbulence statistics relative to a strongly convoluted turbulent/nonturbulent interface (Holzner et al.2006;Westerweel et al.2005,2009).

1.3.Time-Resolved PIV

The temporal evolution of turbulent velocity ?elds can be measured by dynamic,high-speed,or time-resolved PIV (TR-PIV)systems using double-pulsed PIV images in rapid succession.TR-PIV is conceptually straightforward,and its technical achievements are determined principally by the performance of available light sources and cameras.In particular,frequency-doubled pulsed Nd:YLF lasers are capable of generating 10–20mJ per pulse and up to 10,000pulses per second.These are typically combined with CMOS (complimentary metal-oxide semiconductor)cameras that can record 1024×1024-pixel images at framing rates up to 5,000frames per second.(Higher frame rates are achieved for reduced-image formats.)

Hori &Sakakibara (2004)used a stereoscopic TR-PIV system combined with a scanning light sheet to record a volumetric domain of a low–Reynolds number free jet.This was one of the ?rst PIV results at high image density that provided spatially and temporally fully resolved data of a ?ow in a volumetric domain.In addition to measuring the time evolution,TR-PIV time sequences of stereoscopic PIV data and Taylor’s frozen-?eld hypothesis can be used to reconstruct the

414

Westerweel

·

Elsinga

·

Adrian

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

quasi-instantaneous 3D vortical structures in the ?ow,such as for localized turbulence (or puffs)in transitional pipe ?ow (van Doorne &Westerweel 2009)or vortex packets in a turbulent boundary layer (Dennis &Nickels 2011).

TR-PIV offers the possibility of investigating strongly unstationary ?ows (e.g.,Hain et al.2009,Violato &Scarano 2011).Furthermore,the measured distributions can also be used to evaluate the different velocity terms in the continuity and Navier-Stokes equations and solve them for pressure (Liu &Katz 2006,de Kat &van Oudheusden 2012),making it possible to estimate sound sources related to the turbulent ?ow (Koschatzky et al.2011a,b).

The rapid succession of double images offers new possibilities for the improved analysis of PIV images,an area referred to as multiframe PIV.A number of methods have been proposed to exploit multiple images in the form of successive pairs of PIV images provided by current high-speed PIV cameras.These approaches improve the double-pulsed PIV measurements by using images at multiple times to optimize parameters (Legrand et al.2012),averaging over successive measurements assuming constant velocity (A.Sciacchitano,F.Scarano &B.Wieneke,private communication)or using adjustable time delays between double pulses (Hain &K¨a hler 2007).

1.4.Statistical PIV

There is a growing body of experimental studies in which PIV is used not to measure the instanta-neous ?elds,but instead to measure the ?ow statistics such as mean velocity or various components of the Reynolds stress tensor.This approach is particularly well suited to inhomogeneous turbu-lent ?ow in complicated geometries,such as the backward facing step or the separating boundary layer,wherein measurements need to be made on a 2D grid to document the ?ow.Data of this kind are used to validate the results of computational ?uid dynamics codes.The character of the experiments differs from measurements of instantaneous ?elds in that accuracy is of greater impor-tance.The use of PIV usually requires much less time in comparison to traversing a single-point probe to many points in the ?ow.This makes it possible to complete measurements in less time (saving operational costs)and relaxes the requirement of maintaining steady ?ow conditions over extended times.

One may conduct experiments for statistics by performing a series of conventional PIV velocity ?eld measurements and averaging over the ensemble of realizations,producing effectively a time average.But there is a different approach in which the local spatial correlation functions of the images in interrogation spots are averaged over the ensemble and then the mean statistics are extracted from the mean correlation function at each interrogation spot.This method builds on the correlation averaging technique developed by Delnoij et al.(1999)for multiphase ?ow and Meinhart et al.(2000)for microscopic PIV.It can be shown that the mean correlation is a convolution of the probability density function of the image displacements with the particle image function (Adrian 1988,Westerweel 2008).Therefore,either by deconvolution or by neglecting the effect of the image function,one can measure the probability density function,and hence the mean velocity and in-plane components of the Reynolds stress tensor.This approach is executed with very high spatial resolution by employing the single-pixel correlation method (Adrian &Westerweel 2011,Billy et al.2004,Westerweel et al.2004),and it has been utilized with impressive

results (K¨

a hler et al.2006,Scharnowski et al.2012).2.TOMOGRAPHIC PIV

Tomographic PIV (Figure 4)was recently introduced as a promising new method for measuring

the instantaneous 3D velocity ?eld (Elsinga et al.2006).In this approach,the tracer particles are

https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

415

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

Image pairs

Reconstructed volumes Velocity field Flow structures

Tomographic Cross-correlation

Gradients and Camera with lens on Scheimpflug mount

Measurement volume

Thick laser light sheet

Mirror

Light sheet–forming optics

Double-pulsed laser

X Z Y

X

Z Y

X Z Y

Figure 4

Supplemental Material

n n u . R e b y

Window

deformation:

approach in which the second window is

deformed according to the (estimated)

displacement ?eld to match the particle image pattern in the ?rst window

the cross-correlation analysis with the iterative multigrid (3D)window-deformation technique (Scarano &Riethmuller 2000)is usually favored as it produces less noise.

While maintaining the advantages of the simple optical arrangement of 3D PTV,the new tomographic PIV approach proved to be more robust for two main reasons.First,tomography does not rely on any particle detection,thereby relieving constraints on the particle image density and image quality that are inherent to 3D PTV.As a result,the particle image seeding density achieved typically increased by an order of magnitude from 0.005to 0.05particle images per pixel,and this resulted in substantial improvement in the spatial resolution.Second,the cross-correlation analysis of the particle displacement typically has lower noise compared to particle tracking methods.The overall measurement uncertainty for the particle image displacement is reported to be approximately 0.2–0.3voxel units (Elsinga 2008,Humble et al.2009,Atkinson et al.2011,K ¨

uhn et al.2011),which is comparable to stereoscopic PIV and only slightly above the noise level for standard planar PIV.

2.1.Applications

The variety of applications achieved to date clearly demonstrates the versatility of tomographic PIV,perhaps explaining its growing importance.The technique has been used in different ?ow facilities covering a velocity range from only a few micrometers per second in a water tank (Worth &Nickels 2011)to 510m s ?1in a supersonic wind tunnel (Humble et al.2009,Elsinga et al.2010).Furthermore,it can measure anywhere from millimeter-sized volumes when employing a microscope objective with four ports for imaging (Kim et al.2011)to meter-sized volumes,as demonstrated in a large-scale convection cell by K ¨uhn et al.(2011).Extensions to tomographic TR-PIV (Section 1.3)have been achieved in air and water by employing high-speed lasers and imaging systems (Hain et al.2008;Schr ¨

oder et al.2008,2011;Scarano &Poelma 2009).A two-phase ?ow investigation into drop coalescence has been reported by Ortiz-Due ?nas et al.(2010),who matched the refractive index of the phases to avoid optical distortions.They conducted measurements in only one of the two phases,but measurements of both are possible.Refractive index matching also allows internal ?ows to be accessed,such as in a human artery model (Buchmann et al.2011).Recently,Jeon &Sung (2012)applied tomographic PIV to study ?uid-structure interactions.The structures were made from a transparent sheet so that the particles on both sides could be recorded.By putting large marker dots on the surface,the authors could retrieve the position and deformation of the structure simultaneously with the ?ow velocity distribution on both sides.Many of the earlier studies focused primarily on the quantitative visualization of the coherent structures in various turbulent ?ows,based on data similar to those presented in Figure 5.Studies of turbulent boundary layers revealed the spatial organization of 3D vortical structures at different scales (Elsinga et al.2010)and the rapid growth of an ejection event (Schr ¨

oder et al.2008).Such data have been used to validate or to improve existing structural models for the speci?c turbulent ?ow.In all these cases,the 3D approach avoids the ambiguities inherent to the inference of 3D structures from planar data.More recently,however,tomographic PIV has been used to actually quantify the relevant ?ow variables related to the velocity gradients.For example,Worth &Nickels (2011)determined the joint probability density function of enstrophy and dissipation in homogeneous isotropic turbulence,and Elsinga &Marusic (2010)evaluated the material derivative of the invariants of the velocity gradient tensor,which de?ne a characteristic topology timescale.We anticipate that the quanti?cation of the different terms in the continuity and Navier-Stokes equations will become more common in the near future.This will likely produce valuable new insights into the interaction between the terms.In this context,one can think of,for instance,the balance between turbulence production and dissipation,especially in spatially evolving ?ows or

https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

417

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

b

5

Anaglyphs showing(a)vortical structures(lighter gray)as detected by the Q criterion in the transitional

following a zigzag boundary-layer trip(Elsinga&Westerweel2012)and(b)a fully turbulent

boundary layer(Elsinga et al.2007).The darker gray shading in panel b indicates a low-speed region.

the?ow is away from the observer.Depth can be perceived using stereoscopic glasses(i.e.,by placing

cyan?lter in front of the left and right eye,respectively),which deliver a clearer picture of the Supplemental Material

Self-calibration:

approach to correct for the disparity between camera calibrations based on the recorded particle images

fully 3D and turbulent ?ows that cannot be easily represented in DNS.However,direct compar-isons of the velocity gradient statistics between results from numerical simulations and experiments are still of interest,even in canonical cases,and are presently sparsely available.We note that suf-?cient spatial resolution is critical with regard to the accuracy of the measured velocity gradients,as shown in simulations of tomographic PIV measurements (Worth et al.2010)as well as in actual measurements (Tokg ¨oz et al.2012).

2.2.New Algorithm Developments

Advances related to the tomographic reconstruction algorithm have aimed at improving accuracy or mitigating the two main limitations of the original system,which were (a )the sensitivity of the result to the calibration accuracy and (b )the high computational costs of the reconstruction.For an accurate reconstruction of the particles,which are only a few voxels in diameter,the accuracy of the calibration needs to be better than 0.4voxels (Elsinga et al.2006).Although not impossible,this level of accuracy is very dif?cult to achieve routinely.Therefore,Wieneke (2008)introduced a volume self-calibration technique,which ?rst establishes the accuracy of the initial calibration and then corrects it.

A plain MART reconstruction is expensive in terms of the computation time and computer memory,as each reconstructed volume contains on the order of 109voxels,and there are approx-imately 1016weighting coef?cients relating the recorded image intensity to the volume intensity.Luckily,most of the volume is void,i.e.,not occupied by a tracer particle,and consequently most (more than 95%)voxel intensities are zero.The sparseness of the volumes can be exploited to signi?cantly reduce the number of operations in the reconstruction (Worth &Nickels 2008,Atkinson &Soria 2009).Although the exact implementation differs,the basic principle is to de-termine which voxels must have zero intensity and then skip them in the subsequent MART iterations because they do not contribute to the recorded image intensities.This approach re-sults in signi?cant speedup of a reconstruction by approximately a factor of 8(Atkinson &Soria 2009).Additionally,hardware-related speedups have been achieved employing computer clusters with parallelized codes (K ¨

uhn et al.2012)or using GPU computing.Furthermore,some tomo-graphic reconstruction strategies have been proposed recently that have the potential to reduce the computational load even more.These include the multiresolution approach,in which the spatial resolution of the reconstructed volume is progressively increased with the iterations (Discetti &Astarita 2012),and the so-called simultaneous MART algorithm (Atkinson &Soria 2009),which updates the reconstructed volume considering all pixels at the same time as opposed to updating on individual pixels in the standard MART algorithm.Both new methods yield an accuracy similar to MART;however,more iterations may be required for each volume reconstruction.

Improving the accuracy of the tomographic reconstruction may make it possible to operate at higher seeding densities,leading to higher spatial resolution of the ?nal measured velocity distribution.An example of a more advanced tomographic approach is motion tracking en-hancement (Novara et al.2010),which effectively removes the noise peaks that do not reappear in both reconstructed volumes used in the cross-correlation.Subsequent MART iterations then further enhance the reconstructed volume intensities.Motion tracking enhancement reduces random errors,but whether it can reduce bias errors due to so-called ghost pairs (see Section 2.3)remains unclear.Further improvements are expected from taking into account variations of the optical transfer function across the ?eld of view (Schanz et al.2012).New blends of PTV-tomographic reconstruction algorithms are currently also being investigated (Wieneke 2012).These last two methods are at relatively early stages of development,but appear promising.

https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

419

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

Ghost particle:spurious particle in a reconstructed tomographic PIV volume

SNR:signal-to-noise ratio

Source density:number of scattering sites within one particle image area;N S >O (1)indicates speckle

2.3.Accuracy Assessment

The ?nal measurement accuracy of a tomographic system depends on many experimental param-eters,most notably on the particle image density,the volume depth,and the number of cameras.These dependencies have been explored by means of simulated experiments (Elsinga et al.2006,Worth &Nickels 2008,Atkinson &Soria 2009).

The experimental conditions for which tomographic PIV yields accurate results can also be predicted theoretically by means of basic estimates for the number of ghost particles in the to-mographic reconstruction.They can affect the velocity measurement if the same ghost particles reoccur in both volumes used in the cross-correlation analysis,forming so-called ghost pairs.Such ghost pairs may introduce signi?cant bias errors in the measured velocity,which deteriorate the magnitude of the velocity gradients in the depth direction (Elsinga et al.2011).To avoid such bias errors,a signal-to-noise ratio (SNR),de?ned as the ratio of the number of actual particles N p over

the number of ghost pairs N ?

g

,needs to exceed a certain threshold value.The SNR is estimated from the relevant experimental parameters by

SNR ≡N p N ?g

～

=ppp 1?N

A p

?

z

?N

?1z

=

N S

z ?

z 0

1?N π

4d 2τd 2

e ?1

z ?d e /M 0

?1

,

(1)

where ppp is the particle image density in particles per pixel;N is the number of cameras;A p (=πd 2τ

/d 2e )is an effective particle image area (in pixel units),which is typically between 2and 3;and z and z 0are the volume depth in pixel units and physical units,respectively (Elsinga et al.2011).The ratio ?/ z ,or z ?/ z 0,is the relative part of the volume thickness over which the actual particle displacement is coherent,i.e.,varies within one particle image diameter d τ.Its value depends on the details of the ?ow.Furthermore,N S =ppp ×A p is the source density,M 0is the image magni?cation,and d e is the pixel pitch.We note that the same linear dimensions of a pixel and a voxel element are assumed in the above.The three terms between parentheses on the right-hand side of Equation 1represent an effective source density,A P ,and z ?expressed in pixel units,respectively.

One can then obtain an operational envelope for the SNR considering that the ratio ?/ z is at most unity ( ?= z )and bounded on the low end by ?=W S .The latter results from the requirement for nearly uniform displacement within a correlation window of linear dimension W S (in pixel units),for which the minimum size is given by the volume depth,the seeding density,and the required mean number of about seven actual particle pairs per window (Keane &Adrian 1992,Adrian &Westerweel 2011).The resulting SNR dependencies are shown in Figure 6.The region where tomographic PIV yields accurate results is bounded by the threshold value SNR =1[for MART reconstruction (Elsinga et al.(2011)]and the speckle limit N S ～=0.4(Adrian &Westerweel 2011,?gure 3.31).The SNR threshold value is only of order unity because the peak intensity of the ghost particles is lower on average than that of the actual particles (Elsinga et al.2006),which causes them to contribute less to the cross-correlation.Beyond the speckle limit,the fundamental assumption made in the tomographic reconstruction (i.e.,that the image intensity directly represents the particle distribution)is violated,which introduces signi?cant uncertainty.Speckle imaging may be avoided by using an incoherent light source,such as a light-emitting diode,but a limit in N S will exist nonetheless when too many pixels have nonzero intensity and the tomographic reconstruction is highly underdetermined.Still the SNR threshold will likely limit most experiments,as the ?ow ?eld is often unknown beforehand,and the conservative estimate ?= z thus applies.From the ?gure,it also becomes evident that improvements over the current reconstruction accuracy can increase the particle image density by a factor of 2–3at most before

420

Westerweel

·

Elsinga

·

Adrian

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

DSR:dynamic spatial range

DVR:dynamic velocity range Diffraction-limited spot:smallest image of a point source for a perfect lens with a ?nite aperture Fill factor:fraction of the area of a pixel in a digital image sensor that is actually light sensitive

one would actually need to improve the performance of all other system components to achieve signi?cant total improvement.

The spatial resolution and velocity measurement error for a given exposure time delay t between the two recordings are de?ned relatively to the linear dimension x of the ?eld of view and the maximum allowable particle displacement x p ,max ,which are represented as the dynamic spatial range (DSR)and dynamic velocity range (DVR);i.e.,

DSR =

x x p ,max =

L X /M 0

x p ,max

and

DVR =

u max σu =M 0 x p ,max

c τ

d τ

,(2)

where M 0is the image magni?cation,L X (= x M 0)is the linear dimension of the image sensor,u max (= x p ,max / t )is the maximum measurable velocity within the interrogation domain,and c τd τ(=σu t )is the minimum resolvable difference in the particle image displacement for a given particle image diameter d τ(Adrian 1997).c τis a dimensionless measure for the ability of the interrogation algorithm to accurately determine the particle image displacement,and it is typ-ically of the order of 0.05–0.2(Adrian &Westerweel 2011).From these expressions,it follows immediately that the product of DSR and DVR is a constant,

DSR ×DVR =

L X

c τ

d τ

,(3)

which de?nes the performance of a given PIV system.This uncertainty principle states that a PIV system can achieve high spatial resolution only with the penalty of reduced relative measurement accuracy,and vice versa (Adrian 1997).We note that the size of the interrogation domain does not appear in Equation 3;the expression is equally valid for PTV,in which the ?uid motion is determined from individual particle images.

The image diameter cannot be made smaller than the diffraction-limited spot size with current technology,so it is not a practical means of improving performance.The physical size L X of digital camera arrays has been increasing slowly from 1Mpixel to 14Mpixel over the past 20years (see Section 1.1),and further slow improvement seems likely to continue.The value of c τdepends on the algorithm used to measure the displacement (including the peak interpolator),the image noise and background noise,particle image aberration,the ?ll factor of the pixels,particle image overlap,and particle image clipping by interrogation window edges.Currently,the best value achieved for c τin real 3D ?ows is approximately 0.05.The development of many new algorithms to determine displacement has not resulted in any substantial reduction of c τ.Of the effects that determine the value of c τ,the ?ll factor probably offers the most room for improvement,but it may be dif?cult to approach a 100%?ll ratio using PIV cameras,and a new design is probably needed.

Table 1summarizes a selection of representative PIV measurements over recent years.It can be seen that the performance of modern planar PIV approaches that of photographic PIV more than a decade ago.Although tomographic PIV provides volumetric ?ow data,it can only fully resolve turbulent ?ows at very low Reynolds numbers or it under-resolves ?ows at higher Reynolds numbers.This is also evident from Figure 7.We note that the actual values of DSR ×DVR are typically a factor of 2–3lower than the theoretical values given by Adrian &Westerweel (2011,table 7.3),implying that in practice the maximum range of displacements is not fully utilized.Although PIV would not be the primary choice to determine a turbulence spectrum (HWA provides far better capabilities for this),it is instructive to analyze the performance of PIV in relation to the characteristics of a turbulence spectrum.For (nearly)homogeneous and isotropic turbulence,the energy spectrum E (κ)is given by (Tennekes &Lumley 1972)

E (κ)=βε2/3κ?5/3

for

2π

L ≤κ≤2π,(4)

422

Westerweel

·

Elsinga

·

Adrian

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g

b y M i

c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

Table 1Selection of representative PIV measurements of turbulent ?ows

x (mm)

x p ,max (mm)M 0d τ(μm)c τDSR DVR DSR ×DVR Re turb

Cylinder wake (Re =180)with tomographic PIV (Scarano &Poelma 2009)100

0.55

0.14

46.8

–

181

59

6,795

180

Turbulent TC ?ow (Re S =3,800)with

tomographic PIV (Tokg ¨oz et al.2012)

400.900.20––45672,975136

TBL (Re θ=2,460)with six-camera tomographic PIV (Schr ¨

oder et al.2011)68 1.060.3319.00.2064925,925800

Pipe ?ow (Re =5,300)with stereoscopic PIV (van Doorne &Westerweel 2007)

400.320.229.00.12126658,148360

Jet ?ow (Re =1,000)with scanning-plane PIV (Hori &Sakakibara 2004)1000.880.1733.10.06114748,475250

TBL (Re θ=1,900)with tomographic PIV (Elsinga et al.2007)

320.960.1812.00.31332608,640633

Jet ?ow (Re =2,000)with digital planar PIV

(Fukushima et al.2002)450.320.27 6.60.20143659,231500

Pipe ?ow (Re =5,300)with digital planar PIV

(Westerweel et al.1996)400.490.30 5.10.248212210,000362

Pipe ?ow (Re =10,000)with high-speed stereoscopic PIV (unpublished data)400.320.4810.70.1212612015,097565

TBL (Re θ=2,370)with large-format photographic PIV (Adrian et al.2000)

1510.550.8350.00.102739125,000790

Experimental conditions are speci?ed by the ?eld-of-view x ,maximum particle displacement x p ,max ,image magni?cation M 0,and particle-image diameter d τ.c τrepresents the measurement error for the particle-image displacement relative to d τ.The dynamic spatial range (DSR)and dynamic

velocity range (DVR)are de?ned in Equation 2.The turbulent Reynolds number Re turb is based on the wall friction velocity for wall-bounded ?ows [i.e.,pipe,turbulent boundary layer (TBL),and Taylor-Couette (TC)];for jet ?ow,it is based on the turbulence level.

where βis a constant,κis the wave number,L is the integral length scale,and ηis the Kolmogorov length scale,de?ned as

η= ν3/ε 1/4with L

～Re 3/4,(5)where νis the kinematic viscosity,and εis the (mean)dissipation rate per unit mass.The conven-tional view is that in a turbulent ?ow,energy is transferred from the macroscale to the microscale through the breakup of larger eddies into smaller eddies,i.e.,the so-called energy cascade.Eddies have a ?nite extent;i.e.,eddies of size are associated with a wave packet of width κ ～κ

https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

423

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g

b y M i

c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

Super-resolution:approach that

combines correlation analysis with particle tracking

full-scale value.Thus errors are 5–10%for a velocity that is only one-tenth of the full scale (under optimal conditions).

As mentioned in Section 3,Equation 3indicates that improving the performance of current two-pulse PIV requires either increasing the camera array dimension L X or decreasing c τ,which are both not expected to improve by a signi?cant magnitude.At this point,it appears that a factor of 2might be achievable by improving cameras and ?ne-tuning everything else.Clearly,to achieve a more dramatic improvement,we need a signi?cant innovation.

The DVR given by Equation 2is optimistic because the analysis does not take into account accelerations due to temporal and spatial variations of the ?ow that cause velocity changes dur-ing the time between laser pulses.The magnitude of the velocity change increases if x p ,max is made larger by increasing t ,which places another constraint on the maximum achievable DVR.Balancing the random error in the measurement of image displacement against the acceleration error,Boillot &Prasad (1996)have shown that there is an optimum time between pulses given by

t opt =

2c τd τ

M 0 ˙v p max .(7)The resulting DVR becomes limited by acceleration,DVR

M 0

c τ

d τ ˙v

p max .(8)

For example,estimating ˙v p

max ≈u 2max / and taking M 0=0.25,c τd τ=2.5μm,and =10mm,

one obtains a DVR ～=32,appreciably below the value of 100–200that is normally estimated when acceleration is ignored.The remedy clearly calls for a new measurement method that eliminates the acceleration error by directly measuring the acceleration.

4.1.Triple-Pulse PIV

If one uses multiple pulses to create images at more than two times,it is possible to measure the

velocity with greater time accuracy and to measure the acceleration directly.Moreover,one can use the additional position sample(s)to interpolate for particle position and velocity as a function of time,thereby improving the accuracy with which the measurement of the Lagrangian particle velocity is assigned to an Eulerian point in the ?uid.Increasing the number of pulses also reduces the required number of particles,so super-resolution by particle tracking improves (Keane et al.1995),and many other effects,such as particle image overlap and background noise due to particles,are also attenuated.

We consider here the improvements that may be possible by adding a single additional light pulse to the conventional double-pulsed PIV.Three light pulses separated by time delays as small as 10ns can be provided by a combination of beams from three solid-state laser oscillators,and current light-emitting-diode technology is capable of economically providing triple pulses with the pulse duration approaching 1μs (Willert et al.2010).

Haranandani (2011)recently analyzed the degree to which multipulse PIV can improve per-formance,?nding that the move from two to three pulses improves the velocity resolution and the spatial resolution.The improvement depends on the acceleration experienced by the ?uid and the tracer particles.If the acceleration results from path-line curvature at constant speed,the third pulse does little to improve the velocity accuracy and mainly improves the spatial resolution.If the acceleration results from speed variation along the path line,the velocity resolution is improved more than the spatial resolution.Because the resolution depends on the ?ow,it is dif?cult to

https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

425

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

R θ0 Δt / R

.

t ' / Δt

1.4

1.2

1.00.8

0.6

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

3.2

3.4

3.6

0.8

0.6

0.1

0.2

0.30.4

0.5

0.10.20.3

0.40.5

0.6

0.70.80.91.0

Figure 9

The cost function K (γ),de?ned in Equation 12,for 50%position error and 50%velocity error given a noise

level of σX p =2.25μm,R =10mm,and an acceleration factor ¨θ

0 t /2˙θ=1.0.circular path by ˙θ

=˙θ0+¨θ0t .The arc length R θ( t )=R ˙θ

0 t

1+¨θ0 t 2˙θ

0 (11)

consists of the constant velocity displacement plus the added displacement due to acceleration.For

example,if the acceleration parameter ¨θ

0 t /2˙θ0equals unity,the velocity varies by 200%relative to the constant velocity component.Under such conditions,the linear interpolation of the two-pulse data,which assumes constant velocity,makes substantial errors.The smallest interpolation error for position occurs when t =α t ,as the location of the particle is known within measurement error at that time.Interestingly,the best estimate of the velocity is found at different times.Good

performance of the interpolator is achieved when the second pulse is located at α=13

,and the location and velocity are evaluated at t

t =0.74.Figure 9plots contours of the cost function

K (γ)=γσV R ˙θ+(1?γ)σR

R ,

(12)where σV and σR are the root-mean-square errors in the magnitude of the particle velocity and

the radial location of the particle at t ,taking γ=0.5and α=0.33.This function gives a useful representation of the combined errors in position and velocity.The horizontal axis is the displacement of the particle as a fraction of the radius of curvature.For values less than 0.1,the trajectory is nearly a straight line,but for larger values,the curvature becomes important.Over a wide range of particle displacements,the cost function is close to a minimum at approximately t / t =0.74and 0.15.

In similar plots for the acceleration parameter ¨θ

0 t /2˙θ0ranging down to ?0.4and noise ranging down to 0.0,the shapes of the contours vary substantially,but they retain the same

general topology,and there is a relatively broad region at approximately 0.06≤˙θ

0 t ≤0.45,https://www.360docs.net/doc/be184906.html, ?PIV for Complex and Turbulent Flows

427

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .

Table 2Comparison of triple-pulse (3-PIV)and double-pulse (2-PIV)errors in velocity and position for various combinations of displacement,acceleration,and noise Case

Displacement Acceleration

factor Image

noise Velocity error (%)

Position error

(%)

R ˙θ

0 t /R ¨θ

0 t /2˙θ0σX P /R ×1043-PIV 2-PIV 3-PIV 2-PIV 10.100.2250.089?0.060.0040.1320.10.10.2250.0780.260.0090.2930.20 2.250.45?0.280.0340.5040.20.25 4.50.65 1.330.12 1.4750.200.450.090.170.120.5060.30 2.250.300.400.05 1.1270.30.25 2.250.24 1.960.23 2.6080.60 4.50.31 1.500.29 4.4790.60.25 4.50.75 4.38 1.047.83100.80.25 4.50.75 6.40 2.0213.111 1.0000.38 4.50 1.4713.412

1.0

0.25

1.32

9.41

3.90

21.5

All quantities are made dimensionless by the radius R of the curvature of the particle trajectory and the initial velocity R ˙θ

0at t =0.An asterisk marks cases in which double-pulse PIV yields a smaller error than triple-pulse PIV.

for which t / t =0.74yields a cost factor close to the minimum.Thus one can conclude that by placing the second light pulse one-third of the way between the ?rst and third pulse and calculating the interpolated value at a time that is 74%of the way from the ?rst and third pulse,one obtains a combined error in position and velocity that is close to the minimum over a range of ?ow conditions.

Using this recipe for triple-pulse estimation,we compare in Table 2the errors of triple-pulse PIV,calculated as the differences between the interpolated speed or radial position of the particle at time t minus the true values,to double-pulse PIV.In two cases (case 1and 3),the double-pulse PIV yields a smaller error than triple-pulse PIV.This occurs when the displacement relative to the radius of curvature is small,implying a nearly straight-line trajectory,and the acceleration is small,implying nearly constant velocity.These are the conditions for which double-pulse PIV was designed,and it performs better under these circumstances than triple-pulse because the estimated particle location from the third pulse adds noise.However,if acceleration is added (cases 2and 4),the noise is decreased (case 5),or the displacement is increased (case 6),then triple-pulse PIV performs better than double-pulse PIV.With increasing displacement relative to the radius,as would occur for smaller diameter vortices,the errors of double-pulse PIV become quite large,up to one order of magnitude larger than triple-pulse PIV (cases 9–12).This is the situation in which the chord between the ?rst and last images is a poor approximation to the trajectory,even when the acceleration along the path line is zero.

The accuracy with which triple-pulse PIV locates the particle at time t is always better than that of double-pulse PIV,often by a substantial margin.This amounts to an improvement in spatial resolution.Measurement of the acceleration also enables one to make a correction for particle lag,provided that the size of the particle is known.This is another signi?cant and often undetermined error in PIV measurements (Adrian &Westerweel 2011).

428Westerweel

·

Elsinga

·

Adrian

A n n u . R e v . F l u i d M e c h . 2013.45:409-436. D o w n l o a d e d f r o m w w w .a n n u a l r e v i e w s .o r g b y M i c h i g a n S t a t e U n i v e r s i t y L i b r a r y o n 01/03/14. F o r p e r s o n a l u s e o n l y .