PIV and hot wire measurements in the far field of turbulent round jets

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Meas. Sci. Technol. 21 (2010) 125402 (15pp) doi:10.1088/0957-0233/21/12/125402

PIV and hot wire measurements in the farfield of turbulent round jetsP Burattini1, M Falchi2, G P Romano3 and R A Antonia1

1 Discipline of Mechanical Engineering, University of Newcastle, NSW 2308, Australia2 Propulsion and Cavitation Lab, INSEAN, Italian Ship Model Basin, Rome, Italy3 Department of Mechanics and Aeronautics, University ‘La Sapienza’, Rome, Italy

E-mail: paolo.burattini@newcastle.edu.au

Received 28 February 2010, in final form 30 September 2010Published 4 November 2010Online at stacks.iop.org/MST/21/125402

AbstractWe present correction procedures that compensate for the limited spatial resolution of particleimage velocimetry (PIV) and hot wire anemometry (HWA) measurements. Both techniquesprovide two components of the velocity vector. The correction method, which is based onDNS velocity fields, is then applied to measurements in the far field of two round jets at aturbulent Reynolds number of about 179, for PIV, and 370, for HWA. Both large- andsmall-scale quantities are discussed. Amongst the latter, the correction of the velocityderivative variance is larger for the PIV data than for the HWA data. However, the opposite istrue for the correction of the isotropy ratios of the velocity derivatives. After compensation,the two techniques provide similar values of the mean energy dissipation rate and also of oneisotropy ratio.

Keywords: PIV, hot wire anemometry, turbulent round jet

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Particle image velocimetry (PIV, hereafter) is the fastestgrowing experimental technique for fluid mechanics. Therelative ease of set-up and data processing makes it suitablefor both fundamental and applied research. Until recently,however, the majority of small-scale turbulence measurementswere collected using hot wires. Their exceptional spatial andtime resolution has made them for decades the instrument ofchoice in the laboratory.

Of course, both techniques may suffer from limitedresolution, especially when measuring the smallest velocityscales. Methods for correcting HWA data acquired withdifferent types of probes have been discussed in the literaturesince the earlier seminal work of Wyngaard (1968, 1969) andthe more recent contributions of Zhu and Antonia (1995,1996). With the development of PIV, the analysis of itsaccuracy is also being addressed in more detail. Antonia(1993) compared hot wire data in a fully developed channelflow with LDV, PIV and DNS data in the same flow atapproximately the same Reynolds number. He noted that,although the hot wire overestimates the rms wall-normal

velocity especially close to the wall, it has the potentialof satisfactorily reproducing the DNS distributions of wall-normal rms velocity derivatives right across the channel.

Although PIV does not require Taylor’s hypothesiswhen generating streamwise velocity derivatives, the latterare nonetheless approximated by finite differences betweenvectors and are therefore subject to attenuation. The accuracycan be improved by decreasing the error in the computation ofthe finite differences and by reducing the distance between thevectors to the order of a few Kolmogorov scales (Antonia andMi 1993). In this context, a significant improvement has beenachieved with image deformation algorithms, which allow themeasurement of the velocity components with an error in theorder of 0.1 pixel (Scarano 2002, 2003, Nogueira et al 2001,2002). Although sufficient for large-scale quantities, this limitcan still be unsuitable for measurements of the small-scalemotion (SSM, hereafter). Further attempts have been madein order to improve the PIV resolution (Nobach and Tropea2005, Poelma et al 2006) or to compensate for it a posteriori(Lavoie et al 2007, Henning and Ehrenfried 2008, Foucautet al 2004, Foucaut and Stanislas 2002, Tanaka and Eaton2007).

0957-0233/10/125402+15$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

In broad terms, the finite resolution of the interrogationwindow has the effect of a moving average filter on the velocityfield to be measured. Assuming a square window of side δ, thewave-number spectrum of this filter is proportional to the sincfunction, sin(kδ)/kδ, where k is the wave number. Velocityfluctuations at wave numbers larger than 1/δ are severelyattenuated, thus mainly affecting the velocity derivatives,whose spectral content at large wave numbers is particularlyhigh.

To overcome this limitation and increase the stabilityof the image deformation algorithm, Nogueira et al (1999)and Scarano (2004) proposed using functions that weight theinterrogation windows. This improves the analysis in thepresence of large velocity gradients. The frequency responseof PIV depends on the selected weighting function applied, onthe window shift and on the interpolation functions used in theimage deformation procedures. Nogueira et al (2005a, 2005b)and Astarita (2007) (see also Astarita (2006, 2008)) performedthorough investigations of the effect of various weightingfunctions on the stability of the image deformation procedureand on the frequency response. Astarita (2007) and Schrijerand Scarano (2008) also tested the effect of the position of theweighting functions within the iterative window deformationalgorithm. Further analyses regarding the resolution of PIVare reported in Raffel et al (2002) and in Wanstrom et al (2007),with particular reference to circular jets.

In principle, the lack of resolution inherent to anymeasurement can be assessed, once the true velocity field isknown. In the laboratory, this would require an independentand more accurate measurement. Normally, this is difficultto obtain. A viable alternative is to resort to direct numericalsimulations or DNS data in order to mimic the causes leadingto the loss of the resolution of a particular technique. Byreversing the procedure, correction factors for any turbulentquantity can be derived. Therefore, the bias error canbe compensated for, thus improving the accuracy of themeasurement.

In this work, we carry out such a procedure for PIV, inanalogy to what was done earlier by Lavoie et al (2007). Theseauthors compared measurements taken with HWA and PIV ingrid turbulence. Then, they corrected the experimental datausing analytical spectra. Here, we repeat the same approach,but use DNS velocity field data instead. Further, we applythe method to measurements in the round jet. For this flowtype, the large-scale properties of turbulence in the far fieldare well documented, e.g. Mathieu and Scott (2000). For one,the intensities of the velocity fluctuations are unequal—thatis the streamwise direction being about 25% larger than that inthe radial direction, on the jet axis. For the SSM, however, thepicture is less clear. Undoubtedly, these are harder to measure,because they are affected by the spatial resolution limitation ofvirtually all experimental techniques. Nevertheless, the SSMcharacteristics in the jet far field are relevant not only from apractical point of view, but also in a more fundamental respect:since relatively large values of the turbulent Reynolds number(Rλ) can be readily achieved in the jet, it is instrumental instudies of the asymptotic behaviour of turbulence.

In this work, we present data taken with two differenttechniques, PIV and HWA, in the far field of two round

jets. We then compensate the measurements according tothe physical size of the actual probing volumes. The mainobjective is to compare the relative strengths of PIV andHWA, when measuring the SSM. Such comparisons are scarcein the literature, but nonetheless desirable, because they canhelp explain some of the discrepancies reported in differentexperiments. Note that, although a flying hot wire can limitthe rectification problem in large turbulence intensities such asthose found in the jet, in this work we used the more commonstatic configuration.

1.1. Basic definitions

The present correction method is aimed at the measurement ofthe following turbulent variables: the variance of the velocitycomponents

αx = u2x (1)

αy = u2y, (2)

which are large-scale quantities, and the (co)variance of thevelocity derivatives

βx,x =(

∂ux

∂x

)2

(3)

βy,x =(

∂uy

∂x

)2

(4)

βx,y =(

∂ux

∂y

)2

(5)

βy,y =(

∂uy

∂y

)2

(6)

βxy,yx =(

∂ux

∂y

)(∂uy

∂x

), (7)

which are small-scale quantities. Hereafter, a subscript beforethe comma denotes the velocity component, while a subscriptafter the comma refers to the direction of the derivative; theoverbar represents space or time averaging. Using these basicquantities, several ratios can be defined: the velocity rms ratio

Ix =(

αx

αy

)1/2

= I0, (8)

and the velocity derivative ratios

Iy,x ≡ βy,x

βx,x

= (∂uy/∂x)2

(∂ux/∂x)2(=I1) (9)

Ix,y ≡ βx,y

βx,x

= (∂ux/∂y)2

(∂ux/∂x)2(=I2) (10)

Iy,y ≡ βy,y

βx,x

= (∂uy/∂y)2

(∂ux/∂x)2(=I3) (11)

Ixy,yx ≡ −βxy,yx

βx,x

= − (∂ux/∂y)(∂uy/∂x)

(∂ux/∂x)2(=I4), (12)

2

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Figure 1. Schematic of the main parameters in PIV measurement, including the imaged area �x × �y , the interrogation window δx × δy ,the laser thickness δz and the spatial increments εx, εy of the derivatives. The dots represent the locations where single displacement vectorsare available and the symbol ⊗ denotes the centre of the interrogation window where the velocity is estimated.

in the last four definitions, the subscripts of I refer to thenumerator only, as the denominator βx,x is the same for allratios. In isotropic turbulence, it holds I0 = 1, I1 = 2, I2 = 2,I3 = 1 and I4 = 1/2.

The turbulent kinetic energy is

K =∫ ∞

0E(k) dk, (13)

where E(k) is the 3D energy spectrum and k is the wavenumber. The integral length scale is defined as

L = 3π

4K

∫ ∞

0

1

kE(k) dk, (14)

or, in terms of the 1D spectra Ex(kx) and Ey(kx),

Lx = Ex(0)

πu2x

(15)

Ly = Ey(0)

πu2y

; (16)

in isotropic turbulence

L = Lx = 2Ly. (17)

As usual, the turbulence Reynolds number is computed as

Rλ =√

20

3νεK, (18)

where ν is the kinematic viscosity and ε is the mean dissipationrate of K. In isotropic turbulence,

ε = 15νβx,x . (19)

The Kolmogorov scale is given by

η =(

ν3

ε

) 14

. (20)

2. Correction procedure

This section describes the correction method for some velocitystatistics measured by PIV and HWA. The general procedureis based on that of Wyngaard (1968, 1969), which wasoriginally developed for single-wires (SW) and cross-wires(XW). The velocity field is by hypothesis homogeneous, butnot necessarily isotropic. However, the DNS data, used inthe following to obtain the numerical values of the correctionfactors, comply also with isotropy.

2.1. PIV measurements

In PIV, at least three different factors affect the measurementof the velocity field (figure 1):

(i) the size of the interrogation window δx × δy ;(ii) the thickness of the laser sheet δz;

(iii) the spatial resolution of the grid εx, εy .

The individual effects of these factors are described below,before being compounded.

2.1.1. Interrogation window. The algorithm at the baseof PIV relies on the cross-correlation of two images (orexposures) within an interrogation window of size δx × δy ,in order to obtain the velocity vector on the plane x, y of thelaser sheet. This amounts to an average of the true and localvelocity vectors within the window. In previous work, Lavoieet al (2007) assumed a uniform average over the window. Inphysical space, it corresponds to

ux(x; δx, δy) = 1

δxδy

∫ y+δy/2

y−δy/2dy ′

∫ x+δx/2

x−δx/2dx ′ ux(x

′, y ′, z),

(21)

whereby an original true velocity component at x = x, y, z,say ux , is replaced by ux . Hereafter, the tilde denotes a quantityaffected by limited resolution, while the prime indicates a

3

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

−1−0.5

00.5

1

−1

0

10

0.5

1

x/δy/δ

Figure 2. 2D rectangular (Hxy , with δx = δy = δ) and Gaussian (G,with σ/δ = 1/3) weighting kernels in physical space. The functionsare normalized so that the integral under the surfaces is unity.

dummy integration variable. Equation (21) can be equivalentlywritten as a convolution integral

ux(x; δx, δy) = 1

δxδy

∫ ∫ ∞

−∞dx ′dy ′ Hxy(x − x ′, y − y ′)

× ux(x′, y ′, z), (22)

where

Hxy(x − x ′, y − y ′)

={

1 for |x − x ′| < δx/2, |y − y ′| < δy/2,

0 otherwise(23)

is the 2D rectangular function, see figure 2. The integral in(22), in contrast to that in (21), has the advantage of beingdefined over an infinite range and is therefore suitable to beFourier transformed.

The uniformity of the rectangular function produces asevere attenuation, especially at the smallest scales. Toovercome this problem, PIV algorithms typically use anonuniform averaging function that increases the weight of thevelocity vectors near the centre of the interrogation window.Nogueira et al (2002), Astarita (2007) performed parametricstudies for the effects of the shape of the weighting function.Here, we consider a Gaussian weighting of the interrogationwindow, i.e.

G(x − x ′, y − y ′) = C exp

[− (x − x ′)2 + (y − y ′)2

2σ 2

],

(24)

where C is a normalizing factor such that the integral ofG is unity and σ defines the width of the Gaussian; here,σ/δ = 1/3, where δ is the size of the square interrogationwindow. The functions Hxy and G are compared in figure 2.

2.1.2. Thickness of the laser sheet. The finite thickness δz

of the laser sheet causes an averaging of the velocity field inthe direction perpendicular to the sheet. The laser intensity

distribution can be considered, to a close approximation,as Gaussian. The averaging of the velocity field thereforeamounts to

ux(x; δz) = 1

δz

∫ ∞

−∞dz′ Gz(z − z′)ux(x, y, z′), (25)

where

Gz(z − z′) = Cz exp

[− (z − z′)2

2σ 2z

], (26)

with Cz the normalizing factor such that the integral of Gis unity and σz defines the width of the Gaussian; here,σz/δz = 1/3.

2.1.3. Spatial resolution of the grid. Spatial derivatives of thevelocity field are approximated as finite differences on the grid.The grid points are defined by the centre of the interrogationwindows, and the spatial increments on the grid are εx and εy ,see figure 1. Assuming a central difference scheme, a velocityderivative, say ux,x , is approximated by

ux,x(x; εx) = ux(x + εx/2) − ux(x − εx/2)

εx

. (27)

It can also be expressed as a convolution integral

ux,x(x; εx) =∫ ∞

−∞b(x − x ′; εx)ux(x

′, y, z) dx ′, (28)

where

b(x − x ′; εx) = 1

εx

[δD(x ′ − x + εx/2) − δD(x ′ − x − εx/2)]

(29)

and δD is the Dirac delta.

2.1.4. Compounded effect. The total attenuation effect on themeasured velocity field is given by the combined contributionsof the laser thickness, the interrogation window size and thespatial resolution of the grid. To compute the attenuation onthe variance of ux and ux,x , expressions in (24), (26) and (29)are convoluted with the true velocity field. To this end, it isuseful to consider their 3D Fourier transforms (denoted by ahat), i.e.

G(k1, k2; δx, δy) = exp

[−σ 2 (kxδx)

2 + (kyδy)2

2

](30)

Gz(k3; δz) = exp

[−σ 2

z

(kzδz)2

2

], (31)

(where the normalizing factors have been dropped, forsimplicity) and

b(kx; εx) = 1

εx

[exp(ikxεx) − 1]. (32)

Note that, while G and Gz are real, b is complex and thereforeintroduces a phase shift. The combined effect of the PIVresolution on the measured velocity field is given by theproducts

ˆux(k; δx, δy, δz) = G(k1, k2; δx, δy)Gz(k3; δz)ux(k) (33)

ˆux,x(k; εx) = b(kx; εx) ˆux(k), (34)

4

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

where k = (k1, k2, k3) is the velocity vector. By usingParseval’s theorem, the variance of an attenuated velocitycomponent is

αx(δx, δy, δz) = u2x =

∫ ∫ ∫ ∞

−∞G2G2

zux u∗xdk, (35)

while the variance of an attenuated velocity derivative along xis

βx,x(δx, δy, δz, εx) = u2x,x =

∫ ∫ ∫ ∞

−∞G2G2

zB2k2

xux u∗xdk,

(36)

where

B(kx; εx) = sin(kxεx/2)

kxεx/2=

√b(kx; εx)b∗(kx; εx), (37)

which is real. Similar expressions can be obtained for theother velocity components and the other directions. Thecombined effect of the PIV resolution, as given by (35)–(36),is in some cases a simplification of more complex nonlinearbehaviour, which is introduced by the image deformationalgorithms. This simplification is especially suitable for smalldisplacements of the interrogation window (with respect to theparticle image size) and homogeneous flows.

2.1.5. Correction factors. From a known velocity field, theattenuation of the velocity can be computed according to (35)and (36). The correction factors of the velocity variance αx

are

rαx (δ, δz) = αx

αx

(38)

rαy (δ, δz) = αy

αy

(39)

(and similarly for αy), while those of the velocity derivativesare

rβx,x (δ, δz, ε) = βx,x

βx,x

(40)

rβx,y (δ, δz, ε) = βx,y

βx,y

(41)

rβy,x (δ, δz, ε) = βy,x

βy,x

(42)

rβxy,yx (δ, δz, ε) = βxy,yx

βxy,yx

, (43)

where it is assumed that the interrogation window and the gridare square, i.e. δx = δy = δ and εx = εy = ε. In the case thatthe velocity field is isotropic, it holds

rαx = rαy (44)

rβx,x = rβy,y (45)

rβx,y = rβy,x . (46)

Of course, with perfect resolution all these correction factorsare unity. Note that rαx and rαy do not depend on ε, because theestimate of the variance does not require spatial derivatives.

0 0.2 0.4 0.60

0.1

0.2

0.3

0.4

0.5

0.6

0.70.93

0.94

0.95

0.96

0.97

0.98

0.99

δz/L

δ/L

Figure 3. Contours of rαx , see (38), for PIV.

0

10

20

0

10

20

0

5

10

δ†zδ†

ε†

0.6

0.7

0.8

0.9

1

Figure 4. Distribution of rβx,x , see (40), for PIV. The factor iscomputed over the entire domain, but only the data on the threeplanes defined by δ† = 0, δ†z = 0 and ε† = 0 are plotted for clarity.(Largest values are at the origin. Contours correspond to thenumerical values on the right scale.)

The correction factors in (38)–(43) are shown in figures 3–6, as a function of the normalized resolution parameters. Thevalues are computed from an isotropic DNS velocity field, andtherefore the terms on the right-hand sides of (44)–(46) arenot reported. For αx , the attenuation effect of δ is smaller thanthat of δz. The axes of figure 3 are normalized by L, whichis the relevant length scale for the kinetic energy (e.g. at largescales, the velocity power spectra collapse when rescaled withL and αx). If the axes were normalized by η, then the resultswould depend on the Reynolds number. For PIV interrogationwindows smaller than about 0.4 integral length scales, thecorrection factor is less than 1%.

Figures 4–6 report the correction factor for the varianceof the velocity derivatives. Here, the normalization of the axes(indicated by the dagger) uses the true value of η. In fact, thisis the relevant length scale for the small-scale quantities, such

5

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

0

10

20

05

1015

20

0

5

10

δ†zδ†

ε†

0.5

0.6

0.7

0.8

0.9

Figure 5. Distribution of rβx,y , see (41), for PIV.

0

10

20

05

1015

20

0

5

10

δ†zδ†

ε†

0.5

0.6

0.7

0.8

0.9

Figure 6. Distribution of rβxy,yy , see (43), for PIV.

as ε. The correction factors in figures 4–6 are less than unity,as the resolution worsens.

For velocity derivatives, it would be necessary to have aPIV interrogation window as small as 5η to have corrections ofless than 1% (for the present measurements, this correspondsto about 5 pixels).

The correction factors for the variance ratio in (8) aredefined as

rIx = I x

Ix

. (47)

In the present case, it is 1 because the interrogation windowis square and does not distinguish between one velocitycomponent and the other. The correction factors for thederivative ratios in (9)–(12) are

rIy,x = I y,x

Iy,x

(48)

rIx,y = I x,y

Ix,y

(49)

rIy,y = I y,y

Iy,y

(50)

0

10

20

0

10

20

0

5

10

δ†zδ†

ε†

0.9

0.95

1

1.05

1.1

Figure 7. Distribution of rIy,x , see (48), for PIV. (Largest values areon the δz-axis.)

0

10

20

0

10

20

0

5

10

δ†zδ†

ε†

0.85

0.9

0.95

1

1.05

1.1

Figure 8. Distribution of rIx,y , see (49), for PIV. (Largest values areon the δz-axis.) Note that, in isotropic turbulence, rIx,y is identical torIy,x in statistical terms.

rIxy,yx = I xy,yx

Ixy,yx

, (51)

see figures 7–10. Unlike the correction factors of thesingle derivatives, (48)–(51) can be nearly unity, even forrelatively large values of the resolution parameters. Therefore,by carefully choosing (δ, δz, ε), one can minimize theneeded compensation. For the isotropy ratios, it wouldbe necessary to have a PIV interrogation window as smallas 15η, in order to have corrections of less than 1%(for the present measurements, this corresponds to about32 pixels).

The correction factor for the Kolmogorov scale is reportedin figure 11. The values of rη are always > 1, meaning thatη is overestimated. rη is useful in the experiments, wherethe initial (i.e. uncorrected) estimate of the Kolmogorov scale,η(0), is affected by the resolution and should not be used as aninput in the graphs above. Instead, as a preliminary step, η(0)

should be corrected iteratively to obtain the true value η. Thiscan be done by using

rη(δ, δz, ε) = η

η= η† (52)

6

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

0

10

20

0

10

20

0

5

10

δ†zδ†

ε†

0.99

0.992

0.994

0.996

0.998

Figure 9. Distribution of rIy,y , see (50), for PIV. (Largest values areat the origin.)

0

10

20

0

10

20

0

5

10

δ†zδ†

ε†

0.92

0.94

0.96

0.98

1

1.02

1.04

Figure 10. Distribution of rIxy,yx , see (51), for PIV. (Unity valuesare at the origin.)

0

10

20

0

10

20

0

5

10

δ†zδ†

ε†

1

1.05

1.1

1.15

1.2

Figure 11. Distribution of rη, see (52), for PIV. (Smallest value = 1is at the origin.)

and proceeding as follows:

(i) compute a provisional corrector rη(0) from figure (11) with(δ/η(0), δz/η

(0), ε/η(0));(ii) update the value of the Kolmogorov scale η(1) =

η(0)/rη(0);

(iii) iterate steps (1) and (2) above, until η(i+1) = η(0)/rη(i)

converges to a fixed value, which is the corrected η.

In this case, this required five to six iterations. With suchestimate of η, figures 4–10 can be used to determine thecorrection factors for the SSM quantities.

2.2. XW measurements

The correction factors for the XW measurements are describedin Wyngaard (1969), Zhu and Antonia (1996), Burattini et al(2008b), Burattini (2008), and only a brief account is givenhere. The XW is composed of two wires, A and B, inclined at±45◦ with respect to the incoming flow (see figure 12). Thewires have the same length la = lb = l and are separatedby the distance d. The sampling rate fs of the velocity signaldetermines the increment εx used when computing the velocityderivative, as in (27). Assuming Taylor’s hypothesis, it holdsεx = Ux�t , where fs = 1/�t and Ux is the mean streamwisevelocity. Note that, while for HWA, εx is independent ofthe spatial resolution but depends on the temporal samplingresolution, for the PIV the spatial increment, εx is related to theinterrogation window size δ and resolution (number of pixels).The XW correction factors rαx (l, d), rαy (l, d), rIx (l, d),rβx,x (l, d, εx), rβy,x (l, d, εx) and rIy,x (l, d, εx) are shown infigures 13–18.

The streamwise velocity was also measured with a SW(see figure 12), in order to have an independent check. Thecorrections for a SW are detailed in Wyngaard (1968) andonly the factors αx and βx,x are shown here, figures 19 and20. They are functions of the wire length and of the samplingrate.

2.3. Numerical details

The correction factors described in the previous sectionsare computed using DNS velocity fields. These reproducehomogeneous isotropic turbulence of an incompressible fluid.The numerical code is pseudo-spectral where thecomputational domain—a cubic box, periodic in the threedirections—is discretized by N = 5123 Fourier modes, seetable 1. Aliasing is removed using the phase-shifting method,while the computations are performed in single precision(CFL = 0.5) on a cluster of PCs equipped with 64-bitCPUs. Statistical stationarity is achieved through large-scaleforcing; it injects energy into the Fourier modes within athin shell in the low wave number range 1.2 < k < 3.1(more details can be found in Burattini et al (2008a), Burattini(2008)).

The correction factors reported in the following arecomputed from a velocity field (called s0512e, see table 1)having a turbulent Reynolds number of Rλ = 147. Theresolution in terms of the maximum wave number is kMη =2.05, or δ/η = π/kMη = 1.53 in physical space, whereδ is the numerical grid increment step. The results werechecked by performing simulations at different kMη and Rλ,see table 1. In all cases, kMη was larger than 1.5, which isdeemed to provide a well-resolved DNS.

7

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Figure 12. Sketch of the HWA probes. Wires A and B form the XW (left); the SW is on the right.

0 0.05 0.1 0.150

0.05

0.1

0.15

0.950.9550.960.965

0.97

0.975

0.98

0.9850.990.995

l/L

d/L

Figure 13. Distribution of rαx for the XW.

3. Experimental details

The air jet (called AJ, hereafter) located at the Universityof Newcastle is generated by an open circuit wind tunnelequipped with a variable speed centrifugal blower. The tunnelcomprises a diffuser, a settling chamber and a contractionwith an area ratio of 85:1. A flexible connection linksthe blower to the rest of the wind tunnel, so as to reducemotor-induced vibrations. Screens and honeycomb are fittedinside the settling chamber to reduce the turbulence level andstraighten the flow. The jet exits through a nozzle having adiameter D = 55 mm in a relatively large laboratory room,whose temperature remains nearly constant throughout everysingle set of tests. The traversing system allows three degreesof freedom in the streamwise (x), lateral (y) and vertical(z) directions with a resolution of 1, 0.025 and 0.01 mm,respectively. Calibrations are made at the jet exit withinthe potential core, while data are taken at the axial locationx = 32D downstream of the exit.

0 0.05 0.1 0.150

0.05

0.1

0.15

0.97

5

0.980.

985

0.99

0.99

511.0051.0

11.015

l/L

d/L

Figure 14. Distribution of rαy for the XW.

The velocity is measured with a XW and a SW. Theseparation d of the XW is 0.8 mm and the length l of the wiresis 200 times their diameter dw = 2.54 μm, see table 2. Theprobes are operated by an in-house anemometer at an overheatratio of 1.5. The effective angle calibration is used to relatethe tension of the wires to the mean velocity which coveredthe range of 0–14 m s−1 in magnitude and ±40◦ in angle.For the SW, the response is fitted to a third-order polynomial.A Pitot tube connected to a pressure transducer (Setra model239, full scale 0.5 H2O or 125 Pa, accuracy ±0.14% offull scale) provides the reference velocity. Calibrations arerechecked at the end of an experiment, which comprisesseveral single measurements and lasts several hours. Thesignal from the anemometers is first conditioned through buck-and-gain modules, then low-pass filtered at a frequency ff(corresponding to the rise of electronic noise and close to theKolmogorov frequency fK = U/2πη), and finally sampled atfs = 2ff ; the resolution of the digitization is 16 bit. The totalsampling time T is 60 s. The uncertainty in the measurement

8

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Table 1. Details of the DNS data (dimensional quantities are expressed in numerical units, NU).

Name δ/η Rλ ε (NU) K (NU) res (NU) ν (NU) kMη

s0256a 1.64 90 0.540 1.40 256 3 × 10−3 1.91s0512e 1.53 147 0.491 1.42 512 1.26 × 10−3 2.05s0512f 1.34 129 0.52 1.41 512 1.54 × 10−3 2.34s0512g 0.87 93 0.55 1.37 512 2.71 × 10−3 3.62

Table 2. Measurement details for AJ at r/D = 0 and x/D = 32 (see figure 12).

d (mm) l (mm) fs (kHz) εx = Uc/fs (mm) ηa (mm) d/η l/η εx/η d/Lux l/Lux T (s)

0.8 0.5 20 0.275 0.114 7.0 4.4 2.3 0.01 0.007 60

a Computed from βx,x of SW.

0 0.05 0.1 0.150

0.05

0.1

0.15

1.005

1.01

1.015

1.02

1.025

1.03

1.0351.04

l/L

d/L

Figure 15. Distribution of rIy for the XW.

05

1015

0

10

0

5

10

15

d†l†

ε†x

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Figure 16. Distribution of rβx,x for the XW.

of the mean squared velocity derivatives is less than 6.9%(Burattini et al 2005a).

The water jet apparatus (WJ, hereafter) is located at theUniversity La Sapienza and consists of a 50:1 contraction

0

5

10

15

05

1015

0

5

10

15

d†l†

ε†x

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 17. Distribution of rβy,x for the XW.

05

1015

0

10

0

5

10

15

d†l†

ε†x

0.75

0.8

0.85

0.9

0.95

Figure 18. Distribution of rIy,x for the XW.

nozzle (exit diameter D = 20 mm) placed at the end of a pipe.Downstream of the nozzle, the flow enters a tank 30D wideand 60D long. Measurements are taken in this tank in the far-field region, from about 25D to 30D. The outlet jet velocityin the present measurements is 1 m s−1, which yields an exitReynolds number, ReD , of 20 000 and a jet centreline velocity

9

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

0 0.02 0.04 0.06 0.08 0.1 0.120.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

l/L

rαx

Figure 19. Distribution of rαx for the SW.

0 5 10 15 20 250

2

4

6

8

10

120.4

0.50.6

0.70.80.9l†

ε†x

Figure 20. Distribution of rβx,x for the SW.

of about 0.206 m s−1 at 30D. The Reynolds number based onthe Taylor microscale,

Rλ = αx

ν(βx,x/rβx,x )1/2, (53)

is 183, after correcting the velocity derivative. In thearea investigated, the Kolmogorov scale is nearly constant(0.123 mm, after correction, see figure 21). The velocity ismeasured using a 2D PIV system, with illumination providedby a double Nd-YAG laser with wavelength 532 nm and amaximum of 120 mJ per pulse with 7 ns pulse duration.Standard optical components are used to generate the laserlight sheet, which has a thickness of 2 mm. Image acquisitionis performed using a cross-correlation CCD camera (SensicamQE, double shutter, PCO 12 bits, BW) with 1376 × 1040 pixelsresolution, 6.45 μm each. A standard Nikon objective, NikkorAF 50 mm (f /1.8), is used. To perform the acquisitions, thef number is set to 5. The repetition rate of the laser is 10 Hz,which is the time resolution of the system. The time delaybetween two image pairs is optimized to 1 ms, with about80 mJ on each pulse. Hollow glass spheres, diameter 10 μm,

0 0.2 0.4 0.6 0.8 10.1

0.11

0.12

0.13

0.14

0.15

0.16

r/R0.5

η,η

(mm

)

Figure 21. Radial profiles of the Kolmogorov scale for WJ,corrected (◦) and uncorrected (+) values.

almost neutrally buoyant are used as tracers. The contraction,pipe and tank are made of transparent Perspex, to allow fulloptical access with the video camera.

The instantaneous vector fields are computed with across-correlation algorithm (DaVis by LaVision), which alsoperforms the window offset and image deformation correctionand achieves sub-pixel accuracy. A detailed description ofthe features and performances of the algorithm is given inStanislas et al (2008). The interrogation window size is32 × 32 pixels, equal to 1.8 × 1.8 mm2, see table 3. Byconsidering the laser sheet thickness, the velocity vectorsare determined over an almost cubic interrogation volume.The use of 75% overlapping between interrogation windowsreduces the spacing between velocity vectors down to 8 pixels,which corresponds to about 0.46 mm. This is also the distanceover which velocity gradients are evaluated, correspondingto 3.8 Kolmogorov length scales. Image pre-processing isperformed by subtracting the minimum intensity at each pixel,the minimum being evaluated over the entire acquisition set(Falchi and Romano 2009). The uncertainty in the meansquared velocity derivatives is less than 11% (95% accuracy).This error is inherently random, in contrast to the accuracyerror discussed in the rest of the paper, and therefore cannot becompensated for by any correction method (Carr et al 2009).With the optical parameters and seeding tracer used, peak-locking errors are negligible, the particles having a diameterlarger than 1 pixel; a detailed account of this is presented inFalchi and Romano (2009).

The main parameters of the two jets are compared intables 4 and 5. In the latter table, R0.5 is the half-velocityradius, which is defined as the radial location where U = Uc/2,and Uc being the centreline velocity. In WJ, R0.5 is estimatedafter the mean velocity is fitted to the self-similar profile.Figure 22 shows the mean velocity as a function of the radialdistance for the two jets. The data agree well with each otherand also with the simulations of Boersma et al (1998). Notethat, in order to achieve high resolution with PIV, in this work

10

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Table 3. Measurement details for WJ (see figure 1); uncorrected values in round brackets.

�x × �ya (mm2) δx × δy

b (mm2) δzc (mm) εd (mm) fs (Hz) η (mm) δ/η δz/η ε/η δ/Lux δz/Lux

4D × 3D 1.8 × 1.8 1.5 0.46 ∼5 0.123 14.6 12.3 3.8 0.25 0.21(0.138)e (12.9) (10.8) (3.32)

a Imaged area centred on the jet axis.b Interrogation window, square δx = δy = δ.c Thickness of the interrogation window.d Grid resolution. Note that ε/δx = 0.25, i.e. the overlap between windows is 75%.e From uncorrected value of βx,x .

0 1 2 3 4 5 6 70

0.2

0.4

0.6

0.8

1

1.2

1.4

r/D

U/Uc (PIV)U/Uc (HWA)U/Uc (DNS)

Figure 22. Radial profile of the streamwise mean velocity. DNSfrom Boersma et al (1998).

Table 4. Main exit conditions for AJ and WJ.

x/D D (mm) U0 (m s−1) Re0 ν (m2 s−1)

AJ 0 55 26.9 9.5 × 104 1.56 × 10−5

WJ 0 20 1 2 × 104 1.0 × 10−6

we have focused attention on the central part of the jet (r/D <

2), see figure 22. In doing so, we have also avoided the outerradial region of the flow, where the inhomogeneity is moreimportant.

4. Results

4.1. Large-scale quantities

This section reports the correction factor for the large-scaleturbulent quantities and their value after correction.

4.1.1. Correction factors. The correction factors (38), (39)and (47) are computed using the actual size of the measurementvolumes (see section 3) and the distributions of figure 3 forPIV, and the distributions of figures 13–15 for HWA. Thecorrection factor for the variance of the velocity componentsis nearly unity for both WJ and AJ, see table 6. The correctionfactors for the integral length scale (not reported) are alsonearly 1.

0 1 2 3 4 5 6 70

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

r/D

u /Uc (PIV)v /Uc (PIV)uv0.5/Uc (PIV)u /Uc (HWA)v /Uc (HWA)uv0.5/Uc (HWA)u /Uc (DNS)v /Uc (DNS)uv0.5/Uc (DNS)

Figure 23. Radial profiles of the turbulence intensities. DNS fromBoersma et al (1998).

4.1.2. Corrected values. The corrected values of the large-scale quantities are obtained by applying the correction factorsin table 6 to the measured values, see figure 23. It also showsthat the present experimental data agree well with the DNSdata of Boersma et al (1998).

4.2. Velocity derivatives and mean dissipation rate

This section reports the correction factors for the small-scaleturbulent quantities, such as the mean turbulent dissipationrate, and their value after correction.

4.2.1. Correction factors. The correction factors in (40)–(43) are computed using the physical size of the measurementvolumes of PIV and HWA, along with the distributions offigures 4–11 (for PIV) and of figures 16–18 and 20 (forHWA). The physical size of the volumes has to be normalizedusing the corrected Kolmogorov scale. This is computed as apreliminary step following the procedure outlined at the endof section 2.1.5.

In the present case, the correction of the velocityderivatives is larger for PIV than for HWA, table 7. Thevariance of the streamwise velocity derivative is attenuatedby almost 40% in WJ (similar attenuation is observed forother derivatives), while only by 3–7% (depending on theprobe used) in AJ. However, the correction to the isotropyratios is negligible for PIV, while being sizable for the XW,

11

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Table 5. Comparison of main axial conditions for AJ and WJ; uncorrected values in round brackets. The prime denotes the fluctuationrms value.

x/D Uc (m s−1) R0.5/D Lx (mm) Ly (mm) Rλ u′x (m s−1) u′

y (m s−1)

AJ 32 5.3 2.895 70.9 34.2 370a (395) 1.37 1.05WJ 29 0.206 2.5 14.2 7.5 179a (234) 0.0566 0.0447

a From βx,x and (53).

Table 6. Correction factors of the variance of the velocitycomponents and of their ratio (r/D = 0).

x/D rαx rαy rIy

AJ 32 1 1 1WJ 29 0.98 0.98 1

Table 7. Correction factors of the variance of the velocityderivatives (r/D = 0). In round brackets, initial estimates using theuncorrected value of η. Note that, in isotropic turbulence, rβx,y andrβy,x are statistically identical.

x/D rβx,x rβx,y rβy,x rβxy,yx

AJ 32 0.93a, 0.97b n.a. 0.80 n.a.WJ 29 0.59 (0.64) 0.56 (0.62) 0.57 (0.62) 0.57 (0.63)

a XW.b SW.

Table 8. Correction factors for the derivative isotropy ratios (r/D =0; uncorrected values in round brackets).

x/D rIy,x rIx,y rIy,y rIxy,yx

AJ 32 0.86 (0.86) n.a. n.a. n.a.WJ 29 0.96 (0.96) 0.95 (0.96) 0.99 (0.99) 0.97 (0.97)

table 8. The reason for this difference lies in the shape of thetwo measurement volumes: in PIV, the interrogation windowis square and therefore it attenuates equally the gradients in thex and y directions. The XW, on the other hand, preferentiallyattenuates the derivative of uy .

Note that for the WJ, these correction factors are nearlyconstant over the radial distance, since the Kolmogorovscale does not vary significantly in the measured range (seefigure 21). For AJ, they depend slightly on the radialcoordinate (figure 24), as the Kolmogorov scale increaseswith r.4.2.2. Corrected values. The corrected values of the small-scale quantities are obtained by applying the correction factorsof the previous section to the measured values. As expected,in WJ the rectified values of βx,x, βy,x, βx,y, βy,y, βxy,yx aresubstantially augmented (see table 9).

In order to compare directly the variances of the velocityderivatives of the two experimental set-ups, we computedthe mean dissipation rate coefficient Cε . There are severalalternatives for the normalization of ε, depending not only onthe flow type but also on the length and velocity scales chosenfor the normalization, e.g. Burattini et al (2005b). For theround jet, a common choice is

Cε = εR0.5

U 3c

. (54)

0 0.5 1 1.5 2 2.50.5

0.6

0.7

0.8

0.9

1

1.1

r/R0.5

rβx, x

rβy, x

rIy, x

Figure 24. Radial profiles of the correction factors rβx,x , rβy,x andrIy,x for the XW.

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

r/R0.5

C=

βR

0.5

/U3 c

βx,x, XWβy,x, XWβx,x, SWβx,x, PIVβy,x, PIVβx,y, PIVβy,y , PIVβxy,yx, PIV

Figure 25. Uncorrected mean dissipation rate coefficients, see (54),comparison between PIV (WJ) and HWA (AJ) data.

Figures 25 and 26 report the uncorrected and corrected valuesof Cε , respectively. Several distributions are shown, whichdiffer in the velocity derivative used to compute ε. Theagreement between the values of Cε computed with βx,x (oralternatively with βy,x) in both experiments lend support to thecorrection method developed here.

For PIV, the derivatives’ correction is quite large, butis almost identical among them, see table 7. Therefore, the

12

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Table 9. Corrected values of the variance of the velocity derivatives (r/D = 0; uncorrected values in round brackets).

x/D βx,x (s−2) βx,y (s−2) βy,x (s−2) βy,y (s−2) βxy,yx (s−2) εa (m2 s−3)

AJ 32 1.02 × 105 (9.9 × 104)a n.a. 1.6 × 105 (1.28 × 105) n.a. n.a. 23.2 (22.4)a

WJ 29 316 (187) 566 (317) 517 (295) 306 (181) 119 (68) 305 (180)b

a From βx,x of SW.b From βx,x .

0 0.5 1 1.5 2 2.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

r/R0.5

C=

βR

0.5

/U3 c

βx,x, XWβy,x, XWβx,x, SWβx,x, PIVβy,x, PIVβx,y, PIVβy,y , PIVβxy,yx , PIV

Figure 26. Corrected mean dissipation rate coefficients, see (54), acomparison between PIV (WJ) and HWA (AJ) data.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

r/R0.5

Iy,x XWIy,x PIVIx,y PIVIy,y PIVIxy,yx PIV

Figure 27. Uncorrected ratios of the velocity derivative variances(see table 10). The horizontal dashed lines indicate the isotropicvalues, see table 10.

derivative ratios, which are far from the isotropic values beforecorrection, remain so even after correction, table 10. We alsonote that Iy,x , the only isotropy ratio obtainable from the XW,agrees reasonably well with that of PIV, after correction. Theradial profiles of the isotropy ratios are reported in figures 27and 28 for the uncorrected and corrected cases.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

r/R0.5

Iy,x XWIy,x PIVIx,y PIVIy,y PIVIxy,yx PIV

Figure 28. Corrected ratios of the velocity derivative variances (seetable 10). The horizontal dashed lines indicate the isotropic values,see table 10.

Table 10. Corrected values of the derivative isotropy ratios (r/D =0; uncorrected values in round brackets). Isotropic values are listedon the bottom line.

x/D Iy,x Ix,y Iy,y Ixy,yx

AJ 32 1.58 (1.35) n.a. n.a. n.a.WJ 29 1.66 (1.6) 1.82 (1.73) 0.97 (0.96) 0.39 (0.38)ISO 2 2 1 0.5

5. Conclusions

In general, HWA and PIV have very different shapes ofthe measurement volume and sensitivity to the velocitycomponents. When measuring large-scale quantities—muchlarger than the measurement volume—no particular problemwith respect to the spatial resolution arises. However, if oneis interested in length scales of the size of the measurementvolume, or smaller, a significant distortion of the velocityfluctuations is expected. In fact, the real velocity field getsfiltered over the size of the volume. In order to comparemeasurements taken with the two techniques, corrections forspatial resolution need to be applied.

Here, we have compared such corrections starting fromDNS velocity fields. The correctors were then applied tomeasurements in the far field of turbulent jets, focusingattention on a small region around the axis. As expected,the large-scale quantities (such as the velocity variance)estimated by both techniques were almost unaffected by the

13

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

measurement volume size. The small scales, however, weremuch more attenuated. For example, the variance of thestreamwise velocity derivative was reduced by 40% in PIV andby 7% in HWA. After applying the corrections, the normalizeddissipation rates estimated from the streamwise velocity withthe two techniques were found to agree, within experimentaluncertainty. In this sense, this paper extends the earlier workof Lavoie et al (2007) and underlines the need to apply fairlymajor corrections to the small-scale data, especially for PIV.

Further, different degrees of attenuation for differentvelocity derivatives were noted: in PIV, the velocityderivatives were affected more uniformly, compared to HWA.The difference apparently originates in the shape of themeasurement volumes: the PIV interrogation window has asquare symmetry, and therefore does not discriminate betweenone velocity component and another. For the XW, on theother hand, the inclination of the wires seems to affectthe transverse velocity component more than the streamwisevelocity component. As a consequence, the isotropy ratiosof the velocity derivative measured by PIV required only amodest correction, with respect to HWA. The compensationprocedure was nevertheless able to reconcile the values of theisotropy ratios obtained with the two techniques.

This work aims at providing the experimentalist ready-to-use correction factors derived from numerical data. As shownabove, DNS gives the unique opportunity to systematicallyvary one resolution parameter of the measurement at a time,leaving the others unaffected and providing full access toall flow variables. This is difficult or tedious to performexperimentally. Furthermore, our results could serve as aguideline when designing a new experiment. For example,they suggest the appropriate resolution needed to resolve thedissipation scales with a certain accuracy. We conclude byremarking that this work should be considered as a firststep towards the correction of experimental measurementsin inhomogeneous flows. As more DNS studies becomeavailable, it would be possible and desirable to have correctionfactors computed for any specific flow (e.g. plane and roundjet, pipe flow).

Acknowledgments

RAA is grateful to the Australian Research Council. PB isgrateful to the Statistical and Plasma Physics group of theUniversite Libre de Bruxelles for the use of the computingresources.

References

Antonia R 1993 Direct numerical simulations and hot wireexperiments: a possible way ahead? New Approaches andConcepts in Turbulence ed T Dracos and A Tsinober (Basel:Birkhauser) pp 349–65

Antonia R A and Mi J 1993 Corrections for velocity andtemperature derivatives in turbulent flows Exp. Fluids 14 203–8

Astarita T 2006 Analysis of interpolation schemes for imagedeformation methods in PIV: effect of noise on the accuracyand spatial resolution Exp. Fluids 40 977–87

Astarita T 2007 Analysis of weighting windows for imagedeformation methods in PIV Exp. Fluids 43 859–72

Astarita T 2008 Analysis of velocity interpolation schemes forimage deformation methods in PIV Exp. Fluids 45 257–66

Boersma B J, Brethouwer G and Nieuwstadt F T M 1998 Anumerical investigation on the effect of the inflow conditionson the self-similar region of a round jet Phys. Fluids10 899–909

Burattini P 2008 The effect of the X-wire probe resolution inmeasurements of isotropic turbulence Meas. Sci. Technol.19 115405

Burattini P, Antonia R A and Danaila L 2005a Similarity in the farfield of a turbulent round jet Phys. Fluids 17 025101

Burattini P, Kinet M, Carati D and Knaepen B 2008a Anisotropy ofvelocity spectra in quasistatic magnetohydrodynamicturbulence Phys. Fluids 20 065110–1

Burattini P, Lavoie P and Antonia R A 2005b On the normalizedturbulent energy dissipation rate Phys. Fluids 17 098103

Burattini P, Lavoie P and Antonia R A 2008b Velocity derivativeskewness in isotropic turbulence and its measurement with hotwires Exp. Fluids 45 523–35

Carr Z R, Ahmed K A and Forliti D J 2009 Spatially correlatedprecision error in digital particle image velocimetrymeasurements of turbulent flows Exp. Fluids 47 95–106

Falchi M and Romano G P 2009 Comparison between PIVmeasurements with high-speed and cross-correlation camerasin a jet Exp. Fluids 47 509–26

Foucaut J, Carlier J and Stanislas M 2004 PIV optimization for thestudy of turbulent flow using spectral analysis Meas. Sci.Technol. 15 1046–58

Foucaut J and Stanislas M 2002 Some considerations on theaccuracy and frequency response of some derivative filtersapplied to particle image velocimetry vector fields Meas. Sci.Technol. 13 1058–71

Henning A and Ehrenfried K 2008 On the accuracy of one-point andtwo-point statistics measured via high-speed PIV 14th Int.Symp. on Applications of Laser Techniques to Fluid Mechanics(Lisbon, 7–10 July)

Lavoie P, Avallone G, Gregorio F D, Romano G P and Antonia R A2007 Spatial resolution of PIV for the measurement ofturbulence Exp. Fluids 43 39–51

Mathieu J and Scott J 2000 An Introduction to Turbulent Flow(Cambridge: Cambridge University Press)

Nobach H and Tropea C 2005 Improvements to PIV image analysisby recognizing the velocity gradients Exp. Fluids 39 614–22

Nogueira J, Lecuona A and Rodrıguez P 1999 Local field correctionPIV: on the increase of accuracy of digital PIV systems Exp.Fluids 27 107–16

Nogueira J, Lecuona A and Rodriguez P 2001 Identification of anew source of peak locking, analysis and its removal inconventional and super-resolution PIV techniques Exp. Fluids30 309–16

Nogueira J, Lecuona A, Rodrıguez P A, Alfaro J A and Acosta A2005a Limits on the resolution of correlation PIV iterativemethods. Practical implementation and design of weightingfunctions Exp. Fluids 39 314–21

Nogueira J, Lecuona A, Ruiz-Rivas U and Rodrıguez P 2002Analysis and alternatives in two-dimensional multigrid particleimage velocimetry methods: application of a dedicatedweighting function and symmetric direct correlation Meas. Sci.Technol. 13 963–74

Nogueira J, Lecuona1 A and Rodrıguez P A 2005b Limits on theresolution of correlation PIV iterative methods. FundamentalsExp. Fluids 39 305–13

Poelma C, Westerweel J and Ooms G 2006 Turbulence statisticsfrom optical whole-field measurements in particle-ladenturbulence Exp. Fluids 40 347–63

Raffel M, Willert C and Kompenhans J 2002 Particle ImageVelocimetry: A Practical Guide (Berlin: Springer)

14

Meas. Sci. Technol. 21 (2010) 125402 P Burattini et al

Scarano F 2002 Iterative image deformation methods in PIV Meas.Sci. Technol. 13 R1–R19

Scarano F 2003 Theory of non-isotropic spatial resolution in PIVExp. Fluids 35 268–77

Scarano F 2004 On the stability of iterative PIV image interrogationmethods 12th Int. Symp. on Applications of Laser Techniquesto Fluid Mechanics (Lisbon, Portugal, 12–15 July) paper 27.2

Schrijer F and Scarano F 2008 Effect of predictor–corrector filteringon the stability and spatial resolution of iterative PIVinterrogation Exp. Fluids 45 927–41

Stanislas M, Okamoto K, Kahler C, Westerweel J and Scarano F2008 Main results of the third International PIV ChallengeExp. Fluids 45 27–71

Tanaka T and Eaton J K 2007 A correction method for measuringturbulence kinetic energy dissipation rate by PIV Exp. Fluids42 893–902

Wanstrom M, George W, Meyer K and Westergaard C 2007Identifying sources of stereoscopic PIV measurements errorson turbulent round jets 5th Joint ASME/JSME FluidsEngineering Conf. (San Diego, USA, 30 July–3 August)FEDSM2007-3725

Wyngaard J C 1968 Measurement of small-scale turbulencestructure with hot wires J. Phys. E: Sci. Instrum. 1 1105–8

Wyngaard J C 1969 Spatial resolution of the vorticity meterand other hot-wire arrays J. Phys. E: Sci. Instrum.2 983–7

Zhu Y and Antonia R A 1995 The spatial resolution of twoX-probes for velocity derivative measurements Meas. Sci.Technol. 6 538–49

Zhu Y and Antonia R A 1996 The spatial resolution of hot-wirearrays for the measurement of small-scale turbulence Meas.Sci. Technol. 7 1349–59

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