Analysis of errors in the measurement of energy dissipation with two-point LDA

Noname manuscript No.(will be inserted by the editor)

Analysis of errors in the measurement of energy dissipation with

two-point LDA

A. Ducci, M. Yianneskis

Experimental and Computational Laboratory for the Analysis of Turbulence (ECLAT), Department of Mechanical Engineering,King’s College London, Strand, London, WC2R 2LS, United Kingdom

Preprint of the article published in Exp. Fluids, (2005), 38, 449-460.

Abstract In the present study an attempt has beenmade to identify and quantify with a rigorous analyticalapproach all possible sources of error involved in the es-timation of the fluctuating velocity gradients (∂ui/∂xj)2

when a two-point LDV technique is employed. Measure-ments were carried in a grid-generated turbulence flowwhere the local dissipation rate can be calculated fromthe decay of kinetic energy. An assessment of the cumu-lative error determined through the analysis has beenmade by comparing the values of the spatial gradientsdirectly measured with the gradient estimated from thedecay of kinetic energy. The main sources of error werefound to be related to the length of the two control vol-umes, and to the fitting range as well as the functionused to interpolate the correlation coefficient when theTaylor length scale (or (∂ui/∂xj)2) are estimated.

List of symbols

Roman Symbols

Cf Calibration factor (m)D Grid wire diameter (mm)L Test section length (m)Lij Integral length scale in the i th direction

of the uj velocity component (m)M Grid mesh size (m)N Number of points in the fitting range (-)Ntot Number of particles in coincidence (-)Rii(∆xj) Correlation coefficient of the i velocity

component in the j direction (-)Rmii Measured correlation coefficient of the i

velocity component (-)Rrii Real correlation coefficient of the i

velocity component (-)Re Reynold number (-)

Correspondence to: M. Yianneskis

ReM Reynolds number based on M (-)t Particle arrival time (s)Ui Velocity component in the i direction (ms−1)ui Turbulent velocity component in the i

direction (ms−1)Ui Mean velocity in the i direction (ms−1)W Width of the test section (m)x0 Position where the two control volumes overlap

completely (m)xi Coordinate in the i direction (m)xM Non-dimensionalised distance from the grid (-)

Greek symbols

∆xj Displacement in the j direction (m)∆xmax Upper limit of the fitting range (m)∆xmin Lower limit of the fitting range (m)ǫ Viscous dissipation rate of turbulent kinetic

energy (m2s−3)η Kolgomorov length scale (m)Λ Control volume length (m)λij Taylor length scale in the i direction

of the uj velocity component (m)ν Kinematic viscosity (m2s−1)σ Grid solidity ratio (-)τw Coincidence window (s)

Abbreviations

HWA Hot wire anemometryLDA/LDV Laser Doppler anemometry/velocimetryPIV Particle image velocimetry

1 Introduction

Accurate knowledge of the dissipation rate of turbu-lence kinetic energy (ǫ) is essential for the design ofmany engineering processes, especially those involvingmixing operations. The ability to obtain measurementsat the small scales where the dissipation takes place has

2 A. Ducci, M. Yianneskis

however been curtailed by limitations of the measure-ment techniques that are available. Hot-wire anemom-etry (HWA) has been extensively and successfully em-ployed to measure the small scale flow field in a widerange of flows (see, e.g. Simmons and Salter, 1934; Corrsin,1943; Townsend, 1948; Browne et al., 1987; Schenk andJovanovic, 2002), notwithstanding the shortcomings as-sociated with the intrusive nature of the anemometer.On the other hand unobtrusive techniques such as laserDoppler anemometry (LDA) and particle image velocime-try (PIV) are well suited for the study of fluctuating ve-locity gradients provided sufficient spatial resolution canbe achieved to resolve the dissipative scales. In particulartwo-point LDA has the potential to resolve the smallestscales of most flows. Early studies on the application oftwo-point LDA are those of Nakatani et al. (1985); Johnset al. (1986); Fraser and Bracco (1988); Absil (1988);Cenedese et al. (1991); Gould et al. (1992); Benak et al.(1993); Romano (1995); Eriksson and Karlsson (1995).The most significant limitation that prevents the use oftwo-point LDA application for the study of the smallestscales is presented by the dimensions of the control vol-ume that have to be reduced to the order of magnitudeof the local Kolgomorov scale. This problem has beenwell analysed by Benedict and Gould (1999) who investi-gated a backward-facing step flow. Besides the influenceof the control volume dimensions, they also determinedthe effect of the time coincidence window (τw) on the cor-relation coefficient Rii(∆xj) directly estimated using atwo-point LDV technique. The time coincidence windowwas found to affect significantly the correlation functionfor small displacements ∆xj in the main direction of theflow. In this case a high proportion of the velocity pairscollected from the two measuring locations are due tosingle particles that were sufficiently fast to cross thetwo control volumes within the time coincidence win-dow (single particle burst pairs). Moreover, they foundthat geometric bias due to the lateral dimension of thecontrol volume tends to affect span-wise correlation mea-surements.Belmabrouk (2000) has investigated in depth the effect ofthe control volume dimensions on the accuracy of Taylorlength scale estimates using a theoretical approach basedon different models that should represent the real corre-lation function R(∆x). He concluded that there are twomain parameters that determine the shape of the cor-relation function. The first parameter is the ratio L/λof the integral and Taylor length scales. This ratio givesan indication of the energy range contained between thelarge and the small eddies. The second parameter is theratio ∆xmax/L, where ∆xmax is the upper limit of therange of displacement around the point of measurementwhere the correlation function can be approximated witha parabola. Johansson and Klingmann (1994) carried outtwo-point LDV measurements in a low Reynolds num-ber (Re) circular air jet flow with a Kolgomorov lengthscale (η) of around 100 µm. They concluded that two pa-

rameters are the main sources of error in the estimationof the dissipation rate tensor of the Reynolds stresses:misalignment of the laser beams that results in a mis-alignment of the direction of the velocity measured byeach control volume with respect to the directions of theCartesian coordinate system, and the determination ofthe origin x0 where the two control volumes are supposedto overlap completely. The total error involved in thedetermination of the homogeneous part of the Reynoldsstress dissipation rate tensor was found to vary from 5 -10 %. The error due to the finite dimension of the con-trol volume was found to be negligible because a 900 sidescatter optical arrangement was employed, resulting in areduction of the longest dimension of the two control vol-umes (to around 0.9 η). Comparisons of two-point LDAdata with HWA data and with DNS and HWA data havebeen made by Belmabrouk and Michard (1998) in a gridgenerated turbulent flow and by Gan et al. (1996) in aconstant head recirculating water channel. Both worksshow a good agreement between the small scale struc-tures determined employing the different techniques.Most of the works mentioned so far employ a time co-incidence window τw to determine the particles in coin-cidence between the two channels. However other tech-niques based on resampling have been used by Romano(1995) and Benak et al. (1993). Muller et al. (1998) havedeveloped a sample and hold resampling technique thatallows to estimate the correlation coefficient also in therange of displacement ∆x where the two control volumespartially overlap, eliminating in this way the time coin-cidence window bias.The present paper aims to determine and quantify througha rigorous analysis the main sources of error that affectthe estimates of the spatial gradient (∂ui/∂xj)2 and thedissipation rate of kinetic energy ǫ directly measuredwith two-point LDA. Moreover the accumulated errordetermined analytically is assessed by comparing the dis-sipation rate directly measured in grid turbulent flowwith the dissipation calculated from the decay of kineticenergy.

2 Flow configuration and LDA system

The rig consists of a single loop pipe arrangement with aby-pass and a centrifugal pump, which directs the work-ing fluid (water) into an expansion-contraction sectioncontaining an hexagonal honeycomb. As suggested byGroth and Johansson (1988), a screen (in the presentwork a 2 mm perforated plate) is placed downstreamof the contraction to reduce the turbulence levels be-fore the flow reaches the grid. The measurements havebeen carried in a transparent acrylic test section locatedsufficiently downstream of the grid to allow the flow tobecome locally isotropic and homogeneous for a ReM of5500, based on the mesh size. A heat exchanger jacketin the tank of the rig maintains the temperature of the

Analysis of errors in the measurement of energy dissipation with two-point LDA 3

water at a constant value. The dimensions of the gridand of the test section used are shown in table 1. In

Test section Grid

W × W × L M D σ

72 x 72 x 201 5.5 1 0.33

Table 1 Test section and grid dimensions (apart from σ allother quantities are in mm).

the coordinate system used, x1 is aligned to the mainflow direction while the other two axes are perpendicu-lar to the axis of the test section. The LDA employed is aDantec system that comprises three probes mounted ona transverse that can be moved in all three directions.The probe arrangement used to determine the spatialgradient related to the u3 velocity component is shown infigure 1. The estimation of the fluctuating spatial gradi-ent (∂u3/∂x3)2 requires the simultaneous measurementof the same velocity component u3 with two differentchannels. The central probe, denoted as number 2, is thefirst channel and collects in back-scatter the light scat-tered by the 10 µm diameter particles crossing a controlvolume of dimensions 0.034 mm x 0.034 mm x 0.22 mm.The other two probes (second channel) are placed sym-metrically on either side of probe 2, with one probe usedto generate the control volume (number 1) and the other(number 3) collecting the scattered light (and vice versawhen the gradients of the U1 velocity component areconsidered). An exact calculation of the effective size ofthe control volume when operating in side scatter is notpossible because, even though the diameter of the fiberoptic cable (50 µm), which in this system has an equiv-alent role to the pinhole usually placed in front of thephotomultiplier, is known, the lens magnifying power isnot known. From geometrical considerations, taking into account that the angle between the optical axes of theside probes and of the central probe (number 2) is about220, the size of the control volume in side scatter can beestimated to be 0.05 mm x 0.05 mm x 0.2 mm. The cen-tral probe can be moved with respect to the other twoin all the three directions with a minimum displacementof 10 µm. The probes are initially aligned in air usinga pinhole of 50 µm diameter. Once the beams are in-serted in the water, to overcome the effect of refraction,it is necessary to adjust the relative position of the threecontrol volumes along the x2 axis by a known distance.As suggested by Benedict and Gould (1999) a furtheroptimisation of the alignment is achieved by computingthe spatial correlation coefficient Rii(0) for different rel-ative positions of the probes until a maximum is found.

3 Theoretical background and processing

procedure

A review of previous published studies on two-point LDVmeasurements has highlighted three main procedures todetermine the spatial gradients (∂ui/∂xj)2.

The first procedure, which is also the most widelyused (e.g. Trimis and Melling, 1995; Benedict and Gould,1999; Belmabrouk, 2000), is based on the direct estima-tion of the correlation coefficient Rii(∆xj), defined inequation (1), for different displacements ∆xj betweenthe measuring volumes.

Rii(∆xj) =ui(x0, t)ui(x0 + ∆xj , t)

√

ui(x0, t)2√

ui(x0 + ∆xj , t)2(1)

The first derivative of the correlation coefficient is zero atthe origin (see Hinze, 1975) and consequently its Taylorseries expansion has the form shown in equation (2):

Rii(∆xj) = 1 −1

λ2ji

∆x2j + A ∆x3

j + O(∆x4j ) (2)

with a parabolic shape for small values of ∆xj aroundthe origin. The Taylor length scale along the jth direc-tion of the ui velocity component (λji) can be estimatedfrom the coefficient of the second-order term of equation(2), by interpolating a parabolic curve between the datapoints (∆xj , Rii(∆xj)). The spatial gradient (∂ui/∂xj)2

is subsequently calculated from equation (3).

(

∂ui

∂xj

)2

=2 u2

i

λ2ji

(3)

The second procedure, employed by Michelet (1998),is based on the calculation of the coefficient fii(∆xj)defined in equation (4) for different positions ∆xj aroundthe origin x0:

fii(∆xj) = (ui(x0, t) − ui(x0 + ∆xj , t))2 (4)

Once fii(∆xj) is known, the spatial gradients are derivedfrom the limit in equation (5).

(

∂ui

∂xj

)2

= lim∆xj⇒0

(ui(xj , t) − ui(xj + ∆xj , t))2

∆xj2

(5)

The spatial gradients are evaluated by finding the slopeof the straight line which best fits the points (∆xj

2,fii(∆xj)) for ∆xj values close to zero. In the remain-der of the paper the range of points (∆xj

2, fii(∆xj)),that were used in the interpolation to estimate the gra-dient, will be referred to as the fitting range.In the third procedure, suggested by Johansson and

Klingmann (1994), the spatial gradients are determinedfrom equation (6), which has been derived employing


Fig. 1 Probe arrangement to measure the velocity gradients(∂u1/∂x1)2, (∂u1/∂x3)2, (∂u3/∂x3)2, (∂u3/∂x1)2.

Taylor series expansions and central differentiation aroundthe origin.

(

∂ui

∂xj

)2∣

∣

∣

∣

∣

x0

=

1

4

∂2(u2i )

∂x2j

∣

∣

∣

∣

∣

x0

−∂2[ui(

∆xj

2) ui(−

∆xj

2)]

∂∆xj2

(6)The first term on the right hand side is the second deriva-tive of a one-point measurement function around theposition of interest x0, while the second term is the sec-ond derivative of a two-point measurement function. Ifthe turbulence is homogeneous, the first term can beneglected because u2

i does not vary by a large amountaround the reference point x0. The three procedures de-scribed above are analogous and can be interchangeablyused to estimate the spatial gradients as long as smalldisplacement ∆x are considered. It should be noted thatfor all the procedures described, the velocity spatial gra-dients are estimated from the first two terms of equation(2), while the other terms are neglected. In the rest ofthe paper the uncertainties due to the different sourcesof error have been estimated when using the first andthe second procedure. When computing the correlationcoefficient Rii(∆xj) and the coefficient fii(∆xj) definedin equation (1) and equation (4) respectively, the ve-locity fluctuations ui in the two different measurementpoints must originate from particles which cross the con-trol volumes at the same time (it should be noted thatthe two BSAs were used in private-private mode). Thiscondition is imposed by identifying the pairs of particlesthat satisfy the following simultaneity criterion:

|t1 − t2| < τw (7)

where the time coincidence window (τw) is of the orderof magnitude of the transit time taken by the particles

-0.05 0.00 0.05 0.10 0.15

20

40

60

80

100

120

-0.05 0.00 0.05 0.10 0.15

0.86

0.88

0.90

0.92

0.94

0.96

0.98

1.00

x0

Dat

arat

e in

coi

ncid

ence

x [mm]

Datarate

Unbiased zone

Noisew biased zone

R(

x)

[-]

Correlation

Fig. 2 Variation of the data rate in coincidence and of thecorrelation coefficient R(∆x) with ∆x.

to cross the control volume (see, for example Kang et al.,2001).

4 Results and discussion

The variations of the correlation coefficient R(∆x) andof the data rate of particles in coincidence between thetwo channels are shown in figure 2. The maximum ofR(∆x) is smaller than 1 which is the theoretical valuethat the correlation coefficient is expected to assume atthe origin. As suggested by Trimis and Melling (1995)this discrepancy has to be mainly attributed to the levelof noise present in the measurements. However, Trimisand Melling (1995) concluded that if the rms of the noiseof each channel does not vary for different displacements∆x, the measured correlation coefficients R(∆x) are allbiased by the same amount. The above condition canbe considered to be satisfied in two-point LDA measure-ments. In fact the need to estimate the smaller scalesof the flow (η and λ) limits the investigation to a smallrange of displacements around the origin. For examplein the grid turbulence flow studied in the present paperthe maximum displacement ∆x was 2 mm and the in-terpolation procedure has been carried up to a distanceof ∆xmax=0.5 mm. Consequently, because of the small∆x, it can be concluded that the estimates of the spatialgradient are not affected by the level of noise present inthe measurements.

Figure 2 has been divided into three main zones. Thefirst zone (-0.02 mm≤ ∆x ≤0.01 mm) indicates the po-sitional error that is made when the origin x0 is located.This zone represents the range of displacements wherethe two control volumes are supposed to overlap com-pletely. In this case all the velocity pairs detected fromthe two channels belong to the same particle (single par-


ticle burst pair). As suggested by Benedict and Gould(1999), x0 can be determined by finding the point wherethe correlation and the data rate in coincidence reachtheir maxima. From figure 2 it can be concluded thatthe origin is approximately located in the range of dis-placements ∆x = 0 +0.01

−0.02 mm. This source of error ismainly affected by the lateral dimension of the controlvolume and by the value of the mean velocity componentin the direction of ∆x.

The second zone (0.01 mm≤ ∆x ≤0.08 mm) is af-fected by the time coincidence window bias. In this re-gion the number of single particle burst pairs present ineach set of data in coincidence is rapidly decreasing asthe two control volumes are displaced by ∆x. This as-pect is well shown by the significant decrease of the datarate in coincidence and of the correlation coefficient as∆x increases. The region is wider if either τw is longeror the mean velocity component in the direction of ∆xis higher.

The third zone (∆x ≥ 0.08 mm) corresponds to theregion where no single particle burst pairs affect the es-timation of R(∆x). In this region the data rate of parti-cles in coincidence is approximately constant. The lowerboundary of this zone will be denoted as ∆xmin. Theinterpolation procedure described in section 3 has to becarried in this region.

In the following sections the main sources of error inthe estimation of the (∂ui/∂xj)2 have been analysed inthis order:

– Coincidence window bias τw;– Positional error of the location of the origin x0;– Error due to the longer dimension of the control vol-

ume Λ (geometric bias);– Error due the fitting range selected (∆xjmin, ∆xjmax);– Error due to the interpolating function;– Error due to the calibration factor; and– Statistical uncertainty.

4.1 Effect of the time coincidence window τw and of the

accuracy of location of the origin x0 on the estimation

of (∂ui/∂xj)2

The time coincidence window has to be selected to pre-vent the spatial correlation function becoming a space-time correlation. Benedict and Gould (1999) noted thatthe time coincidence window acts as a filter on the veloc-ity pairs selected to measure the correlation coefficient.If the window is too narrow the sample of data in coin-cidence does not represent the total set of data collectedby each probe. They showed that the mean velocity andrms calculated from the velocities of the particles in co-incidence vary with ∆x and differ from those calculatedusing the total velocity data set. In figure 3 the varia-tion of the mean and rms values calculated with the twomethods is shown for both channels (for τw=0.03ms).Although the rms of probe 2 is slightly smaller than the

-0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.036

0.038

0.040

0.042

0.0440.85

0.90

0.95

1.00

1.05

Statistics of the velocity component U

1 due to particles in coincidence:

Probe 1 Probe 2

Statistics of the velocity component U1

due to all particles: Probe 1 Probe 2

Vel

ocity

rms [

m/s

]

x1 [mm]

Mea

n ve

loci

ty [

m/s

]

Fig. 3 Moments of the velocity component U1 estimatedfrom all the particles that crossed each control volume andfrom only those particles that were in coincidence. (a) Meanvelocity; (b) Rms.

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16-5

0

5

10

15

20

25

30

35

40 ( u1 / x1)

2

( u3 / x3)

2

Gra

dien

t per

cent

age

erro

r %

w [ms]

Fig. 4 Variation of the percentage error of the gradient es-timation with the coincidence window τw.

one measured by probe 1, no major difference betweenthe values calculated with the total and partial (filteredby τw) velocity data set can be observed.

However Benedict and Gould (1999) suggested thatthis effect should be more significant for flow fields withhigher turbulence levels. The variations of the percent-age error made on the estimation of (∂u1/∂x1)2 and(∂u3/∂x3)2 for different τw are shown in figure 4. Consid-ering (∂u1/∂x1)2 it is possible to see that the gradient isoverestimated as the time coincidence window increases.


0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

14

Resampled u

3( x

3)2

u3( x

1)2

In coincidence u

3( x

3)2

u3( x

1)2

u 32

[cm

2 s-2]

x3

2, x1

2 [mm2]

Fig. 5 Comparison of R33(∆x3) and R33(∆x1) obtained byapplying a coincidence window and by resampling.

This can be explained by taking into account that inthis case the probes are displaced along ∆x1 which isalso the main direction of the flow. A coincidence win-dow of 0.15 ms increases significantly the number of sin-gle particle burst pairs used to estimate the correlationat the location ∆x1 = 0.1 resulting in a higher correla-tion R11(0.1) and a steeper fitting line in the proceduredescribed in section 3. This behavior is not present in thevariation of (∂u3/∂x3)2 which remains almost constantwith τw. In this case the mean velocity is orthogonal tothe displacement ∆x3. In the rest of the data analysisthe coincidence window has been set to 0.03 ms as thisvalue minimises the error and gives an acceptable datarate of particles in coincidence.As previously mentioned in section 1 a resampling tech-nique procedure has sometime been preferred to a timecoincidence window technique (see for example Romano(1995); Benak et al. (1993)) because it should allow toremove the time coincidence window bias (Muller et al.,1998), and to reduce the experimental time. In this re-spect a comparison between the coefficient fii(∆xj) ob-tained by time coincidence filtering and by resamplingis given in figure 5. It should be noted that the data ofthe two channels were resampled with a sampling fre-quency equal to half of the lowest datarate obtainedfrom the two probes (the datarate were comparable ≈ 1kHz). It is evident that for both the gradients considered(∂u1/∂x3)2 and (∂u3/∂x3)2 the resampling proceduretends to underestimate the gradient (i.e. the slope of thecurve fitting the points (∆x2

j , fii(∆xj)) is less steep for∆xj ⇒ 0). Intuitively the underestimation is expected asthe resampling procedure (‘nearest’ method) put in cor-relation velocity pair that were collected with a time lag∆t (≈ 0.001 s) higher than the time coincidence win-dow (0.00003 s, corresponding to an ideal datarate of33.333 kHz to obtain a similar resolution from resam-pling). This results in higher values of the coefficient

fii(∆xj) obtained from resampling (see figure 5), and asmoother variation of fii(∆xj) with the associated un-derestimation of the gradient. Furthermore, despite thesmoothing effect described, the velocities of the particlesthat were in coincidence between the two channels arelikely to be correlated also after the resampling proce-dure as the arrival times of the two particles are mostlikely to be both the ‘nearest’ to the sampling time con-sidered. This is evident in figure 5 where the value offii(0) is lower and therefore more correlated than thevalues assumed by fii(∆xj) for ∆xj 6= 0. In conclusionon the one hand a resampling procedure technique de-termines an underestimation of the gradient and on theother hand does not eliminate completely the time coin-cidence window bias as the region denoted as zone 2 infigure 2 will still be present.The effect of the positional error on the measurement

0.0 0.5 1.0 1.5 2.0 2.50

2

4

6

8

10

12

Positional error of x0

0 mm -0.015 mm 0.015 mm

u 32 [cm

2 s-2]

x3

2 [mm]

Fig. 6 Variation of the slopes of fitting line with differentorigin x0.

of the spatial gradient (∂u3/∂x3)2 is shown in figure 6.As mentioned earlier in this section the positional errorof x0 has been estimated to be ± 0.015mm. In figure 6the origin x0 and the other data points (∆x2

3, ∆u23) have

been shifted by ± 0.015mm. The difference of the slopesof the fitting lines is evident. The error in the estimationof the gradient is ± 4.5%.

4.2 Effect of the control volume dimension and of rmax

on the Taylor length scale estimation

The size of the control volume is a critical parameterwhen a two-point LDA method is employed to determinethe Taylor length scale. Benedict and Gould (1999) con-cluded that the control volume dimensions should be assmall as the Kolgomorov length scale in the measure-


ment location. However, the latter condition is physi-cally difficult to achieve even when the LDA optics arearranged to collect the light in side-scatter with an angleof 900 between the transmitting probe and the receivingphotomultiplier. For example this optical arrangementwas employed by Benedict and Gould (1999) who es-timated the Kolgomorov scale to be at least 5 timesshorter than the longest dimension of the control vol-ume. Control volume dimensions of the order of η canbe met in relatively simple flows or in large scale rigs andwind-tunnels where the rate of dissipation is small andconsequently the Kolgomorov length scales are relativelylarge. The theoretical approach employed by Belmabrouk(2000) has been used in this study to determine the errordue to the finite dimension of the control volume in theestimation of λ. With Rrii denoting the actual correla-tion function of the velocity component ui, the measuredcorrelation coefficient Rmii, biased by the control volumedimensions, can be estimated from equation (8)

Rmii(r) =1

Λ2

∫ Λ2

−Λ2

∫ Λ2

−Λ2

Rrii(r + (x2 − x′

2) e2) dx2 dx′

2

(8)where x2 and x′

2 are the coordinates of the locationswhere the seeding particles are crossing the control vol-umes, and e2 is a unit vector directed as the longestdimension of the two control volumes. When the mov-able probe (number 1) is displaced orthogonally to thedirection of the axis of the control volume (r = ∆xiei

with i 6= 2) equation (8) becomes:

Rmii(∆xi) =1

Λ2

∫ Λ2

−Λ2

∫ Λ2

−Λ2

Rrii(σ) dx2 dx′

2 (9)

where σ is equal to√

(∆x2i + (x2 − x′

2)2).

This type of analysis is based on three main assump-tions:

– The probe volume is supposed to be one dimensional.– The seeding particles cross the control volume in a

location x2 along the main axis with a uniform prob-ability.

– The turbulence inside the control volume is homoge-neous and isotropic.

Depending on the model selected to represent the realcorrelation function Rrii(∆xj) the integral of equation(9) can be very complicated to solve. To circumvent thisproblem the series shown in equation (10) has been usedwith an equal number (N) of particles crossing the twocontrol volumes:

Rmii(∆xi) =1

N

N∑

j=1

Rrii

√

(∆x2i +

(

ajΛ

2− bj

Λ

2

)2

(10)where the coefficients aj and bj (the subscript refers tothe j th pair of particles) assume random values in a

range of ±1 from a uniform distribution. To minimisethe statistical uncertainty in the estimation of the errordue to the control volume dimension, N was set at 30000.Recently Belmabrouk (2001) tested different models, mo-stly exponential functions, to simulate the correlationcoefficient Rrii(∆xj). The uncertainty on the measuredTaylor length scale depended significantly on the func-tion chosen, as the presence of the third order term in theTaylor expansion series shown in equation (2) can am-plify the deviation from the parabolic curve used to esti-mate the Taylor scale. Belmabrouk and Michard (1998)have demonstrated that the third order term is nil ornegligible when the flow is homogeneous or inhomoge-neous, respectively. Taking into account the above con-siderations, two correlation models have been selected:

R(∆xj) = eB

[

1−

(

1+∆x2

j

A B C

)C]

(11)

R(∆xj) =A

1 + B ∆x2j

+1 − A

1 + C ∆x2j

(12)

Both models depend on three parameters and their Tay-lor expansion series have a nil third order term. Thefirst model (termed number 1 hereafter) has been takenfrom Belmabrouk (2001), while the second one (termednumber 2) is proposed from the results of the presentstudy. The variations of R11(∆x1) and R33(∆x3) with∆x1 and ∆x3 are shown in figures 7 (a) and (b) respec-tively. The normalised and non-normalised correlationsare also shown in figure 71. The normalisation has beencarried by dividing all the non-normalised correlationcoefficients Rii(∆xi) with the value assumed by the fit-ted parabola where it intercepts the ordinate axis. It isevident that the normalisation does not affect the esti-mate of the Taylor scale λii which in the figure is shownby the interception of the fitted parabolas of the nor-malised and non-normalised correlations with the hor-izontal axis. A similar normalisation has been alreadyemployed in the past when the Taylor and integral lengthscales have been estimated from the autocorrelation co-efficient employing Taylor’s hypothesis of frozen turbu-lence (see Benedict and Gould, 1998). In order to esti-mate the error due to the control volume length Λ itis necessary to assume a real spatial correlation func-tion by giving some values to the parameters A, B andC of equations (11) and (12). In his theoretical study,Belmabrouk (2001) has determined the parameters ofthe models used, assuming L/λ = 3. This choice wasconsidered to be realistic as in practice the ratio L/λvaries in range between 2-5. In the present study theparameters A B and C have been determined by in-terpolating the normalised data points (∆xi, Rii(∆xi))with the two models mentioned above. On the one handthis procedure can raise some doubts on the validity of

1 The perfect symmetry of the correlation curve is causedby the fact that Rii(∆xi) has been set equal to Rii(−∆xi).


-2 0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1 Not-normalised R

11 Parabolic curve

Normalised R11

Parabolic curve

L11

=3 mmNorm FunctionModel 1

Normalised Function (Model 1) (Model 2)

L11

=2.8 mmNorm FunctionModel 2

11

R 11(

x 1) [-

]

x1 [mm]

(a)

-2 0 2 4 6 8 100.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1Normalised Function (Model 1) (Model 2)

Not-normalised R11

Parabolic curve

Normalised R11

Parabolic curve

L33


L33


33

R 33(

x 3) [

-]

x3 [mm]

(b)

Fig. 7 (a) Variation of R11(∆x1) with ∆x1:(empty circles) non-normalised correlation, (empty triangles) normalised correla-tion; (b) Variation of R33(∆x3) with ∆x3:(empty circles) non-normalised correlation,(empty triangles) normalised correlation.

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15

20

25

30

35

Model 1

Parabolic interpolation/ =

= 0 = 2 = 4 = 6

(t2 -

m2 )/

t2 %

[-

]

x3max

/ [-]

(a)

0 1 2 3 4 5 6 7 8 9

0

5

10

15

20

25

30

Model 2

Quadratic interpolation / = 0

Parabolic interpolation/ =

= 0 = 2 = 4 = 6

(t2 -

m2 )/

t2 %

[-

]

x3max

/ [-]

(b)

Fig. 8 Variation of the percentage error in the estimation of λ2 with increasing ∆x3/η and for different control volumedimensions Λ/η: (a) Model 1; (b) Model 2.

the real correlation function assumed, as the correlationcoefficients measured are inevitably biased by the con-trol volume dimension. On the other hand the functionsfound are undoubtedly a good representation of the flowand of its scales, especially considering that Λ was onlyslightly longer than 2 η (η ≃ 0.1 mm). The integral un-derneath the correlation curves gives the local integrallength scale Lii of the velocity component ui in the ith

direction. In figure 7 (a), the curves found using the twomodels of equations (11) and (12) are almost identical,and consequently the percentage difference in the esti-mates of L11 is only 7%. The same cannot be said forthe two functions determined by interpolating the datapoints (∆x3,R33(∆x3)) in figure 7 (b). There is a signif-

icant difference between the two curves at ∆x3 > 3 andthe percentage difference in the two estimates of L33 is18%. This discrepancy can be explained by consideringthat the exponential function of model 1 decreases morerapidly than the function of model 2 and a more accu-rate estimation of L33 can only be performed if moredata points are available for higher values of the dis-placement. Nevertheless the shape of the two curves inthe proximity of the origin is identical and consequentlythe estimates of the Taylor scale calculated using thetwo models differ only by a small amount. Figures 8(a) and (b) show the variations of the percentage er-ror in the estimation of λ33 with ∆x3max for the curvesdetermined using model 1 and model 2 respectively. It


should be pointed out that ∆x3max is the upper bound-ary of the parabolic fitting range. In both figures thereare four curves for increasing control volume dimensions.The arrow points in the direction of increasing Λ. Theanalogous graphs representing the variation of the errorinvolved in the estimation λ11 are not shown for brevityof presentation. Considering figure 8 (a), it is evidentthat an increase of the error in the estimation of theTaylor scale corresponds to an increase of Λ. Howeverit should be noted that for a control volume dimensionΛ up to 2 η there is only a very small difference withthe ideal curve representing a size-less control volume.For small control volume sizes the error is rather moreaffected by the parameter ∆x3max used in the parabolicfitting procedure. Considering in figures 8 (a) and (b)the ideal curves for a size-less control volume, it can beconcluded that, independently of the model used, an er-ror of 10 % affects the Taylor scale estimation when aparabolic interpolation is carried up to 4-5 η. This sig-nificant increase in the error can be explained by con-sidering that a parabolic curve approximates the realcorrelation function only in a narrow range around theorigin and, as the the displacement ∆x increases, the 4th

order term of the Taylor expansion series shown in equa-tion (2) tends to cause a deviation of the real correlationfrom the parabolic curve. Belmabrouk (2001) has drawnsimilar conclusions. It should be also noted that Ganet al. (1996), using both hot wire anemometry (HWA)and two-point LDA, have concluded that the correlationhas a parabolic shape within a range of 2-5 Kolgomorovscales around the origin.However a 5 η range around the origin limits the num-ber of points that can be used in the fitting procedureand consequently increases the uncertainties of the Tay-lor scale estimated. A different approach, interpolatinga 4th order polynomial function, can increase the num-ber of data points (∆x3,R33(∆x3)) used in the fit asthe real correlation would deviate from the quadraticpolynomial only when the 6th order term in the Taylorexpansion series becomes significant. The Taylor scale isagain computed from the coefficient of the second orderterm. In figure 8 (b) the variation of the percentage er-ror of λ33 with ∆x3max when a quadratic polynomial wasused in the interpolation procedure is shown. The con-trol volume dimension was set to zero. For ∆x3max/η ≃8 the error is still below 5 %. Johansson and Klingmann(1994) have interpolated a quadratic function among 5

points included in a range of 2.5 <∆xj

η < 13 and they

estimated the error to be 10 %. It is worth mentioningthat their measurements were not affected by geometricbias due to the control volume dimension as Λ ≃ 0.9 η.

4.3 Error due to the statistical uncertainty and to the

calibration factor Cf

The mean squared velocity gradients (∂ui/∂xj)2 havebeen calculated through fitting a straight line betweenthe points (∆xj

2, ∆u2i ) for ∆x approaching zero. To de-

termine the error in the estimation of the gradient itis necessary to recall the equations of the linear leastsquares method used in the fitting procedure. Consider-ing points of coordinates (xi, yi) (where xi and yi cor-respond to ∆x2 and ∆u2 of the ith displacement respec-tively) the slope β of the line best fitting the points canbe determined through the following equation:

β =−N

∑

xi yi + yi

∑

xi

N2 x2 − N∑

x2(13)

where N is the number of points used in the fitting range(at least five were employed in this study). The deriva-tive of β respect to yi will give the error in the estimationof β due to the error in the estimation of yi.

∂β

∂yi=

−N xi +∑

xi

N2 x2 − N∑

x2(14)

Once all the errors of the coordinates yi are known, thetotal error in the estimation of β can be calculated from(see Kline and McClintock, 1953):

dβ =

√

(

∂β

∂y1

dy1

)2

+

(

∂β

∂y2

dy2

)2

+ .. +

(

∂β

∂yidyi

)2

(15)The statistical uncertainty of the quantity ∆u2 was es-timated using some basic equations of statistics. Bendatand Piersol (1987) state that if a variable x is normallydistributed with a mean µx and a variance σ2

x then equa-tion (16) can be used to estimate a confidence intervalaround the real value of σ2

x where the variance s2, cal-culated from Ntot observations, will be contained with a1-α probability.

n s2

χ2n; α/2

≤ σ2x ≤

n s2

χ2n; 1−α/2

(16)

where χ2n; α/2

is a Chi-square distribution with n degrees

of freedom (n = Ntot − 1) and α is the confidence co-efficient. The distribution of the variable ∆u which isnot reported for economy of presentation, has a Gaus-sian shape with a mean ∆u = 0 as the mean velocitieshave been removed before the time coincidence windowfiltering. Therefore equation (16) can be applied to esti-mate the percentage error made on the variance of ∆u(i.e. ∆u2). Using equation (16) for Ntot=10000 (n=9999)with a 95 % confidence level the variance s2 was found tobe affected by an error of ± 2.8 % (0.9725 ≤ s2

σ2x≤ 1.278).

It should be noted that for n ≥ 120 the Chi-square func-tion can be approximated as follows

χ2n;α = n

(

1 +2

9n+ zα

√

2

9n

)3

(17)


Source of error (∂u1/∂x1)2 (∂u3/∂x3)2

τw 4 % 1.3 %x0 4.5 % 4.5 %Λ 1 % 1 %

∆xmax 13 % 10 %Statistical error 4 % 4 %

Cf 3 % 3 %

Total 15.3 % 12.3 %

Table 2 Summary of all the sources of error analysed

with zα being the value of the standard normal distri-bution for a confidence level of α.The calibration factor Cf depends on the wavelengthof the laser beams and the angle between the beams.The error due to the accuracy of the wavelength can beneglected while, in agreement to Johansson and Kling-mann (1994), the percentage error in the estimation ofthe angle between the beams has been estimated to be 1%. The percentage error of ∆u(∆xi)2 is calculated fromthe percentage error in Cf according to the followingequation:

d∆u2

∆u2= 2

dCf

Cf(18)

Finally the percentage errors in the estimation of thespatial gradients (∂u1/∂x1)2 and (∂u3/∂x3)2 due to sta-tistical uncertainty and to the calibration factor werefound to be 4% and 3 % respectively. In table 4.3 the dif-ferent percentage errors in the estimation of (∂u1/∂x1)2

and (∂u3/∂x3)2 are summarised and the total error hasbeen calculated from the square root of the sum of the in-dividual errors squared. The total errors are estimated tobe 15.3 % and 12.3 % for the (∂u1/∂x1)2 and (∂u3/∂x3)2

gradients, respectively.

4.4 Grid-generated turbulence experiments

The measurements have been carried in a grid-generatedturbulence flow to assess the two-point LDA methodol-ogy employed in the estimates of the spatial gradients(∂ui/∂xj)2 and of the related Taylor length scale λji.In fact the conditions of local homogeneity and isotropythat can be reached in this flow allow the determina-tion of the kinetic energy viscous dissipation rate ǫ fromthe decay of the kinetic energy along the main flow di-rection (Tennekes and Lumley, 1973). Moreover a re-liable estimate of the local Kolgomorov scale can beachieved. For brevity of presentation the different meth-ods usually employed to determine the beginning of theisotropic region (i.e. Mohamed and LaRue, 1990; Tressoand Munoz, 2000) will not be discussed, but is worth

18 20 22 24 26 28 30 32

400

800

1200

1600

2000

2400

2800

3200

3600

4000

-15 %

+ 15 %

-15 %

+15 %

U1 ( q2/ x

1)(1/15 )

0.5 U

1 ( q2/ x

1)(1/15 )

( u1/ x

1)2 ( u

1/ x

3)2

( u3/ x

3)2 ( u

3/ x

1)2

(1/U12)( u

1/ t)2

(

u i/x j)2

[ s-2

]

x1/M [-]

Fig. 9 Comparison between the values assumed by the spa-tial gradients directly measured and the gradient (∂u1/∂x1)2

calculated from the kinetic energy decay and from Taylor’shypothesis.

20 22 24 26 28 30 320.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0.022

- 10 %

+ 10 %

[m

2 s-3]

x1/M [-]

0.5 U1 ( q2/ x

1)

Direct measurement

Fig. 10 Decay of kinetic energy viscous dissipation ratealong the centreline behind the grid using two different meth-ods: dissipation derived from the kinetic energy decay (filleddots); direct measurement of ǫ (empty triangles).

mentioning that the region started at a distance of 20M from the grid and that the kinetic energy decayed ina power law manner with an exponent of 1.3. A com-parison between the spatial gradients directly measured(the gradients (∂u1/∂x1)2, (∂u1/∂x3)2, (∂u3/∂x1)2 and(∂u3/∂x3)2 were measured separately) and the gradient(∂u1/∂x1)2 calculated from both the kinetic energy de-cay and from Taylor’s hypothesis is shown in figure 9 atsix locations along the duct centerline. xM is the distancefrom the grid normalised with the mesh size M of thegrid. It should be noted that in a locally isotropic flowthe gradients (∂ui/∂xj)2 and (∂ui/∂xi)2 are equal to

2(∂u1/∂x1)2and (∂u1/∂x1)2 respectively (Hinze, 1975).


It can be observed that the gradients (∂u1/∂x1)2 and(∂u3/∂x3)2 assume very similar values and that both thegradients (∂u1/∂x3)2 and (∂u3/∂x1)2 are around twicethe value of (∂u1/∂x1)2, as expected for grid turbulenceflow. In agreement with the error analysis carried on inthe previous sections of this paper, the highest percent-age error between the gradients directly measured andthe one estimated from the kinetic energy decay is 14%. The variation of the viscous dissipation of turbulentkinetic energy along the centerline has been plotted infigure 10. To estimate the dissipation rate, the unknownspatial gradients have been substituted according to thelocal isotropy relations and the following equation hasbeen derived:

ǫ = ν

[

(

∂u1

∂x1

)2

+ 2

(

∂u3

∂x3

)2

+ 3

(

∂u1

∂x3

)2

+ 3

(

∂u3

∂x1

)2]

(19)The agreement between the dissipation rate calculatedfrom equation (19) and the one calculated from the ki-netic energy decay is excellent. In this case the highestpercentage difference between the two estimates is 7 %.This value is smaller than the calculated total error ofthe spatial gradient measurement. This can be explainedby considering that in the estimation of the dissipationrate the errors due to the spatial gradient tend to aver-age out.

5 Conclusions

In the present study an attempt has been made to iden-tify and quantify through a rigorous analytical approachall possible sources of error involved in the estimationof the fluctuating velocity gradients (∂ui/∂xj)2 whena two-point LDA technique is employed. Measurementswere carried in a grid-generated turbulence flow wherethe local dissipation rate can be calculated from the de-cay of kinetic energy and an assessment of the error de-termined through the theoretical analysis could be made.In agreement with Belmabrouk (2001), the main sourceof error was found to be related to the fitting range usedto estimate the spatial gradient. As the upper limit ofthe fitting range increases, the error also increases reach-ing a value of 10 % for ∆xmax ≈ 4.5 η, when a paraboliccurve is interpolated through the correlation coefficientsRii(∆xj). However this estimation is strictly related tothe type of curve used in the interpolation. In the presentstudy a quadratic curve has also been employed to deter-mine the error in the estimation of the spatial gradient.In this second case the error was found to be less than 5% for values of ∆xmax/η up to 8. The error due to thecontrol volume length (≈ 2 η in the flow studied) wasestimated to be around 1 %. However it should be notedthat for control volume as long as 6 η the error due tothe control volume dimension increases by up to 10 %.The above error estimates were found using two differ-ent functions modelling the real correlation coefficient

function. Despite the major differences in the expres-sion of the two correlation functions, the estimates ofthe error due to the fitting range and to the control vol-ume dimension were only slightly affected by the modelused. The error caused by the time coincidence windowτw tends to affect mostly the gradient in the main flowdirection. In this case a longer coincidence window in-creases the number of single particle burst pair with anincrease of the error in the estimation of the gradientup to 30 %. However if τw is selected correctly the errordue to the time coincidence window is less than 5 % forboth the longitudinal (in the main flow direction) andtranverse (perpendicular to the main flow direction) gra-dients. The positional error made in the selection of theorigin x0 where the two control volumes are supposed tooverlap completely, can be used to determine an error inthe gradient estimation of ±4.5 %. Finally the statisticalerror and the error due to the calibration factor are 4 %and 3 % respectively. The cumulative errors made in theestimation of the gradients (∂u1/∂x1)2 and (∂u3/∂x3)2

were found to be 15.3 % and 12.3 % respectively. Theabove values are in good agreement with those foundwhen the spatial gradients directly measured were com-pared with the gradients estimated from the decay of thekinetic energy along the main flow direction. Moreoverit was observed that the errors made in the estimationof the spatial gradients tend to average out when thedissipation rate of the turbulent kinetic energy ǫ is com-puted, with a 7% difference found between the ǫ valuesdetermined from the fluctuating velocity gradient mea-surement and the decay of the kinetic energy.

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Analysis of errors in the measurement of energy dissipation with two-point LDA

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