Experimental Thermal and Fluid Science 39 (2012) 71–78

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Experimental Thermal and Fluid Science

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Estimate of turbulent dissipation in a decaying grid turbulent flow

A. Liberzon a,⇑, R. Gurka b, P. Sarathi c, G.A. Kopp c

a School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israelb Department of Chemical Engineering, Ben Gurion University, Beer Sheva, Israelc Boundary Layer Wind Tunnel Laboratory, University of Western Ontario, London, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Received 3 October 2011Received in revised form 4 December 2011Accepted 10 January 2012Available online 21 January 2012

Keywords:DissipationDecaying turbulenceGrid turbulenceParticle image velocimetryKinematic relationsStructure functions

0894-1777/$ - see front matter � 2012 Elsevier Inc. Adoi:10.1016/j.expthermflusci.2012.01.010

⇑ Corresponding author. Tel.: +972 3 640 8928; faxE-mail address: alexlib@eng.tau.ac.il (A. Liberzon).

The dissipation rate of kinetic energy is a key quantity in turbulent flows. The number of measured com-ponents and the resolution of the measurement techniques limit dissipation rate estimates; thus, theyare derived by surrogates of dissipation. We examine the validity and accuracy of these estimates byinvestigating decaying grid turbulence using particle image velocimetry and laser Doppler velocimetry.Dissipation rates are computed and compared via three different methods (i) using the decay rate of tur-bulent kinetic energy, (ii) direct calculation using measured velocity gradients, and (iii) using second andthird order structure functions. Discrepancies have been found between the surrogate methods; specif-ically, the structure function method requires correction terms. The known factors leading to bias are vis-cous correction which is significant at low Reynolds numbers and inhomogeneity of the decaying flow.Furthermore, we demonstrate inaccuracy in calculations of the third order structure function whichare related to inter-scale dependencies. Test procedures are suggested for decaying and inhomogeneousflows to determine susceptibility to these sources of error.

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1. Introduction

The dissipation rate of the turbulent kinetic energy (e � 2msijsij,hereinafter is called dissipation for the sake of brevity, sij is the rateof strain tensor) is a fundamental quantity in the characterizationof turbulence. It is essential to estimate or calculate the total dissi-pation accurately as it represents the total amount of energy loss ina given system. In turbulent flows, the fluctuating part of the dis-sipation rate is larger than the dissipation rate stemming fromthe mean flow by several orders of magnitude [1]. The dissipationrate, as it appears in the turbulent kinetic energy equation, is threedimensional, regardless of the quasi-two-dimensional approxima-tion of the mean flow (in flows such as boundary layers, planar jetsand wakes, decaying grid turbulence, etc.). There are numerousprocedures to estimate turbulent dissipation using experimentaltools, which have been extensively reviewed in the handbook ofexperimental fluid mechanics [2] and in the recent books on turbu-lence (e.g. [3,4]).

Among the experimental methods the ones of note are thosethat access turbulent dissipation rate via all of its components, fol-lowing the definition: e � 2msijsij. These include the three-dimen-sional hot-wire arrays, e.g. [5,6], and optical methods such as 3Dparticle tracking velocimetry (PTV) [7], tomographic particle imagevelocimetry (PIV) [8], holographic PIV [9], scalar imaging [10],among others, e.g. [2]. However, some techniques are limited to

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moderate Reynolds number (e.g. PTV) or limited due to the lackof spatial resolution (such as the hot-wire technique which is alsoan intrusive method).

Thus, the majority of experimental data are obtained using toolsthat are typically limited by resolution either in time or in space oronly enable the measurement of partial components of velocityand velocity derivatives. Dissipation rate estimates based on a lim-ited number of components are commonly known as surrogates ofdissipation rate; i.e, expressions derived from a set of assumptionsrelating to the magnitude of the missing components.

Inhomogeneity in turbulent flow has been known to bias esti-mates of dissipation. For instance, Folz and Wallace [11] measuredall of the components of the velocity gradients in the atmosphericboundary layer. The authors estimated various contributions todissipation rate using multi-hot-wire sensor data and comparedthree surrogates: (i) e � 2m

Pi

Pjð@ui=@xjÞ2, which ignores inhomo-

geneous cross products of velocity derivatives, (ii) e � 1/mxixj and(iii) e � 15m(@u/@x)2, the isotropic surrogate, where x, u is thestreamwise coordinate/velocity respectively, and ui, xj are 3D vec-tors, i.e. i,j = 1, 2, 3. The three surrogates were compared to the di-rect measure of 2msijsij, which showed relatively large discrepanciesbetween the different techniques, pointing out the importance ofthe inhomogeneous components. Browne et al. [12] used hot-wiresto measure squared gradient terms in the cylinder wake. The termswere found to be anisotropic and the isotropic estimatee � 15m @u

@x

� �2 underestimated the mean dissipation rate by 45 to80 percent. Balint et al [13] using a nine-sensor hot-wire probe,found that the isotropic estimate is lower by 85–60% in the near

72 A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78

wall region of the boundary layer, at y+ = 11 and y+ = 72, respec-tively. Without cross-product terms, values were underestimatedby 25% and 10%, respectively. It is noteworthy that the largest erroris in the near wall region of boundary layers, close to the viscoussublayer, i.e. a low Reynolds number turbulent flow. This is animportant notion for analysis of decaying flows or for flows nearsolid boundaries, where low velocities are prominent.

De Jong et al. [14] reviewed five (direct and indirect) methods toestimate dissipation in a zero-mean turbulent flow apparatus. Theauthors recommended use of the second order structure functionmethod, which was found to be most robust and accurate. Directmethods are typically discarded as not accurate and prone to dis-cretization and resolution errors. Lavoie et al. [15] compared PIVresults with Hot Wire Anemometry, (HWA) results in decaying gridturbulence at the same range of Reynolds numbers as presented inthis manuscript. Their comparison has shown that HWA is moreaccurate in terms of resolution but with proper corrections, PIVcan provide a comparable second order structure function valuesto meet the ones obtain by the HWA. The immediate output forthe end-user is to use PIV with the corrections suggested by theauthors [14–16], among others and apply second order structurefunctions for the estimation of dissipation. The present work addsto this point of view with terms that allow for correction of the biasin inhomogeneous shear flows.

We report on a similar experiment in which a grid turbulencedecays in a water channel. We measured the flow using PIV, esti-mated dissipation using direct and indirect methods and comparedwith LDV measurements. Our results show that, in order to definedissipation properly, one has to measure the decay of turbulentkinetic energy at several locations and use the power law of decayfor the estimation of dissipation rate as a function of streamwisecoordinate. Although the direct method compares favorably withthe LDV and indirect methods, in high Reynolds number turbulentflows, PIV resolution becomes insufficient and we expect the errorsdue to interrogation window size and differentiation to becomesignificant. The two-point statistics, such as second-order andthird-order structure functions are useful to estimate these typesof errors. By applying the kinematic relations (defined by Hosokawa[17]) and conditional averaging, we demonstrate that the results aredependent on the large scale flow. The bias is therefore non-linearand changes with the streamwise distance from the grid, accordingto the growth of spatial turbulent scales (L / x) where L is theintegral length scale and the decay of turbulent velocity (u / 1/x).

The objective of this manuscript is to investigate new toolsavailable for the assessment of key quantities in measured turbu-lent flows, mainly the so-called ’kinematic relations’. In the flowunder investigation, we demonstrate that at typical Reynolds num-ber used in water channels past a grid (e.g. Ref. [18]) the flow con-ditions should not be oversimplified. The presence of both largeand small scale dynamics in turbulent flows such as grid turbu-lence is shown using the kinematic relations. The paper does notsolve the problem but presents an additional method to assessthe turbulent flow characteristics.

The paper is organized as follows. The experimental facility andmethods are outlined in the following Section 2. We present the re-sults regarding the turbulent quantities in the decaying turbulentflow past a grid in Section 3. The analysis of various componentsand the sources of errors are continued in the framework of kine-matic relations in Section 4. Finally, we summarize the main find-ings and draw some conclusions regarding the applicability of thesurrogates in various turbulent flows.

2. Experimental method

Experiments were performed in the water tunnel at the Bound-ary Layer Wind Tunnel Laboratory of the University of Western On-

tario shown in Fig. 1. The facility consists of an inlet reservoir,where water is introduced into the tunnel, followed by a settlingchamber consisting of a honeycomb and screens, an 8:1 contrac-tion, the test section, and a 90 degree turn to return the water tothe sump through a control valve. The present set-up allows amaximum flow rate of 0.036 m3/s, with a uniform mean velocityprofile (within 1%) and a turbulence intensity of less than 1% inthe test section. The length of the test section was chosen to be10 times the height. The width, W, and height, H, of the test sectionare 600 mm and 300 mm, respectively.

Measurements were carried out in turbulent flow generated bya square mesh grid interwoven with stainless steel rods of6.35 mm diameter and a spacing of M = 25.4 mm, placed perpen-dicular to the flow at the beginning of the test section. (The solidityratio of the grid was 64%). Measurements were made at five loca-tions downstream of the grid (i.e. x = 100, 500, 600, 800 and900 mm) with a mean flow velocity of 0.20 m/s. The flow behindthe grid evolves from a highly non-homogeneous velocity fieldclose to the grid where it is generated, to a state where it becomeshomogeneous and isotropic (within a close approximation). Thedissipation rate estimate at the closest measurement point,x = 100 mm, is in the developing region and we disregard it fromthe analysis in the following sections. All the results will be pre-sented for the streamwise locations associated with the developeddecaying grid flow, namely at 500, 600, 800 and 900 mm.

Measurements were conducted using Particle Image Velocime-try (PIV). The measured field of view was in the center of the watertunnel, in the streamwise-wall normal plane, where for eachstreamwise location, the system was traversed. The PIV systemused for the current study makes use of a double pulse Nd:YAG la-ser operating at 15 Hz with energy of 120 mJ/pulse that produces asheet of light at a wavelength of 532 nm illuminating a flow fieldthat is seeded with Silicon Carbide particles with an average diam-eter of 2 lm and a density of 3200 kg/m3. Despite the density dif-ference, the particles are small enough to follow the water flow,fulfilling the requirements suggested by Melling [19]. The scat-tered light from the particles is collected into a CCD camera located90 degrees to the light sheet. The CCD has a pixel array of1600 � 1200with a dynamic range of 12 bits operating in doubleexposure mode. The time interval between two sequential imageswas set to be 2.5 � 10�3 sec. During each PIV experiment 4000images were acquired per batch resulting in 2000 vector maps.The cross-correlation analysis was performed using OpenPIV(http://www.openpiv.net, Taylor et al. [20]) for interrogation win-dows of 32 � 32 pixels with a 50% overlap. Special filters such asmean, standard deviation and local median, removed the errone-ous vectors. In total, about 5% of the vectors were removed.

To complement the PIV results and verify the estimate of thedissipation rate, we have used a single component Laser DopplerVelocimetry (LDV) system operated in back-scatter mode. Thetransmitting lens had a focal length of 350 mm in air, resulting ina measuring volume diameter and length of 0.046 mm and1.2 mm, respectively. A two-axis motor, driving the traversing unit,was used to move the LDV probe in two directions. The flow wasseeded using the same particles as for the PIV.

3. Results

3.1. Dissipation estimate based on energy decay

This section describes the various estimates of the dissipationrate in the decaying turbulent flow past a grid. The results are gi-ven at the four streamwise locations (x = 500, 600, 800 and900 mm from the grid).

The equation of turbulent kinetic energy transport as derivedfrom the Navier–Stokes equations is (see e.g. [1]):

(a)

(b)

Fig. 1. Top: overview of the water tunnel facility. Bottom: Schematic description of the measurement setup and location of measurements.

A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78 73

Uj@

@xj

12

u1u2

� �¼ � @

@xj

1q

ujpþ12

uiuiuj � 2muisij

� �� uiujSij

� 2msijsij ð1Þ

In grid turbulence the mean shear, Sij is zero. Hence, there is no pro-duction of turbulent kinetic energy, i.e. the term �uiujSij � 0. Theenergy budget for streamwise mean flow velocity component, U1 is:

U1@

@x1

12

u1u2

� �¼ � @

@x1

1q

u1pþ 12

uiuiu1

� �� e ð2Þ

Typically, several terms are neglected. It is of note that the vis-cous transport term can be strictly neglected only for high Rey-nolds number flows. Furthermore, only if it is assumed thatu� U, then the turbulent transport term is also negligible com-pared to the transport by the mean flow. The outcome of theseassumptions is a surrogate for the dissipation term (e.g. [1,3]):

U@

@x12

uiui

� �¼ �e ð3Þ

Turbulent kinetic energy ðhq2i ¼ uiuiÞ in grid turbulence experi-ments is typically estimated using the root-mean-square of thestreamwise component of fluctuating velocity, u0 = hu � Ui1/2, esti-mated mostly from single hot-wire measurements of the stream-wise fluctuations. Here in after, the averaging operator h�i denotes

both the temporal (ensemble in the case of PIV) averaging overthe flow realizations and spatial averaging along y, the homoge-neous direction of the flow. Instead of measuring the rate of decay,@hq2i@x , we can use the fact that velocity fluctuations decay in the flow

past a grid in a self-similar manner according to the power law [3]:

hq2iU2 ¼ A

x� x0

M

� ��nð4Þ

where x0 is a virtual origin, n the power-law exponent and A a con-stant of proportionality. It is noteworthy that the value of A varieswidely depending on the geometry of the grid and the Reynoldsnumber (e.g. Ref. [21]). This power law fit can be applied to PIVand LDV data as shown in Fig. 2. In our case the two-componentLDV system was employed in such a way that in both experimentalmethods we obtained the streamwise and the transverse, velocitycomponents. Therefore, we can estimate a surrogate for turbulentkinetic energy, q2, assuming that the two transversal components,v, w are statistically equal,

hq2i ¼ hu02i þ hv 02i þ hw02i � hu02i þ 2hv 02i ð5Þ

The results are presented in Fig. 2 for the decay rate of the meanturbulent kinetic energy, hq2i. Although there is a small differencebetween the quantitative values, the trend and the slope are

Fig. 2. (Left) Decay of the turbulent kinetic energy hq2i estimated using two-component LDV (circles) and PIV (squares) as a function of distance x/M. Curves represent thepower law fit, q2 / x�1.2. (Right) Dissipation rate e which is estimated using Eq. (3) based on the PIV data (symbols), while the curve is the power law relation e / x�2.2 from Eq.(6).

Fig. 3. Comparison of the dissipation rates estimated using the PIV and LDVmeasurements; Eq. (3) using PIV (circle); direct estimation, 2lhsijsiji, (square); Eq.(4) using LDV (triangle); Eq. (7) using LDV (diamond).

74 A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78

preserved. The discrepancies between the two systems when esti-mating the kinetic energy results from the procedure we appliedfor the PIV. In order to compare between the LDV and PIV, we havefirst ensemble averaged the instantaneous velocity maps, resultingin the spatial distribution of the kinetic energy at the measuredfield of view. Following that, we have spatially averaged the kineticenergy values over the entire field of view (200 � 200 mm) obtain-ing a representative value for the specific location. The spatial aver-aging serves as a filter to the data and, therefore, underestimatesthe measured value at any given data point. Since in the equationswe have the unknown constant, A, we use a comparative approachin order to obtain the most accurate estimate of dissipation. For thesake of comparison, we use the results obtained from the PIV andLDV data with different approaches.

The slopes of the two curves on the left panel of Fig. 2 providethe value of the decay exponent, n, which is found to be approxi-mately 1.2 in the present study (virtual origin is set to x0 = 0 fol-lowing the work of Mohamed and Larue [22]). Previous studiesreport values of the decay component between 1.15 and 1.45,e.g. [21,23], confirmed by the DNS study in decaying homogeneousisotropic turbulence by Burattini et al. [24].

Combining the Eq. (3) with the decay rate in Eq. (4), one obtainsthe estimate of the dissipation rate as:

hei ¼ nAU3

2Mx� x0

M

� ��ðnþ1Þð6Þ

The result is shown in the right panel of Fig. 2: the data obtainedusing PIV in symbols and the power law fit as a curve, accordingto Eq. (3).

An additional method that can be applied to the single point,time-resolved measurements from LDV experiment is using thetemporal derivative of velocity being used as a surrogate for thedissipation [21]:

@hu02i@t

¼ hei ð7Þ

This result is shown in Fig. 3, along with the results using thedecay of turbulent kinetic energy and direct measurements of dis-sipation using PIV and velocity derivatives, @ui/@ xj.

Flow realizations obtained from the PIV measurements containspatial information and can be differentiated directly to obtain fiveout of nine velocity derivative components, namely @u/@x, @u/@y,@v/@x, @v/@y, and @w/@z (from continuity). Calculating dissipationfrom PIV data adds an error due to the numerical differentiationof the velocities [25,26]. In order to minimize this error, we applythe least squares numerical differentiation scheme, which waschosen following a previous study and an extensive error analysis[27].

The PIV data acquired in the four different location in thestreamiwse direction were used to estimate the dissipation. Foreach location we have averaged the dissipation over the entire fieldof view in order to obtain a representative value at each location. Inaddition, we have compared the direct calculation with dissipationvalues as estimated from Eq. (6), using the same set of data andvalues obtained by the LDV using Eq. (7).

Fig. 3 depicts a comparison between the dissipation values asestimated from PIV and LDV directly and indirectly along thestreamwise direction. It is plausible to assume that the discrepan-cies between the techniques are due to the lack of the componentsof the third dimension and the respective derivatives. In addition,consideration of the uncertainty involved in instantaneous gradi-ents that are applied in the direct calculation causes an additionaldiscrepancy. Comparing the dissipation as estimated with LDV andPIV, one can observe that the values obtained by LDV (using Eq. (7))are roughly 1.5 times higher than estimated using PIV. The directand indirect methods (through Eq. (3) and Eq. (7)) provided similarestimates, with a variation of about 15% close to the grid, while atdistances further from the grid, all values collapse. This result canbe related to the increase of homogeneity and isotropy as the fluidmoves away from the grid.

3.2. Dissipation estimate based on two-point statistics

The above methods calculate dissipation using single pointquantities, either velocity derivatives in the direct method or

Fig. 4. Schematic definition of u1 and u2 at distance r. Only the vertical,homogeneous components are used. Fig. 6. The third order structure function, according to Eq. (9), normalized by the

h�ir. The sum of the LHS (circles), second term in Eq. (9) (squares) and the third term(triangles) on the RHS are shown as compared to the straight line at 4/5. The plot isfor the data at x = 600 mm.

A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78 75

turbulent kinetic energy values indirectly calculated using decaylaws. In this section we estimate the dissipation using two-pointstatistics, namely using the second- and third-order structurefunctions.

The structure function is defined via velocity differences,Du = u(x + r) � u(x) between two points at distance r. Only verticalvelocity component (v) along the homogeneous direction (y) isused to calculate the Du(y) = u1 � u2, as shown schematically inFig. 4. Homogeneity of the flow in the vertical direction allowsfor spatial averaging of the increments at a given distance r (lim-ited to the half-height of the observation field). In addition, we uti-lize time averaging at any given streamwise coordinate, x, meaningthat h�i denotes an average over all the vectors at distance r.

The second order structure function derived from the isotropicsurrogate of the dissipation rate for small r is related to the dissi-pation rate as:

hðDuÞ2i ¼ 115

r2

mhei ð8Þ

Second order structure functions for various streamwise loca-tions are plotted in Fig. 5. The left panel shows the second orderstructure function, (Du)2 for various streamwise locations andthe right panel shows the dissipation rate estimate using the Eq.(8). The two subfigures show the respective curves as functionsof distance r between the two points, normalized with the Kol-mogorov length scale, g, estimated using the dissipation valuesin Section 3.1.

Fig. 5. (Left) Second order structure functions for various streamwise locations,

We proceed to the dissipation rate estimate using the terms ofthe following extension of the Kolmogorov 4/5 law for inhomoge-neous flows [15]:

hðDuÞ3i ¼ �ð4=5Þh�irþ6mdhðDuÞ2i

drþ3ðU=r4Þ

Z r

0s4 dhðDuÞ2i

dxds ð9Þ

For decaying turbulent flow past a grid, the third-order structurefunction h(Du)3i is compared to the �4/5h�ir with the followingterms (the second and third terms on the right hand side): (i) vis-cous term due to the relatively low Reynolds number and (ii) anadditional term due to inhomogeneity of the decaying turbulence,see e.g. Refs. [23,28,29]. The result is shown in Fig. 6. These resultsare in agreement with those previously obtained by Antonia andBurratini [23] in a wind tunnel using an array of hot-wires and byLavoie et al. [15] who used both PIV and hot-wires. It should benoted the largest component for large values of r/g as shown inFig. 6, is the term associated with inhomogeneity, which is derivedfrom the decay of Du2 along the streamwise coordinate (see Fig. 2).

Our results confirm the results obtained in previous studies, e.g.[15,23] amongst others. The important extension, however, is thedirect estimate of the additional terms appearing in the 3rd orderstructure function method of dissipation rate estimation, in which

(Du)2 as a function of r/g. (Right) Dissipation rate estimate applying Eq. (8).

Fig. 7. Conditional averages: (left) hu2�ju1i and (right) h(Du)2ju+i showing dependence of the two-point statistics of small scale velocity increments, plotted for growing r. On

the left panel the solid symbols are curves of dependence on the u2. On the right panel the solid symbols are of the shortest measured distance r = 5g and the longest r = 80g.The values are for the x = 500 mm.

(a) (b)

Fig. 8. (a) Second order kinematic relation according to Eq. (13). Left hand side h(Du)2i is shown as open symbols while the right-hand side as the closed symbols. (b) Curvesfor different streamwise locations x collapsed using the value of the curve at large separations r ?1. Inset shows the decay power law of the values at large r.

76 A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78

second order structure functions were estimated at differentstreamwise locations and differentiated along x. The significanceof the inhomogeneous term for the estimation of the dissipationrate using the structure function, and its growing importance withr is peculiar. (Du) is a local quantity in terms of r, i.e. it is com-monly assumed that it depends only on the velocity at scale r. Thisis the basic assumption used in construction of the 4/5 law. Theimportance of the values at different distances, apparently largerthan r, leads to a more careful assessment of the inter-scale depen-dence in what follows.

4. Discussion

The two-point statistics presented in the previous sectionemphasized the need for further analysis in order to understandthe discrepancies between the various direct and indirect methods.One could infer that there is some interference between the largescale quantities (e.g. velocity) and the small scale quantities (e.g.velocity derivatives).

The inter-scale dependence might not describe the inaccuracyof the dissipation estimates but can point out the possible sourcesof uncertainty and ways to improve it. For this purpose, we apply

the kinematic relations derived by Hosokawa [17] based on thetwo-point average, u+(r) = (u(x + r) + u(x))/2 and two-point differ-ence u�(r) = (u(x + r) � u(x))/2 (note that u� = 0.5Du), and the con-ditional sampling analysis. In the context of our analysis, it isimportant to note that the average is a large-scale quantity associ-ated with the field of velocity, while the difference is a small scalequantity, associated with the field of velocity increments. In Ref.[17] it is shown that the quantity hu2

þu�i, which involves both largeand small scale quantities, in locally isotropic turbulent flows, isequal to the third order structure function:

ðDuÞ3 ¼ �45h�ir ð10Þ

hu2þu�i ¼ �

130h�ir ð11Þ

The relation shown in Eq. (11) is a consequence of the Kolmogo-rov 4/5 law, Eq. (10), and a kinematic relation, which is valid underthe assumption of local isotropy:

hu3�i ¼ �3hu2

þu�i ð12Þ

The above relations could be conceptualized as surrogates forthe dissipation rate, using two-point statistics, therefore, differentfrom the surrogates using single-point statistics. There is an

(a) (b)

Fig. 9. Third order kinematic relations prescribed by Eq. (14), collapsed together using the peak value of each curve in the range of r = 20–40g. (b) Decay and the power law ofthe peak values of the structure functions. Circles are for the (Du)3 and squares for the �3h u2

1 þ u22

� �Dui.

A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78 77

additional advantage in using these terms as surrogates for the dis-sipation rates – the difference term, Du or u� appears only once inthe kinematic relations as compared to the term raised by the sec-ond or third power in the structure functions, thus it significantlyreduces the error due to the PIV discretization errors.

Applying this type of kinematic relations also addresses one ofthe goals that is preeminent in turbulence research: applying theKolmogorov 4/5 law (Eq. (10)), which is strictly valid only for glo-bal homogeneous and isotropic flow to all various type of flows,releasing the constraints to the local homogeneity and isotropyconditions, where local refers to a spatial scale on the order of r,arising from the apparent scaling law of hðDuÞ3i ¼ � 4

5 rh�i. Weshow here that for decaying turbulence, the Hosokawa relationhu2þu�i allows one to obtain the Kolmogorov 4/5 law analogy from

transverse velocity component, v, which is only locally homoge-neous [30].

4.1. Conditional averaging

In order to probe the scale-dependence of the quantities, weutilize conditional averaging. In the following Fig. 7, we show u�(left panel) and (Du)2 (right panel), conditionally averaged on thetwo quantities. The left panel is conditioned on the velocity atone of the two points, denoted u1. The right panel is conditionedon the two-point average, u+. In addition, in both panels the datais shown for different values of r. As the conditional average valuesincrease with increasing r we can infer that there is a dependenceof quantities like u� on the values of velocity, e.g. u+ at various dis-tances. Such dependence cannot be neglected in the point-wiseestimates of turbulent quantities or even in two-point statisticsat different r. For example, spatially under-resolved PIV measure-ment will lead to a biased result partially due to this dependency.

4.2. Mixed type kinematic relations

There is a long list of kinematic relations introduced in Ref. [30]which we partially reproduced for the current flow in Ref. [31].Generally, they can be divide into several types, for example themixed-type relations mix the single-point and two-point quanti-ties such as h(Du)2i = �2h u1Dui = 2hu2Dui. These can appear assymmetric or asymmetric relations (i.e. using only one end veloc-ity, u1 or both ends u1 and u2). There are also two-point relationssuch as Hosokawa’s hu3

�i ¼ �3hu2þu�i, which comprise only two-

point quantities.

We use the mixed-type relations relating to the questions of theinter-dependence of the small-scale turbulent flow statistics, suchas dissipation rate and second order structure functions (Du)2,using the local instantaneous velocity, which is chosen as u1 oras a two-point average, u+. The correlation between the large-scalecharacteristic, u1 and small-scale ones (Du)2 is identical toh(Du)3i = �2hu1(Du)2i. These correlations emphasize that there isan intrinsic difficulty in constructing a power law for all grid turbu-lence flows that are slightly different from each other in terms ofthe velocity field.

Fig. 8 shows the results for the asymmetric kinematic relation,defined as:

hðDuÞ2i ¼ �2hu1ðDuÞi ð13ÞFig. 8a shows the left-hand-side (LHS, open symbols) and the

right-hand-side (RHS, closed symbols) of Eq. (13) as obtained fromthe experimental data. The curves of LHS and RHS are similar at allstreamwise locations, pointing toward a self-similarity of the kine-matic relations. On the right panel Fig. 8b the curves collapse atlarge r. The values at large r obtained from Fig. 8a are shown inthe inset, depicting a power decay law; similar to the resultsshown in Fig. 2a. This results is in agreement with the self-similar-ity of the longitudinal structure functions demonstrated by Moninand Yaglom [32, p. 192].

The symmetric third order kinematic relations presented inFig. 9 is defined by the following:

hðDuÞ3i ¼ �3h u21 þ u2

2

� �Dui ð14Þ

The third order relation in the grid turbulence exhibits a plateau inthe range 10 < r/g < 50.

This kinematic relation exhibits a similar trend as the secondorder one; the curves collapse once normalized by the peak value(appearing in the range of r = 20–40g). However, in this case thedecay law is different from the one obtained in Fig. 2b. The resultsdemonstrate that although the similarity is preserved qualitatively,the quantitative estimate is not sufficient.

We would like to stress that the kinematic relations of velocityincrements comprise a manifestation of non-local effects, e.g. largeand small scale quantities are correlated. This feature invalidatesthe so-called random sweeping hypothesis that large and smallscales are statistically independent and hinders, in some sense,the use of Taylor’s hypothesis. It is noteworthy that, kinematicrelations emphasize the non-local effects thereby become dynam-ically significant. As presented here, kinematic relations could beused for the validation of experimental results.

78 A. Liberzon et al. / Experimental Thermal and Fluid Science 39 (2012) 71–78

5. Summary and conclusions

Different methods to estimate the dissipation are presented andcompared. Three methods were utilized: calculation based on thedecay rate of the turbulent kinetic energy, direct estimate using5 velocity derivatives out of nine, and two-point statistics. Discrep-ancies have been found based on the surrogate methods that wereused. The direct method and the decay rate have provided similarestimates of the dissipation, with variation of about 15% close tothe grid while at further distances from the grid, all values collapse.The third method, two-point statistics, utilizes second and third or-der structure functions to estimate the dissipation. These invokethe use of Kolmogorov 4/5 law and locally isotropy and homoge-neous assumptions. As part of this study, we have shown biasesin the calculations due to two factors: (i) the viscous correctionsdue to the low Reynolds numbers contribute a non-negligible partof the bias and (ii) the flow inhomogeneity is the main source ofbias introduced into the surrogate methods. Further examinationprovided an additional source of error resulting from the inter-scale dependency between the large and small scales. This wascharacterized through the use of certain types of kinematic rela-tions: mixed-type ones where velocity averages and velocity dif-ference are correlated. In addition, variations of the correlation asfunction of conditional averaging provided an additional evidenceof the error embedded in the dissipation estimates due to the flownature rather than the technique itself. The analysis is limited tothe moderate Reynolds number range. Higher Reynolds numberexperimental results are needed to examine if the trends are Rey-nolds number dependent or qualitatively similar to the presentedhere. We suggest some of the procedures shown here to be amongthe standard test procedures in the decaying and other inhomoge-neous turbulent flows.

Acknowledgments

The authors wish to thank Arkady Tsinober for the invaluablecontribution of ideas, discussions and support at all stages of theresearch. RG is thankful to NSERC Discovery grant for the supportin this research.

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