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11th International Conference on Fluid Control, Measurements and VisualizationFLUCOME 2011 Paper No. 186

December 5-9, 2011, National Taiwan Ocean University, Keelung, Taiwan

1

SIMULTANEOUS MEASUREMENTS OF ENERGY AND TEMPERATURE DISSIPATION RATES IN

THE FAR FIELD OF A HEATED CYLINDER WAKE

N.M. SoumehsaraeiSchool of Mechanical and Chemical

EngineeringThe University of Western Australia

35 Stirling Highway,Crawley, WA,6009, Australia

Z. HaoCollege of Logistics Engineering,

Shanghai Maritime University,Shanghai

China 200135

T. ZhouSchool of Civil and Resource

EngineeringThe University of Western Australia

35 Stirling Highway,Crawley, WA, 6009,Australia

ABSTRACTUsing a probe consisting of four X hot wire probes and

two pairs of parallel cold wires, simultaneous measurements of

full energy and temperature dissipation rates, denoted as f

andf

, are obtained. The performance of the probe in

measuring velocity and temperature derivatives and vorticityhas been checked satisfactorily by comparing with isotropy

calculations. The differences between the present

measurements of the two dissipation rates and those obtainedby using isotropic relations are also discussed in terms of their

mean values and corresponding spectra. The correlations

between the two dissipation fields are also examined. The

present results show that the use ofiso and

iso underestimates

rr

ln,lnby about 20% in the scaling range. The large scatter of

rr

ln,lnin the literature indicates that this correlation

coefficient may depend on the magnitude of

R as well as the

nature or the initial conditions of the flows.

INTRODUCTIONThe turbulent energy dissipation rate )2(

ijijfss , the

temperature dissipation rate )(,, iif

k and the enstrophy

)(2

ii are important characteristics of the small-scale

properties of turbulence, where 2/)(,, ijjiij

uus is the rate of

strain; is the kinematic viscosity and ;/, jiji

xuu k is the

thermal diffusivity of the fluid, ii x /, and )3,2,1( ii is the vorticity component; subscript f denote the full

expressions of two dissipation rates. Measurements of eitherenergy or temperature dissipation rates have been made by a

number of researchers in different turbulent flows (e.g. [1-4]).

While it is fraught with difficulties to measure the above two

dissipation fields simultaneously, direct numerical simulations(DNS) complement experiments in some respects and provide

new information on both statistical and structural aspects of the

small-scale structures since they can resolve complete

information on three-dimensional vorticity and full energy and

temperature dissipation fields. The correlation of the two above

dissipation fields can therefore be analyzed using DNS results[5,6]. To our best knowledge, there are not many experimentalresults reported previously on the simultaneous measurements

of the full energy and temperature rates except the study by

Zhou et al [7] in a grid turbulent flow. The most common

method used to measure the two dissipation rates is employing

a single hot wire and single cold wire probes by invoking

Taylors hypothesis i.e.

1,1215 u

iso (1)

and2

1,3 k

iso (2)

However, significant differences of the above substitutions with

their full expressions have been reported (Zhou and Antonia

[8]; Hao et al. [9]).The first objective of this paper is to measure the full

energy and temperature dissipation rates simultaneously in acylinder wake by using a multiple hot and cold wire probe

(Figure 1), which contains eight hot wires and four cold wires

The performance of the probe will be assessed by comparingthe present results with those reported previously. On the basis

of these measurements, the correlations between the two

dissipation fields and the vorticity will be further examined

(second objective). These results will be compared with those

obtained preciously by using various approximations to the

dissipation rates so that appropriateness of theseapproximations can be assessed. Differences of the correlation

are compared when the full expressions for and or their

isotropic counterparts iso andiso are used.

EXPERIMENTAL SET UPThe experiments were conducted in a wind tunnel a

Nanyang Technological University with a cross-section of 1.2m

(width) 0.8m (height) and 2 m long. The free stream velocityU is about 3.6 m/s, corresponding to a Reynolds number Re

( /dU ) of 1520, or a Taylor microscale Reynolds number

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R of about 50, where mm35.6d is the diameter of the

brass cylinder. The measurement location is at 240/ dx . At

this location, the Kolmogorov length scale [ 4/13 )/( ]

is about 0.66 mm. The local turbulence intensity Uui /' is about

2%, favoring the use of Taylor's hypothesis (where a primedenotes rms value). A heating wire with diameter of about 0.5

mm was wrapped and inserted into a ceramic tube, which was

put inside the brass cylinder as a heating element. The

resistance of the heating wire is about 30. The wire washeated electrically with a power consumption of about 200W.The mean temperature increase T on the centerline of the

wake at the measurement location, relative to the ambient, is

about 1.0C. This value is small enough to avoid any buoyancy

effect and large enough to assure high signal-to-noise ratio.

A probe consisting of four X-probes and a pair of parallelcold wires (Figure 1) was used to approximate the energy and

temperature dissipation rates. The four X-probes also allow the

calculation of all components of the vorticity vector using the

measured velocity signals u1, u2,and u3. With the two pairs of

parallel cold wires (Fig.1), the three temperature derivatives

ixi(/ 1, 2, 3) can be measured simultaneously. The

included angle of the X-wire is about 100

. The separations x2andx3of the two X-probes in the opposite direction are 2mmand 2.7mm, respectively, and the separations x2c and x3c

between the opposite cold wires are about 2.5mm and 2.2 mm.

Figure 1: Sketches of the multi hot and cold wire probe.

(a) Side view; (b) Front view.

The hot and cold wires were etched from Wollaston (Pt-

10%Rh) wires. The active lengths are about 200dw and 800dw

for the hot and cold wires respectively (where dw is the

diameter of the wires and equals to 2.5 m for the hot wires and1.27 m for the cold wires). The hot wires were operated with

in-house constant temperature circuits at an overheat ratio of0.5. The cold wires were operated with constant current (0.1

mA) circuits. The probe was calibrated at the centerline of the

tunnel against a Pitot-static tube connected to a MKS Baratronpressure transducer. The yaw calibration was performed over

20C. The output signals from the anemometers were passedthrough buck and gain circuits and low-pass filtered at a

frequency fc of 800 Hz, which is close to the Kolmogorov

frequencyfK(U/2). The filtered signals were subsequently

sampled at a frequency fs = 2fc using a 16 bit A/D converter

The record duration was about 60 s.

RESULTS AND DISCUSSIONThe full experission for the mean dissipation rate

ijijss 2 can be written as

.22

2

222

2,33,21,33,1

1,22,1

2

2,3

2

3,2

2

1,3

2

3,1

2

1,2

2

2,1

2

3,3

2

2,2

2

1,1

uuuu

uuuu

uuuu

uuu

(3)

Except for the second and third terms all velocity derivatives on

the right hand side of Eq. (3) can be measured with the present

probe. The quantities 2,2u and 3,3u can, however, be estimated

by assuming incompressibility, viz.

1,1u + 2,2u + 3,3u = 0 (4)

or .42223,32,2

2

1,1

2

3,3

2

2,2 uuuuu (5)

Assuming homogeneity the last term on the right hand of Eq

(5) can be replaced by 2,33,2

4 uu since

2,33,2

uu = 3,32,2

uu . (6)

Substitution of Eqs. (6) and (5) into (3) then yields

.22

2

4

2,33,21,33,1

1,22,1

2

2,3

2

3,2

2

1,3

2

3,1

2

1,2

2

2,1

2

1,1

uuuu

uuuuu

uuuu

(7)

With the four X-wires the three vorticity components were also

obtained from the measured ,iu viz.

3

2

2

3

3,22,31 x

u

x

uuu

(8)

1

3

3

1

1,33,12x

u

x

uuu

(9)

2

1

1

2

2,11,23x

u

x

uuu

, (10)

where3

u and1

u in Eqs. (8) and (10) are velocity differences

between X-wires a and c, respectively (Fig. 1);2

u and1

u inEqs. (8) and (9) are velocity differences between X-wires b and

d, respectively. Derivatives in the x direction were estimated

using Taylors hypothesis, i.e. txU //1

. With the two

pairs of parallel cold wires (Fig. 1), the three temperatur

derivatives ixi (/ 1, 2, 3) can be measured. The ful

temperature dissipation rate can be obtained, viz.

.)/()/()/( 23

2

2

2

1 xxxk (12)

As for ,/1

xui

1/ x can be determined using Taylors

hypothesis.

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Basic performance checks of the probe

Measurements of and are not straightforward since the

velocity and temperature derivatives have to be resolved

adequately by using finite differences. It is therefore important

to ensure the performance of the probe before any further

analysis can be conducted. Fig.2 shows the distribution of the

root-mean-square values (indicated by a prime) of u across the

wake normalized by the maximum velocity deficit0

U and

wake half-width L. As it can be seen in Fig.2. the 'u values

from 4 different X-probes agree very well with each other and

also shows the same level of agreement with the results from

Hao et al. [ 9 ] measured using 2-X wire probes in the

streamwise direction ( results are not shown here).

Fig3. Shows the distributions of ' , normalized by the

temperature increase0

T at the centerline, across the wake

from the four cold wires. The differences of the rms values of

temperature fluctuations measured by two cold wires are small

enough to be regarded as the experimental uncertainty ( %4 ).The measured values of the vorticity and energy and

temperature dissipation rates have to be corrected due to the

effect of imperfect spatial resolution of the probe on

measurement of the velocity gradients. The spectra of thevelocity derivatives and the vorticity will be attenuated due tofine wire length and wire separation. This attenuation can be

corrected based on the local isotropy assumption. Details of the

correction procedures can be found in Zhu and Antonia [4]. Asa brief check on the performance of the probe, Figure 4 shows

the comparison of the measured vorticity spectra with

calculated based on isotropic assumption. For isotropic

turbulence, the spectra )(1k

i can be calculated from

1,1u (Antonia et al. [10]):

1

1,1

1,11

)(4)()(

11 k

u

u

caldk

k

kkk

(13)

and

1

11

111

)(

2)(

2

5)()( 1,1

1,132 k

kkkkk

u

u

calcal

dkk

kk

u

1

1,1

)(2

1

(14)

where1,1u

is the spectrum of the streamwise velocity

derivative. The agreement between measured and calculated

spectra is very good, especially for wavenumber larger than

0.05 except for )(11k

x over the wavenumber range of

0.010.05. In the figure, the asterisk * denotes normalization by

the Kolmogorov length scale , velocity scale /Ku and/ortemperature scale 2/1)/(

KKu ].

Fig.5 compares the rms values of1

,2

and3

obtained

in the present study with those previously for the wake flow.

These values are corrected for the spatial resolution of theprobe based on the corrected spectra shown in Figure 4 using

0

*

1

*

1

*2* )( dkki

c

i , where the superscript c represents

corrected. The rms values ofi

are normalized based on L

and U0. To account for the Reynolds number effect, Antonia e

al. [11] suggested that ReL ( /0LU ) should be included, i.e

0

'2/1'/)(Re UL

iLi

. The present rms values of the three

vorticity componentsc

i

' after correction for spatial resolution

are significantly smaller than those obtained by Antonia et al

[12] and Zhu and Antonia [4]. The present values of c

i

' agree

more favorably with those reported by Lfdahl et al. [13].

0

0.1

0.2

0.3

0 0.5 1.0 1.5 2.0 2.5

X-probe 1

X-probe 2

X-probe 3

X-probe 4

y/L

u'/U

0

Fig. 2. Rms values of velocity component fluctuationsmeasured using four X-probes across the wake.

0

0.1

0.2

0.3

0.4

0 0.5 1.0 1.5 2.0 2.5

cold wire 1

cold wire 2

cold wire 3

cold wire 4

y/L

'/T

0

Fig3.Rms valuses of temperature fluctuations across the wake

Fig4. Comparison of the measured vorticity spectra with their

calculated counterparts.

0.001

0.01

0.1

1

10

0.0001 0.001 0.01 0.1 1

*

z

(k1

*)

*

y

(k1

*)

*

x

(k1

*)

k1

*

*

x(k

1*),

* y

(

k1*

),

* z

(k1*

),

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0

0.04

0.08

0.12

0.16

0 0.5 1.0 1.5 2.0 2.5

Antonia et al. (1988)

Lofdahl et al. (1995)

Zhu and Antonia

Present

(a)

y/L

(ReL-1

/2)x'

L/U

0

0

0.04

0.08

0.12

0 0.5 1.0 1.5 2.0 2.5

Antonia et al. (1988)Zhu and Antonia (1999)

Lofdahl et al. (1995)

Present

(c)

y/L

(ReL-1

/2)z'

L/U

0

Figure 5. Distributions of rmsvalues of the three vorticitycomponents.

(a)x

; (b)y

; (c)z

.

Statistics of energy and temperature dissipation rates

The mean energy dissipation rates obtained using Eqs. (1) and

(7) are compared in Fig.6. All the velocity derivatives have been corrected according to procedures outlined by Zhu and

Antonia [4] to account for the effect of spatial resolution. Since

more derivative correlations are included in Eq. (7) than that in

Eq. (1), where isotropy is assumed, it is expected that the

former should provide more reasonable values for than when

iso(Eq. 1) is used. Figure 6 shows that the magnitude of

f

is about 10% larger than that ofiso over the range ofy/L 0.8, there agreement betweenf and

iso is

satisfactory, indicating that the one-dimensional surrogate can

represent the full mean energy dissipation rate properly. The

present values of f and iso are about 20~50% smallerthan those by Zhu and Antonia [4], Lofdahl et al [13] and

Browne et al. [14], which could be caused by the difference in

Reynolds numbers among different experiments.

The mean temperature dissipation rates obtained using Eqs.(2) and (12 ) are compared in Fig.7. The magnitude of

iso

and full

collapse across the whole wake, indicating

that the iso

is a good approximation to full

, at least in

terms of the mean values. This result seems in agreement with

that reported previously by Hao et al. [9] using other simplerapproximations.

0

0.01

0.02

0.03

0 0.5 1.0 1.5 2.0 2.5

Browne et al. (1988)

Lofdahl et al (1995)

Present (full

)

Present (iso

)

y/L

L/U

03

Figure 6. Distribution of energy

dissipation rates.

0

0.01

0.02

0 0.5 1.0 1.5 2.0 2.5

(iso)

(full)

y/L

L/(T2U

0)

Figure 7. Distribution oftemperature dissipation rates.

The results shown in Figures 6 and 7 seem to indicate that the

mean energy and temperature dissipation rates estimated using

the isotropy relations can represent the full dissipation rate

properly. However, this does not necessarily mean that they can

represent the full dissipation rates in other aspects. Fig. 8 (a,b)shows the spectra corresponding to the energy and temperature

dissipation rates, respectively, obtained on the wake centerline

using both the full expressions and the relations based on

isotropic assumption. These spectra, )(1k

f and )(

1k

f are

obtained by summing all the spectra of the velocity andtemperature derivatives involved in Eqs. (7) and (12)

respectively. Since the velocity derivative spectra are

normalized by the Kolmogorov scales, the areas under thevarious distributions of the energy dissipation rate should be

equal to 1. This is true both for )(1k

f and )(

1k

iso . The

distribution of )(1k

f shows significant departure from that of

)(1k

iso for 5.0

*

1k , even though the agreement between these

two distributions is reasonable for 5.0*1k . Since

)(15)( 1/1kk

xuiso , the spectrum for

iso is actually the same(multiplied by a factor of 15) as that of xu / while thespectrum of the full energy dissipation rate )(

1k

f is the sum

of the spectra of the velocity gradients involved in Eq. (7)

which are different from that of xu / . The departure o)(

1k

iso from that of )(

1k

f in Figure 8 is therefore no

surprising. This result indicates that the use of the instantaneous

values ofiso as a substitute of the instantaneous values of, as

usually used in experimental studies, may cause significanterrors, at least in the spectral domain. The satisfactory

agreement between )(1k

f and )(

1k

iso at high wavenumbers

indicates that local isotropy is satisfied by variousapproximations to energy dissipation rate, even at a relatively

low Reynolds number. The measured distribution )(1k

iso

departs significantly from )(1k

f [ )()()(

111 3,2,1,kkk

over the entire wavenumber range, indicating the

inappropriateness ofiso

as a substitute off

, at least in the

spectral domain.

10-2

10-1

100

101

10-4

10-3

10-2

10-1

100

*c

(full)

*c

(iso)

k*

1

*c

full

(k* 1),

*c

iso

(k* 1)

Figure 8. Comparison of the

spectra )(1k

f and )(

1k

iso

10-3

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

,iso

*c(k

1

*)

,full

*c(k

1

*)

k1

*

,iso

*c

(k1*),

,full

*c

(k1*

)

Fig. 9. Comparison of the spec

)(1k

f and )(

1k

iso

0

0.04

0.08

0.12

0 0.5 1.0 1.5 2.0 2.5

Lofdahl et al. (1995)

Antonia et al. (1988)

Zhu and Antonia (1999)

Present

(b)

y/L

(ReL-1

/2)y'

L/U

0

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To further examine the difference between the signals ofiso

andful or between

iso and

ful , the spectra of the above

quantities, denoted by )(1k

to differentiate with those shown

in Figures 8 and 9, which are obtained by summing or the

velocity (or temperature) gradients involved in their

expressions, calculated from their instantaneous signals areshown in Figure 10. These spectra are defined such that

1)(10 1

dkk

. The difference between )(

1k

iso and )(

1k

ful

or )( 1kiso and )( 1kful are apparent. It seems that there are

more contributions to )(1k

iso or )(

1k

iso than to )(

1k

ful or

)(1k

ful at high wavenumbers. This result suggests that the

isotropic substitutesiso and

iso are more related to small

scale structures thanful and

ful .

The correlation betweenr

ln andr

ln is important in

quantifying the behavior of the temperature structure function

n)( in the inertial range, perhaps in a more physical

sense, ascertaining the interdependence between the two

dissipative fields through their relation to the vortical structures

of the flow, where )()( 11 xrx is the temperaturestructure function, the subscript r means averaging over alength r. Van Atta [5 noted that

rr

ln,lnshould vary only

weakly with r across the inertial range. By using the present

probe, the energy and temperature dissipation rates can be

measured simultaneously and therefore, this correlation can bequantified. In the present study, an inertial range is not expected

due to the low

R (about 50). A scaling range instead, centered

on the value ofr* at which 1*3* )( ru is maximum can be

defined, where )()( xurxuu is the velocity increment in

the streamwise direction. The values ofrr

ln,ln

calculated

using either (ff

, ) or (isoiso

, ) on the wake centerline are

shown in Fig. 11. The magnitudes ofrr

ln,ln

obtained using

(ff

, ) and (isoiso

, ) increase with r in the scaling range,

with the former combination being about 20% larger than the

latter in the range 10030 * r . This finding suggests that forthe present flow, the isotropic substitutes (

isoiso , ) cannot

represent (ff

, ) properly. Comparing with other reported

data, the present magnitudes ofrr

ln,ln

are consistently larger

than that reported by Meneveau et al. [15] in a heated wake at

higher values of

R . This may suggest that the correlation

coefficientrr

ln,ln

decreases with the increase of

R in the

same flow, which is in contrast with the R dependence of

rr

ln,lnreported by the DNS data of Wang et al. [6] in a forced

box turbulence. The present measured values ofrr

ln,ln

differ

significantly from those for a heated turbulent jet reported by

Antonia and Van Atta [16] and the atmospheric surface layer

reported by Antonia and Chambers [17]. Meneveau et al. [15]

noted that the apparent discrepancy ofrr

ln,ln

may be due to

the influence of the hot wire on the cold wire signals. This

effect may be significant in a turbulent jet because of the high

turbulent intensity of the flow. However, this explanation maynot be adequate, especially when the separation between the ho

and cold wires is sufficient to avoid any interference from the

hot wire to the cold wire signals. The large scatter in Fig. 11

suggests that the magnitude ofrr

ln,ln

may depend on the

relations used to approximate and . A more likely

possibility is the effect of

R . Further, the dependence of

rr

ln,lnon the nature of the flow or the initial conditions

cannot be dismissed.

10-2

10-1

100

101

10-4

10-3

10-2

10-1

100

iso

(k1

*)

ful

(k1

*)

iso

(k1

*)

ful

(k1

*)

k1

*

(k1*

)

Figure 10. Spectra of the energy and temperature dissipation rates

calculated from the their instantaneous signals.

0.1

0.2

0.5

1

100

101

102

103

r*

ln

r,

lnr

Figure 11. Comparison of

rr

ln,lnmeasured on the wake

centerline with previous results. Present: , (ff

, ); +

(isoiso

, ). , Meneveau et al. [15]; , Antonia and Van Atta

[16];

, Antonia and Chambers [17]; , Wang et al. [6]. The

arrows indicate the scaling range.

CONCLUSIONS

A twelve-wire probe consisting of eight-hot wires and four

cold wires was used to measure the full energy and temperature

dissipation rates simultaneously in the present study. Theperformance of the probe in measuring the two dissipation rates

and also the three-component vorticity has been verified

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properly. While the isotropic approximations (isoiso

, ) can

represent the full expressions (ff

, ) properly in the mean

values, the spectral analysis shows that there is large departure

of the spectrum ofiso (or

iso ) from that of (or ) at low

wavenumbers, indicating that the use of the instantaneous

values ofiso (or iso ) as a substitute of the instantaneous values

off (or

f ), as usually used in experimental studies, may

cause significant errors, at least in the spectral domain. This is

supported by the spectra calculated from the instantaneoussignals of

iso (or

iso ) and

f ( or

f ). Previous experimental

results obtained usingiso and

iso yields rather disparate

values forrr

ln,ln

in different flows and at different

R . The

present results show that the use ofiso and

iso underestimates

rr

ln,lnby about 20% in the scaling range. The large scatter of

rr

ln,lnin the literature indicates that the correlation

coefficient may depend on the magnitude of

R as well as the

nature or the initial conditions of the flows.

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