The influence of Reynolds number on a plane jetRavinesh C. Deo,a� Jianchun Mi,b� and Graham J. NathanFluid Mechanics, Energy and Combustion Group, School of Mechanical Engineering,The University of Adelaide, South Australia 5005, Australia

�Received 31 January 2008; accepted 17 June 2008; published online 22 July 2008�

The present study systematically investigates through experiments the influence of Reynoldsnumber on a plane jet issuing from a radially contoured, rectangular slot nozzle of large aspect ratio.Detailed velocity measurements were performed for a jet exit Reynolds number spanning the range1500�Reh�16 500, where Reh�Ubh /� with Ub as the momentum-averaged exit mean velocity, has the slot height, and � as the kinematic viscosity. Additional centerline measurements were alsoperformed for jets from two different nozzles in the same facility to achieve Reh=57 500. Allmeasurements were conducted using single hot-wire anemometry to an axial distance �x� ofx�160h. These measurements revealed a significant dependence of the exit and the downstreamflows on Reh despite all exit velocity profiles closely approximating a “top-hat” shape. The effect ofReh on both the mean and turbulent fields is substantial for Reh�10 000 but becomes weakerwith increasing Reh. The length of the jet’s potential core, initial primary-vortex shedding frequency,and far-field rates of decay and spread all depend on Reh. The local Reynolds number,Rey0.5�2Ucy0.5 /�, where Uc and y0.5 are the local centerline velocity and half-width, respectively,are found to scale as Rey0.5�x1/2. It is also shown that, for Reh�1500, self-preserving relationsof both the turbulence dissipation rate ��� and smallest scale ���, i.e., ��Reh

3�x /h�−5/2 and��Reh

−3/4�x /h�5/8, become valid for x /h�20. © 2008 American Institute of Physics.�DOI: 10.1063/1.2959171�

I. INTRODUCTION

Following the early investigation of Schlichting,1 planejets have received significant attention from the turbulence

research community.2–10 A practical plane jet is produced bya rectangular slot of dimensions w�h, with w�h �where wand h are the slot’s long and short sides� bounded by twoparallel plates, known as sidewalls, attached to the slot’sshort sides. The presence of sidewalls restrict the jet fromdeveloping in the direction normal to the short sides, so thatthe flow develops statistically two dimensionally up to a cer-tain downstream distance, provided that the nozzle aspectratio, AR��w /h�, is sufficiently large.11 The primary objec-tive of the present work is to investigate the effect ofReynolds numbers on the downstream development of aplane jet.

Studies undertaken over many years12–16 have shownthat the downstream development of a jet depends on theflow at the jet origin, often termed initial conditions. Theanalytical work of George12 and experimental assessmentson round and plane jets �e.g., Refs. 13–16� pointed out thattheir asymptotic states depend on the initial conditions. Theinitial conditions of a jet are conventionally defined to be theexit Reynolds number �Reh�, nature of exit lateral profiles of

mean velocity and turbulence intensity, aspect ratio �AR��noncircular jets�, nozzle-exit geometric profiles, and globaldensity ratio of the jet fluid to ambient fluid. The down-stream development of a jet is also dependent on the bound-ary conditions, e.g., the presence or absence of sidewalls fora plane jet. In our previous work on a plane jet, we havecharacterized the effects of AR,11 nozzle geometric profile,16

and the sidewalls.17 Some insight into the effects of Reh on aplane jet can be gleaned from the limited investigations re-viewed below. However, no detailed assessment of this in-fluence in both the near and far fields is available. Thepresent work aims to fulfill this need.

The influence of Reynolds number on a jet’s develop-ment is challenging to understand. This is partly because theReynolds number affects the development of the boundarylayer through the nozzle, making it difficult to independentlyassess jet development and exit boundary layer thickness.Zaman18 found that the initial turbulence intensity andboundary layer growth of a round jet are functions of jet exitReynolds number �Red�. His work showed that round jets areinitially laminar when measured at Red�1.0�105 but be-come fully turbulent when Red5.0�105. Hence Reynoldsnumbers exerts both a direct influence on the turbulent mix-ing field by controlling the significance of viscosity and anindirect influence by modifying the initial conditions �e.g.,boundary layer thickness�.

It is well known that at sufficiently high Reynolds num-ber, the jet decay and spread rates of round jets are almostindependent of Re.19,20 However, there is some variation inthe reported values of jet exit Reynolds number at which thisoccurs, suggesting that it depends on initial conditions. In a

a�Present address: The Center for Remote Sensing and Spatial InformationSciences, School of Geographical Sciences and Planning, The Universityof Queensland, Brisbane 4072, Australia.

b�Author to whom correspondence should be addressed. Also at Departmentof Energy and Resource Engineering, College of Engineering, Peking Uni-versity, Beijing 100871, China. Electronic mail: jcmi@coe.pku.edu.cn.Telephone: 86 10 6276 7074.

PHYSICS OF FLUIDS 20, 075108 �2008�

1070-6631/2008/20�7�/075108/16/$23.00 © 2008 American Institute of Physics20, 075108-1

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

related review for round jets, Dimotakis21 concluded that afully developed turbulent field occurs when Red�104. Theexperimental work of Dimotakis et al.,22 which assessed thescalar-mixing fields of round jets at Red=2.5�103 and 104,found the asymptotic state to occur for Red=104. Miller andDimotakis23 reported that the scalar fluctuations decreasewith an increase in Red and converge to an asymptotic statefor Red�2.0�104. Likewise, Gilbrech24 found that theasymptotic state of the mixing field occurs near Red�2.0�104. Koochesfahani and Dimotakis25 used scalar imagesof laser-induced fluorescence at Red=1.75�103 and2.30�104 to show more qualitative and well-mixed state atRed=2.30�104 relative to Red=1.75�103. Similar resultswere found by Michalke,26 who found that the growth ofinstability waves in jet shear layers can be reduced dramati-cally when the Reynolds number is increased sufficiently.

Oosthuizen27 measured the flow of a round jet at lowReynolds number and found that both the mean and turbu-lence fields depend strongly on Red for Red�104. Bogey andBailly28 used large eddy simulations of a transitional roundjet �1.70�103�Red�4.0�105� and showed that as Red de-creases, a jet develops more slowly within the potential core,but more rapidly further downstream. Their round jetsachieved self-similarity closer to the exit plane at low valuesof Reh, a finding, which agrees well with Pitts29 for turbulentaxisymmetric jets measured between 3950�Red�11 880.Moreover, Mi et al.14 found that the scalar decay rate in theself-similar far field depends strongly on Red when the jetissues from a smooth contraction, but this dependence be-comes much weaker and may be negligible for jets originat-ing from a long pipe, in which the flow is fully turbulent.

Less detail is available for the transition to an asymptoticstate for a plane jet. The investigation by Lemieux andOosthuizen30 over the range 700�Reh�4200 found a sig-nificant influence of Reh on the shear stress, jet decay,spreading rates, and mean velocity profiles. While their studyconcluded that jet properties become invariant of Reh forReh4200, other investigations have found a different Reh

threshold for this invariance. Namar and Ötügen,31 who mea-sured a high-AR rectangular jet over the range 1000�Reh

�7000, showed that Reh has significant influence on themixing field for Reh�7000. Likewise, Everitt andRobbins,32 whose measurements for a plane jet spanned1.6�104�Reh�7.5�104, reported a large spread in veloc-ity decay rates. However, it is not possible to separate theeffects of Reh from AR in their investigation since this wasalso varied between AR=21 and AR=128.11 Klein et al.33

conducted a direct numerical simulation �DNS� investigationof a plane jet in the near and transition fields �i.e., forx /h�20� but for lower ranges of Reynolds numbers,Reh�6000. It is evident that none of these investigationsreached the range Reh104 identified by Dimotakis21 for around jet.

Further insight into the Re dependence of a plane jet issought by assembling data from previous investigations withclosely matched initial conditions. A careful examination,however, reveals significant inconsistencies even for compa-rable Reh and initial conditions. Figure 1 displays the center-line evolutions of locally normalized turbulence intensity,

uc� /Uc, reported in literature. Here, uc� and Uc are the fluctu-ating and mean components of the centerline velocity.Among these data sets, the closest match is between those ofGutmark and Wygnanski10 and Heskestad,7 who measured aplane jet at similar Reynolds numbers but different AR andnozzle contraction profiles, so that they are not directly com-parable. The two data sets agree closely within the midfield,i.e., x /h�60, but exhibit some differences in the far field�for x /h60�. Likewise, the results from smoothly contract-ing nozzles of Gutmark and Wygnanski10 and Bradbury8 forrelatively high nozzle aspect ratios �AR=39 and 40� mea-sured at identical Reynolds numbers �Reh=30 000� differsignificantly. These differences can probably be attributableto the use of a coflow by Bradbury8 �for a coflow to jetvelocity ratio of �16%�. The work of Browne et al.34 andThomas and Goldschmidt35 �Reh=7700 and 6000� reportedmeasurements at different ARs �20 and 44�, consistent withthem exhibiting significant near-field differences. Thomasand Goldschmidt35 and Namar and Ötügen31 used similarvalues of AR and Reh, except that the latter did not employsidewalls. Likewise, the results of Hussain and Clark,36 whostudied plane jets for Reh=32 500 and 81 400, show the low-est midfield turbulence intensity despite this value beingsimilar to that of Browne et al.34 of Reh�7700. Also impor-tantly, their midfield turbulence intensity is significantlylower than those of Namar and Ötügen31 who measuredtheir jet at Reh=7000. In addition to the differences in initialconditions reported, other possible differences from labora-tory to laboratory include surface finish and measurementuncertainties.

Thus the wide range of different operating/experimentalconditions used in previous studies makes it impossible toisolate the effect of Reynolds number from those of othervariables. It is also clear that systematic measurements of theinfluence of Reh on a plane jet for Reh7500 are presentlynot available. We seek to fill these gaps by systematicallymeasuring �i� the flow field of a plane jet for Reh varying

0

0.1

0.2

0.3

0.4

0 40 80 120 160

[36] 44 32,550 No SC[36] 44 81,400 No SC[31] 56 1,000 No SC[35] 48 6,000 Yes SC[34] 20 7,700 Yes SC[10] 39 30,000 Yes SC[7] 120 34,000 Yes OP[8] 40 30,000 Yes SC

Ref. AR Reh sidewalls nozzle profile

x/h

u'c/

Uc

FIG. 1. �Color online� The streamwise evolution of the centerline turbulenceintensity �uc� /Uc� of previous investigations of a turbulent plane jet. Thelegend lists the nozzle AR, jet exit Reynolds number �Reh�, presence orabsence of sidewalls in the x-y plane, and nozzle-exit geometric profiles�SC=smooth contraction; OP=orifice plate�.

075108-2 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

between 1500 and 16 500, issuing from a radially contouredslot of AR=60 in detail, and �ii� the centerline velocity decayrate for Reh varying between 4500 and 57 500, issuing froma radially contoured slot of AR=36 in the same flow facility.Our specific aim is to examine the dependence on Reh of the

mean velocity, the root mean square �rms�, and high-ordermoments of the velocity fluctuation over a greater down-stream distance �0�x /h�160� than has previously been re-ported. We also aim to relate this dependence to the under-lying structures in the flow.

FIG. 2. �Color online� A schematic of the present experimental setup, showing �a� the wind tunnel details and nozzle attachment, �b� the side view showingnozzle parameters and sidewalls, �c� the jet development characteristics, and �d� the measurement apparatus. Note that diagrams drawn are not to scale.

075108-3 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

II. EXPERIMENT DETAILS

A schematic of the present plane jet facility is shown inFig. 2, described in more detail by Deo.15 Briefly, the facilityconsists of an open circuit wind tunnel made from woodenmodules with polished inner surfaces. The tunnel is mountedto the roof and driven by a 14.5 kW aerofoil-type centrifugalfan mounted firmly to the floor to minimize vibrations. Thetunnel employs a 25° wide-angle diffuser of 2100 mm inlength, which expands into a settling chamber. Inside thediffuser are two screens with an open area ratio of �60%mounted at equal intervals of 600 mm. The screens assist inreducing the velocity defect in the turbulent boundary layer.The settling chamber is 1400 mm long and contains a hon-eycomb of drinking straws aligned horizontally with themain flow to reduce swirl. Following the honeycomb arethree screens, each separated by 150 mm. There is followedby a 600 mm long settling chamber prior to the contractionto reduce turbulence intensity. To further enhance flow uni-formity, the wind tunnel contraction has an area ratio of 6:1and employs a smooth contraction profile based on a thirdorder polynomial curve.

A plane nozzle with 12 mm radial contraction profile onits long sides �Fig. 2�a�� was mounted to the tunnel exit. Theparallel sidewalls were flushed with the short sides of the slotand aligned along the x-y plane �Fig. 2�b��. The sidewallsextended 2000 mm downstream and 1800 mm vertically andwere secured tightly by bolts to the ceiling to avoid vibra-tions. The slot height �h�, aligned with the lateral direction�y-plane� of the nozzle, was fixed at 5.60 mm. The slot span�w�, aligned along the z-direction, was 340 mm to provide alarge aspect ratio nozzle of AR=w /h=60. The fan pressurelimited operation to Reh�16 500 for this nozzle. Hence, twoother nozzles of AR=36 were designed, comprised of�h ,w�= �10 360� and �h ,w�= �20 720� mm, respectively,providing exit areas of 3600 and 14 400 mm2, respectively.The latter allowed the maximum value of Reh of 57 500 to beachieved. For all the cases, the exit radius of curvature isr /h�1.80. This, according to our previous findings,15,16 en-sures that the exit mean velocity profile of the jet is approxi-mately uniform.

The laboratory, acoustically isolated from external noise,is 18 000 mm long, 7000 mm wide, and 2500 mm high. Thedistance from the jet exit to the front wall of the room was�1400h and that between the jet and the ceiling/floor was�125h. The plane jet was located horizontally at about themidpoint between the floor and ceiling. The ratio of the roomheight to the nozzle width was �446. Likewise, the ratio ofthe cross-sectional room area �in the same plane as thenozzle area� to the actual nozzle area was �10 000. Underthese conditions, we used a similar model to that proposedby Hussein et al.20 to assess the loss in streamwise momen-tum of the jet at different Reh. It was found that, even atx /h=160 �the maximum downstream distance�, the presentjet sustains 99.2% and 99.5% of its initial streamwise mo-mentum for Reh=1500 and 16 500, respectively. Thisconfirms that the present jet closely resembles a truly uncon-fined jet.

Reh was controlled by adjusting the speed of thewind tunnel fan to provide a momentum-averaged mean dis-charge velocity Ub varying between 3.98 and 44.20 m s−1

and achieve Reh=1500–16 500 for AR=60. For AR=36,w�h=3600, jets with Ub between 3 and 38.5 m s−1 weremeasured, corresponding to 2000�Reh�26 000, and forAR=36, w�h=14 400, jets with 3.37�Ub�42.4 m s−1

were measured, corresponding to 4500�Reh�57 500.The measurements were performed over the flow

region 0�x /h�160 using a single hot-wire anemometer,under isothermal conditions of ambient temperature20.0 °C�0.1 °C. A custom-built �tungsten� hot-wire sensor�diameter dw=5 �m and length lw=0.8 mm� was used, withthe overheat ratio of 1.5. The square wave test revealed thatthe maximum response frequency of the wire was 15 kHz. Toavoid aerodynamic interference of the prongs on the hotwire, the present probe was carefully mounted, with prongsparallel to the plane jet. This alignment is appropriate formeasurement of the streamwise component of the flow ve-locity, although it is not possible to eliminate directional am-biguity completely. The single wire probe, if used with cau-tion, encounters reduced errors when compared with dual ortriple wires because the adjacent probe can probably influ-ence the measured velocity for the later cases. On the cen-terline of a plane jet, the cross-steam velocity is very small�must be zero on average� so that the uncertainty due todirectional ambiguity is insignificant.

Calibrations of the hot-wire were conducted using astandard Pitot static tube, connected to hand-held digital ma-nometer, located side by side with the hot-wire probe at jetexit plane �where �u21/2 /Uc�0.5%� prior to and after eachset of measurements. Data points were converted from volt-ages to velocities using a fourth order polynomial curve.Both calibration functions were checked for discrepancies,and the experiment was repeated if the velocity drift ex-ceeded 0.5%. The average accuracy of each calibration func-tion was found to be within �0.2%. Velocity signals ob-tained were low pass filtered with an identical cutofffrequency of fc=9.2 kHz at all measurement locations toeliminate excessively high-frequency noise. Then the voltagesignals were offset to within 0–3 V as a precautionary mea-sure to avoid signal clipping37 and amplified appropriatelythrough the circuits. They were digitized on a personal com-puter at fs=18.4 kHz via a 16 channel, 12 bit PC-30F A/Dconverter with signal input range of 0–5 V. The samplingperiod was approximately 22 s, during which about 4�105

data points were collected at each measured location.Using the inaccuracies in calibration and observed scat-

ter in present measurements, the uncertainties are estimatedto have a mean error of �4% at the outer edge of the jet and�0.8% on the centerline. The errors in the centerline meanvelocity, rms velocity, the skewness, and the flatness werefound to be approximately �0.8%, �1.8%, �2%, and�1.5%, respectively. The errors in the momentum integralquantities and jet virtual origins are estimated to be �3%.

075108-4 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

III. RESULTS AND DISCUSSION

A. Characterization of the exit conditions

The exit conditions were characterized by measuring thevelocity profiles in each plane jet at x /h=0.5 along the lat-eral �y� direction over the range −0.70�y /h�0.70 �Figs.3�a� and 3�b��. Dependence of the exit flow on Reh is evi-dent. All cases produce an approximately top-hat mean exitvelocity profile. However, the extent of uniformity in theprofiles differs significantly. As Reh is increased from 1500to 16 500, the exit velocity profiles become flatter, and thehorizontal region of uniformity widens. A consistent trend inthe initial turbulence intensity u� is evident from Fig. 3�b�.The peak value of u� /Ub decreases with Reh while the cen-terline value remains almost constant at about 1.0%. Also,the mean exit velocity profiles for Reh=10 000 and 16 500are nearly indistinguishable, although some differences areevident in the peak values of u� /Ub within the shear layers.

The initial boundary layer characteristics may beestimated from the measurements at x /h=0.50 �the profilesclosest to the nozzle�. The displacement thickness � 0.5�,the momentum thickness ��0.5�, and the shape factor�H0.5= 0.5 /�0.5� were calculated approximately from thesemean velocity profiles and the momentum integral equations,viz.,

0.5 = 0

�

�1 − U/Ub�x=0.5hdy �1�

and

�0.5 = 0

�

�U/Uo,c��1 − U/Ub�x=0.5hdy . �2�

Table I presents the averages of the above parameterscalculated independently on each side of the profile using“best-fit” spline curves and numerically integrating Eqs. �1�and �2�. Although the measurement resolution is somewhatcoarse, clear trends are evident. As Reh is increased from1500 to 16 500, both 0.5 and �0.5 decrease from approxi-mately 0.133h to 0.097h and 0.068h to 0.039h, respectively.This agrees qualitatively with other measurements for around jet18,38 and also that for a high-AR rectangular jet.31

B. The mean velocity field

Figure 4 presents the axial variation of centerline meanvelocity �Uc� normalized by the bulk mean velocity at theexit �Ub� for 1500�Reh�16 500 in log-log form. First weconsider the near field. It is evident that the length of thepotential core, xp, in which Uc is constant, exhibits a Reh

dependence. This can be seen more clearly in Fig. 5, whichpresents the dependence of xp, defined to be the axial loca-tion at which Uc�x��0.98Uo,c on Reh. This demonstratesthat xp decreases asymptotically with Reh, consistent withprevious results for a round jet39 and for a high-AR rectan-gular jet.31 This implies that increasing Reh will increase jetentrainment in the near field. The comparison in Fig. 6 of thelateral profiles of Uc at x /h=5 for the different Reh lends

0

0.02

0.04

0.06

0.08

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

(b)

y/h

u'/U

b

0

0.2

0.4

0.6

0.8

1.0

1.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6

Reh = 150030007000

1000016500

U/U

b

(a)

FIG. 3. �Color online� Lateral profiles of �a� the normalized mean velocityand �b� the turbulence intensity measured x /h=0.5 for plane jets with dif-ferent Reh values.

TABLE I. A summary of the normalized boundary layer characteristicsestimated from mean velocity profiles at x /h=0.5 for different Reh.

Reh Displacementthickness 0.5

Momentumthickness �0.5

Shape factorH0.5

1500 0.133h 0.068h 1.95

3000 0.116h 0.056h 2.09

7000 0.109h 0.045h 2.43

10 000 0.098h 0.040h 2.45

16 500 0.097h 0.039h 2.49

0.1

0.2

0.5

1

1 10 100

Reh= 150030007000

1000016500

x-1/2

Reh

x/h

Uc/

Ub

FIG. 4. �Color online� The evolution of mean centerline velocity �Uc /Ub�for a plane jet with different Reh values.

075108-5 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

support to this deduction. Evidently the jet spread in the nearfield increases with Reh. From Table I, this near-field in-crease is associated with a decrease in the exit boundarylayer thickness.

Next we return to Fig. 4 to assess the influence of Reh onthe far field using the well-known similarity relationship fora plane jet,

�Ub/Uc�2 = Ku�x/h + x01/h��3�

or Uc/Ub = �Ku�x/h + x01/h��−1/2.

This becomes valid from the transition region following thepotential core. Here, the constant Ku is the slope of �Ub /Uc�2,which is a measure of the decay rate of Uc, and x01 is thex-location of the virtual origin of �Ub /Uc�2. Although thepresent data for AR=60 appear to converge asymptotically,the asymptotic value of convergence cannot be determinedfrom these data because it exceeds 16 500. However, itis evident from the jets from the other two nozzles,w /h=360 /10 and 720/20, that Ku converges asymptoticallyfor each jet at Reh�25 000 �Fig. 7�. However, the jets fromthe different nozzles do not converge onto the same line dueto their different inflow boundary conditions. Figure 7 alsoreveals a consistent trend that the decay rate of plane jet

decreases as Reh is increased. The dependence of x01 on Reh

follows a similar trend to that of Ku, although with greaterscatter �as is typical for this measurement�. Once again, thedata do not collapse onto a single curve due to the differ-ences in initial conditions.

Figure 8 presents the lateral profiles of U /Uc at selecteddownstream locations for the AR=60 jet. Here, they-coordinate is normalized using the velocity half-width y0.5,determined from the location at which U�x ,y�= 1

2Uc�x�. As

0

p

4

6

8

0 5000 10000 15000 20000

Reh

x p/h

FIG. 5. �Color online� The dependence of the jet’s potential core lengths�xp� on jet exit Reynolds number �Reh�.

0

0.2

p

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0

3,0007,00010,000

Sym. Reh

y/h

U/U

c

FIG. 6. �Color online� The dependence of lateral profiles of U /Uc on Reh

measured at x /h=3.

0

0.05

0.10

0.15

0.20

0.25

0.30

0 1x104 2x104 3x104 4x104 5x104 6x104

-2

0

2

4

6

8

10

AR = 36, w × h = 3600AR = 36, w× h = 14400AR = 60, w× h = 14400

x01

Ku

Reh

Ku

x 01/h

FIG. 7. �Color online� Dependence on Reh of Ku �filled symbols� and x01 /h�open symbols� for AR=60 �r /h=2.14� and AR=36 �r /h=1.80� for twovalues of w and h.

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

(c)

yn= y/y0.5

U/U

c

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

(b)

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

x/h = 235102050100

(d)

yn= y/y0.5

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

~ exp [-ln2 (yn)2]

Gaussian

U/U

c

(a)

0

0.2

0.4

0.6

0.8

1.0

0 0.5 1.0 1.5 2.0 2.5 3.0

(e)

yn= y/y0.5

U/U

c

FIG. 8. �Color online� Lateral distributions of U /Uc for AR=60: �a� Reh

=1500, �b� Reh=3000, �c� Reh=7000, �d� Reh=10 000, and �e� Reh

=16 500.

075108-6 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

expected, the velocity profiles become self-similar at ashorter downstream distance when the Reynolds number ishigher. For instance, self-similarity is attained at x /h5 forReh=16 500 and at x /h20 for Reh=1500. The self-similarvelocity profiles conform closely to the Gaussian distributionUn=e−ln 2�yn�2

, where Un=U /Uc and yn=y /y0.5.The influence of Reh on the mean spreading rate of the

AR=60 jet is assessed from the half-width y0.5. Figure 9�a�shows that the half-widths vary linearly with x for x /h10for all the present Reynolds numbers. This linear variationcan be represented by

y0.5/h = Ky�x/h + x02/h� , �4�

where the slope Ky is a nominal measure of the jet spreadingrate and x02 is the virtual origin of jet spread. Figure 9�b�demonstrates that as Reh increases from 1500 to 16 500, Ky

decreases asymptotically from about 0.14 to 0.09, while x02

increases from about −3 to 0. This trend of Ky is consistentwith that of Ku �Fig. 5�. Taking them together, we concludethat an increase in Reh leads to the reduced entrainment ofthe jet in the far field. This is opposite to the near-field trend�see Fig. 6�, as will be discussed below.

Next we consider the local Reynolds number defined byRey0.5�2Uc�x�y0.5�x� /�. Figure 10�a� shows the streamwisevariation of Rey0.5 for different values of Reh. It is clear that

Rey0.5 increases with axial distance for each value of Reh.This occurs because the far-field decay rate of Uc is lowerthan the growth rate of y0.5, as is evident from Eqs. �3� and�4�. Combining Eqs. �3� and �4�, the local Reynolds numberin the self-similar region should vary as

Rey0.5= 2 RehKyKu

−1/2��x − xo�/h�1/2. �5�

To check relation �5�, we plot in Fig. 10�b� the results ofP=0.25�Rey0.5

/Reh�2KuKy−2 versus x /h for Reh=1500, 7000,

and 16 500. Good collapse of the data is evident, within ex-perimental uncertainties. That is, Eq. �5� is approximatelyvalid with xo�0. Note that Eq. �5� appears to work even inthe very near field where Eqs. �3� and �4� are not valid. Wecan thus conclude that Rey0.5 increases with the downstreamevolution of a plane jet and must eventually become veryhigh no matter how low its initial value is. This is in contrastto the case for the axisymmetric jet or plane wake where thelocal Reynolds number remains constant in the far field. Alsoof interest, the opposite occurs in the axisymmetric wake,where the local Reynolds number decreases with increasingx.40 Note that this increase in Rey0.5 with x /h does not meanthat plane jets will eventually converge to the sameasymptotic state with axial distance since the dependence offar-field state on Reh has already been described above. It isalso interesting to note that at x /h=160, Rey0.5 reaches theorder of 105 for Reh=16 500 but only of 104 for Reh=1500.

C. The fluctuating velocity field

Figure 11 presents the centerline evolution of the turbu-lence intensity �uc

�=uc� /Uc� for different values of Reh. Evi-dently, as x is increased, uc

� grows rapidly from the exit valueof about 1% to the local maximum �uc,max

� �, then decreases togradually converge to an asymptotic value �uc,�

� � in the farfield. However, the magnitudes and axial locations �xmax

� ,x�� �

of both uc,max� and uc,�

� are dependent on Reh. Several obser-vations can be made. First, as Reh is increased from 1500 to16 500, both uc,max

� and xmax� decrease together from 0.31 to

0.22 and from 18 to 12, respectively �see Fig. 12�. However,uc,max

� and uc,�� exhibit opposite dependence on Reh, so that

the lowest Reh exhibits the highest uc,max� and the lowest uc,�

� .This results in the near-field peak being overwhelmingly

0

3

6

9

12

15

18

0 20 40 60 80 100

Reh = 150030007000

1000016500

(a)

x/h

y 0.5/

h

0

0.05

0.10

0.15

0.20

0 5000 10000 15000 20000-3

-2

-1

0

1

2

3

4

(b)

Reh

Ky

x 02/h

FIG. 9. �Color online� �a� Streamwise evolution of y0.5 /h for Reh

=1500–16 500. �b� Dependence of the spreading rate �Ky� and virtual origin�x02 /h� on Reh.

0

50

100

150

200

0 40 80 120 16

P = x/h

7000

16500

Reh= 1500

(b)

x/h

P=

0.25

(Re y 0.

5/Re h)

2 KuK

y-2

0

0.2x105

0.4x105

0.6x105

0.8x105

1.0x105

0 40 80 120 160

(a)

10000

7000

3000

1500

Reh= 16500

x/h

Re y 0.

5≡2

Uc

y 0.5

/υ

FIG. 10. �Color online� Streamwise evolution of �a� the local Reynoldsnumber �Rey0.5� for Reh=1500–16 500 and �b� the data scaled according toEq. �5�.

0

0.05

0.10

0.15

0.20

0.25

0.30

0 60 120 180

Reh= 150030007000

1000016500

x/h

u'c/

Uc

FIG. 11. �Color online� Streamwise evolution of u� /Uc for Reh

=1500–16 500.

075108-7 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

dominant for low Reh, while for high Reh the near-field peakis almost nonexistent and the far-field “peak” is dominant.Finally, there is general consistency in the asymptotic valuesof the present data and those presented earlier �Fig. 7�, withboth uc,�

� and x�� becoming independent of Reh at about

Reh�25 000 �Fig. 13�. However, there is some scatter, withuc,�

� reaching the asymptotic state at Reh�40 000.The increase of present uc,�

� with Reh found here standsin contrast to the observation of Namar and Ötügen31 that anincrease in Reh led to a decrease in uc,�

� . The probable expla-nation for this apparent discrepancy is that their study did notemploy sidewalls, so that their jet exhibits a different behav-ior, such as axis switching. Our previous investigation17 hasshown that although a high-AR-rectangular nozzle withoutsidewalls produces a statistically two-dimensional plane-jet-

like region, the overall magnitudes of flow statistics are sig-nificantly different from those in the same rectangular nozzlewith sidewalls.

Figures 14 and 15 display the centerline evolutionsof the skewness Su���u3 / �u23/2� and flatness Fu���u4 /�u22� at the different values of Reh. These provide measuresof the symmetry and flatness, respectively, of the probabilitydensity function �PDF� of u. Both Su and Fu were estimatedfrom a large data sample ��400 000 data points� to achievethe convergence of the PDF. During measurements a voltageoffset was applied to the analog-to-digital range of the A/Dboard to ensure that neither factors were truncated due toclipping of the tails of the PDFs, as can occur for a finiteinput range of the sampling board. Also note that a higherscattering of the far-field data of Su for Reh=1500 is due toits relatively low velocity ��1 m/s for x /h70� and thus a

0.20

0.24

p8

0.32

0.36

0 10000 20000 30000 40000

0

5

10

15

20

AR = 60, w× h = 14400AR = 36, w × h = 3600AR = 36, w× h = 14400

Reh

u*c,

max

x* max

FIG. 12. �Color online� Dependence on Reh of uc,max� and its axial location�xmax

� �.

0.15

0.20

0.25

0.30

0 0.6x104 1.2x104 1.8x104

40

80

120

Reh

u*c,

∞

(a) AR = 60

0.15

0.20

0.25

0.30

0 2x104 4x104 6x104

15

20

25

30

35w × h = 3600w × h = 3600w × h = 14400w × h = 14400

Reh

x*∞

(b) AR = 36

FIG. 13. �Color online� Reh dependence of the asymptotic value of uc,�* and the axial location �x

�*� at which the turbulence intensity asymptotes for the cases

�a� AR=60 and �b� AR=36. Note that for AR=36, two nozzles of different exit areas �w�h=3600 and 14 400� were tested.

1 10 100

16500

10000

7000

3000

Reh=1500

0

0

0

1

0

0

-1

Gaussian

x/h

S u=<u

3 >/(

<u2 >

)3/2

FIG. 14. �Color online� Streamwise evolutions of Su for Reh

=1500–16 500. Symbols are identical to Fig. 11.

075108-8 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

high uncertainty for this case; the estimation of Fu is moreaccurate than that of Su since Fu� �Su�, and generally Su issmall.

Both factors vary dramatically in the near-field region�x /h�30�, presumably owing to the dominance of large-scale coherent motions in the near-field region. Movingdownstream from the exit, both Su and Fu increase fromnearly Gaussian values of �0, 3� to highly non-Gaussian localmaxima in the vicinity of the end of the potential core. Thislocal maximum in Su occurs between 4�x /h�12 �indicatedby arrows on the plot� while that for Fu occurs around5�x /h�10 �Fig. 15�. They then decrease to a local mini-mum before increasing gradually to their asymptotic valuesby 30�x /h�40. In general, as Reh increases, the near-fieldmaxima increase, and their x-locations translate upstream.The latter is consistent with the shortening of xp �Fig. 5�,increased near-field entrainment �Fig. 6�, and a reduction inxmax �Fig. 12�. More coherent motions in the shear layers areevidenced by less random fluctuations, i.e., by more depar-ture of the PDF from the Gaussian distribution and of Su andFu from the values of 0 and 3, respectively. In addition, Figs.14 and 15 show that Su and Fu approach their respectiveasymptotic values, Su

c,� and Fuc,�, at x /h�40 nearly for all

values of Reh. However, both Suc,� and Fu

c,� exhibit a consis-tent, albeit weak dependence on Reh. Overall, as Reh is in-creased, Su

c,� decreases and Fuc,� increases �Fig. 16�.

Next we assess the effect of Reh on the small-scale tur-bulence by examining the centerline dissipation rate ��� ofthe turbulent kinetic energy at different values of Reh. Forhigh-Re flows, it is usually considered that the dissipation ofturbulent kinetic energy by the smallest-scale structures isequal to the supply rate of the turbulence energy from thelarge-scale structures, which is of order Uo

3 /Lo, where Uo andLo are the local characteristic velocity and length scales �see,e.g., Ref. 41�. Based on this argument, we obtain the center-line dissipation ��Uc

3 /y0.5 by taking Uo=Uc and Lo=y0.5 fora plane jet. It follows from Eqs. �3� and �4� that self-preservation of the flow further requires

��hUb−3� = C��x/h�−5/2, �6�

and thus the smallest scale or Kolmogorov scale, defined by����3 /��1/4, to follow

�/h = C�−1/4 Reh

−3/4�x/h�5/8, �7�

where C� is determined by experiments. It is evidentfrom Eq. �7� that the smallest scale � decreases with in-creasing Reh. Here we check the above relations and espe-cially the effect of Reh on C�. To estimate � from thetime derivative of uc, as in literature, both the isotropicassumption �=15����uc /�x�2 and Taylor’s hypothesis�uc /�x=Uc

−1�uc /�t are invoked. Figure 16 illustrates thepresent results, together with that of Antonia et al.42 forReh=22 000. As is well known, accurate dissipation mea-surements require full spatial and temporal resolutions of thesmallest-scale fluctuation in velocity.43,44 Estimation of theKolmogorov frequency fK �corresponding to the smallest-scale fluctuation� based on the iterative scheme developed byMi et al.45 suggests that the present measurements �low-passcutoff at 9.2 kHz� can resolve the smallest-scale fluctuationsof u at x /h�0, 20, and 70 for Reh=1500, 3000, and 7000,respectively, but cannot do so for Reh=10 000 and 16 500within the measured range of x /h. Accordingly the presentdata of � and � shown in Fig. 17 are only for Reh=1500,3000, and 7000.

Figure 17 indicates that both relations �6� and �7� areapproximately valid in the far field of a plane jet at least forReh�1500. Interestingly this validity appears to occur justdownstream from x /h=20 for all the Reynolds numbers in-cluding the lowest Reh=1500. This is beyond the conven-tional consideration since the distance of 20h is shorter thanthose required for achieving the asymptotic values of u� /Uc

or of the skewness and flatness factors �see Figs. 14 and 15�.Similar observations can be also made from Figs. 2–4 and 6of Ref. 42 for both plane and circular jets. It is well knownthat the dissipation rate reflects the statistical behavior of thesmallest-scale turbulence while other properties shownabove are for the large-scale turbulence. This seems to sug-gest that the asymptotic state is more easily achieved bysmall-scale turbulence than by large-scale turbulence.Clearly further investigations are needed to address this issuemore comprehensively.

Figure 17 also shows that as Reh is increased, both C�

and � decrease. However, C� depends not only on Reh but

2.5

p

2.7

2.8

0 0.5x104 1.0x104 1.5x104

-0.10

-0.05

0

0.05

0.10

0.15

Reh

F∞ u

S∞ u

FIG. 16. �Color online� Dependence of the asymptotic value of Su� and Fu

�

on Reh.

1 10 100

16500

10000

7000

3000

Reh= 1500

Gaussian3

4

5

3

3

3

3

2

x/h

Fu=

<u4 >

/(<

u2 >)2

FIG. 15. �Color online� Streamwise evolutions of Fu for Reh

=1500–16 500. Symbols are identical to Fig. 11.

075108-9 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

also on other initial and/or boundary conditions, so that itsvalue for Reh=22 000 estimated from the measurements ofAntonia et al.42 is higher than that deduced from the presenttrend. Note that the impact of C� on the Reh dependence ofthe dissipation rate is relatively small. This can be seen in thealternative form of Eq. �6�, i.e.,

� = C� Reh3�3h−4�x/h�−5/2. �8�

That is, � increases cubically �i.e., at a rate much higher thanthat of decrease of C�� as Reh �or Ub� increases. For example,when Reh is increased from 1500 to 7000, � rises by approxi-mately 4100% while C� reduces by only 144%. The depen-dence of � on x /h is shown in Fig. 17�b� for constant valuesof Reh. The good agreement with Eq. �8� confirms that thedata are well resolved for these Reh.

Figure 18 assesses those data, both from the presentmeasurement and past investigations that do not adequatelyresolve the smallest scales of turbulence. This is done byscaling the data according to Eq. �6�. The departure of our

data from the scaling by �x /h�−5/2 is evident for Reh

=10 000 and 16 500, as discussed above, consistent with thestrong dependence of � on Reh �Fig. 17�b��. Figure 18 alsoshows that the results of � reported in Refs. 7 and 10 forReh=36 000 and 30 000, respectively, also do not resolve thesmallest scales of turbulence. Both of these studies had a toopoor frequency response for adequate resolution of �.

IV. FURTHER DISCUSSION

It is evident that the statistical properties of the stream-wise velocity field of plane jets depend strongly on Reh up tothe asymptotic value, both in the near field and the far field,with nonidentical states of self-similarity. To provide someexplanations, we examine the Re dependence of the powerspectra of u and use these to assess the underlying coherentstructures at different values of Reh.

A. Spectral results of different Reynolds numbers

The existence of large-scale coherent structures in thenear field of a turbulent plane jet has been well demonstratedby previous studies.6,46–49 These near-field structures can oc-cur symmetrically or antisymmetrically with respect to thejet centerline, termed the “symmetric mode” and “antisym-metric mode,” respectively. As revealed by Sato,6 Rockwelland Niccolls,48 and Thomas and Goldschmidt,49 the symmet-ric mode occurs in a jet issuing from a smooth contractionnozzle, whose exit mean velocity is nearly uniform or in a“top-hat” shaped profile. The long channel or pipe nozzleproduces a power-law profile of the exit mean velocity andleads to the initial dominance of antisymmetric modes for thejet.50 The presence of these structures can be detected in

10-6

10-5

10-4

10-3

10 20 50 100 200

Reh= 150030007000

22000 [42]

1.2

Cε= 2.2

0.9

1.5

Lines: ε h U -3b = Cε (x/h)-5/2

x/h

0.002

0.005

0.01

0.02

0.05

0.1

20 50 100 200

Reh=1,5003,0007,000

22000 [53]

0.9

1.5

Cε = 2.2

1.2

Lines : η / h =Cε-1/4Reh

-3/4(x / h)5/8

x / h

η/h

(a)

(b)

εh

U-3

b

FIG. 17. �Color online� Streamwise evolution of �a� turbulent kinetic energydissipation ��hUb

−3� and �b� Kolmogorov scales �� /h� for the present casesReh=1500, 3000, and 7000, as well as those of Ref. 42 �Reh=22 000�.

10-6

10-5

10-4

10-3

10-2

20 50 100 200

16500

Re = 10000

30000 [10]

34000 [7]

~(x/h)5/8

~(x/h)-5/2

εε εε(h

Ub-3

)

ηη ηη/h

x / h

FIG. 18. �Color online� Streamwise evolutions of �hUb−3 and � /h for the

present cases of Reh=10 000 and 16 500 and for Ref. 10 �Reh=30 000� andRef. 7 �Reh=34 000�.

075108-10 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

power spectra of the fluctuating velocity ��u�. Figure 19presents the centerline evolution of �u from x /h=1 tox /h=10, where �u is defined in �u2=�0

��udf and the nor-malized frequency is f�= fh /Ub. Figure 19�f� illustrates sche-matically how the peak values of �u can be used to assesswhether the symmetric and/or antisymmetric modes aredominant. This is based on previous observations6 that thepassage frequency of the primary vortices detected by theprobe on the centerline should be lower for the antisymmet-ric mode than for the symmetric mode. In this context, it is

important to note that the values of f� obtained on and off thecenterline are identical, so that we present only the centerlinedata here.

Figure 19 demonstrates that, very close to the nozzle atx /h�1, no fundamental �or dominant� peaks in �u can beidentified in the range 0.1� f��0.3, indicating that this isupstream from the location where primary vortices form.Further downstream at x /h�2, such peaks are present, dem-onstrating that these vortices are present within the shearlayers and are thus detected by the probe on the centerline.

10-2 10-1 100

1

2

3

4

5

6

8

x/h=10

(c) Reh= 7000

f*=fh/Uo,b

10-2 10-1 100

1

x/h=10

8

6

5

4

3

2

(b) Reh= 3000

f*=fh/Uo,b

10-7

10-5

10-3

10-1

101

103

10-2 10-1 100

1

23

4

5

8

x/h=10

(a) Reh= 1500

f*=fh/Uo,b

Φu(

f* )

10-2 10-1 100

1

2

3

4

5

6

8

x/h=10

(e) Reh= 16500

f*=fh/Uo,b

10-7

10-5

10-3

10-1

101

103

10-2 10-1 100

1

x/h=10

8

6

5

4

3

2

(d) Reh= 10000

f*=fh/Uo,b

Φu(

f* )

SymmetricAnti-symmetric

Both modes

SymAnti-sym

Single mode

CL

(f)

(a) (b) (c)

(d) (e)

FIG. 19. �Color online� Centerline spectra ��u� of the velocity fluctuation �u� measured between x /h=1 and x /h=10 for �a� Reh=1500, �b� Reh=3000, �c�,Reh=7000, �d� Reh=10 000, �e� Reh=16 500, and �f� a schematic of �u, showing the symmetric and antisymmetric modes based on previous observation ofRef. 6. Note that CL denotes the centerline.

075108-11 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

However, distinct characteristics of �u are evident fordifferent Reynolds numbers. Evidently, when Reh�7000,�u is bimodal over the range 0.1� f��0.3, whereas forReh�10 000, it is broadly unimodal at f��0.22. It is hencededuced that both modes occur for Reh�7000 and that theirrelative importance varies with Reh. For Reh=1500, the an-tisymmetric mode appears to dominate the near field fromx /h=1 to 5. Further downstream at x /h=8 and 10, the pair-ing of primary vortices occurs, and the peak in �u shifts tolower frequencies �Fig. 19�a��. That the first shift in f� for theReh�7500 is due to a transition between the mode typerather than vortex pairing can be deduced from the findingthat this transition does not occur for Reh�10 000. Furtherdownstream, the primary vortices break down and less co-herent structures develop. It is deduced below that these far-field structures take the antisymmetric mode.

For Reh�10 000, the near-field, streamwise evolution of�u suggests that the symmetric mode becomes increasinglyimportant over the range x /h�8 as Reh is increased. That is,although both modes of the primary vortical structures coex-ist in the near field, the symmetric mode becomes over-whelmingly prevailing as Reh is increased. This differs fromthe finding of Sato.6 He concluded that the symmetric andantisymmetric modes occur, respectively, in plane jets withuniform profiles �from a smooth contraction� and parabolicprofiles �from a long channel� of the exit mean velocity. Onereason for this apparent discrepancy may be that Sato6 didnot use sidewalls for his plane jets. It is also possible that hissmooth contraction produced more uniform exit velocity pro-files than the present nozzles.

The jets of Reh=10 000 and 16 500 have highly uniformexit velocity profiles �see Fig. 3�. Under these circumstancesthe primary vortices in the two mixing layers are expected tosolely exhibit the symmetric mode. This is indeed the case,as reflected by a significant single broad peak in �u atf��0.22 for all cases other than a weak secondary peak atx /h=2. Moreover it is important to note that the distributionsof �u are quite similar at all the x locations in the two jets,suggesting the statistical similarity of their near-field under-lying structures. It also demonstrates that no vortex pairingoccurs for these jets.

Figure 20 demonstrates the Re dependence of theStrouhal numbers for the two modes �Stsym,Stant�. The valuesof Stsym and Stant are those of f� corresponding to obviouspeaks of �u at x /h=2–6, as indicated in Fig. 19 by dashedlines. �Note again that the results of Stsym and Stant obtainedon and off the centerline are identical so that we present herethe centerline data only.� Figure 20 also shows the results ofSato6 for comparison. It appears that an increase in theReynolds number from Reh=1500 to Reh=7000 causes bothStsym and Stant to increase slightly, implying a weak depen-dence on Reh of the vortex shedding. This contrasts theprevious results of Sato,6 who reported that Stsym=0.23and Stant=0.14 over the range Reh=1500–8000, and Namarand Ötügen,31 who reported that Stsym=0.273 forReh=1000–6000. Again we note that it is possible that thisdifference is due, at least in part, to the fact that both of thoseinvestigations dealt with quasiplane jets without sidewalls

and also used nozzles with different contraction profiles. Fig-ure 20 also shows that Stsym is nearly independent of Reh

when Reh�10 000.Now we examine the far-field u-spectra �u and their

centerline evolution. Figures 21�a�–21�d� present the far-fielddata of �u obtained at x /h=20, 40, and 80 on the centerlinein the form of f�u versus log�fy0.5 /Uc� to identify the peakof �u due to the presence of the far-field coherent structuresand their quasiperiodic passage.50 Note that the results forReh=1500 were not reliable due to too few data points of �u

at low frequencies �f��0.1� even for x /h=20 and are thusnot reported here. Note that this corresponds to the productof a local Strouhal number and the frequency, in contrast tothe exit Strouhal number reported earlier. It is evident thatthe spectral peak of the turbulent kinetic energy is centeredaround fp

� = fpy0.5 /Uc�0.1 to 0.22 �depending on Reh�. Wealso estimated the results of �u at y0.5 from measurementsof u across the jets. Figure 22 shows the results forReh=16 500. It is demonstrated by the dashed lines of Figs.21�c� and 22 that the value of fp

� for y=y0.5 is about a half ofthe centerline value. This is consistent with previous findingsthat the far-field vortex structures are antisymmetrically lo-cated on either side of the centerline of a plane jet,50–53 sothat the local structural frequency recorded on the centerlineshould be twice that obtained at y=y0.5 or beyond. The samefinding was also observed by Thomas and Prakash.50 It fol-lows that the local passage normalized frequency of the vor-tical structures should be �fp

��true=0.5�fp��CL. Figure 23 shows

the present result of �fp��true versus Reh, together with those

previously reported in Refs. 50–53. Note that the relation�fp

��true=0.5�fp*�CL was used to determine the value of �f

p*�true

for Antonia et al.51 since their measurements only provided�fp

��CL �see their Fig. 2�.Figure 23 demonstrates that the normalized frequency

�fp*�true increases noticeably with Reh. This Reh dependence

is consistent with the near-field result of Fig. 20. This obser-vation, based on both the present and previous measure-ments, does not support the claim of Cervantes de Gortari

0

0.1

0.2

0.3

0.4

0 5000 10000 15000 20000

Symbols: presentLines: Sato [6]

symmetric, Stsym

antisymmetric, Stant

Reh

St=

f ph/U

b

FIG. 20. �Color online� Dependence of the Strouhal number �St� fph /Ub�of the symmetric �Stsym� and antisymmetric �Stant� modes on Reh. Note thatthe values of Stsym and Stant are those of f� corresponding to peaks of �u

between x /h=2 and x /h=6 as indicated in Fig. 19 by dashed lines.

075108-12 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

and Goldschmidt52 that �fp*�true��0.11� is independent of the

Reynolds number in the range of their investigation, i.e.,Reh=7900–15 100 �see Fig. 23�. The suggestion of theseauthors that the apparent “flapping” is a universal feature ofthe plane jet, i.e., caused by a bulk displacement of the meanvelocity profile, is also ruled out by, e.g., Antonia et al.51 Theflapping is actually associated with the quasiperiodic passageof the far-field coherent structures.

B. Interpretation of differences in statisticalproperties due to varying Reh

It is evident from the above analysis that, in the near-field region, the arrays of the vortical structures in the mixinglayers on either side of the Reh=1500 jet can exhibit eitherthe symmetrical or the antisymmetrical modes of motion�and growth� with respect to the centerline. As Reh is in-creased, the symmetric mode becomes more predominant, asevidenced by a higher passage rate of primary structures�StsymStant, see Fig. 20�. This change in mode is associatedwith an increase in near-field spread and entrainment, as evi-

0

0.4

0.8

1.2

0.001 0.01 0.1 1 10

p = 204080

(a) Reh= 3000

f y0.5 / Uc

fΦu(

f)

0

0.4

0.8

1.2

0.001 0.01 0.1 1 10

x/h=204080

(b) Reh= 7000

f y0.5/Uc

fΦu(

f)

0

0.25

0.50

0.75

1.00

1.25

0.001 0.01 0.1 1 10

x/h=204080

(c) Reh= 10000

f y0.5/Uc

fΦu(

f)

0

0.4

0.8

1.2

0.001 0.01 0.1 1 10

x/h=204080

(d) Reh= 16500

f y0.5/Uc

fΦu(

f)

FIG. 21. �Color online� Evolution of the centerline �u obtained at x /h=20, 40, and 80 in the form of f�u vs log�fy0.5 /Uc�.

0

0.4

0.8

1.2

0.001 0.01 0.1 1 10

x/h=102050

100

f y0.5/Uc

fΦu(

f)

FIG. 22. �Color online� Streamwise evolution of �u for y=y0.5 obtained atx /h=20, 40, and 80 in the form of f�u vs log�fy0.5 /Uc�.

0

p

0.10

0.15

0 5000 10000 15000 20000

[53]

[50]

[52]

[51]

present

Reh

[fp*

] true

FIG. 23. �Color online� Dependence of �fp*�true on Reh.

075108-13 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

denced by the reduction in potential core length with increas-ing Reh �Figs. 4–6�. It therefore implies that the symmetricmode is more effective at entrainment in the near field than isthe antisymmetric mode. This change in dominant mode typecan also explain why the local maximum �uc,max

� � of the tur-bulence intensity uc

�=uc� /Uc decreases significantly with in-creased Reh since the antisymmetric mode can be expectedto cause higher centerline fluctuations, whereas its x-location�xmax

� � shifts upstream �see Fig. 12�, since this peak is asso-ciated with the end of the potential core, being caused by thecollision of the vortices on either side of the layer. This alsocan explain the Re-dependent discrepancies of variations ofthe near-field skewness and flatness factors of u �see Figs. 14and 15�. Furthermore, for Reh�10 000, the asymmetricmode is no longer present, as evidenced by the Strouhalnumber Stsym being nearly independent of Reh �see Fig. 20�.Importantly, this is associated with no obvious Re-dependentdifferences in the various statistical properties being ob-served for Reh�10 000.54

Next we seek an explanation for why the trends in theinfluence of Reh in the near-field and far-field rates of spreadand decay are opposite. The investigation of Antonia et al.51

�and others� reveals that the far-field coherent structures areestablished at a location well downstream from the nominalmerging point of the mixing layers but upstream from theonset of self-preservation. This was observed for a jet withReh�7700, where the symmetric mode can be expected todominate. It is possible that the interaction of symmetricalprimary structures from the opposite mixing layers results intheir own destruction when they collide near to the end of thepotential core, which would require new structures to formfurther downstream. Since the antisymmetric mode is preva-lent in the far field, this can also explain why the modes aredifferent in the near and far fields. However, such destructionis less likely to occur for the antisymmetric mode. Further-more, the antisymmetric mode can propagate directly intothe far field, without losing as much vortical energy in themerging process of the two layers, explaining why the far-field spreading rate is higher for jets with an initially anti-symmetric mode. The good match of near-field and far-fieldStrouhal numbers gives further support that this direct propa-gation at low Reh is possible.

We also note from Fig. 23 that the scaling passage fre-quency of the far-field structures increases with Reh. Thisimplies that the effect of Reh on the far-field organized mo-tion is not negligible, in contrast to the constant value of�f

p*�true reported earlier.52 Significantly, for a fixed value of

Reh, the dimensionless �fp*�true is approximately constant in

the self-similar region �of the mean velocity�, i.e., indepen-dent of x. This means that the frequency �fp� associated withthe passage of coherent structures decreases with down-stream distance �x� as fp�x−1.5, as derived from the ob-served relations Uc�x−1/2 �Fig. 4� and y0.5�x �Fig. 9�. Like-wise their scale varies directly with x. Antonia et al.51

estimated their size by integrating the maximum coherencebetween the lateral fluctuations of velocity in the y-directionand obtained that the streamwise extent of the structures is atleast two to three times as large as the lateral ��y0.5� andspanwise ��0.7y0.5� extents. This suggests that the sizes of

the far-field structures decrease with increasing Reh. Never-theless, it is recognized that the present spectral measure-ments are not sufficient to fully explain the Reh dependenceof the jets, and further work is required to confirm thesedeductions.

V. CONCLUSIONS

Previous experimental studies of a plane jet were con-fined to the high Reynolds number regime �Reh�10 000�and mostly limited to a single Reynolds number, each withdifferent initial and/or boundary conditions. Only the worksof Namar and Ötügen31 and Lemieux and Oosthuizen30 of-fered a systematic investigation of Reh. However, Namar andÖtügen’s31 jets were not constrained by sidewalls and did notextend into the asymptotic regime �1000�Reh�7000�,while that of Lemieux and Oosthuizen30 spanned an evenmore limited range, 700�Reh�4200. The recent DNS of aplane jet by Klein et al.33 was similarly limited in their rangeof Reynolds number Reh�6000 and axial extent x /h�20.Accordingly, the effects of Reynolds number on a two-dimensional plane jet cannot be adequately assessed from theexisting literature.

The present measurements have found the present planejet to depend on Reh over the range Reh�25 000. Within thisrange, changes within both the near and far fields are foundin the mean velocity decay rate and half-width, and in theevolution of turbulence intensity and skewness and flatnessfactors of the fluctuating velocity �u�. These changes are as-sociated with differences in the exit velocity profiles and inthe frequency spectra, implying different behaviors of theunderlying coherent structures. More specifically, the mainfindings of the influences of Reh over this range are summa-rized as follows.

�1� The details of exit flows, measured at x /h=0.5, dependon Reh, although the mean velocity profiles closely ap-proximate a top hat in each case. The lateral extent ofuniform flow widens, and the thickness of exit boundarylayer decreases as Reh is increased. This is associatedwith a reduction in the turbulence intensity profilesthere. These differences decrease asymptotically to be-come small, but not insignificant, for Reh=10 000.

�2� An increase in Reh causes the length of the potentialcore to decrease and the near-field spreading rate to in-crease. It also causes a decrease in the magnitude of thenear-field hump in the streamwise distribution of turbu-lence intensity, uc,max� , and also in its axial location fromxc,max�18h to 10h.

�3� Over the range 1500�Reh�10 000, the near-field, pri-mary coherent structures are found by spectral analysisto occur bimodally in both the symmetric and antisym-metric modes with respect to the jet centerline. Whilethe symmetric mode dominates at Reh=1500, the preva-lence of the antisymmetric mode increases with Reh tobecome dominant at Reh�10 000. The correspondingStrouhal numbers are found to increase slightly, but dis-cernibly, with increasing Reh. It is found that the sym-metric mode results in higher rates of near-field entrain-ment and growth than does the antisymmetric mode.

075108-14 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

�4� Secondary coherent structures are present in the far fieldand their passage frequency �fp� follows the relationfp�x−1.5, as shown in previous studies. The characteris-tic size of these structures thus obeys the requirement forself-preservation. It is the new finding of the presentwork that the local Strouhal number St= fpy0.5 /Uc variesbetween 0.05 and 0.11 when Reh is varied from 1500 to16 500 �Fig. 23�. This finding differs from the previousclaim that St=0.11 independent of the magnitude ofReh.52

�5� The far-field rates of mean velocity decay and spreadexhibit the opposite dependence on Reh to the near field.That is, they decrease asymptotically as Reh is increased.Their rates of asymptotic convergence are comparablewith that in the near field.

�6� The underlying structure in the far field is confirmed tobe in the antisymmetric mode, as found previously.50–52

It is hypothesized that the opposite trend in spreadingrates in the near and far field �above� may be caused bythe cancellation of much more of the coherent motionsfor the symmetric mode than for the antisymmetricmode when the coherent motions collide near the end ofthe potential core. That is, some of the near-field anti-symmetric vortices can propagate directly into the farfield, while the symmetric vortices largely cancel andmust re-establish in the antisymmetric mode.

�7� The local Reynolds number, Rey0.5�2Ucy0.5 /�, is foundto increase with downstream distance as Rey0.5�x1/2, forall cases of investigation, and this scaling begins almostimmediately downstream from the end of the potentialcore. This implies that all jets must eventually attain ahigh Rey0.5 no matter how low is the initial Reh. Never-theless, each jet converges to a different asymptoticstate.

�8� In the self-similar field, the asymptotic value of stream-wise turbulence intensity increases with an increase inReh, as do the asymptotic values of the skewness andflatness factors.

�9� For Reh�1500, both self-preserving relations of the ki-netic energy dissipation ��� and Kolmogorov scale ���,i.e., ��Reh

3�x /h�−5/2 and ��Reh−3/4�x /h�5/8, become

valid from x /h=20, a location, which is significantlycloser to the jet exit than those required for achievingthe asymptotic values of the turbulence intensity, theskewness, and flatness factors of u.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the support of theFluid Mechanics, Energy and Combustion Group of TheUniversity of Adelaide, the Endeavor Postgraduate ResearchScholarship, and ARC Linkage Scheme. We are thankful toProfessor W. K. George �Chalmers University of Technol-ogy� for insightful comments on R.C.D.’s doctoral thesis,particularly of discussions on the local Reynolds number ef-fect. Finally, we thank the reviewers for insightful com-ments, which have strengthened the final draft of this manu-script.

1H. Schlichting, “Laminare strahlausbreitung,” ZAMM 13, 260 �1933�.2E. Forthmann, “Über turbulente strahlausbreitung,” Ing.-Arch. 5, 42�1934� �translated as “Turbulent jet expansion,” National Advisory Com-mittee for Aeronautics Technical Memorandum No. 789, National Aero-nautics and Space Administration �1936��.

3W. Bickley, “The plane jet,” Philos. Mag. 23, 727 �1937�.4D. R. Miller and E. W. Comings, “Static pressure distribution in a freeturbulent jet,” J. Fluid Mech. 3, 1 �1957�.

5B. G. Van Der Hegge Zijnen, “Measurements of the distribution of heatand matter in a plane turbulent jet of air,” Appl. Sci. Res., Sect. A 7, 277�1958�.

6H. Sato, “The stability and transition of a two-dimensional jet,” J. FluidMech. 7, 53 �1960�.

7G. Heskestad, “Hot-wire measurements in a plane turbulent jet,” ASMETrans. J. Appl. Mech. 32, 721 �1965�.

8L. J. S. Bradbury, “The structure of self-preserving turbulent planar jet,” J.Fluid Mech. 23, 31 �1965�.

9P. E. Jenkins and V. W. Goldschmidt, “Mean temperature and velocitymeasurements in a plane turbulent jet,” ASME Trans. J. Fluids Eng. 95,581 �1973�.

10E. Gutmark and I. Wyngnanski, “The planar turbulent jet,” J. Fluid Mech.73, 465 �1976�.

11R. C. Deo, J. Mi, and G. J. Nathan, “The influence of nozzle aspect ratioon a plane jet,” Exp. Therm. Fluid Sci. 31, 825 �2007�.

12W. K. George, Recent Advances in Turbulence �Hemisphere, New York,1989�, pp. 39–73.

13P. Bradshaw, “Effects of external disturbances on the spreading rate of aplane turbulent jet,” J. Fluid Mech. 80, 795 �1977�.

14J. Mi, D. S. Nobes, and G. J. Nathan, “Influence of jet exit conditions onthe passive scalar field of an axisymmetric free jet,” J. Fluid Mech. 432,91 �2001�.

15R. C. Deo, “Experimental investigations of the influence of Reynoldsnumber and boundary conditions on a plane air jet,” Ph.D. thesis, TheUniversity of Adelaide, Australia, 2005. Available at Australian DigitalThesis Program, http://thesis.library.adelaide.edu.au/public/adt-SUA20051025.054550/index.html.

16R. C. Deo, J. Mi, and G. J. Nathan, “The influence of nozzle-exit geomet-ric profile on statistical properties of a turbulent plane jet,” Exp. Therm.Fluid Sci. 32, 545 �2007�.

17R. C. Deo, J. Mi, and G. J. Nathan, “Comparison of turbulent jets issuingfrom rectangular nozzles with and without sidewalls,” Exp. Therm. FluidSci. 32, 596 �2007�.

18K. B. M. Q. Zaman, “Far-field noise of a subsonic jet under controlledexcitation,” J. Fluid Mech. 152, 83 �1985�.

19N. R. Panchapakesan and J. L. Lumley, “Turbulence measurements inaxisymmetric jets of air and helium, Part 1, air jet,” J. Fluid Mech. 246,197 �1993�.

20H. J. Hussein, S. P. Capp, and W. K. George, “Velocity measurements in ahigh Reynolds number momentum conserving axisymmetric turbulentjet,” J. Fluid Mech. 258, 31 �1994�.

21P. E. Dimotakis, “The mixing transition in turbulent flows,” J. Fluid Mech.409, 69 �2000�.

22P. E. Dimotakis, R. C. Miake-Lye, and D. A. Panpantoniou, “Structure anddynamics of round turbulent jets,” Phys. Fluids 26, 3185 �1983�.

23P. L. Miller and P. E. Dimotakis, “Reynolds number dependence of scalarfluctuations in a high Schmidt number turbulent jet,” Phys. Fluids A 3,1156 �1991�.

24R. J. Gilbrech, “An experimental investigation of chemically reacting gas-phase turbulent jets,” Ph.D. thesis, California Institute of Technology,1991.

25M. M. Koochesfahani and P. E. Dimotakis, “Mixing and chemical reac-tions in a turbulent liquid mixing layer,” J. Fluid Mech. 170, 83�1986�.

26A. Michalke, “Survey on jet instability theory,” Prog. Aerosp. Sci. 21, 159�1984�.

27P. H. Oosthuizen, “An experimental study of low Reynolds number turbu-lent circular jet flow,” ASME Applied Mechanics, Bioengineering andFluids Engineering Conference, Houston, TX, 20–22 June 1983, ASMEPaper No. 83-FE-36.

28C. Bogey and C. Bailly, “Large eddy simulations of transitional round jets:Influence of the Reynolds number on flow development and energy dissi-pation,” Phys. Fluids 18, 065101 �2006�.

29W. M. Pitts, “Reynolds number effects on the mixing behavior of axisym-metric turbulent jets,” Exp. Fluids 11, 135 �1990�.

075108-15 The influence of Reynolds number on a plane jet Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp

30G. P. Lemieux and P. H. Oosthuizen, “Experimental study of behavior ofplanar turbulent jets at low Reynolds numbers,” AIAA J. 23, 1845 �1985�.

31I. Namar and M. V. Ötügen, “Velocity measurements in a planar turbulentair jet at moderate Reynolds numbers,” Exp. Fluids 6, 387 �1988�.

32K. W. Everitt and A. G. Robbins, “The development and structure ofturbulent plane jets,” J. Fluid Mech. 88, 563 �1978�.

33M. Klein, A. Sadiki, and J. Janicka, “Investigation of the influence of theReynolds number on a plane jet using direct numerical simulation,” Int. J.Heat Fluid Flow 24, 785 �2003�.

34L. W. B. Browne, R. A. Antonia, S. Rajagopalan, and A. J. Chambers,“The interaction region of a turbulent plane jet,” in Structure of ComplexTurbulent Shear Flows, IUTAM Symposium, Marseille, 1982, pp.411–419.

35F. O. Thomas and V. W. Goldschmidt, “Structural characteristics of adeveloping turbulent planar jet,” J. Fluid Mech. 163, 227 �1986�.

36A. K. M. F. Hussain and A. R. Clark, “Upstream influence on the nearfield of a planar turbulent jet,” Phys. Fluids 20, 1416 �1977�.

37J. Tan-Atichat, W. K. George, and S. Woodward, Handbook of Fluids andFluids Engineering, edited by A. Fuhs �Wiley, New York, 1996�, Vol. 3,Sec. 15.15, pp. 1098–1116.

38F. H. Champagne, “The fine-scale structure of the turbulent velocity field,”J. Fluid Mech. 86, 67 �1978�.

39S. J. Kwon and I. W. Seo, “Reynolds number effects on the behavior of anon-buoyant round jet,” Exp. Fluids 38, 801 �2005�.

40P. V. Johansson and W. K. George, “Equilibrium similarity, effects ofinitial conditions and local Reynolds number on the axisymmetric wake,”Phys. Fluids 15, 603 �2003�.

41H. Tennekes and J. L. Lumley, A First Course in Turbulence �MIT Press,Cambridge, MA, 1972�.

42R. A. Antonia, B. R. Satyaprakash, and A. K. M. F. Hussain, “Measure-

ments of dissipation rate and some other characteristics of turbulent planeand circular jets,” Phys. Fluids 23, 695 �1981�.

43J. Mi and G. J. Nathan, “The influence of probe resolution on the mea-surements of a passive scalar and its derivatives,” Exp. Fluids 34, 687�2003�.

44R. A. Antonia and J. Mi, “Corrections for velocity and temperature de-rivatives in turbulent flows,” Exp. Fluids 14, 203 �1993�.

45J. Mi, R. C. Deo, and G. J. Nathan, “Fast-convergent iterative scheme forfiltering velocity signals and finding Kolmogorov scales,” Phys. Rev. E71, 066304 �2005�.

46I. Wygnanski and E. Gutmark, “Lateral motion of the two-dimensional jetboundaries,” Phys. Fluids 14, 1309 �1971�.

47G. S. Beavers and T. A. Wilson, “Vortex growth in jets,” J. Fluid Mech.44, 97 �1970�.

48D. O. Rockwell and W. O. Niccolls, “Natural breakdown of planar jets,”ASME J. Basic Eng. 94, 720 �1972�.

49F. O. Thomas and V. W. Goldschmidt, “Structural characteristics of devel-oping turbulent plane jets,” J. Fluid Mech. 163, 227 �1986�.

50F. O. Thomas and K. M. K. Prakash, “An experimental investigation of thenatural transition of an untuned plananr jet,” Phys. Fluids A 3, 90 �1991�.

51R. A. Antonia, W. B. Browne, S. Rajagopalan, and A. J. Chambers, “Onorganized motion of a turbulent planar jet,” J. Fluid Mech. 134, 49 �1983�.

52J. Cervantes de Gortari and V. W. Goldschmidt, “The apparent flappingmotion of a turbulent plane jet—further experimental results,” ASMETrans. J. Fluids Eng. 103, 119 �1981�.

53V. W. Goldschmidt and P. Bradshaw, “Flapping of a plane jet,” Phys.Fluids 16, 354 �1973�.

54J. Mi, R. C. Deo, and G. J. Nathan, “Characterization of turbulent jetsfrom high-aspect-ratio rectangular nozzles,” Phys. Fluids 17, 068102�2005�.

075108-16 Deo, Mi, and Nathan Phys. Fluids 20, 075108 �2008�

Downloaded 22 Jul 2008 to 192.43.227.18. Redistribution subject to AIP license or copyright; see http://pof.aip.org/pof/copyright.jsp