deo et al. 2006 - the influence of nozzle aspect ratio on plane jets

www.elsevier.com/locate/etfs

Experimental Thermal and Fluid Science 31 (2007) 825–838

The influence of nozzle aspect ratio on plane jets

R.C. Deo *, J. Mi, G.J. Nathan

Turbulence Energy and Combustion (TEC) Research Group, School of Mechanical Engineering, The University of Adelaide, SA 5005, Australia

Received 19 March 2006; received in revised form 14 August 2006; accepted 30 August 2006

Abstract

This paper reports a systematic investigation of the effect of nozzle aspect ratio (AR) on plane jets. The aspect ratio AR (� w/h, whereh and w are the nozzle height and width) was varied from 15 to 72. The present velocity measurements were performed using single hot-wire anemometry, over a downstream distance of up to 85h and at a nozzle-height-based Reynolds number of Reh = 1.80 · 104. Resultsobtained reveal that both the extent and character of statistical two-dimensionality of a plane jet depend significantly on AR. Mostaspects of the near field flow exhibit an asymptotic-like dependence on AR, but do not become independent of AR within the rangeof AR investigated. A region of statistically two-dimensional (2-D) mean velocity field is achieved only for AR P 20, and its axial extentincreases with AR. However, the centerline turbulence intensity in the far field displays an asymptotic-like convergence only forAR P 30. In the self-similar region, both the mean decay and spreading rates of the jet increase as AR increases and do not reachan asymptotic value, even at AR = 72. The aspect ratio of the local jet (w/local velocity half-width) at the end of the 2-D region becomesasymptotically independent of nozzle aspect ratio, for approximately AR P 30. That is, the plane jet ceases to be statistically 2-D at afixed value of local jet aspect ratio for nozzle aspect ratios greater than 30. The skewness and flatness factors also depend on AR. Theseresults imply that independence of AR, even in the near field, will require very much larger aspect ratios than have been investigatedpreviously.Crown Copyright � 2006 Published by Elsevier Inc. All rights reserved.

Keywords: Plane jet; Turbulent mixing; Effect of nozzle aspect ratio; Hot-wire measurement

1. Introduction

Since the work of Schlichting [1], the plane jet hasreceived significant attention (e.g., Heskestad [2], Bradbury[3] and Gutmark and Wygnanski [4]). In experiments, aplane jet is produced by a slender rectangular slot andtwo parallel plates, known as sidewalls, attached to theslot’s short sides. The presence of sidewalls restrict the jetfrom developing in the spanwise direction (normal to theshort sides) so that it is expected to achieve a statisticallytwo-dimensional (2-D) flow over a sufficiently great down-stream distance into the far field. It follows that the nozzleaspect ratio, AR (� w/h, where w and h are the long and

0894-1777/$ - see front matter Crown Copyright � 2006 Published by Elsevie

doi:10.1016/j.expthermflusci.2006.08.009

* Corresponding author. Previous address: School of Engineering andPhysics, Faculty of Science and Technology, The University of the SouthPacific, Fiji. Tel.: +61 8 83035460; fax: +61 8 8303 4367.

E-mail address: [email protected] (R.C. Deo).

short sides of the slot), if sufficiently high, is believed tohave negligible influence on the downstream developmentof a plane jet. Because of this, previous studies, e.g., Bashirand Uberoi [5], Van der Hegge Zijnen [6] have all claimedthat they investigated the 2-D plane jet, although usingquite different values of AR, varying from 20 to 144.

The dominant mean motion of a plane jet is in thestreamwise (x) direction and its secondary spread is inthe lateral (y) direction. No entrainment can occur in thespanwise (z) direction due to the presence of sidewalls,placed in the x–y plane, that are necessary to achieve thestatistical two-dimensionality of a plane jet [13,14], over areasonably large axial distance. These sidewalls differenti-ate them from ‘‘rectangular’’ jets, a term that is commonlyused for the case when the jet has no sidewalls, althoughsuch jets are only truly ‘‘rectangular’’ at the exit plane.

The analytical investigations of George [7] and Georgeand Davidson [8] and the experimental work on a round

r Inc. All rights reserved.

mailto:[email protected]

Nomenclature

AR nozzle aspect ratio (AR = w/h)ARlocaljet local jet aspect ratio (� w/y0.5 local) at x2-D

using local half width, y0.5local

Fu centerline flatness factor, F u ¼ hu4i=ðhu2iÞ2F max

u maximum value of centerline flatness factorF1u asymptotic value of centerline flatness factorh height of a plane nozzleKu decay rates of mean centerline velocityKy jet spreading (widening) rateRe Reynolds number Reh � Uo,ch/tSu centerline skewness factor, Su = hu3i/(hu2i)3/2

Sminu minimum value of centerline skewness factor

S1u asymptotic value of centerline skewness factoru fluctuation component of mean velocity in

streamwise (x) directionu 0 root-mean-square (rms) of the velocity fluctua-

tion, u 0 = hu2i1/2

u0n;c normalized centerline turbulence intensity,u0n;c ¼ u0c=Uc

u0c;1 asymptotic value of centerline turbulence inten-sity

Uc local mean velocity on the centerlineUo,b exit bulk mean velocityUo,c mean exit centerline velocityUn,c normalized centerline mean velocity (Un,c =

Uc/Uo,c)w width of a plane nozzlex01 virtual origin of the normalized mean centerline

velocityx02 virtual origin of the normalized velocity half-

widthxp length of the jet’s potential corex2-D downstream distance that represents the end of

the 2-D regiony0.5 velocity half-width, calculated at the y-location

at which UðxÞ ¼ 12 U cðxÞ

m kinematic viscosity of the air, m � 1.5 · 10�5

m2 s�1 at 20 �C ambient conditionsx, y, z streamwise or axial (x), lateral (y) and spanwise

or transverse (z) coordinate

826 R.C. Deo et al. / Experimental Thermal and Fluid Science 31 (2007) 825–838

and plane jets [9–11] have shown that the asymptotic stateof turbulent flows, e.g., a round or a plane jet, depend upontheir initial and boundary conditions. By normal conven-tion, the choice of the value of AR becomes a boundarycondition of a plane jet. Gouldin et al. [12] have assumedthat the magnitude of AR may govern aspects of the down-stream behavior. However, to our best knowledge, no sys-tematic examination of the effect of aspect ratio on eitherthe behavior or extent of the statistically 2-D flow regionis currently available.

Much more information on the effects of nozzle aspectratio on downstream development is available for rectan-gular jets than for plane jets. This can provide usefulinsights into plane jets, since both these jets have an initialquasi-planar (2-D) flow [15], although the exit conditionsof both flows are fundamentally different. Nevertheless,within this quasi-planar region, rectangular jets appear tobehave broadly similarly to plane jets, thus they may wellbe used to study a plane jet behavior of varying AR. Tren-tacoste and Sforza [16] pioneered the investigation ofaspect ratio effect on rectangular jets over the range2.5 6 AR 6 100. Their study found an increase in lengthof the jets’ potential cores, a reduction in the decay of meancenterline velocity and a reduction in jet spreading ratewith an increase in nozzle aspect ratio. A decade later, Sfeir[17] measured jets from rectangular nozzles of aspect ratiosover the range 10 6 AR 6 60. He obtained similar resultsto Trentacoste and Sforza [16]. Marsters and Fothering-ham [18] also obtained similar trends for smaller nozzleaspect ratios between 1.88 and 3.39. Then followed manyother studies, such as those of Krothapalli et al. [19],

Tsuchiya et al. [20] and Quinn [21], all of which focusedon aspect ratio issues in rectangular jets. Based on hisvelocity measurements, Quinn [21] found that the near-fieldspreading rate increases with increasing nozzle aspect ratiofrom AR = 5 to AR = 20. He attributed a deducedincrease in jet mixing to an increase in three-dimensionalityof the entire jet. The flow visualization of rectangular jets(of AR ranging from 2 to 10) by Tsuchiya et al. [22] founda shorter potential core and an increase in spreading ratefor AR = 10 relative to AR = 2. By contrast, the measure-ments of the mass flow rate of a rectangular jet at exitMach number 0.95 by Zaman [23] for AR = 2–38suggested that the entrainment rate increases significantlywith nozzle aspect ratio only for AR P 8. More recently,Mi et al. [15] found that rectangular jets of AR = 15–120can be characterized by three distinct zones: an initialquasi-plane jet zone, a transition zone and a final quasi-axisymmetric-jet zone. Importantly, the extent of thequasi-plane zone was found to depend on nozzle aspectratio. In synopsis, previous investigations of rectangularnozzles reveal a distinct dependence of the flow evolutionon nozzle aspect ratio.

There are only a few previous investigations that directlyinvestigated the effect of nozzle aspect ratio on the evolutionof plane jets. Bashir and Uberoi [5] provided by far the bestindication of the effect of nozzle aspect ratio in a plane jet.Their measurements in the flow from a plane nozzle with asmoothly contoured exit at Re = 2770 and AR = 20, 44 and144 show that the AR = 144 jet decays at the highest rate(Fig. 1) and its asymptotic value of turbulence intensity isthe lowest of these jets (Fig. 2). However, they provide no

0

5

10

15

20

25

0 20 40 60 80 100

Heskestad [2] 120 36.900Hitchman et al [26] 60 7.000Bashir & Uberoi [5] 144 2.700Bashir & Uberoi [5] 44 2.700Bashir & Uberoi [5] 20 2.700Bradbury [3] 48 30.000Thomas & Goldschmidt [27] 46 6.000Browne et al. [13] 20 7.700

Sym. Authors AR Re

x/h

(Uo,

b/Uc)2

Fig. 1. Streamwise evolutions of centerline velocity of previous investi-gations of plane jets.

0

0.1

0.2

0.3

0 20 40 60 80 100 120

Heskestad [2] 120 18.000Bashir & Uberoi [5] 20 2.700Bashir & Uberoi [5] 40 2.700Bashir & Uberoi [5] 144 2.700Bradbury [3] 48 30.000Thomas & Goldschmidt [27] 46 6.000Browne et al [13] 20 7.700

Sym. Authors AR Re

x/h

u c' / Uc

Fig. 2. Streamwise evolutions of centerline turbulence intensity ofprevious investigations of plane jets.

R.C. Deo et al. / Experimental Thermal and Fluid Science 31 (2007) 825–838 827

data in the near field, and their Reynolds number is wellbelow the value of 10000 argued by Dimotakis [24] to benecessary to achieve a fully turbulent state. A similar trendwas found by Van der Hegge Zijnen [6], but over a muchsmaller range of aspect ratio for AR = 20 and 25, and againtheir data were limited to the far field. These papers also donot provide other details, such as the effects of aspect ratioon the higher order statistics. There are also some apparentdiscrepancies. The plane nozzle of Heskestad [2] measuredat AR = 120 produced the highest mean velocity decay(Fig. 1) and the largest asymptotic turbulence intensity(Fig. 2) of all previous investigations, although it is notquite the largest nozzle aspect ratio. This apparent inconsis-tency with Bashir and Uberoi’s [5] findings is probablyexplained by the use of a sharp-edged orifice slot by Heskes-tad [3], but this difference and others prevents a definitivecomparison. In addition, although there is reasonably goodagreement between the mean velocity decay of Bashir and

Uberoi [5] and Browne et al. [13] (both measured atAR = 20, Fig. 1), their turbulence intensities are discerniblydifferent (Fig. 2). This is primarily due to differences in Rey-nolds number (Namar and Otugen [25], Deo [9]), althoughdifferences in other initial conditions could also play a sig-nificant role. For instance, note that Bradbury [3] used thehighest Reynolds number but obtained second lowest tur-bulence intensity. The apparent discrepancy may be attrib-utable to the use of a co-flow (co-flow to jet velocity�16%),use of a higher nozzle aspect ratio (AR = 48) or any associ-ated differences between the nozzle surface finish (forinstance, rough or smooth). Given the expected dependenceof a jet flow on all boundary conditions [7–10], and therange of different boundary conditions used by previousinvestigators, it is therefore not possible to obtain a detailedpicture of AR effects on plane jets from previous investiga-tions. Hence a primary aim of the present investigation is todetermine in detail, the dependence of the evolution of aplane jet on nozzle aspect ratio by a systematic variationof only this parameter.

Further issues that remain unexplored are deduced fromcareful examination of results of previous investigations.Fig. 1 shows the decay of mean centerline velocity of planejets, as reported by previous investigators. As is wellknown, a relationship of the form Uc � x�1/2 (where theexponent �1/2 is a necessary indicator of a 2-D jet) is evi-dent in the self-similar region. However, a close examina-tion reveals that axial extent over which data is reportedincreases with nozzle aspect ratio. For instance, Heskestad[2], Hitchman et al. [26], Bradbury [3] and Thomas andGoldschmidt [27] used AR > 45, and present velocity datafor the range x/h 6 100. In contrast, the low-aspect ratioplane jets of Bashir and Uberoi [5] and Browne et al.[13], for AR = 20, only report data to the range x/h = 55and 40, respectively. This suggests that they have onlyreported data for the axial range where the jet was statisti-cally two-dimensional, perhaps prior to the secondaryinfluence of sidewalls becoming significantly large. Basedon the review of previous data, Gouldin [12] noted thatthe axial extent of two-dimensionality is expected todepend on AR. This notion is particularly relevant forthe data of Bashir [28] and Bashir and Uberoi [5], who per-formed measurements using three nozzles of aspect ratios20, 40 and 144, so their measurement range obviouslyappears not to be limited by experimental apparatus. Forinstance, for the case of AR = 20, their data extends upto x/h = 30 whereas for AR = 144, their data extends upx/h = 80. Were a dependence of the maximum planar flow(x2-D) region on AR be confirmed to be genuine, one couldestablish a possible relationship between the width of thelocal plane jet relative to the spacing between the sidewallsat which the transition from 2-D to 3-D flow occurs. Anysuch previously unidentified relationship would be of widegeneral significance to other planar flows, e.g., 2-D wakes,and would be a useful test for two-dimensional models.Hence another aim of the present research is to providesome insight into this issue.


2. Experiment details

The plane nozzle facility, shown schematically in Fig. 3,consists of an open circuit wind tunnel driven by a 14.5 kWaerofoil-type centrifugal fan, a wide angle diffuser, withflow straightening elements (including a honeycomb andscreens) and a smooth contraction exit of area ratio 6:1.The honeycomb has its cells aligned in the flow directionand attempts to reduce mean or fluctuating variations inspanwise velocity while producing minimal effect onstreamwise velocity because of the small pressure drop.Likewise, the screens help reduce the velocity defect inthe turbulent boundary layer that passes through it, furtherstreamlining the incoming airflow. The present screenshave an open area ratio of �60% and the smooth contrac-tion is based on a polynomial curve.

Two flat plates were mounted to the end of the wind-tunnel contraction, with radially contracting long-sides,with two parallel plates, known as sidewalls, attached tothe slot’s short sides, to create a plane nozzle. The heightof the nozzle was fixed at h = 10 mm for these measure-ments and the width (separation between the sidewalls)was varied from w = 150 mm to w = 720 mm to achieveseven aspect ratios of AR = 15, 20, 30, 40, 50, 60 and 72.

Fig. 3. A schematic view of the present experi

The inner radius of the long side of nozzle is r = 36 mmso that r/h = 3.60.

The facility was mounted horizontally, with the planenozzle located near to the mid point between the floorand ceiling in an experimental laboratory of dimensions18 m (long) · 7 m (wide) · 2.5 m (high). The distance fromthe jet exit to the front wall of the laboratory was �1400h

and between the jet and the ceiling/floor was �125h, allow-ing the unheated jet to discharge into still air freely. Basedon the approach of Hussein et al. [29], the effects of roomconfinement is estimated to produce less than 0.5%momentum loss for AR = 15–72 at a downstream distanceof 85h. Hence the present jets closely approximate planejets in an infinite environment. For all cases of the presentinvestigation, the jet exit centerline mean velocity was fixedat Uo,c � 27 m s�1 which results in a Reynolds numberReh � 1.80 · 104.

Velocity measurements were performed over the region0 6 x/h 6 85 using single hot-wire anemometry in an iso-thermal laboratory of ambient temperature 20 �C ±0.1 �C. To avoid aerodynamic interference of the prongson the hot wire senor, the present probe was carefullymounted perpendicular with prongs parallel to the planejet. In addition, the hot wire was aligned properly so that

mental setup. Dimensions are not to scale.

0

0.2

0.4

0.6

0.8

1.0

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

15 20 30 50 72

Sym ARU /

Uo,

c

0

0.05

0.10

0.15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7y/h

u'/U

o,c

y/h

Fig. 4. Lateral profiles of the (a) normalized mean velocity and(b) turbulence intensity, measured x/h = 0.2.


it corresponds closely to the streamwise component of theflow velocity to minimize directional ambiguity, although itis not possible to eliminate this effect completely. The singlehot-wire (tungsten) sensor was 5 lm in diameter and�0.8 mm in length, aligned in parallel to the long nozzlesides. The overheat ratio of the hot-wire sensor was 1.5,and a square wave test revealed a maximum frequencyresponse of 15 kHz. Hot-wire calibration was conductedusing a standard Pitot tube, placed side by side with thehot-wire probe, at the jet’s exit (x/h = 0), where the turbu-lence intensity was 60.5%, before and after measurementsof each AR. Both calibration functions were tested for dis-crepancies, and if velocity drift exceeded 0.5%, the experi-ment was repeated. No further corrections were applied,thus it is expected that measurements away from the cen-terline (in the outer region of the jet) are significantly inerror, because of high-velocity fluctuations relative to themean value. Nevertheless, the central aim here is tocompare the measurements of one experiment withanother, with most data taken on jet centerline for eachcase of AR. While converting data points from voltagesto velocities using a fourth order polynomial curve similarto the one proposed by George et al. [31], the average accu-racy of each calibration function was ±0.2%.

Signals obtained were low-pass filtered with an identicalcutoff frequency of fc = 9.2 kHz to eliminate high-frequency noise at all the measured locations. The voltagesignals were offset to a range of 0–3 V (as a precautionarymeasure that no clipping of the signal occurs [32]). Theywere then amplified appropriately through the circuits,and then digitized on a personal computer at fs = 18.4 kHzvia a 16 channel, 12-bit PC-30 F A/D converter of signalinput range 0–5 V. The sampling duration was about22 s, in which at approximately 400000 (instantaneous)data points were collected. Using the inaccuracies in cali-bration data and observed scatter in the measurements,the present random uncertainties correspond to a meanerror of ±4% at the outer edge of the jet and ±0.8% onthe centerline. The errors in the centerline rms velocitieswere found to be ±1.8%, and in the skewness (Su) and flat-ness (Fu) factors up to 3% and 8%, respectively.

3. Jet exit conditions

The exit velocity profiles were obtained at x/h = 0.2 ineach isothermal air jet using nozzles of AR = 15–72.Fig. 4(a) and (b) show that measurements near the exitplane confirm that these nozzles, like conventionalsmoothly-contracting ones, produce a quasi-‘top-hat’ meanvelocity profiles, which are uniform (U/Uo,c � 1) for all ARwithin 5% and a central-region turbulence intensitybetween 0.8% and 1.8%. Using these figures, the exitconditions are defined by calculations of the boundary-layer displacement thickness, dd ¼

R h=2

0 ½1� U=Uo;c�dy andmomentum thickness, hm ¼

R h=2

0 ½1� U=Uo;c�U c=U o;cdy. AsAR is varied from 15 to 72, the present values of dd andhm vary from 0.2h to 0.1h and 0.08h to 0.14h, respectively.

Likewise, the corresponding shape factors, H = dd/hm,which often represents the flatness (uniformity) of the meanvelocity profiles [33] lie between 2.4 and 2.5 for AR = 15–72, compared with a value of �2.6 for a Blasius exit veloc-ity profile. Thus the present plane nozzles may be charac-terized as having an initially laminar boundary layersince the shape factors are close to that of a Blasius profile(Schlichting [33]).

4. Results and discussion

Fig. 5 presents, in log–log form, the mean streamwisevelocity, Un,c = Uc/Uo,c along the jet centerline for all noz-zle aspect ratios. Note that successive profiles are verticallyoffset for clarity. The length of the jet’s potential core, xp,estimated by the maximum axial (x) distance at whichUc(x) = 0.98Uo,c is a function of nozzle aspect ratio.Apparently, as AR is increased from 15 to 72, xp increasesasymptotically (as shown later in Fig. 18) from xp � 2h toxp � 5h. Since the magnitude of xp is one measure of thenear-field entrainment rate, this implies that the rate of

1 10 100

AR = 15 20 30 40 50 60 72

Uc~ x

-1/2

x/h

Un,

c= U

c / U

o,c

[arb

itrar

y sc

ale]

Fig. 5. The evolution of centerline mean velocity, Un,c = Uc/Uo,c fromnozzles of different AR. Note the y-ordinate is arbitrary.

0

0.2

0.4

0.6

0.8

1.0

0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

AR = 15 30 50 72

y/h

U/U

c

Fig. 6. Lateral profiles of the mean velocity, U/Uc for AR = 15–72measured at x/h = 3.

10

30

50

70

90

110

130

150

170

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

Heskestad [2]Bashir & Uberoi [5]Hitchman et al. [26]Thomash & Goldschmidth [27]Bradbury [3]Miller and Comings [35]Browne et al. [13]present data

AR

x 2-D

/ h

Fig. 7. The normalized maximum downstream distance, x2-D/h, up towhich the mean velocity field retains statistical two-dimensionality. Valuesfrom previous investigations are estimated from axial extent of datapresented. (See also Refs. [2,5,26,27,3,35,13].)


entrainment of the present plane jets increases as ARdecreases. This is further confirmed by lateral profiles ofthe mean velocity in the potential core region (at x/h = 3), shown in Fig. 6, where a higher jet spreading rateis obtained for AR = 15 than for AR = 72.

Further interesting observations are made from Fig. 5,which shows that the jets of AR P 20 all exhibit fourdistinct regions of decay in Uc. Closest to the nozzle isthe well-known non-decaying potential core region. Thejet then undergoes a transition into the power-law-decayingregion, where Uc � x�1/2. This is followed by the fourthregion in which Uc does not appear to obey any systematicdependence on x. The departure from Uc � x�1/2 in the

fourth region is evidence of emerging three-dimensionaleffects, presumably induced directly and indirectly by side-walls. For AR = 15, there appears to be no obvious regionin which Uc � x�1/2. This reveals that an aspect ratio of 15is perhaps insufficient to achieve a statistically 2-D jet. Thatthe present exponent (=1/2) is independent of AR stands incontrast to previous measurements of rectangular jets,since both Trentacoste and Sforza [16] and Sfeir [17] foundthe magnitude of their exponent to be dependent on boththe nozzle geometry and aspect ratio. The reason for thisapparent discrepancy is presently unclear, but it seems tounderline the need for sidewalls to achieve a statistically2-D jet.

One must note from Fig. 5 that the axial extent of the1/2-power-law region increases with AR, as indicated bythe length of the dashed lines. In addition, it is also shownthat AR has an influence on the extent of other flowregions. To establish a relationship, if any, between themaximum planar flow region (denoted as x2-D) on AR,Fig. 7 plots x2-D/h against various AR. Note that for pre-vious investigations, we have estimated x2-D from the axialrange of their data, assuming that their Uc � x�1/2 wasprovided up to this downstream distance. It is clear fromFig. 7 that the present data set, in which only AR is varied,exhibits an excellent fit to a linear scaling of x2-D/h � AR.That is, the present data is well represented by a linearequation of the form x2-D

h ¼ m½ARþ C�, where m and C

are deduced to be 1.40 and 2.10, respectively. Some scatterbetween the present and previous measurement is evident,although this is quite expected, given that no previousinvestigators directly measured x2-D. Nevertheless, theagreement between present and previous data is sufficientto establish confidence that the scaling of x2-D � AR is gen-eric. Further, the actual values of m and C are unlikely tobe universal, but rather can be expected to depend upon on

4

6

8

10

10 20 50 100

present data

Bashir [5]

Heskestad [2]

Van der Hegge Zijnen [6]

AR (= w/h)

AR

loca

l jet

= w

/ y 0.

5 lo

cal

Fig. 8. The variation of local jet aspect ratio ARlocaljet against nozzleaspect ratio AR, calculated at x2-D for the present plane jets.

0

2

4

6

8

10

12

14

16

18

0 10 20 30 40 50 60 70 80 90

AR = 15 20 30 40 50 60 72

x/h

(Uo,

c / U

c)2

Fig. 9. The normalized profiles of centerline mean velocity for nozzleaspect ratios 15–72.


initial conditions [7–9], which perhaps contributes to thescatter between the present and past values of x2-D.

Next we assess the AR dependence of aspect ratio of thelocal jet at the end of the 2-D region (x2-D). To facilitatethis, we characterize the jet width by its half-width, y0.5,the location at which Uðx; y0:5Þ ¼ 1

2U cðxÞ, and define the

local half-width as the value of y0.5local at the locationx2-D. Hence the local aspect ratio of the jet at which itceases to be 2-D is ARlocaljet = w/y0.5local at x2-D. Fig. 8presents the dependence of ARlocaljet on AR. Two distinctregimes, separated by a clearly defined ‘elbow point’become immediately evident. As the nozzle aspect ratio isincreased from 20 to 30, ARlocaljet decreases rapidly from9.08 to 7.16, with ARlocaljet achieving an approximatelyconstant value for AR P 30. Some comparisons of thepresent measurements with previous investigations are pos-sible, if we retain the assumption used previously to obtainFig. 7 (i.e., that ARlocaljet is obtained from their furthestdownstream data-point). These measurements includeVan der Hegge Zijnen [6] for a nozzle of AR = 20, Heskes-tad [2] for a nozzle of AR = 120 and Bashir and Uberoi[5,28] for a nozzle of AR = 144. The present values ofARlocaljet are in good agreement with two of the three esti-mates derived from previous work [5,6] and moderateagreement with the other. The observed scatter is consis-tent with the rather crude method for extracting x2-D fromprevious measurements. Hence it is clear from Fig. 8 that,for AR P 30, the plane jet undergoes a transition from sta-tistically 2-D to a 3-D behavior at a fixed value of local jetaspect ratio. This physically explains that the transitionfrom a 2-D to a 3-D jet occurs when the width of the jet,which grows with axial distance, reaches a critical value rel-ative to the spacing between the sidewalls. In plane jets, the2-D roller-like counter-rotating large-scale structures,which are symmetric about the jet centerline, dominatethe mixing layers which bound the potential core, the inter-action region and the self-preserving region (Browne et al.

[34]). The transition from a 2-D to a 3-D jet perhapsimplies that at downstream distances greater than x2-D,these large-scale structures, which naturally scale with thewidth of the jet, undergo a substantial change, say frombeing roller-like 2-D to more three-dimensional (e.g., simi-lar to those found in rectangular jets). Such a change willprobably occur when the size of the large-scale structuresmatches the spanwise extent of the plane jet. If we considerthat the ratio of the half-width to the ‘‘outer edge’’ of a jetis about three times at x2-D, then the transition from 2-D to3-D occurs when the spanwise extent of the jet and lateralextent become almost equal. In other words, the spacingbetween the sidewalls controls this transition, which occurswhen the actual aspect ratio of the local jet is approximatesto unity.

Although Uc � x�1/2 within the 2-D region for all testedcases, the decay of centerline mean velocity for various ARare discernibly different (Fig. 9). Clearly, an inverselysquare relationship of the form (Uo,c/Uc)

2 = Ku(x/h +x01/h) is demonstrated, where Ku is a measure of the center-line mean velocity decay rate and x01 is the jet’s virtualorigin. The mean centerline velocity becomes self-similarwhen x/h P 20 for AR P 20. Consistent with Fig. 5, noobvious 2-D region is evident for AR = 15. The presentdependence of Ku on AR is plotted in Fig. 10, along withKu of some previous investigations. The present values ofKu reveal a clear trend. As AR is increased from 20 to72, Ku increases approximately linearly from 0.115 to0.180, with subtle, but definite differences in Ku forAR = 60 and 72. Such a trend was also found by Bashirand Uberoi [5] for AR = 20–144, although their depen-dence of the Ku on AR appears weaker. The obvious highervalues of Ku from Bashir and Uberoi [5] at similar aspectratios are, at least in part, explained by differences inReynolds number. Both Namar and Otugen [25] and Deo[9] found that a low-Reynolds number plane jet producesa higher value of Ku than a high-Re jet when measured

0.05

0.10

0.15

0.20

0.25

0.30

0 15 30 45 60 75 90 105 120 135 150

Heskestad [2]

present

Bashir & Uberoi [5]

Hitchman etal. [26]

AR

Ku

Fig. 10. The decay rate of the centerline mean velocity for aspect ratios15–72.


using identical nozzles. Hence, both the present and pastmeasurements reveal a consistent trend that the meanvelocity decay depends upon nozzle aspect ratio, even atAR = 72.

Lateral profiles of the mean velocity, normalised by thecentreline value are shown in Fig. 11(a)–(e). The maximumaxial location of the measured profiles span across till x/h 6 20 for AR = 15 and x/h 6 40 for AR = 20 and 30.For AR = 50 and 72, the profiles are measured up to themaximum achievable distance of x/h = 80. In the self-sim-ilar region, the profiles conform closely to the Gaussianform Un = exp[�ln2(yn)2]. Fig. 11 clearly demonstratesthat the distance at which self-similarity is reacheddecreases with increasing nozzle aspect ratio. For example,the mean velocity profiles become self-similar at x/h = 20for AR = 15, much further downstream than the x/h = 5for AR = 72. As mentioned previously, and revealed fromFigs. 5 and 7, the increasing influence of the sidewalls, withdecreasing nozzle aspect ratio, causes three-dimensionaleffects to dominate at axial locations closer to the nozzleexit for nozzles of smaller aspect ratios. Correspondingly,a higher-AR jet is associated with a larger axial distanceover which the self-similar state is achieved. Fig. 12 pre-sents the streamwise variations of the normalised velocityhalf-widths, y0.5/h. The half-width y0.5 varies linearly withx, a trend consistent with well-established self-similar prin-ciples [2]. The half-widths agree closely with a far field rela-tion of the form y0.5/h = Ky(x/h + x02/h), where Ky is thespreading (widening) rate and x02 is the virtual origin ofjet spread. As AR is decreased from 72 to 15, the jetsspread at decreased rates such that AR = 15 has the small-est spreading rate (see insert of Fig. 12). This is consistentwith the trend in decay rate of Uc discussed above (Figs. 9and 10). This finding is consistent with Zaman [23] whofound that a jet issuing from a nozzle of higher aspect ratiorectangular nozzle entrains ambient fluid at a higher ratethan a nozzle of lower aspect ratio. Since Zaman [23]

attributed increased entrainment at higher aspect ratiosto the streamwise vortical structures, it is likely thatchanges to the nozzle aspect ratio modify the large-scalestructures emanating from the present plane jet.

Fig. 13 presents the evolution of centerline turbulenceintensity, u0n;c ¼ u0c=Uc in the present jets. The profiles areoffset vertically for clarity, so that an arbitrary ordinatescale is used. A near field plateau in u0n;c is clearly visiblefor plane jets of AR P 30, although with a greater scatterthan for Un,c, as one might expect. There is a flat region inthe far field for AR P 30, leading to the widest plateau forthe highest AR. That is, the width of the plateau increaseswith aspect ratio. Over the entire x/h, the values of u0n;c arestrongly dependent on AR. Between x/h = 8 – 9, possiblehumps in centerline turbulence intensity are evident forboth, AR = 15 and 20, although it is most pronouncedfor the smaller aspect ratio. Further downstream fromthe hump, u0n;c approaches an asymptotic value forAR P 30, whose magnitude (Fig. 14), u0c;1 is strictly depen-dent on AR. Consistent with Fig. 5, the case of AR = 15shows that u0n;c does not achieve a constant value at all, fur-ther indicating strong three-dimensional effects from thesidewalls. Since u0n;c is found not to be constant forAR = 20, this indicates that an aspect ratio of 20 is alsoinsufficient to achieve a fully developed 2-D turbulent flow.This is broadly consistent with Figs. 5, 7 and 10, whichreveal a very small region of x/h for which the mean veloc-ity field remains statistically two-dimensional. ForAR P 30, u0n;c maintains asymptotic values of u0c;1 to thepresent ranges of x/h. Taken together, these observationssuggest that a fully developed 2-D flow is truly achievedonly for AR P 30 for the present ranges of x/h.

Fig. 15(a)–(e) presents the lateral profiles of turbulenceintensity, u 0/Uc for jets of different AR. As clearly evident,the turbulence intensity profiles for the cases AR = 15 and20 never become truly congruent, consistent with the otherdata presented so far. However, for AR = 30, turbulenceintensity profiles become congruent for x/h P 20. As withthe case of mean velocity profiles, the distance requiredfor the turbulence intensity profiles to become congruentdecreases with AR, so that for AR = 50 and 72, the profilesbecome congruent for x/h P 10. This, again, is further evi-dence that the two-dimensionality of the fully developedflow increases with AR throughout the investigated regime.

The dependence of flow statistics on AR may be furtherinvestigated using the moments of higher order fluctua-tions. Figs. 16 and 17 present the centerline evolutions ofthe skewness, Su = hu3i/(hu2i)3/2, and flatness, Fu = hu4i/(hu2i)2. Note that the values of Su and Fu were determinedfrom �400000 data points of U (t) at each axial location,so the convergence of the calculations is reasonably satis-factory, and an appropriate offset applied to the A/D sam-pling board ensured that Su and Fu were not truncated dueto clipping effects arising from the finite range of our sam-pling board (Tan-Atichat et al. [32]). Clear trends emerge,both Su and Fu appear to be AR-dependent. The factorsevolve from nearly Gaussian value (Su,Fu) = (0, 3) at the

0

0.25

0.50

0.75

1.00

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40

0

0.25

0.50

0.75

1.00

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40 80

0

0.25

0.50

0.75

1.00

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20

Gaussian

yn= y / y0.5

yn= y / y0.5

Un=

U /

Uc

Un=

U /

Uc

0

0.25

0.50

0.75

1.00

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40 80

yn= y / y0.5

Un=

U /

Uc

yn= y / y0.5

Un=

U /

Uc

yn= y / y0.5

Un=

U /

Uc

0

0.25

0.50

0.75

1.00

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40

Fig. 11. Lateral profiles of the normalized mean velocity U/Uc for (a) AR = 15, (b) AR = 30, (c) AR = 50, (d) AR = 60, and (e) AR = 72. A Gaussiancurve is also shown in each plot.


nozzle exit to highly non-Gaussian values around x/h = 3–5. Such were the observations of Browne et al. [34], whosepassive temperature fluctuations at the exit of a plane jetwere Gaussian, while at the potential core region (between3 and 5h) were highly non-Gaussian. The present large val-ues of Su and Fu around the potential core region (say

between x/h = 3 and 5) perhaps reflects the occasionalintrusions of low-velocity ambient fluid into the jet(Browne et al. [34]), due to the growth of the large-scaleroller-like structures in the mixing layers. An estimate ofthe relationship between flatness and intermittency is givenby Batchelor and Townsend [30] as Fu = hu4i/(hu2i)2 � 3/c,

0

2

4

6

8

10

0 10 20 30 40 50 60 70 80 90

AR = 15 20 30 50 72

x/h

y 0.5

/ h

0.10

0.15

0.05

Ky

AR0 20 40 60 80

Fig. 12. Streamwise evolutions of mean velocity half-width for AR = 15–72. Note that inserted figure shows the dependence of jet spreading rates,Ky on AR.

0 20 40 60 80 100

AR = 15 20 30 40 50 60 72

x/h

u'c/U

c [ar

bitr

ary

scal

e]

Fig. 13. Streamwise evolutions of locally normalized turbulence intensity,u0c=Uc for AR = 15–72. Note that the y-axis values are arbitrary.

0.15

0.17

0.19

0.21

0.23

0.25

0 15 30 45 60 75

AR

u'c,

∞

Fig. 14. Variations of far field asymptotic turbulence intensity, u0c;1 withAR.


where c is the intermittency factor, which defines theproportion of time for which the velocity signal is turbu-lent. This derivation assumes that all the ‘‘turbulent’’ fluidhas a flatness of 3, which may not necessarily be true. How-ever, it suggests that the high-near-field values of Fu areassociated with low-turbulence intermittency. Specificobservations show that Su exhibit a local maximum,jSmax

u j, and a local minimum, jSminu j, (Fig. 16 insert) at

x/h = 3 and 5, respectively, while Fu reaches a maximumjF max

u j at x/h = 5 (Fig. 17 insert). Figs. 16 and 17 alsoshow that the magnitudes of jSmin

u j and jF maxu j at x/h � 5

depend on AR. The reason for this stems from the findingin Figs. 5 and 18. Note from those figures that the length ofthe potential core decreases with AR. A shorter potentialcore implies a more rapid development of large-scale,coherent structures through its shear-layer, and increased

fluid entrainment and, so too, perhaps more coherentlarge-scale structures. As with earlier results, while thenear field values of jSmin

u j and jF maxu j appear to exhibit an

asymptotic-like dependence on AR, they retain subtle butsignificant differences in their magnitudes. Moving down-stream from x/h = 5, both factors Su and Fu approach nearGaussian (0, 3) in the far field, albeit, not reaching trulyGaussian values. This again is consistent with Browneet al. [34], whose far field passive temperature distribu-tions did not approximate a Gaussian value. Greater devi-ations of both factors from their Gaussian values arenotable for the cases AR 6 20, as confirmed by the proba-bility density functions (pdf) of the centerline velocityfluctuations [9].

To examine whether or not, there is an asymptotic-likeconvergence of near field flow, Fig. 18 plots the virtual ori-gins, x01 and x02 and the potential core lengths xp for var-ious AR. Although with some scattering, there is sufficientevidence to conclude that the near field mean statisticsshow a consistent dependence on AR, however, the flowretains subtle differences from being truly asymptotic, for-bidding a complete elimination of the effect of exit condi-tions. The effect of the notable departures from its trueasymptotic state in the near field is confirmed by the signif-icant differences in flow properties in the far field. In otherwords, the near field effect does propagate into the far field,causing the flow to be significantly dependent upon thenozzle aspect ratio.

It is revealed the maximum axial distance (x) requiredfor the attainment of self-preservation depends on AR,and more specifically, on the spanwise dimension (w) ofthe nozzle if h is constant. It is thus deduced that a low-AR-rectangular jet (without sidewalls) will be initiallyquasi-2D but will eventually become axisymmetric at asmall value of x relative to a high-AR-rectangular jet which

0

0.1

0.2

0.3

0.4

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20

u'/U

c

0

0.1

0.2

0.3

0.4

0 0.5 1.0 1.5 2.0 2.5

x/h = 35102040

yn= y / y0.5

0

0.1

0.2

0.3

0.4

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40

u'/U

c

0

0.1

0.2

0.3

0.4

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40

0

0.1

0.2

0.3

0.4

0 0.5 1.0 1.5 2.0 2.5

x/h = 35

10 20 40 80

u'/U

c

yn= y / y0.5

yn= y / y0.5 yn= y / y0.5

u'/U

c

u'/U

c

yn= y / y0.5

Fig. 15. Lateral profiles of the turbulence intensity, u 0/Uc for (a) AR = 15, (b) AR = 20, (c) AR = 30, (d) AR = 50, and (e) AR = 72.


will be quasi-2D up till a larger value of x and it becomesaxisymmetric too. However, with sidewalls; the latter jet isexpected to experience the influence of the wall-inducedsecondary flow into the jet centerline, causing increased

three-dimensionality. This is a useful subject for furtherinvestigation and can be undertaken by comparison oftwo jets from identical rectangular nozzles but configuredwith and without sidewalls.

0 10 20 30 40 50 60 70 80 90

Gaussian-1

2

0

1

0

0

0

0

0

0

-2

x/h

S u= <

u3 > /

(<u2 >

)3/2

0.6

0.7

0.8

0.9

1.0

0 15 30 45 60 75

0.10

0.15

0.20

0.25

0.30

AR|S

u min

|

|Su∞|

Fig. 16. Streamwise evolutions of the skewness, Su, for AR = 15–72. The insert shows the AR-effect on the near-field minimum of Su and on far fieldasymptotic Su. Note that each profile is shifted vertically by unity relative to its neighbour for clarity, and symbols are identical to Fig. 13.


5. Conclusions

In summary, the present systematic study reveals that allthe normalized mean and turbulent properties of a plane jetare dependent on nozzle aspect ratio even at AR = 72. It isalso suggested that asymptotic independence from AR willrequire much larger aspect ratios than have been investi-gated previously. The present results are consistent withpreviously reported data, but provide a much greater detailthan has previously been available. An asymptotic-likeconvergence of many near-field aspects of the velocity field,as AR approaches 72, is noted, but substantially high-values of AR are required to achieve true convergence. Inthe near field, an increase in AR leads to an asymptotic-likeincrease in the length of the jet’s potential core, the jetvirtual origins, x01 and x02, and the magnitude of the nearfield troughs in skewness and spikes in flatness. In contrast,a true asymptotic state is not attained in the far field forpresent ranges of AR, with subtle, but definite differencesfound even for the cases of AR = 60 and 72. This is indi-cated by rates of jet-spreading and centerline velocitydecay, which continue to increase approximately linearlywith AR and by the far field skewness and flatness factors.Since the presence of large-scale structures, if any, typically

control the overall-spreading and decay rate, this depen-dence implies an AR-dependence of the underlying large-scale structures on nozzle aspect ratio.

It is proposed that in the self-similar field, a statistically2-D jet is attained only for AR P 30 over an axial distanceof up to 85h. This is evidenced, for example, by the meancenterline velocity scaling as Uc � x�1/2, by the normalizedturbulence intensity approaching a constant value and byskewness and flatness factors of velocity fluctuationsapproaching near-Gaussian values for these cases. Further,the axial extent of the region of statistical 2-D jet increasesapproximately linearly with AR. It must be noted thatthere is often a misconception when differentiating betweena large AR-rectangular jet and a true plane jet. It must berecognized that the latter is only achievable using a rectan-gular nozzle with sidewalls, however, if sidewalls are notused, the flow field of the former case will be significantlydifferent from the latter case, even in the near field quasi-2D region, where aspect ratio effects are believed thoughtto be small. For plane jets of AR P 30, the axial locationat which the jet undergoes a transition from 2-D to 3-Dis found to occur at a fixed local jet aspect ratio. Thiscorresponds approximately to the condition where theouter edge of the jet equals the sidewall spacing, i.e., when

0 10 20 30 40 50 60 70 80 90

6

5

4

3

Gaussian

3

3

3

3

3

3

2

x/h

Fu=

<u4 >

/ (<

u2 >)2

Fu

∞

3.0

2.8

2.6

Fu

max

AR0 15 30 45 60 75

12

8

4

Fig. 17. Streamwise evolutions of the flatness, Fu for AR = 15–72. The insert shows the AR-effect on the near-field maximum of Fu. Note that each profileis shifted vertically by unity relative to its neighbour for clarity, and symbols are identical to Fig. 13.

-12

-8

-4

0

0 15 30 45 60 75 900

2

4

6

8

10x02x01xp

r2 = 0.95

r2 = 0.95

AR

virt

ual o

rigi

ns, x

01, x

02

pote

ntia

l cor

e le

ngth

, xp

Fig. 18. The dependence of the virtual origins, x01, x02 and lengths of thepotential core, xp on AR.


the actual aspect ratio of the local jet is approximatelyunity.

Taken together, the statistical dependence of the velocityfield on AR is attributable to differences in the underlyinglarge-scale structures that propagate downstream, emanat-ing from nozzles of differing boundary conditions. The

present study, along with the recent measurements byDeo [9] therefore provides further support for the analyti-cal hypotheses proposed by George [7] and George andDavidson [8], that the downstream development of a planejet is entirely governed by initial conditions. Hence, consis-tent with established knowledge for round jets, the classicalhypothesis, which argues that all jets should becomeasymptotically independent of the source conditions, atsufficiently large distances from the source, does not holdtrue for a plane jet.

Acknowledgements

We acknowledge that this original research articlereports work from the first author’s Ph.D. thesis completedthrough the support of Endeavor International Postgradu-ate Scholarship, Adelaide Achievers Scholarship and anARC Discovery Grant in collaboration with FCT. We alsothank the two anonymous reviewers whose comments haveimproved the overall clarity of our paper.

References

[1] H. Schlichting, Laminare strahlausbreitung, Z. Angew. Math. Mech.13 (1933) 260–263.


[2] G. Heskestad, Hot-wire measurements in a plane turbulent jet, Trans.ASME, J. Appl. Mech. 32 (1965) 721–734.

[3] L.J.S. Bradbury, The structure of a self-preserving turbulent planarjet, J. Fluid Mech. 23 (1965) 31–64.

[4] E. Gutmark, I. Wygnanski, The planar turbulent jet, J. Fluid Mech.73 (3) (1976) 465–495.

[5] J. Bashir, S.M. Uberoi, Experiments on turbulent structure and heattransfer in a 2-D jet, Phys. Fluids 18 (4) (1975) 405–410.

[6] B.G. Van der Hegge Zijnen, Measurements of the distribution of heatand matter in a plane turbulent jet of air, Appl. Sci. Res. A7 (1958)277–292.

[7] W.K. George, The self preservation of turbulent flows and its relationto initial conditions, in: Recent Advances in Turbulence, Hemisphere,New York, 1989, pp. 39–73.

[8] W.K. George, L. Davidson, Role of initial conditions in establishingasymptotic behaviour, AIAA J. 42 (3) (2004) 438–446.

[9] R.C. Deo, Experimental investigations of the influence of Reynoldsnumber and boundary conditions on a plane air jet. Ph.D. thesis.School of Mechanical Engineering. The University of Adelaide, SouthAustralia, 2005.

[10] P. Bradshaw, The effect of initial conditions on the development of afree shear layer, J. Fluid Mech. 26 (1966) 225–236.

[11] P. Bradshaw, Effects of external disturbances on the spreading rate ofa plane turbulent jet, J. Fluid Mech. 80 (1977) 795–797.

[12] F.C. Gouldin, R.W. Schefer, S.C. Johnson, W. Kollmann, Non-reacting turbulent Mixing flows, Prog. Energy Combust. Sci. 12(1986) 257–303.

[13] L.W.B. Browne, R.A. Antonia, S. Rajagopalan, A.J. Chambers,Interaction region of a two-dimensional turbulent plane jet in still air,in: Proc. of Structure of complex turbulent shear flows, IUTAMSymp., Marseille, 1982, pp. 411–419.

[14] S.B. Pope, Turbulent Flows, Cambridge University Press, UK, 2002.[15] J. Mi, R.C. Deo, G.J. Nathan, Characterization of turbulent jets from

high-aspect-ratio rectangular nozzles, Phys. Fluids 17 (6) (2005).[16] N. Trentacoste, P. Sforza, Further experimental result for three-

dimensional free jets, AIAA J. 5 (5) (1967) 885–890.[17] A. Sfeir, Investigation of three-dimensional turbulent rectangular jets,

AIAA J 17 (10) (1979) 1055–1060.[18] G.F. Marsters, J. Fotheringham, The influence of aspect ratio on

incompressible turbulent flows from rectangular slots, Aeronaut.Quart. J 31 (4) (1980) 285–305.

[19] A. Krothapalli, D. Bagano, K. Karamcheti, On the mixing ofrectangular jet, J. Fluid Mech. 107 (1981) 201–220.

[20] Y. Tsuchiya, C. Horikoshi, T. Sato, On the spread of a rectangularjets, Expts. Fluids 4 (1985) 197–204.

[21] W.R. Quinn, Turbulent free jet flows issuing from sharp-edgedrectangular slots: The influence of slot aspect ratio, Expl. Therm.Fluid Sci. 5 (1992) 203–215.

[22] Y. Tsuchiya, C. Horikoshi, T. Sato, M. Takahashi, A study of thespread of rectangular jets: (The mixing layer near the jet exit andvisualization by the dye methods), JSME Int. J. 32 (Series II (1))(1989) 11–17.

[23] K.B.M.Q. Zaman, Spreading characteristics of compressible jets fromnozzles of various geometry, J. Fluid Mech. 383 (1999) 197–228.

[24] P.E. Dimotakis, The mixing transition in turbulent flows, J. FluidMech. 409 (2000) 69–98.

[25] I. Namar, M.V. Otugen, Velocity measurements in a planar turbulentair jet at moderate Reynolds numbers, Expts. Fluids 6 (1988) 387–399.

[26] G.J. Hitchman, A.B. Strong, P.R. Slawson, G. Ray, Turbulent planarjet with and without confining walls, AIAA J. 28 (10) (1990) 1699–1700.

[27] F.O. Thomas, V.W. Goldschmidt, Structural characteristics of adeveloping turbulent plane jet, J. Fluid Mech. 163 (1986) 227–256.

[28] J. Bashir, Experimental study of the turbulent structure and heattransfer of a two dimensional heated jet. Doctoral Dissertation,University of Colorado, 1973.

[29] H.J. Hussein, S.P. Capps, W.K. George, Velocity measurements in ahigh Reynolds number, momentum-conserving axisymmetric turbu-lent jet, J. Fluid Mech. 258 (1994) 31–60.

[30] G.K. Batchelor, A.A. Townsend, Proc. R. Soc. Lond. A 199 (1949)238–255.

[31] W.K. George, P.D. Beuther, A. Shabbir, Polynomial calibrations forhot wires in thermally varying flows, Expl. Therm. Fluid Sci. 2 (1989)230–235.

[32] J. Tan-Atichat, W.K. George, S. Woodward, Use of Computer forData Acquisition and Processing, in: A. Fuhs (Ed.), Handbook ofFluids and Fluids Engineering, vol. 3, Wiley, NY, 1996, pp. 1098–1116, Sec. 15.15.

[33] H. Schlichting, Boundary Layer Theory, McGraw-Hill, 1968 (Chap-ter 7).

[34] L.W.B. Browne, R.A. Antonia, S. Rajagopalan, A.J. Chambers, Theinteraction region of a turbulent plane jet, J. Fluid Mech. 149 (1984)355–373.

[35] D.R. Miller, E.W. Comings, Static pressure distribution in a freeturbulent jet, J. Fluid Mech. 3 (1957) 1–16.

deo et al. 2006 - the influence of nozzle aspect ratio on plane jets

Documents