The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

Accepted Manuscript The influence of nozzle-exit geometric profile on statistical properties of a tur bulent plane jet R.C. Deo, G.J. Nathan, J. Mi PII: DOI: Reference:

10.1016/j.expthermflusci.2006.08.010 ETF 6858

To appear in:

Experimental Thermal and Fluid Science

Received Date: Accepted Date:

19 June 2006 30 August 2006

S0894-1777(06)00139-7

Please cite this article as: R.C. Deo, G.J. Nathan, J. Mi, The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet, Experimental Thermal and Fluid Science (2007), doi: 10.1016/j.expthermflusci. 2006.08.010

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ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet R C Deo1, G J Nathan and J Mi Turbulence, Energy and Combustion [TEC] Research Group School of Mechanical Engineering The University of Adelaide, SA 5005, Australia Phone +61 8 8303 5460

Facsimile +61 8 8303 4367

Corresponding Author1 [email protected]

ABSTRACT The paper reports an investigation of the influence of geometric profile of a long slot nozzle on the statistical properties of a plane jet discharging into a large space. The nozzle profile was varied by changing orifice plates with different exit radii (r) over the range r = (0 – 3.60) times the slot height (h). The present measurements were made at a slot-height based Reynolds number (Reh) of 1.80 × 104 and slot aspect ratio of 72. The results obtained suggest that both the initial flow and the downstream flow are dependent upon r/h. A “top-hat” mean exit velocity profile is closely approximated when r/h approaches 3.60. The decay and spread rates of the jet’s mean velocity decrease in an asymptotic-like manner as r/h is increased, with the differences becoming small as r/h approaches 3.60. A decrease in r/h results in a higher formation rate of the primary vortices. The far-field values of the centerline turbulence intensity are higher for smaller r/h, and display asymptotic-like convergence as r/h approaches 3.60. Overall, the effect of r/h on the mean and turbulence fields becomes small as r/h becomes closer to 3.60. The statistical differences observed are deduced to result from differences in the underlying near-field structure produced by the different nozzle-exit geometric profiles.

KEYWORDS plane jet, turbulence structure, nozzle-exit geometry, turbulence statistics

Submitted to: Experimental Thermal & Fluid Sciences

ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

NOMENCLATURE AR

nozzle aspect ratio, AR = w / h

Fu

centerline flatness factor, F u = < u 4 > (< u 2 > )

2

Fumax maximum value of centerline flatness factor h

slot-height of a plane nozzle

H

shape factor of the initial velocity profile, H = δd / θm

Ku

decay rate of centerline mean velocity

Ky

jet spreading (widening) rate

(

S u ψψ centerline skewness factor, S u = < u 3 > < u 2 >

)

3/ 2

S umin

minimum value of centerline skewness factor

Reh

Reynolds number based on slot-height (h), Re h ≡ U o,b h υ

Re θ m

Reynolds number based on momentum thickness, Re θ m ≡ U o,b θ m / ν

u

fluctuation component of jet velocity

u'

root-mean-square (rms) of the velocity fluctuation, u '= u 2

u c',∞

asymptotic value of centerline turbulence intensity

u 'p

peak fluctuation component of the jet velocity within jet’s shear layers

u n',c

normalized centerline turbulence intensity, u n',c = u c' / U c , where the subscript “c” stands

1/ 2

for centerline value Uc

centerline local mean velocity

Uo,b

exit bulk mean velocity

Uo,c

mean exit centerline velocity

Um,c

maximum centerline velocity

U n ,c

normalized centerline mean velocity, U n,c = U c U o,b

w

slot-width of a plane nozzle

x01

virtual origin of the normalized mean centerline velocity

x02

virtual origin of the normalized velocity half-width

xp

length of the jet’s potential core

x, y, z streamwise (x), lateral (y) and spanwise (z) coordinate system y 0.5 2

velocity half-width, calculated at the y-location at which U ( x) = 12 U c ( x)

ACCEPTED MANUSCRIPT Deo, Nathan and Mi

δd

Submitted to Experimental, Thermal and Fluid Sciences: June 2006

boundary layer displacement thickness, δ d =

h/2

[1 − U / U ] dy o,c

0

θm

boundary layer momentum thickness, θ m =

h/2

[1 − U / U ] [U / U ] dy o ,c

o,c

0

ν

kinematic viscosity of the air (≈ 1.5 × 10-5 m2 s-1 at 20 °C ambient temperature)

INTRODUCTION Plane jets have received significant attention after the seminal work of Schlichting [1], e.g. Heskestad [2], Bradbury [3] and Gutmark & Wygnanski [4]. This is mainly because of their two-dimensional nature, leading to potential applications in numerical modeling and validation of turbulence models (Gouldin et al. [5]), heat and mass transfer applications in air curtains (Stephane et al. [6]) and ventilation and air conditioning units (Moshfegh et al. [7]). In a laboratory experiment, a plane jet is produced by a slender rectangular slot of dimensions w × h and two parallel plates, known as sidewalls, attached to the slot’s short sides. The configuration ensures a mean jet propagation in streamwise (x) direction, spread in the lateral (y) direction and no entrainment in the spanwise (z) direction due to the presence of sidewalls parallel to the x-y plane. Such a configuration has been found to result in statistical two-dimensionality over a

reasonably large downstream distance, depending upon the nozzle aspect ratio, AR ≡ w / h (Deo et al. [8]). In most cases, smoothly contoured plane nozzles have been used to produce a “top-hat” velocity profile [3,4] and a laminar flow state at the nozzle exit, while some have adopted a sharp-edged orifice-like slot [2, 9], which produces a saddle-backed velocity profile. However, there are very few studies using a plane jet issuing from a sharp-edged orifice-plate (Wilson & Danckwerts [10]) perhaps due to its initial and near field flow structure being far more complex (e.g. existence of a vena contracta) than that from a smoothly contracting plane nozzle. Nevertheless, due to the simplicity of its design and manufacture, investigations on most nonplanar and non-circular jets have employed sharp-edged orifice-plates [11 – 13].

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Experimental evidence reveals that nozzles of different geometry produce significantly different downstream flows [14] and the choice of each configuration depends on the application. In addition, although the impacts of initial conditions on downstream flow are becoming well-known [15, 16], seldom, if ever, has any study investigated plane nozzles of different geometry in identical flow facilities. To address this need, the present study aims to report the statistical behavior of a plane jet by exploring the impacts of varying nozzle contraction profile, as characterized by the parameter r/h, where r is the nozzle’s inner-wall contraction radii. Our choice of varying the nozzle-exit geometric profile is motivated by a number of round and plane jet investigations, which provide substantiating evidence that a jet‘s downstream behavior is significantly governed by upstream (exit) conditions. While comparing the scalar mixing fields from three types of (axisymmetric) nozzle geometries, Mi et al. [13] found the highest mean scalar decay and highest frequency of the primary vortex formation from a sharp-edged orifice plate, followed by a smoothly contoured nozzle and the lowest for a pipe-jet. Likewise, in another comparison of the same three flow types, Mi & Nathan [17] concluded that the highest velocity decay rate occurs in the flow from a sharp-edged orifice plate, following by a smoothly contoured nozzle and the lowest for a pipe-jet. Other investigators, e.g. Antonia & Zhao [16], Hussain & Zedan [18] studied smoothly contracting axisymmetric nozzles and pipe-jets, arriving at similar conclusions. Further information is available from Mi et. al. [19], whose measurements using jets from nine different nozzle geometries revealed that the breakdown of axisymmetry of the initial jet generally results in an increase in both the mean velocity decay rate and in the root-mean-square of velocity fluctuations. From measurements in a plane jet, Hussain & Clark [20] found that an initially laminar boundary layer results in a higher rate of change of the mass flux (spreading rate) and the attainment of asymptotic state closer to the exit plane. In a similar investigation using Schlieren 4

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photography and spectral analysis, Chambers et al. [21] found more organized and symmetric large-scale shear layer structures to be dominant in the initially laminar jet than in the initially turbulent one. While a higher mixing rate was noted for the laminar case, the initially turbulent case produced more three-dimensional and asymmetric structures, indicating that the nature of initial boundary layer controls the development of large-scale structures. Goldschmidt and Bradshaw [22] studied the effect of exit centerline turbulence intensity on the flow field of a plane jet. Importantly, they found a larger jet-spreading angle for jets of higher exit turbulence intensity. This stands in contrast to the spreading rate of a pipe-jet, which has higher exit turbulence intensity than a smooth contraction nozzle, but a lower spreading rate. Likewise, Hussain [23] noted some dependence of primary vortex formation on nozzle geometry from flow visualization of an initially laminar plane jet. Similarly, Russ and Strykowski [24] found that as the boundary layer thickness at exit plane increases, vortex shedding and pairing occurs further downstream, leading to a reduction in both the potential core lengths and the transport of jet momentum. Likewise Eaton and Johston [25] provided an evidence of the influence of initial boundary layer thickness on downstream development of free shear flows. A more recent study conducted by Ali and Foss [26] found that the geometric design of plane nozzles produce an influence on the discharge properties of submerged plane jets. Their study revealed that for plane jets having a Reynolds numbers greater than 1500, the shape of the downstream portion of the nozzle produces an influence on the entrainment rate of the plane jet by altering its pressure field at the exit plane, although they did not attempt to quantify the extent of this influence by using different nozzle-exit geometric profiles. From the above discussion, it is evident that the nozzle shape influences the initial boundary layer characteristics to an appreciable extent, so causing significant differences in the evolving flow properties of both plane and round jets. However, no systematic study appears to have been undertaken of the influence of nozzle contraction profile on a plane jet, such as by changing the parameter r/h. This information is of fundamental and practical relevance for the

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development and validation of turbulence models and for the optimization of nozzle designs. In addition, it is useful to determine what, if any, radius of contraction is necessary to closely approximate a “top-hat” exit velocity profile, since radially contoured nozzles are easier to design, construct and configure than are conventional smoothly contoured ones. With this view, we assess the influence of r/h on the mean flow and turbulent statistics of a plane jet up to a downstream distance of 85h using five nozzles of different r but identical h.

EXPERIMENT DETAILS The plane nozzle facility, shown schematically in Figure 1, consists of an open circuit wind tunnel, flow straightening elements including a honeycomb and screens, and a smooth contraction exit of dimensions 720 mm × 340 mm. The honeycomb composed of drinking straws, with its cells aligned perpendicular to the main stream, and assists in reducing velocity fluctuations along the transverse direction, while producing minimum effect on streamwise velocity because of the small pressure drop. Likewise, the screens help reduce the velocity defect in the turbulent boundary layer that passes through it, further streamlining the incoming airflow. The present screens have an open area ratio of approximately 60% and the smooth contraction is based on a polynomial curve. Two flat plates were mounted to the end of the wind tunnel contraction (Fig. 1b), with radially contracting long-sides of four different exit radii and two parallel plates, as sidewalls, attached to the slot’s short sides to create a plane nozzle. The slot height of the nozzle was fixed at h = 10 mm and the width (separation between the sidewalls) was kept at w = 720 mm, producing a large aspect ratio plane nozzle of w/h = 72. The inner-wall radius of the nozzle exit (r) was varied from r = 4.5 to 36 mm by a factor of 2 for each case, as shown in Fig. 1(b); resulting in four radially contoured nozzles of r h = 0.45, 0.90, 1.80 and 3.60. During measurements, the nozzle contraction faced upstream for these four cases. In addition to these four nozzles, a fifth configuration of r/h ≈ 0 was achieved by

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reversing the orientation of the orifice plates of r/h = 0.45, so that it opens out downstream. The symbol “≈” is used in recognition that this configuration is somewhat different from the conventional sharp-edged (45° beveled) configurations more commonly employed (Heskestad [2], Van der Hegge Zijnen [9], Quinn [11]). Since it is not possible to achieve any configuration with an identically zero radius, and, given the sensitivity of a flow to inlet conditions, it is probable that subtle differences may exist for all types of “sharp-edged” orifice plates. Further, the presently chosen configuration does have a sharp right-angle opposite the contraction side of the plate, and does give consistent (though probably not unique) trends in the data. For these reasons, we have also chosen to use dashed lines (---) to connect data points (shown later in results) between the configurations r/h = 0.45 and r/h ≈ 0. The plane jet facility, located in an acoustically noise-free, fluid mechanics laboratory of dimensions 18 m (long) × 7 m (wide) × 2.5 m (high), was mounted horizontally, with the plane nozzle located at the mid point between the floor and ceiling. Throughout the present investigation, great care was taken to ensure that the experimental facility remained isolated from any external disturbances. The distance from the jet exit to the front wall of the laboratory was approximately 1400h and between the jet and ceiling/floor was approximately 125h, allowing the unheated jet to discharge freely into still air. Based on the approach of Hussein et al. [27], the effects of room confinement is estimated to produce less than 0.5% momentum loss for all plane jets at a downstream distance of 85h. Hence the present jets closely approximate plane jets in an infinite environment. For all cases of the present investigation, the jet discharge bulk mean velocity was kept fixed at U o,b ≈ 27 ms-1, resulting in a Reynolds number based on slot-height (h) and kinematic viscosity (ν) of Reh ≈ 1.80 × 104. The velocity measurements were performed over the flow region 0 ≤ x/h ≤ 85 using a single hot-wire anemometer, under isothermal conditions of ambient temperature 20.0 °C ± 0.1 °C. To avoid aerodynamic interference of the prongs on the hot wire, the present probe was carefully mounted perpendicular with prongs parallel to the plane jet. In addition, the hot wire 7

ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

was aligned properly so that it corresponds closely to the streamwise component of the flow velocity to minimize directional ambiguity, although it is not possible to eliminate this effect completely. The single hot-wire anemometer, if used with caution, encounters reduced errors when compared with dual or triple wires, in which the adjacent probe can probably influence the measured velocity (Jorgensen [28]). The hot-wire (tungsten) sensor was 5 µm in diameter and 0.8 mm in length, aligned in parallel to the long nozzle sides. The overheat ratio of the wire was 1.5 and square wave test revealed a maximum frequency response of 15 kHz. Hot-wire calibrations were conducted using a standard Pitot-static tube, placed side by side with the hotwire probe, at the jet’s exit (x/h ≈ 0), where the turbulence intensity was approximately 0.5%, before and after measurements of each case of r/h. The low initial turbulence intensity, as documented by Stainback and Nagabushana [29], is a necessity for an accurate calibration procedure. Both calibration functions were tested for discrepancies, and if velocity drift exceeded 0.5%, the experiment was repeated. No further corrections to the velocity measurements were applied. Thus it is expected that measurements away from the centerline (towards the outer region of the jet) are significantly in error, because of high velocity fluctuations relative to the mean value. Nevertheless, the central aim here is to compare the measurement of one case of r/h with another, with most data taken on the jet centerline. While converting data points from voltages to velocities using a fourth order polynomial curve similar to the one proposed by George et al. [30], the average accuracy of each calibration function was found to be ± 0.2%. The signals obtained were low-pass filtered with an identical cut-off frequency of fc = 9.20 kHz to eliminate high frequency noise at all the measured locations. The voltage signals were offset to within 0 – 3V (as a precautionary measure to avoid signal clipping [31]) and amplified appropriately through the circuits, and then digitized on a personal computer at fs =18.4 kHz via a 16 channel, 12-bit PC-30F A/D converter (Fig. 1c) of signal input range 0 – 5V. The sampling duration was approximately 22 s, during which 400,000 instantaneous data points 8

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were gathered. Using the inaccuracies in calibration and observed scatter in present measurements, the uncertainties are estimated to have a mean error of ± 4% at the outer edge of the jet and ± 0.8% on the centerline. The errors in the centerline root-mean-square (rms) velocities were found to be ± 1.8% and in the skewness (Su) and flatness (Fu) factors up to 3% and 8% respectively. The errors in the momentum integral quantities and jet virtual origins are estimated to be up to 8%.

CHARACTERIZATION OF JET EXIT FLOW The exit flows of each plane nozzle were characterized by measuring the velocity profiles at x/h ≈ 0.2 for r/h = 0.45 – 3.60 and at x/h = 1 for r/h ≈ 0 along the lateral (y) direction over the range –0.60

ξ

0.60, where ξ = y/h. These are presented in Figure 2(a-e). Herein, Uo,c

is the exit centerline mean velocity. Clearly, the exit velocity profiles depend on r/h and specifically undergo a substantial transition from being saddle-backed for r/h ≈ 0 (Fig 2a) to closely approximate a “top-hat” profile for r/h = 1.80 and 3.60 (Figs. 2d & e). Although the present case for r/h ≈ 0 cannot be classified as a conventional sharp-edged orifice plate, the saddle-backed velocity profiles indicate that the reversing of the nozzle plates of smallest contraction radius (Fig 1 b (i), r/h ≈ 0) produce an exit flow that behaves closely to that of a sharp-edged orifice plate. It is also interesting to note that the exit velocity profiles for the cases

r/h ≤ 0.90 have the highest velocity located towards the edge of the jet, resulting in the observed saddle-back. Apparently the cases r/h ≤ 0.90 generate vena contractas immediately downstream from the exit plane. Interestingly, the smaller the value of r/h, the greater is the velocity deficit on the centerline relative to the maximum value close to edge of the jet. This trend suggests that the smaller the radius of contraction, the more closely will the nozzle configuration approach a sharp-edged orifice plate. The presence of vena contractas for nozzles of r/h ≤ 0.90 are also broadly consistent with a number of previous investigations such as Quinn [11], Mi & Nathan [12], Mi et al. [13] and Tsuchiya [32], all of which found vena contractas in jets issuing from 9

ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

sharp-edged orifice plates. The exit flow appears to be uniform within the region |ξ| ≤ 0.45 for the cases r/h = 1.80 and 3.60. There is also a consistent trend in the initial turbulence intensity profiles, u n' = < u 2 >1 / 2 / U o,c , with changes in r/h (Fig. 2, Table 1). As r/h is increased from 0.45 to

3.60, the turbulence intensity in the shear layer decreases from about 17% to 4%, as does that in the middle of the jet, from un'≈ 2.0% for r/h = 0.45 to un'≈ 1.5% for r/h = 3.60. This observation stands in contrast to round orifice plates (Mi et al. [33]), which typically produce weaker velocity fluctuations and thus lower rms values. Such a difference underlines the fundamental influence of nozzle exit geometry (i.e. planar versus round) on the exit velocity field of these two jets. An estimate of the initial boundary layer characteristics is indispensable for an assessment of the veracity of the downstream flow statistics, since jet development has previously been found to be dependent on exit conditions (Antonia & Zhao [18], Hussain and Zedan [20], Hussain & Clarke [20]). Using Figure 2, the exit conditions are characterized by estimating the boundary-layer displacement (δd) and momentum (θm) thickness using momentum integral equations; δ d =

y =h / 2

y =h / 2

y =0

y =0

[1 − U / U o,c ] dy and θ m =

[1 − U / U o,c ] [U / U o,c ] dy

(Table 1). To compensate for the comparatively low measurement resolution and paucity of data points through the boundary layer, a best-fit spline curve was used to perform the numerical integration of two momentum equations, on both sides of each velocity profile (i.e. from y = 0 to

h/2 and y = - h/2 to 0), yielding two independent values of δd and θm. Both were then averaged to further reduce errors. Note also that the case r/h ≈ 0 has an entirely different geometry, which required that the measurements be conducted further downstream (x/h = 1). This prevented a reliable estimate of its boundary layer thickness.

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Table 1 reveals that as r/h is increased from 0.45 to 3.60, the boundary layer thickness,

δd increases from 0.054h to 0.151h, θm increases from 0.030h to 0.061h and so does the Reynolds number based on θm. It is thus expected that these variations in the exit flow, which control the shear layer development, have a significant influence on downstream behavior of present plane jets. This is borne out in the data presented in the next section. Table 1

Initial boundary layer characteristics obtained at x = 0.2h.

r/h

δd

θm

H = δd / θm Re θm = U o ,b θ m / ν uc'/ U c (%)

u 'p / U c (%)

0

-

-

-

-

-

-

0.45

0.054h 0.030h

1.80

551

2.0

16.8

0.90

0.068h 0.035h

1.95

642

1.9

8.1

1.80

0.127h 0.052h

2.44

955

1.9

5.5

3.60

0.151h 0.061h

2.48

1120

1.5

3.9

The maximum cumulative error in momentum integral quantities is 8%. This, combined with the consistent trends evident in Table 1, gives confidence in the results. However, it must be noted that the measurements were performed at x/h = 0.2, so that the values will differ somewhat from the actual exit values. The corresponding shape factors (H = δd / θm), which are often used to determine the flatness (uniformity) of the mean velocity profiles (Pope [34]), possess values between 1.80 and 2.48, compared with a value of 2.60 for a true Blasius exit velocity profile. Thus the present plane nozzles of r/h = 1.80 and 3.60 may be characterized as having an initially laminar boundary layer, since the shape factors closely resemble those of a Blasius velocity profile (Schlichting [35]).

STATISTICAL PROPERTIES OF THE DOWNSTREAM FLOW Figure 3 shows the near field evolution of mean centerline velocity, Uc, normalized by exit bulk mean velocity, Uo,b. As expected, there is a consistent dependence of U c / U o,b on 11

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r h , in particular, the case of r h ≈ 0 being discernibly different from that for other cases. A

hump in U c / U o,b (at x h ≈ 2 for the cases 0 ≤ r h ≤ 0.90) becomes obvious, although its magnitude tends to decrease as r h increases. This hump is yet another feature which unambiguously supports the existence of vena contractas for nozzles of r h ≤ 0.90 and is

consistent with Fig 2(a-c). Figure 3 also shows that, for x h > 6, the decay rate of the mean

centerline velocity depends on r h , with the jet issuing from r h ≈ 0 decaying at the highest

rate. This trend appears to be consistent with previous findings of round orifice-jets (Mi et al. [13]). An assessment of the asymptotic-like dependence of the near field hump of U c / U o,b on r h (at x h ≈ 2) is shown in Figure 4. That is, as r h is increased from 0 to 3.60, the ratios of

the maximum centerline velocity, U m,c and exit bulk mean velocity, U o,b are found to decrease asymptotically from approximately 1.30 to 1.00. Interestingly, the nozzle profile with r h ≈ 0 produces U m,c U o,b ≈ 1.30. This value is lower than U m,c U o,b ≈ 1.55 found by Quinn [11]

from a sharp-edged orifice-plate, although the trends are consistent (Fig. 3 and 4). Taken together, Figures 2 – 4 provide sufficient evidence to show that the present case of r h ≈ 0

produces an exit and near-field flow structure that is qualitatively similar to other sharp-edged orifice plate flows, with the subtle differences attributable to the differences in the nozzle designs. To investigate how the primary vortex shedding in the near field varies with changes in

r/h for the present cases, we have analyzed the centerline velocity spectra, Φ u ( f * ) measured at x h = 3, where the normalized vortex shedding frequency, f * ≡ f h / U o ,b and the integral Φu ( f * ) d f * = 1.

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The spectra (Figure 5) reveal that there are clear differences in the underlying flow structure of all five jets. Each jet exhibits a broad peak in Φ u ( f * ) , revealing the periodic passage of primary vortices in the near-nozzle region. The processes of vortex formation and growth in the near region of plane jets are well established. For instance, it is well-known that 2D roller-like counter-rotating vortices dominate the shear layers which bound the potential core (Browne et al. [36]). Thus the present spectra clearly confirm the regular occurrence of primary vortices from all the tested nozzles. Similarly, flow visualizations of a round jet from a sharpedged orifice-plate by Mi et al. [13] revealed well-defined coherent vortices along their potential core region. In their smoke visualization experiments, Tsuchiya et al. [32] noted axially symmetric vortices within 0 – 4 nozzle widths downstream. The mechanism leading to vortex formation immediately downstream from the nozzle exit is a known feature (Namar & Otugen [37]) and so is the fact that the unstable laminar shear layers roll up to produce the primary vortices. During their streamwise propagation, the vortices convect the irrotational ambient fluid into the jet. Early observations of plane jets by Brown [38] and Beavers & Wilson [39] found that the symmetrical vortices occur on alternate sides of a plane jet. Their successive growth into larger and larger vortices through coalescence with adjacent vortices (Rockwell & Niccols [41]) causes them to eventually breakdown as they propagate downstream. The process of coalescence typically depends on the exit conditions, as can be seen from the work of Sato [41], who found that an externally driven noise of a frequency close to that of the natural vortex shedding frequency causes vortices to grow and coalesce closer to the nozzle exit. The dependence of vortex dynamics on initial conditions is also well-known, e.g. Gutmark & Grinstein [42]. Collating from the past and present work, it becomes apparent that as the exit conditions are varied by changing r h from 3.60 to 0, the normalized vortex shedding frequency, St h increases from approximately 0.24 to 0.39. Recall from Figure 2 and Table 1 that the boundary layer gets thicker as r h increases. Therefore, a thinner boundary layer, with more

concentrated vorticity, results in a higher formation rate of the primary vortices for the case r h 13

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≈ 0 (Mi et al. [13]). As evidenced, different nozzle exit-geometric profiles probably generate structurally different vortices, which convect downstream at different rates. Also note that the exit and near-field centerline turbulence intensity is higher for smaller values of r h (Fig. 2 and

Fig. 12 shown later). This indicates that a higher formation rate of the primary vortices (found for smaller r h ), is associated with larger velocity fluctuations. However, a higher frequency is typically associated with a smaller scale of the vortex motion, indicating that the higher turbulence intensity suggests a greater variability in the instantaneous location of the vortex cores. Investigation

Geometry

Nozzle Profile

Re

AR

f*

Beavers & Wilson [39]

plane

orifice-plate

500-3,000

-

0.43

Tsuchiya et al. [32]

rectangular

orifice-plate

3,500

5

0.40

Sato1 [41]

plane

channel

1,500-8,000

10-67

0.23

Namar and Otugen [37]

rectangular

contoured

1,000-7,000

56

0.27

present, r/h ≈ 0

planar

orifice-like

18,000

72

0.39

present, r/h = 1.80, 3.60

planar

radially contoured 18,000

72

0.24

Beavers & Wilson [39]

round

orifice-plate

500-3,000

-

0.63

Johansen [42]

round

orifice-plate

200-1,000

-

0.60

Mi et al. [13]

round

orifice-plate

16,000

-

0.70

Ko and Davis [43]

round

contoured

-

-

0.20

Crow & Champagne [44]

round

contoured

10,500-30,900 -

0.30

Mi et al. [13]

round

contoured

16,000

0.40

Table 2

1

The normalized vortex shedding frequency, configurations.

-

f* for previous jets of round, rectangular and plane

Note: Sato [41] used a contoured planar nozzle with an upstream channel of length between 300-1100 mm.

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Table 2 assembles the f* data from previous investigations of round and plane nozzles. Also listed are the key initial conditions. The present measurement of f* = 0.39 using r h ≈ 0 is in good agreement with the values f* = 0.43 and 0.40 measured by Beavers & Wilson [39] and Tsuchiya et al. [32] using plane and rectangular nozzles, respectively. This close comparison provides further support that the geometry of r h ≈ 0 produces a flow structure similar to those

from other sharp-edged orifice-plates. However, all the f* values are significantly higher than a value of 0.23 measured by Sato [41] for a channel (analogous to a pipe). This difference reflects the key role that a nozzle’s geometry plays in vortex formation. Importantly, the values of 0.27 and 0.24 measured by Namar and Otugen [37] using smoothly contracting nozzles and the present value for r h = 1.80 and 3.60 are in good agreement too, confirming that they closely

approximate other types of smoothly contoured plane nozzles. Our analysis has, so far, unambiguously established that there are several distinctions in the initial and near field flow structure of plane jets with changes in r/h. This prompts us to assess the influence of nozzle-exit contraction profiles on the far field flow of the present plane jets. Figure 6 presents the far field mean centerline velocity, Uc, normalized by the exit bulk mean velocity, Uo,b. It is revealed that in the self-similar region, U c ~ x −1 / 2 , leading to the usual relationship of the form U o ,b Uc

2

= Ku

x x 01 + h h

[1],

where K u is the velocity decay rate and x01 is its virtual origin location. As with the near field case, the decay rates of the far field mean centerline velocity reveal a consistent dependence on r h , with the nozzle of r h ≈ 0 exhibiting the highest far field velocity decay. The velocity

decay rates, shown explicitly in Figure 7, exhibit an asymptotic-like convergence toward a single curve as r/h approaches 3.60. While the differences between the cases r h = 1.80 and

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ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

3.60 are within experimental uncertainty, the trend is consistent, both internally and with other data presented later. Next presented are the lateral distributions of the normalized mean velocity at a few downstream locations. These are shown in Figure 8 for r h ≈ 0, r h = 0.45, 0.90 and 3.60. The

mean velocity profiles become approximately self-similar at x h = 20 for r h ≈ 0, which is

significantly further downstream than x h = 5 for r h = 3.60. That is, the downstream distance

required for the lateral profiles of the mean velocity to achieve self-similarity decreases as r h is increased. A larger downstream distance is thus required for the velocity profiles to collapse on to each other for the nozzle which closely approximates to an orifice plate (i.e. the case r h

[

]

≈ 0). All the self-similar profiles conform closely to a Gaussian relation, U n = exp − ln 2 ( y n )2 .

Likewise, the streamwise variations (Fig. 9) of the normalized velocity half-widths, y 0.5 / h , conform to a far-field relationship of the form; y 0.5 x x 02 = Ky + h h h

[2],

where K y is the spreading rate and x02 is the virtual origin of the half-width. Clearly, the different values of r/h produce different values of y 0.5 / h , confirming that the jet spreading angles differ for each nozzle geometry. Figure 10 indicates that the spreading rate appears to decrease asymptotically as r h is increased from 0 to 3.60, with the highest spread rate for r h ≈ 0. This trend, in turn, coincides with the measured trends in the decay of Uc (Figure 7), and

thus shows internal consistency of the present data. The magnitude of the virtual origins, x01 and x 02 (each with uncertainty of approximately 8%) is found to increase asymptotically with r h (Figure 11), although with greater scatter as expected. That is, the nozzle of r h ≈ 0 has the smallest of these virtual

origins, consistent with the presence of a vena contracta (Figs. 3, 4) for this case. A dependence 16

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of virtual origins on initial conditions was also revealed by Gouldin et al. [5]. Importantly, Flora & Goldschmidt [45] noted that their virtual origin moved upstream with a relatively modest increase in exit turbulence intensity from 1.06% to 1.28%. This trend again, is consistent with the present results, where an increase in the initial turbulence intensity from approximately 1.7% to 2.3% leads to a translation of the virtual origins from x01/h ≈ 3.9 to ≈ 0.4 and x02/h ≈ 4.7 to ≈ 2.0. There appears to be a small difference between the values of x01 for the cases r/h = 1.80 and r/h = 3.60, confirming that the flows from these configurations are very similar. At this point in discussion, one may ask how distinct the turbulent velocity field is for plane jets measured at different values of r h . Figure 12 addresses this question by presenting

the streamwise evolutions of locally normalized turbulence intensity, u n',c = u c' / U c , for different values of r h , where u c' = < u 2 >1 / 2 . As for the mean velocity field, the turbulent field exhibits

a consistent dependence on r h , providing a clear indication that the nozzle exit geometric

profile has an influence on the turbulent velocity field. This dependence is, as expected, greatest in near field but does not vanish in the far field. In general, the centerline turbulence intensity decreases as r h is increased. The initial rapid increase of u n',c is a distinct feature of all plane jets, reflecting the streamwise growth of the shear-layer instability (Antonia et al. [47]) due to the large-scale structures, perhaps similar to those evidenced from the plane jet flow visualizations of Gordeyev1 & Thomas [47] and Shlien & Hussain [48]. It is these large-scale structures which are responsible for large-scale engulfment of the ambient fluid, higher velocity fluctuations and higher decay of mean velocity, and thus high turbulence intensity. It is also deduced from previous work that the far field flow is influenced by propagation of these structures (Namar & Otugen [37]), and the dominance of large-scale structures diminish as they convect downstream due to the generation of a broader range of smaller eddies. The different shape in the evolution of u n',c for jets of different r h implies differences in the underlying large-scale structures of these jets. A distinct `hump'in u n',c is found at x h ≈

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13 for r h ≈ 0. A hump in turbulence intensity is probably associated with increased,

intermittent incursions of low-velocity, predominantly ambient, fluid at this location, causing higher velocity fluctuations relative the mean values. In the self-similar far field ( x h > 30), the centerline turbulence intensity clearly

depends on r h . This dependence is highlighted in Figure 13, which plots u c',∞ against r h .

Despite scattering, a consistent trend emerges; as r h is increased from 0 to 3.60, u c',∞ decreases

in an asymptotic-like manner with r/h. This systematic dependence of u c',∞ on r/h indicates that the state of fully developed turbulent flow is dependent on the nozzle-exit geometric profile. Further evidence of the dependence of the turbulent velocity field on nozzle-exit contraction profile is given by the lateral distributions of turbulence intensity, u n' = < u 2 >1 / 2 / U c as shown in Figures 14(a-d). There are clear differences resulting from changes in the values of r/h. Consistent with the trends in the lateral profiles of the mean velocity, the axial distance at which turbulence intensity profiles become self-similar increases with r h . For instance, when r h ≈

0, the axial distance required for the attainment of self-similarity of u ' n is at x/h = 40, whereas for r h = 3.60, this distance is reduced to x/h = 10. This indicates that the development of the large-scale structures in the outer shear layers depend on the nozzle-exit geometric profile as well. The dependence of the flow statistics on r/h may further be examined using the moments of higher order velocity fluctuations. Figures 15 and 16 present the centerline evolutions of the

(

skewness, S u = < u 3 > < u 2 >

)

3/ 2

(

and flatness (kurtosis), Fu = < u 4 > < u 2 >

)

2

of all five jets.

Note that S u and Fu were determined from a large sample of approximately 400,000 data points of the instantaneous velocity, so the convergence of the calculations is good and an appropriate voltage offset applied to the analogue-to-digital range ensured both factors were not truncated due to clipping effects arising from finite input range of our sampling board (Tan18

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Atichat et al. [31]). The profiles are also vertically offset by unity, and the ordinate is drawn on a logarithmic scale for clarity. Both factors evolve from nearly Gaussian values ( S u , Fu ) = (0, 3) at the origin, to highly non-Gaussian values, around 4 < x h < 6, consistent with previous work. For example, see Browne et al. [36], who found that their passive temperature fluctuations at the exit of a plane nozzle were Gaussian, while those within the potential core region (between 35h) were highly non-Gaussian. A departure from Gaussian values is typically interpreted to result from the presence of coherent, non-random motions due to the growth of the large-scale roller-like structures in the shear layers. The near field trends of S u and Fu , which are governed by r/h, reflect a dependence of the downstream convection of the underlying large-scale shear layer structures, on source (exit) conditions. Also importantly, the absence of “potential cores” for cases of small r/h (e.g. r/h = 0, 0.45 and 0.90) implies more rapid development of large-scale structures through its shear-layer, increased fluid entrainment and quite possibly, more coherent large-scale structures. To inspect the variations of the minima (Sumin) and maxima (Fumax) of both factors due to changes in r/h, we have plotted their relative magnitudes in Figure 17. Despite some scatter, a clear asymptotic-like dependence of both values on r/h is evident, with the cases

r/h = 1.80 and 3.60 occupying closer values. The present nozzles of small r/h register larger values of skewness and kurtosis, indicating that the near field flow encounters higher instabilities, perhaps due to greater incursion of low-velocity ambient fluid, when compared with nozzles of larger r/h. In the interaction and fully developed region (i.e. x/h > 20), both factors approach, but do not reach, truly Gaussian values. The departure of the moments of higher order statistics from their respective Gaussian values are consistent with Browne et al. [36], whose passive scalar measurements were non-Gaussian in the self-similar field.

CONCLUSIONS

In summary, the statistical properties of the present jets were found to depend systematically on the nozzle-exit contraction profiles measured over the range 0 < r/h < 3.60. The results provided

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consistent differences that were evident from the exit velocity profiles, from the near-field, into the far field, and are deduced to be associated with differences in the underlying flow structure of jets with different nozzle-exit geometric profiles. The exit velocity profiles were found to depend systematically upon r h , with a gradual

transition from being saddle-backed for the nozzle contraction profile close to a “sharp-edged” orifice plate (e.g. for the case r/h ≈ 0) to approximately top-hat for the “radially” contoured nozzles (for the cases r/h = 1.80 & 3.60). For the cases r h

0.90, the mean exit velocity

profiles exhibit a “saddle-back”, a shape which characterizes sharp-edged orifice plates. Importantly, these configurations exhibit vena contractas, indicating upstream flow separations of the main stream. However, the extent of the departure of these velocity profiles from being top-hat, as characterized by the ratio of the maxima in Uc, to the exit bulk mean velocity, Uo,b, decreased in an asymptotic-like manner with r h . When the exit-contraction radii was sufficiently large, which occurs for the cases r h = 1.80 and 3.60, the mean exit flow was found

to conform closely to a top-hat velocity profile. Likewise, the thickness of the initial boundary layer was found to increase monotonically with an increase in r h , while its peak turbulence intensity decreased. The power spectra of the

centerline velocity fluctuations revealed that the near field vortex shedding frequency decreases monotonically, from having value of f* = 0.39 for r h ≈ 0 to f* = 0.24 for r h = 1.80 and 3.60. In the self-similar far field, the rates of centerline velocity decay and jet spread were found to decrease in an asymptotic-like manner with an increase in r h , so that the differences between

r h = 1.80 and 3.60 were small. The streamwise turbulence intensity revealed a distinct hump for the case r h = 0 near to x h = 13, while no significant hump was found for the radially

contoured nozzles ( r h = 1.80 and 3.60). The far field values of turbulence intensity also

decreased in an asymptotic-like manner as r h was increased to 3.60. However, one would probably expect that a further increase in r/h towards infinity would cause the jet properties to 20

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depart from those of a top-hat exit flow, to converge towards a fully-developed channel flow. This study, although a useful subject for further investigation, was beyond the scope of the present work, and therefore awaits another independent study. The collective findings from present work, together with the proposed hypothesis of George [15], experimental work of George & Davidson [49] and recent measurements of Deo [50] confirm that the downstream development of any plane jet is entirely controlled by its exit boundary conditions. The classical theory, which argues that the influence of initial conditions decays with downstream distance, and eventually becomes unimportant, is therefore questionable for a plane jet. In other words, even in the fully developed state, a plane jet does not ‘forget’ its origin. Therefore, at sufficiently large distances from the jet source, the perception that all jets should become asymptotically independent of source conditions and that the jet properties will depend only on the rate at which momentum is added and the distance from its source, is not valid for a plane jet.

ACKNOWLEDGEMENTS The paper is taken from the primary author’s PhD thesis undertaken at the School of Mechanical Engineering at The University of Adelaide, South Australia. It was supported by an international postgraduate funding, The Adelaide University Achievers Scholarship and an ARC Linkage Grant in partnership with FCT-Combustion.

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REFERENCES 1. H Schlichting, “Laminare strahlausbreitung”, Z. Angew. Math. Mech. 13, 260–263, (1933). 2. G Heskestad, “Hot-wire measurements in a plane turbulent jet”, Trans. ASME, J. Appl. Mech. 32, 721-734, (1965). 3. L J S Bradbury, “The structure of a self-preserving turbulent planar jet”, J. Fluid Mech. 23, 31–64, (1965). 4. E Gutmark and I Wygnanski, “The planar turbulent jet”, J. Fluid Mech. 73(3), 465–495, (1976). F C Gouldin, R W Schefer, S C Johnson and W Kollmann, “Non-reacting turbulent mixing flows”. Prog. Energy Combust. Sc, 12, 257-303, (1986). M Stephane, S Camille and P Michel, “Parametric analysis of the impinging plane air jet on a variable scaled-down model”, In Proc: ASME/JSME Fluids Engineering Division Summer Meeting, Vol. F-227, Boston, Massachusetts, (2000). 7. B Moshfegh, M Sandberg and S Amiri, “Spreading of turbulent warm/cold plane air jet in a well-insulated room”, http://www.hig.se/tinst/forskning/em/spreading-of-turbulent.htm,

Supported by: University of Gavle and K K-Foundation, (2004). 8. R C Deo, G J Nathan and J Mi, “The influence of nozzle aspect ratio on plane jets”, Expl. Therm. Fluid. Sci. Accepted in May, In Press, (2006). 9. B G Van der Hegge Zijnen, “Measurements of the distribution of heat and matter in a plane turbulent jet of air” Appl. Sci. Res. A7, 277–292, (1958). 10. R A M Wilson and P V Danckwerts, “Studies in turbulent mixing – II. A hot air jet”, Chem. Eng. Sci. 19, 885-895, (1964) 11. W R Quinn, “Development of a large-aspect ratio rectangular turbulent free jet”. AIAA J. 32(3), (1994). 12. J Mi and G J Nathan, “Mean velocity decay of axisymmetric turbulent jets with different initial velocity profiles”, In Proc: 4th Int. Conf. on Fluid Mechanics, Dalian China, (2004). 22

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13. J Mi, G J Nathan and D S Nobes, “Mixing characteristics of axisymmetric free jets from a contoured nozzle, an orifice plate and a pipe”. J. Fluids. Eng. 123, 878-883, (2001). 14. E J Gutmark and F F Grinstein, “Flow control with non-circular jets”, Ann. Rev. Fluid. Mech. 31, 239-272, (1999) 15. W K George, “The self-preservation of turbulent flows and its relation to initial conditions”, In “Recent Advances in Turbulence”, Hemisphere, New York, 39-73, (1989). 16. R A Antonia and Q Zhao, “Effects of initial conditions on a circular jet”. Exp. Fluids. 31, 319-323, (2001). 17. J Mi and G J Nathan, “Effect of small vortex generators on Scalar Mixing in the Developing Region of a Turbulent Jet”, Intl. J. Mass Heat Transfer. 42, 3919-3926, (1999). 18. A K M F Hussain and M F Zedan, “Effect of the initial conditions of the axisymmetric free shear layer: effect of initial momentum thickness”. Phys. Fluids. 21, 1100-1112, (1978). 19. J Mi, G J Nathan G J and R E Luxton, “Centerline mixing characteristics of jets from nine differently shaped nozzles”, Exp. Fluids. 28, 93-94, (2000). 20. A K M F Hussain and A R Clark, “Upstream Influence on the near field of a planar turbulent jet”, Phys. Fluids. 20(9), (1977). 21. A J Chambers, R A Antonia and L W B Browne. “Effect of symmetry and asymmetry of turbulent structures on the interaction region of a plane jet”. Exp. Fluids. 3, 343-348, (1985). 22. V W Goldschmidt and P Bradshaw. “Effect of nozzle exit turbulence on the spreading (or widening) rate of plane free jets”, In Joint Engineering, Fluid Engineering and Applied Mechanics Conference, ASME, 1-7, Boulder, Colarado, June 22-24, (1981). 23. A K M F Hussain. “Coherent structures - reality and myth”, Phys. Fluids. 26, 2816-2850, (1983). 24. S Russ and P J Strykowski. “Turbulent Structure and Entrainment in Heated Jets: The Effect of Initial Conditions”, Phys. Fluids. A 5(12), 32 16-3225, (1993).

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25. J K Eaton and J P Johston, A review of research on subsonic turbulent flow reattachment, AIAA J., 19(9), 1093 – 1100, (1981). 26. S K Ali and J S Foss, “The discharge coefficient of a planar submerged slit-jet”, ASME J Fluids Eng., 125, 613-619, (2003). 27. H J Hussain, S P Capp and W K George. “Velocity measurements in a high Reynolds number momentum conserving axisymmetric turbulent jet”, J. Fluid Mech., 258, 31-75, (1994). 28. F E Jorgensen. “How to measure turbulence using hot wire anemometers – a practical guide”, Dantec Dynamics, Pub. No. 9040U6151., Skovlunde, Denmark, (2000). 29. P C Stainback and K A Nagabushana. “Review of hot-wire anemometry techniques and the range of their applicability for various flows”, Electr. J. Fluids Eng. Trans. ASME., (1993). 30. W K George, P D Beuther and A Shabbir. “Polynomial calibrations for hot wires in thermally varying flows”, Expl. Thermal Fluid. Sci., 2, 230-235, (1989). 31. J Tan-Atichat, W K George and S Woodward, “Use of computer for data acquisition and processing”, Handbook of Fluids and Fluids Engineering (A.Fuhs, ed.), 3, sec. 15.15, 10981116, Wiley, NY, (1996). 32. Y Tsuchiya, C Horikoshi, T Sato and M Takahashi. “A study of the spread of rectangular jets: (The shear layer near the jet exit and visualization by the dye methods)”. JSME Int. J. 32, Series II (1), 11-17, (1989). 33. J Mi, P Kalt and G J Nathan. “Velocity measurements in a circular jet from an orifice plate with high initial turbulence intensity”, Dynamics of Continuous, Discrete and Impulsive Systems. Accepted, (2006). 34. S B Pope. “Turbulent Flows”, Cambridge University Press, London, pp 303, (2000). 35. H Schlichting. “Boundary Layer Theory”, McGraw-Hill, Chap 7, (1968). 36. L W B Browne, R A Antonia, S Rajagopalan and A J Chambers. “The interaction region of a turbulent plane jet”. J. Fluid. Mech., 149, 355-373, (1984). 24

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37. I Namar and M V Ötügen. “Velocity measurements in a planar turbulent air jet at moderate Reynolds numbers”, Expts. Fluids., 6, 387-399, (1988). 38. G B Brown. “On vortex motion in gaseous jets and origin of their sensitivity in sound”, Proc. Phys. Soc. London. 47, 703-732, (1935). 39. G S Beavers and T A Wilson. “Vortex growth in jets”. J. Fluid Mech. 44(1), 97-112, (1970). 40. D O Rockwell and W O Niccolls. “Natural breakdown of planar jets”, Trans. ASME J. Basic. Eng. 94, 720-730, (1972). 41. H Sato. “The stability and transition of a two-dimensional jet”. J. Fluid Mech., 7, 53-80, (1960). 42. F C Johansen. “Flow through pipe orifices at low Reynolds numbers”. Proc. Roy. Soc. 126, 231-245, (1929). 43. N W M Ko and P O A L Davies. “The near field within the potential core of subsonic cold jets”, J. Fluid. Mech., 50, 49-78, (1971). 44. S C Crow and F H Champagne. “Orderly structure in jet turbulence”. J. Fluid. Mech., 48(3), 547-591, (1971). 45. J J Flora and V W Goldschmidt. “Virtual origins of a free plane turbulent Jet”. AIAA J., 7(12), 2344-2446, (1969). 46. R A Antonia, W B Browne, S Rajagopalan and A J Chambers, “On organized motion of a turbulent planar jet”, J. Fluid Mech. 134, 49–66, (1983). 47.

S V Gordeyev1 and F O Thomas, “Visualization of the topology of the large-scale structure in the planar turbulent jet”, In Proc: 9th International Symposium on Flow Visualization, 13, (2000).

48.D

J Shlien and A K M F Hussain “Visualization of large-scale motions of a plane Jet”, Flow

visualization III; In Proc: Third International Symposium, Ann Arbor, MI, September 6-9, Washington DC, 498-502, (1985).

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49. W K George and L Davidson, “Role of initial conditions in establishing asymptotic behavior”, AIAA J. 42(3), 438–446, (2004). 50. R C Deo, “Experimental investigations of the influence of Reynolds boundary conditions on a plane air jet”, PhD Thesis, School Engineering, The University of Adelaide, South Australia, (2005).

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Figure Captions Fig. 1

Schematic view of experimental setup showing (a) the wind tunnel and nozzle attachment (b) nozzles of contraction profiles denoted by (i) r/h ≈ 0, (ii) r/h = 0.45, (iii) r/h = 0.90, (iv) r/h = 1.80, (v) r/h = 3.60 and (c) other experimental apparatus. Note that sidewalls have been omitted for clarity, and diagrams drawn are not to scale.

Fig. 2

Lateral profiles of the mean velocity, U/Uo,c (denoted by the symbol -- --) and turbulence intensity, u’/Uo,c (denoted by the symbol –Ο–) for (i) r/h ≈ 0 at the x h = 1; (ii) r h = 0.45, (iii) r/h = 0.90, (iv) r/h = 1.80 and (v) r/h = 3.60 at the

x h = 0.2. Fig. 3

Near field evolution of the normalized centerline mean velocity, U c / U o,b for various cases of investigation.

Fig. 4

A dependence of the ratio of mean centerline velocity maximum, U m,c and exit bulk mean velocity, U o,b on r h obtained at x h = 3.

Fig. 5

Power spectra, Φ u (f*) of the centerline velocity fluctuations measured at x h =

3. Fig. 6

The normalized profiles of centerline mean velocity for different values of r h .

Fig. 7

The decay rates of the centerline mean velocity for r/h = 0 – 3.60.

Fig. 8

Lateral profiles of the mean velocity, U / U c for (a) r h ≈ 0 (b) r h = 0.45 (c) r h = 0.90 and (d) r h = 3.60.

Fig. 9

Streamwise evolutions of mean velocity half-width for different values of r h .

Fig. 10

The jet spreading rates for r h = 0 – 3.60.

Fig. 11

The jet’s virtual origins for r h = 0 – 3.60.

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ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

Fig. 12

Streamwise evolutions of locally normalized turbulence intensity, u c' / U c , for different values of r h .

Fig. 13

Variations of far field asymptotic turbulence intensity, u c',∞ with r h .

Fig. 14

Lateral profiles of the turbulence intensity, u '/ U c for (a) r h ≈ 0 (b) r h = 0.45 (c) r h = 0.90 and (d) r h = 3.60.

Fig. 15

Streamwise evolutions of the skewness, Su for different values of r h . Note that

each profile is shifted vertically by unity relative to its neighbour for clarity, and symbols are identical to Figure 12. Fig. 16

Streamwise evolutions of the flatness, Fu for different values of r h . Note that

each profile is shifted vertically by unity relative to its neighbour for clarity, and symbols are identical to Figure 12. Fig. 17

The dependence of the near-field minima in skewness, Sumin, and maxima in flatness, Fumax, on r/h.

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Figures (a)

(c)

Fig. 1 29

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(b)

(a) 1.25

0.25

1.00

0.20

1.00

0.20

0.75

0.15

0.75

0.15

0.50

0.10

0.50

0.10

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0.05

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0.05

U/Uo,c

U/Uo,c

0 0 -0.6 -0.3 0 0.3 0.6

0 0 -0.6 -0.3 0 0.3 0.6

ξ = y/h

ξ = y/h

(c)

(d)

0.25

1.25

0.25

1.00

0.20

1.00

0.20

0.75

0.15

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u' /Uo,c

u' /Uo,c

1.25

U/Uo,c 30

u' /Uo,c

0.25

u' /Uo,c

1.25

0 0 -0.6 -0.3 0 0.3 0.6

0 0 -0.6 -0.3 0 0.3 0.6

ξ = y/h

ξ = y/h

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(e) 1.25

0.20

0.15

0.75 0.10

0.50

u' /Uo,c

U/Uo,c

1.00

0.05

0.25

0

0

-0.6 -0.3

0

0.3

0.6

ζ = y/h Fig 2 (a-e)

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1.6

r/h = 0 0.45 0.90 1.80 3.60

1.4

Uc/Uo,b

1.2

Quinn [10] orifice-plate Quinn [11], orifice-plate (rectangular) (rectangular) nozzle

1.0 0.8 0.6 0.4

0

10

5

x/h Fig. 3

32

15

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1.6

present data Quinn [11]

Um,c/Uo,b

1.4

1.2

1.0

0.8

0

0.5

1.0

2.0

1.5

2.5

3.0

3.5

r/h Fig. 4

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0.1

(r/h, Sth) ≅ (3.60, 0.24) ≅ (1.80, 0.24) ≅ (0.90, 0.26) ≅ (0.45, 0.28)

φu(f *)

0.01

0.001

0.0001

(r/h, Sth) ≅ (0, 0.39)

0.00001

0

1.0

0.5

f * = f h / Uo,b Fig.5

34

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25

r/h = 0 0.45 0.90 1.80 3.60

(Uo,b/Uc)

2

20 15 10 5 0

0

10

20

30

50

40

60

70

80

90

x/h Fig. 6

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0.25 0.23

Ku

0.21 0.19 0.17 0.15

0

0.5

1.0

1.5 Fig. 7

36

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2.5

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(a) 2

Un= exp[-ln 2 (yn) ]

Un= U / Uc

1.0

0.8

0.6

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0.4

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0.2

0.2

0

0 0.5 1.0 1.5 2.0 2.5

0

yn= y/y0.5

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0.6

0.6

0.4

0.4

0.2

0.2

0

0

0.5

0.5 1.0 1.5 2.0 2.5

yn= y/y0.5

1.0

0

x/h = 3 5 10 20 40 80

1.0

0.8

0

Un= U / Uc

(b)

1.0

1.5

2.0

2.5

x/h = 3 5 10 20 40 80

0

0.5

1.0

1.5

2.0

2.5

yn= y/y0.5

yn= y/y0.5 Fig. 8 (a-d)

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ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

10

r/h = 0 0.45 0.90 3.60

8

y0.5/h

6 4 2 0

0

10

20

30

50

40

x/h Fig. 9

38

60

70

80

90

ACCEPTED MANUSCRIPT Deo, Nathan and Mi

Submitted to Experimental, Thermal and Fluid Sciences: June 2006

0.11

Ky

0.09

0.07

0.05

0.03

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

r/h Fig. 10

39

ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

-5

6

-3

-1 2

1

3

0

5 0

2

1

r/h Fig. 11

40

3

4

x02/h

x01/h

4

ACCEPTED MANUSCRIPT Deo, Nathan and Mi

Submitted to Experimental, Thermal and Fluid Sciences: June 2006

u' = u' / Uc n,c c

0.3

0.2

r/h = 0 0.45 0.90 1.80 3.60

0.1

0

10

20

30

40

50

60

70

80

90

x/h Fig. 12

41

ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

0.30

u' c,∞

0.28

0.26

0.24

0.22

0

2 r/h

1 Fig. 13

42

3

4

ACCEPTED MANUSCRIPT Deo, Nathan and Mi

Submitted to Experimental, Thermal and Fluid Sciences: June 2006

(b)

u' = u' /Uc n

(a)

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

0

0.5

1.0

1.5

2.0

x/h = 3 5 10 20 40 80

0

2.5

0.5

u' = u' /Uc n

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0 1.0

2.0

2.5

(d)

(c)

0.5

1.5

yn= y / y0.5

yn= y /y0.5

0

1.0

1.5

2.0

2.5

yn= y / y0.5

x/h = 3 5 10 20 40 80

0

0.5

1.0

1.5

2.0

2.5

yn= y / y0.5 Fig. 14 (a-d)

43

ACCEPTED MANUSCRIPT The influence of nozzle-exit geometric profile on statistical properties of a turbulent plane jet

1

0

3

2

Su= / ()

3/2

0

0 0 0 -1

Gaussian

-2 10

1

x/h Fig. 15

44

100

ACCEPTED MANUSCRIPT Deo, Nathan and Mi

Submitted to Experimental, Thermal and Fluid Sciences: June 2006

6

4

3

Gaussian

3

2

Fu= / ()

4

2

5

3 3 3 2 100

10

1

x/h Fig. 16

7 2

5

Fu

min

Su

-2

max

6

0

-4

4

-6

3

-8

0

2

1

3

4

r/h Fig. 17 45