Exp Fluids (2007) 43:17–30 DOI 10.1007/s00348-007-0308-0
RESEARCH ARTICLE
Performance of a 12-sensor vorticity probe in the near field of a rectangular turbulent jet A. Cavo Æ G. Lemonis Æ Th. Panidis Æ D. D. Papailiou
Received: 18 June 2006 / Revised: 1 April 2007 / Accepted: 5 April 2007 / Published online: 24 June 2007 Springer-Verlag 2007
Abstract The design and operational characteristics of a 12-sensor hot wire probe for three-dimensional velocity– vorticity measurements in turbulent flow fields is described and discussed. The performance of the probe is investigated in comparison with X-sensor probe measurements in the near field of a rectangular turbulent jet with aspect ratio 6. Measurements have been conducted at Reynolds number ReD = 21,000 at nozzle distances of x/D = 1, 3, 6 and 11, where D is the width of the nozzle. The results obtained with the 12-sensor probe compare well to the results of the X-sensor probe. Distributions of mean and fluctuating velocity–vorticity fields are presented and discussed. Among the results the most prominent is the experimental confirmation of the high levels of fluctuating vorticity in the shear layers. Keywords Rectangular jet Vorticity Hot wire anemometry (HWA) Multi-sensor HWA
1 Introduction All turbulent flows are characterized by the presence of fluctuating vorticity. This is a key feature that distinguishes turbulence from other stochastic fluid motions which are not categorized as turbulent (Tennekes and Lumley 1972). For this reason, vorticity plays an essential role in the
A. Cavo Th. Panidis (&) D. D. Papailiou Laboratory of Applied Thermodynamics, University of Patras, 26504 Patras-Rio, Greece e-mail:
[email protected] G. Lemonis Centre for Renewable Energy Sources, Pikermi, Greece
description of turbulence. Despite the general acceptance of the importance of vorticity in flow fields, there is considerable knowledge deficit on vorticity dominated processes as compared to the accumulated knowledge on velocity, which is mostly employed in flow research and flow phenomena description. The reasons hindering progress towards understanding vorticity should be mainly attributed to existing serious difficulties prohibiting its direct measurement, as this demands ‘spatially well resolved, simultaneous recording of all three velocity components and their spatial derivatives. During the last decades the major techniques suitable to measure the velocity field in turbulent flows, namely: hot wire anemometry (HWA), laser doppler velocimetry (LDV, Agui and Andreopoulos 2003) and particle image velocimetry (PIV, Adrian 1991) are competing and complementing each other to fill the gap. The present work is based on recent advances for the development and use of multi-sensor HWA probes for vorticity measurements. Simultaneous measurement of all three components of the velocity and the vorticity vectors with HWA was first reported by Wallace (1986). Wallace et al. used a nine-wire probe consisting of three three-wire arrays with thick, common central prongs. In the course of development of their technique, Wallace et al. have separated the central prong and have added a fourth wire to each three-wire array (Marasli et al. 1993; Vukoslavcˇevic´ and Wallace 1996). Tsinober et al. (1992) used three-wire arrays with separate inner prongs. They found that by adding a fourth wire to each three-wire array, higher accuracy could be achieved. The velocity vector components at the centroid of each array was computed using fourth order Chebychev polynomials to approximate the three-dimensional array calibration data. Velocity derivatives in cross-stream
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directions were estimated by finite difference schemes of first order accuracy. Streamwise velocity derivatives were calculated by means of Taylor’s hypothesis. With their twelve-wire probe, consisting of three 4-wire arrays, they made experimental investigations of the velocity–vorticity relations in turbulent grid flow and in the outer region of a turbulent boundary layer. They have shown important alignment properties of the vorticity vector in these flows. Subsequently, they added two further 4-wire arrays to the twelve-wire probe to obtain a twenty-wire probe (Kit et al. 1993). The six cross-stream derivatives of the velocity components were estimated from a five-point central difference scheme of second order accuracy. The wire arrangement in this probe brings the advantage that all nine-velocity derivatives are evaluated at the same point. The discretization distance in this arrangement, however, is considerably larger than in three-array probes. With this probe, vorticity measurements in turbulent jet flow were carried out along with studies concerning the vorticity alignment and enstrophy transport properties. The probe designed by Lemonis and Dracos (1995, 1996) is an improved version of the five-array probe developed by Kit et al. (1993). The size of the arrays has been kept the same, but due to the new arrangement it was possible to reduce the overall size of the probe and to decrease in that way the discretization increment from 2.8 to 1.4 mm improving the spatial resolution of the probe. Furthermore, Lemonis and Dracos (1995) developed a novel calibration and data reduction technique which exploits the entire uniqueness domain of the 4-sensor arrays. The technique uses look-up tables from angular probe calibrations at various calibrating velocities to determine the three-dimensional velocity vector components at the centroid of each array by three-dimensional spline interpolation. This technique brings the advantage that it does not require any knowledge about the probe geometry; furthermore, no assumptions are being made about the directional sensitivity coefficients of the wires. Multi sensor hot wire anemometry has been utilised in the first stages of development for measurements of the velocity and vorticity field primarily in simple flow configurations such as grid turbulence and boundary layers (Vukoslavcˇevic´ et al. 1991; Balint et al. 1991; Tsinober et al. 1992; Lemonis 1995; Honkan and Andreopoulos 1997a). More recent work includes high shear flows such as mixing layers (Balint et al. 1989; Foss and Haw 1990), turbulent jets (Kit et al. 1993), cylinder wakes (Marasli et al. 1993) and vortex dominated flows like those over delta wings (Honkan and Andreopoulos 1997b). The probe used in the present study consists of three 4-wire arrays in triangular arrangement. This presentation is focused on the performance of the 12-wire probe in the near field of a rectangular turbulent jet with aspect ratio six.
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Rectangular turbulent free jets have practical interest in propulsion, combustion and environmental flows. Most of the early work concentrated on the influence of the aspect ratio of the nozzle exit on the distribution of statistical velocity field properties (Sfeir 1976, 1979; Sforza et al. 1966; Sforza and Stasi 1979; Marsters and Fotneringham 1980). The development of a planar jet in the near field region has been the focus of detailed studies by Wier et al. (1981) and Browne et al. (1984). Studies on jets issuing from sharp-edged rectangular slots have also been presented by Tsuchiya et al. (1986), Quinn (1992) and Lozanova and Stankov (1998). The flow field of a rectangular jet is characterized by the presence of three distinct regions, defined by the centreline mean streamwise velocity, Uc, decay (Krothapalli et al. 1981), namely: (a) the potential core region (x/D = 0 to 4– 5), with almost constant Uc, which ends when the two shear layers along the short dimension of the nozzle meet, (b) the two-dimensional region (x/D = 4–5 to 25–30) in which the velocity decays at a rate roughly the same as that of a planar jet, and (c) the axisymmetric region, in which the velocity decays as in an axisymmetric jet, originating at x/D = 25–30, the location where the two shear layers along the long dimension of the nozzle meet. The measurements of the present investigation have been conducted in the first and second regions which are roughly the same as those of a planar jet (for x/D = 1 up to 11). Among the numerous investigations of turbulent rectangular jet flow few experimental results have been published on the behaviour of a limited number of vorticity components.
2 Experimental arrangement The jet nozzle used for the present experiments is shown schematically in Fig. 1. A centrifugal blower air supply system, of 1.5 kW power, was used to provide the airflow to a diffuser which decelerates the flow and converts the dynamic pressure to static in order to minimize the energy losses and preserve the uniformity of the flow field. Before reaching the nozzle, the air is passed through a settling chamber which contains a honeycomb and five screens (1 mm square mesh, 0.3 mm diameter wire) at 5 cm spacing to reduce disturbance at the inlet of the nozzle. The honeycomb installation in the settling chamber (not shown in Fig. 1) helps the smoothing and alignment of the flow resulting to turbulence reduction. Large scale eddies break to smaller ones and the mean velocity cross variations are minimized in this section. A 2D Boerger nozzle is used to accelerate the flow. The nozzle exit is rectangular of length L = 300 mm and width D = 50 mm. The aspect ratio is AR = L/D = 6.
Exp Fluids (2007) 43:17–30
19
Front View
0.354
0.247
Long prong Short prong
0.1
75
BOERGER NOZZLE
5 SCREENS
1.2
tube teflon insulation
epoxy
prong 45°
DIFFUSER
~13
AIR
Fig. 2 Schematic diagram showing the 12-sensor probe front view and a 45-inclined hot-wire with supporting prongs Fig. 1 Experiment set-up
The experiment was conducted at a Reynolds number Re = U0 D/m = 21,000, where U0 is the exit velocity from the nozzle, D is the nozzle’s width and m is the kinematic viscosity of air at environmental temperature 23C. The longitudinal turbulence intensity distribution at the nozzle exit was uniform and had a constant value of about 1%, except in the boundary layers adjacent to the nozzle lips, where the maximum value was about 2%. Based on the shape factor and the mean velocity profile the initial boundary layers are characterised as laminar (Hussain and Clark 1977; Lozanova and Stankov 1998). The 12-sensor probe manufacturing, probe calibration and data reduction techniques rely upon previous work of Lemonis (1995) and Lemonis and Dracos (1995), and have been further improved and refined at the Laboratory of Applied Thermodynamics of the University of Patras. The accuracy, sensitivity and reliability of the technique have been established in grid turbulence and boundary layer measurements. The probe used in the present study is inhouse manufactured consisting of three 4-wire arrays (wire diameter 2.5 lm, length 0.5 mm) in triangular arrangement (Fig. 2). To minimise wire cross talk and thermal interference no common central prong is used in this arrangement. The ends of each wire are welded on two teflon insulated tungsten prongs with a base bare diameter of 125 lm uniformly sharpened along their length to 20 lm at the tip to minimize blockage, without acting as sensors themselves. Bare prongs’ length is of the order of 13 mm keeping the sensor array at a distance more than ten times the sensor array diameter from the probe base, minimizing
blockage and aerodynamic interference with the flow. A tiny droplet of epoxy resin at midpoint of prongs’ length gives to the prongs’ structure of each array the necessary rigidity to avoid vibrations. The pyramidal shape of the sensors’ array minimizes thermal radiation between the sensors and ensures that no sensor is in the wake of another sensor for flow angles less than almost 45. An overheat ratio of 1.4 is used as a good compromise to avoid thermal interference without increasing probe sensitivity to ambient temperature variations. Ambient temperature variations never exceeded ±0.2C during a calibration and measuring period. The probe measures simultaneously all three velocity vector components at the centroids of each 4-wire array on a cross plane of the flow field. Twelve A.A. Lab Systems Constant Temperature Anemometers were used to control the wire voltage. Array separation is constrained by resolution and accuracy criteria. Noise and calibration uncertainties as well as accuracy of the separation distance measurement result in large overestimation of the measured velocity gradients (Antonia et al. 1993; Wallace and Foss 1995). On the other hand for reasonable resolution the sensor separation should be comparable to the minimum scales of the flow. According to previous investigations optimum separation for determining velocity gradients appears to be 2–4 Kolmogorov scales, satisfying both constraints. The Kolmogorov scale of the present flow based on the isotropic relation for the dissipation was estimated in the range g = 0.47–0.14 mm. The dimensions over which the velocity differences are determined are 1.04 and 1.2 mm which range between 2.2g-7.4g and 2.6g-8.6g. Thus, the probe is able to resolve all but the smallest vorticity scales.
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Probe calibrations are performed in the core region of a round jet. One novelty in relation to the work of Lemonis and Dracos (1995) is the use of a rotatable nozzle for calibration, which is driven by two computer-controlled step motors in pitch and yaw directions. The jet is positioned in front of the probe and it is removed after the calibration is completed. Thus, the probe is calibrated insitu with a fully automated calibrating facility enabling positioning accuracy of the calibration jet of better than 0.01, and rapid angular calibrations, lasting ~2–2.5 min for a typical ±45, 9 pitch · 9 yaw positions grid. Finer calibration grids (13 · 13 and 17 · 17) have also been tested but gave only marginal differences in the estimation of velocity and vorticity components. Analysis of measured data is based on the approximation of the calibration response of the wires by three-dimensional B-splines according to the technique developed by Lemonis and Dracos (1995). Velocity estimates are obtained assuming that the velocity does not change substantially across a 4-wire array. The 4-wire array is treated as a combination of four independent ‘‘three wire’’ sub-arrays leaving out one wire each time. The three components of the velocity vector corresponding to a subarray are determined solving a system of three nonlinear equations. The four different sub-array solutions are used for the estimation of the velocity vector at the centroid of each 4-wire array. The technique has been described extensively elsewhere (Lemonis 1995; Lemonis and Dracos 1995, 1996) and no further details will be given here. Velocity measurements presented in the following refer to the velocity vector estimation of the first 4-wire array. The uniqueness domain of 4-wire arrays has been investigated analytically by Rosemann (1989). For 45slanted wires he found that uniqueness is guaranteed at any pitch, a, and yaw, b, angles satisfying (a2 + b2)0.5 < /, where / was estimated in agreement with Pailhas and Cousteix (1986) to be at least 30. Similar results were found by Lemonis and Dracos (1996) examining anemometer voltage data sets obtained by probe calibration for pitch and yaw angles in the range –30 to +30, with a probe similar to that used in the present study. Spatial cross stream velocity derivatives are estimated from a forward difference scheme of first order accuracy. Streamwise velocity derivatives cannot be obtained directly from multi hotwire sensors with all sensors positioned on the flow cross plane. The Taylor’s hypothesis of frozen turbulence is used in the majority of experiments in simple form using the mean velocity as the convection velocity (Vukoslavcˇevic´ et al. 1991; Balint et al. 1991; 1989; Kit et al. 1993) although more refined forms have also been employed (Honkan and Andreopoulos 1997a). The validity of Taylor’s hypothesis in high turbulent shear flows is still an open issue in flow research. Zaman
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and Hussain (1981) investigating the vorticity field, in circular air jets with coherent structures organized through controlled excitation, concluded that an average velocity across the shear region is the least questionable choice for the convection velocity. Regarding streamwise derivatives of a scalar field, Werner and Southerland (1997) reported a correlation of derivatives obtained from Taylor’s frozen flow hypothesis of 0.74 with the true streamwise derivative of a scalar in the self similar region of an axisymmetric turbulent jet, while Mi and Antonia (1994) concluded in a similar flow that the usual form of Taylor’s hypothesis applies only near the jet axis. In the present work, the local average has been chosen as the Taylor’s hypothesis convection velocity since it is the most straightforward, directly applicable option allowing direct comparison with previous results. It has been used by other investigators in turbulent shear flows (Balint et al. 1989; Kit et al. 1993). The high sampling rate, used in the presented experiments, corresponds to a Taylor’s hypothesis spatial resolution in the worst case of ~0.5 mm (always smaller than the corresponding probe dimensions over which velocity differences are determined in the spanwise and lateral directions), in agreement with the suggestions of Piomelli et al. (1989) and Cenedese et al. (1991) who concluded that for high turbulence levels Taylor’s hypothesis is substantially correct for small separation distances. The results are reasonable indicating that all three components of the rms vorticity are of about the same magnitude on the jet centreline, while in the high shear region the streamwise vorticity fluctuations are consistently larger than the spanwise and lateral components (Stanley et al. 2002). Hot wire anemometry with the use of a Dantec X-wire probe was also employed for measuring the mean and the fluctuating velocity field, to characterize the jet and provide reference measurements for 12-wire probe performance assessment. The X-wire sensors were 5 lm in diameter and 1 mm in length. The X-wire was calibrated at nozzle exit under identical inlet flow conditions with the measurements. The anemometer signals were low passed filtered to prevent aliasing and discretized on a 14-bit A/D converter for subsequent data processing in a computer. The 12-wire data sets comprise time series of 100,000 samples at 12 kHz, while X-wire ones 16,000 samples at 2 kHz. Hot wire anemometry measurements may suffer due to calibration drift, which undermines accuracy and can considerably affect derivative estimation. Care was taken during measurements to identify, exclude and resample pathological data sets. It has to be noted that hot-wire measurements near the edge of a jet, where very small or even negative streamwise velocities may be present, should be considered with some reservation.
Exp Fluids (2007) 43:17–30
21
A video camera is used to perform visualization of the flow field. A Quantel, Nd Yag double pulse laser and suitable Dantec optics were used to generate the light sheet. Recent PIV measurements (not presented here) further support the present study results.
3 Mean velocity–vorticity fields Experimental results based on measurements with 12-wire and X-wire probes are presented in the following, compared with several experimental data of other researchers in the near field of planar and rectangular jets of various aspect ratios AR = L/D and Reynolds numbers (see Table 1). Since rectangular jets’ properties especially in the near field depend on initial conditions, nozzle geometry, aspect ratio and room turbulence (Quinn 1992; Krothapalli et al. 1981; Stanley and Sarkar 2000), the results are rather complementing than directly comparable to those of other investigators. Similarity should be looked for in properties’ trends and relative magnitudes. The data of Gutmark and Wygnanski (1976) have been used in particular to illustrate self similar behaviour. The mean streamwise velocity, U, profiles normalized by the centreline velocity, Uc, at different stations, are shown in Figs. 3, 4. The velocity at the exit plane of the nozzle presented an almost top hat profile as evidenced by x/D = 1 measurements. Further downstream the profiles assume a bell shaped distribution and are in agreement with corresponding profiles of Quinn (1992) and Browne et al. (1984). Figure 5 shows the mean profiles of the lateral velocity component, V, normalized by U0, at two stations in the jet, x/D = 6, 11. The experimental results of Gutmark and Wygnanski (1976) and DNS results of Stanley et al. (2002) indicating similar trends are also included in the figure. Browne et al. (1984) conclude that measured mean– velocity profiles in a plane jet appear to be self-preserving Table 1 Initial conditions for planar and rectangular jets
Source
beyond x/D = 5. For a rectangular jet, Everett and Robins (1978) reported that the downstream distance where the profiles first assume self-similarity appears to be directly related to the aspect ratio. Krothapalli et al. (1981) observed that similarity, for nozzle with aspect ratio 16.7, was found in mean velocity distributions, beyond x/D = 30. According to these findings, similarity should be expected in the axisymmetric region of rectangular jets. The presented results at x/D = 6 and 11 indicate that planar jet similarity as far as the mean velocity distributions are concerned may be claimed at least for a limited range, within the two-dimensional region of rectangular jets. Figure 6 depicts the downstream growth of the jet halfvelocity width, yc, defined as the distance from the jet centreline to the point at which the mean streamwise velocity is half of the centreline velocity. Beyond x/D = 5 the half-width of the jet varies linearly with streamwise coordinate, x, the slope depending on aspect ratio and geometry of the nozzle (Krothapalli et al. 1981). This linear evolution is usually described with a relation of the form: hx i yc þ A2 ¼ A1 D D As shown in Table 2, the value A1 for the current results is lower than results of other experimental studies. For the planar jets, the value of A1 varies between 0.09 and 0.12 (Kotsovinos 1976). There is a large variation in the virtual origins A2of the experimental studies, which makes a comparison difficult. The analysis of planar jets predicts an inverse-squared relationship between the mean centreline velocity and the downstream coordinate, x, for x/D ‡ 5:
U0 UC
2 ¼ B1
hx i þ B2 D
where Uc is the centreline mean velocity and U0 is the velocity at the nozzle exit (Fig. 7). Table 2 shows a Re
AR = L/D
Exit type
Present results (X-w and 12-w)
21,000
6
2D nozzle
Quinn (1992)
1.11 · 105
5
Sharp-edged slot
Tsuchiya et al. (1986)
15,000
5
Various
Krothapalli et al. (1981)
12,000
16.7
Rectangular channel
10
Brown et al. (1983)
7,620
20
2D nozzle
Gutmark and Wygnanski (1976)
30,000
38.5
2D nozzle
32,550
44
2D nozzle
81,400
44
Hussain and Clark (1977) Stanley et al. (2002)
3,000
–
‘‘2D nozzle’’
Thomas and Prakash (1991)
8,000
16
2D nozzle
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Exp Fluids (2007) 43:17–30
1.0
AR=6 x/D=1 X-wire AR=6 x/D=1 12-wire AR=6 x/D=6 X-wire AR=6 x/D=6 12-wire AR=5 x/D=5 Quinn (1992) AR=10 x/D=5 Quinn (1992) AR=20 x/D=7 Browne et al. (1984)
0.8
3
yc / D
U / UC
0.6 0.4
AR=6 X-wire AR=6 12-wire AR=5 X-wire Tsuchiya (1986) AR=10 X-wire Quinn (1992)
4
2
0.2 1
0.0
yc /D=0.085(x/D+2.35)
0.0
0.5
1.0
1.5
2.0
2.5
0
0
y / yc
U / UC
0.6 0.4 0.2 0.0 0.0
0.5
1.0
1.5
2.0
2.5
y / yc
x/D=6 X-wire x/D=11 X-wire x/D=6 12-wire x/D=11 12-wire x/D=11 Stanley et al. (2002) x/D=140 Gutm. & Wygn. (1976)
0.04
V / U0
0.02
•
• 0.00 -0.02 -0.04 -0.06 0.0
0.5
1.0
1.5
2.0
2.5
3.0
y/yc
Fig. 5 Mean lateral velocity profiles (x/D = 6, 11)
comparison of B1 and B2 values with results from several studies. As with A2 values, the values of B1, B2 for the different experiments also vary. This is not surprising since jet development in the near field is expected to be influ-
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enced by aspect ratio, jet exit formation, Reynolds number and room turbulence (Krothapalli et al. 1981; Stanley and Sarkar 2000). Based on the mean results presented so far, 12-wire probe measurements are in good agreement with X-wire ones, attaining similar values and trends. In general measurements with both probes compare favourably and are considered as equivalent. Larger but still acceptable variations will be found primarily in velocity rms values to be presented and discussed in the following. The deviations that appear are mainly expected, and can be attributed to (see Fig. 8): •
Fig. 4 Mean streamwise velocity profiles (x/D = 11)
0.06
15
Fig. 6 Jet half-velocity width in streamwise direction
AR=6 x/D=11 X-wire AR=6 x/D=11 12-wire AR=5 x/D=12.6 Quinn (1992) AR=10 x/D=12.6 Quinn (1992)
0.8
10
x/D
Fig. 3 Mean streamwise velocity profiles (x/D = 1, 6)
1.0
5
(a) different behaviour exhibited by pyramidical 4-wire sensors and X-wires subjected to flows with considerable gradients oU~i =oxj (i, j = 1–3, U~i : instantaneous value of the velocity component in direction i), (b) Different sensing wire dimensions: 2.5 lm diameter and 0.5 mm length for the 12-wire probe, and 5 lm diameter and 1 mm length for the X probe, (c) reduced ability of X-wire probes to measure three dimensional flows as will be further illustrated in following paragraphs.
The ability of the 12-wire probe to directly measure vorticity is assessed next. In rectangular jets, downstream derivatives of mean velocity components are expected to be small compared to cross-stream derivatives, ¶/¶x << ¶/¶y. Assuming a 2D plane jet flow with a Gaussian longitudinal mean velocity profile and the jet development constants A1, A2, B1 and B2 of the present results it can be shown analytically that the longitudinal derivative ¶V/¶x is always less than ±3% of the cross stream derivative ¶U/¶y for three standard deviations span of the distribution. Under these conditions, the dominant mean vorticity component Wz = ¶V/¶x-¶U/¶y can be estimated approximately as
Exp Fluids (2007) 43:17–30
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Table 2 Streamwise half-width growth and centreline velocity decay rates for planar and rectangular jets Source
A2
A1
B1
B2
0.8
0.8
x/D=3
0.6
0.6
Present results X-w
0.085
2.35
0.085
8.32
0.4
0.4
Quinn (1992) AR = 5
0.12
–0.96
0.09
5.26
0.2
0.2
–1.51
Tsuchiya et al. (1986)
0.125
1.7
Browne et al. (1983)
0.104
–5.00
0.143
9.00
Stanley et al. (2002)
0.92
2.63
0.201
1.23
0.0 -0.2
yc / UC
0.12
0.0 -0.2
3.0
1.0
z
AR = 10
2.5
0.8
0.0
0.5
1.0
1.5
2.0
2.0
x/D=1
0.5
1.5
1.0
2.0
x/D=6
0.2
0.5 0.0
0.0
-0.5
-0.2 0.0
0.5
1.0
1.5
2.0
0.0
y/yc
0.5
1.0
1.5
2.0
y/yc
Fig. 9 Mean spanwise vorticity distributions
2
(U0 / UC)
2
0.0
0.4
1.0
3
x/D=11
0.6
1.5
AR=6 X-wire AR=6 12-wire AR=5 Quinn (1992) AR=20 Browne et al. (1983) AR=20 DNS Stanley et al. (2002)
differetiating U(y) 12-wire vorticity mean
1.0
1.0
1
2
(U0 /UC ) =0.085(x/D+8.32)
0 2
0
4
8
6
10
12
14
x/D Fig. 7 Jet centreline mean velocity decay
y u, v x
derivative, differentiating U(y) measurements curve. Using either X-wire or 12-wire probe measurements produces similar results. The second Wz estimate is based on averaging the time series of local instantaneous spanwise vorticity results obtained with the 12-wire probe. Both estimates present a satisfying convergence supporting the capability of the 12-wire probe to measure local instantaneous vorticity. It has to be noted that since streamwise derivatives are calculated based on Taylor hypothesis the mean values of all measured streamwise derivatives, ¶/¶x (including ¶V/¶x), are zero. Since dimensional analysis indicates that ¶V/¶x is negligible, no appreciable error is expected.
Ø2.5 µm ~1 mm
4 Turbulent velocity–vorticity fields
(a) u, v
Ø5 µm ~0,8 mm
(b) Fig. 8 Differences between pyramidical sensors and X hot-wires
-¶U/¶x and a reasonable estimate of Wz may be obtained by differentiating the distribution U(y). Figure 9 depicts lateral profiles of mean spanwise vorticity, Wz, at different locations downstream. Two kinds of estimates are presented. Based on the above analysis Wz is first estimated calculating the ¶U/¶y
The behaviour of the planar jet due to the merging of the mixing layers is consistent with the eventual breakdown of the structures generated from the rollup and pairing of spanwise vortices (0 £ x/ D £ 4 ‚ 5) to strongly threedimensional turbulence. Figure 10 shows a characteristic flow visualization image of the rectangular jet in the near field region. Rockwell and Niccolls (1972) and Antonia et al. (1983) results, as well as the present visualization, indicate that near the nozzle exit the large-scale structures in the flow field are predominately symmetric for flat exit profiles. When the shear layers interact downstream, these structures reorganize into an asymmetric configuration in the fully developed region of the jet. The turbulence intensities, the root mean square (rms) values u¢, v¢ and w¢, on the centreline, normalized by Uc, based on X-wire and 12-wire probe measurements are presented in Figs. 11, 12. The u¢/Uc distribution is close to
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u'/UC X-wire v'/UC X-wire w'/UC X-wire u'/UC 12-wire v'/UC 12-wire w'/UC 12-wire u'/UC Krothapalli et al. (1981) v'/UC Krothapalli et al. (1981) w'/UC Krothapalli et al. (1981)
u' / UC, v' / UC, w' / UC
0.25
0.20
0.15
0.10
0.05
0.00 0
4
8
12
16
20
x/D
Fig. 10 Flow visualization of a rectangular jet in the near field region
Fig. 12 Turbulent intensity component variation along the jet centreline
0.30 AR=6 X-wire AR=6 12-wire AR=5 Quinn (1992) AR10 Quinn (1992)
0.25
L/D d1 (mm) h (mm) d1/h
Source
Re
0.15
Present study
21,000 6
0.10
Hussain and 32,550 44 Clark (1977) 81,400 44
0.20
u' / UC
Table 3 Initial boundary layer characteristics measured at x/D = 0.04 (Hussain and Clark results at x/D = –0.05) Condition
0.564
0.216
2.61
0.472
0.175
2.6774 Laminar
Laminar
0.373
0.135
2.7791 Laminar
0.05 0.00 0
10
20
30
40
x/D
Fig. 11 Streamwise turbulent intensity variation along the jet centreline
Quinn (1992) results for AR = 5. His results for AR = 10 as well as those of Krothapalli et al. (1981) indicate a steeper increase of the turbulence intensity components in the initial region which can be attributed to the different aspect ratio and shape of the nozzle according to Quinn (1992). Krothapalli’s u¢ /Uc distribution shows a small peak at x/D = 11. In the literature, it is inferred, but not fully confirmed, that the values of u¢/Uc along the centreline depend on the nature of the boundary layers at the nozzle lips. If these are turbulent then a monotonic increase in u¢/Uc occurs, while if they are laminar, a peak is observed (Browne et al. 1984). The mean characteristics of the initial boundary layer for the present study are outlined in Table 3 along with those of Hussain and Clark (1977). According to Schlichting (1968) the shape factor d1/h (d1 is the displacement thickness and h the momentum thickness) suggests that the boundary layers of the present study at the nozzle exit, x/D = 0.04, are laminar. Quinn (1992) has provided evidence that sharp peaks are found only in jets
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with aspect ratio greater than ten, while in a jet with aspect ratio five, close to six of the present data, a peak is barely discernible. Figures 13, 14, 15 show the profiles of longitudinal, lateral and spanwise velocity rms at x/D = 6, 11. The data of Thomas and Prakash (1991) and Gutmark and Wygnanski (1976) are also included. Quantitative differences with Gutmark and Wygnanski (1976) data reveal the difference of the flow under study with the fully developed self-similar planar jet region. Differences with Thomas and Prakash (1991) measurements should be attributed to the different turbulent levels of the initial conditions and also different aspect ratios. The largest differences between 12-wire and X-wire probe measurements are observed in these figures. The Xwire allows simultaneous measurement of the two velocity components in a plane midway between the planes of the two hot wires. On the other hand, 12-wire probes have the capability to measure simultaneously the three velocity components. Since the velocity field in shear layers is three-dimensional, 12-wire probe measurements are expected to be more reliable than the X-wire ones. The pitch and yaw angles of the mean velocity vector measured with the 12-wire probe are presented in Fig. 16, and the corresponding rms values in Figs. 17, 18. Mean
Exp Fluids (2007) 43:17–30
25
x/D=6 x/D=11 x/D=6 x/D=11 x/D=12 x/D=140
0.30
X-wire X-wire 12-wire 12-wire Thom. & Pra. (1991) Gutm. & Wyg. (1976)
0.20
0.25 0.20 0.15
0.10
0.05
0.10 0.05 0.0
0.5
1.0
1.5
2.0
2.5
0.00 0.0
3.
0.5
1.0
1.5
Fig. 13 Turbulent intensity distributions of the streamwise velocity component
x/D=6 x/D=11 x/D=6 x/D=11 x/D=12 x/D=140
0.25
2.5
3.0
Fig. 15 Turbulent intensity distributions of the spanwise velocity component
30
X-wire X-wire 12-wire 12-wire Thom. & Pra. (1991) Gut. & Wygn. (1976)
x/D=1 pitch x/D=3 pitch x/D=6 pitch x/D=11 pitch x/D=1 yaw x/D=3 yaw x/D=6 yaw x/D=11 yaw
25 20
a, b (deg)
0.20
2.0
y/yc
y/yc
v ' / UC
X-wire X-wire 12-wire 12-wire Gutm. & Wygn. (1976)
0.15
w ' / UC
u '/ UC
x/D=6 x/D=11 x/D=6 x/D=11 x/D=140
0.15
0.10
15 10 5 0
0.05
-5 0,0
0.00 0.0
0.5
1.0
1.5
2.0
0,2
0,4
0,6
y/yc Fig. 14 Turbulent intensity distributions of the lateral velocity component
1,0
1,2
1,4
1,6
Fig. 16 Lateral profiles of the mean values of the pitch, a, and yaw, b, angles
18
x/D=1 x/D=3 x/D=6 x/D=11
16 14 12
a' (deg)
yaw angle values remain close to zero illustrating the twodimensional character of the flow field. Mean pitch angle values increase with the distance from the flow axis at a decreasing pace as the distance from the exit is increasing in accordance with the mean velocity profiles. In most cases, the range of the angles spanning three times the rms value around the mean value remains within the ±30 range where the accuracy of the probe is well established. Even at the furthest from the axis locations measured angle values remain well within the ±45 calibration range. The high values of angle fluctuations in the shear layers illustrate the limitations of X-wire probe capability to measure the turbulent velocity field in this region. Figure 19 depicts the lateral profiles of the three vorticity fluctuations rms components at different stations.
0,8
y/yc
2.5
10 8 6 4 2 0 0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
y/yc
Fig. 17 Lateral profiles of the rms values of the pitch angles
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26
Exp Fluids (2007) 43:17–30
This behaviour can be associated with the turbulence level of the nozzle boundary layers and the roll up of the outer streakline of the jet into an apparent vortex (Rockwell and Niccolls 1972) within the mixing layer (see Fig. 10). Downstream the vorticity fluctuations are redistributed initially to a larger area due to vortex merging (x/D = 3, Fig. 10), and further downstream spread out towards the centre and the edge of the jet due to the merging of the two mixing layers. The probability density distributions of the three vorticity components on the centreline at x/D = 1 and at y/yc = 0.85, x/D = 6 depicted in Fig. 20 give further support to the quality of vorticity fluctuations estimation. Since the residual real vorticity at x/D = 1 on the centreline should be very small the width of the distributions provide a high limit of the noise level of the measuring setup (Vukoslavcˇevic´ et al. 1991; Balint et al. 1991). As shown in Fig. 20 the widths of the distributions at x/D = 1 are all of the same order of magnitude and significantly lower than the widths of the distributions at y/yc = 0.85, x/D = 6 indicating a high signal to noise ratio of the measurements. Average rms values at x/D = 1 on the centreline correspond in the worst case to 8.4, 10.3 and 6.9% of the rms vorticity fluctuations x¢x, x¢y, and x¢z, measured in the shear layer region. These accuracy estimates are in agreement with the results of Lemonis (1997) who found a
x/D=1 x/D=3 x/D=6 x/D=11
21 18
b' (deg)
15 12 9 6 3 0 0,0
0,2
0,4
0,6
0,8
1,0
1,2
1,4
1,6
y/yc
Fig. 18 Lateral profiles of the rms values of the yaw angles
The profiles have been non-dimensionalized either with the constant U0/D to depict the redistribution of vorticity fluctuations or with Uc/yc to reveal the increase of vorticity fluctuations compared to the local shear. In the potential core region the vorticity fluctuations reflect their dependence on the interaction of the jet core flow with the surrounding air, i.e. the values are quite low on the centreline of the jet and quite high in the shear layers. Fig. 19 Lateral profiles of the rms values of the three fluctuating vorticity components
x/D=1 x/D=3 x/D=6 x/D=11
7
'z D/U0
6 5
4
'z yc / UC
24
4 3 2
3 2
x/D=1 x/D=3 x/D=6 x/D=11
1
1 0
0 1.0
1.5
8 7 6 5 4 3 2 1 0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
4 3 2 1 0
0.5
1.0
1.5 6
8
5
'x yc / UC
10
6 4
4 3 2
2
1 0
0 0.0
123
0.0 5
0.0
'x D/U0
0.5
'y yc / UC
'y D/U0
0.0
0.5
1.0
1.5
y/yc
y/yc
(a)
(b)
Exp Fluids (2007) 43:17–30
27
0,0015
X/D=1 y/yc=0
PDF (ωx)
X/D=6 y/yc=0.85 0,0010
0,0005
0,0000 –3000
–2000
–1000
0
–2000
–1000
0
ωx
1000
2000
3000
1000
2000
3000
PDF (ωy)
0,0015
0,0010
0,0005
0,0000 –3000
4 ' x
yc / UC
3
' y ' z
2
c
deviation of 8% on the average between measured and theoretical estimates of the velocity gradients in grid turbulence. The downstream variation of vorticity rms components at the centreline is depicted in Fig. 21 non-dimensionalized with the local mean shear. Once the interaction effects have reached the centreline all values increase monotonically due to vorticity redistribution and the decrease of the local mean shear. The streamwise vorticity fluctuations at the centreline remain higher than the spanwise and lateral components. The evolution of turbulent quantities in the shear layers deserves special attention since turbulence and vorticity are generated initially there. Variations of streamwise velocity turbulence intensity u¢s(x) in the shear layer along the plane defined by y/yc 1 are shown in Fig. 22. According to Hussain and Clark (1977) and Sato (1960) laminar initial boundary layers result in rapid initial turbulence growth
1
0 0
2
4
6
8
10
12
x/D Fig. 21 Variation of the rms values of the three fluctuating vorticity components along the jet centreline
rate due to the initial lack of growth retarding non-linear interactions. Thus, due to instability the growth rates are initially higher for the laminar initial boundary layer than turbulent cases. Figure 23 shows the downstream variations of shear layer rms values of the three fluctuating vorticity components. In contrast to the evolution along the centreline, vorticity fluctuations in the shear layer attain high intensities close to the nozzle exit and decrease downstream. This behaviour, very similar to that of velocity fluctuation intensities, is illustrative of the turbulence diffusion from the shear layer towards the centre of the jet in these developing phases. It is worth noting that the streamwise component of vorticity fluctuation intensities remains in all cases higher than the other components. Stanley et al. (2002) have also reported higher value of x¢x, although their computations refer to significantly different conditions and the general downstream trend is different. 0.20
ωy
0.16
0,0015
us / U0
0,0010
'
PDF (ωz)
0.12
0,0005
0,0000 –3000
0.08 X-wire 12-wire Re=32550 Hussain & Clark(1977) Re=81400 Hussain & Clark(1977)
0.04
0.00 –2000
–1000
0
ωz
1000
2000
3000
Fig. 20 Probability density functions of fluctuating vorticity components measured at y/yc = 0, x/D = 1, and y/yc = 0.85, x/D = 6
0
5
10
15
20
25
30
35
x/D Fig. 22 Streamwise turbulent intensity variation along the plane defined by y/yc 1
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28
Exp Fluids (2007) 43:17–30 2.5 9
AR=6 AR=6 AR=5 AR=10 AR=20
' x
8
2.0
7
/ Uc x 100
' y
X-wire 12-wire Quinn (1992) Quinn (1992) Browne et al. (1984)
1.5
2
' z
x/D=11 x/D=11 x/D=12.6 x/D=12.6 x/D=12
6 5
1.0
0.5
4 3
0.0 0.0 0
2
4
6
8
10
0.5
1.0
1.5
2.0
2.5
y/yc
12
x/D Fig. 25 Reynolds shear stress profiles (x/D = 11) Fig. 23 Variation of the rms values of the three fluctuating vorticity components along the plane defined by y/yc 1
In Figs. 24, 25 the Reynolds shear stresses measured with X wire and the 12 sensor probe, normalized by U2c , show similar trends with the data of Quinn (1992) and Browne et al. (1983). Quantitative differences should be attributed to differences in aspect ratio and initial conditions. A more complete picture is depicted in Fig. 26 where all Reynolds shear stresses measured with the 12-wire probe are presented. It is well known that turbulence depends on the shear due to the mean streamwise velocity gradient ¶U/¶y for the production of Reynolds stress . Figure 26 shows that the major component of the Reynolds shear stresses is always . The spatial distribution of is compatible with the evolution of the jet, the high values associated with the presence of the Kelvin-Helmholtz eddies developing at the edges within the first region, which grow due to successive vortex merging and migrate towards the centre after the merging of the shear layers, alternating thereafter on opposite sides of the centreline in
an antisymmetric mode. The dominance and the profile of indicate that the basic large-scale structures consist of roller-like eddies with axes aligned with the spanwise direction of the planar jet.
5 Conclusions The performance of a 12-wire HWA probe has been assessed in a rectangular jet of aspect ratio 6, at Re = 21,000. Measurements with an X-wire probe have been used as a reference. Results referring to measurements of the velocity with the two sensors are in good agreement, except in locations where the steep velocity gradients and the three dimensionality of the velocity field undermine X-wire probe measurements. The ability of the 12-wire probe to measure vorticity is illustrated by the good convergence of the mean lateral
1.2
1.2
1.4 X-wire 12-wire Quinn (1992) Quinn (1992) Browne et al. (1984)
0.8
0.4
0.6 0.4 0.2
0.8 0.4 0.0
-0.4
-0.4
0.0 1.0
x/D=11
x/D=3
0.0
2
2
/ U C x 100
1.0
x/D=6 x/D=6 x/D=5 x/D=5 x/D=5
0.8
/ U C X 100
AR=6 AR=6 AR=5 AR=10 AR=20
1.2
12-wire 12-wire 12-wire
0.5
1.0
0.0
x/D=1
0.5
1.0
x/D=6
0.8
0.5
0.4
0.0
0.0
0.0
1.5
-0.4
0.0
0.5
1.0
1.5
y/yc Fig. 24 Reynolds shear stress profiles (x/D = 6)
123
2.0
2.5
0.0
0.5
1.0
y/yc
1.5
0.0
0.5
1.0
y/yc
Fig. 26 Reynolds shear stress profiles using 12-wire probe
1.5
Exp Fluids (2007) 43:17–30
vorticity distributions estimated either by differentiating the mean velocity distribution U(y) or averaging direct vorticity measurements of the 12-wire probe. Limited presentation of vorticity results confirm that vorticity plays an essential role in the description of turbulence in the rectangular jet. Prominent results include the experimental confirmation of the high levels of fluctuating vorticity in the shear layers. It is evident that a more detailed, careful study of the role of vorticity and its dynamics in jets will vastly enhance understanding of flow phenomena promoting also technological advances. Multi-sensor HWA is a unique tool for turbulent flow vorticity measurements which however, needs further improvement, in order to overcome limitations such as (a) inherent construction difficulties of the sensing probe, (b) achieving improved level of performance and reliability, and (c) satisfactory functional simplicity.
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