Measurements in the near flow field of an isosceles triangular turbulent free jet

Abstract

The near field mean flow and turbulence characteristics of a turbulent jet of air issuing from a sharp-edged isosceles triangular orifice into still air surroundings have been examined experimentally using hot-wire anemometry and a pitot-static tube. For comparison, some measurements were made in an equilateral triangular free jet and in a round free air jet, both of which also issued from sharp-edged orifices. The Reynolds number, based on the orifice equivalent diameter, was 1.84×10⁵ in each jet. The three components of the mean velocity vector, the Reynolds normal and primary shear stresses, the one-dimensional energy spectra of the streamwise fluctuating velocity signals and the mean static pressure were measured. The mean streamwise vorticity, the half-velocity widths, the turbulence kinetic energy and the local shear in the mean streamwise velocity were obtained from the measured data. It was found that near field mixing in the equilateral triangular jet is faster than in the isosceles triangular and round jets. The mean streamwise vorticity field was found to be dominated by counter-rotating pairs of vortices, which influenced mixing and entrainment in the isosceles triangular jet. The one-dimensional energy spectra results indicated the presence of coherent structures in the near field of all three jets and that the equilateral triangular jet was more energetic than the isosceles triangular and round jets.

1 Introduction

Jets issuing from triangular orifices can be used in many technical applications but especially so in combustion systems in which large-scale mixing, initiated at the flat sides of the slot, and small-scale mixing, in the corner regions, are required. The combustion process needs large-scale mixing to promote bulk mixing of the fuel and oxidizer and small-scale mixing to facilitate chemical reactions. Expedient manufacturing and installation ease will dictate that a triangular orifice be made by machining flat work pieces, with resulting sharp edges, which are then appropriately assembled to obtain the orifice shape. A sharp-edged triangular orifice is, therefore, used in the current study.

No study in the extant archival literature is entirely devoted to the turbulent jet issuing from an isosceles triangular slot. Schadow et al. (1988) present some evidence of coherent structures emanating from the base side of an isosceles triangular orifice, with a vertex angle of 30°, and disordered structures from the vertex of the orifice. Gutmark et al. (1989) found that flames issuing from isosceles triangular burners had the cross-sectional shape of the burner exit plane in the very near field (X/D_e=0.26). The shape of the flame cross-section then became oval at X/D_e=1.3 and quasi-circular at X/D_e=2.1 and further downstream. They also found, in agreement with cold flow measurements they had made earlier, large-scale vortices at the flat sides of the burner and an unstructured flame at the vertex. Miller et al. (1995) performed a numerical study, using direct numerical simulation, of jets issuing from circular, elliptic, square, rectangular, equilateral triangular and isosceles triangular nozzles into co-flowing surroundings at a Reynolds number of 8×10² based on the nozzle equivalent diameter and the velocity difference between the jet and the co-flowing stream. The aspect ratio of the elliptic and rectangular nozzles was 2:1. The isosceles triangular jet, in terms of mixing enhancement relative to the circular jet, was found to be the most efficient. The mean streamwise velocity decay on the jet centerline, mass entrainment and product formation amount, in the case of reacting jets, were used as measures to arrive at the conclusion made in the study. Mi et al. (2000) studied experimentally the mean streamwise velocity decay and streamwise turbulence intensity behavior along the jet centerline for the circular and a number of noncircular turbulent free jets at a Reynolds number of 1.5×10⁴ based on the orifice equivalent diameter. The study concluded that the jet from the isosceles triangular orifice had the best mixing enhancement characteristics relative to the circular jet.

The present study was undertaken to investigate further the claim that the turbulent isosceles triangular jet has the best mixing enhancement characteristics among noncircular turbulent jets and to provide detailed mean flow and turbulence data which can be used to facilitate the numerical computation of the flow. Mass entrainment, half-velocity widths and the mean streamwise vorticity, which have been obtained from the mean flow data, and mean static pressure data are also presented, along with the one-dimensional energy spectra of the fluctuating streamwise velocity signals.

2 Experimental details

The blow-down jet flow facility used for the present study is shown in Fig. 1. It consists of a small centrifugal fan, a diffuser, a settling chamber, a three-dimensional contraction and a screen cage. The fan, supported on anti-vibration neoprene pads, drew air from a room adjacent to the laboratory and delivered it to the isosceles triangular orifice via the diffuser, settling chamber and contraction. The diffuser was fitted with a baffle at its upstream end, honeycomb and mesh-wire screens. The settling chamber, a plywood box of 0.762×0.762 m cross-section and 1.054 m in length, was also fitted with mesh-wire screens. The three-dimensional contraction had a contour, which is a third-degree polynomial that had zero derivatives as end conditions. The contraction, 0.523 m in length, had a circular cross-section, with 0.762-m diameter at its upstream end, and a 0.305×0.305 m square cross-section at its downstream end. The isosceles triangular orifice, shown in Fig. 2, capped the downstream end of the contraction, which was flush with a 2.438×2.438 m plywood wall. The coordinate system used is shown in Fig. 2. The streamwise coordinate, which is not shown, is perpendicular to the spanwise and lateral coordinates and forms a right-hand system with them. The contraction ratio was 283. The plywood wall formed one side of a screen cage, which extended 3.658 m downstream from the wall. The experiments were performed in a 7.70×7.01×2.87 m room.

Fig. 1

Plan view section of the flow facility

Fig. 2

Isosceles triangular orifice

A three-dimensional traversing system was used for moving the sensing probes in the flow field. The system consisted of a rack and pinion for traversing in the X direction and lead screws for traversing in the Y and Z directions. The base of the traversing system was, like the fan, also supported on anti-vibration neoprene pads. Traversing in all three coordinate directions was effected by microcomputer-controlled stepping motors. Positioning accuracy of the sensing probes was 0.3 mm in the X direction and 0.01 mm in both the Y and Z directions. The data were acquired on a grid in the Y–Z plane at each X location. The grid spacing, which was kept the same in the Y and Z directions, varied from 2.54 mm close to the jet exit to 12.7 mm further downstream.

The mean velocity and turbulence data were acquired with DANTEC P51 X-array probes (Dantec Dynamics, Denmark). These probes consist of two 5-μm diameter platinum-plated tungsten slant wire-sensing elements about 1-mm long and about 1 mm apart. The hot-wire probes, operated by DANTEC constant temperature anemometers at a resistance ratio of 1.8, were calibrated on-line close to the exit of the jet against the output of a pitot-static tube which was connected to a pressure transducer and a Barocel electronic manometer. The calibration data were fitted to the exponent power law: E²=A+BU ⁿ_eff and A, B and n were optimized with a linear least-squares goodness-of-fit procedure. A “cosine law” response to yaw was assumed and the effective angle was found from a yaw calibration following Bradshaw (1975). Temperature variations from the calibration temperature were monitored with a thermocouple placed in proximity to the hot-wire probe and corrections for such variations were made using the procedure in Bearman (1971) in the data-reduction software. The mean flow and turbulence data were corrected for the effects of the mean velocity gradients on the spacing between the two slant sensing-wire elements of the X-array probes using the formulae given in Bell and Mehta (1989).

The hot-wire signals were linearized by the laboratory microcomputer and digitized, along with the signals from the thermocouple, with the National Instruments AT-A2150 dynamic signal acquisition board. This board consists of four analog input channels, each of 16-bit resolution. Each of the analog input channels is preceded by a third-order Butterworth low-pass analog anti-aliasing filter with an 80 kHz cut off. The filtered signal is sampled with a 1-bit delta–sigma modulating analog sampler at 64 times the chosen sampling rate. This reduces the quantization noise considerably. The output of the sampler is then fed to a digital anti-aliasing filter, which is built into the A/D converter chip, and the output of this filter re-samples the signal to the data rate, namely, 16-bit digital samples. It should be noted that all four analog input channels can be sampled simultaneously and, therefore, no sample-and-hold units, which are required for successive approximation and dual slope A/D converters, are needed here. The input range of the AT-A2150 board is ±2.828 V (or 2 V rms) so amplification was only needed for the thermocouple signal and not for the hot-wire signals. The mean velocity and turbulence data were obtained from records containing 8,192 samples obtained at a sampling rate of 4 kHz.

The mean static pressure measurements were made with a 2.3-mm diameter pitot-static tube, made of stainless steel with an ellipsoidal head and four circumferentially located static pressure holes, connected to a DATAMETRICS (USA) pressure transducer and an electronic manometer. These signals were also digitized with the AT-A2150 board but a voltage divider was needed to bring the signals into the input range of the board.

The mean streamwise velocity at the center of the slot exit plane was 61 m/s and this resulted in a Reynolds number of 1.84×10⁵ based on the orifice equivalent diameter D_e=45.3 mm. The streamwise turbulence intensity at the orifice exit plane was 0.4%.

3 Results and discussion

3.1 Mean streamwise velocity decay on the jet centerline

The decay of the mean streamwise velocity along the centerline of the isosceles triangular jet is shown in Fig. 3a. The data, acquired in our laboratory, for an equilateral triangular jet and for a round jet, both issuing from sharp-edged orifices, and those of Mi et al. (2000) are included for comparison. It should be noted that the diameter of the sharp-edged round orifice is the same as the equivalent diameter of the isosceles triangular orifice. The data of Mi et al. (2000) were, as stated previously, acquired at a Reynolds number of 1.5×10⁴ while those for the isosceles triangular, the equilateral triangular and round jets in the present study were obtained at a Reynolds number of 1.84×10⁵. The data in Fig. 3a have been fitted to

$$ \frac{{U_{\max } }} {{U_{{\text{cl}}} }} = K_u \left( {\frac{X} {{D_{\text{e}} }} + C_u } \right) $$

(1)

Fig. 3

a Mean streamwise velocity decay on the jet centerline. b Near field mean streamwise velocity decay on the jet centerline

In the region EquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacaaIWa % GaeyizIm6aaSaaaeaacaWGybaabaGaamiramaaBaaaleaacaWGLbaa % beaaaaGccqGHKjYOcaaI1aGaaGOmaaaa!3F12! EquationSource $$ 20 \leqslant \frac{X} {{D_{e} }} \leqslant 52 $$ , the K _u values are 0.207, 0.196 and 0.205 for the present isosceles triangular jet, equilateral triangular jet and round jet, respectively. The corresponding C _u values are 0.181, 0.389 and −2.167. For the data of Mi et al. (2000) K _u =0.199 and C _u =2.108 in the regionEquationSource% MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbb % a9q8WqFfeaY-biLkVcLq-JHqpepeea0-as0Fb9pgeaYRXxe9vr0-vr % 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaiaaikdacaaIWa % GaeyizIm6aaSaaaeaacaWGybaabaGaamiramaaBaaaleaacaWGLbaa % beaaaaGccqGHKjYOcaaIZaGaaGOnaaaa!3F14!EquationSource$$ 20 \leqslant \frac{X} {{D_{e} }} \leqslant 36 $$. The far field velocity decay rate of the isosceles triangular jet of the present study is slightly larger than those of the other jets considered here. The fact that the kinematic virtual origins of the jets are not the same can be attributed to the difference in the initial geometrical shapes of the orifices. It should be noted that the kinematic virtual origins of the triangular jets are located behind the orifice exit plane while that of the round jet is located ahead of the orifice exit plane. The mean streamwise velocity decay rates of the three jets in the near flow field are presented in Fig. 3b to shed some light on the mixing characteristics of the jets of the present study. The near flow field is a region of interest in practical jet applications, such as in combustion. The mean streamwise velocity decay of both of the triangular jets is significantly faster than that of the round jet and the equilateral triangular jet has the fastest mean streamwise velocity decay rate among the three jets considered here in the near flow field. It is clear that the far field mean streamwise velocity decay rate is not a good measure of near field mixing efficiency in a jet. The length of the potential core, among other metrics, is a better indicator of near field mixing effectiveness in a jet. The potential core lengths in the present study are 2.91D_e, 3.14D_e and 3.50D_e for the equilateral triangular jet, the isosceles triangular jet and the round jet, respectively. The potential core lengths also indicate that the fastest near field mixing occurs in the equilateral triangular jet.

3.2 Mean static pressure distribution on the jet centerline

The mean static pressure distribution on the jet centerline for the isosceles triangular jet is shown in Fig. 4. The data for an equilateral triangular jet and for a round jet are also shown for comparison. All three jets exhibit similar behavior in the very near flow field, namely, the mean static pressure drops from a positive value (i.e., above atmospheric pressure) at the slot exit plane to zero (i.e., atmospheric pressure) as a result of the acceleration of the jet fluid brought about by the vena contracta effect. The further decrease in the mean static pressure to negative values is triggered by the rapid production of turbulence from mean flow shear and its redistribution, via the pressure fluctuations, in the near flow field. Mixing in a free jet is initiated in the shear layers at the periphery of the jet. The mixed jet fluid then advances towards the center of the jet with downstream distance until the jet fluid and the surrounding fluid are fully mixed. The static pressure is then the same, namely, atmospheric pressure, throughout the jet. In this regard, the speed with which the mean static pressure in a jet recovers from negative values to atmospheric pressure can perhaps be taken as a measure of how fast mixing is taking place within the jet. Based upon this, near field mixing in the equilateral triangular jet, in agreement with the aforementioned near field mean streamwise velocity decay rate, is the fastest among the three jets considered. This result is at variance with that of Miller et al. (1995), but it should be recalled that the Reynolds number in their study was 8×10² and the jets studied, unlike those of the present study, were forced.

Fig. 4

Mean static pressure distribution on the jet centerline

3.3 Mean streamwise velocity contour maps

The contour maps for the mean streamwise velocity are shown in Fig. 5. Initially, at X/D_e=0.5, the contours have the isosceles triangular shape of the slot and are very closely spaced, indicating that very little mixing has taken place at this location. At X/D_e=1.0, the contours have acquired a diamond (or oval) shape and the spacing between the contour levels has increased. Further downstream, at X/D_e=2.0, the contours have the inverted shape of the isosceles triangle at the slot exit plane; this has been referred to as axis switching by various investigators of noncircular jets. The spacing between the contour levels has increased further at this location, implying that mixing is taking place. Any memory of the initial isosceles triangular shape is completely lost at X/D_e=10.0, as revealed by the shape of the contours at this location. The observations of the mean streamwise velocity field in the present study are in agreement with those made by Gutmark et al. (1989) for the average product formation field of a flame issuing from an isosceles triangular burner at the corresponding locations.

Fig. 5

Mean streamwise velocity (U/U_cl) contour maps

3.4 Mass entrainment into the jet

Mass entrainment into the jet has been calculated from the mean streamwise velocity data which were acquired with a pitot-static tube in entire Y–Z planes at several X/D_e locations from:

$$ Q = \int\limits_A {\rho U{\text{d}}A} $$

(2)

by numerical quadrature using Simpson’s 1/3 rule. The results are shown in Fig. 6 for the isosceles triangular, equilateral triangular and round jets. At each location, the lowest velocity considered was about 1% of the local mean streamwise velocity on the jet centerline. The equilateral triangular jet clearly has the largest mass entrainment, implying the fastest rate of mixing, among the three jets examined in the present study. This result is not in agreement with that of Miller et al. (1995) but one is reminded that the initial conditions in the two studies were vastly different. The influence of the initial conditions on jet evolution is fairly well known.

Fig. 6

Mass entrainment into the jet

3.5 Development of the jet half-velocity widths

The development of the jet half-velocity widths in X–Y and X–Z planes, taken through the origin of the coordinate system in Fig. 2, is shown in Fig. 7 for the three jets studied. The half-velocity width of a jet is defined as the distance from the centerline of the jet to the point where the mean streamwise velocity is half its value on the jet centerline. The half-velocity widths were calculated from data obtained from Y-direction and Z-direction pitot-static tube traverses across the entire jet at several X/D_e locations and at each location the average of the Y_1/2 and Z_1/2 values on either side of the jet centerline is presented in Fig. 7. The half-velocity widths of the isosceles triangular and equilateral triangular jets show similar behavior in both the X–Y and X–Z planes. In the X–Y plane, the half-velocity widths of both triangular jets decrease initially, due to the vena contracta effect, and then increase monotonically with downstream distance triggered by the large-scale structures emanating from the sides, which are inclined to the base in both triangles. The half-velocity widths of the triangular jets in the X–Z plane increase initially and then decrease for some distance downstream before they start increasing in a monotonic manner. The axis-switching phenomenon observed by others who have studied noncircular jets is clearly present in the half-velocity width development of the present triangular jets. While several axis-switches were observed for the equilateral triangular jet, only two axis-switches, at X/D_e=3.0 and at X/D_e=30.0, were found in the isosceles triangular jet. Axis switching is known to facilitate mixing in a jet and the fact that this phenomenon is observed several times in the present equilateral triangular jet suggests faster mixing in this jet compared to its isosceles triangular counterpart.

Fig. 7

Development of the jet half-velocity widths

The geometric mean of the half-velocity widths, B_e=(Y_1/2Z_1/2)^0.5, is used to facilitate a clear comparison of the spread of the three jets of the present study. The data are shown in Fig. 8 from which it is clear, as has already been indicated by the mean static pressure data on the jet centerline shown in Fig. 4 and the mass entrainment data shown in Fig. 6, that the equilateral triangular jet spreads faster than the other two jets.

Fig. 8

Geometric mean of the jet half-velocity widths

3.6 Mean static pressure contour maps

The mean static pressure contour maps are shown in Fig. 9. The mean static pressure distribution along the jet centerline presented in Fig. 4 showed positive values up to about X/D_e=2 and negative values up to about X/D_e=20. It is, therefore, not surprising to find positive mean static pressures in the central regions of the jet at X/D_e=0.25, 1.0 and 2.0 and negative mean static pressures throughout the jet at X/D_e=10.0 in Fig. 9. It is clear that ambient fluid will generally be pumped into the jet as a result of the difference in pressure between the ambient and the jet. This is the case at X/D_e=10.0. At the other three locations shown in Fig. 9, regions of positive mean static pressure within the jet will pump jet fluid to regions of negative mean static pressure. At X/D_e=0.25, for example, jet fluid will be pumped from the region inside the isosceles triangle to the corners which will, in turn, receive ambient fluid.

Fig. 9

Mean static pressure (2(P_s–P_atm)/ρU²_cl) ×100 contour maps

3.7 Mean streamwise vorticity contour maps

Mean streamwise vorticity was calculated from the V-data and W-data by applying a central differencing procedure to the formula: $\omega _x = (\partial W/\partial Y) - (\partial V/\partial Z)$ and the results are shown as contour maps in Fig. 10. The uncertainty in the mean streamwise vorticity data is ±19.5% at 20:1 odds. The reader is reminded that since vorticity cannot be created within the core of a homogeneous fluid, vortices in the unbounded flow considered here must necessarily exist as counter-rotating pairs. Also, positive and negative ω _x indicate counter-clockwise and clockwise rotation, respectively, consistent with the right-hand system used here.

Fig. 10

Mean streamwise vorticity (ω _x D_e/U_cl) contour maps

At X/D_e=0.5, the mean streamwise vorticity field consists of an inflow pair, aligned with the top corner of the isosceles triangle, and two outflow pairs, aligned with the bottom corners of the isosceles triangle, of counter-rotating vortices. The mutual induction of these inflow and outflow vortex pairs, as they are referred to in the literature, results in the sideways movement of the two longer sides of the isosceles triangle and in the downward movement of the base side of the triangle. The dynamics of these vortex pairs ultimately produces the diamond (or oval) shape of the mean streamwise velocity contour map X/D_e=1.0 in Fig. 5.

At X/D_e=1.0, three pairs of inflow vortices, aligned with the top and sides of the aforementioned oval shape, and an outflow vortex pair, at the bottom of the oval shape, of counter-rotating vortices can be identified in the mean streamwise vorticity field. The mutual induction of these inflow and outflow vortex pairs causes the outward movement of the sides of the oval shape of the mean streamwise velocity contour map shown at X/D_e=1.0 in Fig. 5 and thus produces the 180° rotation of the initial isosceles triangular shape of the mean streamwise velocity contour map at X/D_e=2.0, also shown in Fig. 5. This has been referred to as the axis-switching phenomenon in the literature on noncircular jets, as mentioned previously.

The mean streamwise vorticity contour map at X/D_e=2.0 can be explained in the same way as has been done for the maps at the two upstream locations, namely, in terms of inflow and outflow vortex pairs. It should be noted, as pointed out by Quinn (1992) for the square jet, that the counter-rotating vortex pairs have moved closer to the jet centerline at this location. Viscosity has clearly diffused the vorticity at X/D_e=10.0, as the mean streamwise vorticity contour map at this location shows.

3.8 Evolution of the turbulence intensities along the jet centerline

The evolution of the streamwise, spanwise, and lateral turbulence intensities, along with streamwise turbulence intensity data for a round jet, is shown in Fig. 11. A steep initial increase in all the turbulence intensities is observed as turbulence, produced by mean flow shear in the shear layers emanating from all three sides of the isosceles triangle, is transported by diffusion and convection to the jet centerline. The spanwise and lateral turbulence intensities show the same behavior as the streamwise turbulence intensities but generally have lower values. This observation is not surprising since the spanwise and lateral turbulence intensities are not produced directly from the local shear in the mean velocity but from the streamwise turbulence intensity via the pressure fluctuations. The turbulence intensities in the isosceles triangular jet peak at X/D_e=5.72 before the streamwise turbulence intensity in the round jet reaches its peak value, an indication of enhanced mixing in the isosceles triangular jet. It should be recalled that the mean static pressure along the jet centerline in the isosceles triangular jet, shown in Fig. 4, also starts to recover from negative values towards atmospheric pressure at X/D_e=5.72 before the static pressure in the round jet does.

Fig. 11

Evolution of the turbulence intensities along the jet centerline

3.9 Turbulence kinetic energy variation on the jet centerline

The turbulence kinetic energy variation along the jet centerline is shown in Fig. 12a. The turbulence kinetic energy is normalized by the local centerline mean streamwise velocity. The turbulence kinetic energy increases quickly and significantly with increase in X/D_e in all three jets as turbulence is produced in the jet shear layers and transported by convection and diffusion to the jet centerline. In the near flow field, as Fig. 12b shows, the highest values of the centerline turbulence kinetic energy are found in the equilateral triangular jet flow and this, along with the previously mentioned shortest potential core length, fastest near field centerline mean streamwise velocity decay, fastest spreading rate, etc., again implies that the fastest near field mixing in the three jets considered here occurs in the equilateral triangular jet.

Fig. 12

a Turbulence kinetic energy variation on the jet centerline. b Near field turbulence kinetic energy variation on the jet centerline

Beyond X/D_e=15, the highest centerline turbulence kinetic energy values are found in the isosceles triangular jet. The three jets do not show any self-preserving behavior of the turbulence quantities up to X/D_e=65. It should, for comparison, be noted that the spanwise turbulence intensity in the round air jet studied by Panchapakesan and Lumley (1993) was not self-preserving even at X/D=150.

3.10 Streamwise Reynolds normal stress contour maps

Contour maps for the streamwise Reynolds normal stress $(\overline {u^{'2} } )$ are shown in Fig. 13. The shapes of these contour maps correspond very closely to those of the mean streamwise velocity shown in Fig. 5 at the corresponding streamwise locations. Large or small values of the streamwise Reynolds normal stress are found, as is to be expected and will be shown later, in regions where the local shear in the mean streamwise velocity (∂U/∂Y, ∂U/∂Z) is large or small. The spanwise $(\overline {v^{'2} } )$ and lateral $(\overline {w^{'2} } )$ Reynolds normal stresses were also measured but the results are not presented here for space reasons. The values of these quantities are, as expected, smaller than those of the Reynolds normal stress at the corresponding locations in the flow field.

Fig. 13

Streamwise Reynolds normal stress $(\overline {u^{'2} } /U^2 _{{\text{cl}}} \times 100)$ contour maps

3.11 Spanwise Reynolds primary shear stress contour maps

The spanwise Reynolds primary shear stress $(\overline {u'v'}$ or, strictly, ρ$\overline {u'v'} )$ contour maps are shown in Fig. 14. The spanwise Reynolds primary shear stress represents the mean rate of transfer of the spanwise component of the linear momentum through a unit area normal to the streamwise direction. Contour maps for the local spanwise shear (∂U/∂Y) in the mean streamwise velocity are shown in Fig. 15 to facilitate the discussion of the spanwise Reynolds primary shear stress results. It is clear from Figs. 14 and 15 that the spanwise Reynolds primary shear stress is well correlated with the spanwise local shear in the mean streamwise velocity. This is not surprising since the dominant term in the generation of $\overline {u'v'}$ by the mean flow is $\overline {v^{'2}}$ $\partial U/\partial Y.$ The close correspondence between $\overline {u'v'}$ and ∂U/∂Y suggests that the stress-strain relationship: $- \overline {u'v'} = \nu _t \partial U/\partial Y$ is valid in the current flow, implying that an isotropic turbulence model, such as the k–ε model, can be used to close the set of governing equations in the numerical computation of this flow.

Fig. 14

Spanwise Reynolds primary shear stress $(\overline {u'v'} /U^2 _{{\text{cl}}} \times 100)$ contour maps

Fig. 15

Spanwise shear in the mean streamwise velocity (∂U/∂Y) contour maps

3.12 Lateral Reynolds primary shear stress contour maps

The lateral Reynolds primary shear stress $(\overline {u'w'} )$ contour maps are shown in Fig. 16 and those for the local lateral shear (∂U/∂Z) in the mean streamwise velocity are shown in Fig. 17. The stress $(\overline {u'w'} ),$ which is produced from the mean flow mainly by the term $\overline {w^{'2}}$ ∂U/∂Z, is clearly well correlated with the strain (∂U/∂Z) as was the case with the spanwise Reynolds primary shear stress data, which were discussed previously.

Fig. 16

Lateral Reynolds shear stress $(\overline {u'w'} /U^2 _{{\text{cl}}} \times 100)$ contour maps

Fig. 17

Lateral shear in the mean streamwise velocity (∂U/∂Z) contour maps

The major contributions to the production of the streamwise normal stress (u^’2) from the mean flow are made by: $- \overline {u'v'} \partial U/\partial Y$ and $-\overline { u'w'} \partial U/\partial Z.$ This is why a close relationship exists between the normal stress contour maps shown in Fig. 13 and the contour maps for the spanwise and lateral local shear in the mean streamwise velocity shown in Fig. 15 and in Fig. 17, respectively.

3.13 One-dimensional energy spectra of the streamwise fluctuating velocity on the jet centerline

The one-dimensional energy spectra of the streamwise fluctuating velocity on the jet centerline in the near flow field of the isosceles triangular jet, the equilateral triangular jet and the round jet are shown in Fig. 18a–c, respectively. At each location, the power spectra density (ϕ _u (f)) has been normalized by the square of the streamwise velocity fluctuation $(\overline {u^{'2}}).$ All of the one-dimensional energy spectra exhibit discrete peaks at a frequency of about 645 Hz at X/D_e=0.5, 1.0 and 2.0. The magnitude of the spectral peaks reaches a maximum at X/D_e=2.0 in all the three jets but the largest magnitude of the peaks is found in the equilateral triangular jet flow, again indicating that the fastest near field mixing among the three jets occurs in this jet. The spectral peaks indicate the passage of coherent structures. The one-dimensional energy spectra of the streamwise fluctuating velocity at X/D_e=10.0 in all three jets is broadband, a characteristic of fully turbulent flow.

Fig. 18

One-dimensional energy spectra of the streamwise fluctuating velocity on the jet centerline. a The isosceles triangular jet. b The equilateral triangular jet. c The round jet

4 Conclusions

The flow field of a turbulent free jet of air issuing from a sharp-edged isosceles triangular orifice into still air surroundings has been studied experimentally using hot-wire anemometry and a pitot-static tube. For the purpose of comparison, some measurements were made in an equilateral triangular jet and in a round jet, which also issued from sharp-edged orifices. The following conclusions are drawn from the results:

• 1. Mixing in an isosceles triangular jet, as measured by the entrainment of ambient fluid, the spread of the jet, the centerline mean streamwise velocity decay, the centerline turbulence kinetic energy, and the recovery of the mean static pressure on the jet centerline, is faster than in the round jet but slower than in the equilateral triangular jet.

• 2. The mean streamwise vorticity field in an isosceles triangular jet is dominated by counter-rotating pairs of vortices aligned, initially, with the corners of the isosceles triangle. These vortices influence mixing and entrainment in the jet.

• 3. Large-scale coherent structures, which facilitate mixing in the jet, are present in the near field of the jet; the equilateral triangular jet is the most energetic of the three jets considered here.