Experimental study of a lenticular jet

Abstract

Non-circular jets, particularly elongated jets and jets with sharp corners, are of interest due to their enhanced large- and small-scale mixing and combustion stability. Non-circular jet geometries such as elliptic, triangular, rectangular, and square have been extensively studied. The non-uniform curvature and elongated design of elliptic and rectangular jets has been shown to facilitate axis switching. Additionally, jets with sharp corners have been shown to facilitate small-scale mixing and further vortex ring deformation, as has been observed in triangular, square, and rectangular jets. There are, however, currently no data on the flow characteristics of a jet that combines these two features, i.e., it enhances both large- and small-scale mixing. The jet is issuing from a nozzle shape defined by two arcs of a circle connecting at sharp corners in the shape of an eye (hereafter termed ‘lenticular’). The current study uses experimental techniques to characterize the lenticular jet and compare its behavior to previously studied circular and non-circular jets. Additionally, snapshot POD analysis was used to identify coherent structures and their dynamic features. The lenticular jet was found to have higher entrainment and stronger mixing than a traditional round jet, and comparable mixing and entrainment characteristics as previously studied non-circular jet geometries. It is particularly interesting to compare the jet behavior of the lenticular jet to an elliptic jet. The geometry of these two jets is extremely similar, the difference being sharp corners along the major axis of the lenticular jet. By comparing the lenticular jet and the elliptic jet, the effect of sharp corners, which have been shown to increase small-scale mixing and vortex ring deformation, can be observed. It was found that along the minor axis, where geometry was similar, shear layer turbulence intensity and spreading rate of the jets were also similar. Along the major axis, however, introducing corner features in the lenticular jet lowered the spread rate, resulted in a faster breakdown of turbulence in the shear layer, increased exit turbulence, and resulted in anisotropic centerline turbulence.

Graphic abstract

Geometry of a jet issuing from a novel exit shape defined by two arcs of a circle connecting at sharp corners. The exit shape has been termed lenticular and possesses characteristics known to increase both large- and small-scale mixing. The ensuing jet sharply contracts along the major axis and rapidly expands along the minor axis resulting in an axis switch.

1. Introduction

Jets are widely used in various mixing and thrust-producing applications. Altering the geometry of the jet has been identified as a low-cost technique of passive flow control. Non-circular jets, particularly elongated jets and jets with sharp corners, are of interest due to their enhanced large- and small-scale mixing and combustion stability. Non-circular jet geometries such as elliptic, triangular, rectangular, and square have been extensively studied. A summary of previously studied non-circular jets and their flow characteristics can be found in Table 1. There are, however, currently no data on the flow characteristics of a jet issuing from a nozzle shape defined by two arcs of a circle connecting at sharp corners in the shape of an eye (hereafter termed ‘lenticular’). This shape is inspired by the shape vocal folds take during phonation. The shape is similar to an elliptic jet, which has been well researched (Ho and Gutmark 1987; Husain 1983; Quinn 1989), but includes sharp corner features along the major axis.

Table 1.

Mean streamwise velocity decay parameters on the jet centerline for referenced jets

Nozzle shape	1st investigator	Nozzle type	Re	AR	De (mm)	Lpc (De)	Kd	Ck	Range of x/De
Lenticular	Present study	Contour	36,000	2.5	13.7	3.6	.174	−.07	8–30
			72,000			3.01	.177	.3	5–20
Circular	Wygnanski and Fiedler (1969)	Contour	86,400	1	26.4	–	.169	−3	10–63
	Gutmark et al. (1991)	Contour	650,000		19	5.22	–	–	–
	Mi and Nathan (2009)	Contour orifice	15,000		14.2	5.6	.16	−1.7	16–40
						4.2	.208	−.6	11–40
	Quinn (2006)	Contour orifice	184,000		45.3	4.26	.164	−2.15	18–55
						3.5	.167	−3.65	18–55
	Xu and Antonia (2002)	Orifice	86,000		12	4	.16	1.58	10–17
Ellipse	Ho and Gutmark (1987)	Contour	78,000	2	34	2.9	.19	.37	5–20
	Hussain (1983)		100,000		50.4	3.0	.164	−.03	15–57
	Mi and Nathan (2009)	Orifice	15,000		14.7	3.1	.202	.8	8–40
	Quinn (1992)		184,000		45.3	2.83	.202	.421	8–60
Square	Gutmark et al. (1991)	Contour	650,000	1	19.2	3.94	–	–	–
	Grinstein et al. (1995)		42,000		49.2	2.21	–	–	–
	Mi and Nathan (2009)	Orifice	15,000		14.5	3.2	.203	−.5	8–40
	Quinn and Militzer (1988)		184,000		45.3	3.43	.185	−.15	8–64
Rectangular	Mi and Nathan (2009)	Orifice	15,000	2	14.5	2.9	.2	.9	8–40
	Quinn (1992)		208,000		45.3	2.44	.199	.753	18–60
Equilateral triangle	Gutmark et al. (1991)	Contour	650,000	1	18.7	4.09	–	–	–
	Mi and Nathan (2009)		15,000		14.5	3	.203	1.0	5–40
	Quinn (2005)	Orifice	184,000		45.3	2.91	.207	.389	20–52
	Xu et al. (2014)		50,000		12	3.05	.18	2.5	10–17
Isosceles triangle	Gutmark et al. (1991)	Contour	650,000	1.9	19.6	2.82	–	–	–
	Mi and Nathan (2009)	Orifice	15,000	2.5	14	1.1	.2	3.6	5–40
	Quinn (2005)		184,000	1.7	45.3	3.14	.196	.181	20–52
Diamond	Xu et al. (2013)	Orifice	50,000	1.7	12	2.12	.204	1.44	10–17

Previous studies have shown that non-circular jets enhance mixing by altering the natural development of the coherent structures and their breakdown into turbulence (Gutmark and Grinstein 1999). This enhances the three dimensionality of the flow and thus entrainment and mixing. In non-circular jets, the vortical structures can evolve through shapes similar to those of the jet nozzle, but with axes successively rotated at angles characteristic of the jet geometry. The result is a more rapid spreading of one axis relative to the other, leading to an axis switch (Gutmark and Grinstein 1999).

Axis switching is regarded as the main mechanism responsible for enhanced entrainment in non-circular jets. The underlying mechanism for the enhanced entrainment properties results from self-induced Biot-Savart deformation of vortex rings with non-uniform azimuthal curvature and interaction between azimuthal and streamwise vorticity. Portions of the vortex, such as corner portions, move downstream faster resulting in vortex ring deformation. The resulting deformation redistributes the energy between azimuthal and streamwise vortices whose subsequent interaction increases the small-scale content of the jet (Gutmark and Grinstein 1999). Vortex self-induction effects and axis switching are observable in elliptical (Ho and Gutmark 1987; Husain 1983; Quinn 1989), rectangular (Grinstein 2001; Quinn 1992), and triangular jets (Gutmark et al. 1991; Koshigoe et al. 1989; Quinn 2005; Schadow et al. 1988). These jets were shown to have entrainment rates significantly larger than those of a circular jet.

The occurrence of axis switching is sensitive to initial conditions such as non-uniform initial shear layer thickness, turbulence level, and jet forcing. Work done on triangular jets (Koshigoe et al. 1989) and square jets (Grinstein et al. 1995) showed that in pipe jets, shear layer growth rate at the vertex and flat sides were similar, and subsequently no axis switch occurred. However, in orifice jets with the same geometry, the flat sides grew faster than the vertex sides resulting in an axis switch. The lack of axis switching in pipe jets can also be attributed to the fact that in pipe jets, secondary flows inside of the nozzle are characteristically directed away from the jet axis and toward the corners, opposing spreading associated with axis switching. In nozzles with a smooth contraction, secondary flows are directed toward the jet axis from the corners and thus promote spreading associated with axis switching (Gutmark and Schadow 1987; Quinn 1992).

Corner features also increase mixing by increasing the small-scale turbulent content of the jet. Past studies have shown that sharp corners introduce high instability modes into the flow (Grinstein and Devore 1996; Schadow et al. 1988; Toyoda et al. 1993). Nozzles with corner features still develop large coherent structures at the flat sections, but at the corner sections, flow is predominantly three dimensional with small-scale turbulence (Grinstein and Devore 1996; Gutmark et al. 1989; Koshigoe et al. 1989; Schadow et al. 1988). The nonlinear interactions between turbulent fluctuations in the mean flow result in higher spreading at the flat sides relative to the corner sections. This behavior can be observed in triangular (Gutmark et al. 1991; Koshigoe et al. 1989; Quinn 2005; Schadow et al. 1988), square (Grinstein et al. 1995; Grinstein and Devore 1996; Gutmark et al. 1991), and rectangular jets (Grinstein 2001; Quinn 1992).

Jets that are able to improve both large- and small-scale mixing are of particular interest. These jets are advantageous in particular to combustion systems where entrainment provides the necessary amount of air to sustain combustion, and small-scale mixing at corner regions helps to start the reaction closer to the nozzle’s exit. Triangular and rectangular jets are classic examples of jets that enhance both large- and small-scale mixing. Large scale is enhanced through axis switching, while small-scale mixing is enhanced near corner regions and downstream due to the breakdown of vortex ring coherence and rapid transition to turbulence.

The lenticular geometry has features previously shown to enhance large- and small-scale mixing. The non-uniform curvature and elongated design has been shown to facilitate axis switching, as has been observed in elliptic and rectangular jets. Additionally, the corners along the major axis have been shown to facilitate small-scale mixing and further vortex ring deformation, as has been observed in triangular, square, and rectangular jets. The current study uses experimental techniques to characterize the lenticular jet and compare its behavior with previously studied non-circular jets.

2. Methodology

The lenticular jet was produced from a nozzle that is defined by two intersecting arcs that form an eye shape (Fig. 1a). Each arc is a portion of a circle with a radius of 13.8 mm and a central angle of 87.2°. The shape formed measures 19.05 mm along the major axis and 7.62 mm along the minor axis, with an aspect ratio of 2.5 and an equivalent diameter of D_e = 13.7 mm. The nozzle smoothly contracts from a circular chamber with an inner diameter of 100 mm to the lenticular shape over a fifth-order polynomial.

Fig. 1.

Nozzle exit geometry

Upstream the flow was monitored and regulated using a flow controller (Parker, MPC series) and flow meter (MicroMotion Inc, CMF025 Coriolis Flow Meter). The flow entered the nozzle through a settling chamber that was fitted with a honeycomb and screens to condition the flow. Measurements were taken at two Reynolds numbers, Re = U₀D_e/v = 36,000 and 72,000, based on the equivalent diameter D_e and exit velocity U₀.

Velocity measurements were taken using 2-D particle image velocimetry (PIV). This was accomplished using a dual cavity Nd:YLF laser (Litron Lasers, LDY304) synchronized with a high-speed video camera (Photron, miro 340). The camera was equipped with a Nikon 105-mm lens, yielding a spatial resolution of 13.9 pixels/mm. Images were taken in full resolution at a rate of 400 Hz. The position of the nozzle was controlled using a traverse system, allowing data sets to be taken up to x/D_e = 30 downstream. The laser sheet was focused to a 1 mm thickness and aligned with either the major or minor axis when collecting data.

PIV data were processed with DAVIS 8.3 software (LaVision GmbH) using multipass interrogation windows with decreasing size (64 × 64 to 32 × 32) and 50% overlap. In total, 1000 average image pairs were used based on a convergence study. The convergence study found that peak turbulence intensity at x/D_e = 20 changed less than 5% when averaging over 900 images. For POD analysis, an additional data set of 3500 images was taken up to x/D_e = 6 downstream of the nozzle exit.

Due to reflections off of the nozzle, PIV data could not be collected closer than x/D_e = .1, so a single hotwire probe (Dantec 55P11—1.25 mm long and 5 μm diameter) was used to measure the mean and turbulent velocities at the nozzle exit. Calibrations were made in the center of the exit plane of a smooth contraction circular jet fitted with a Honeywell pressure transducer. Hotwire data were fitted to a fourth-order polynomial to obtain a calibration curve. Data were then sampled at 5000 Hz at a constant overheat ratio of 1.6.

Uncertainty of PIV data was analyzed using DAVIS 8.3 software (LaVision GmbH). Velocity uncertainty (u_d) was less than .28 m/s resulting in at most a 2% error in velocity measurements. Additionally, PIV data were compared to hotwire data at x/D_e = 1, 2, and 3. Average difference between PIV and hotwire data was 3.8% (Fig. 2).

Fig. 2.

Comparison of streamwise velocity measured using PIV to hotwire data at x/D_e = 1, 2, and 3

Snapshot proper orthogonal decomposition (POD) analysis was used to identify coherent structures in the flow. POD is a mathematical tool that decomposes turbulent flow into modes that capture dynamically significant structures (Lumley et al. 1993). The snapshot POD method used in the current work was developed by Sirovich (1987). The method involves a sufficiently large number of snapshots of the flow (u′, v′, w′)_i = 1 ….N where N = number of snapshots. The variables u′, v′, and w′ represent the velocity fluctuations and are calculated by subtracting the mean flow from the instantaneous velocity field. The snapshots are assembled into a matrix M. The correlation matrix is represented by C_i = M^TM. The POD mode, U_i, is then found by solving the eigenvalue problem C_ia = λ_ia. The POD mode can then be expressed as follows:

$$U=\sum _{i=1}^N{a}_{i}U(x,t_i)$$

With the energy of each mode expressed as follows:

$$E_i=\frac{{\lambda }_i}{{\sum }_{i=1}^N{\lambda }_i}.$$

3. Results

The mean streamwise velocity (U) along the major and minor axes is plotted in Fig. 3 at Re = 36,000. The spreading rate in the two axes planes is noticeably different. Along the minor axis (Fig. 3a), the jet continuously spreads downstream of the nozzle exit. Along the major axis (Fig. 3b), the jet initially contracts and then begins to spread. These spreading trends are also observable in the normalized mean streamwise velocity profiles (U/U₀) (Fig. 4). At increasing axial distance from the nozzle exit along the minor axis, the shear layer spreads to the quiescent surrounding (Fig. 4a), while the shear layer shrinks toward the potential core along the major axis (Fig. 4b).

Fig. 3.

Contour of mean streamwise velocity along the minor (a) and major axes (b) at Re = 36,000

Fig. 4.

Mean streamwise velocity profiles along the major (a) and minor axes (b) at Re = 36,000

To quantify the spreading trends of the lenticular jet, the velocity half widths at Re = 36,000 and Re = 72,000 are plotted along the major and minor axes (Fig. 5a, b, respectively). The velocity half 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. Here y_1/2 is the velocity half width along the major axis, and z_½ is the velocity half width along the minor axis.

Fig. 5.

Half-width velocity of the lenticular jet at Re = 36,000 compared to two contoured circular jets (Wygnanski and Fiedler 1969; Xu and Antonia 2002) (a) and at Re = 72,000 compared to a contoured elliptical jet (Ho and Gutmark 1987) an orifice diamond jet (Xu et al. 2013) and two contoured circular jets (Wygnanski and Fiedler 1969; Xu and Antonia 2002) (b)

The lenticular jet grows continuously along the minor axis. Along the major axis, the lenticular jet contracts until x/D_e ≈ 5 and then begins to spread. Following x/D_e = 25 the major axis and minor axis begin to collapse and spread at the same rate as a circular jet (Fig. 5a). The spreading trends are likely tied to the dynamics of the circumferential K-H vortices formed near the jet exit as observed in elliptical (Ho and Gutmark 1987), rectangular (Grinstein 2001), and square jets (Grinstein and Devore 1996).

The first axis rotation of the jet cross section is due to self-deformation of the circumferential vortices (CV) due to non-uniform azimuthal curvature at the initial jet shear layer. Koshigoe et al. (1988) showed, using stability theory, that deformation of these vortices is the result of eigenmode instabilities. Eigenmodes were tracked as a function of core eccentricity as the jet-core cross section evolved from a circle to an ellipse with AR = 2. It was found that as core eccentricity increased, eigenmodes became localized with sufficiently large differences in phase speeds and comparable amplification rates. Additionally, regions with higher curvature had higher phase speed resulting in deformation of these vortex sections downstream. Grinstein and Devore (1996) demonstrated this deformation in square jets. At the corner regions of square jets, where curvature is high, the CV deformed downstream and toward the centerline, causing the jet to contract along the diagonal axis. Like the square jet, there is an observable contraction of the lenticular jet where corners are present. It is hypothesized here that the high curvature present along the major axis of the lenticular jet caused the CV to deform downstream and is responsible for the shrinking observed along the major axis. In an elliptic jet the stretching of the vortices downstream of areas of high curvature is accompanied by deformation of the transverse section. Ho and Gutmark (1987) showed through flow visualization that in an elliptical jet, self-induction of the vorticity azimuthally distorts the elliptic CV. The result is that the major axis initially contracts inward, and the minor axis spreads outward. This same outward spread is observable on the minor axis of the lenticular jet. The difference in growth rates along the respective axes produced one observable axis switch in the lenticular jet at x/D_e = 1.9. Axis switching is known to facilitate mixing; thus, the occurrence of an axis switch in the lenticular jet is indicative of strong entrainment.

The shape of the lenticular jet is characterized by elongated rounded edges along its minor axis and sharp corner features along its major axis. This geometry is a hybrid of an elliptical and diamond shape. For this reason, it is interesting to compare the spread of the jet along the respective axes. Since jet growth rate depends on several parameters including Re, initial turbulence, and aspect ratio (Gutmark and Grinstein 1999; Husain 1983; Quinn 1992), comparisons are made at Re = 72,000 to better match the Re of the elliptical and diamond jet (78,000 and 50,000, respectively). Additionally, jets being compared had similar aspect ratios of 2.25, 2, and 1.7 for the lenticular, elliptical, and diamond jet, respectively. Both the elliptical and lenticular jet emanated from contoured nozzles, whereas the diamond jet emanated from a sharp edged orifice plate which would yield a different distribution of initial shear layer thickness, and thus, a different jet growth rate (Gutmark and Schadow 1987).

The spreading behavior of the lenticular jet shows similarity along the respective axes. Along the minor axis the growth of the lenticular jet is nearly identical to an elliptical jet. In an elliptical jet it is the distortion of the CV along the minor axis that causes a large amount of the surrounding fluid to be induced toward the axis and entrained in the jet (Ho and Gutmark 1987). The similarly high growth rate along the minor axis of the lenticular jet is further evidence of the strong entrainment of this jet. Along the major axis, all three jets contract before beginning to spread again between x/D_e = 4 and 5. The spreading rate of the lenticular and diamond jet, however, is lower than that of the elliptical. This is due to interactions between the turbulent fluctuations and the mean flow caused by the corner regions (Grinstein et al. 1995; Grinstein and Devore 1996). This same reduction in spread rate is observable along the minor axis when comparing the spread of the diamond jet with its sharp corners to the lenticular and elliptical, which are both rounded.

The basic mechanism for the first axis rotation of the jet cross section is the self-deformation of the CV; however, subsequent axis rotations of the jet are strongly effected by interactions between large-scale vortices, as has been observed, in elliptic and square jets (Grinstein and Devore 1996; Gutmark and Grinstein 1999; Ho and Gutmark 1987). Interaction between large-scale vortices enhances fluid entrainment, induces further CV deformation, and triggers azimuthal instabilities (Gutmark and Grinstein 1999). The similarity in testing conditions for these jets suggests the main difference in spread rates is the result of differences in the large-scale structures responsible for axis switching and entrainment.

The lenticular jet had the greatest difference in spread rates between the major and minor axes due to the contrasting geometric features, and thus, the axis switching is observed further upstream (x/D_e = 1.9) compared to both the elliptical and diamond jets (x/D_e = 2.1 and x/D_e = 2.6, respectively). If the spreading rate of the jet is viewed as an indication to degree of entrainment, then the implication of these trends is that the entrainment of the lenticular jet is higher than a circular jet in both the near and far field.

Previous studies have shown that POD energy patterns provide a good representation of coherent structures (CS) in the flow field. Figure 6a shows the cumulative energy distribution of the POD modes along the major and minor axes. In total, 70% of the cumulative energy is contained in roughly the first 20 modes. These modes capture the essential large-scale structures of the jet. Eigenvalues above roughly 135 contain less than 5% of the energy and can therefore be neglected. The majority of energy is held in the first two modes along the minor axis, 12.2% and 11.8%, respectively, followed by a large drop in energy for subsequent modes (Fig. 6b). This indicates strong CS formation along the minor axis. Along the major axis, the energy of modes 1 and 2 is lower than that of the minor axis; however, there is less of a drop in energy for subsequent modes. Consequently, following mode 3, the energy along the major axis is higher than the minor axis until mode strength begins to converge at mode 10. This indicates higher small-scale fluctuations along the major axis, likely due to the corner features. Higher energy along the minor axis in low mode numbers, as well as higher energy along the major axis in high mode numbers, is also observable in an elliptic jet (Zhang et al. 2018).

Fig. 6.

Percent energy of first N modes (a) POD, eigenvalue spectrum (b)

The POD energy patterns for various modes are presented in Fig. 7, along the major and minor axes. There are clear alternating patterns of positive and negative energy, indicating the formation of CS. Along the major axis, CS contract toward the centerline and merge downstream. Along the minor axis, CS are strong and symmetric near the nozzle exit. CS on the minor axis spread toward the quiescent surrounding where they proceed breakdown slightly downstream of the axis switch, around x/D_e = 2.5. On both axes, following the first two modes, there are a decrease in energy near the nozzle exit and an increase in energy downstream. In subsequent modes, the CS become smaller and weaker, and the energy patterns become less symmetric.

Fig. 7.

POD modes along the major (a) and minor (b) axes

The decay of the mean streamwise velocity along the centerline of the lenticular jet at Re = 36,000 and Re = 72,000 along with other jets emanating from contoured nozzles is plotted in Fig. 8. The decay rate of the lenticular jet and previously studied non-circular jets has been fitted using linear regression to:

$$U_{\max }/U_{\mathrm{cl}}=K_{\text{d}}(x/D_{\text{e}}+C_{\text{K}})$$

where U_max is the maximum mean streamwise velocity on the jet centerline, K_d is the decay rate of the jet, and C_k is the kinematic virtual origin. The values of K_d, C_k, and range x/D_e used to fit the curves for this study and past studies are given in Table 1. The decay rate of the lenticular jet was calculated at Re = 36,000 and 72,000, but was approximately the same (K_d = 1.74 and 1.77, respectively).

Fig. 8.

Decay of mean streamwise centerline velocity compared to a contoured circular jet (Xu and Antonia 2002) and a contoured elliptical (Ho and Gutmark 1987)

Higher streamwise velocity decay rates imply higher entrainment of ambient fluid and consequently better mixing. The past studies have shown the decay rate of noncircular jets to be higher than round jets (Koshigoe et al. 1988; Mi and Nathan 2009; Xu and Antonia 2002). The decay rate of the lenticular jet (K_d ≈ 1.75) falls in the middle range of previously studied non-circular jet’s decay rates (K_d = 0.15–0.207) (Grinstein et al. 1995; Ho and Gutmark 1987; Husain 1983; Mi and Nathan 2009; Quinn and Militzer 1988; Quinn 1989, 2006; Schadow et al. 1988; Xu and Antonia 2002). This can be attributed to the fact that the majority of previously studied jets emanated from orifice nozzles, which have been shown to have higher centerline decay than contoured nozzles (Gutmark and Schadow 1987; Mi et al. 2000; Mi and Nathan 2009; Quinn 2006). Comparing the lenticular jet to only contoured nozzles, the lenticular jet showed higher decay rates than contoured circular nozzles (K_d = 0.16–0.169) (Mi and Nathan 2009; Quinn 2006; Wygnanski and Fiedler 1969). Additionally, at Re = 72,000, the centerline velocity decay of the lenticular jet (K_d = .177) was approximately the same as the contoured elliptical jet (Ho and Gutmark 1987) (K_d = .19), which was shown to have substantially higher mass entrainment than a circular and two-dimensional jets.

A strong indicator of near-held mixing is the length of the potential core (Gutmark and Grinstein 1999). A shorter potential core is indicative of stronger, faster near-held mixing. This region is of practical interest in jet applications, specifically combustion. Here we define the length of the potential core (L_pc) as the location in which the centerline velocity is 98% of the maximum centerline velocity (U_cl = .98 Umax). A summary of the L_pc for the lenticular jet and jets used for comparison can be found in Table 1. It is notable from this table that nozzle design, exit shape, and Reynolds number all strongly affect the length of the potential core. Previous studies have shown L_pc to be shorter in non-circular jets, orifice jets, and jets with higher Re (Mi and Nathan 2009; Quinn 2006). As expected in the lenticular jet, L_pc decreased with increasing Re (L_pc = 3.6 and 3.01 for Re = 36,000 and 72,000, respectively). Relative to previously studied non-circular jets, the L_pc of the lenticular jet is on the high end of the range (L_pc = 1.1–3.94) (Grinstein et al. 1995; Ho and Gutmark 1987; Husain 1983; Mi and Nathan 2009; Quinn and Militzer 1988; Quinn 1989, 2006; Schadow et al. 1988; Xu and Antonia 2002). This can be attributed to differences in nozzle design (contoured or orifice) and Re. Comparing the lenticular jet to contoured circular nozzles, the L_pc is shorter than jets tested at lower and higher Reynolds numbers (L_pc = 5.6 and 4.26 at Re = 15,000 and 184,000, respectively) (Mi and Nathan 2009; Quinn 2006), indicating stronger near-held mixing than a traditional round jet. Additionally, at Re = 72,000 the potential core length of the lenticular jet (L_p = 3.01) was approximately the same as the contoured elliptical jet (Ho and Gutmark 1987) at Re = 78,000 (L_pc = 2.9).

Contour plots of the normalized streamwise, spanwise, and lateral turbulence intensity (u′/U₀, v′/U_0, and w′/U₀, respectively) are presented in Fig. 9 at Re = 36,000. All profiles begin as a two peak profiles with maximums in the shear layer of the jet and a low turbulence intensity in the core of the jet due to the contoured nozzle design. The peak streamwise turbulence intensity is measured in the shear layer at x/D_e = .5 in both the major and minor axes (u′/U₀ = 0.21 and 0.23, respectively) (Fig. 10a, b). The spanwise turbulence intensity starts at a lower magnitude and increases up to x/D_e = 1 along the minor axis and x/D_e = 2 along the major axis (Fig. 10c, d). The initial high shear turbulence level and acceleration of the jet resulted in a reduction of turbulence level downstream. Along the major axis the peak streamwise turbulence moved toward the core up to x/D_e = 8 and then began to flatten out. Along the minor axis, the peak streamwise turbulence intensity constantly spreads to the quiescent surroundings and begins to flatten out after x/D_e = 5.

Fig. 9.

Contours of mean streamwise turbulence intensity along major (a) and minor axes (b). Mean spanwise turbulence intensity along major axis (v′/U₀) (c) and lateral turbulence intensity along the minor axis (w′/U₀) (d) at Re = 36,000

Fig. 10.

Streamwise turbulence intensity profiles along the major (a) and minor (b) axes. Spanwise turbulence intensity profiles along the major axis (c) and lateral turbulence intensity profiles along the minor axis (d)

Contour plots of the Reynolds stress along the major ($u^{\prime }{v}^{\prime }/U_0^2$) and minor ($u^{\prime }{w}^{\prime }/U_0^2$) are presented in Fig. 11 at Re = 36,000. In the major axis plane, the Reynolds stress was almost zero beyond y/D_e = 1 (Fig. 12a). The high Reynolds stress between y/D_e = 0 and y/D_e = 1 indicates most of the transverse momentum transfer occurred in the high-speed region of the jet. Along the minor axis, the Reynolds stress mainly occurred beyond z/D_e = 0.5 (Fig. 12b). The location of the high Reynolds stress on the minor axis indicates that the momentum transfer occurred where the jet spread outward.

Fig. 11.

Contour of Reynolds shear stress along the minor ($u^{\prime }{w}^{\prime }/U_0^2$) (a) and major ($u^{\prime }{v}^{\prime }/U_0^2$) (b) axes at Re = 36,000

Fig. 12.

Profiles of normalized Reynolds shear stress along the major ($u^{\prime }{v}^{\prime }/U_0^2$) (a) and minor ($u^{\prime }{w}^{\prime }/U_0^2$) (b) axes

Turbulence intensity in the shear layer along each axis is plotted in Fig. 13. Due to reflections off of the nozzle, turbulence data were not obtainable closer than x/D_e = 0.5 using PIV. Nozzle exit data presented here were collected using hotwire anemometry. Hotwire data showed higher exit turbulence intensity at the vertex along the major axis than the minor axis (u′_max/U₀ = 0.1 and 0.06, respectively). Further downstream, amplification of turbulence resulted in a higher peak turbulence intensity along the flatter minor axis of the jet (0.23 and 0.21 along the minor and major axis, respectively). These trends reflect those previously observed in square and triangular jets. In these jets, exit turbulence intensity was higher at the corner regions (Grinstein et al. 1995; Gutmark et al. 1989; Schadow et al. 1988) and peak streamwise turbulence intensity was higher at the flat side (Grinstein et al. 1995; Gutmark et al. 1989). Gutmark et al. (1989) showed that in square and isosceles triangular jets, the exit turbulence is higher at the vertex due to small-scale turbulent content. Moving downstream, there is a greater amplification of turbulence along the flat side resulting in higher peak turbulence relative to the vertex side.

Fig. 13.

Peak turbulence intensity variation along the shear layer of the major and minor axes of the lenticular jet and the elliptic jet (Ho and Gutmark 1987)

It is interesting to compare the shear stress components of the lenticular jet to an elliptic jet (Ho and Gutmark 1987) to observe the impact of adding corners along the major axis (Fig. 13). In the elliptical jet the shear stresses at the exit are the same on the major and minor axes (u′_max/U₀ = .04), whereas in the lenticular jet, corner features resulted in a higher exit turbulence along the major axis than the minor axis (u′_max/U₀ = 0.1 and 0.06, respectively). Downstream, along the minor axis, turbulence intensity in the shear layer is similar between the lenticular and elliptic jets. This is to be expected, due to the similarities in AR and nozzle geometry. Along the major axis however, the addition of a corner feature in the lenticular jet results in an initial higher turbulence intensity in the shear layer relative to the elliptic jet, similar to trends observed in square and triangular jets (Grinstein et al. 1995; Gutmark et al. 1989; Schadow et al. 1988).

Along the jet centerline all components of turbulent fluctuations increase along the potential core (Fig. 14a). The steep initial increase is observed as turbulence, produced in the shear layers, is transported by diffusion and convection to the jet centerline. A peak turbulence intensity in the lenticular jet can be observed at x/D_e = 7 for Re = 36,000, which is approximately twice the length of the potential core. In a circular jet the peak turbulence intensity would occur at the end of the potential core. A peak turbulence intensity downstream of the end of the potential core is a characteristic of non-circular jets. The streamwise turbulence intensity (u′/U₀) was consistently higher than the spanwise (v′/U₀) and lateral turbulence intensity (w′/U₀). At the jet exit, spanwise and lateral turbulence intensities were similar up to x/D_e = 4. At this point, spanwise and lateral turbulence intensities became anisotropic up to x/D_e = 16. In this range spanwise turbulence intensity, v′/U₀, was higher in magnitude than the lateral turbulence intensity, w′/U₀. Similar anisotropic centerline turbulence is observable along the jet centerline in other jets with corner features (Gutmark et al. 1991; Quinn 2005).

Fig. 14.

Evolution of turbulence intensities along the jet centerline in the lenticular jet (a). A comparison of the streamwise turbulence intensity, (b) streamwise relative turbulence intensity (c) and spanwise and lateral turbulence intensity (d) of the lenticular jet to previously studied circular and non-circular jets

Due to differences in nozzle design and Re, it is difficult to compare the turbulence of the lenticular nozzle to previously studied non-circular jets. There are, however, some trends worth noting. The peak turbulence intensity occurs at a similar location to the elliptical jet (Ho and Gutmark 1987), which, like the lenticular jet, emanated from a contoured nozzle. The peak location of both of these contoured non-circular jets occurred downstream of non-circular orifice jets used for comparison in Fig. 14b. Previously, it has been shown that orifice nozzles have peak turbulence values further upstream than contoured nozzles (Gutmark and Schadow 1987; Mi and Nathan 2009; Quinn 2006). The peak turbulence location of the lenticular jet does, however, occur upstream of circular jets emanating from both contoured and orifice nozzles at Re= 184,000 (Quinn 2006). A peak turbulence location upstream of circular jets implies faster near-held mixing. The difference in peak turbulence intensity values can be attributed to differences in initial conditions and nozzle geometry.

Due to a rapid centerline velocity decay, the relative turbulence intensity, defined as the streamwise turbulence normalized by the local centerline velocity (u′/U_cl), quickly increases until it begins to approach an asymptotic value of ~ 0.25 downstream (Fig. 14c). In the near field, the relative turbulence intensity of the lenticular jet is higher than a circular jet from both a contoured nozzle and orifice nozzle (Quinn 2006), indicating faster near-field mixing. In the far field, the lenticular jet appears to be approaching an asymptotic value similar to the other jet geometries. Differences in asymptotic turbulence intensity values can be attributed to differences in initial conditions and jet geometries.

The peak spanwise and lateral turbulence intensity occur at a similar location in all of the non-circular and upstream of both the contoured and orifice circular jets (Fig. 14d). In comparing spanwise to lateral turbulence intensity values, it is interesting to note that in an elliptical jet, where both the major and minor axes are similar, the spanwise and lateral turbulence intensity values are isotropic. However, in the lenticular jet, as well as in the triangular jet, the turbulence intensity associated with the axis that possesses corner features shows higher turbulence intensity than the smooth or flat side.

4. Conclusions

Non-circular jets have been extensively studied due to their enhanced large- and small-scale mixing. The current study investigates experimentally the velocity and turbulence fields of a jet emanating from a lenticular geometry. The lenticular shape has sharp corners, to increase small-scale turbulence and mixing, and is elongated, to enhance large-scale mixing and entrainment. Enhanced entrainment relative to a traditional round jet was shown through an observable axis switch, a higher jet spreading rate, and a higher centerline decay rate. Additionally, the length of the potential core and centerline turbulence intensity were indicative of stronger faster mixing relative to traditional round jet. It is difficult to compare the lenticular jet to previously studied non-circular jets due to differences in nozzle design and initial conditions. However, the centerline decay, potential core length, and centerline turbulence intensity were comparable to previously studied non-circular jet geometries known to have stronger mixing and higher entrainment.

The geometry of the lenticular jet closely mimics that of an elliptical jet, which has been extensively studied, the difference being the corners along the major axis of the lenticular jet, which are known to enhance small-scale mixing. It is interesting to compare the lenticular jet of this study with the elliptical jet (Ho and Gutmark 1987) to observe the effect the corners have. Both jets emanated from contoured nozzles, with similar aspect ratios and exit Re. The length of the potential core, centerline velocity decay, and centerline streamwise turbulence intensity were extremely similar. Additionally, jet behavior along the minor axis was also similar. Along the minor axis, where the jets had a similar curvature, the spreading rate, measured by the half width of the jet, and the streamwise turbulence intensity in the shear layer, were nearly identical. Along the major axis, however, where the lenticular jet had sharp corners, and the elliptic jet had a rounded profile, there were several observable differences. The streamwise turbulence intensity in the shear layer of the jet along the major axis was lower in the lenticular jet due to a faster breakdown of streamwise turbulence intensity. Additionally, the spreading rate of the lenticular jet along the major axis was lower than the elliptical jet. This decrease in spreading rate, due to corner features, was further exemplified by comparing both the elliptic and lenticular jet to a diamond jet. Along both the major and minor axes, jets with corner features had a lower spreading rate. Along the major axis the diamond and lenticular jet spread slower than the elliptic jet; along the minor axis the diamond jet spread slower than the lenticular and elliptic jet. Koshigoe et al. (1988) showed, using stability theory, that regions of higher curvature, such as corners, have a higher phase speed resulting in stretching of the circumferential vortices downstream. This stretching of these vortices at corner regions causes the jet to shrink, thus decreasing the spreading. Further the vortex dynamics, such as hairpin vortices and braid vortices, are likely also involved in the lower spreading rate of the jet at corner regions, as has been observed in square and rectangular jets. Further work is needed to fully understand the vortex dynamics of the lenticular jet.

The addition of corner features along the major axis of the lenticular jet also introduced differences in behavior along the major and minor axes that were not observable in an elliptic jet. In the elliptic jet, streamwise exit turbulence in the shear layer of the major and minor axes was the same (u′/U₀ = 0.04). In the lenticular jet, however, streamwise exit turbulence was higher in the shear layer of the major axis than the minor axis (u′/U₀ = 0.1 and 0.06, respectively). This difference is due to small-scale turbulence structures created at the corner regions, as has been observed in both square and triangular jets. In addition to exit turbulence, the spanwise and lateral turbulence intensities along the jet centerline showed different behaviors in the lenticular jet than the elliptic jet. In the elliptic jet, spanwise and lateral turbulence intensity was isotropic. In the lenticular jet, however, the spanwise and lateral turbulence intensity was anisotropic. The spanwise turbulence intensity, associated with the corners along the major axis, was higher than the lateral turbulence intensity, associated with the smooth rounded edges of the minor axis. Anisotropic centerline turbulence intensity resulting from corners in the jet has previously been observed in triangular jets.

In conclusion, the lenticular jet demonstrated higher entrainment and stronger mixing than a traditional round jet. The lenticular jet’s mixing and entrainment characteristics were comparable to previously studied non-circular jet geometries known to have higher entrainment. The most interesting comparisons came from comparing the lenticular jet to an elliptic jet. The geometries of these two jets are extremely similar, the difference being corner features along the major axis of the lenticular jet. By comparing the lenticular jet and the elliptic jet, it was observed that along the minor axis, where geometry was similar, trends were similar. Along the major axis however, introducing corner features lowered the spread rate, resulted in a faster breakdown of turbulence in the shear layer, increased exit turbulence, and resulted in anisotropic centerline turbulence.