Proper orthogonal decomposition analysis of near-field coherent structures associated with V-notched nozzle jets

Experimental Thermal and Fluid Science 112 (2020) 109972

Contents lists available at ScienceDirect

Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs

Proper orthogonal decomposition analysis of near-field coherent structures associated with V-notched nozzle jets

T

H.D. Lima, Junfei Dingb, Shengxian Shib, T.H. Newa,

⁎

a b

School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Republic of Singapore School of Mechanical Engineering, Shanghai Jiao Tong University, 200240 Shanghai, China

ARTICLE INFO

ABSTRACT

Keywords: Indeterminate-origin jets Time-resolved particle-image velocimetry Proper orthogonal decomposition Jet-mixing

Time-resolved particle image velocimetry measurements and proper orthogonal decomposition (POD) analysis were conducted on freely-exhausting V-notched nozzle jets at Re = 5000. Energy redistributions from low order modes to higher order modes are observed, particularly for the first two POD modes typically associated with large-scale coherent flow structures. Furthermore, analysis of the first two POD modes reveals highly cyclical large-scale coherent flow structures formed along the nozzle peak-to-peak (PP) planes, while non-cyclical incoherent flow structures are observed along the trough-to-trough (TT) planes. POD mode coefficients reveal mode pairing behaviour along the PP-planes and reduced peak frequencies in their power spectral densities. In contrast, no mode pairing behaviour is observed along the TT-planes and multiple instances of the same frequency peak transcending two adjacent POD modes are observed instead. This suggests an energy cascade process whereby large-scale flow structures are broken down into smaller-scale ones at a common frequency. Finally, a comparatively sharper nozzle leads to earlier formations of flow structures along both PP- and TTplanes but does not significantly impact upon the periodicity or coherence of the flow structures.

1. Introduction Free jet flows have been the subject of academic and industrial interest for the past century [1–4], motivated by the desire to further understanding in the origin, evolution and decay of jet flows, as well as current and future engineering applications. Earlier studies revealing the initiation, evolution and decay of large-scale coherent structures demonstrated their influence in the dynamics of free jets [5,6], and it is now widely accepted that controlling these coherent structures is effective in achieving better entrainment and mixing characteristics. The origins of these structures may be described with linear stability theory [7], whereby perturbations in the form of initial turbulence, acoustic waves or mechanical excitation are amplified. If the perturbations are unstable, they will lead to growths in the initial velocity perturbations and formations of coherent structures as the jets convect downstream. These flow structures form at frequencies that are dependent upon the initial shear layer thickness [8,9], and may undergo vortex-pairing behaviour which decreases the vortex formation frequency and increases the structure wavelength [5,10]. Further downstream from the jet exits, three-dimensional effects dominate the jet vortex structures, where self-induction, vortex stretching, as well as disconnection and reconnection [11] can dominate the fundamental jet flow dynamics. ⁎

Aside from conventional round jets, a significant number of studies on jet flow control via nozzle modifications had also been reported. In particular, nozzle exit modifications achieved by varying the azimuthal axial lengths of the nozzle lips in a systematic manner was first termed as indeterminate-origin (IO) nozzles [12], and schematics of selected circular IO nozzles are provided in Fig. 1. The earlier study focused on the initial azimuthal flow instabilities and energy redistributions of the jet shear layer for inclined and stepped nozzles. The inclined nozzles were observed to modify the jet shear layer spread rates through an azimuthally propagating self-excitation mechanism, while the stepped nozzles developed two shear layers independently in both halves of the jet before undergoing cross-excitations to form vortices of the second subharmonic frequency. Although the nozzle modifications were effective in achieving azimuthal energy redistributions in the jet shear layers, axial developments of instability frequencies remain invariant to these physical modifications. A subsequent study on inclined nozzles was conducted with a particular emphasis on the effects of forcing frequency and nozzle inclination angles upon the behaviour of the near-field coherent structures [13]. No pairings were observed at the preferred Strouhal number and the initial inclination angles of ring-vortices gradually deviate from the nozzle inclination angle as they convect downstream. Ring-vortices

Corressponding author. E-mail address: [email protected] (T.H. New).

https://doi.org/10.1016/j.expthermflusci.2019.109972 Received 6 January 2019; Received in revised form 23 October 2019; Accepted 26 October 2019 Available online 28 October 2019 0894-1777/ © 2019 Elsevier Inc. All rights reserved.

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 1. Circular IO nozzles with (a) inclined, (b) stepped, (c) crowned (or chevroned), (d) A-notched and (e) V-notched exit geometries.

As opposed to crown-shaped or chevron nozzles where there are typically four or more peaks and troughs, V- and A-notched nozzles only have two peaks and two troughs [20]. Note that the former possesses smooth peaks and sharp troughs, while the latter has sharp peaks and smooth troughs. A flow visualization study on V-notched jets [21] revealed mutual induction and interactions of the bent vortex ring (following the contour of the nozzle exit geometry), leading to development of streamwise vortex pairs at both peak and trough locations, which spread outwards by riding on the primary ring vortices. These flow structures were found to be very similar to those observed in crown-shaped nozzles [14] where four peaks and four troughs exist. A subsequent study on jets issuing from V-notched nozzles with different sharpness was conducted [22], which served to clarify the flow development and interpretations of the vortex dynamics in V-notched jets. Earlier pairings of the V-shaped vortex filament sections at the troughs lead to formation of streamwise vortices and as a result, a vortex disconnection and reconnection process occur which leads to a “bent” vortex filament that proceeds to undergo further flow developments reminiscent of axis-switching behaviour seen in elliptic vortex filaments. At the same time, streamwise vortices formed at the peaks disconnected vertically from the vortex filament as orphaned vortex loops. Vorticity results indicate vortex filaments displayed higher vortex strength at the peak as compared to the trough locations, and this difference can be accentuated with the use of sharper V-notched nozzles. This suggests that the nozzle relative sharpness may be used to control the vortex dynamics in order to achieve improved jet-mixing with the ambient fluid. At this point in time, much of the information gathered on nozzles modified with peaks and troughs are based on visualizations of the resulting vortical structures, as well as instantaneous and mean flow characteristics captured through various experimental techniques. In contrast, modal decomposition analysis of the key coherent flow structures, flow instabilities and modes underpinning the flow dynamics observed in the past studies remains lacking. Better understanding of the flow energy content distributions and their correlations to the different fundamental flow modes will shed light upon how jetmixing is augmented through such nozzles. To address that, a timeresolved particle image velocimetry (TR-PIV) and proper orthogonal decomposition (POD) analysis was performed on jets issuing from Vnotched nozzles used in earlier studies [22] to better understanding on how the transient flow behaviour can be related to the jet-mixing characteristics.

with large initial inclination angles undergo breakdowns much closer to the nozzle and were hypothesized to be due to higher growth rate of longitudinal instabilities in the strained braid regions between the ringvortex cores. Above the preferred Strouhal number however, inclination angles of the ring-vortices remain relatively invariant where vortex-pairings and vortex breakdowns were observed for small and large nozzle inclination angles respectively. Other IO nozzle types such as crown-shaped nozzles with varying number of sharp peaks and troughs were also investigated [14] to determine their effects on the behaviour of the jet shear layers. As opposed to conventional round jets whereby natural hydrodynamic instabilities lead to secondary streamwise structures at non-specific azimuthal locations, crown-shaped nozzles introduce azimuthal instabilities according to the nozzle exit geometrical designs and dictate the locations where secondary streamwise structures are formed. Furthermore, entrainment of ambient fluid into the jets was observed at the nozzle troughs, while jet fluid moves radially away from the nozzle peaks. As such, spreading of a round jet may be modified by trough and peak placements to achieve specific jet-mixing and control strategies. These observations are further reinforced by more work on tapered crown-shaped nozzles [15,16], whereby periodic formation of incursion streamwise vortex-pairs at the nozzle troughs were observed to reorganize through a pairing process to form excursion vortex-pairs at the nozzle peaks. The excursion vortex pairs are directly responsible for an enhanced spread rate along the nozzle peak planes and lead to much faster increments in the shear layer momentum thickness as compared to that along the nozzle trough planes or conventional round nozzle at the same working conditions. Chevron nozzles which are analogous to crown-shaped nozzles have also received considerable attention due to their efficacy in jet noise reduction. A parametric study based on the number of chevrons, amount of chevron penetration, chevron length and chevron symmetry had been conducted by examining their effects on the flow field and jet acoustics [17]. The results for Mach = 0.9 hot and cold jets indicated deeper chevron penetration and lower number of chevrons lead to more rapid centreline velocity decay, and therefore enhancing mixing. The chevron length had limited impact on the flow field while introducing asymmetry to chevrons reduces the effectiveness of the chevrons. While deeper chevron penetration has benefits in terms of enhanced mixing and low-frequency noise attenuation, thrust penalty and an increase in high-frequency noise have also been observed. The benefits of chevrons at a specific chevron penetration depth was also highly dependent on the velocity difference across the high-speed and low-speed stream [18], indicating that chevron designs are application specific and has to be utilized at near design conditions. Subsequently, streamwise vortices educed by chevrons in a Mach = 0.9 jet was examined [19], and results indicate a higher increase in the axial vorticity particularly in the initial region of the jet, hence improving local entrainment and mixing. When compared with streamwise vortices produced by microjets, chevron axial vorticity was stronger initially. However, it also decayed more rapidly with downstream distance, while generating small-scale structures that increases the high-frequency noise.

2. Experimental setup and procedures 2.1. Experimental setup and jet apparatus The general experimental setup was based on the one adopted by New and Tsovolos [22] and used recently by Shi et al. [23,24], hence its description will only be briefly covered here. For more details, readers are encouraged to refer to the above-mentioned studies. All experiments were conducted in an octagonal closed-loop vertical water tank 2

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 2. Schematic of experimental setup used in the present study.

Fig. 3. Schematics of (a) AR2 and (b) AR4 V-notched nozzles.

constructed from 20 mm thick Perspex panels, with internal dimensions of 360 mm and a height of 800 mm. Water from a small reservoir was redirected into a flexible rubber hose via a water pump, where it was subsequently regulated by an electromagnetic flow meter and a needle valve to control the flow rate. Thereafter, it entered a jet apparatus where a series of flow-conditioning devices such as a diffuser, honeycomb section, three layers of fine screens and a smooth circular-tocircular contraction section with 20:1 contraction-ratio were installed, before exhausting out of the test nozzle into the water tank. The rig was designed such that all test nozzles were located at the centre and bottom of the water tank, for a more axisymmetric setup in the subsequent experiments. Based on the nominal jet exit velocity of approximately 0.2 m/s, ambient temperature of 25 °C and the nozzle inner diameter, the Reynolds number used for all test jets was estimated to be approximately Re = 5000. Note that all jet flows exhausted freely from the test nozzles without any external artificial perturbations such that the vortex structures and dynamics occurred naturally, as opposed to jet studies where artificial mechanical forcing was introduced to the jets using speaker-driven piston [13,20]. A schematic of the experimental setup is provided in Fig. 2.

2.2. Test nozzle designs One circular baseline and two V-notched nozzles were investigated here and followed the conventions made use of by New and Tsovolos [22] previously. The notches were designed such that for every V-notch that was formed by two nozzle “cuts”, each cut was designed according to a half-ellipse of a selected aspect-ratio (AR). As such, the two notched nozzle configurations were named after the aspect-ratio of the half-ellipse used (i.e. AR = 2 and 4) and will be known as AR2 and AR4 nozzles hereafter. Consequently, the AR4 nozzle will be relatively sharper than its AR2 counterpart. Furthermore, to ease understanding, 2D measurement planes that are aligned with both nozzle peaks and troughs are termed as PP- and TT-planes respectively, similar to the naming conventions adopted by the above-mentioned study. All three test nozzles were also designed with common mean heights of H = 30 mm (i.e. mean of the longest and shortest nozzle axial lengths), inner diameters of D = 20 mm and nozzle wall thickness of t = 1 mm. Lastly, they were fabricated out of stainless-steel to avoid material deterioration during the study. Schematics of the V-notched nozzles are provided in Fig. 3.

3

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

2.3. Time-resolved particle-image velocimetry

direction (streamwise), and v denotes the velocity in the y-direction. The fluctuating matrix, uv , is then constructed by subtracting the time¯ , from uv . averaged velocity field, uv

For TR-PIV experiments, both the water jets and water tank were seeded with 50 µm polyamide seeding particles with a nominal density of ρ = 1.03 g/cm3 from Dantec Dynamics. Note that during the course of the experiments, the reservoir water was replaced periodically to remove undesirable agglomerated seeding particles suspended in the water. Additionally, seeding particles that gradually accumulated on the Perspex panels were removed on a regular basis to avoid any possible optical contamination as well. Illumination was provided by a 1 W, 532 nm wavelength diode-pumped solid-state (DPSS) continuouswave laser, where an approximately 1 mm thick laser sheet was produced with a plano-concave lens and a slit filter. For each test case, 4998 sequential particle-images were captured at 700 frames-persecond (FPS) with an IDT NX8-S1 camera fitted with a Nikon 50 mm f 1.4D lens, with an initial measurement area (streamwise × radial) of approximately 5D × 7D which was reduced to 4.7D × 4D when presenting the results. Experiments were conducted in an air-conditioned environment with the room temperature maintained at 25 °C and the nozzle exterior surfaces were spray-painted with matt black paint to reduce laser reflection. To process the particle images, a two-step refinement multi-grid cross-correlation analysis of the particle-images were conducted with initial and final interrogation windows of 128px × 128px and 32px × 32px respectively. An overlapping ratio of 50% was used throughout. Spurious vectors were rejected via global and local rejection criteria and replaced by a 3-point by 3-point neighbourhood scheme. Based on the actual measurement window size, the final velocity vector fields have a resolution of 1.47–1.60 mm (approximately 0.07–0.08D), depending on the exact nozzle. Note that the differences in the resolution were due to constraints imposed by the experimental setup. For more information on the TR-PIV technique and post-processing procedures, readers are advised to refer to relevant earlier studies [25,26].

Next, the two-point correlation matrix is constructed by:

CNxN = uv T × uv The eigenvectors A and eigenvalues solving the eigenvalue problem:

1 vM

(5)

= uv × A

Finally, the POD mode coefficients or temporal coefficients may be obtained by projecting the fluctuating velocity field on the POD modes by: (6)

a = uv T ×

Order reduction can then be achieved by reconstructing the fluctuating flow field based on selected POD modes:

uvrecon =

(7)

× aT

By selecting POD modes that are the most representative of the flow field, this effectively filters out the influence of experimental noise and fine-scale turbulence. It is also possible to reconstruct the fluctuating flow field based on phase-sorted snapshots [30], to obtain phase-resolved fluctuating flow fields of a cyclical flow phenomenon. For POD analysis in the x-z plane, the velocity matrix is redefined by snapshots taken in the x-z plane:

u11 uw =

u1N

1 uM

w11

w1N

1 wM

(8)

where w denotes the velocity in the z-direction. Subsequent procedures follow the same as described earlier and will be omitted for the sake of brevity. 3. Results and discussions 3.1. Mean jet flow characteristics Before moving to the POD results and analysis, it will be insightful to have an appreciation of the mean jet flow characteristics first. This will help in discerning the important mean flow differences that manifested as a result of the underlying flow dynamics, which POD analysis will be able to shed more light upon later on. Fig. 4 shows the mean jet velocity profiles taken across the nozzle exit at x/D = 1.1 location, turbulence intensity profiles, and centreline velocities normalized by the jet centreline exit velocity, U (taken at x/D = 1.1), as well as the streamwise evolution of the vorticity thicknesses. Fig. 4(a) shows that the baseline jet exhibits a relatively “top-hat” mean velocity profile. On the other hand, those for the notched nozzles demonstrate different changes to the velocity profile, depending on whether it is along the PPor TT-plane. For instance, velocity profiles along PP-planes show slight but consistent increments to the velocity magnitudes closer to the two nozzle peaks. In contrast, velocity profiles along TT-planes show more significant and consistent reductions to the velocity magnitudes closer

u1N

1 uM

v11

may then be obtained by

The eigenvalues represent the relative amount of energy associated with each eigenvector and can be used as a criterion to rank the eigenvectors. The spatially orthogonal POD modes may then be obtained by:

The use of POD approach for fluid flow analysis was first introduced by Lumley [27] and subsequently improved upon by Sirovich [28], Berkooz et al. [29] and many others. The approach is capable of extracting information from experimental data by approximating a highdimensional fluid problem with a lower dimension energy-efficient system, hence allowing the identification of the most important and energetic structures that are dominant within the flow field that is being studied. There have been many successful applications of POD analysis in a wide variety of flow scenarios, such as flow past cylinders [30], turbulent jets [31–35], flow past backward-facing step [36], laminar separation bubble [37] and time-resolved schlieren images of a supersonic ramp nozzle [38]. In this present study, implementation of the POD analysis was based on an earlier study where statistically independent and time-resolved datasets were used [32]. The domain of POD application was to the initial PIV measurement area of 5D × 7D (streamwise × radial). In order to perform POD analysis in the x-y plane, the instantaneous velocity fields (snapshots) are arranged in single column vector format to obtain the velocity matrix,

uv =

(3)

(4)

CA = A

2.4. Proper orthogonal decomposition analysis

u11

(2)

¯ uv

uv = uv

v1N (1)

where M denotes the number of velocity vectors in a single snapshot, N denotes the total number of snapshots, u denotes the velocity in the x4

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 4. Mean jet characteristics depicting the (a) velocity profiles at x/D = 1.1, (b) streamwise evolution of vorticity thicknesses (c) streamwise evolution of turbulence intensity profiles and (d) centreline velocities of all test nozzles.

to the two troughs, such that they are more “parabolic”. While the magnitude changes between the AR2 and AR4 nozzles are not too significant, it can nevertheless be discerned that a higher nozzle AR (i.e. sharper nozzle) leads to bigger differences in the velocity profiles between the two planes. These observations are further supported by the normalized vorticity thicknesses ( /D ) plotted in Fig. 4(b). The vorticity thickness was based on the definition:

=

Ucl |du/ dr|max

(9)

where Ucl represents the local centreline velocity and |du/dr|max represents the maximum slope of the velocity profile (for PP-plane, r = y, and for TT-plane, r = z). The maximum slope was obtained based on lines fitted to two adjacent points within each profile. For the initial vorticity thickness (x/D = 1.1), the AR4 PP-plane produces the smallest value, followed by AR2 PP-plane, baseline, AR2 TT-plane and AR4 TT5

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

plane. Although a higher nozzle AR leads to larger deviations from the baseline, the difference between the AR2 and AR4 nozzles for the same plane is not too significant (see Fig. 4(b)) and this is consistent with what was observed in Fig. 4(a) earlier. In contrast, differences between the PP- and TT-planes of the same AR nozzle are much more significant, implying that the nozzle peak and troughs have a larger impact on the initial flow field than the nozzle AR. As the streamwise distance increases, the vorticity thickness of the baseline is always the smallest, while the AR4 nozzle is always the largest. While the vorticity thicknesses along TT-planes of both notched nozzles start off larger than their PP-plane counterparts, the rate of increase for the PP-planes is higher than that of the TT-planes, resulting in a crossover point at approximately x/D = 3.8 for the AR2 nozzle and x/D = 3.3 for the AR4 nozzle. Beyond the crossover point, the vorticity thickness is larger along the PP-plane as compared to the TT-plane. For the initial turbulence intensity profiles (x/D = 1.1) depicted in Fig. 4(c), the baseline nozzle attained a peak level of approximately 10% along the jet shear layers. However, peak turbulence intensity levels of the notched nozzles are discernibly lower with clear dependency upon the exact plane. In particular, peak turbulence intensity levels within the jet shear layers along the PP-planes are lower than those along the TT-planes. More intriguingly, turbulence intensity characteristics associated with the jet shear layers along the TT-planes exhibit a “twin-peak” phenomena, with the outer peaks having higher peak magnitudes than the inner peaks. It is also interesting to note that the baseline nozzle produces narrow peaks, as opposed to the notched nozzles which produce peaks that are relatively wider. This indicates that the notched nozzles could have wider mixing layers that lead to overall better mixing and faster erosion of the jet potential cores. As the downstream distance increases, the turbulence intensity levels within the jet shear layers along the PP-planes exceed those along the TTplanes, while maintaining a similar but wider peak profile to the baseline. In contrast, the profiles along the TT-planes become much flatter with eroded peaks and are no longer symmetric about the jet axis. From these results, it is possible that the notches have introduced earlier development of three-dimensionalities. To ascertain the above conjecture, the jet centreline velocity profiles are extracted and presented in Fig. 4(d). For a more representative overall comparison, results along the PP- and TT-planes for the same notched nozzle were averaged, and it can be discerned that the use of notched nozzles lead to more rapid decay of the centreline velocities, particularly for the higher AR nozzle. This result is consistent with

earlier observations of thicker mixing layers and earlier development of three-dimensionality. Mean vorticity contour plots are presented in Fig. 5, where they are plotted based on the same colour legend to ease comparisons. It can be observed that the vorticity regions associated with the jet shear layers produced by the notched nozzles are significantly reduced as compared to the baseline case. In particular, prominent “forking” of the jet shear layers can also be observed along the PP-planes of both notched nozzles, indicative of the streamwise vortices produced by the nozzle peaks [22]. This shows that the jet flow behaviour for the notched nozzles based on the working conditions here remain relatively similar to those observed in an earlier study based on the same nozzles and similar test conditions [22]. The vorticity region along the jet shear layers of both TT-planes can also be observed to intrude deeper into the jet potential core region, consistent with the relatively flatter turbulence intensity profile previously observed. Lastly, the mean turbulent kinetic energy (TKE) results are presented in Fig. 6. It can be seen that the notched nozzles are able to reduce the TKE associated with the jet shear layers. This is usually indicative of enhanced mixing where large-scale coherent jet shear layer vortices are broken down into less coherent ones with smaller velocity fluctuations. Together with the rapid decay in the centreline velocity seen in Fig. 4(d) and the rapid decrement of vorticity magnitude observed in Fig. 5, evidences presented so far point towards the present notched nozzles being viable devices for passive jet-mixing enhancement purposes. Note that the lack of symmetry in the results of the TTplanes of both notched nozzles can be attributed to the use of TR-PIV, where the sampling time per run is significantly lower as compared to standard 15 Hz double-frame PIV, resulting in lesser number of flow cycles captured per run. However, since the focus here is look into the unsteady flow characteristics, the mean flow characteristics remains reasonable in terms of providing indicative flow trends. 3.2. Flow energy distributions To start off the POD analysis, fractional energy distributions in terms of the eigenvalue spectrum up to the first twenty POD modes of all test configurations are presented in Fig. 7. Additionally, the cumulative energy content for the first twenty and first two POD modes are included in Table 1 as well. For the baseline nozzle, the cumulative energy content of 61.3% for the first twenty modes and 21.3% for the first two modes compare well with the POD analysis of a Re = 3000 plane jet earlier by Shim et al. [32]. As for the notched nozzles, their

Fig. 5. Mean vorticity contour plots determined for (a) baseline, (b) AR2 PP-plane, (c) AR2 TT-plane, (d) AR4 PP-plane and (e) AR4 TT-plane. 6

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 6. Normalized mean turbulent kinetic energy distributions associated with the (a) baseline, (b) AR2 PP-plane, (c) AR2 TT-plane, (d) AR4 PP-plane and (e) AR4 TT-plane.

use leads to significantly lower total cumulative energy content of approximately 46% and 55% for AR2 and AR4 nozzles respectively. These reductions indicate that there is significant energy redistribution to POD modes higher than the first twenty and suggest that the more large-scale and coherent flow structures typically associated with lower POD modes are being rendered more incoherent and smaller-scale by the notched nozzles. This will also be consistent with the observations made in the mean jet flow characteristics earlier. In fact, the largest fractional energy reductions occur for POD modes 1 and 2, which would correspond to the large-scale coherent flow structures. Interestingly though, results show that the AR2 nozzle appears to redistribute the energy content better than the relatively sharper AR4 nozzle, though tangible differences will become more negligible beyond POD mode 6.

Table 1 Percentage of flow energy content for first twenty and first two POD modes. First twenty POD modes PP-plane Reference AR2V AR4V

45.9% 53.5%

61.3%

First two POD modes

TT-plane

PP-plane

46.1% 56.8%

11.5% 14.2%

TT-plane 21.3% 10.5% 11.9%

and 2 coefficient correlations are determined and plotted in Fig. 8(a)–(e). These scatter plots have been used in past studies to illustrate cyclical relationship within mode pairs [25,30], and will be used here to examine the cyclical behaviour of the structures that are dominant within the jet flows encountered here. Each point in the scatter plot represents a single snapshot and the mode coefficient correlation scatter plot is in fact a visual representation of the contribution of the first two POD modes on the overall flow field. In the work of New et al. [25] where POD analysis was applied to flow past a cylinder problem, vortex-shedding behaviour described by POD modes 1 and 2

3.3. POD mode coefficient correlations Since flow energy content results presented earlier demonstrate that the use of notched nozzles impacts upon POD modes 1 and 2 significantly, it will be appropriate to take a closer look at these two POD modes, particularly their correlations with each other. POD modes 1

Fig. 7. Fractional energy distributions associated with (a) PP- and (b) TT-planes of the notched nozzles. Results for the baseline nozzle are included for comparison purposes. 7

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 8. POD modes 1 and 2 coefficient correlation scatter plots determined for (a) baseline, (b) AR2 PP-plane, (c) AR2 TT-plane, (d) AR4 PP-plane and (e) AR4 TTplane. Plot (f) shows the correlation between POD modes 2 and 3 of AR4 TT-plane.

Fig. 9. POD mode 1 results for (a) baseline nozzle+, (b) AR2 PP-plane+, (c) AR2 TT-plane#, (d) AR4 PP-plane+ and (e) AR4 TT-plane#. Note: +based on v’/U, #based on w’/U.

take up more than 50% of the total flow energy, and the scatter points generally lies on the perimeter of a circle, away from the origin. For jet flows, the first two modes typically contain a much lower fraction of the total flow energy [32], at 21.3% for the current study (baseline test case). Hence, many of the snapshots have significant contributions from higher order modes and this is illustrated by the scatter points that are

located near the origin where both POD modes 1 and 2 coefficients are small (see Fig. 8(a)). Nonetheless, cyclical behaviour can still be inferred from circular distributions within the scatter plots. For the baseline nozzle shown in Fig. 8(a), clear circular distributions indicative of highly cyclical flow patterns can be readily observed. As such, the first two POD modes are expected to display a strong POD 8

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 10. POD mode 2 results for (a) baseline nozzle+, (b) AR2 PP-plane+, (c) AR2 TT-plane#, (d) AR4 PP-plane+ and (e) AR4 TT-plane#. Note: # based on w’/U.

+

based on v’/U,

depictions of large-scale, coherent flow structures that are symmetrical about the jet axis. On the other hand, POD modes 1 and 2 results determined along the TT-planes of the notched nozzles (shown in Figs. 9(c), (e), 10(c) and (e)) show otherwise. They are consistently asymmetrical, more disorganized and weaker regardless of the exact nozzle AR. Between the two POD modes, they do not appear to share similar structures with a π/2 phase difference. Hence, POD mode pairing between modes 1 and 2 along the TT-plane is not significant, or at least a lot weaker than what was observed along PP-plane earlier. This observation agrees well with what was discerned from the mode coefficient correlation scatter plots earlier. In fact, similar trends can be observed in Figs. 11 and 12, where POD modes 1 and 2 in terms of u’/U are presented. Again, the baseline nozzle and notched nozzle PP-planes produce results that are cyclical and highly coherent, while more incoherent and asymmetric flow structures are produced by notched nozzles along their TT-planes. In this case however, the distributions centred more around the jet potential core regions rather than along the jet shear layers. This is because the vortex cores are located along the jet shear layer edges, hence the magnitude of u’/U is expected to be the highest adjacent to the shear layer edges, near the centreline region where contributions from both sides of the jet shear layers cumulate. Similar features have been observed in the study of jets using POD analysis [35]. Finally, comparisons between the two notched nozzles indicate that the AR4 nozzle leads to earlier formations of flow structures along the PP-plane than the baseline and AR2 nozzle, with no significant changes to the periodicity or the coherence of the flow structures. From the POD mode 1 and 2 results based on u′/U, v′/U and w′/U parameters seen earlier, it can be concluded that flow structures and behaviour along the TT-planes are significantly more irregular due to the nozzle troughs imposed by the notches. The flow cycles along the TT-planes are also more significantly affected and appear to be much less cyclical than those along the PP-planes. This will tie in with the earlier observation from the mean jet flow characteristics, whereby more rapid reductions in the vorticity magnitudes along the jet shear layer regions, as well as consistently lower TKE levels along the TTplanes than PP-planes, were observed. As such, the POD results thus far

mode pair relationship, with both POD modes expected to display similar structures with the only difference being a π/2 phase shift. Note that this has also been observed in earlier POD based studies where mode pair relationships were found [30,32]. Along the PP-planes of the notched nozzles, the mode coefficient correlation scatter plots shown in Fig. 8(b) and (d) can also be observed to produce circular distributions. Hence, highly cyclical flow structures and POD mode pair relationships similar to that of the baseline nozzle can be expected. On the other hand, the scatter plots along the TT-planes of the notched nozzles do not produce clear circular distributions in Fig. 8(c) and particularly so for Fig. 8(e). This indicates less cyclical behaviour of the coherent structures, of which they are expected to manifest in the further results of POD modes 1 and 2 later. Interestingly, when the scatter plot analysis was extended to POD modes 2 and 3 for the AR4 TT-plane as shown in Fig. 8(f), an off-centric circular distribution can be observed, suggesting the possibility of limited cyclical behaviour in the higher order POD modes. 3.4. Analysis of POD modes 1 and 2 Figs. 9–12 show POD modes 1 and 2 results plotted using streamwise u’/U and radial v’/U and w’/U results for all three test nozzles and test configurations. Starting with the baseline nozzle POD modes 1 and 2 results based on v’/U as presented in Figs. 9(a) and 10(a), it can be seen that it comprises of highly cyclical flow behaviour due to the observation of similar structures between the two modes, with the only difference between them being a π/2 phase difference that arises due to the spatially orthogonal property of POD modes. The results also indicate coherent and symmetrical vortex structures along the jet shear layer regions. As such, these flow structures will be the large-scale and coherent vortex roll-ups along the baseline jet shear layers. Since the plotted parameter is v’/U where radial entrainment plays a dominant role, it is unsurprising that the jet shear layer vortex roll-ups will be highlighted in Figs. 9 and 10. As for the notched nozzles, the corresponding POD modes 1 and 2 results determined along their PP-planes (shown in Figs. 9(b), (d), 10(b) and (d)) demonstrate highly cyclical flow behaviour, with clear

9

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 11. POD mode 1 results based on u’/U component for (a) baseline nozzle, (b) AR2 PP-plane, (c) AR2 TT-plane, (d) AR4 PP-plane and (e) AR4 TT-plane.

indicate that the vortex dynamics are very different between the PPand TT-planes, and that the nozzle troughs are able to break down coherent structures into fine-scale structures and hasten transition to turbulence much better than the nozzle peaks. While it is possible that nozzle misalignment gave rise to the flow asymmetry seen in the notched nozzle TT-plane results in Figs. 9 and 10, it seems unlikely since the flow structures and behaviour along PP-plane remains highly symmetric and coherent despite using similar nozzle alignment procedures during the present study. Hence, a more plausible explanation is that these observations are due to the random and asymmetrical variations in the entrainment of ambient fluid at the nozzle troughs associated with every cycle of the jet shear layer vortex formation.

acquisition time to extract the dominant peak frequencies. Subsequently, the data was subjected to the Welch’s power spectral density (PSD) estimate function in MATLABTM. This methodology is preferred over the standard fast Fourier transform (FFT), as the spectra averaging process reduces spurious influences coming from the signal noise. The dataset of 4997 time-resolved velocity field with a sampling frequency of 700 Hz was divided into two equal-length segments with segment lengths of 90% of the dataset (rounded to nearest whole number). From segment to segment, the number of overlapping samples was 80% of the dataset (rounded to nearest whole number). Each segment was windowed with a Hamming window function, and the modified periodograms were averaged to obtain the PSD estimate. From the PSD estimate, the peak amplitudes were identified and nondimensionalized to Strouhal number based on the equation,

3.5. POD frequencies To look into the effects of the notched nozzles upon the flow frequency behaviour, POD mode coefficients are plotted against its

St =

fD U

(10)

Fig. 12. POD mode 2 results based on u’/U component for (a) baseline nozzle, (b) AR2 PP-plane, (c) AR2 TT-plane, (d) AR4 PP-plane and (e) AR4 TT-plane. 10

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 13. PSD results for (a) baseline nozzle+, (b) AR2 PP-plane+, (c) AR2 TT-plane#, (d) AR4 PP-plane+ and (e) AR4 TT-plane#. Note: +for POD modes 1 and 2, #for POD modes 1, 2 and 3.

where D is the inner diameter of the nozzle and U is the jet centreline exit velocity (taken at x/D = 1.1). For the PSD estimate, 4096 discrete Fourier transform (DFT) points were used and the frequency resolution was 0.171 Hz. This translates to a Strouhal number resolution of 0.011 to 0.014 depending on the test case. For each nozzle-plane test case, all PSDs were normalized by the same peak amplitude value. These results are presented in Fig. 13, with Table 2 consolidating the Strouhal numbers associated with the peak frequencies determined from this approach for comparison purposes. A systematic study to assess the

sensitivity of the spectral results to the signal processing procedures can be found in Appendix A. The frequencies captured based on POD modes represent the global frequencies pertaining to a selected POD mode, which effectively filters out the frequencies associated with fine-scale flow structures and/or turbulence captured by higher order POD modes. This is more advantageous over typical frequency extraction techniques based on the “point-probe” principle by conducting FFT analysis on time-series data, since the extracted signal will retain characteristics associated with

Table 2 Strouhal number associated with the peak frequencies determined from PSD results presented in Fig. 14, ranked according to the PSD normalized amplitudes.

11

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. 14. Comparison of signal extraction approaches via (a) POD mode coefficient and (b) point-probe method at location of maximal radial velocity fluctuations [(x/ D, y/D, z/D) = (2.5, −0.55, 0)]; for the baseline jet.

flow turbulence and is hence inherently noisier. For comparison purposes, Fig. 14(b) illustrate signals extracted via the point-probe approach at the location of maximal radial velocity fluctuations [(x/D, y/ D, z/D) = (2.5, −0.55, 0)], and can be observed to be noisier than the signal obtained from the POD based approach shown in Fig. 14(a). A detailed comparison between the point-probe and POD mode coefficient methods to extract frequency information can be found in Appendix B. For the baseline nozzle and both notched nozzles along their PPplanes, it is clear from Fig. 13(a), (b) and (d) that the PSD results for POD modes 1 and 2 display excellent matches. The strong spectral coherence between the two POD modes and the requirement that they must be spatially orthogonal is indicative of the two POD modes representing the same flow structures but with a phase difference of π/2 [32]. Since phase information is lost in PSD computations, a strong match in the spectral results is well within expectations. In terms of flow behaviour, it signifies that these three test cases produce highly cyclical flow structures, which further reinforce similar notions implied by the earlier results based on the coefficient correlations scatter plots. In contrast, PSD results for notched nozzles along their TT-planes do not exhibit such good matches between POD modes 1 and 2. In fact, no good matches are obtained even when the PSDs for POD mode 3 are included in Fig. 13(c) and (e) for comparisons. However, as earlier results have also indicated poor POD mode pairing outcomes for these test cases, these PSD results are not that surprising. In terms of the actual peak frequencies extracted from Fig. 13 and consolidated in Table 2, two well-defined peak frequencies with similar amplitudes were identified at St1 = 0.4 and 0.44 for the baseline nozzle (see Fig. 13(a)). These frequencies agree well with the range of Strouhal number of the preferred mode for round jets, which ranges from St = 0.3 to 0.64 [8]. Note that multiple peaks can co-exist in a single POD mode because POD modes are not temporally orthogonal, hence multiple phenomena characterized by unique frequencies can co-exist in a single POD mode. The observation of two peaks with similar frequencies and amplitudes is suggestive that they correspond to the same flow phenomenon (preferred mode of the jet), and the discrepancy could possibly be due to slight variations in the initial turbulence of the jet. Since the experiments involved jets that exhausted freely without any form of external artificial forcing, slight variations in the results can be expected. As for the notched nozzles along their PP-planes, there are some distinct differences between the PSDs along the AR2 and AR4 PPplanes. For the PSDs along the AR2 PP-plane as shown in Fig. 13(b), the maximum amplitude peak occurs at St1 = 0.33, while the next highest

peak occurs at St2 = 0.35 with an amplitude value of 0.29. The relatively well-defined and distinct single peak indicates a strong likelihood that there is a single flow phenomenon (the preferred mode at St1 = 0.33) described by the first two POD modes for the AR2 PP-plane. In contrast, for the PSDs along the AR4 PP-plane as shown in Fig. 13(d), the maximum amplitude peak occurs at St1 = 0.25, while the next two highest peaks occur at St2 = 0.21 and 0.34 with a relatively comparable amplitude of approximately 0.68. Based on this observation, it is likely that there are multiple flow phenomena described by the first two POD modes along the AR4 PP-plane, with the preferred mode at St1 = 0.25 being the most dominant one. At this point in time, it is unclear what type of flow phenomena St2 = 0.21 and 0.34 may possibly correspond to. As compared to the baseline test case, the reduction in the peak frequencies for notched nozzles along their PP-planes is indicative of increased coherence time and length scales, which agrees well with earlier POD mode results shown in Figs. 9 and 10 where a smaller number of coherent flow structures appears to exist in the same streamwise range as compared to the baseline nozzle. This in turn could be due to the interactions between the jet shear layer roll- ups and the streamwise vortices induced by the nozzle peaks (shown in Fig. 5), leading to roll-up behaviours that deviated from those observed in the baseline test case. This will also agree well with the observations made in an earlier study [22], where streamwise vortices were consistently observed to produce along the nozzle peaks and affect the overall jet spread. For the notched nozzles along their TT-planes, the associated PSD plots are considerably noisy and multiple peaks at relatively high amplitudes can be observed. The peak frequency situations are even more intriguing. For instance, the peak frequencies detected for POD mode 1 in AR4 nozzle along TT-plane occur at St1 = 0.15 and St2 = 0.04, ranked according to their amplitudes. For POD mode 2 however, the peak frequencies ranked in terms of amplitudes are detected at St1 = 0.04 and St2 = 0.29, while it is St1 = 0.29 for POD mode 3. The situation is similar for the AR2 nozzle and hence, the details will not be repeated for the sake of brevity and the readers can refer to the peak frequencies in Table 2. Even though no strong cyclical relationships are exhibited between adjacent POD modes in notched nozzles, multiple instances of one peak frequency co-existing in two adjacent POD modes can be observed. These results suggest the possibility of an energy cascade process whereby large-scale structures are broken down into smaller-scale ones (POD modes 1 and 2 respectively) at St = 0.04, and the small-scale structures continue to break down into even smallerscale structures (POD modes 2 and 3 respectively) at St = 0.29. Such an

12

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

explanation will agree well with the observations made earlier, where results along the TT-planes were observed to exhibit non-cyclical, incoherent and asymmetrical flow structures about the jet axis, with strong changes in the radial velocity fluctuation strengths of adjacent POD modes. As such, the present results indicate that the nozzle troughs are significantly more capable in breaking down large-scale coherent structures than the nozzle peaks, and that the breaking-down process could possibly involve a systematic energy cascade process.

notched nozzles and indicate breaking down of large-scale coherent jet shear layer vortices into more incoherent ones. POD analysis shows that there is an energy redistribution from loworder modes to higher-order modes along both PP- and TT-planes of both notched nozzles. The first two POD modes along the PP-planes are observed to have highly cyclical, coherent and large-scale flow structures that are symmetrical along the jet centrelines. While this is largely similar to the baseline nozzle, the associated peak Strouhal numbers are reduced. This is postulated to be due to the interactions between the jet shear layer vortices and the streamwise vortices induced by the nozzle peaks. In contrast, notched nozzle TT-planes lead to non-cyclical flow structures that were significantly more irregular, incoherent and asymmetric along the jet axis. A higher AR (and hence comparatively sharper) nozzle produces flow structures earlier with minimal impact on their flow periodicity or coherence. At least one peak Strouhal number co-exists in two adjacent POD modes along the TT-planes, which suggest an energy cascade process in which large-scale structures are broken down into smaller-scale ones at frequencies that transcends adjacent POD modes. The preceding results suggest that the peak planes of V-notched nozzles exhibit and maintain flow coherence much alike to what was observed for the baseline jet but at a lower Strouhal number, while the trough planes is dominated by breakdown of structures through a systematic energy cascade process due to much earlier transition to turbulence.

4. Conclusions Free jets issued from V-notched nozzles were investigated using TRPIV and POD techniques to study the flow dynamics underpinning the differences in the flow and mixing behaviour observed along their PPand TT-planes. Mean jet characteristics revealed a “top-hat” velocity profile produced by the baseline nozzle and sharper peaks in the turbulence intensity profiles. In contrast, both notched nozzles produce slight velocity magnitude increments near the nozzle peaks along the PP-planes and more “parabolic” velocity profiles along the TT-planes. The vorticity thicknesses of both notched nozzles are consistently larger than that of the baseline nozzle within the streamwise measurement range, with AR4 nozzle always displaying larger values for the same nozzle plane. They also produce broader peaks in the initial turbulence intensity profiles, indicative of wider mixing layers. As the downstream distance increases, the turbulence intensities of the PP-planes maintain similar but wider peak profiles to the baseline, while that of the TTplanes became much flatter and are no longer symmetric, which suggest the possibility of wider mixing layers that intrude into the jet potential core. This agrees well with observations of earlier reductions in the mean jet shear layer vorticity magnitudes and faster decays in the jet centreline velocity profiles for notched nozzles. Furthermore, peak TKE levels along the TT-planes are reduced more significantly for the

Acknowledgment The authors gratefully acknowledge support for the study through a Singapore Ministry of Education AcRF Tier-2 grant (Grant number: MOE2014-T2-1-002), support for the first author through an NTU Nanyang President Graduate Scholarship, and facility support from Shanghai Jiao Tong University.

Appendix A. Sensitivity of spectral results to signal processing procedures A systematic study on the different parameters of the signal processing procedures was performed to illustrate the sensitivity of the spectral results to the procedures. The choice of the window length is deemed to be the most critical parameter as it can determine both the frequency resolution and the number of blocks for averaging, hence it will be the main focus in this systematic study. For a dataset with fixed total length of 4997 (number of velocity fields) and fixed sampling frequency of 700 Hz (image acquisition rate), the independent variables (window length and overlap length) were varied to produce varying values of the dependent variables (overlap percentage, number of blocks and frequency resolution) for analysis. Note that for the purpose of illustration, the window length was selected based on powers of 2 such it is the same as the number of DFT points. Hence, the frequency resolution can be computed by simply taking the sampling frequency divided by the window length. It is then converted to the Strouhal number based on the definition provided in Eq. (10). Table A1 summarizes the parameters that were varied to produce the spectral results while Fig. A1 shows the spectral results. From Fig. A1, it can be observed that choosing short windows can lead to more blocks for averaging. However, the frequency resolution is poorer and may lead to poor results. For example, in test #4 as shown in Fig. A1(d), the frequency resolution of ΔSt = 0.043 is clearly insufficient in resolving the twin peaks which can be observed in Fig. A1(a). In contrast, choosing long windows can lead to finer frequency resolution of the PSD as shown in Fig. A1(a). However, it is also visibly noisier (see Fig. A1(a)) due to lesser number of blocks used in the averaging. Hence, there is a tradeoff to consider when deciding on the window length. For the current study, it was necessary to select a window length that is as large as possible in order to improve the frequency resolution. This is because the frequencies of interest are around 7 Hz which is relatively low as compared to the sampling frequency of 700 Hz. Table A1 Parameters for signal processing procedure. Test no.

Window length

Overlap length

Overlap percentage

No. of blocks

Frequency resolution (ΔSt)

#1 #2 #3 #4

4096 2048 2048 1024

3277 1024 1536 512

80% 50% 75% 50%

2 3 6 8

0.011 0.022 0.022 0.043

13

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. A1. Normalized PSD results based on POD approach for test number (a) 1, (b) 2, (c) 3 and (d) 4.

Appendix B. Comparison with point-probe method for frequency extraction Time-series radial fluctuating velocity data were extracted from four locations along the jet shear layer as shown in Table B1. The locations were selected by examining the POD mode 1 v’/U contour map and picking locations with the highest radial fluctuating velocity magnitude. This is shown in Fig. B1. The time-series data was then processed with the same signal processing procedure and parameters as those described in test #1 of Appendix A (see Table A1). The spectrum results are shown in Fig. B2 below. As compared to Fig. A1(a), these spectrums are visibly noisier as the signal at each point-probe location consists of all 4997 POD modes. Depending on the point-probe location, the frequencies associated with the peak amplitude ranges between St = 0.4, 0.41, 0.44 and 0.51. From the results, it appears that there is a decrease in frequency with streamwise evolution. Note that similar observation of decreasing frequency with streamwise evolution has also been observed in plane jet studies by Lee and Hassan [39]. The St = 0.4 and 0.44 peaks are consistent with those that were obtained from the POD spectral results as shown in Fig. A1(a). Since POD is a global method which considers the entire 2D velocity field, the spectral peaks do not show a natural bias to any particular location. In contrast, the spectrum of the raw data is a 1D “point-probe” method that can only extract the frequency information at a specific point location, hence the spectral peaks can differ based on the selected location (due to signal noise, insufficient averaging, etc). Table B1 Point-probe location for spectral analysis. Coordinates

B1

B2

B3

B4

x/D y/D z/D

1.90 0.57 0

2.42 0.57 0

3.09 0.57 0

3.84 0.42 0

14

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al.

Fig. B1. Point-probe locations for data extraction.

Fig. B2. Normalized PSD results based on the point-probe method at location (a) B1 (b) B2 (c) B3 and (d) B4.

31 (1) (1999) 239–272. [3] K. Jambunathan, E. Lai, M. Moss, B. Button, A review of heat transfer data for single circular jet impingement, Int. J. Heat Fluid Flow 13 (2) (1992) 106–115. [4] C.K. Tam, Supersonic jet noise, Annu. Rev. Fluid Mech. 27 (1) (1995) 17–43. [5] S.C. Crow, F. Champagne, Orderly structure in jet turbulence, J. Fluid Mech. 48 (3)

References [1] A. Glezer, M. Amitay, Synthetic jets, Annu. Rev. Fluid Mech. 34 (1) (2002) 503–529. [2] E. Gutmark, F. Grinstein, Flow control with noncircular jets, Annu. Rev. Fluid Mech.

15

Experimental Thermal and Fluid Science 112 (2020) 109972

H.D. Lim, et al. (1971) 547–591. [6] G.L. Brown, A. Roshko, On density effects and large structure in turbulent mixing layers, J. Fluid Mech. 64 (4) (1974) 775–816. [7] H. Schlichting, K. Gersten, Boundary-layer theory, Springer Science & Business Media, 2003. [8] E. Gutmark, C.M. Ho, Preferred modes and the spreading rates of jets, Phys. Fluids 26 (10) (1983) 2932–2938. [9] A. Michalke, On spatially growing disturbances in an inviscid shear layer, J. Fluid Mech. 23 (3) (1965) 521–544. [10] C.D. Winant, F.K. Browand, Vortex pairing: the mechanism of turbulent mixinglayer growth at moderate Reynolds number, J. Fluid Mech. 63 (2) (1974) 237–255. [11] A.F. Hussain, Coherent structures and turbulence, J. Fluid Mech. 173 (1986) 303–356. [12] R. Wlezien, V. Kibens, Passive control of jets with indeterminate origins, AIAA J. 24 (8) (1986) 1263–1270. [13] D.R. Webster, E.K. Longmire, Vortex dynamics in jets from inclined nozzles, Phys. Fluids 9 (3) (1997) 655–666. [14] E.K. Longmire, J.K. Eaton, C.J. Elkins, Control of jet structure by crown-shaped nozzles, AIAA J. 30 (2) (1992) 505–512. [15] F. Shu, M.W. Plesniak, P.E. Sojka, Indeterminate-origin nozzles to control jet structure and evolution, J. Turbul. 6 (2005) N26. [16] S. Poussou, M.W. Plesniak, Near-field flow measurements of a cavitating jet emanating from a crown-shaped nozzle, J. Fluids Eng. 129 (5) (2007) 605–612. [17] J. Bridges, C. Brown, Parametric testing of chevrons on single flow hot jets, 10th AIAA/CEAS aeroacoustics conference, (2004). [18] B. Callender, E.J. Gutmark, S. Martens, Far-field acoustic investigation into chevron nozzle mechanisms and trends, AIAA J. 43 (1) (2005) 87–95. [19] M.B. Alkislar, A. Krothapalli, G. Butler, The effect of streamwise vortices on the aeroacoustics of a Mach 0.9 jet, J. Fluid Mech. 578 (2007) 139–169. [20] T.H. New, H.M. Tsai, Experimental investigations on indeterminate-origin V-and Anotched jets, AIAA J. 45 (4) (2007) 828–839. [21] T.H. New, K.M.K. Lim, H.M. Tsai, Vortical structures in a laminar V-notched indeterminate-origin jet, Phys. Fluids 17 (5) (2005) 054108. [22] T.H. New, D. Tsovolos, Influence of nozzle sharpness on the flow fields of V-notched nozzle jets, Phys. Fluids 21 (8) (2009) 084107. [23] S. Shi, J. Ding, T.H. New, J. Soria, Light-field camera-based 3D volumetric particle image velocimetry with dense ray tracing reconstruction technique, Exp. Fluids 58 (7) (2017) 78. [24] S. Shi, J. Ding, C. Atkinson, J. Soria, T.H. New, A detailed comparison of single-

camera light-field PIV and tomographic PIV, Exp. Fluids 59 (3) (2018) 46. [25] T.H. New, S. Shi, Y. Liu, On the flow behaviour of confined finite-length wavy cylinders, J. Fluids Struct. 54 (2015) 281–296. [26] S. Shi, T.H. New, Some observations in the vortex-turning behaviour of noncircular inclined jets, Exp. Fluids 54 (11) (2013) 1614. [27] J.L. Lumley, The structure of inhomogeneous turbulent flows, Atmospheric Turbulence and Radio Wave Propagation, (1967). [28] L. Sirovich, Turbulence and the dynamics of coherent structures. I. Coherent structures, Q. Appl. Math. 45 (3) (1987) 561–571. [29] G. Berkooz, P. Holmes, J.L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech. 25 (1) (1993) 539–575. [30] B. Van Oudheusden, F. Scarano, N. Van Hinsberg, D. Watt, Phase-resolved characterization of vortex shedding in the near wake of a square-section cylinder at incidence, Exp. Fluids 39 (1) (2005) 86–98. [31] O. Semeraro, G. Bellani, F. Lundell, Analysis of time-resolved PIV measurements of a confined turbulent jet using POD and Koopman modes, Exp. Fluids 53 (5) (2012) 1203–1220. [32] Y.M. Shim, R.N. Sharma, P.J. Richards, Proper orthogonal decomposition analysis of the flow field in a plane jet, Exp. Therm Fluid Sci. 51 (2013) 37–55. [33] B. Zang, T.H. New, On the wake-like vortical arrangement and behaviour associated with twin jets in close proximity, Exp. Therm Fluid Sci. 69 (2015) 127–140. [34] B. Weitao, S. Yasuhiko, O. Koji, M. Haruki, Time-resolved proper orthogonal decomposition of the near-field flow of a round jet measured by dynamic particle image velocimetry, Meas. Sci. Technol. 14 (8) (2003) L1. [35] B. Malla, D. Cuppoletti, J. Kastner, E. Gutmark, Proper orthogonal decomposition on a subsonic jet exhausted from an axisymmetric nozzle and chevron nozzles with varying penetration, 49th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition, (2011). [36] R. Sampath, S. Chakravarthy, Proper orthogonal and dynamic mode decompositions of time-resolved PIV of confined backward-facing step flow, Exp. Fluids 55 (9) (2014) 1792. [37] D. Lengani, D. Simoni, M. Ubaldi, P. Zunino, POD analysis of the unsteady behavior of a laminar separation bubble, Exp. Therm Fluid Sci. 58 (2014) 70–79. [38] M.G. Berry, M.Y. Ali, A.S. Magstadt, M.N. Glauser, DMD and POD of time-resolved schlieren on a multi-stream single expansion ramp nozzle, Int. J. Heat Fluid Flow 66 (2017) 60–69. [39] S. Lee, Y.A. Hassan, Experimental study of flow structures near the merging point of two parallel plane jets using PIV and POD, Int. J. Heat Mass Transf. 116 (2018) 871–888.

16