Proper Orthogonal Decomposition Applied to Liquid Jet Dynamics

Transcription

1 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May Proper Orthogonal Decomposition Applied to Liquid Jet Dynamics M. Arienti *, M. Corn, G. S. Hagen, R. K. Madabhushi and M. C. Soteriou United Technologies Research Center East Hartford, CT 618 USA Abstract Sequences of consecutive line-of-sight images of a liquid jet in crossflow taken in the near field at a sampling rate of 14,35 Hz are analyzed using Proper Orthogonal Decomposition (POD). While the number of POD modes necessary to achieve a good reconstruction of the spray rapidly increases with the jet unsteadiness, low-frequency column fluctuations and higher-frequency surface waves are extracted from the first few modes by standard power spectral analysis of the associated time series. Spectral analysis of individual or combined orthogonal modes provides an insight of the dominant wavelengths that emerge at the jet interface. After a simple demonstration test, this procedure is applied to four test conditions with momentum flux ratio between 7.1 and 13 and gas Weber number between 9.8 and 49. Depending on injection conditions, the liquid jet dynamics appears to be composed by low-frequency column bending motions, high-frequency sharp peaks related to traveling surface waves, and moderate-frequency broad bands related to the amplification of column waves near the point of column break-up. * corresponding author: arientm@utrc.utc.com

2 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May Introduction Because of the short time and length scales involved in primary atomization, the experimental studies of liquid jet break-up have been until recently mostly limited to the analysis of long-exposure photographs or of uncorrelated snapshots of the overall spray. While that information can characterize the spray in a time-averaged sense, it is insufficient to understand the essentially dynamic process of atomization at the liquid column. To advance the field of study, high-frequency imaging in the near field as well as appropriate analysis techniques are necessary. Reviews of the physics of liquid jet in crossflow (LJIC) atomization can be found in Refs. [1] and [2]. Briefly, drops can either be shed directly from the jet surface or from ligaments that form when the jet begins to break. As the liquid to gas momentum-flux ratio increases, the jet penetrates farther into the crossflow until the primary atomization mode shifts from column break-up-dominated to surface stripping-dominated; see Ref. [3] for a recent discussion on the break-up regime classification. Spray penetration h b typically scales with the momentum flux ratio of the liquid to gas phase. The time a fluid element would take at velocity v jet to reach the top of the liquid column depends on h v b l b (1) t jet d u ρ ρ and is typically of the order of.1 ms for gas and liquid injection values of interest for industrial applications. The time scales of primary atomization are comparable or smaller than t b. Similarly, the waves that develop and grow at the liquid interface range from a few orifice diameters d to fractions of d, according to the measurements of column and surface waves reported in [4]. Few examples of dynamic analysis exist that are based on information acquired from high sampling rate imaging of a liquid jet. In their experiment of the break-up of a planar liquid sheet, Lozano et al. [5] used time-averaging of an image sequence to determine the oscillation amplitude of the sheet. However, the dominant frequency of the temporal instability was lost during the time-averaging process. Studies by Refs. [5-7] characterized the frequency of the oscillating interface of a liquid sheet by using a photodiode and coherent light source arrangement to probe the liquid sheet at various locations. The dominant frequency of the waves was found by spectral analysis of the time variation of the photodiode signal measurement. Images were g processed in a semi-automated fashion by selecting one or several lines in the ligament formation zone and analyzing the light intensity signal. Ligament boundaries were detected with a technique that applied a threshold to the gradient evolution of the luminance signal, but human validation was occasionally required. In another liquid sheet breakup study [8], image processing was used to quantify the spatial growth rate of the interface instability prior to the break-up with maximum spatial resolution of 27 µm/pixel. More recently, the dynamics of a thin annular water sheet injected between two annular coflowing air streams was investigated by using near-field snapshots acquired at up to 3 frames per second with maximum spatial resolution of 66 µm/pixel and 15 to 3 µs exposure time [9]. The dominant frequency of interface oscillations prior to break-up was obtained by fast Fourier transform of the time ensemble reconstructed at given axial locations. In this paper, we examine images acquired with a spatial resolution of 5 µm/pixel and a temporal resolution of 14,35 frames per second. In particular, we focus on extracting the frequency and wavelength content of the interface features that can be obtained from these line-of-sight images. Of particular interest is the windward side of the jet, which is unencumbered by the surrounding spray. Unsteady vortices on the windward side of the jet that potentially contribute to droplet stripping have been observed from velocity fields acquired with 2D Particle Image Velocimetry [1]. Windward waves were measured by Faeth s group [4] from pulsed shadowgraph images of non-turbulent liquid jet injected in crossflow. Windward waves along the column were classified as either column or surface waves. Column waves were associated with the nodes of bag break-up, while surface waves were shorter than the orifice diameter. For crossflow Weber numbers (We g ) below 2, the measurements showed a correlation between wavelengths and We g [4]. High-speed video cameras acquire gigabytes of video that are captured at a high sampling rate. The higher the sampling rate, the more images are available to the researcher, begging the question whether automated procedures are available to capture the dynamics of the interface. Proper Orthogonal Decomposition (POD) can successfully capture bulk flow structures in a fluid, but, to the authors knowledge, has not yet been applied to analyze gas-liquid interfaces. POD was first introduced to investigate the coherency of turbulent flow structures [11]. Since that seminal contribution, POD has been a popular tool to extract systematically hidden, but deterministic, structures in turbulent

3 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May flows. The idea is to start with an ensemble of data, or snapshots, collected from an experiment or a numerical simulation of a physical system. POD then produces a set of basis functions which spans the snapshot collection. The set is optimal in the sense that it captures the maximum amount of energy (pixel illumination variation, in our case) among all possible truncations of the same order After a brief description of the experimental facility, we proceed by illustrating the main points of the POD procedure. The technique is demonstrated on a video of a laminar jet subjected to a very low crossflow of air and then applied to four test conditions with jet to crossflow momentum flux ratios between 7.1 and 13 and gas Weber numbers between 9.8 and 49. Experimental facility and matrix of test conditions The experiments were conducted by Energy Research Consultants in an ambient test facility. The test fluid, Mil-C-PRF-Type II, was injected into a 7 mm x 7 mm wide test section from a.48 mm internal-diameter orifice with length-to-diameter ratio of 4. The orifice was preceded by a 118 deg. contraction from a 7.5 mm orifice. Additional details of the test facility can be found in Refs. [3] and [12]. Near-field images of the jet were acquired with a Camera Phantom 7.3 installed with a 15 mm lens set at an f-stop of 2.8. Four 5 W halogen lamps illuminated the jet from the front and the rear of the test section. Based on the camera lens, light, and flow conditions, a 6 µs exposure was selected as the best compromise between using a sufficient proportion of the 8-bit dynamic range of the camera and freezing the spray motion. An image size of 512x384 pixels and a sampling rate of 14,35 fps were used throughout the tests. All cases were acquired with a 5 µm/pixel resolution (equivalent to a field of view of 25.6 mm x 19.2 mm) except in test Case, which was acquired with a 25 µm/pixel resolution (equivalent to a field of view of 12.8 mm x 9.6 mm). Test conditions are listed in Table 1; see Figure 1 for a gallery of snapshots of jets at these conditions. Table 1. Test conditions Case v inj [m/s] u s [m/s] q We g Proper Orthogonal Decomposition Analysis We briefly outline the construction of Proper Orthogonal Modes (POMs) by the method of snapshots. For further details, see Ref. [13]. Consider a collection of data with elements {x i}, i = 1, 2,, N, belonging to a linear metric space Ω with inner product ( s) y( s) x, y = x ds, x, y Ω. (2) In the context of image analysis, the elements are a sequence of N consecutive camera snapshots of n by m pixels, and, is the L 2 inner product j k ( j k) y( j k) x,,. The -th POM is the mean N 1 φ = x i. (3) N i The remaining modes are calculated from the N x N correlation matrix K with entries K = x, x, (4) i, j i j where x i = x i φ expresses the snapshot deviation with respect to the N-averaged image φ. The solution of the eigenvector problem Kv = e v (5) r gives N eigenvalues, e1 e2 K en. From each eigenvalue/eigenvector pair, the mode φ r is constructed by linear combination of the N images, N r = i= 1 r r φ v x r 1K N. (6) = r, i i The projection of { x i } onto φr generates the series a = x, φ i 1K N. (7) r, i i r = If the snapshots are consecutives, a r,i can be interpreted as forming a time series for mode φ r, that is, {a r (i t s )}, with t s the sampling time interval, t s = 2π i/f s. Data reduction can be achieved by taking only the first M modes, M < N, M, i = + r. iφr r= 1 z M φ a. (8) The relation between the POMs and a reconstructed image z M,,i at time i is illustrated in Figure 2 for M = 12. Note that the modes are independent from spatial variations in the background intensity level (as long

4 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May as they remain constant in time) because the mean image φ has been subtracted from each snapshot. The truncation error of z M with respect to a single image deviation xi can be expressed in normalized form as ε x z φ, x z x, x i M, i i M, i = M, i. (9) i i φ Again, this measure is independent from the timeaveraged image φ. For the overall time series, the truncation error can be formulated as E M = N i= 1 ε N. (1) M, i As we will see in the following section, higher liquid jet and crossflow velocities produce increasingly unsteady interface dynamics, which in turn requires an increasingly large number of orthogonal modes to reach a prescribed truncation error. Figure 3 illustrates the relation between eigenvalues of the correlation matrix (with eigenvalues normalized to sum to 1) and the truncation error E M. A clear separation is only visible amongst the first few eigenvalues, after which the distribution appears almost continuous in the log-log plot. E M decreases by an order of magnitude also for cases 1 to 4, but with much greater number of modes. Overall, the residual with all the N modes included is of the order of 1-4 due to numerical truncation. The individual POM can be characterized in terms of dominant time and length scale. The time series a r can be immediately analyzed by plotting its Power Spectral Density (PSD). A dominant frequency f r then corresponds to a repetition period of the particular pattern represented by mode φ r. This point is further discussed in the results section. Patterns of negative and positive variations of pixel luminosity of a POM lead to the question of whether a characteristic length can be identified in an automated way. The following simple approach is proposed, where φ r matrix is scanned by rows and by columns. The arrays from the maximum and the minimum matrix entries along every column, b and along every row, b ( j) Maxφ ( j, k) + minφ ( j k ) =, (11), rk r k r k ( k) Maxφ ( j, k ) + minφ ( j k) =, (12) rj r r, j j can then be treated similarly to the time series a r. Additions of row-wise (or column-wise) max and min values are found to be beneficial in making a periodic pattern stronger. If power spectral analysis reveals a strong peak, the corresponding wavelengths λ rk or λ rj can be taken as the dominant row- or column-wise length scale of φ r. At the end of this section, the most important question is what feature of jet dynamics, if at all, f r and λ r correspond to. This question is considered in the following section. Results and Discussion A summary of the test conditions for each case is listed in Table 1, and corresponding snapshots of the liquid jet are presented in Figure 1. Case corresponds to a laminar liquid injection into a very low crossflow velocity. Such conditions produce a jet in which surface waves are easily observed. Case 1 illustrates column instability caused by increasing the crossflow velocity while maintaining the jet velocity of Case. Column dynamics is fully sampled at these conditions. The effect of increasing the jet velocity at the same crossflow velocity can be observed by comparing Cases 1 and 2. The jet in Case 2 penetrates farther as a result of the increased jet momentum. Finally, Cases 2, 3 and 4 form a set of conditions in which the crossflow Weber number varies at a constant jet-to-crossflow momentum flux ratio. As We g increases, droplet formation from the column surface initiates closer to the injection point and becomes more vigorous. Demonstration: Case The conditions corresponding to Case can be used to demonstrate the POD methodology. In the images, the jet displays a glassy quality of the liquid interface and a slight column bend, consistent with the low jet and crossflow velocities. Clearly visible are waves traveling along the interface at approximately the injection velocity. Figure 4 shows the first twelve POMs from the processed snapshot sequence. Modes φ 3 and φ 4 are characterized by a sequence of positive and negative variations of luminosity with respect to the timeaveraged image. These blobs are spaced by a distance of the order of the orifice diameter. The cross-correlation of the two modes indicates that they are 9 deg. out of phase at the maximum crosscorrelation amplitude. Their temporal and spatial spectral densities display a very strong peak at f 3 = f 4 = 6565 Hz and at λ 3 = λ 4 = 552 µm, see Figure 5. This analysis supports the conclusion that modes 3 and 4 describe the same traveling wave with phase velocity f 3 λ 3 = f 4 λ 4 = 3.67 m/s. This is shown distinctly by animated sequences of only the two combined POMs, a 3 ( t) φ 3 + a 4 ( t) φ 4. The estimated phase velocity is close to the nominal jet velocity, 3.1 m/s.

5 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May Similarly to φ 3 and φ 4, the modes φ 14 and φ 15 (not displayed here) are characterized by positive and negative blobs spaced apart by nearly one-half d. The peaks are at f 14,15 = 96 Hz with λ 14,15 = 288 µm. It is reasonable to assume that the spectral analysis in this case is affected by aliasing, and that the real frequency is in fact f 14,15 = f s 96 Hz = 1313 Hz, giving f 14 λ 14 = f 15 λ 15 =3.78 m/s, similarly to the previous mode couple. Modes 1 and 2 do not exhibit a detectable characteristic length, and show a low-frequency peak located at less than 4 Hz. This frequency is too low to be associated with the column shedding frequency of 1741 Hz, which is based on the given crossflow velocity, orifice diameter, and a Strouhal number of.2. Animation sequences of a 1 ( t) φ 1 + a 2 ( t) φ 2 show a back-and-forth slow flapping motion of the column in the direction of the crossflow. An examination of the video from the experiment indeed reveals a lowamplitude, low-frequency bending of the column. Since injection conditions are nominally constant, this motion could be caused by either an undesired perturbation of the air inlet, or a natural response of the jet itself to the crossflow. In any case, it appears that the POD decomposition has captured both slow and fast dynamics of the interface, at least as revealed by the liquid surface reflection in the experiment. Effect of Crossflow Velocity: Case vs. Case 1 The analysis that was demonstrated in Case is now applied to the rest of the cases to study the behavior of the jet at different conditions. The first comparison is with Case 1, in which the crossflow velocity is substantially increased while the liquid jet velocity is maintained near 2-3m/s as in Case. The increased velocity is sufficient to cause the jet to break up within the field of view of the camera, and at this low q and We g conditions the bag break-up mode dominates. Figure 6 shows the first twelve POMs. Similar to the previous case, modes 1 and 2 display a spatial structure that suggests a bending mode, while modes 3 and 4 show a repeating wave-like pattern which appears to be related to the large-scale wavy motion of the column after it bends in the crossflow. In the snapshots, the wave becomes progressively longer away from the orifice. Amongst the first twelve modes, this structure is best captured by modes 3 and 4, and again by modes 11 and 12, see Figure 7b. According to these results, the dominant lengths identified by the column-wise scan vary from λ/d = 14 to 6.6. However, the column wavelengths reported in [4] are considerably smaller, approximately 2.7d at We g = 9.8, suggesting that the measurements by Faeth et al. may have been obtained much closer to the jet orifice. Spectral analysis of the POM coefficients (Figure 7a) confirm that modes 1 and 2 appear as a low-frequency bulk oscillation. POMs 3 and 4 show a definite 63 Hz peak, while modes 11 and 12 have peak at 121 Hz. These frequencies are considerably lower than the 6565 Hz frequency found in Case. Effect of Liquid Velocity: Case 1 vs. Case 2 As the liquid jet velocity is increased at constant crossflow velocity, the jet begins to exhibit smallerscale structures. These are reflected for the POMs above the sixth mode in Figure 8. The higher liquid velocity in Case 2 preserves the jet momentum in the crossflow and appears to maintain the coherence of the surface waves. In particular, the periodic structures seen in POMs 3 and 4 in Case 1 begin to occur at later modes in Case 2. The column bending mode still occurs at POM 1, but becomes increasingly intricate in POM 2, which shows a crossover point. POMs 3-5 are a hybrid that bridges the column bending mode and the column traveling wave. This latter mode manifests itself more clearly in POMs 6-8. The frequency of this wave structure is around 26-27Hz (Figure 9a) and the nondimensional wave number kd o varies from.83 to.95 (Figure 9b). Increasing the liquid velocity produces a column wave with a higher frequency and a shorter wavelength. Beyond mode 12, the spatial patterns become more complex with multiple wave number peaks. Effect of Weber Number: Cases 2-4 The effect of increasing We g while maintaining the same q can be seen in the plots for Cases 2-4 (Figures 8-13). In each of these cases, POMs 1-4 have a structure associated with a bulk column motion, while distinct periodic structures are observed in the higher-order modes. There is no clear correspondence between particular POM patterns at the different conditions. For example, the twelfth mode across Cases 2, 3, and 4 are different. This can be attributed to the different order of the eigenvalues from Eqn. (5), since the e r are all very close to each other (Figure 3a). While each specific POM cannot be compared directly, we can observe trends within each case. Taking Case 4, conducted at the highest We g condition, as an example, we can observe large-scale periodic structures in POMs 5-8 and small-scale periodic structures in POMs 9-12 (see Figure 12). The POD coefficients associated with the large-scale structures in POMs 5-8 have frequencies in the range Hz and normalized wave numbers kd in

6 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May the range.36 to.83 (see Figure 13). The POD coefficients associated with the small-scale structures in POMs 9-12 have a higher frequency range that peaks beyond the Nyquist frequency f s /2. Summary The considerable amount of information that is made available by recent advances in high-sampling rate imaging techniques requires a corresponding advancement in analysis. For the first time in jet primary atomization investigation, Proper Orthogonal Decomposition has been applied to line-of-sight snapshots of liquid jet in crossflow. The methodology relies on capturing a sufficiently defined interface, but does not attempt to identify the jet interface, nor is it affected by a (constant) non-uniform background. The output is a series of orthogonal modes (in the L 2 norm) whose structure can be analyzed in the space domain. Here are the conclusions of this paper. In the regime of moderate gas Weber numbers examined, the projection of the snapshot series on the first few modes shows a broad frequency band with a well-defined non-zero peak. This means that some of the POMs recur periodically in the snapshot sequence. These same modes display alternating negative and positive variations of illumination that could be interpreted either as surface or column waves, depending on their dominant wavelength. The remaining POMs correspond to low-frequency bending modes of the column. The periodic patterns are observed even past column break-up and suggest a coherent release of droplets in the crossflow. This observation supports the role of fluid dynamics instabilities in selecting a particular set of frequencies for break-up in this regime. A cascade of finer temporal and spatial structures are found with increasing mode number (decreasing variation intensity), until the aliasing limit is reached (7,35 Hz temporally and 1 µm spatially). Analysis indicates that the number of modes necessary to reconstruct each snapshot below a certain threshold error increases very rapidly with crossflow velocity. However, the number of distinct wavelengths and frequencies of interest may be limited to few principal modes with several subharmonics. This hypothesis needs to be confirmed by ruling out features that may be introduced by the image acquisition technique used here. In conclusion, POD appears as a promising technique for systematic analysis of data that can be used to shed light on the mechanism of column break-up. Acknowledgements The authors wish to thank Energy Research Consultants (ERC) for their expertise in acquiring the videos that were used in this analysis. Nomenclature d q v jet u gas We g σ ρ f s k K e r, v r φ r x i z Mi ε Mi Ε Μ jet orifice diameter ρ l v 2 jet/ρ g u 2 g, momentum flux ratio jet injection velocity at the orifice crossflow velocity ρ g u 2 g d /σ, gas Weber number surface tension density sampling rate 2π /λ, wave number correlation matrix eigenvalue / eigenvector pair proper orthogonal mode (POM) i-th snapshot from video camera reconstruction of the i-th snapshot using the first M modes truncation error for i-th snapshot truncation error for N snapshots References [1] Wu, P. K., Kirkendall, K. A., Fuller, R. P., and Nejad, A. S., Journal of Propulsion and Power 13:64-73 (1997). [2] Sallam, K. A., Ng., C.L., Sankarakrishnan, R., Aalburg, C., and Lee, K, 44th Aerospace Sciences Meeting & Exhibit, Reno, NV, AIAA Paper [3] Madabhushi, R. K., Leong, M. Y., Arienti, M., Brown, C. T., and McDonell, V. G., 19 th Annual Conference of ILASS-Americas 26, Toronto, Canada. [4] Mazallon, J., Dai, Z., and Faeth, G.M., Atomization and Sprays, 9: (1999). [5] Lozano, A., Barreras, F., Hauke, G., Dopazo, C., J Fluid Mech. 437: (21). [6] Mansour, A., and Chigier, N., Phys. Fluids A 2: (199). [7] Berthomieu, P., and Lavergne, G., J. Vis. 4(3): (21). [8] Park J., Huh K.Y., Li X., and Renkizbulut, M., Phys. Fluids 16(3): (24). [9] Wahono, S., Honnery, D., Soria, J., and Ghojel, J., Experiments in Fluids 44: (27). [1] Elshamy, O.M., Tambe, S., Cai, J., and Jeng, S- M., 44th Aerospace Sciences Meeting & Exhibit, Reno, NV, AIAA Paper , 26.

7 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May [11] Lumley, J.L., Atmos. Turbu. Radio Wave Propagat (1967). [12] Brown, C.T. and McDonell, V.G., 19 th Annual Conference of ILASS-Americas 26, Toronto, Canada. [13] Holmes, P., Lumley, J. L., and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press, Cambridge, (a) (b) (c) (d) (e) Figure 1. Single 512x384 pixel frame from the high-frequency videos. (a) Case ; (b) Case 1; (c) Case 2; (d) Case 3; (e) Case 4. Actual Frame t Reconstructed Frame t N-Frame Average POM 1 POM 2 POM 12 = + a 1 (t) + a 2 (t) a 12 (t) Figure 2. Example illustrating the reconstruction of a single image using the first twelve Proper Orthogonal Modes. normalized eigenvalues 1 case 4 case 3 case case 1 case reconstruction error (a) # reconstruction modes Figure 3. (a) POD eigenvalues; (b) Truncation error as a function of the number of POMs used in data reconstruction. (b)

8 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May POM Figure 4. Snapshot sample and POD for Case (q = 357, We g =.4). POM1 POM2 POM3 POM4 POM 1 POM 2 POM 3 POM Hz Hz POM5 5 POM6 5 POM7 5 POM8 5 1 POM POM POM POM POM9 5 POM1 5 POM11 5 POM POM POM POM POM Frequency (Hz) Wave Number Kd o = (2π/λ) d o 5 1 (a) (b) Figure 5. PSD analysis of the POD coefficients from Case (q = 357, We g =.4) based on (a) the fluctuating timeseries component of the images, and (b) the column-wise spectral content of the POM.

9 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May POM 1 POM 2 POM 3 POM 4 POM 5 POM 6 POM 7 POM 8 POM 9 POM 1 POM 11 POM 12 Figure 6. Snapshot of the jet and first twelve Proper Orthogonal Modes for Case 1 (q = 7.1, We g = 9.8), shown with 5 mm gridlines. 1 5 POM1 1 5 POM2 1 5 POM3 63Hz 1 5 POM4 63Hz.1.5 POM POM POM POM POM5 POM POM6 POM POM7 POM11 121Hz POM8 POM12 121Hz POM POM POM POM POM POM POM POM Frequency (Hz) (a) (b) 5 1 Wave Number kd o = (2π/λ) d o 5 1 Figure 7. PSD analysis of the POD coefficients from Case 1 (q = 7.1, We g = 9.8) based on (a) the fluctuating timeseries component of the images, and (b) the row-wise spatial content of the POM.

10 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May POM 1 POM 2 POM 3 POM 4 POM 5 POM 6 POM 7 POM 8 POM 9 POM 1 POM 11 POM 12 Figure 8. First twelve Proper Orthogonal Modes for Case 2 (q =1, We g =9.8), shown with 5 mm gridlines. 5 POM1 5 POM2 5 POM3 521Hz 5 POM4 438Hz.1 POM POM POM POM POM5 658Hz 5 5 POM6 2686Hz 5 5 POM7 2686Hz 5 5 POM8 2577Hz POM POM POM POM POM9 3755Hz 5 5 POM1 381Hz 5 5 POM POM POM POM POM POM Frequency (Hz) Wave Number Kd o = (2π/λ) d o 5 1 (a) (b) Figure 9. PSD analysis of the POD coefficients from Case 2 (q=1, We g =9.8) based on (a) the fluctuating timeseries component of the images, and (b) the row-wise spatial content of the POM.

11 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May POM 1 POM 2 POM 3 POM 4 POM 5 POM 6 POM 7 POM 8 POM 9 POM 1 POM 11 POM 12 Figure 1. First twelve Proper Orthogonal Modes for Case 3 (q =99, We g =3), shown with 5 mm gridlines. 5 POM1 5 POM2 55Hz 5 POM3 74Hz 5 POM4 93Hz.1.5 POM POM POM POM POM5 37Hz 5 5 POM6 37Hz 5 5 POM7 5 5 POM POM POM POM POM POM9 5 5 POM1 5 5 POM POM POM POM POM POM 12 14Hz 21Hz.5.36, , , Frequency (Hz) Wave Number Kd o = (2π/λ) d o 5 (a) (b) Figure 11. PSD analysis of the POD coefficients from Case 3 (q=99, We g =3) based on (a) the fluctuating timeseries component of the images, and (b) the row-wise spatial content of the POM.

12 ILASS Americas, 21 st Annual Conference on Liquid Atomization and Spray Systems, Orlando, Florida May POM 1 POM 2 POM 3 POM 4 POM 5 POM 6 POM 7 POM 8 POM 9 POM 1 POM 11 POM 12 Figure 12. First twelve Proper Orthogonal Modes for Case 4 (q = 13, We g = 49), shown with 5 mm gridlines. 5 POM1 5 POM2 52Hz 5 POM3 118Hz 5 POM4.1.5 POM POM POM POM POM5 19Hz 5 5 POM6 32Hz 5 5 POM7 37Hz 5 5 POM8 52Hz POM POM POM POM POM9 5 5 POM1 5 5 POM POM POM POM POM POM Frequency (Hz) Wave Number Kd o = (2π/λ) d o 5 (a) (b) Figure 13. PSD analysis of the POD coefficients from Case 4 (q = 13, We g = 49) based on: (a) the fluctuating timeseries component of the images, and (b) the row-wise spatial content of the POM.