## Abstract

Liquid fuel jet in crossflow (LJIC) is a vital atomization technique significant to the aviation industry. The hydrodynamic instability mechanisms that drive a primary breakup of a transverse jet are investigated using modal and traveling wavelength analysis. This study highlights the primary breakup mechanisms for aviation fuel Jet-A, utilizing a method that could be applied to any liquid fuel. Mathematical decomposition techniques known as POD (proper orthogonal decomposition) and Robust MrDMD (multiresolution dynamic mode decomposition) are used together to identify dominant instability flow dynamics associated with the primary breakup mechanism. Implementation of the Robust MrDMD method deconstructs the nonlinear dynamical systems into multiresolution time-scaled components to capture the intermittent coherent structures. The Robust MrDMD, in conjunction with the POD method, is applied to data points taken across the entire spray breakup regimes: enhanced capillary breakup, bag breakup, multimode breakup, and shear breakup. The dominant frequencies of breakup mechanisms are extracted and identified. These coherent structures are classified with an associated time scale and Strouhal number. Three primary breakup mechanisms, namely, ligament shedding, bag breakup, and shear breakup, were identified and associated with the four breakup regimes outlined above. Further investigation portrays these breakup mechanisms to occur in conjunction with each other in each breakup regime, excluding the low Weber number enhanced capillary breakup regime. Spectral analysis of the Robust MrDMD modes’ entire temporal window reveals that while multiple breakup mechanisms are convolved, there is a dominant breakup route for each breakup regime. An associated particular traveling wavelength analysis further investigates each breakup mechanism. Lastly, this study explores the effects of an increased momentum flux ratio on each breakup mechanism associated with a breakup regime.

## 1 Introduction and Objectives

The injection of a liquid fuel jet into an incoming crossflow is a widely prevalent fuel injection technique used in various systems, including gas turbine combustors, airbreathing propulsion engines, liquid rockets engines, ramjet, scramjet, and other high-speed vehicles. Liquid jet in crossflow is a highly effective method that produces high-quality atomization and high evaporation rates [14]. The liquid jet dynamics in the crossflow process are exceptionally turbulent and are usually associated with multiple convolved instabilities such as the Kelvin-Helmholtz, Rayleigh-Taylor, and the Plateau-Rayleigh instabilities [5]. These instabilities lead to the breakup process, which involves microscopic-scale motions and intricate interfacial physics, which become hard to characterize with traditional experimental techniques [6].

The breakup mechanisms of a liquid jet in crossflow are generally divided into two categories, i.e., the Primary breakup mechanism and the Secondary breakup mechanism. The Primary breakup mechanism is the process where the disintegration of the liquid column jet occurs via traveling wavelengths, which acts as a manifestation of the hydrodynamic instabilities. These instabilities can act separately or simultaneously to produce these waves within a liquid jet. There are two types of traveling waves associated with a transverse jet: a column and a surface wave. It was found by Wu et al. [7] that there exist four different breakup mechanisms within the primary breakup mechanism. These breakup mechanisms were found to be dictated by a range of the Weber number, these include the enhanced capillary (Wec < 4), bag (4 < Wec < 60), multimode (60 < Wec < 110), and shear breakup mechanism (Wec > 110) [713]. Figure 1 presents raw snapshots captured in this study of the primary breakup regimes. The secondary breakup mechanism is the further fragmentation of breakup structures caused by the primary breakup mechanisms, leading to droplets. A review of various research conducted on LJIC and its phenomenology can be found in reviews conducted by Broumand and Birouk [14], Karagozian [15], and Mahesh [16].

Fig. 1
Fig. 1
Close modal

The inherent dynamics and coherent features associated with a transverse jet's primary breakup mechanism are yet to be clearly understood due to its multiscale turbulence behavior. However, with the advent of data-driven science, it became possible to identify, extract, and analyze deterministic coherent dynamic structures more robustly hidden within turbulent flows. Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have been used in past studies to characterize LJIC and the coherent dynamic structures associated with its primary breakup mechanisms. Meyer et al. [17] were the first to apply the method of POD to a transverse flow. They considered transverse flows with a jet-to-crossflow velocity ratio R ranging from 1.3 to 3.3. R is a nondimensional parameter ratio of the velocity divided by the crossflow velocity which can be used to characterize flow structures within a LJIC system due to the fact that flow structures are strongly dependent to it [18,19]. From the utilization of POD large flow structures associated with the dominant POD modes were detected, these large flow structures correlated with waking vortices for R = 3.3 and shear vortices for R = 1.1. Arienti and Soteriou [6] demonstrated the feasibility of using POD on a time-resolved snapshot of a liquid in crossflow. Arienti and Soteriou extracted normal modes, which captured wavelike characteristics along with the jet column profile. These traveling wave modes were interpreted as manifestations of the Kelvin-Helmholtz instability and observed in a bag and multimode breakup. However, it was noticed that this method becomes redundant at a high Weber number and does not provide any comprehensible, coherent structures associated with breakups. Rowley et al. [20] first demonstrated the utility of using DMD to analyze a numerically simulated jet in crossflow and decompose modes associated with the shear layer. These modes were shown capable of capturing the significant frequencies more precisely than global eigenmodes of the linearized system. Schmid [21] later demonstrated that DMD could be used for photographic images by performing Schlieren imaging analysis on a helium jet. Iyer and Mahesh [22] used DMD to study the shear layer instability of transverse jets for R = 2 and R = 4. They discovered the dominant structures of both those cases to be the shear layer.

These past studies demonstrate the influence of nondimensional parameters on the breakup mechanisms and signify the breakup regimes’ critical parameters. POD and DMD have been used to study these regimes; however, only global modes associated with the entire time window were decomposed. It is to be understood that the utilization of POD for a highly nonlinear system such as a transverse jet, as studied in this experiment, has several shortcomings. POD is a linear statistical method that operates under the condition that each mode determined can be superimposed together to reconstruct the original signal; this means that complex nonlinear cases can cause the temporal coefficients of a POD mode to be convolved with multiple frequencies, which can describe several relevant contributing mechanisms of the flow [23]. The DMD algorithm, as demonstrated in these previous studies, extracts temporal orthogonal modes, which, unlike POD, conveys a discrete frequency and, as such, a single mechanism [23]. The challenge with DMD modes is pinpointing the most relevant modes; unlike POD DMD, modes are not ranked by their energy content but rather by their persistence of dynamics [2325]. This ranking makes it challenging to isolate modes that are related to pertinent physical mechanisms. The current optimal selection of DMD modes relies on sorting the modes with the highest amplitudes; however, as Jovanović et al. [26] discovered, keeping only a subset of DMD modes with the largest amplitude is prone to a reduced decomposition of relevant, coherent structures. In this paper, the method to obtain relevant DMD modes is based on a procedure established by Roy et al. [27], which identifies and extracts reproducible modes that persist in dynamics in multiple realizations of an experiment [22,23]. This method is implemented by comparing reproducible and nonreproducible characteristics of the flow by comparing the DMD spectra of different subgroupings of the snapshots pertaining to the experiment [28]. These modes, commonly referred to as Robust modes, are excellent in pinpointing global mechanisms that persist through the entire time window. However, Robust modes are unable to capture intermittent transient local dynamics within a flow. These dynamics are highly prevalent within turbulent flows and have a significant influence on the breakup mechanisms of a jet in crossflow.

The primary purpose of this study is to characterize coherent structures and their associated dynamics with each breakup regime and connect them to critical nondimensional parameters. The current approach implements proper orthogonal decomposition and robust methodology of dynamic mode decomposition, in conjunction with a novel technique called the multiresolution dynamic mode decomposition (MrDMD) to identify the intermittent coherent structures of the captured liquid surface. MrDMD was developed by Kutz et al. [29]. This method is based on applying the concept of multiresolution analysis in time with DMD [30]. It is an extension of DMD, which parses nonlinear dynamical systems into multiresolution time-scaled components [31]. The utility of this method is its ability to capture intermittent mechanisms within a system. We believe that intermittent mechanisms play a significant role in the dynamics of a transverse jet and influence its behavior within specific breakup regimes.

In this study, the application of the MrDMD algorithm is conducted to a series of time-resolved column region snapshots of various breakup regimes. Using MrDMD and the Robust mode technique constructed by Roy et al. [27], significant local modes are identified by the extraction of recurring frequencies. These local modes are classified for each specific breakup regime. The role and influence of these local coherent structures associated with the extracted local modes are defined for each break regime. In addition, the influence of the Weber number and momentum flux ratio onto each of these mechanisms is characterized by analyzing how they change and evolve across the breakup regime map. These coherent structures are classified with an approximate time scale and an associated Strouhal number. The dynamics and spatial constructs of these coherent structures are also qualitatively analyzed. The extraction of these relevant dominant modes which capture these sparse dynamics is imperative to the study of transverse jets due to the fact that fluid dynamics evolve on a low-dimensional attractor structured by coherent spatiotemporal [22,25,29,3234]. Each of these modes is further characterized by linking them to the associated traveling column and surface wavelength.

The rest of the paper is outlined as follows: Sec. 2 contains a discussion of the experimental conditions and setup. Section 3 provides a brief overview of the theory of MrDMD and covers the methodology behind extracting the relevant modes of MrDMD. This section also considers the advantage of Robust MrDMD extraction methodology in contrast to the traditional MrDMD for highly turbulent flow captured through time-resolved snapshots. Section 4 utilizes the technique developed to study the transverse jet at different breakup conditions. The paper is concluded in Sec. 5 with an overview of the significant findings of the research.

## 2 Experiment and Diagnostics

### 2.1 Experimental Facility.

Experimental testing is conducted at the Propulsion and Energy Research Laboratory (perl) at the University of Central Florida. The blow-down Ramjet facility and design specifications can be found in Geikie and Ahmed publication [35]. Air from the ramjet enters the fifth-order polynomial nozzle to create a uniform top hat flow field with low turbulence distribution effects, as seen in Fig. 2. A top hat flow field at low turbulence level was characterized to ensure the experimental measurements are unbiased. This uniform flow is established prior to entering the test section, where the interaction of LJIC begins. The optical viewing walls and LJIC top plate section were combined and attached to the transitional nozzle of the Ramjet facility, as shown in Fig. 3. To optimize visualization of the liquid jet in crossflow interaction, quartz glass viewing windows were installed on the bottom and at the sides. The design maximizes LJIC viewing and space for illumination of the liquid fuel. An axial length of 241 mm and a height of 121 mm were utilized for the side viewing windows. The bottom window was identical in length but was designed with a width of 91 mm. The test section has a rectangular cross section of 127 × 45 mm2; a similar configuration is found in Geikie and Ahmed [35]. A new top plate was designed with a threaded hole for the injector to be screwed in and flush-mounted. The injector was flush-mounted to the top plate's interior face, which has a polished finish to eliminate boundary layer effects and a thickness of 12.7 mm. An injector was inserted into the threaded top plate hole with an orifice diameter of Dj = 1.07 mm and an eccentricity of zero. To ensure uniform fuel flow prior to entering the test section, the design has an orifice L/D = 47.47. The orifice hole is located 63.5 mm on the center of the top plate and 57.2 mm axially downstream of the test section entrance.

Fig. 2
Fig. 2
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Fig. 3
Fig. 3
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### 2.2 Flow Controls.

The facility utilizes control valves and actuators to hold experimental flow points, ensuring precise flow measurements. In Fig. 4, a flow control schematic shows the components used to modulate and hold the flow conditions. A valve regulates the main air flow actuated DM4500 series from JFlow, which is pneumatically modulated and measured with a Preso venturi flowmeter CV-300-65. A feedback signal from the Dwyer Series 626 pressure transducer is sent to the operator's computer; the operator uses LabView software to monitor instrumentation, trigger to the camera, and acquire data through an NI-6413 Data Acquisition device. In order to inject the fuel into the crossflow, a compressed air system was used to pressurize the fuel storage tank. A Dwyer pressure regulator was placed inline before the fuel storage tank to control the fuel flowrate.

Fig. 4
Fig. 4
Close modal

### 2.3 Fuel and Flow Conditions.

A commonly found fuel known as Jet-A (POSF 10325) was used to conduct the experimental studies. The fuel was injected at ambient conditions of 293.15 K, resulting in a density of ρ = 795.7 kg/m3, a viscosity of μ = 4.77 mm2/s, and surface tension of σ = 0.023 N/m for composition properties. The Ramjet facility supplied a uniform crossflow profile and Mach numbers ranging from M = 0.012 to 0.093 to introduce hydrodynamic instabilities. Flow conditions for the crossflow and liquid fuel jet can be found in Table 1. Fuel flows range from Vj = 0.91 to 32.23 m/s; in each spray regime, two data points are measured. The study consisted of a total of nine cases within the four spray regimes specified in Fig. 5.

Fig. 5
Fig. 5
Close modal
Table 1

Experimental flow conditions

Experimental caseMomentum flux ratio (q)Weber number (Wec)Vc (m/s)Vj (m/s)
23037.411.58
3301516.573.53
4306534.507.35
53020060.5012.90
64037.411.82
7601516.575.00
8806534.5012.00
910020060.5023.54
1012530074.1032.23
Experimental caseMomentum flux ratio (q)Weber number (Wec)Vc (m/s)Vj (m/s)
23037.411.58
3301516.573.53
4306534.507.35
53020060.5012.90
64037.411.82
7601516.575.00
8806534.5012.00
910020060.5023.54
1012530074.1032.23

The droplet or spray rain impact across the range of conditions investigated due to the fuel injection from the top would be negligible, as the dominant properties for the injection force are the injection pressure and density of the fuel. In addition, LJIC breakup is a primary function of momentum flux ratio and Weber number. The Weber number is a function of drag force with respect to the cohesion force, which is composed of density, velocity, and surface tension of the fluid.

Analyzing the atomization of the fuel tested is based on the Weber number with respect to the momentum flux ratio. The first five cases of the study remain at a constant momentum flux while the Weber number is being varied for the subsequent cases. Referring to Fig. 5, case 2 demonstrates the enhanced capillary region. In this region, the curving and breakup of the fuel are due solely to hydrodynamic forces experienced on the shear layer between the fuel and crossflow air. Case 3 belongs to the bag breakup region with column breakup.

With a low momentum flux and a small Weber number, case 4 falls in the multimode breakup region with column breakup. Case 5, distributed after the transition line, falls in the shear side of the Ohnesorge line within the surface breakup regime. The Ohnesorge number is a function that represents the ratio of viscous force to the surface tension and inertial forces. The Ohnesorge number line seen in Fig. 5 lies at the value of 0.0104 and it dictates the column and surface breakup regimes. Cases 6 through 10 maintain the prior cases’ Weber number and incorporate the momentum flux further to analyze its effect on the atomization of the jet. Cases 6 and 7, with slightly larger q than their corresponding case, remain in the same region. Cases 8, 9, and 10, with the increase of q, crosses over the transition line and have a surface breakup.

### 2.4 Time-Resolved Imaging.

The atomization of the test fuel at different conditions was captured via high-speed broadband imaging using a Photron SA-Z monochromatic high-speed camera. The time-resolved images were captured at 40,000 frames per second at an image resolution of 768 × 432 pixels. The single-pixel resolution for the images captured is 21.94 pixels/mm. A Nikon 50 mm FL lens was used to capture the imaging and focus on the spatial domain. High-intensity lighting was necessary to illuminate the jet spray in the facility fully. Four 500 W and 250 W halogen lamps were focused and positioned to radiate through optical viewing windows. Two halogen lights were located a positive and negative 45 deg from the center of the side viewing window. Another was added directly across from the negative 45 deg halogen light. The high-speed camera was positioned normal to the side viewing window.

Four regions of interest were investigated pertaining to each case. The regions were segmented based on the column breakup location as determined by Wu et al. [7] for the enhanced capillary breakup case. The locations determined from this case served as the baseline for all the other cases. The region of interests is segmented as the entire spatial window, upstream, downstream, and shear layer as shown in Fig. 6, xb is the axial distance to column fracture point and yb is the transverse height of the column fracture point. The dominant modes pertaining to each case were determined using the Strouhal number. The frequency used is the shedding frequency associated with the relevant mode, and Dj and Vj are the diameter and velocity of the jet, respectively. The shedding frequency is the dominant frequency calculated via the Robust MrDMD technique; this technique is highlighted in Sec. 4. A definition of the Strouhal number is provided in Eq. (1)
$St=fDjVj$
(1)
Fig. 6
Fig. 6
Close modal

## 3 Modal Analysis

### 3.1 Multiresolution Dynamic Mode Decomposition.

Multiresolution dynamic mode decomposition or MrDMD, developed by Kutz et al. [31], is a recursive application of DMD. This method aims to segment the convoluted dynamics within a system into its separate time scales. This technique relies on recursively computing the DMD to separate the slow modes from the fast modes. The modes are sorted into levels within a given range of frequency. The MrDMD algorithm uses an iterative, recursive process to extract the low-frequency modes. At each iteration, it divides the time domain in half and performs the DMD within that halved time domain [29,34,36]. It then proceeds to extract the slow modes of that resolution level. The process is repeated until the threshold limit of the resolved sampling frequency is approached. The three different parameters to construct the complete solution of MrDMD are the number of levels (L), which segments the frequency, rank, or the time bin (J) for each level, and the modes preserved at each level concerning the time window [24,29,31,32].

l = 1,2,…, L decomposition of frequency levels

j = 1,2,…, J time bins per level (J = 2(l-1))

m = 1,2,…, M number of modes extracted at specific (L, J)

The MrDMD solutions expansion:
$x(t)=∑l=1L∑j=1J∑m=1Mfl,j(t)bm(l,j)ϕ(l,J)e(ωk(l,J)t)$

The fl,j(t) is the indicator function, the $bm(l,j)$ amplitude, and ϕ(l,J) is the mode. The MrDMD solutions expansion includes all the information on the particular level, time bin, and the number of modes extracted [29,30,34,36,37].

### 3.2 Modal Extraction.

The extraction of relevant modes within the typical MrDMD framework performs in conjunction with the Modal frequency-amplitude map, which plots the modes with respect to amplitude and frequency through time. The basis of this extraction method relies purely on an amplitude-magnitude basis and does not consider active modes that are repeated across the time-frequency bins. Investigation of the modes with the largest amplitudes is a procedure maladapted for turbulent flow, captured via time-resolved snapshots. This study investigates repeated modes pertaining to each breakup regime as they make up the coherent structures fundamental to the flow. We follow the conjecture that robust modes represent features of a system that are reproducible and thus significant. Modes that are not reproducible with multiple realizations of an experiment are noise or nonreproducible modes outside the boundary of relevancy. These modes considered nonreproducible or noise are modes that depict a spatial structure not seen consistently within the entire temporal window. While these modes are physical structures connected to the physics of the flow, they are not coupled to the repeated active modes or robust modes which dictate the dominant breakup mechanisms associated with each breakup regime, and thus within the context of extracting a dominant breakup mechanism; we consider these nonreproducible modes as not relevant. In this experiment, we utilize the method of robustness developed by Roy et al. [27] in the extraction of repeated active-based modes. In this experiment, 17,245 frames of pixel intensity information were captured at a rate of 40 kHz. The entire time domain of data is 0.431 s. Post-processed spray images were flipped along the y-axis to show positive jet penetration. The computation of the singular value decomposition (SVD) of the full reconstructed data matrix is computationally inefficient and thus not feasible. In order to circumvent the cost of expensive computation, the data were sparsed by skipping four snapshots in between; every fifth image was considered.

The subgroupings of the data set are configured into two different libraries; the first is the alternative configuration which changes the starting point of each set; the first group starts at the first snapshot, the second group starts at the second snapshot on, and so forth. The methodology is shown in Fig. 7. This method effectively captures the whole dynamics of the system. The second method is the segmentation of the data set into equal parts. Segmentation of data allows a higher resolution inspection of the data. Following the subgroupings, we restructure each subgrouping into a data matrix, where each snapshot is reshaped into a single column vector.

Fig. 7
Fig. 7
Close modal

The POD computation on a system provides an orthogonal set of basis vectors with minimal dimensions. This feature is useful when determining the optimal hard thresholding value τ as a rank for computation of Robust MrDMD. The significant concentration of energy throughout the POD modes provides an effective measure to determine τ. It was determined that the thresholding at the 80th percentile of cumulative energy of the POD modes was a good marker for conducting SVD. Higher thresholding percentage provided redundant information and simply increased the computational time. The 80th percentile was determined using an iterative method and is illustrated in Fig. 8. The MrDMD is applied to the eight subgroups for each case after applying the appropriate thresholding value attained by the implementation of POD. Frequency is resolved based on the acquisitional frequency adjusted to each subgroup and the implementation of the Nyquist-Shannon sampling theorem. Systematically using the sampling framework as shown in Fig. 9, we remove the spatiotemporal features of our jet in crossflow with a recursive refinement of the sampling data. Within this recursive framework, there may usually exist more than one mode indexed by m. This behavior is typical in complex nonlinear systems. As the system analyzed in this study is highly chaotic, each segmented mode within a specific level and time bin has more than one mode. The only exception being mode $ϕ1(1,1)$.This mode is a zero-mode component, and it has a period of T = ∞, infinity being the entire time domain captured for a particular experiment. The current implementation of MrDMD to extract relevant modes lies purely in the method of inspection of modes with the largest amplitude.

Fig. 8
Fig. 8
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Fig. 9
Fig. 9
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The theory is that selection of modes with the largest amplitude can be interpreted as the selection of Koopman modes, which have the strongest influence on the system [3840]. However, in Sec. 3.3, we show that keeping only a subset of modes with the largest amplitude in complex turbulent flows (captured through a temporal sequence of images) leads to a poor approximation of the dynamics and the coherent structures associated. In this study, we follow the conjecture that repeated active modes that are consistent within all the subgroups are, in fact, the Koopman modes. That is, these modes characterize the most dominant features of each breakup regime and influence its overall behavior. The modes are referred to as Robust MrDMD modes. Robust MrDMD modes are obtained through the application of MrDMD of each segmented case with an associated ranking threshold value attained from POD. The typical method of modal extraction based on amplitude magnitude is relinquished, and instead, the MrDMD cell matrix is deconstructed. All the modes from each case are segmented and sectioned based on individual frequency range, time bin, and modes specific to that particular frequency and time bin. Each segmented case contains ten libraries, which are based on the frequency decomposition levels. Each library is indexed by the mode numbers and the time bin. The process is portrayed in Fig. 9 with the flow.

The frequency contents of the modes for each level are inspected through time as seen in Fig. 10, to observe how local frequencies change throughout the duration of the flow. The spectral contents of the MrDMD modes are clustered to find the dominant frequency content of that level. Dominant frequency content can usually be found by inspecting the frequency behavior through time for less complicated transverse flows. However, as the Weber number increases and the dominant breakup mechanism transitions toward the shear region, the time-frequency inspection method is not sufficient to find the dominant frequency, and the clustering method must be utilized. The frequency clustering method utilized in this research is based on a similar method used by Dylewsky et al. [36]. The use of a sliding window DMD technique in conjunction with frequency clustering is utilized to study various multiscale systems [36].

Fig. 10
Fig. 10
Close modal

The clustering of frequencies for each level particular to each indexed mode is concatenated into a single vector to inspect the frequencies through time. The frequencies clustered are squared to inflate the separation of higher frequencies and compress the lower frequencies. The frequency clustering method is applied to each subgroup from which the dominant frequencies are extracted. The dominant frequencies are inspected across all eight subgroups, and the frequencies that fall within a 3% margin of each other are considered Robust MrDMD modes. The 3% margin is a consideration used by Roy et al. [27] to determine the modes which are robust. The inspection of frequency through time and its spectral clustering is shown in Figs. 10 and 11, respectively, for Case 3.

Fig. 11
Fig. 11
Close modal

Based on the spectral clustering and qualitatively ascertaining the spatial modes associated with the spectral clustering group, it was found that there exist, two strong bands of frequency associated with the bag breakup regime case, one is large ligament shedding, and the other is bag formation. The strong band of frequency found within a specific subgroup is compared across all the subgroups to identify whether it is robust, i.e., common among all the subgroups within a 3% margin. If that criterion holds true, the average is taken for the frequency cluster residing within that margin. The robust frequency that is found is considered to be the most relevant one. Each mode is associated with the frequency cluster, inspected for structural similarity, corresponds to specific spatial modes within a dynamically evolving jet breakup structure, and is referred to as a dominant mode in this literature. These modes are compared with the breakup structures described by Wu et al. [7] and Sallam et al. [9]. The histogram seen in Fig. 11 shows three dominant peaks. These dominant peaks are associated with robust modes. The highest peak, which lies at a Strouhal number of 0.145, is related to bag formation. The second highest peak at an approximate Strouhal number of 0.55 is linked to ligament shedding's breakup mechanism. The third peak associated with the Strouhal number 0.03 also portrays a dominant robust mode; however, upon inspection of the modes’ spatial structure, it is seen that these low-frequency modes are associated with the average jet column behavior, and hence do not capture breakup mechanisms.

### 3.3 Modal Validation.

To verify the proposed modal extraction and analysis method, Robust MrDMD is qualitatively compared with POD, DMD, and MrDMD, as portrayed in Fig. 12. The significant difference between MrDMD and Robust MrDMD is the extraction procedure of the relevant modes. Conventional MrDMD utilizes an amplitude-based method to extract modes, which is based on the fact that the modes considered to influence a system dominantly are the modes with the highest amplitudes. However, an amplitude basis modal extraction can lead to redundancy and poor approximation for a complicated turbulent flow system captured with time-resolved snapshots. The is because the data attained from time-resolved snapshots of a nonreacting flow only gives the intensity of light scattering in a matrix structure. The Robust MrDMD modal extraction technique relies on capturing the most frequent active spatial modes repeated throughout all the subgroups. The methodology is outlined in Sec. 2. POD resolves a dataset based on energy content and structures all the modes based on highest to lowest energy content. The most relevant modes within a POD framework are the modes with the highest energy content, which in a POD context are the first few modes or the first few vectors of the U matrix containing the spatial structure of each mode. DMD modal extraction theory is also based on energy content, and the highest energy content modes or modes with the highest amplitude are the most dominant modes. This extraction method based on selecting the higher amplitude modes can be noted as selecting Koopman modes, which significantly influences a system's dynamics. However, because the dynamics of transverse jet becomes increasingly turbulent as the Weber number and momentum flux ratio increase and the dataset of this experiment are values consisting of intensity magnitudes, the traditional extraction of modes within a DMD framework which outputs global modes becomes insufficient.

Fig. 12
Fig. 12
Close modal

As is portrayed in Fig. 12, Robust MrDMD captures the most dominant coherent structures associated with the ligament breakup, based on the modal comparison between the four methodologies. The primary breakup mechanism of enhanced capillary breakup happens via ligament shedding. This classical breakup mechanism has been thoroughly studied and captured via a fast acquisition of time-resolved snapshots due to its simplistic breakup mechanism, as seen in Fig. 12. Robust MrDMD captures the exact mechanism and dynamics of the ligament shedding behavior with higher accuracy applied to time-resolved snapshots relative to other modal decomposition techniques. Fine characteristics, such as the column waves associated with the deflection of a jet caused by the crossflow and its influence in causing the pinching behavior to shed ligaments from the jet, can be observed in the dominant spatial Robust MrDMD mode.

Comparison of the Robust MrDMD modes to the modes attained by POD, DMD, and MrDMD, as portrayed in Fig. 12, reveals that the first POD mode shows an average behavior, and while the second POD mode shows shedding behavior, other information such as the rollup of the jet and pinching behavior is nonexistent. The periodic behavior captured by the second mode portrays average shedding. The two highest energetic DMD modes show small-scale structures that are nonpertinent to the breakup associated with an enhanced capillary breakup. Conventional MrDMD modal extraction depicts the most dominant modes, which are modes that portray an average behavior with slight corrugations on the jet surface with associated ligament breakup downstream. While MrDMD modes capture some aspects of the capillary breakup mechanisms, robust MrDMD captures the exact mechanism associated with the simple enhanced capillary breakup mechanism. This comparison between the modal analysis techniques using the known enhanced capillary breakup regime serves as a baseline to demonstrate the feasibility of Robust MrDMD.

## 4 Results and Discussion

### 4.1 Breakup Mechanisms

#### 4.1.1 Enhanced Capillary Breakup Regime.

Application of the Robust MrDMD analysis reveals that there is a dominant frequency cluster or a dominant mode present for the enhanced capillary breakup case at a nondimensional St = 0.057. These nondimensional Strouhal number stays constant as the momentum flux ratio is increased from 30 to 40. This dominant mechanism, which corresponds to ligament breakup and shedding from the jet column, is found at a timescale of τ = 6.74 ms for both Case 2 and Case 6. Investigation of this frequency cluster shows the spatial mode associated with them to contain coherent structures portraying ligament shedding from the jet. Qualitatively, this behavior is similar to a falling fluid stream breakup under the influence of gravitational force. It is noticed from these spatial modes in Fig. 13 that the formation of these ligaments happens due to a pinch-off action, which leads to the thinning of the jet. The thinning and pinch-off action of a falling jet occurs due to an interplay between the jet radius and the curvature radius of the traveling column wave, induced by the Plateau-Rayleigh instability [41,42]. The jet radius shrinks at the trough of the jet, causing increased pressure due to surface tension; however, the traveling column wave's curvature's radius decreases the pressure at the trough [41,43]. Likewise, the inside radius of the jet reduces the pressure at the crest of the liquid column, whereas the column wave increases the local pressure. This behavior is observed when studying the spatial modes of both Case 2 and Case 6. The traveling column wavelength for both these cases was calculated by directly counting the pixels between the crests. It is found that as the momentum flux ratio increased from 30 to 40, the traveling column wavelength increased from 2.92 ± 0.2 mm to 3.98 ± 0.5 mm. Slight surface corrugations along the windward side of the jet were detected, but they were dominated by the amplitude of the column wave and shown to decay temporally.

Fig. 13
Fig. 13
Close modal

#### 4.1.2 Bag Breakup Regime.

Cases 3 and 7 pertain to the bag breakup case; this breakup regime is known for forming bag-like structures that rupture to induce atomization. The formation of these bags can be observed in Fig. 14 in the bag column for Cases 3 and 7. Within the bag breakup regime dynamics, three types of intermittent primary breakup mechanisms are responsible for generating the droplets downstream or secondary atomization. Ng et al. [44] noted that the three groups are ligament shedding, bag-like structures, and rings. The difference between bags and ring is that bags are still connected to the jet column, whereas the ring is a bag that has shed away from the liquid column. The rupture of both the bag and ring membrane leads to secondary atomization. From the utilization of robust MrDMD, two dominant modes were extracted from this breakup regime. The ring and bag structures were captured by one dominant mode or one spectral cluster, and another captured the ligament structures. The bag and ring structures convolved within one mode are generally referred to as a bag structure in this paper, as seen in Fig. 14 Case 3, Bag breakup mode. The more dominant route of primary breakup in this breakup regime is the bag breakup mechanism, as there exists more modes associated with the spatial bag structure than the ligament. The mode structures associated with bag breakup are found to occur at an St = 0.144 for both Case 3 and Case 7. Increasing the momentum flux ratio decreases the timescale associated with the bag breakup mechanism in the bag breakup regime case; it is found that structures associated with the bag breakup occur at a timescale of τ = 3.37 ms for Case 3, and τ = 1.68 ms for Case 7, respectively. The ligament structures coupled to ligament shedding occur at St = 0.051, and the timescale associated with a breakup from Cases 3 and 7 is at τ = 6.74 ms. The transition from the enhanced capillary breakup regime to the bag breakup regime and an increase of the momentum flux ratio within the bag breakup regime did not affect the timescale of breakup associated with ligament shedding. The traveling column waves associated with ligament shedding increased from 3.14 ± 0.38 mm for Case 3 to 4.76 ± 0.55 mm for Case 7 as the momentum flux ratio increased. The causation for the bag breakup within this regime was also column waves; however, these column wavelengths were detected to be much smaller in magnitude yet more frequent than the larger column waves associated with ligament shedding. The average column wavelengths associated with the bag breakup mechanism in Case 3 were 1.1 ± 0.13 mm, whereas for Case 7, it is 0.73 ± 0.09 mm. The wavelengths calculated in these two cases closely matched to Sallam et al. [9] findings, in which he noted that bag breakup occurs when the ratio of wavelength to jet diameter is approximately unity.

Fig. 14
Fig. 14
Close modal

#### 4.1.3 Multimode Breakup Regime.

The multimode breakup regime is a transition region comprised three breakup mechanisms: ligament shedding, bag breakup, and shear breakup. These breakup mechanisms are convolved with one another and can intermittently fluctuate between each other or simultaneously exist together. From the Robust MrDMD analysis, it is observed that the ligament breakup associated with this breakup regime occurs sparsely, and while it is a dominant mode, it is not as influential as a mechanism like the bag breakup and shear breakup mechanism. The breakup associated with ligament shedding within this regime originates from large column traveling waves. These traveling waves have increased in wavelength from the previous cases and are responsible for larger breakup structures. The average traveling wavelengths for the multimode breakup is 4.01 ± 0.7 mm for Case 4 and 4.84 ± 0.7 mm for Case 8, where the momentum flux ratio increased, and the Strouhal number for this structure is St = 0.06. Like the bag breakup regime, shorter column wavelengths are associated with a bag breakup mechanism. These bag-like structures occur at a Strouhal number close to the bag breakup case at an average of St = 0.146 with a wavelength of 0.51 ± 0.1 mm and 0.44 ± 0.07 mm for Case 4 and Case 8, respectively. Corresponding to the shorter wavelengths, it can be observed from Fig. 15 that the bag structures are smaller than the bags produced in the bag breakup regime. Inspecting both ligament shedding mode and the bag breakup mode, it can also be seen from Fig. 15 that the surface corrugations observed in the earlier cases have developed into surface waves, which start to strip the small ligaments from the liquid jet.

Fig. 15
Fig. 15
Close modal

These surface waves are small and only last a short duration of time. It can be seen from this case that these surface waves develop on the windward side of the jet and travel alongside the jet. It is noticed qualitatively from the modes associated with the entire spatial window that shear breakups occur more upstream for Case 8, which has a higher momentum flux ratio than Case 4. However, because breakup associated with column waves are more prevalent within the entire spatial window and shear breakup is a small-scale breakup structure, a coherent structure associated with the shear breakup could not be extracted within the upstream entire spatial window. The shear breakup mechanism, as seen in Fig. 16, was extracted at shear layer ROI with surface traveling wavelengths of 0.25 ± 0.07 mm for Case 4 and 0.23 ± 0.02 mm for Case 8 with an St = 0.22. The timescale of the breakup for ligament shedding for both cases occurred at τ = 1.68 ms. The bag breakup structures associated with Case 4 reside in the same temporal window as the ligament shedding structures, but the increased momentum flux ratio shift this bag breakup time scale to τ = 0.84 ms. The shear breakup timescale also occurred at τ = 0.84 ms for both cases.

Fig. 16
Fig. 16
Close modal

#### 4.1.4 Shear Breakup Regime.

The shear breakup regime comprises of two dominant modes: the shear breakup mode and the ligament shedding mode. Like the shear breakup mode in the multimode breakup regime, this breakup mode is associated with surface waves, responsible for stripping small-scale structures from the liquid jet. Contrary to the multimode breakup regime, surface breakup structures and associated surface waves were detected in the upstream ROI; this can be seen in Fig. 18. From this figure, it can be seen that the shear breakup structures are more prevalent upstream in Case 9, which has a higher momentum flux ratio than Case 5. The spatial shear breakup mode, extracted from the shear layer ROI seen in Fig. 17, suggests that one of the major driving factors for generating these surface waves is the relative velocity between the crossflow and liquid, which alludes to the fact that these surface waves could be generated more dominantly from the Kelvin-Helmholtz instability. It is found that the small shear breakup structures are generated at the normalized frequency shedding value of St = 0.23. The surface traveling waves associated with these small-scale structures are at 0.16 ± 0.017 mm and 0.13 ± 0.01 mm for Case 5 and Case 9, respectively. The time scale for both these breakup mechanisms is found at τ = 0.84 ms. The dominant mode associated to ligament shedding is found to occur when the entire spatial window is investigated with Robust MrDMD; from Fig. 18, it is noticed that the ligament has become larger due to the long column traveling waves. Qualitatively, while this large ligament shedding is occurring, there is also shear breakup occurring on the liquid column and the ligament breakup structure happening simultaneously. The column wavelength for Case 5 is 4.76 ± 0.59 mm, and for Case 9 is 5.01 ± 0.64 mm, with a timescale of τ = 1.64 ms, occurring at St = 0.051. Based on inspecting the time-resolved snapshots, it is also seen that there is a less dominant bag breakup and surface stripping mode, as portrayed in Fig. 19. The surface stripping mode is generated at the windward side of the jet orifice exit, whereas bag breakup mode occurs at the leeward side of the jet near the jet exit. It is observed that the generation of these two modes is due to surface waves developing with sufficient amplitude at the jet exit. However, these breakup modes could not be extracted by the Robust MrDMD method, as they are not modes with a significant spectral cluster and thus do not dictate the dominant breakup mechanism leading to the disintegration of the jet column.

Fig. 17
Fig. 17
Close modal
Fig. 18
Fig. 18
Close modal
Fig. 19
Fig. 19
Close modal

The breakup regime for Case 10 is a highly dominant shear breakup case with a Weber number of 300 and a momentum flux ratio of 125. This breakup regime is encompassed by a highly repeated active shear breakup mode with a timescale of τ = 0.84 ms. The onset of the column droplet shear stripping happens immediately right at the exit of the jet orifice for the highly dominant shear breakup case as observed from shear mode spatial reconstruction in Fig. 20 attained at the upstream region of interest.

Fig. 20
Fig. 20
Close modal

Therefore, it can be insinuated that as the Weber number increases, the energy contained within the traveling waves also increases, which leads to the formation of the droplet stripping mechanism closer to the orifice. The shear breakup is associated with an St = 0.239. It is also found that the ligament breakup occurs at the same nominal value of St = 0.055, at a similar timescale of τ = 0.84 ms. This suggests that ligament shedding for Case 10 co-occurs with the shear breakup. The shear breakup mechanism occurs at the small scale due to surface wavelengths of 0.09 ± 0.02 mm, and ligament shedding occurs due to long traveling wavelengths of 5.05 ± 0.35 mm.

### 4.2 Shedding Behavior and Validation of Timescale.

Implementing the Robust MrDMD modal analysis technique resulted in a unique spectral cluster for the dominant breakup mechanisms associated with each breakup regime. It became evident that upon nondimensionalizing these oscillating frequencies with the Strouhal number, each breakup mechanism is linked to a specific Strouhal number. Large coherent structures associated with ligament breakup are associated with an average St = 0.055 ± 0.006, bag structures originating from column waves are linked to St = 0.15 ± 0.004, and small-scale structure associated with the shear breakup in the upstream and shear layer ROI is coupled to St = 0.23 ± 0.008. The identification of these Strouhal numbers leads to the formulation of the three breakup mechanisms, identified by their oscillating frequency as defined in Eqs. (2)(4).

Ligament breakup:
$flig=(0.055±0.006)VjDj$
(2)
Bag breakup:
$fbag=(0.145±0.004)VjDj$
(3)
Shear breakup:
$fshear=(0.23±0.008)VjDj$
(4)

It was found that even though regimes such as bag, multimode, and shear contain multiple dominant breakup mechanisms that are convolved with each other, there exists a peak dominant breakup mechanism route for each of these regimes. For the bag breakup regime, the primary dominant mechanism is the bag breakup, and the secondary is ligament breakup. For the multimode breakup regime, the primary breakup mechanism is bag breakup, followed by shear and ligament breakup. The most dominant breakup mechanism for the shear breakup regime is the shear breakup, followed by ligament breakup, as illustrated in Fig. 21. The dominant peak mechanism is extracted based on the number of counts associated with a particular spectral cluster, as observed previously (cf., Fig. 10).

Fig. 21
Fig. 21
Close modal
The timescales associated with each of the breakup mechanisms were validated by comparing the theoretical timescale formulation by Sallam et al. [9], shown in Eq. (5) for a range 4 < We < 160 with constant momentum flux ratio. The range of Weber numbers for Sallam's theoretical model coincides with Case 3 and Case 4 in this study. The theoretical formulation is compared with the dominant route of breakup mechanism for both Case 3 and Case 4, as these are the primary mode of a breakup for the liquid jet. The dominant breakup route for both Case 3 and 4 is the bag breakup mechanism, as illustrated in Fig. 21
$tb=2.5(ρcρj)12djVc$
(5)

The timescale extracted for the Robust MrDMD analysis of the bag breakup mechanism for Cases 3 and 4 were found to be 3.84 ms and 1.68 ms, respectively, the breakup timescale theoretically calculated for Cases 3 and 4 were 4.15 ms and 1.99 ms, respectively. The percent error associated with the Robust MrDMD timescale was calculated to be 7.5% and 15.5%. The associated reasonable margin of error validates the timescale extracted by the Robust MrDMD methodology.

## 5 Conclusion

The present research introduces a novel modal extraction method utilizing the multiresolution dynamic mode decomposition algorithm for time-recorded snapshots of turbulent-based flows. This study shows that Robust MrDMD can effectively extract important coherent structures associated with a fluid flow. In this study, Robust MrDMD is applied to study the four breakup regimes associated with the primary breakup mechanism of a transverse jet. It was found that a Robust MrDMD method can effectively extract relevant modes pertaining to a liquid jet in crossflow. Based on the Robust MrDMD modal spectral analysis in the entire temporal window, three modes of primary breakup mechanism of the liquid jet in crossflow configuration exist. Characterized for the entire breakup regime, these are ligament shedding, bag breakup, and shear breakup. It was found that there are multiple breakup mechanisms convolved with each other at different timescales for bag breakup, multimode, and shear breakup regimes. For the regimes dominated by bag breakup, the breakup modes are ligament shedding and bag breakup. The multimode breakup dominated regime contains all three breakup mechanisms: ligament shedding, bag breakup, and shear breakup. The shear breakup mode for the multimode breakup regime is found to occur along with the shear layer as well as further downstream in the region of interest. The shear breakup regime contains large intermittent ligament shedding modes convolved with fast timescale shear breakup modes. From the spectral analysis of each regime, it was further discovered that while multiple breakup mechanisms exist in conjunction with another, it was noticed that there is a dominant route of breakup based on the number of modes associated with a specific spectral cluster for each regime. In the enhanced capillary breakup regime, it was found that ligament shedding was the only dominant mode of the liquid jet breakup. The latter mechanism occurred at an approximate Strouhal number of St = 0.055 ± 0.006. This ligament shedding behavior was found to be constant for every case investigated in this study. The traveling column wavelengths associated with the ligament shedding mode increased in magnitude as both the Weber number and the momentum flux ratio also increased. The bag breakup mode, which is observed with its characteristic bag-like structures and rings, was correlated to a specific St = 0.145 ± 0.004. This mode is the dominant route of breakup for both the bag and the multimode breakup regime. The shear breakup mode, the dominant route of breakup for Cases 5, 9, and 10, is associated with small-scale structures shedding from the liquid jet column. That phenomenon was recorded at a nondimensionalized oscillating frequency of St = 0.23 ± 0.008 and a breakup timescale of 0.84 ms.

## Acknowledgment

The authors acknowledge support from the Office of Naval Research (N00014-20-1-2555, Program Officer: Dr. Steven Martens).

## Conflict of Interest

There are no conflicts of interest.

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