High-fidelity simulations of a rotary bell atomizer with electrohydrodynamic effects

High-fidelity simulations of a rotary bell atomizer with electrohydrodynamic e↵ects Venkata Krisshnaa,⇤, Wanjiao Liub,1, Mark Owkesa aMontana State University, Department of Mechanical &Industrial Engineering, P.O. Box 173800, Bozeman, Montana 59717-3800, USA bFord Motor Company, Research and Innovation Center, 2101 Village Road, Dearborn, Michigan 48121, USA Abstract Electrostatic rotary bell atomizers are extensively used as paint applicators in the automobile industry. Paint undergoes atomization after exiting the edge of a high-speed rotating bell. In most setups, the paint is electrically charged and a background electric field is applied between the nozzle and the target surface to increase the transfer efficiency (TE). The atomization process directly determines the droplet size and droplet charge distributions which subsequently control TE and surface finish quality. Optimal spray parameters used in industry are often obtained from expensive trial-and-error methods. In this work, three-dimensional near-bell atomization is computationally simulated using a high-fidelity volume-of-fluid transport scheme that includes electrohydrodynamic (EHD) e↵ects. We find that electrifying the setup results in the production of smaller droplets. The distribution of charge in droplets after atomization is found to be insufficient to cause Coulombic fissive breakup for the range of parameters studied. Additionally, the electric field has a minor e↵ect on primary atomization but a negligible e↵ect on the size and stability of atomized droplets after secondary breakup. This cost-e↵ective method of simulating EHD-assisted atomization allows for the understanding of the e↵ect of the electric field and the extraction of droplet charge characteristics which is otherwise challenging to obtain experimentally. Keywords: NGA, CFD, electric field, charge distribution, electrospray, rotary atomizer 1. Introduction With the world becoming increasingly dependent on automotive means of transportation and motor vehicle production approaching 100 million units per year [66], automobile manufacturers are looking for ways to minimize production costs. In an automobile manufacturing facility, the paint shop can account for up to 70% of the total energy costs [36], demand up to 50% of the electricity and up to 60% of the fossil fuel energy [55] used in the facility. These costs are associated with the energy used in operating and maintaining the HVAC equipment of the painting booth, paint drying, and control of pollutants like volatile organic compounds generated by paint overspray [36]. The overspray leads to significant material waste causing environmental and cost concerns. This makes the paint shop one of the most ex- ⇤Corresponding author Email addresses: [email protected] (Venkata Krisshna), [email protected] (Wanjiao Liu), [email protected] (Mark Owkes) pensive aspects of automobile manufacturing, accounting for up to 50% of its total costs [3, 12]. The global automotive paints and coatings market size was valued at $17.34 bn in 2019 and is expected to be about $26.8 bn by 2027 [78]. Additionally, the paint shop is responsible for over 80% of the environmental concerns in a manufacturing facility [37]. Rotary bell atomizers (RBAs) (Fig. 1) are extensively used as paint applicators in the automobile industry. In addition to automobile paint shops, rotary atomizers are used in several other applications such as agricultural spraying [16], food processing [73], and pharmaceutical drug-delivery [57]. An RBA is a high-speed rotating nozzle that atomizes paint into droplets that range from a few micrometers to tens of micrometers in diameter. Paint is injected onto the inner surface of a bell-shaped nozzle where it spreads into a thin film on the surface due to centrifugal forces. The fluid film, on reaching the edge of the highspeed rotating bell, exits as multiple ligaments which further atomize into droplets [34]. It is common practice to electrically charge the paint and apply a backPreprint submitted to International Journal of Multiphase Flows February 15, 2023 Manuscript File Click here to view linked References Figure 1: A rotary bell atomizer in operation. Photograph courtesy of RISE Research Institutes of Sweden and Fraunhofer-Chalmers Centre. ground electric field to enhance the transfer efficiency (TE) of the device [24, 45]. Atomized droplets, which also carry electric charge, move toward the grounded target surface under the influence of the electric field. Metrics of significant importance such as the droplet size uniformity, surface finish quality, TE, deposition thickness, and environmental impact are directly dependent on the atomization process in the nozzles [24, 2, 15]. The nozzle operating conditions used in industry that optimize these metrics are often obtained from expensive trial-and-error experimental methods. The optimal setup of RBAs can still be improved, for example, painters currently are required to over-coat the target surface to ensure sufficient finish quality using excess paint [83]. Due to the high expense of the painting process, small improvements can result in significant cost savings and waste reduction. Early research conducted in RBAs investigated the physics behind the atomization process [44, 8, 22, 58, 64, 15, 56, 21]. These works identified four crucial processes leading up to atomization - film formation, ligament formation, ligament thinning, and ligament breakup. Several articles have studied and characterized the process of film formation as paint flows along the inner surface of the bell [96, 25, 53, 59, 95] and ligament formation at the edge of the bell [82, 88, 85]. In this work, ligament thinning and ligament breakup are the relevant physical processes. While many RBAs are equipped with shaping air (an annular stream of focused air around the bell) to increase TE, the system modeled here does not include it. Multiple research groups have studied the operation of non-electrified RBAs focusing on various aspects such as the bell geometry [22, 90, 49, 74], the e↵ect of shaping air [92, 18], the breakup process [87, 65, 50, 85], and the droplet size distribution [1]. As mentioned, RBAs are often operated in a background electric field that helps improve their TE. The electric field interacts with the charged liquid and a↵ects its flow near the bell. Electrified flows are described by electrohydrodynamics (EHD) - the science of characterizing interactions between fluid dynamics and electrostatics [11, 35]. EHD has seen several decades of research and is now employed in various engineering applications including food technology [4, 51, 38], inkjet printing [79], electrostatic precipitation [105], powder coating [48], biochemistry [93], microfluidics [6], and biomedical applications [29, 26, 32]. It is common practice to use electric fields to control uncharged liquid jets [100] and charged fuel sprays [33, 89]. EHD-assisted atomization has been an increasingly important means of producing liquid droplets that is well-established to o↵er advantages over other industrial spraying processes [7, 42, 39]. Previous studies on electrified RBAs that have included an electrostatic model have explored its e↵ects as a macroscopic phenomenon focusing on the resulting spray pattern, paint film thickness, over-spray control, and TE [5, 47, 23, 14, 27, 91, 77, 60, 46, 45, 43, 102]. Experimental studies on near-bell atomization in electrified RBAs have been limited in their ability to understand droplet charge characteristics [104]. Recent numerical e↵orts have characterized the e↵ects of various operating parameters on droplet trajectories, droplet size distribution, and charge evaporation in electrified RBAs [13, 28, 75, 76, 80, 40]. Generally, most prior models have given little or no attention to electric effects on multiphase transport phenomena driving atomization. Moreover, high-speed cameras and imaging equipment are seldom placed in electrified RBA setups to prevent electric arcing [41]. For these reasons, experimental studies of the atomization process in electrified RBAs are limited in detail. Additionally, there is currently no experimental data on the charge distribution in atomized droplets in RBAs. Although some research has been conducted on the e↵ect of charge density on the surface finish quality [98] and charge distribution in atomized droplets [61] in electrostatic sprays, they have not been studied for charged rotary atomizers. The process of atomization in the presence of an electric field can be modeled using the incompressible Navier-Stokes and EHD governing equations. In this way, the operation of this device can be studied computationally for a wide range of parameters without the cost of operating the device in a lab. Moreover, numer2 ical analysis allows varying fluid properties with ease thus avoiding the expensive process of meticulously fabricating an appropriate fluid for experimental procedures. In this project, we investigate the microscopic e↵ects of EHD on atomization by computationally simulating three-dimensional RBA near-bell atomization using a high-fidelity volume-of-fluid transport scheme that includes EHD e↵ects. We perform numerical experiments to understand the e↵ect of EHD on atomization to obtain and examine the resulting droplet size distributions and droplet charge distributions in the atomized droplet cloud near the nozzle. Ultimately, this research contributes to the understanding of the underlying physics in electrically-assisted atomization processes and provides insight into the droplet conditions after atomization. These conditions can serve as initial conditions for a Lagrangian flow solver that predicts the trajectories of the atomized charged droplets toward the target surface in a background electric field. The governing equations of the physics modules used to model liquid flows in electrified RBAs are detailed in Section 2. The validation e↵orts of the physics modules, details of the domain geometry, and the mesh sensitivity of the tool are presented in Section 3. Inferences from a parameter study and atomization statistics are addressed in Section 4. Section 5 contains the summary of this project and potential future work that can be conducted with the tool built to study atomization in electrified RBAs. 2. Governing equations The physics modules discussed below have been implemented within a code called NGA - a high-order, fully conservative, variable density, low Mach number Navier-Stokes solver that contains various multiphysics modules implemented in parallel using message passing interface (MPI). The formulation discretely conserves mass and momentum in a periodic domain. NGA uses a conservative unsplit geometric volume-offluid (VOF) scheme as described in the works of Blanquart et al. [10], Desjardins et al. [19, 20] and Owkes and Desjardins [71, 70, 69, 72]. NGA has been developed by several groups to solve multiphase EHD flows, details of which can be found in Van Poppel et al. [101] and Sheehy and Owkes [86]. 2.1. Multiphase fluid dynamics For low-Mach number, variable density, multiphase flows, mass and momentum conservation laws in both phases can be written as follows @⇢i @t+r·(⇢iui)=0 (1) @⇢iui @t+r·(⇢iui⌦ui)=rpi+r·(f i+e i)+ns+fexternal (2) where ⇢is the density, uis the velocity field vector, t is time, and pis the hydrodynamic pressure. The subscript idenotes the fluid phase (gas or liquid). The term nsdenotes surface tension force, where is the surface tension coefficient, is the local curvature of the interface computed using the ACES technique [68], nis the normal vector to the interface, and sis a Dirac-delta function that is nonzero only on the interface [71]. The viscous stress tensor f iin Eq. 2 is given by f i=µi(rui+ruT i)2 3µi(r·ui)I(3) where µis the dynamic viscosity and Iis the identity tensor. These equations form the basis of the fluid dynamics module in this work and further details can be found in Owkes and Desjardins [72, 69], Desjardins et al. [19, 20]. e iis the Maxwell stress tensor which will be described in the next section. fexternal is a combination of the forces experienced by a fluid element in the system. fexternal =fcentrifugal +fCoriolis +fgravity (4) 2.2. Electrohydrodynamics The Maxwell stress tensor e iin Eq. 2 is given by e i="iEi⌦Ei"i 2Ei·Ei 1⇢i "i @"i @⇢i!I(5) where Eis the electric field vector and "is the electric permittivity equal to the product of the relative permittivity iand the vacuum permittivity space "0. Magnetic e↵ects have been ignored since the EHD time scale is several orders of magnitude larger than the magnetic time scale [84] in the atomization process that is of interest here. This electrostatic assumption eliminates the e↵ect of the velocity of the charges (i.e., current) on the electric field thus dictating that the electric field is only influenced by the instantaneous electric charge distribution. The electric force can be expressed as the divergence of the Maxwell stress tensor, r·e i=qiEi1 2E2 ir"i+r 1 2⇢i @"i @⇢i E2 i!(6) 3 where qis the volumetric electric charge density. The first term on the right-hand side of Eq. 6 is the Coulomb (or Lorentz) force. The second and third terms denote the dielectric and the electrostrictive forces respectively, which are only significant in a transient electric field with time scales several orders of magnitude larger than what is encountered in atomization problems [52]. As the electric field vector is irrotational it can be expressed in terms of the scalar electric potential as Ei=r.(7) The electric potential Poisson equation describes the relationship between charge density qiand as r·("ir)=qi.(8) The electric Poisson equation is solved using the hypre library of routines for scalable parallel solutions of linear systems [30, 31]. The accumulation of bulk volumetric charge as a surface charge in a thin electric boundary layer much smaller than the hydrodynamic boundary layer is a commonly made assumption in charged jet simulations [103, 99]. The electric charge has previously been modeled in two ways: either a constant bulk volumetric charge or a fully relaxed surface charge. According to the classic leaky dielectric model [63, 97, 84] that is commonly used to describe the e↵ects of electric charge in dielectric liquids, the fundamental underlying assumption is that bulk volumetric electric charge has sufficient time to fully relax to a surface charge. However, for atomizing flows the advection time scale is similar to the charge relaxation time scale and a fully relaxed charge assumption is invalid. The charge relaxation time ⌧qrepresents the time required for volumetric charge qlto relax to the surface [17, 52]. The fluid advection timescale ⌧fis a characteristic time for a fluid element to move across a relevant length scale L0. The ratio of the two time scales called the electric Reynolds number Reewas formulated in Stuetzer [94] as ⌧q="l lql ,(9) ⌧f=L0 u,(10) Ree=⌧q ⌧f (11) where values of the quantities that correspond to typical automotive paints and RBA operations listed in Table 2 yield a value of Reeapproximately equal to 8.93. This suggests that the advection and charge relaxation timescales are comparable in this application. The above time scale analysis highlights the necessity to model the process of charge relaxation by migration and di↵usion in addition to advection with the fluid velocity. The common assumption of charges to be accumulated on the surface disallows the charge migration dynamics and limits the accuracy of the droplet charge distribution after atomization. In this work, we assume a constant bulk volumetric charge at the injector and allow charges to relax as the jet propagates through the domain. Charge transport is described by the conservation equation @qi @t+r·Ji=0 (12) where the current density Jiis formulated as [62, 101] Ji=qiui+qi iEiDirqi(13) where iis the charge mobility coefficient and Diis the molecular di↵usivity. The three terms that contribute to the current density can be described as advection due to the velocity field, advection due to the electrical velocity, and di↵usion. In summary, the charge density field (q) and boundary conditions (BCs) in electric potential are used to obtain the electric potential field () using Eq. 8. The electric field (E) is then obtained using Eq. 7 after which Eq. 6 allows for the calculation of the Coulomb force (felectric) on the charged fluid. More details on the equations involved in the formulation of the EHD module can be found in Sheehy and Owkes [86]. 2.3. Rotating reference frame In an RBA undergoing ligament breakup mode, each ligament exiting the serrated bell edge behaves similarly with statistically similar droplet size distribution and charge transport behavior. Therefore, we focus on one ligament ejected from the bell edge in our numerical simulations. We attach our reference frame to the rotating nozzle and ligament. To account for the rotating nozzle and the consequent forces, we include equations that describe the rotating reference frame. This subjects the liquid to centrifugal and Coriolis forces which can be formulated as fcentrifugal =⇢i!⇥(!⇥r) (14) fCoriolis =2⇢i!⇥ui(15) where !is the angular velocity and ris the radial distance from the axis of rotation. 4 Figure 2: Validation of centrifugal force: Numerical results of a rotating tank experiment where cases 1, 2, and 3 correspond to rotation rates of 4, 10, and 14 rad/s respectively. The analytical solution of the interface (Eq. 16) at various locations is plotted with red circles. The numerically computed liquid interface and liquid bulk are shown as black lines and blue areas respectively. The dashed line represents the axis of rotation for each case. 3. Numerical simulations of RBA jets 3.1. Validation of physics modules The EHD module in NGA has seen continued development and validation e↵orts have been presented in Sheehy and Owkes [86]. To validate the implementation of centrifugal force (described in Section 2.3) into the EHD formulation, numerical experiments of a rotating tank configuration are conducted. The solution for the steady-state interface profile h(x) of a liquid in a rotating tank is analytically derived as h(x)=h0+1 2g(!x)2(16) where h0is the height of the liquid at the axis of rotation and gis the gravitational acceleration [54]. Numerical results show excellent agreement with the analytical solution for varying rotation rates (Fig. 2). The numerical implementation of the Coriolis force (described in Section 2.3) is tested by simulating a liquid jet in a rotating frame. The path of an object experiencing the centrifugal and Coriolis forces follows a circle with a growing radius, i.e., a spiral. The analytical solution for the position of the object (x,y) at a given time tis obtained as x(t)=(x0+vx0t)cos (!t)+t⇣vy0+!x0⌘sin (!t) y(t)=(x0+vx0t)sin (!t)+t⇣vy0+!x0⌘cos (!t) (17) where vx0and vy0are the xand ycomponents of the initial velocity of the jet and x0is the position of the jet at t=0. The derivation of this analytical solution is provided in Appendix A. This solution is derived for a rigid point-like object with its all mass concentrated at its center. Liquid jets, however, are not point-like approximations and include e↵ects of viscosity and surface tension. Despite these challenges, we proceeded with validating the e↵ect of the Coriolis force on liquid jets in a rotating reference frame and obtained satisfactory results. Numerically simulated liquid jets follow the analytically predicted trajectory for varying rotation rates (Fig. 3). 3.2. Using e-Mesh An addition to the EHD module of NGA developed for this project involves using a domain called e-Mesh. e-Mesh is much larger than the flow-solver domain (henceforth called the NS-Mesh) where the equations described in Section 2 are solved. Since our interest lies in primary and secondary atomization, NS-Mesh lies at the edge of the bell and extends a few millimeters outside of it. NS-Mesh requires a well-defined electric potential field () to obtain the electric field (E). We solve for using appropriate BCs at the boundaries of NS-Mesh. However, Fig. 4(a) highlights that is well defined (labeled Dirichlet) only at the nozzle (bell electric potential) and at the target surface (grounded) but is not readily available at the boundaries of NS-Mesh. Instead of assuming values of BCs on NS-Mesh, a new domain called e-Mesh that spans between regions of well-defined is initialized and used to obtain accurate BCs on NS-Mesh. The volume of e-Mesh is about 8000 times larger than that of NS-Mesh, i.e., the region of interest in this project lies in about 0.0125% of the volume of e-Mesh. Making NS-Mesh this large would require significantly more computational resources to solve the fluid dynamics far from the bell edge. The breakup activity and charge transport within the liquid, which are the processes of interest in this work, are already completed within the NS-Mesh domain. To avoid solving hydrodynamics outside of NS-Mesh, we create 5 Figure 3: Validation of Coriolis force: Numerical results of the liquid jet interfaces in a reference frame rotating counter-clockwise about an axis that points out of the 2D plane. The analytical solution (Eq. 17) is shown as a superimposed solid line for each case. e-Mesh - a domain spanning the entire region between the bell and the target surface - to be used exclusively as an electric potential solver. A Dirichlet BC of the value of the bell electric potential is imposed on the top boundary where the nozzle overlaps e-Mesh. A zero potential boundary condition is imposed on the bottom boundary which represents the target surface. Periodic BCs are imposed on the z+and zboundaries of e-Mesh and Neumann BCs are imposed on its remaining boundaries. The right boundary (x+) was chosen in such a way that it is far enough away that the electric potential field lines are perpendicular to the target surface, allowing for a Neumann BC to be imposed. In addition to the BCs, e-Mesh is populated with the charge density field (q) which is used to solve for (on e-Mesh) (Fig. 4(b)). Values of are interpolated from e-Mesh (which has a stretched grid and is coarser than NS-Mesh) to cells that lie on the boundaries of NS-Mesh using a tri-cubic interpolation method. These interpolated values are then used as BCs to compute on NS-Mesh. 3.3. Domain geometry The domain geometry is represented schematically in Fig. 5. NS-Mesh, where the electrohydrodynamics is solved, has dimensions of 12.96 mm ⇥960 µm⇥ 360 µm. e-Mesh, which spans from the bell edge to the target surface, has dimensions of 0.2 m ⇥0.25 m ⇥ 360 µm. In RBAs, the bell edge plane is angled away from the plane perpendicular to the axis of rotation. In the numerical setup, the jet is injected into the domain from the top wall with initial velocity components that correspond to an edge angle of 25. Some RBAs are equipped with serrations present at the edge of the bell which causes the fluid to be ejected as thin jets. In the absence of these serrations, the fluid forms liquid sheets on the inner surface of the bell. These liquid sheets may (a) e-Mesh shown in context with NS-Mesh and available BCs (b) Electric potential field () on e-Mesh Figure 4: Boundary conditions and electric potential field () on eMesh (not to scale) 6 Table 1: Mesh sensitivity simulation domain parameters Mesh Cells per Total Cell Processor No. of ID diameter cells width count structures 05 CPD 5 2.592⇥10612 µm 16 74 10 CPD 10 20.73⇥1066µm 96 121 15 CPD 15 69.98⇥1064µm 256 705 be subject to interfacial instability and form periodic ligaments at the edge of the bell or break up in sheet mode depending on the operating conditions. In this work, a single jet of diameter 60 µm flowing out of a serrated bell is simulated. The top boundary is a wall that allows slip velocity and the left boundary is a no-flux wall. the bottom and right boundaries are convective outflows. Periodic BCs are imposed on the front and back walls. Although the periodic BCs on a Cartesian mesh would misrepresent jet interaction far from the bell edge, the angular separation of the jets close to the bell edge is not significant enough to account for its curvature. In addition to mass and momentum (and charge) being transported across the periodic zboundaries, the electric field is also computed with the e↵ect of the periodicity. This means that at any instance the single jet simulated in the computational domain is solved with full consideration of its nearest neighboring jets. The axis of rotation, which is the center of the bell, is located one radial length away (in the negative x direction) from the inflow at a distance of 0.025 m. As mentioned previously, the model does not currently include a shaping air setup. Breakup is induced by imparting velocity modulations to the inflow. The inflow velocity (uinflow) is modulated by adding sinusoidal perturbations at a superposition of three (40, 50, and 60 kHz) frequencies (!) at an amplitude (A) which is 9% of its original velocity (u) (Eq. 18). uinflow =u+ 3 X n=1 Asin (2⇡!nt).(18) 3.4. Mesh Sensitivity To determine the e↵ect of mesh resolution on atomization and to help choose an optimal numerical setup, three simulations are performed with varying mesh resolutions (Table 1). We note from Fig. 6 that droplets larger than 10 µm are resolved in a 5 cells-per-diameter (CPD) mesh while the 10 CPD mesh captures droplets smaller than 10 µm. Additionally, the 15 CPD mesh captures the droplets even smaller than 5 µm. Larger droplets are similarly resolved for all meshes. Between the two finer meshes the smaller droplets are increasingly well resolved. We note the necessity for a mesh finer than 5 CPD since the coarseness of the mesh does not allow for charge migration to be captured. All liquid structures in the 5 CPD mesh contain the injected charge density value of 2.879 C/m3. Using 10 and 15 CPD meshes, we are able to resolve the charge migration process and investigate the resulting droplet charge distribution. Most of the droplets contain a charge density close to the initial bulk charge density value. Compared to the initial charge density, it is more probable to find a droplet with a lower charge density. Fig. 7 highlights that charge migration is more resolved in a 15 CPD mesh and is consistent in its trend with the 10 CPD mesh. Although there is uncertainty with the smallest scales of droplet sizes, the number of droplets larger than 20 µm does not vary on increasing the mesh resolution, suggesting that the larger droplets are well resolved in a 15 CPD mesh. Since the goal of this project is to demonstrate the development of the tool, we believe that the underlying physical processes are sufficiently captured when using a 15 CPD mesh. 4. Results and discussion Simulations are performed in a domain as described in Section 3.3. Table 2 contains a list of parameter values used in simulations. The non-dimensional numbers corresponding to the simulations are listed in Table 3. In the interest of understanding the e↵ect of EHD on the flow, we present a brief discussion on the relevant non-dimensional numbers. The electric Reynolds number [94] is the ratio of the charge convection timescale to the charge mobility timescale while the electric P´ eclet number [86] is a measure of the charge mobility timescale to the charge di↵usion timescale. Based on the values of the electric Reynolds and P´ eclet numbers corresponding to this setup, it is reasonable to model all three charge dynamics processes, i.e., convection, di↵usion, and migration, since their timescales are comparable to one another. The electric Bond number [9] quantifies the importance of the deforming electrical force compared to the restoring surface tension force. A low value signifies the dominance of surface tension driven breakup activity. The electro-inertial number [86] denotes the importance of EHD forces compared to inertial advective forces. A low value implies that inertial forces predominantly drive the hydrodynamics of the flow in this setup. Figs. 8 and 9 show the positions of the liquid interface of the jet at di↵erent instances in time as viewed 7 Figure 5: A schematic representation of the domain geometry of the numerical problem showing positions and dimensions (not to scale) of NSMesh and e-Mesh. Table 2: Standard values of parameters in the RBA simulations Property Symbol Value Unit Bell rotation rate !40⇥103RPM Bell radius Rbell 0.025 m Edge angle - 25 deg Inlet jet diameter Djet 60⇥106m Liq. flow rate - 7.96⇥109m3/s Liq. viscosity µl0.1 Pa.s Liq. density ⇢l1000 kg/m3 Surface tension 0.03 N/m Bell electric potential Vbell 80⇥103V Liq. charge density ql2.879 C/m3 Liq. rel. permittivity l50 - Liq. molecular di↵.Dl2⇥106m2/s Liq. ionic mobility l1.79⇥108m2/V.s Figure 6: Probability distribution of droplet sizes for varying mesh resolutions. The lines show a log-normal fit of the data. Table 3: Non-dimensional numbers used in the RBA simulations Number Definition Value Density ratio ⇢l/⇢g830.56 Viscosity ratio µl/µg5546.31 Permittivity ratio l/g50.00 CFL |umax|t/x0.30 Reynolds ⇢l|ujet|Djet/µl1006.24 Weber ⇢l|ujet|2Djet/77.87 Ohnesorge pWe/Re 0.0088 Electric Reynolds "l|ujet|/(Lql l) 8.93 Electric P´ eclet ql lL2/(Dl"l)0.21 Electric Bond "l|E|2L/0.091 Electro-inertial q2 lL2/⇣"l⇢l|ujet|2⌘0.0017 8 (a) 10 CPD (b) 15 CPD Figure 7: Volume weighted charge density probability distributions for varying mesh resolutions. Note the logarithmic scale on the yaxis. from di↵erent viewing planes. We can capture complex and chaotic breakup activity comprising primary and secondary atomization that begins approximately 6 mm away from the bell edge. 4.1. Parameter study A parameter study is performed on a 15 CPD mesh to understand the e↵ects of changing four parameters - nozzle rotation rate, liquid flow rate, liquid charge density, and bell electric potential. Results of the parameter study are shown in Fig. 10 as snapshots of the liquid interface after 200 µs. The standard values of all varied properties are listed in Table 2 unless otherwise stated in the figure. As the nozzle rotation rate increases, centrifugal forces are stronger on the jet and stretch it out faster leading to early elongation and breakup (Fig. 10(a)). Slower rotation rates do not stretch the jet out as much in the same duration. A higher flow rate through the same jet diameter acts in the same way as increasing jet velocity, which initially pushes the jet out further before centrifugal forces take over (Fig. 10(b)). These results are in agreement with experimental observations conducted by Corbeels et al. [15]. An increase in either qin or bell stretches the jet along its downstream direction (Figs. 10(c), 10(d)). This is because the Coulomb force vectors point towards the direction of propagation of the liquid jet, i.e., the electric potential field contour lines are approximately perpendicular to the liquid velocity at that location. 4.2. E↵ect of electrifying the atomizer Fig. 11(a) shows that an electrified jet produces more droplets than a non-electrified one. In this particular simulation after 300µs, an electrified jet contains about 40% more droplets than a non-electrified jet. A lateral shift is evident in the probability distribution of the size distribution between the two cases (Fig. 11(b)), i.e., the most probable diameter in an electrified jet is about 8% smaller than a non-electrified jet. Moreover, both the number and probability distributions show that significantly more droplets of smaller size are produced when the jet is electrified. This behavior is in agreement with the findings presented in Pendar and P´ ascoa [77] where a similar shift towards smaller droplets has been observed in the number distribution for increasing background potentials. According to Pendar and P´ ascoa [77], the presence of an electric field causes a net reduction in the e↵ective surface tension in large droplets and ligaments and therefore an increase in the local Weber number of the structures. This can be the reason behind the increased rate of breakup and the production of smaller droplets in an electrified setup. Fig. 11(c) highlights that droplets are created at a higher rate in an electrified domain. The delay in the first instance of atomization in a charged jet is speculated to be due to a stabilizing force provided by the electric field during the initial dynamics of the liquid core. However, once the breakup begins, the electric field accelerates the process. This phenomenon will be further investigated in future work. It is valuable to know the charge density in droplets after atomization. Fig. 12(a) shows that a majority of the droplets contain a charge density that is very close to the input charge density (shown as a black dashed line). Additionally, compared to the initial charge density, it is more probable to find a droplet with a lower charge density. This can be explained as follows - as charges relax and move to the surface of the ligament, the charge density in the bulk of the liquid reduces. Since a majority of the liquid structures break o↵the bulk volume (simply because of a greater liquid volume in the bulk than near the surface), most droplets contain a charge density slightly lower than the input value. The process of 9 63–78. [35] Fylladitakis, E.D., Theodoridis, M.P., Moronis, A.X., 2014. Review on the History, Research, and Applications of Electrohydrodynamics. IEEE Transactions on Plasma Science 42, 358–375. [36] Galitsky, C., Worrel, E., 2008. 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