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In this paper, we address the challenge of accurately calculating droplet/bubble properties (e.g., volume, number) in diffuse-interface two-phase flow simulations. Currently, flood-fill algorithms can truncate a significant portion of the volume of droplets/bubbles contained within the diffuse interface region or artificially merge multiple droplets/bubbles. This error is also dependent on the volume fraction cutoff value, which is typically chosen to be 0.5 arbitrarily, in the flood-fill algorithms. We propose a simple volume-correction approach that incorporates an analytical approximation of the truncated volume to correct for the missing droplet/bubble volumes. This proposed method results in accurately recovering the dispersed phase volumes with minimal volume error over a wide range of volume fraction cutoff values; and hence, can also accurately recover the number of droplets/bubbles. This can be a valuable tool for accurate calculation of drop/bubble size distributions for analysis and for Eulerian-to-Lagrangian conversion of the dispersed phase in multi-scale modeling approaches., Comment: 15 pages, 12 figures, journal preprint

In this work, we perform numerical simulations of forced two-phase isotropic turbulence to study the stationary states of a two-phase mixture. We first formulate three different approaches to force a two-phase turbulent flow that maintains a constant (a) mixture kinetic energy, (b) mixture kinetic energy + surface energy, and (c) individual phase kinetic energy, and study their effect on the final stationary state of the system. We show that forcing the two-phase system eliminates the arbitrariness associated with the initialization of the second phase in two-phase turbulence simulations. Next, we study the global statistics of the two-phase mixture in the stationary state for varying density ratios ($10^{-3}$ to $10^3$) and viscosity ratios ($10^{-3}$ to $10^3$), void fraction (dilute to dense regimes), and Weber numbers (breakup and non-breakup regimes). We utilize a spatial filtering approach, that is applicable for the general case of incompressible/compressible two-phase flows, to compute the energy spectra of each of the phases and study the turbulent kinetic energy distribution across scales. We find that the interface behaves as a soft wall, inhibiting the accommodation of eddies inside the dispersed phase. Finally, we show that bubbles and drops exhibit different turbulent characteristics for the same global conditions of the carrier phase, with bubbles exhibiting higher turbulent intensities inside them and droplets behaving ballistically, similar to solid particles. This study acts as a foundational work on stationary two-phase turbulence, which can be used for further analysis and the development of subgrid-scale models for larger-scale simulations of two-phase turbulent flows., Comment: 30 pages, 21 figures, journal preprint

Cluster and void formations are key processes in the dynamics of particle-laden turbulence. In this work, we assess the performance of various neural network models for synthesizing preferential concentration fields of particles in turbulence. A database of direct numerical simulations of homogeneous isotropic two-dimensional turbulence with one-way coupled inertial point particles, is used to train the models using vorticity as the input to predict the particle number density fields. We compare autoencoder, U--Net, generative adversarial network (GAN), and diffusion model approaches, and assess the statistical properties of the generated particle number density fields. We find that the GANs are superior in predicting clusters and voids, and therefore result in the best performance. Additionally, we explore a concept of ``supersampling", where neural networks can be used to predict full particle data using only the information of few particles, which yields promising perspectives for reducing the computational cost of expensive DNS computations by avoiding the tracking of millions of particles. We also explore the inverse problem of synthesizing the enstrophy fields using the particle number density distribution as the input at different Stokes numbers. Hence, our study also indicates the potential use of neural networks to predict turbulent flow statistics using experimental measurements of inertial particles.

In this work, we propose a novel scalar-transport model for the simulation of scalar quantities that are confined to the interface in two-phase flows. In a two-phase flow, the scalar quantities, such as salts and surfactants, can reside at the interface and can modify the properties of the interface, in the time scales of interest. This confinement of the scalars leads to the formation of sharp gradients of the scalar concentration values at the interface, presenting a serious challenge for its numerical simulations. To overcome this challenge, we propose a computational model for the transport of scalars that maintains the confinement condition for these quantities. The model is discretized using a central-difference scheme, which leads to a non-dissipative implementation that is crucial for the simulation of turbulent flows. The model is used with the ACDI diffuse-interface method (Jain, J. Comput. Phys., 2022), but can also be used with other algebraic-based interface-capturing methods. Furthermore, the provable strengths of the proposed model are: (a) the model maintains the positivity property of the scalar concentration field, a physical realizability requirement for the simulation of scalars, when the proposed criterion is satisfied, (b) the proposed model is such that the transport of the scalar concentration field is consistent with the transport of the volume fraction field, which results in effective discrete confinement of the scalar at the interface; and therefore, prevents the artificial numerical diffusion of the scalar into the bulk region of the two phases. Finally, we present numerical simulations using the proposed model for both one-dimensional and multidimensional cases and assess: the accuracy and robustness of the model, the validity of the positivity property of the scalar concentration field, and the confinement of the scalar at the interface., Comment: 18 pages, 8 figures

A phase-field method for unstructured grids that is accurate, conservative, and robust is proposed in this work. The proposed method also results in bounded transport of volume fraction, and the interface thickness adapts automatically to local grid size. In addition to this, we present a novel formulation for two-phase flows on collocated grids that is provably energy stable, which is a critical feature for robust simulations of two-phase turbulent flows. The proposed method is first evaluated on canonical test cases, including scenarios like drop advection and drop in a shear flow. The accuracy of the proposed method in the presence of grid transitions, which is important for simulations in complex geometries using unstructured grids, is also evaluated. We assess the robustness of our method by performing simulations of a drop in homogeneous isotropic turbulence at infinite Reynolds number with varying density ratios. Furthermore, as a step toward verification and validation, simulations of test cases in complex geometries and conditions, such as, a damped surface wave, infinitesimal interface distortion on a liquid jet, and the atomization of a liquid jet from the engine combustion network's Spray A nozzle are presented. These simulations illustrate the accuracy, robustness, and applicability of the proposed method in various kinds of complex two-phase flow applications of engineering interest., Comment: 44 pages, 20 figures

In this study, five ice shapes generated at NASA Glenn's Icing Research Tunnel (IRT) are simulated at multiple angles of attack (Broeren et al., J. of Aircraft, 2018). These geometries target different icing environments, both early-time and longer-duration glaze and rime ice exposure events, including a geometry that results from using a thermal ice-protection system. Using the laser-scanned geometries, detailed representations of the three-dimensional ice geometries are resolved on the grid and simulated using wall-modeled LES. Integrated loads (lift, drag, and moment coefficients) and pressure distributions are compared against experimental measurements in both clean and iced conditions for several angles of attack in both pre-and post-stall regions. The relevant comparisons to the experimental results show that qualitative and acceptable quantitative agreement with the data is observed across all geometries. Glaze ice formations exhibit larger and highly nonuniform ice features, such as `horns', in contrast to rime ice formations characterized by smaller, uniformly distributed roughness elements. In wall-modeled LES, it was observed that larger roughness scales in the glaze ice that trigger transition can be accurately resolved. Therefore, it is possible for WMLES to accurately capture the aerodynamics of glaze ice shapes without the need for additional modeling. In contrast, rime ice geometries required additional resolution to accurately represent the aerodynamic loads. This study demonstrates the effectiveness of the wall-modeled LES technique in simulating the complex aerodynamic effects of iced airfoils, providing valuable insights for aircraft design in icing environments and highlighting the importance of accurately representing ice geometries and roughness scales in simulations.

In this work, a localized artificial-viscosity/diffusivity method is proposed for accurately capturing discontinuities in compressible flows. There have been numerous efforts to improve the artificial diffusivity formulation in the last two decades, through appropriate localization of the artificial bulk viscosity for capturing shocks. However, for capturing contact discontinuities, either a density or internal energy variable is used as a detector. An issue with this sensor is that it not only detects contact discontinuities, but also falsely detects the regions of shocks and vortical motions. Using this detector to add artificial mass/thermal diffusivity for capturing contact discontinuities is hence unnecessarily dissipative. To overcome this issue, we propose a sensor similar to the Ducros sensor (for shocks) to detect contact discontinuities, and further localize artificial mass/thermal diffusivity for capturing contact discontinuities. The proposed method contains coefficients that are less sensitive to the choice of the flow problem. This is achieved by improved localization of the artificial diffusivity in the present method. A discretely consistent dissipative flux formulation is presented and is coupled with a robust low-dissipative scheme, which eliminates the need for filtering the solution variables. The proposed method also does not require filtering for the discontinuity detector/sensor functions, which is typically done to smear out the artificial fluid properties and obtain stable solutions. Hence, the challenges associated with extending the filtering procedure for unstructured grids is eliminated, thereby, making the proposed method easily applicable for unstructured grids. Finally, a straightforward extension of the proposed method to two-phase flows is also presented., Comment: 24 pages, 11 figures, Under review in the Physical Review Fluids journal

The predictive capability of two way--coupled point-particle Euler-Lagrange model in accurately capturing particle-flow interactions under grid refinement, wherein the particle size can be comparable to the grid size, is systematically evaluated. Two situations are considered, (i) uniform flow over a stationary particle, and (ii) decaying isotropic turbulence laden with Kolmogorov-scale particles. Particle-fluid interactions are modeled using only the standard drag law, typical of large density-ratio systems. A zonal, advection-diffusion-reaction (Zonal-ADR) model is used to obtain the undisturbed fluid velocity needed in the drag closure. Two main types of interpolation kernels, grid-based and particle size--based, are employed. The effect of interpolation kernels on capturing the particle-fluid interactions, kinetic energy, dissipation rate, and particle acceleration statistics are evaluated in detail. It is shown that the interpolation kernels whose width scales with the particle size perform significantly better under grid refinement than kernels whose width scales with the grid size. Convergence with respect to spatial resolution is obtained with the particle size--based kernels with and without correcting for the self-disturbance effect. While the use of particle size--based interpolation kernels provide spatial convergence and perform better than kernels that scale based on grid size, small differences can still be seen in the converged results with and without correcting for the particle self-disturbance. Such differences indicate the need for self-disturbance correction to obtain the best results, especially when the particles are larger than the grid size., Comment: Submitted to International Journal of Multiphase Flow

Large-amplitude current-driven instabilities in hollow cathode plumes can generate energetic ions responsible for cathode sputtering and spacecraft degradation. A 2D2V (two dimensions each in configuration [D] and velocity [V] spaces) grid-based Vlasov--Poisson (direct kinetic) solver is used to study their growth and saturation, which comprises four stages: linear growth, quasilinear resonance, nonlinear fill-in, and saturated turbulence. The linear modal growth rate, nonlinear saturation process, and ion velocity and energy distribution features in the turbulent regime are analyzed. Backstreaming ions are generated for large electron drifts, several ion acoustic periods after the potential field becomes turbulent. Interscale phase-space transfer and locality are analyzed for the Vlasov equation. The multidimensional study sheds light on the interactions between longitudinal and transverse plasma instabilities, as well as the inception of plasma turbulence., Comment: Center for Turbulence Research Proceedings of the Summer Program 2022

Accurate numerical modeling of compressible flows, particularly in the turbulent regime, requires a method that is non-dissipative and stable at high Reynolds ($Re$) numbers. For a compressible flow, it is known that discrete conservation of kinetic energy is not a sufficient condition for numerical stability, unlike in incompressible flows. In this study, we adopt the recently developed conservative diffuse-interface method (Jain, Mani $\&$ Moin, $\textit{J. Comput. Phys.}$, 2020) along with the five-equation model for the simulation of compressible two-phase flows. This method discretely conserves the mass of each phase, momentum, and total energy of the system. We here propose discrete consistency conditions between the numerical fluxes, such that any set of numerical fluxes that satisfy these conditions would not spuriously contribute to the kinetic energy and entropy of the system. We also present a set of numerical fluxes\textemdash which satisfies these consistency conditions\textemdash that results in an exact conservation of kinetic energy and approximate conservation of entropy in the absence of pressure work, viscosity, thermal diffusion effects, and time-discretization errors. Since the model consistently reduces to the single-phase Navier-Stokes system when the properties of the two phases are identical, the proposed consistency conditions and numerical fluxes are also applicable for a broader class of single-phase flows. To this end, we present coarse-grid numerical simulations of compressible single-phase and two-phase turbulent flows at infinite $Re$, to illustrate the stability of the proposed method in canonical test cases, such as an isotropic turbulence and Taylor-Green vortex flows. A higher-resolution simulation of a droplet-laden compressible decaying isotropic turbulence is also presented, and the effect of the presence of droplets on the flow is analyzed., Comment: 36 pages, 14 figures, under review in the journal of computational physics

In this work, we propose a novel phase-field model for the simulation of two-phase flows that is accurate, conservative, bounded, and robust. The proposed model conserves the mass of each of the phases, and results in bounded transport of the volume fraction. We present results from the canonical test cases of a drop advection and a drop in a shear flow, showing significant improvement in the accuracy over the commonly used conservative phase-field method. Moreover, the proposed model imposes a lesser restrictive Courant-Friedrichs-Lewy condition, and hence, is less expensive compared to other conservative phase-field models. We also propose improvements on computation of surface tension forces and show that the proposed improvement significantly reduces the magnitude of spurious velocities at the interface. We also derive a consistent and conservative momentum transport equation for the proposed phase-field model and show that the proposed model when coupled with the consistent momentum transport equation results in discrete conservation of kinetic energy, which is a sufficient condition for the numerical stability of incompressible flows, in the absence of dissipative mechanisms. To illustrate the robustness of the method in simulating high-density ratio turbulent two-phase flows, we present the numerical simulations of high-density ratio droplet-laden isotropic turbulence at finite and infinite Reynolds numbers., Comment: 29 pages, 12 figures

This work describes three diffuse-interface methods for the simulation of immiscible, compressible multiphase fluid flows and elastic-plastic deformation in solids. The first method is the localized-artificial-diffusivity approach of Cook (2007), Subramaniam et al., (2018), and Adler and Lele (2019), in which artificial diffusion terms are added to the conservation equations. The second method is the gradient-form approach that is based on the quasi-conservative method of Shukla et al., (2010) and Tiwari et al., (2013), in which the diffusion and sharpening terms (together called regularization terms) are added to the conservation equations. The third approach is the divergence-form approach that is based on the fully conservative method of Jain et al., (2020), in which the regularization terms are added to the conservation equations. The primary objective of this work is to compare these three methods in terms of their ability to maintain: constant interface thickness; the conservation property; and accurate interface shape for long-time integration. The second objective of this work is to consistently extend these methods to model interfaces between solid materials with strength. To assess and compare the methods, they are used to simulate a wide variety of problems, including (1) advection of an air bubble in water, (2) shock interaction with a helium bubble in air, (3) shock interaction and the collapse of an air bubble in water, and (4) Richtmyer-Meshkov instability of a copper-aluminum interface. The current work focuses on comparing these methods in the limit of relatively coarse grid resolution, which illustrates the true performance of these methods. This is because it is rarely practical to use hundreds of grid points to resolve a single bubble or drop in large-scale simulations of engineering interest., Comment: 37 pages, 12 figures, submission for journal of computational physics

In this article, we propose a novel scalar-transport model for the simulation of scalar quantities in two-phase flows with a phase-field method (diffuse-interface method). In a two-phase flow, the scalar quantities typically have disparate properties in two phases, which results in effective confinement of the scalar quantities in one of the phases, in the time scales of interest. This confinement of the scalars lead to the formation of sharp gradients of the scalar concentration values at the interface, presenting a serious challenge for its numerical simulations. To overcome this challenge, we propose a model for the transport of scalars. The model is discretized using a central-difference scheme, which leads to a non-dissipative implementation that is crucial for the simulation of turbulent flows. Furthermore, the provable strengths of the proposed model are: (a) the model maintains the positivity property of the scalar concentration field, a physical realizability requirement for the simulation of scalars, when the proposed criterion is satisfied, (b) the proposed model is such that the transport of the scalar concentration field is consistent with the transport of the volume fraction field, which results in the enforcement of the effective zero-flux boundary condition for the scalar at the interface; and therefore, prevents the artificial numerical diffusion of the scalar across the interface. Finally, we present numerical simulations using the proposed model in a wide range of two-phase flow regimes, spanning laminar to turbulent flows; and assess: the accuracy and robustness of the model, the validity of the positivity property of the scalar concentration field, and the enforcement of the zero-flux boundary condition for the scalar at the interface., Comment: 38 pages, 24 figures, submitted to the journal of computational physics

In this article, we propose a novel conservative diffuse-interface method for the simulation of immiscible compressible two-phase flows. The proposed method discretely conserves the mass of each phase, momentum and total energy of the system. We use the baseline five-equation model and propose interface-regularization (diffusion--sharpening) terms in such a way that the resulting model maintains the conservative property of the underlying baseline model; and lets us use a central-difference scheme for the discretization of all the operators in the model, which leads to a non-dissipative implementation that is crucial for the simulation of turbulent flows and acoustics. Furthermore, the provable strengths of the proposed model are: (a) the model maintains the boundedness property of the volume fraction field, which is a physical realizability requirement for the simulation of two-phase flows, (b) the proposed model is such that the transport of volume fraction field inherently satisfies the total-variation-diminishing property without having to add any flux limiters that destroy the non-dissipative nature of the scheme, (c) the proposed interface-regularization terms in the model do not spuriously contribute to the kinetic energy of the system and therefore do not affect the non-linear stability of the numerical simulation, and (d) the model is consistent with the second law of thermodynamics. Finally, we present numerical simulations using the model and assess (a) the accuracy of evolution of the interface shape, (b) implementation of surface tension effects, (c) propagation of acoustics and their interaction with material interfaces, (d) the accuracy and robustness of the numerical scheme for simulation of complex high-Reynolds-number flows, and (e) performance and scalability of the method., Comment: 50 pages, 24 figures

Breakup of a liquid jet in a high speed gaseous crossflow finds wide range of engineering and technological applications, especially in the combustors of the gas turbine engines in aerospace industry. In this study, we present volume-of-fluid method based direct numerical simulations of a liquid jet injected into a swirling crossflow of gas. The liquid is injected radially outwards from a central tube to a confined annular space with a swirling gas crossflow. The essential features of the jet breakup involving jet flattening, surface waves and stripping of droplets from the edges of the jet are captured in the simulations. We discuss the effect of swirl on the spray characteristics such as jet trajectory, column breakup-length, and size, shape-factor and velocity distribution of the drops. Drop size increases with swirl and penetration is slightly reduced. Moreover, the trajectory follows an angle (azimuthal) that is smaller than the geometric angle of the swirl at the inlet. Interestingly, we also observe coalescence events downstream of the jet that affect the final droplet size distribution for the geometry considered in this study., Comment: 30 pages, 26 figures, Published in Atomization and Sprays

Soft solids in fluids find wide range of applications in science and engineering, especially in the study of biological tissues and membranes. In this study, an Eulerian finite volume approach has been developed to simulate fully resolved incompressible hyperelastic solids immersed in a fluid. We have adopted the recently developed reference map technique (RMT) by Valkov et. al (J. Appl. Mech., 82, 2015) and assessed multiple improvements for this approach.These modifications maintain the numerical robustness of the solver and allow the simulations without any artificial viscosity in the solid regions (to stabilize the solver). This has also resulted in eliminating the striations ("wrinkles") of the fluid-solid interface that was seen before and hence obviates the need for any additional routines to achieve a smooth interface. An approximate projection method has been used to project the velocity field onto a divergence free field. Cost and accuracy improvements of the modifications on the method have also been discussed.

We simulate the flow of two immiscible and incompressible fluids separated by an interface in a homogeneous turbulent shear flow at a shear Reynolds number equal to 15200. The viscosity and density of the two fluids are equal, and various surface tensions and initial droplet diameters are considered in the present study. We show that the two-phase flow reaches a statistically stationary turbulent state sustained by a non-zero mean turbulent production rate due to the presence of the mean shear. Compared to single-phase flow, we find that the resulting steady state conditions exhibit reduced Taylor microscale Reynolds numbers owing to the presence of the dispersed phase, which acts as a sink of turbulent kinetic energy for the carrier fluid. At steady state, the mean power of surface tension is zero and the turbulent production rate is in balance with the turbulent dissipation rate, with their values being larger than in the reference single-phase case. The interface modifies the energy spectrum by introducing energy at small-scales, with the difference from the single-phase case reducing as the Weber number increases. This is caused by both the number of droplets in the domain and the total surface area increasing monotonically with the Weber number. This reflects also in the droplets size distribution which changes with the Weber number, with the peak of the distribution moving to smaller sizes as the Weber number increases. We show that the Hinze estimate for the maximum droplet size, obtained considering breakup in homogeneous isotropic turbulence, provides an excellent estimate notwithstanding the action of significant coalescence and the presence of a mean shear.

Breakup of liquid drops occurs in several natural and industrial settings. Fully resolved Volume of Fluid based simulations presented in this study reveal the complete flow physics and droplet dynamics that lead to the breakup of a drop in a particular mode. We have investigated the effects of density ratio and Reynolds number on the dynamics of drop deformation and subsequent breakup. A density ratio-Weber number phase plot is presented that indicates the variation in the deformation of the drop at various density ratios and Weber numbers. We show that the breakup dynamics of the droplets at low density ratios is significantly different to that observed at high density ratios. We also study the temporal characteristics of the droplet deformation and motion.