arXiv subjects

Heinz Pitsch

Publications and source records attributed to Heinz Pitsch.

At least 19 recordsLinked to original sources

arXiv2026

Preferential and differential diffusion in RANS simulation of lean hydrogen flames with tabulated chemistry

Lean hydrogen flames are prone to thermo-diffusive instabilities due to preferential and differential diffusion effects, posing significant challenges for their modeling in computational fluid dynamics simulations. This work extends a tabulated-chemistry (TC) model that includes preferential and differential diffusion effects to a Reynolds-averaged Navier-Stokes (RANS) framework and assesses its performance for a lean premixed $\mathrm{H_2}$-air slot burner at two Reynolds numbers ($\mathrm{Re}=5500$ and $11000$) using direct numerical simulation (DNS) as a reference. The approach is based on transport equations for the progress variable and mixture fraction derived from the species mass transport equations considering mixture-averaged diffusion and Soret effect, and incorporates turbulence--chemistry interaction via a presumed probability density function (PDF) approach. RANS simulations including preferential-differential diffusion are able to correctly reproduce the DNS flame length, heat-release distribution, and the characteristic equivalence-ratio and super-adiabatic temperature branches of the slot flame. Comparisons with (i) a unity-Lewis-number variant and (ii) a model including thermo-diffusive effects only in the flamelet table show the impact of preferential and differential diffusion on the TC model at both the thermochemical and transport levels. Finally, the impact of the turbulence closures for turbulent diffusion, scalar dissipation rate, and Reynolds stresses is assessed. The results presented in this paper demonstrate the capability of the model to include preferential and differential diffusion effects in cost-effective RANS simulations of lean hydrogen flames.

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arXiv2026

Modeling of Reaction Dynamics in a Turbulent Hydrogen-Air Slot Flame Using Resolvent Analysis

This work applies Resolvent Analysis (RA) to study the dynamics of a hydrogen-air slot flame with a Reynolds number of 5500, a Karlovitz number of 20, and an equivalence ratio of 0.4. Direct Numerical Simulations (DNS) data are analyzed using shifted Spectral Proper Orthogonal Decomposition (SPOD), and the resulting structures are compared with optimal resolvent responses obtained from the linearization of a RANS-EBU reaction rate model. Both SPOD and RA show that the flow dynamics are dominated by Kelvin-Helmholtz wave packets over a broad frequency range, particularly between 300 and 1000 Hz. This behavior is reflected in the resolvent gains and SPOD eigenvalues, which exhibit consistent amplification within this range. The velocity fluctuation mode shapes predicted by RA agree well with the SPOD modes. However, the corresponding mode shapes for the progress variable and heat release show weaker agreement. To address this limitation, the study introduces a generalized active-flame closure calibrated with high-fidelity data, which remains compatible with the linearized framework and improves the agreement with SPOD modes. Overall, the results indicate that thermodiffusive instabilities in turbulent hydrogen flames do not hinder the applicability of the active-flame resolvent approach.

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arXiv2026

Physics-guided laminar flame speed correlation for methane-hydrogen-air mixtures with varying dilution

Fuel-flexible, low-carbon combustion systems need to accommodate methane/hydrogen mixtures with air and exhaust-gas dilution. To develop these, we require accurate and efficient correlations for laminar flame speed (LFS). In this work, we introduce a physics-guided LFS correlation that applies to burners, gas engines, and turbines. Our model uses a core-kinetic approach based on flame temperatures, an algebraic function for the equivalence ratio, and a mass-flux-based blending law. This allows for accurate predictions with any methane/hydrogen blend. We set the model parameters using one-dimensional flame simulations with C3Mech v4.0.1, chosen for its high prediction accuracy for a wide range of experimental data, including new results from our spherical combustion chamber. The new correlation provides accuracy comparable to a machine learning approach (Gaussian process regression), yet remains physically consistent, differentiable, and extrapolates well. This makes it suitable for computational fluid dynamics and control of fuel-flexible combustion systems.

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arXiv2026

aPriori: a Python package to process direct numerical simulations

In the field of computational fluid dynamics, direct numerical simulations generate highly detailed data for the analysis of turbulent flows by resolving all relevant physical scales. Yet their large size, complexity, and heterogeneity make systematic post-processing and data reuse increasingly challenging. Despite the growing availability of high-fidelity simulations through public repositories, extracting meaningful physical insight often requires substantial technical effort, specialized workflows, and access to high-performance computing resources. In this article we introduce \texttt{aPriori}, an open-source Python package developed to address these limitations by providing a dedicated, memory-efficient, and user-oriented framework for the analysis of direct numerical simulation data. The software enables streamlined handling of three-dimensional fields, including filtering, scale separation, gradient evaluation, thermochemical analysis, and visualization, using concise and reproducible scripts. Its pointer-based data management strategy allows very large datasets to be processed on standard workstations without excessive memory usage, significantly lowering the barrier to advanced analysis. Beyond basic post-processing, \texttt{aPriori} supports workflows central to modern turbulence and combustion research, such as a priori model assessment, data-driven closure development, and detailed chemical analyses that include computational singular perturbation. By unifying these capabilities within a coherent and extensible software architecture, \texttt{aPriori} enhances productivity, promotes reproducibility, and facilitates broader and more effective use of high-fidelity simulation data within the computational fluid dynamics community.

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arXiv2026

Direct Numerical Simulation of MILD Combustion: Mixing and Autoignition from Non-Premixed Streams

Moderate or intense low-oxygen dilution (MILD) combustion is achieved by strongly diluting and preheating the reactants through mixing with hot combustion products before ignition. To better understand how fuel/air/product mixing and interaction govern MILD combustion dynamics, a novel direct numerical simulation (DNS) dataset of a temporally evolving three-stream mixing layer consisting of fuel, air, and hot combustion products has been performed. In this configuration, both fuel-air and air-hot products mixing processes are considered with varying time scales, through four carefully designed DNS cases, to assess how their combined interaction controls ignition under MILD conditions. It is observed that the cases with higher dilution levels fall within the MILD combustion regime, whereas those with lower dilution correspond to non-MILD conditions. The results show that, as long as MILD conditions are observed, ignition is mainly driven by mixing with hot products. Flame index (FI) combined with chemical explosive mode analysis (CEMA) further identifies the local combustion mode: in MILD cases, ignition occurs predominantly through a premixed-autoignition mode, while in non-MILD scenarios, the premixed-deflagrative contribution to the heat release rate is more substantial. Conditional analysis of scalar dissipation rates shows that the combustion modes in MILD conditions are sensitive to mixing by both the fuel and hot products, whereas the combustion modes in non-MILD conditions are mainly influenced by the mixing of the fuel with the surrounding gases.

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arXiv2026

Unified scaling and shape laws for turbulent premixed methane and hydrogen jet flames

The scaling of turbulent premixed flames is typically described by correlations derived for unity-Lewis-number fuels. However, their validity for hydrogen (H$_{2}$) remains uncertain due to the thermodiffusive effects associated with its low Lewis number. In this study, turbulent premixed H$_{2}$ and methane (CH$_{4}$) jet flames are systematically compared over a wide range of operating conditions. Experiments were conducted for Reynolds numbers between 5000 and 60000 and effective Karlovitz numbers spanning 3-368. Flame structure and global flame geometry were characterized using spatially resolved OH$^{*}$ chemiluminescence imaging, allowing consistent comparison between the two fuels across different turbulence intensities. The results are interpreted via a unified framework that incorporates two thermodynamic- and fuel-dependent parameters: a flame speed factor, $\alpha$, representing the enhancement of local burning rates, and a shape factor, $\gamma$, describing the scaling of mean flame geometry. Despite significant fuel-specific thermodiffusive effects associated with preferential diffusion and intrinsic reactivity, which lead H$_{2}$ flames to exhibit enhanced sensitivity to turbulence and more compact flame configurations, both H$_{2}$ and CH$_{4}$ flames are found to exhibit robust and consistent turbulent scaling behavior when analysed within the proposed unified framework. The resulting correlations provide a generalised description of turbulent burning velocity and flame structure, demonstrating that key turbulence-chemistry interactions can be captured within a common model across fuels with widely different Lewis numbers. Overall, the dataset spans multiple turbulence regimes and flame geometries for both fuels, providing a valuable experimental benchmark for the validation of turbulent combustion models across different regimes.

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arXiv2026

Large-eddy simulations of a lean hydrogen premixed turbulent jet flame with tabulated chemistry

Large-eddy simulations (LES) of a planar turbulent lean hydrogen-air jet flame at Re = 11000 are performed using a tabulated flamelet model based on mixture-averaged diffusion that incorporates detailed transport, including differential and preferential diffusion, wall heat loss, and thermodiffusion. The approach is extended to turbulent combustion in LES using a presumed-shape probability density function formulation that accounts for sub-filter effects. The flame exhibits a highly corrugated front, driven by local variations of mixture fraction induced by strong thermodiffusive transport. These effects significantly alter both the flame structure and morphology. The LES results are systematically compared to a reference direct numerical simulation across varying LES filters through different mesh resolutions to evaluate the predictive capability of the model. The LES accurately reproduces instantaneous flow structures and thermodiffusive effects. Global flame characteristics including flame length, surface area, and consumption speed, are well captured and show limited sensitivity to mesh resolution. The role of thermodiffusion is also examined, showing that its incorporation leads to a more reactive flame and should not be neglected in the formulation. Heat losses are incorporated into the tabulated chemistry framework for completeness but are found to have a negligible impact, consistent with the walls weak influence in the present configuration. Overall, the results demonstrate that the proposed approach provides reliable predictions of the main flame characteristics, with remaining discrepancies primarily associated with unresolved sub-filter effects that deserve further investigation.

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arXiv2025

Dynamic mixed turbulence modeling using a super-resolution generative adversarial approach

A dynamic mixed super-resolution model (DMSRM) for large-eddy simulations (LESs) is proposed, which combines the traditional dynamic mixed model (DMM) formulation with the generation of super-resolved velocity fields from which the subfilter-scale (SFS) stress tensor can be computed. A data-driven super-resolution generative adversarial network (SR-GAN) is employed to upsample the grid-filtered velocity fields by a factor of two, enabling the evaluation of both scale-similarity and the dynamic Smagorinsky contributions. A priori analyses of forced homogeneous isotropic turbulence show that the SR-GAN accurately reconstructs fine-scale flow features and generalizes well across different filter sizes and higher Reynolds number flow configurations, even for unseen input fields. The DMSRM reproduces SFS stresses and energy dissipation more accurately than the traditional DMM. A posteriori LES calculations further confirm that DMSRM predicts the energy spectrum and intermittency more accurately than DMM, even for different LES grid-scale resolutions and for higher Reynolds numbers than those used for training. Unlike DMM, DMSRM yields realistic backscatter and physically consistent SFS energy dissipation. These improvements arise from the physically accurate super-resolved fields generated by the SR-GAN, from which SFS stresses are directly computed. The result is a closure that accurately reproduces stress magnitudes and dissipation while reducing reliance on additional dissipation from the dynamic term. The DMSRM formulation achieves a balance of physical fidelity, robustness, and computational efficiency, offering a promising alternative to traditional DMMs for turbulence LES modeling.

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arXiv2025

Role of enthalpy transport in laminar premixed hydrogen flames at atmospheric and elevated pressures

This work discusses the role of diffusive enthalpy transport in relation to the origin of thermodiffusive instability and the resulting enhanced reactivity. Thermodiffusive effects in premixed hydrogen flames are typically explained and modelled via local equivalence ratio fluctuations. However, it is reiterated here that the imbalance between species and thermal diffusion (differential diffusion), rather than local species-to-species diffusive imbalances (preferential diffusion) is the leading-order effect. Reactant (H$_2$), product (H$_2$O) and intermediate (H) species are demonstrated to all play a role in the transport of enthalpy through an analysis of enthalpy flux divergence terms in unstretched flames. Premixed counterflow flames at various strain rates and pressures are then analysed to demonstrate that enhanced reactivity originates from a combination of enthalpy transport and the broadness of the reaction zone relative to the thickness of the flame. Effects resulting from key pressure fall-off reactions are also discussed to determine the importance of detailed chemistry, and the usage of Zeldovich number. Finally, two-dimensional planar flames are simulated and analysed to demonstrate the role of curvature in addition to strain rate, and the implications of the findings in blends and turbulent flames are discussed.

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arXiv2025

Effects of Intrinsic Flame Instabilities on Nitrogen Oxide Formation in Laminar Premixed Ammonia/Hydrogen/Air Flames

This study investigates the characteristics of nitrogen oxide (NO) formation in two-dimensional (2D) laminar premixed ammonia/hydrogen/air flames and the impact of thermodiffusively driven intrinsic flame instabilities (IFIs). To this end, a set of three highly resolved direct numerical simulations (DNS) at lean ambient conditions and varying hydrogen fraction in the fuel blend are conducted. The analysis of these DNS reveals a significant increase of NO formation in positively curved regions of the flame, particularly for lower hydrogen fuel fractions, while negatively curved areas exhibit reduced NO concentrations. However, despite the strong variations of local mass fractions of NO in the flame sheet, the mean mass fraction in the post-flame region remains close to the solution from a one-dimensional flame. Through a representative flame segment analysis of positively curved, negatively curved, and flat regions, key reactions contributing to NO formation are determined, with the HNO pathway being the predominant production and the deNOx pathway being the predominant consumption pathway across all cases. Thermal NO plays no significant role in the considered cases. Generally, the peaks of NO production shift to lower values of progress variable in the negatively curved regions, leading to an annihilation of the production and consumption terms in the low hydrogen fuel fraction case. The decrease of NO production is found to be mainly driven by changes of the radical concentrations, rather than changes of the temperature-dependent reaction rate coefficients.

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arXiv2024

ZipGAN: Super-Resolution-based Generative Adversarial Network Framework for Data Compression of Direct Numerical Simulations

The advancement of high-performance computing has enabled the generation of large direct numerical simulation (DNS) datasets of turbulent flows, driving the need for efficient compression/decompression techniques that reduce storage demands while maintaining fidelity. Traditional methods, such as the discrete wavelet transform (DWT), cannot achieve compression ratios of 8 or higher for complex turbulent flows without introducing significant encoding/decoding errors. On the other hand, a super-resolution-based generative adversarial network (SR-GAN), called ZipGAN, can accurately reconstruct fine-scale features, preserving velocity gradients and structural details, even at a compression ratio of 512, thanks to the more efficient representation of the data in a compact latent space. Additional benefits are ascribed to adversarial training. The high GAN training time is significantly reduced with a progressive transfer learning approach and, once trained, they can be applied independently of the Reynolds number. It is demonstrated that ZipGAN can enhance dataset temporal resolution without additional simulation overhead by generating high-quality intermediate fields from compressed snapshots. The ZipGAN discriminator can reliably evaluate the quality of decoded fields, ensuring fidelity even in the absence of original DNS fields. Hence, ZipGAN compression/decompression method presents a highly efficient and scalable alternative for large-scale DNS storage and transfer, offering substantial advantages over the DWT methods in terms of compression efficiency, reconstruction fidelity, and temporal resolution enhancement.

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arXiv2024

Comprehensive linear stability analysis for intrinsic instabilities in premixed ammonia/hydrogen/air flames

Two-dimensional direct numerical simulations of planar laminar premixed ammonia/hydrogen/air flames are conducted for a wide range of equivalence ratios, hydrogen ($\rm H_2$) fractions in the fuel blend, pressures, and unburned temperatures to study intrinsic flame instabilities (IFIs) in the linear regime. For stoichiometric and lean mixtures at ambient conditions, a non-monotonic behavior of thermo-diffusive instabilities with increasing ($\rm H_2$) fraction is observed. Strongest instabilities occur for molar ($\rm H_2$) fractions of 40%. The analysis shows that this behavior is linked to the joint effect of variations of the effective Lewis number and Zeldovich number. IFIs in ammonia/hydrogen blends further show a non-monotonic trend with respect to pressure, which is found to be linked to the chemistry of the hydroperoxyl radical $\rm HO_2$. The addition of $\rm NH_3$ opens new reaction pathways for the consumption of $\rm HO_2$ resulting in a chain carrying behavior in contrast to its chain terminating nature in pure $\rm H_2$/air flames. Theoretically derived dispersion relations can predict the non-monotonic behavior for lean conditions. However, these are found to be sensitive to the different methods for evaluating the Zeldovich number available in the literature.

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arXiv2023

Three-dimensional numerical investigation of flashback in premixed hydrogen flames within perforated burners

Predicting flashback represents a pivotal challenge in the development of innovative perforated burners for household appliances, especially for substituting natural gas with hydrogen as fuel. Most existing numerical studies have utilized two-dimensional (2D) simulations to investigate flashback in these burners, primarily to reduce computational costs. However, the inherent complexity of flashback phenomena suggests that 2D simulations may inadequately capture the flame dynamics, potentially leading to inaccurate estimations of flashback limits. In this study, three-dimensional (3D) simulations are employed to examine the impact of the actual slit shapes on the flashback velocities of hydrogen-premixed flames. Steady-state simulations are conducted to compute flashback velocities for three equivalence ratios ($\phi=0.6$, $0.8$, and $1.0$), investigating slits with fixed width $W$ and varying length $L$. Additionally, transient simulations are performed to investigate the flashback dynamics. The results are compared with those from 2D configurations to assess the reliability of the infinite slit approximation. For stable flames, 2D simulations underpredict the burner plate temperature compared to slits with lengths typical of practical devices but match the 3D results as $L\to\infty$. Conversely, flashback velocities are consistently underpredicted in 2D simulations compared to 3D simulations, even as $L\to\infty$. This is due to the critical role of the slit ends in flashback dynamics, where favorable aerodynamics, preferential diffusion, the Soret effect, and higher preheating due to a higher surface-to-volume ratio trigger the initiation of flashback in those regions. These findings underscore the necessity of employing 3D simulations to accurately estimate the flashback velocities in domestic perforated burners.

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arXiv2023

Scalar mass conservation in turbulent mixture fraction based combustion models through consistent local flow parameters

Mixture fraction-based models are widely employed for predicting turbulent non-premixed combustion processes due to their cost-effectiveness and well-established subfilter closure. In these models, the transport of reactive scalars in physical space is decomposed into two components: scalar transport relative to mixture fraction and transport of mixture fraction in physical space. Conventional flamelet models do not consider that these two processes have to be formulated consistently, which can lead to scalar mass conservation errors. In the context of multiphase flows, scalar transport in mixture fraction space is governed by three conditional flow-dependent parameters: the conditional scalar dissipation rate, the conditional scalar diffusion rate, and the conditional spray source term. The evolution of mixture fraction in physical space is typically modeled using the presumed Filtered Density Function (FDF) approach. This paper introduces a novel formulation for the conditional flow parameters that aligns with the presumed FDF approach, thereby ensuring scalar mass conservation. The proposed model is applied to a Large-Eddy Simulation (LES) of the inert ECN Spray A case, with a comparison against a conventional flow parameter model that employs an inverse error function shape for the scalar dissipation rate. The results indicate that the conventional model produces similar conditional dissipation rates to the new model in regions where combustion takes place. However, significant discrepancies are observed in the conditional diffusion rate, highlighting the susceptibility of the conventional model to scalar mass conservation errors for non-unity Lewis number scalars.

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arXiv2023

Influence of adversarial training on super-resolution turbulence reconstruction

Supervised super-resolution deep convolutional neural networks (CNNs) have gained significant attention for their potential in reconstructing velocity and scalar fields in turbulent flows. Despite their popularity, CNNs currently lack the ability to accurately produce high-frequency and small-scale features, and tests of their generalizability to out-of-sample flows are not widespread. Generative adversarial networks (GANs), which consist of two distinct neural networks (NNs), a generator and discriminator, are a promising alternative, allowing for both semi-supervised and unsupervised training. The difference in the flow fields produced by these two NN architectures has not been thoroughly investigated, and a comprehensive understanding of the discriminator's role has yet to be developed. This study assesses the effectiveness of the unsupervised adversarial training in GANs for turbulence reconstruction in forced homogeneous isotropic turbulence. GAN-based architectures are found to outperform supervised CNNs for turbulent flow reconstruction for in-sample cases. The reconstruction accuracy of both architectures diminishes for out-of-sample cases, though the GAN's discriminator network significantly improves the generator's out-of-sample robustness using either an additional unsupervised training step with large eddy simulation input fields and a dynamic selection of the most suitable upsampling factor. These enhance the generator's ability to reconstruct small-scale gradients, turbulence intermittency, and velocity-gradient probability density functions. The extrapolation capability of the GAN-based model is demonstrated for out-of-sample flows at higher Reynolds numbers. Based on these findings, incorporating discriminator-based training is recommended to enhance the reconstruction capability of super-resolution CNNs.

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arXiv2023

Physics-based reduced-order modeling of flash-boiling sprays in the context of internal combustion engines

Flash-boiling injection is one of the most effective ways to accomplish improved atomization compared to the high-pressure injection strategy. The tiny droplets formed via flash-boiling lead to fast fuel-air mixing and can subsequently improve combustion performance in engines. Most of the previous studies related to the topic focused on modeling flash-boiling sprays using three-dimensional (3D) computational fluid dynamics (CFD) techniques such as direct numerical simulations (DNS), large-eddy simulations (LES), and Reynolds-averaged Navier-Stokes (RANS) simulations. However, reduced order models can have significant advantages for applications such as the design of experiments, screening novel fuel candidates, and creating digital twins, for instance, because of the lower computational cost. In this study, the previously developed cross-sectionally averaged spray (CAS) model is thus extended for use in simulations of flash-boiling sprays. The present CAS model incorporates several physical submodels in flash-boiling sprays such as those for air entrainment, drag, superheated droplet evaporation, flash-boiling induced breakup, and aerodynamic breakup models. The CAS model is then applied to different fuels to investigate macroscopic spray characteristics such as liquid and vapor penetration lengths under flash-boiling conditions. It is found that the newly developed CAS model captures the trends in global flash-boiling spray characteristics reasonably well for different operating conditions and fuels. Moreover, the CAS model is shown to be faster by up to four orders of magnitude compared with simulations of 3D flash-boiling sprays. The model can be useful for many practical applications as a reduced-order flash-boiling model to perform low-cost computational representations of higher-order complex phenomena.

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arXiv2023

Efficient and accurate calculation of dispersion relations for intrinsically unstable premixed flames

Premixed flames are susceptible to hydrodynamic and thermodiffusive instabilities that wrinkle the flame front and lead to complex multiscale patterns. They strongly impact the flame propagation and dynamics, increasing the speed of a laminar flame by several folds, easily as large as a factor of five for lean hydrogen flames at high pressure. The dispersion relation, which represents the growth rate of the different harmonic components of the perturbation of the flame front for different wavelengths, is useful to understand the dynamics during the linear phase of flame instabilities. In this work, an efficient and accurate approach based on a Fourier analysis of flame wrinkling is proposed to calculate the dispersion relation. Differently from the typical approach based on perturbing the flame with a single wavelength, the flame is perturbed with a spectrum of sine waves and their growth is followed with a spectral analysis. With the present method, the full dispersion relation is computed with a single simulation; this is significantly more efficient computationally than running a series of simulations with a single-wavelength perturbation for each point of the dispersion relation. It is shown that the presented approach is accurate and also solves an issue encountered when a single perturbation is imposed to compute the growth rate of large wavelengths. Several numerical and initialisation parameters, including resolution, domain size, and amplitude of the initial perturbation, are studied systematically and assessed.

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arXiv2023

Dimensional Analysis of Vapor Bubble Growth Considering Bubble-bubble Interactions in Flash Boiling Microdroplets of Highly Volatile Liquid Electrofuels

Electrofuels (e-fuels) produced from renewable electricity and carbon sources have gained significant attention in recent years as promising alternatives to fossil fuels for the transportation sector. However, the highly volatile e-fuels, such as short-chain oxymethylene ethers are prone to flash vaporization phenomena, which is associated with the formation and growth of vapor bubbles, followed by explosive bursting of the liquid jet. The simulation of a flash boiling spray of such highly volatile liquid fuels in the context of automotive or cryogenic engines is numerically challenging due to several reasons, including (1) the complexity of the bubble growth process in the presence of multiple vapor bubbles and (2) the need to use an extremely small time step size to accurately capture the underlying physics associated with the flash boiling process. In this paper, we first present a bubble growth model in flash boiling microdroplets considering bubble interactions along with the finite droplet size effects. Based on the dimensional analysis of the newly derived Rayleigh Plesset equation, a simplified semi-analytical solution for bubble growth, which also includes the bubble interactions, is then derived to estimate the bubble growth behavior with reasonable accuracy using the larger time step sizes for a wide range of operating conditions. The derived semi-analytical solution is shown to be a good approximation for describing the bubble growth rate over the whole lifetime of the bubble. The bubble interactions are found to delay the onset of droplet bursting due to the slower growth of the vapor bubble compared to the bubble growth without bubble interactions. Furthermore, in a comparison with DNS results, the proposed bubble growth model is shown to reasonably capture the impact of bubble interactions leading to smaller volumetric droplet expansion.

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