The collision kernel of droplets in warm clouds is a crucially important quantity for the parametrization of precipitation in weather and climate models. Nevertheless, its accurate representation remains a challenge, specifically in the bottleneck range 15 µ m < r < 40 µ m , within which turbulence is believed to substantially contribute to droplet growth. In this work, we address this problem by performing direct numerical simulations of polydisperse inertial particles suspended in three-dimensional turbulence at Reynolds number up to Re λ = 418 . Collision statistics are compiled for droplet pairs across the Stokes number range St ∈ [ 0.02 , 2 ] , yielding comprehensive bidisperse maps of collision kernels, radial relative velocities, and radial distribution functions at contact. Our analysis reveals that polydispersity enhances collisions between light droplets through differential sampling, but attenuates collisions at larger Stokes numbers by rapidly reducing the spatial overlap of droplet clusters. By benchmarking existing models, we show that dominant bidisperse errors arise from overpredicted cross-species clustering. In light of these results, we propose an adapted model for the bidisperse radial distribution function, as well as a novel parametrization for the associated collision kernel, applicable to the smallest droplets in the bottleneck range with small settling velocities. Finally, we study the broadening of the droplet size distribution due to collision-coalescence and demonstrate that droplet growth is markedly accelerated in parcels of large local dissipation rate, supporting the hypothesis that turbulent intermittency may help overcome the bottleneck barrier.

This work presents an efficient statistical model to simulate expected scalar transport in fractured porous media below the representative elementary volume scale. We focus on embedded, highly conductive, isolated fractures. The statistical integro-differential fracture model (Sid-FM) solves for ensemble-averaged solutions directly, avoiding computationally expensive Monte Carlo simulation. The expected fluid exchange between isolated fractures and the porous matrix is modelled via a non-local kernel function, leading to a set of integro-differential equations. The model is validated against reference data from Monte Carlo simulations for statistically one-dimensional test cases and shows good agreement.

We advance the understanding of inertial clustering and the role of sling events in high-Reynolds number (Re) particle-laden turbulence. To this end, we perform one-way coupled particle tracking in flow fields obtained from direct numerical simulations (DNS) of forced homogeneous isotropic turbulence. Additionally, we examine the impact of filtering utilized in large eddy simulations by applying a sharp spectral filter to the DNS fields. Our analysis reveals that while instantaneous clustering through the centrifuge mechanism explains clustering at early times, the path history effect—the sampling of fluid flow along particle trajectories—becomes important later on. The filtered fields expose small-scale fractal clustering that cannot be predicted by the instantaneous flow field. We show that there exists a filter-effective Stokes number that governs the degree of fractal clustering and preferential sampling, revealing scale-similarity in the spatial distributions and fractal dimensions. Sling events are prevalent throughout our simulations and impose prominent patterns on the particle fields. In pursuit of investigating the sling dynamics, we compute the relative velocity, averaged over proximal neighboring particles, to identify particles undergoing caustics. As postulated in recent theories, we find that in fully resolved, high-Re turbulence, sling events occur in thin sheets of high strain, situated between turbulent vortices. This behavior is driven by rare, extreme events of compressive straining, manifested by characteristic fluctuations of the flow velocity gradients.

Understanding how bubbles on a substrate respond to ultrasound is crucial for applications from industrial cleaning to biomedical treatments. Under ultrasonic excitation, bubbles can undergo shape deformations due to Faraday instability, periodically producing high-speed jets that may cause damage. While recent studies have begun to elucidate this behaviour for free bubbles, the dynamics of wall-attached bubbles is still largely unexplored. In particular, the selection and evolution of non-spherical modes in these bounded systems have not previously been resolved in three dimensions, and the resulting jetting dynamics has yet to be compared with that observed in free bubbles. In this study, we investigate individual micrometric air bubbles in contact with a rigid substrate and subjected to ultrasound. We introduce a novel dual-view imaging technique that combines top-view bright-field microscopy with side-view phase-contrast X-ray imaging, enabling visualisation of bubble shape evolution from two orthogonal perspectives. This set-up reveals the progression of bubble shape through four distinct dynamic regimes: purely spherical oscillations, onset of harmonic axisymmetric meniscus waves, emergence of half-harmonic axisymmetric Faraday waves and the superposition of half-harmonic sectoral Faraday waves. This stepwise evolution contrasts with the behaviour of free bubbles, which exhibit their ultimate Faraday wave pattern immediately upon instability onset. For the substrate chosen, the resulting shape-mode spectrum appears to be degenerate and exhibits a continuous range of shape mode degrees, in line with our theoretical predictions derived from kinematic arguments. While free bubbles also display a degenerate spectrum, their shape mode degrees remain discrete, constrained by the bubble spherical periodicity. Experimentally measured ultrasound pressure thresholds for the onset of Faraday instability agree well with classical interface stability theory, modified to incorporate the effects of a rigid boundary. Complementary three-dimensional boundary element simulations of bubble shape evolution align closely with experimental observations, validating this method's predictive capability. Finally, we determine the acceleration threshold at which shape mode lobes initiate cyclic jetting. Unlike free bubbles, jetting in wall-attached bubbles consistently emerges from the side not restricted by the substrate.

We study the role of turbulence in enhancing the collisions of droplets within the cloud droplet bottleneck range. To this end, we perform direct numerical simulations of polydisperse inertial particles suspended in three-dimensional turbulence at high Reynolds number. Our analysis reveals that polydispersity enhances collisions of light droplets, but attenuates collisions at larger Stokes numbers by reducing the degree of inertial clustering and weakening the sling effect. We present a novel parameterization for the bidisperse collision kernel which blends the mechanisms related to inertial clustering and the sling effect with polydispersity. We then demonstrate that the droplet size distribution of typical cloud droplets is effectively broadened by turbulence and that droplet growth may be accelerated significantly by turbulent intermittency–a condition that the here provided theoretical contemplations in conjunction with recent measurements in clouds suggest is likely.

Random walk-based particle methods, wherein particles serve as statistical surrogates of fluid volumes, enable incorporation of pore-scale physics into macroscopic models. This work focuses on developing a random walk particle-tracking framework to model multiphase flow and transport in fractured porous media with explicit representations of fractures embedded in a permeable matrix. Specifically, the interest lies in macroscopic multiphase flow with purely advective phase-transport, e.g., in the absence of a macroscopic capillary pressure difference. The key challenge arises from handling discontinuous saturation fronts, and the typically advantageous absence of numerical dispersion in particle schemes does not provide assistance. In this regard, we selectively add artificial diffusion to the system, which regularizes the conservation law governing the evolution of a phase-saturation. The resulting smoother fronts facilitate an efficient coupling of the particle distributions and the Eulerian fields, which influence the fluid flow, through simple 'density estimation' techniques. An explicit coupling between the two is a necessary step for the non-linear problem under consideration. The particle-tracking scheme, incorporating a proposed adaptive diffusion coefficient, is demonstrated to simulate sharp saturation profiles of a 1-D Buckley-Leverett problem. Additionally, the diffusion coefficient is extended for 2-D domains, and anisotropic and isotropic variants of the diffusion coefficient are evaluated for various flow scenarios, including gravitational effects and matrix heterogeneity. Subsequently, we present the particle-based results within the context of a realistic 2-D fracture network embedded in a permeable matrix. The developed framework provides a platform to model non-linear dynamics originating from physical sub-grid phenomena, e.g., dissolution of one phase into the other or precipitation-dissolution type reactions, by means of simple stochastic processes.

Flow in fractured porous media is associated with high uncertainty, particularly regarding fracture properties and their overall configuration within the domain. This is especially pronounced for disconnected fractures of smaller yet comparable size to the domain. Consequently, ensemble averages are often used to capture this statistical variability and predict the expected behavior. This leads to enormous computational costs, as flow simulations of single realizations with millions of fractures are extremely expensive; and much more so full Monte Carlo studies involving hundreds of realizations. Alternatively, a recently introduced model aims to directly estimate expected flow rates and pressure fields. The model involves few degrees of freedom, leading to low-cost computations. This is achieved by using integro-differential equations involving non-local kernel functions that encompass the statistical information of fractures. So far this statistical integro-differential fracture model (Sid-FM) considers only ensembles with identical fractures having constant aperture and lengths. In this paper Sid-FM is extended to account for arbitrary fracture aperture profiles and reservoirs with fractures following specified length distributions, which is a crucial step towards applications with realistic fractured reservoirs. In a series of numerical experiments, it is demonstrated that the Sid-FM's predictions are in excellent agreement with Monte Carlo reference data, which are based on many fracture-resolving simulations. The applicability is demonstrated through statistically one-dimensional cases, laying crucial groundwork for 2D and 3D extensions. Future work will focus on further generalizations and extensions such as transport processes and 2D/3D applications.

In this work we focus on expected flow in porous formations with highly conductive isolated fractures, which are of non-negligible length compared with the scales of interest. Accordingly, the definition of a representative elementary volume (REV) for flow and transport predictions may not be possible. Recently, a non-local kernel-based theory for flow in such formations has been proposed. There, fracture properties like their expected pressure are represented as field quantities. Unlike existing models, where fractures are assumed to be small compared with the scale of interest, a non-local kernel function is used to quantify the expected flow transfer between a point in the fracture domain and a potentially distant point in the matrix continuum. The transfer coefficient implied by the kernel is a function of the fracture characteristics that are in turn captured statistically. So far the model has successfully been applied for statistically homogeneous cases. In the present work we demonstrate the applicability for heterogeneous cases with spatially varying fracture statistics. Moreover, a scaling law is presented that relates the transfer coefficient to the fracture characteristics. Test cases involving discontinuously and continuously varying fracture statistics are presented, and the validity of the scaling law is demonstrated.

Purpose The purposes of this study were to determine whether biomechanical properties of mature oocytes could predict usable blastocyst formation better than morphological information or maternal factors, and to demonstrate the safety of the aspiration measurement procedure used to determine the biomechanical properties of oocytes. Methods A prospective split cohort study was conducted with patients from two IVF clinics who underwent in vitro fertilization. Each patient’s oocytes were randomly divided into a measurement group and a control group. The aspiration depth into a micropipette was measured, and the biomechanical properties were derived. Oocyte fertilization, day 3 morphology, and blastocyst development were observed and compared between measured and unmeasured cohorts. A predictive classifier was trained to predict usable blastocyst formation and compared to the predictions of four experienced embryologists. Results 68 patients and their corresponding 1252 oocytes were included in the study. In the safety analyses, there was no significant difference between the cohorts for fertilization, while the day 3 and 5 embryo development were not negatively affected. Four embryologists predicted usable blastocyst development based on oocyte morphology with an average accuracy of 44% while the predictive classifier achieved an accuracy of 71%. Retaining the variables necessary for normal fertilization, only data from successfully fertilized oocytes were used, resulting in a classifier an accuracy of 81%. Conclusions To date, there is no standard guideline or technique to aid in the selection of oocytes that have a higher likelihood of developing into usable blastocysts, which are chosen for transfer or vitrification. This study provides a comprehensive workflow of extracting biomechanical properties and building a predictive classifier using these properties to predict mature oocytes’ developmental potential. The classifier has greater accuracy in predicting the formation of usable blastocysts than the predictions provided by morphological information or maternal factors. The measurement procedure did not negatively affect embryo culture outcomes. While further analysis is necessary, this study shows the potential of using biomechanical properties of oocytes to predict embryo developmental outcomes.

Turbulent reactive flows laden with droplets appear in various energy systems but are difficult to understand and parametrize. Such flows involve interactions of turbulent fluctuations, phase changes, and chemical reactions that give rise to complex phenomena. To improve our knowledge, we performed direct numerical simulations of a canonical shear flow. It is composed of a hot, quiescent outer layer and a cold, turbulent inner layer that is laden with droplets. Due to the turbulent fluctuations, the droplets form clusters. Due to the high temperatures, the droplets evaporate quickly and flames emerge spontaneously at the interface of the two layers. We observed premixed flames that enclose droplet clusters and diffusion flames that enclose vapor pockets or single droplets. To examine these flame structures in more detail, we varied the droplet size, droplet loading, and shear rate. We found that the droplet size and droplet loading have significant effects, whereas the shear rate has only subtle effects.

In this work, we focus on a multi-hole pressure-probe-based flow measurement system for wind tunnel measurements that provides real-time feedback to a robot probe-manipulator, rendering the system autonomous. The system relies on a novel, computationally efficient flow analysis technique that translates the probe's point measurements of velocity and pressure into an updatable mean flow map that is accompanied by an uncertainty metric. The latter provides guidance to the manipulator when planning the optimal probe path. The probe is then guided by the robot in the flow domain until an available time budget has been exhausted, or until the uncertainty metric falls below a prescribed target threshold in the entire flow domain. We assess the capabilities of our new measurement system using computational fluid dynamics data, for which the ground truth is available in the form of a mean flow field. An application in a real wind tunnel setting is provided as well.

Laser-induced cavitation bubble dynamics at different distances from a rigid boundary is investigated using high-speed synchrotron x-ray phase-contrast imaging. This is achieved through the design of a tailored experimental chamber specifically designed to reduce the x-ray absorption along the path length in water while mitigating boundary effects. The highly resolved undistorted radiographs are able to visualize a sharp bubble interface even upon complex shapes, which can serve as high-quality benchmarks for numerical simulations. Here, the measured bubble shapes are compared to simulations using the incompressible boundary integral method. The direct optical access to the high-speed liquid jet provides accurate measurements of the evolution of the jet speed, which is contrasted to the simulated results. After the jet has impacted the opposite side of the cavitation bubble, the cavity assumes a toroidal shape, the volume of which can be accurately measured from the radiographs and its temporal evolution compared to the bubble-ring model. Thanks to the clear optical access to the cavity lobes throughout the collapse, non-axisymmetric splashing within the bubble resulting from the jet impact, also known as Blake's splashing, is observed and characterized for stand-off parameters of γ<1. Measurements extracted from the highly resolved visualizations provided herein have been validated against scaling laws for droplet impact on a thin liquid film, which contribute to confirm and elucidate the splashing phenomenon.

Upon interaction with underwater shock waves, bubbles can collapse and produce high-speed liquid jets in the direction of the wave propagation. This work experimentally investigates the impact of laser-induced underwater impulsive shock waves, i.e. shock waves with a short, finite width, of variable peak pressure on bubbles of radii in the range 10–500 $\mathrm {\mu }$ m. The high-speed visualisations provide new benchmarking of remarkable quality for the validation of numerical simulations and the derivation of scaling laws. The experimental results support scaling laws describing the collapse time and the jet speed of bubbles driven by impulsive shock waves as a function of the impulse provided by the wave. In particular, the collapse time and the jet speed are found to be, respectively, inversely and directly proportional to the time integral of the pressure waveform for bubbles with a collapse time longer than the duration of shock interaction and for shock amplitudes sufficient to trigger a nonlinear bubble collapse. These results provide a criterion for the shock parameters that delimits the jetting and non-jetting behaviour for bubbles having a shock width-to-bubble size ratio smaller than one. Jetting is, however, never observed below a peak pressure value of 14 MPa. This limit, where the pressure becomes insufficient to yield a nonlinear bubble collapse, is likely the result of the time scale of the shock wave passage over the bubble becoming very short with respect to the bubble collapse time scale, resulting in the bubble effectively feeling the shock wave as a spatially uniform change in pressure, and in an (almost) spherical bubble collapse.

Advection dominated transport processes in sub-surface formations are characterized by discontinuities in the fields of transported quantities, and realistic predictions are challenging for Eulerian transport schemes because they suffer from numerical diffusion. Henceforth, we have focused on developing a Lagrangian particle-tracking scheme for modeling advective solute transport in fractured media. To this end, we adopt an Embedded Discrete Fracture Model (EDFM) for fractured media with a permeable matrix. The flexibility to use non-conformal fracture-matrix discretizations makes EDFMs a compelling choice in field-scale flow problems. Unaffected by the numerical diffusion, Lagrangian transport schemes complement the potential of EDFMs by allowing the use of sufficiently large grid cell sizes for flow field computations. In an EDFM framework, the inter-continuum fluid mass exchange cannot be quantified by the particle trajectories/pathlines due to the unresolved fracture-matrix interfaces and different dimensionalities of the matrix and fracture discretizations. These constraints motivate the use of a stochastic particle-tracking scheme, and thus, we formulated a pathline-specific probability of inter-continuum particle transfer based on mass conservation of an elementary solute/fluid mass. The particle's transfer probability is calculated for the maximum residence time period in its associated fracture/matrix control volume, thus making the scheme time-adaptive. In addition, a conditional residence time distribution was derived, which dictates the timestamp of the particle transfer. First, we showcase that the tracking scheme preserves the initially homogeneous solute concentration field, suggesting that the probabilistic rules are consistent with the inter-continuum fluxes. Additionally, we illustrate the estimation of an evolving solute plume and compare the results with those of an Eulerian counterpart. The presented stochastic approach enables straightforward formulations of Lagrangian models for dynamic and sub-grid processes, e.g., solute interactions with the solid phase, the mapping of their effects onto large scale transport and additionally, be included in random walk models for dispersion.

In recent years, energy prices have become increasingly volatile, making it more challenging to predict them accurately. This uncertain market trend behavior makes it harder for market participants, e.g., power plant dispatchers, to make reliable decisions. Machine learning (ML) has recently emerged as a powerful artificial intelligence (AI) technique to get reliable predictions in particularly volatile and unforeseeable situations. This development makes ML models an attractive complement to other approaches that require more extensive human modeling effort and assumptions about market mechanisms. This study investigates the application of machine and deep learning approaches to predict day-ahead electricity prices for a 7-day horizon on the German spot market to give power plants enough time to ramp up or down. A qualitative and quantitative analysis is conducted, assessing model performance concerning the forecast horizon and their robustness depending on the selected hyperparameters. For evaluation purposes, three test scenarios with different characteristics are manually chosen. Various models are trained, optimized, and compared with each other using common performance metrics. This study shows that deep learning models outperform tree-based and statistical models despite or because of the volatile energy prices.

Solute transport observed in subsurface formations shows complex behavior, particularly in the presence of fractures. In this work, we focus on formations with fractures that are smaller compared to the domain of interest and which are distributed in a heterogeneous matrix at densities below the percolation threshold. We developed a simplified Lagrangian approach to characterize and predict advective transport in 2-D domains containing differently oriented fractures. To this end, we performed Monte Carlo simulation (MCS) studies using ensembles of random domain realizations and gathered/analyzed tracer particle displacement statistics. The series of displacement steps were defined by the locations of particles entering the matrix after traversing through one or several connected fractures. Then, we identify key correlation structures in the evolution of and between displacement step coordinates, namely, the step length, its orientation and the traverse time. Subsequently, a correlated random walk model was derived which is able to accurately reproduce longitudinal and transverse macrodispersion as recorded in the MCS. Finally, we explored the predictive capabilities of our stochastic model by simulating macrodispersion in stratified media composed of heterogeneous zones with different fracture orientations.

Summary An accurate prediction of transport in sub-surface formations is challenging for Eulerian schemes, especially if advection dominates. Lagrangian particle-tracking schemes, not being hindered by numerical dispersion, are compelling alternatives and have increased relevance in the modelling of non-linear transport, e.g., saturation transport in a multi-phase setting [1]. In this work, we have focused on the development of a particle-tracking scheme for multiphase flows in fractured media with a permeable matrix. To this end, we adopted an Embedded Discrete Fracture Model where fractures are treated as lower dimensional manifolds [2]. The fracture-matrix interfaces are not resolved at the sub-grid level, thereby necessitating an alternative modeling strategy for inter-continuum interactions. In [3], we presented a stochastic particle-tracking scheme for advective solute transport in single-phase flow. A particle’s continuum state is modeled as a two-state Markov chain with transition probabilities for inter-continuum particle transfer. The probabilities are pathline-specific and scale with the particle’s modeled travel time through the grid cell. In a first step, we aim to improve the efficiency of the above-mentioned scheme, especially for the scenarios involving high contrast in matrix-fracture pore-volumes. Then, the scheme is extended to model saturation evolution in two-phase immiscible flows. Following the work of [1], saturation is modeled as a statistical quantity and is estimated with two distinct particle ensembles, i.e., one for each phase. In the absence of dispersive effects, e.g., in flows with high Péclet numbers and without capillary pressure differences, saturation discontinuities are a typical feature. Therefore, near the fronts, inaccurate estimates and instabilities may result due to finite-sized particle ensembles in cells of the computational domain. To render such inaccuracies and instabilities insignificant, an adaptive diffusivity is added to the system, which selectively acts near the front and attains negligible values away from it. To quantify the diffusivity, a Smagorinsky-type [4, 5] model is proposed, which scales with the magnitude of the saturation gradient. In the pursuit of efficient yet accurate large-scale modeling, the proposed Lagrangian scheme can be extended to connect dynamic and locally unresolved sub-grid processes, e.g., phase dissolution, with the macroscopically observed effects.

This paper examines droplets that cluster and evaporate in reactive turbulence with direct numerical simulations. The flows are statistically homogeneous and isotropic with mass loadings of about 0.1, Stokes numbers of about 1, and Taylor-scale Reynolds numbers of about 40. Our simulation results reveal diffusion and premixed flames. When the mass loading is small or the Stokes number is large, clusters contain few droplets such that diffusion flames surround single droplets. However, when the mass loading is large or the Stokes number is small, clusters contain many droplets such that premixed flames propagate through clusters and diffusion flames surround clusters.