Collaborative Research on Droplet Dynamics Under Extreme Ambient Conditions

Abstract

A fundamental understanding of droplet dynamics is important for the prediction and optimization of technical systems involving drops and sprays. The Collaborative Research Center (CRC) SFB-TRR 75 was established in January 2010 to focus on the dynamics of basic drop processes, and in particular on processes involving extreme ambient conditions, for example near thermodynamic critical conditions, at very low temperatures, under the influence of strong electric fields, or in situations involving extreme gradients of the boundary conditions. The goal of the CRC was to gain a profound physical understanding of the essential processes, which is the basis for new analytical and numerical descriptions as well as for improved predictive capabilities. This joint initiative involved scientists at the University of Stuttgart, the TU Darmstadt, the TU Berlin, and the German Aerospace Center (DLR) in Lampoldshausen. This first chapter provides a brief overview of the overall structure of this CRC as well as a summary of some selected scientific achievements of the subprojects involved. For further details the reader is referred to the subsequent chapters of this book related to the individual subprojects.

1 Introduction

Many processes in nature and technology are influenced by droplet dynamics. These include ubiquitous phenomena and applications, such as rain clouds or fuel injection into combustion chambers, but also highly advanced devices such as rocket engines or spray based production processes in the pharmaceutical industry. Due to their omnipresence, droplets have been the focus of scientific interest for a long time: Plateau [34] and Lord Rayleigh [38] for example studied the generation of droplets from liquid jets under the influence of surface tension in the 19th century. The books of Pruppacher and Klett [37] and of Lefebvre [23] can be seen as seminal works in the field of droplets in nature or engineering, respectively. While there are many scientific results available about droplet processes in complex systems, they are often restricted to moderate ambient conditions. However, many processes take place under extreme ambient conditions. This holds for the near critical or supercritical conditions in modern combustion engines as well as for thunderstorm clouds with their high electric fields or supercooled droplets impinging on airplane wings, to name just a few examples. Much less research has been conducted in these areas, which is due to the increased effort in theoretical and numerical modelling or the necessary complexity of experimental equipment. Therefore, detailed research in this area not only demands extensive methodological development or improvement of experimental techniques and numerical models, but also very close interaction between numerical and experimental research. Thus, the consequential involvement of expertise from fields like thermodynamics, fluid mechanics, mathematics, computer sciences, electrical engineering and visualization is required. The SFB-TRR 75 combined the knowledge and expertise of scientists from all the above-mentioned disciplines. The multitude of thermodynamic processes with droplets under extreme ambient conditions can be divided into a few classes: The CRC set itself the goal of investigating the following extreme ambient conditions in droplet dynamic processes in more detail:

• Droplets near the critical point and at high pressures

• Droplets in strong force fields (e.g. electric fields)

• Droplets at extreme temperatures in thermodynamic non-equilibrium

• Droplet processes with extreme gradients (e.g. at the three-phase contact line)

According to the basic insight that complex droplet dynamic processes can be determined by the interaction of numerous elementary processes, the CRC concentrated since 2010 on illuminating the physics behind such basic processes in a generic manner. In order to give these rather abstract extreme ambient conditions a concrete reference to individual processes and systems, the investigations within this CRC have been carried out in connection with selected guiding examples. These guiding examples are:

• 1. Phase transition of supercooled and potentially electrically charged droplets in clouds

• 2. Impact of supercooled large droplets on aircraft components

• 3. Behaviour of water droplets on insulating surfaces in high voltage power transmission systems

• 4. Behaviour of droplets in rocket combustion chambers

• 5. Behaviour of fuel sprays in future combustion systems

In each of these guiding examples, one or more processes under extreme ambient conditions can be observed. Selected results of the work in the SFB-TRR 75 are summarized in [54, 55, 57, 58]. The CRC comprises 18 subprojects in three research areas, which are listed below together with the individual principle investigators of the subprojects:

Research Area A: Methods and Fundamentals

• SP-A1: Interactive visualization of droplet dynamic processes (T. Ertl)

• SP-A2: Development of numerical methods for the simulation of compressible droplet dynamic processes under extreme conditions (C.-D. Munz)

• SP-A3: Analysis and numerical simulations of front and phase field models for droplet dynamics (C. Rohde)

• SP-A4: Molecular dynamics simulation of droplet evaporation in the non-linear response regime (F. Müller-Plathe)

• SP-A5: Simulation of the mechanical deformation and movement of droplets under the influence of strong electric fields (E. Gjonaj, S. Schöps)

• SP-A7: Modelling and simulation of droplet collisions at modified ambient pressures, with high velocity and concentration gradients and for immiscible fluids (D. Bothe, K. Schulte)

Research Area B: Free Droplets

• SP-B1: Investigation of the behaviour of supercooled droplets concerning evaporation, condensation and freezing at different boundary conditions (B. Weigand)

• SP-B2: Experimental investigation of droplet evaporation under extreme conditions using temporally highly resolved laser diagnostic methods (G. Lamanna, A. Dreizler)

• SP-B3: Modelling and simulation of droplet evaporation in different gas environments under supercritical conditions (A. Sadiki)

• SP-B4: Experimental investigation of transient injection phenomena in rocket combustors at vacuum conditions with flash evaporation (M. Oschwald)

• SP-B5: Modelling and simulation of the flash evaporation of cryogenic liquids (A. Kronenburg)

• SP-B6: Droplets subjected to temperature and velocity gradients using atomistic simulations (J. Vrabec)

Research Area C: Droplets with Wall-Interactions

• SP-C1: Numerical simulation of the transport processes during drop impingement onto heated walls with special consideration of the evaporating three-phase contact line (T. Gambaryan-Roisman, P. Stephan)

• SP-C2: Highly resolved measurements of heat transfer during drop impingement onto a heated wall with particular consideration of evaporation at the three phase contact line (P. Stephan, T. Gambaryan-Roisman)

• SP-C3: Impact of supercooled droplets onto cold surfaces (S. Jakirlic, C. Tropea)

• SP-C4: Interaction of a single drop with a heated wall at high ambient pressures (I.V. Roisman, C. Tropea)

• SP-C5: Mechanical and electrical phenomena of droplets under the influence of high electric fields (V. Hinrichsen)

• SP-Z: Administration of the SFB-TRR 75 (B. Weigand)

The declared goal of the SFB-TRR 75 was to gain a deeper physical understanding of processes with droplets under extreme ambient conditions. Based on this, methods to describe them experimentally, analytically and/or numerically have been identified and implemented. The better understanding of the basic underlying processes enables more precise predictions of their behaviour and dynamics and, thus, also an improvement of the prediction of processes in larger and more complex systems in nature or in engineering applications.

2 Research Area A: Methods and Fundamentals

Many challenges exist in multiphase flows under extreme ambient conditions. There are not only open questions in engineering applications, but also in the fundamental physical, mathematical and numerical modelling. Projects to address these challenges with respect to methods and fundamental tools were combined together in Research Area A. Numerical and analytical methods were developed, which were prerequisites for conducting the work in the other research areas. The topics in Research Area A have been linked to the three main areas:

• Numerical methods for the direct numerical simulation of droplet dynamics

• Thermodynamic modelling of surface phenomena

• Visualization

2.1 Numerical Methods for the Direct Numerical Simulation of Droplet Dynamics

The improvement and extension of the mathematical and numerical modelling of multiphase interfaces were the focus of subprojects in Research Area A. The development of simulation tools for the direct numerical simulation of droplets in the fully compressible regime was the objective in SP-A2 and SP-A3.

Fig. 1

Velocity magnitude distribution of a 2D shock/-drop interaction. The phase interface is depicted in white

To emphasize the consistency of the thermodynamic and hydrodynamic modelling, a sharp interface approximation of the phase transition front was applied in SP-A2 (see Jöns et al. in this volume). The phase interface is considered as a discontinuity in the macroscopic continuum equations. Direct numerical simulations of flows with droplets require several components for the simulation tool: A solver for compressible multicomponent flows with an equation of state for real fluids, methods for coupling the different fluids and algorithms for tracking the material or phase boundaries. In SP-A2 the sharp interface coupling is based on the ghost fluid method and on the solution of the multiphase Riemann problem [8]. The tracking of the material boundary is performed by a level-set approach. Level-set and the flow equations are all calculated with a discontinuous Galerkin method, in which locally the resolution can be improved by h-p adaptation. The thermodynamical modelling for the two-phase flow scenarios in a high pressure and temperature environment was considered in [9]. Phase transition effects are included by the solution of the two-phase Riemann problem at the interface, supplemented by a phase transition model based on classical irreversible thermodynamics. An exact as well as an approximate Riemann solver were constructed. Numerical results compared to molecular dynamics data for evaporation in a shock tube and steady state evaporation exhibited a good agreement with the reference data from molecular dynamics, see [15, 16]. Figure 1 shows the distribution of the velocity magnitude for a 2D shock/-drop interaction, where the interface is depicted by a white line.

The modelling of liquid–vapour flow with phase transition requires accurate physical and mathematical models and the implementation into efficient numerical methods and programs. In SP-A3 (see Magiera and Rohde in this volume), compressible models for the dynamics of single droplets and droplet ensembles were developed and studied analytically and numerically. Both, a sharp interface approach and a diffuse interface approach were considered. There was a close cooperation between SP-A2 and SP-A3, with the focus on the numerical modelling and application and into the mathematical foundations, respectively (see [8, 17]). Analytical Riemann solvers for basic isothermal two-phase flow scenarios were developed in SP-A3. For complex thermodynamical settings, an interface solver was constructed based on molecular dynamics simulations [17]. It was shown to be even applicable for two-phase flows with multiple components. To understand merging and splitting phenomena for droplet ensembles, diffuse interface models were considered [30]. For the Navier–Stokes–Korteweg systems variants, numerical schemes were developed by reformulating the capillarity operator by a relaxation procedure such that the hyperbolicity of the full set of equations are recovered. Figure 2 shows the evolution of a group of droplets towards a single large droplet by coalescence [28].

Fig. 2

(Source [17], reprinted with permission from Elsevier under No. 5039260334246.)

Evolution of a droplet ensemble and trend to (spherical) equilibrium.

In the subproject SP-A5 (see Gjonaj et al. in this volume) droplet dynamics in strong electric fields has been investigated numerically by means of an electrohydrodynamic (EHD) simulation model [33]. The fluid flow is in the incompressible regime and has been simulated via OpenFOAM, coupled to an electro-quasi-static field computation based on the charge conservation equation. This coupling allows the charge convection and relaxation phenomena to be included in the simulations, which are typical for droplet and spray processes. Furthermore, the model contains a detailed description of contact angle dynamics for capillary flows in the presence of electric fields and charges. One class of applications was the simulation of oscillating droplets on the surface of high voltage insulators. For this investigation, there was close cooperation between SP-A5 and SP-C5 focusing on the experimental validation of simulation results obtained with the EHD model. The contact angle characteristics resulting from this analysis provide the basic measure for the estimation of the inception field thresholds for electrical discharges occurring on the insulation layer of high voltage devices [31]. The second class of applications was the numerical characterization of electrosprays. Extensive electrospray simulations for various liquids with different electrical and mechanical properties provided quantitative insights into the electrospray dynamics. In particular, scaling laws for the charge-radius droplet characteristics were determined for different applied voltages and capillary flow rates in the transient as well as in the steady state electrospray regime [32]. The simulation results in Fig. 3 illustrate exemplarily the effect of electrical conductivity on the droplet size and density distributions at the onset of electrospray for two selected liquids.

Fig. 3

Adapted figure with permission from [32], Copyright 2020 by the IEEE

Onset of electrospray from a sessile droplet shown at different time instants. The colour map depicts electric field strength. Left: methanol. Right: a heptane mixture. The electrical conductivity of the heptane mixture is chosen to be nearly one order of magnitude smaller than that of methanol.

2.2 Thermodynamic Modelling and Surface Phenomena

An important topic in multiphase flows is the accurate thermodynamic modelling of the surface phenomena. Such research topics were considered by the projects SP- A4 and SP-A7. This work was important for the further development of simulation tools, but also for the analysis and interpretation of experimental results. Molecular dynamic simulations with many particles performed in SP-A4 contributed to the study of evaporation of nanometer droplets at hot surfaces. Here, non-equilibrium molecular dynamics were developed, by which steady evaporation processes were observed in detail at different complex surfaces.

The understanding of basic mechanisms in heat transfer, adhesion, and phase transition requires insight into the local phenomena of liquid gas and liquid–solid contact. Since the characteristic length at these interfaces measures only a few molecular diameters, one needs microscopic simulations to gauge thermodynamic and continuum models. Molecular dynamics simulations investigate these phenomena on a molecular scale. Simulations in SP-A4 with many fluid particles contributed to the study of evaporation of nanometer-size droplets from hot surfaces. For this purpose, new non-equilibrium techniques were developed, by which evaporation processes can be observed in detail at different complex surfaces in a steady state for better statistics [62]. The evaporating droplet is replenished by rapidly duplicating molecules at its center, while removing molecules in the gas phase far away from its surface. Incidentally, this splitting method can also be used to calculate mutual diffusion coefficients by reverse non-equilibrium molecular dynamics [4].

The models and algorithms were applied to investigate phenomena as different as the jump pinning/unpinning motion of the contact line during droplet evaporation from a nanopatterned surface (see Fig. 4) and the strong influence of the contact line curvature on the contact angles and the contact area of a drop, both in equilibrium and during evaporation [60]. SP-A4 investigated the rebound of impinging drops from solvophobic surface and its suppression by small amounts of polymer additives. Two molecular modes of action have been shown to work side by side [22] (see also Lee et al. in this volume).

Fig. 4

Evaporation of a nanodrop from a surface with contact-line pinning and unpinning. a Both contact lines reside on the solvophilic surface regions (blue, contact angle $\sim$67 ${}^{\circ }$). b When the drop becomes too small to span the distance between the solvophobic regions, one contact line unpins and quickly traverses the solvophobic region (white, contact angle $\sim$115 ${}^{\circ }$). Such jumps are characteristic for chemically nanopatterned surfaces. c The drop is too big for one solvophobic region and therefore squeezed into an unfavourable shape with a large contact angle. d The drop has shrunk enough to fit onto one solvophilic region, the contact angle is back to its equilibrium value [61]

Collision processes play a major role in different science and engineering problems ranging from cloud dynamic processes to technical spray applications. A realistic description of the droplet dynamics needs the understanding of the elementary process of binary droplet collisions. The focus of SP-A7 (see Potyka et al. in this volume) was the proper modelling of the collision process according to the collision outcome bouncing, coalescence or drop disintegration, as well as the collision of different liquids. The program Free Surface 3D (FS3D) has been further developed with respect to the predictive description of these phenomena and to capture the complete physics of the droplet collision numerically. In particular, an integrated subgrid-scale modeling based on an enhanced lubrication approximation allows to account for the rarefied flow dynamics inside the thin gas layer between colliding droplets [25]. High-resolution simulations predicted droplet collisions in the spattering regime and provided detailed insights into the evolution of the rim instability [24]. For the case of collision of immiscible liquids, the complexity of the process strongly increases due to the occurrence of a triple line. An enhanced continuous surface stress model was introduced for an accurate surface force computation, which is also applicable to thin films [35]. Moreover, a new face-based interface reconstruction has been developed and extended to the case with appearance of triple lines [19]. For collisions of fully wetting liquids, excellent agreement with experimental data was achieved in different collision regimes; an example is shown in Fig. 5.

Fig. 5

Morphological comparison of simulation results (bottom) for an off-center collision of a silicon oil M5 (light) and a 50% glycerol-water solution (dark) with experimental results (top) (see [35] for further details). Pictures of experiments provided by courtesy of C. Planchette. A tool developed in SP-A1 (see Heinemann et al. in this volume) has been used for the visual analysis

2.3 Visualization

The third topic in Research Area A was visualization, which contributed to nearly all other projects. The main goal of SP-A1 (see Heinemann et al. in this volume) was the development of new and improved algorithms especially for interactive visualization of droplet dynamic processes from large, time-dependent simulations and experimental data. These research activities included the development of new methods for the analysis of single droplets and droplet groups, but at the end also for the investigation of sprays. Here, the challenge was the analysis of the enormous amount of data. Novel visualization techniques were required to handle the increasing complexity of simulation and experimental data and the complex physical phenomena under investigation. The visualization of phase transitions constituted an essential research field. Techniques were developed to analyse multiphase processes such as evaporation, icing and drop impingement. Coalescence and break-up events needed to be visually presented in an interactive explorable way. As an example a visualization of a jet breakup is shown in Fig. 6. Tools for the visual analysis of energy transport were designed to obtain information about phase transitions, also in the context of the three-phase contact line. New visual tools were necessary for the analysis of large droplet groups and sprays, as well as for various physical phenomena investigated in SFB-TRR 75. SP-A1 spanned visualization techniques for observing physical phenomena, such as energy transport or collisions for single droplets, to the analysis of large-scale simulations with sprays and jets. In addition, the task of finding relevant structures, features of interest or a general dataset overview was addressed in this subproject.

Fig. 6

©2017 IEEE. Reprinted, with permission, from [18]

Jet dataset with selected time steps shown. (Top) Jet (purple) is injected from left, and due to its high velocity at the nozzle, it splatters into various-sized droplets (blue). (Bottom) Parametric graph shows the space-time evolution of the jet stream. Edge clusters represent droplets with similar area to volume ratio. The box in the middle shows an enlarged node with volume distribution indicated on the clustered edges. A node cluster can be unfolded (left box) and folded back to reveal the actual edge connectivity and drop geometry. Here, highly deformed ligaments with large area to volume ratio are visible.

3 Research Area B: Free Droplets

In Research Area B, different phenomena of free drops under different extreme thermodynamic and fluid dynamics conditions were investigated. The overall scientific objective was to understand the physical mechanisms underlying these droplet dynamic processes. Based on this physical understanding, models were developed and validated. The necessary validation data were provided in this research area, partly using new laser diagnostic approaches. The mathematical models formed the necessary basis for numerical simulations of systems defined via the CRC guiding examples. The investigations in Research Area B focused on

• Phase transitions of supercooled droplets (evaporation, melting and freezing processes) that are in thermodynamic non-equilibrium

• Transcritical evaporation and mixing processes

• Flash evaporation of cryogenic liquids under near vacuum conditions

Fig. 7

Temporal evolution and rendered simulation of freezing ice particle (left); Experimental results for the time evolution of a levitated drop from liquid to frozen state (right)

3.1 Phase Transition of Supercooled Droplets

The formation of ice crystals of arbitrary structures, e.g. also by collisions within clouds, is still very poorly understood. However, this is of importance for a correct description of cloud dynamics. SP-B1 focused on this topic for free droplets, whereas SP-C3 in Research Area C focused on the wall interaction of supercooled droplets. The dynamics of supercooled droplets is characterized by their metastable state and the resulting phase change. An improved understanding and a more accurate prediction of the phase changes of and in supercooled droplets is of fundamental interest, e.g. for the understanding of cloud dynamics at high altitudes and for the development of improved climate models. The focus in SP-B1 (see Reutzsch et al. in this volume) was therefore on studying evaporation, sublimation and freezing of supercooled water droplets. On the one hand, physically consistent numerical models, capable of calculating all three phase transitions, were developed [43, 45]. On the other hand, the numerical work within the program FS3D was validated and compared with different experimental results [44, 47]. In an optical levitator, the dynamics of supercooled water droplets was investigated by light scattering and shadowgraphy methods for ambient conditions corresponding to temperatures appearing in clouds at high altitudes. From this, sublimation and evaporation rates of frozen and supercooled droplets were determined and compared with direct numerical simulation (DNS) results. Figure 7 shows on the left the temporal development of an ice crystal from a spherical seed and a rendered picture of the simulation result. On the right in Fig. 7, the time evolution of a levitated drop is shown from the liquid to the frozen state.

3.2 Transcritical Evaporation and Mixing Processes

The transcritical and supercritical region of fluids involve extremely high pressures and temperatures. These conditions, which are important e.g. for energy conversion systems, are still poorly understood with respect to evaporation and mixing. This topic has been investigated in SP-B2, SP-B3 and SP-B6.

The focus of SP-B2 [20, 21] is on dynamics of single droplets at near critical conditions, see also Lamanna et al. in this volume. Figure 8 exemplarily shows the mixture fraction of n-hexane in the wake of an evaporating droplet and gas temperatures assuming adiabatic mixing [2]. Contrary to sprays, droplets provide a simplified configuration, for which analytical solutions can be obtained to describe the simultaneous exchange of mass and energy at high pressure conditions. For an experimental investigation, a high-pressure, temperature controlled test rig, equipped with an electrical droplet generator, was built to perform experiments under well-controlled conditions. Several laser diagnostic methods have been developed and improved. Spontaneous Raman scattering [2] and combined laser induced fluorescence and phosphorescence [36] enabled the measurement of gas composition in the wake of an evaporating droplet, gas temperatures and, for the case of acetone, mean droplet temperatures. Laser-induced thermal acoustics has been extended to enable the measurement of the acoustic damping rates in the supercritical region of pure fluids [56]. Based on these measurements, the importance of bulk viscosity in the liquid–like supercritical region was assessed and closure models for describing momentum transport and dissipation effects in supercritical fluids were evaluated. For the example of the onset of single-phase mixing, the analysis revealed that in the presence of large temperature and concentration gradients evaporation processes are highly important.

Fig. 8

n-hexane droplets in nitrogen atmosphere. Single droplet mixture fraction result for nitrogen atmosphere temperature of 533 K, injector temperature of 473 K and pressure of 6 MPa (left). Number density data estimated adiabatic mixing temperature curves dependent on mixture fraction. Varying ambient nitrogen temperature as shown, injector temperature fixed at 473 K and a pressure of 6 MPa (right)

In SP-B3 (see Kuetemeier and Sadiki in this volume), the dynamics of sprays under transcritical conditions was investigated using large-eddy simulation (LES). Multi-component real fluid systems were considered. Based on models for the simulation of evaporating single droplets, the goal was to further develop innovative models for sprays. For this purpose, a regime spanning evaporation model and a subgrid scale model taking anisotropy into account were integrated into an Euler-Euler as well as an Euler-Lagrange simulation. As a new approach, the entropy inequality (second law of thermodynamics) was considered in the large-eddy simulation. Thus, on the one hand, the physical consistency of the subgrid scale modeling was ensured and on the other hand, the potential of entropy production as an analytical tool for the assessment of fluid dynamic processes was explored. Obtained by means of the developed Eulerian-Eulerian method, Fig. 9 shows exemplarily the field of mass density evolvement during the disintegration of an elliptic jet of supercritical fluoroketone injected into a supercritical helium environment. The numerical model includes a description of the two-phase flow fluid with phase change as multi-component mixtures in which the real fluid properties are accounted for by a composite Peng-Robinson equation of state. It reproduces correctly the jet disintegration regimes as observed experimentally (see [29]) in terms of penetration length along with mass density and jet spreading angle. The effect chain of the evolving processes is especially consistently reproduced.

Fig. 9

Instantaneous field of mass density at mid-plane section of the fluoroketone jet: Eulerian-Eulerian-based LES results of a grid resolution with ca. 3 million control volumes. Jet spreading angle: 19 ${}^{\circ }$ (Experiment: 20 ${}^{\circ }$) and effect chain of processes evolving

Fig. 10

Left: Temperature dependence of the intra-, Maxwell-Stefan and Fick diffusion coefficients for the mixtures (top) $\text {CH}_{4} + \text {CO}_{2}$, (center) $\text {C}_{6}\text {H}_{6} +\text {CO}_{2}$ and (bottom) $\text {C}_{7}\text {H}_{8} + \text {CO}_{2}$ at $p=$ 9 MPa and $x_{\text {CO2}}$ = 0.99 mol  mol-1. Right: Simulation results for the Fick diffusion coefficient (green squares) are compared with predictive equations (lines) by Sassiat et al. (black) [48], Wilke and Chang (blue) [59], Catchpole and King (green) [3], He and Yu (red) [14] and Scheibel (cyan) [49]

The behaviour of droplets under strong non-equilibrium conditions is poorly understood. In the associated heat and mass transfer phenomena, the phase boundary between droplets and their environment plays a crucial role. Due to its typically very small spatial extent, the phase boundary in non-equilibrium can be directly simulated using atomistic molecular dynamics, providing detailed insight on a sound physical basis. In SP-B6 (Heinen et al. in this volume), liquids and gases interacting with each other across a phase boundary, such as liquid nitrogen and gaseous hydrogen, were studied under non-equilibrium conditions using atomistic simulations. One focus was on the question under which conditions the atomization process transitions from a two-phase behaviour to dense-fluid mixing. The influence of strong temperature and composition gradients was considered with respect to heat and mass transfer as well as the physical quantities of the phase boundary. In addition, for trans- and supercritical fluid conditions, diffusion coefficients for binary solute systems were predicted by equilibrium molecular dynamics simulation and the Green-Kubo formalism. Several predictive equations for the Fick diffusion coefficient [3, 14, 48, 49, 59] were compared with present simulation results. As shown in Fig. 10, although some equations were able to reasonably predict the temperature dependence of the Fick diffusion coefficient at the studied conditions, none of the studied equations could be established as reliable.

3.3 Flash Evaporation of Cryogenic Liquids Under Near-Vacuum Conditions

Although important phenomena of spray dynamics and flash boiling have been described in the literature, valid and physically consistent modelling is lacking. Furthermore, similar to the studies of supercritical droplets, only very few validation data under well-controlled boundary conditions were available. Therefore, a combined experimental and numerical approach has been followed by SP-B4 and SP-B5.

At high altitude, liquid fuels are injected under cryogenic conditions into rocket combustion chambers where very low pressures prevail. The associated pressure drop leads to superheating and subsequently to eruptive vaporization (flash boiling) with rapid expansion. To investigate this complex phenomenon, a test rig was set up in SP-B4 (see Rees and Oschwald in this volume) with a special cryogenic injection system for the injection of molecular nitrogen under flash boiling conditions [41]. High-speed shadowgraphy and phase Doppler measurements were used to study the spray jet topology, droplet size and droplet velocity for a wide range of parameters. With increasing superheat, the transition from narrow and turbulent sprays to wide opened, fine and well atomized sprays was observed. Based on the experimental results, a new break-up regime called wide flashing regime for highly superheated jets was found [42]. The transition to this regime is visualized in Fig. 11 using shadowgraphy images. The phase Doppler measurements in wide flashing liquid nitrogen sprays revealed fast and large droplets close to the injector, while the sprays become more monodisperse with slow and small droplets for an increasing distance to the injector [39, 40]. This extensive database is available for the validation of numerical simulations.

Fig. 11

LN2 sprays with different degrees of superheat for the atomization regimes a aerodynamic break-up, b transition regime and c fully flashing regime and the new d wide flashing regime

Numerical simulation of flash boiling is of great importance e.g. for the design of rocket combustors. For this a detailed phenomenological understanding of bubble growth and bubble-bubble interactions and their effect on spray breakup, droplet dynamics and subsequent mixing processes is needed. Complementary to the experimental studies in SP-B4, DNS [5, 6, 26] and RANS [10] simulations were conducted in SP-B5 (see Gärtner et al. in this volume). RANS simulations were also performed for the overall process to investigate fluid dynamic effects and shock wave formation downstream of the injector [10]. Figure 12 shows an example of the temporal development of the spray break-up process and droplet generation for selected boundary conditions [26]. As an important result for this aspect, it was found that only bubbles close to the jet surface grow significantly. Bubbles at distances greater than ten bubble diameters from the jet interface can be neglected in the dynamics of jet expansion and break-up [5].

Fig. 12

Time sequence of the spray breakup process and droplet generation. Break-up in the ligament stretching regime with conditions $T_L=$ 120 K, $R_f^*=$ 5, We $=$ 3.62, Oh $=$ 0.104

4 Research Area C: Droplets with Wall-Interactions

In Research Area C drops with wall interactions have been studied. The extreme ambient conditions were related to elevated pressure, hot and/or cold walls, and applied electric fields. The overall goal in all of these studies was to better understand the physical phenomena involved in the respective interactions with the aim to improve predictive capabilities through physics based models. All of the subprojects included in the research program involved experiments, theoretical analyses and numerical simulations, yielding well verified and advanced models. The investigations in Research Area C focused on

• Droplet and spray interactions with hot walls

• Droplet/-wall interactions with cold walls

• Droplet/-wall interactions under the presence of strong electric fields

4.1 Droplet and Spray Interactions with Hot Walls

The applications in mind when conceiving this research area were diverse, but timely and challenging. Three subprojects (SP-C1, SP-C2 and SP-C4) involved drops impinging onto hot walls, a phenomenon encountered in numerous situations, but especially when the drops are used for cooling, either with or without phase change. At higher wall temperatures of course, the drops evaporate, greatly increasing the heat flux from the wall, since the latent heat of vaporization is exploited. However, this phase change also introduces a strong influence of the thermodynamic conditions on the hydrodynamics of drop impact. This is a particular challenge, extending beyond the well-established knowledge base of isothermal drop impact onto walls. Furthermore, at high heat flux levels, the heat removed from the substrate must also be taken into account, becoming a conjugate heat transfer problem. In combusting systems this occurs often at elevated pressures; hence, this extreme condition was also investigated. At extremely high wall temperatures, typical of transient cooling and/or quenching, the challenge is further to predict the Leidenfrost point, i.e. the transition from film boiling to nucleate boiling.

SP-C1 (see Sontheimer et al. in this volume) and SP-C2 (see Gholijani et al. in this volume) address the same physical phenomenon, the former in terms of numerical simulations and the latter in terms of laboratory experiments. In both cases the drop impingement is onto a substrate at a temperature above the saturation temperature but below the Leidenfrost temperature of the impinging liquid, i.e. the liquid is wetting the substrate surface and evaporating at the same time. This generic situation applies e.g. for spray cooling at moderate heat flux levels. The heat transfer mechanism is a combination of single phase convective heat transfer and phase-change heat transfer. The hydrodynamics of isothermal drop impact are already well known; however, in the non-isothermal case with a superheated wall, there is a strong interaction of heat transfer and hydrodynamics, and this is specifically addressed in SP-C1 and SP-C2. The associated heat transfer represents a particular challenge due to numerous factors. For one, drop impacts are characterized by an inertia dominated spreading phase followed by a surface tension dominated retracting phase. The internal flow in the drop, which is totally different in these two phases, strongly affects the thermal boundary layers in both, the drop and the substrate. Hence, the exact internal flow behaviour and the velocity of the three-phase contact line during the spreading and retracting phases become important for the convective and the evaporative heat transfer and vice versa. A particular emphasis in SP-C1 and SP-C2 is placed on correctly measuring and modelling this “micro region” flow and heat transfer. The numerical treatment of this problem requires modelling at both, the macro and the micro scale, and an appropriate connection between the two. A very revealing physical insight afforded by SP-C1 is illustrated in Fig. 13, in which the heat transfer paths from substrate to drop to gas is graphically shown for the spreading and receding phase of a drop impact. These exemplary results, obtained through numerical simulation, underline the importance of the contact line micro region.

The heat transfer from single drop impacts was characterized in dependence of impact parameters and material parameters, expressed in dimensionless form [12]. This work was then extended to the case of multiple drop impacts, either vertically coalescing or side-by-side. Furthermore, the influence of elevated pressure on the heat transfer was quantified. These results then represent a first step to extend basic knowledge about single drop impacts to spray impingement by supplying appropriate scaling parameters of the net heat transfer.

SP-C2, the experimental pendant to SP-C1, addresses the challenge of measuring the heat transfer in the micro region around the three-phase contact line. For this, two dedicated and novel experimental facilities were designed and constructed. In one facility the concept of a moving wall and stationary contact line in laboratory coordinates was introduced. This simplifies the observation of phenomena at the contact line. The substrate was heated using Joule heating of a chromium layer: A chromium nitride layer with high emissivity was used together with an infrared camera with high spatial and temporal resolution to monitor the evolution of the temperature profile near the contact line. In this manner the transient heat flux at the three-phase contact line could be computed.

The experimental results confirm the associated numerical simulations in SP-C1. The heat flux at the forward moving contact line reached levels twice as high as at a receding contact line. This is due to the stronger micro-convection at the contact line and the thinner boundary layer inside the drop. Additional experiments were performed at elevated pressures, confirming a decrease in overall heat flux due to the reduction in latent heat of vaporization, which specifically reduces the evaporation in the receding phase.

The peak heat flux observed at the steady state three-phase contact line was also observed for the transient phases during drop impact. Besides varying the impact parameters of drop size and impact velocity, also the influence of structured and porous substrates was investigated. Finally, experimental data for vertical and horizontal coalescing drops were collected for comparison with the accompanying numerical simulations [1, 11]. An example measurement result of the time resolved impact of two drops captured with a side view camera and the associated computed heat flux distributed across the wetted area is shown in Fig. 14.

Fig. 13

Heat transfer paths during the a spreading phase and b receding phase of the drop impact. The isotherms with a spacing of $\Delta T=$ 2 K and streamlines are shown in a moving reference frame close to the contact line

Fig. 14

Side view images of an impact and merger of two drops (upper line) and bottom view computed images of heat flux of a horizontal coalescence of two drops (lower line). ($D_0={0.93\,\textrm{mm}}$, $U_0={0.53\,\mathrm{ ms^{-1}}}$, $\Delta T= {7.3\,\mathrm{\text {K}}}$, $p={0.9\,\textrm{bar}}$)

When a drop impacts onto a heated wall, the associated heat transfer depends on numerous quantities, above all the temperature of the substrate. In general the heat transfer is categorized according to the so-called boiling curve regimes: convective heat transfer, nucleate boiling, transition boiling and film boiling, whereby the latter two are separated by the Leidenfrost point. In the framework of SP-C4 (see Schmidt et al. in this volume) the transition regime was further elucidated, revealing the phenomenon of thermal atomization, in which the drop makes contact with the substrate resulting in a large number of small secondary droplets, followed by a levitation of the remaining drop liquid and a break-up of the liquid fragments into larger drops through the Rayleigh-Plateau instability. For all of the above-mentioned regimes, theoretical models were developed and validated to predict the heat transferred upon drop impact and the drop lifetime until complete evaporation. A further insight was gained about the threshold temperature $T^*$, from which on drops rebound and the prediction of this point for a given drop impact and substrate material. The threshold point separates film boiling, in which the heat transfer is very low, from nucleate boiling, which exhibits very high heat transfer; thus, at the Leidenfrost point the heat transfer takes a minimum value. In Fig. 15a–c different outcomes of drop impact, deposition, partial and complete rebound, are shown after impact of the substrate at various initial temperatures. The dependence of the drop residence time on the wall temperature is shown in Fig. 15d. The novel insight gained is related to the importance of the substrate material, in particular the thermal effusivity of the substrate. The heat transfer at these high temperatures is strongly a conjugate heat transfer problem; hence, the rate at which the thermal boundary in the substrate develops becomes an important factor in defining this point [50].

Fig. 15

Typical outcome phenomena of a drop impact in the regimes drop deposition, drop dancing and drop rebound are shown in a, b and c. In d the residence time of the drops at the surface is shown as a function of the surface temperature in comparison with the theoretical models. The temperature $T^*$ indicates the onset of drop rebounds

While the initial research program was concerned with the impact of single drops onto heated surfaces, later work investigated the impact of sprays, as would be found in spray cooling scenarios. This transient spray cooling deviates in some manner from the steady state boiling curve view mentioned above, since the thermal history of the substrate becomes influential. In particular, this work revealed the relative duration in which the various heat transfer regimes were applicable.

4.2 Droplet/-Wall Interactions with Cold Walls

With reference to icing phenomena in the aviation industry, SP-C3 (see Gloerfeld et al. in this volume) examined the impact of supercooled drops onto cold surfaces and their subsequent solidification and accretion. Although the prediction of ice accretion is a standard step in the aircraft certification process, most presently used computational models are highly empirical and are not completely recognized as a confirmation of airworthiness. This is especially true for more recent regulatory stipulations, in which also icing from supercooled large droplets (SLD) must be considered.

Although the degree of supercooling can be quite substantial ($< -$10 ${}^{\circ }\text {C}$), in practice SLD can remain in a meta-stable liquid state, even after impacting onto a solid surface. The freezing is initiated by heterogeneous nucleation, which is influenced by numerous factors, including the surface topology, temperature, possible contamination, all which may lead to a nucleation embryo. This is an extreme challenge to predict, since the microscopic boundary conditions at work are not all known. Thus, in SP-C3 a statistical approach was taken to formulate nucleation models expressing the freezing delay between impact and first nucleation. Once nucleation occurs, the freezing continues with dendrite propagation through the bulk, by which a small volume of the liquid solidifies and the remainder warms to the melting temperature. Models for this dendritic phase have also been developed [51]. At this stage, the drop then solidifies according to the Stephan problem.

The complexity is increased with impacting drops because the wetted area on the surface changes with time as the drop spreads and possible retreats, depending on the wettability. Within the drop both a hydrodynamic and thermal boundary layer develop and this then also influences the material parameters, especially the viscosity. These influences on the hydrodynamics can alter the impact outcome, but also the final iced area on the surface [53]. Also for this interaction between thermodynamics and hydrodynamics, appropriate models have been proposed and validated with experimental data [52].

A quantity of great interest and importance upon drop impact is the residual volume, i.e. what portion of the impacting liquid remains on the substrate? This directly influences the rate of ice accretion, which is an elusive quantity to measure and in SP-C3 a novel method of measuring residual volume, also for drops which have solidified on the substrate, has been developed and applied to deliver first data on residual mass as a function of impact parameters [13]. The associated models which have been developed are directly appropriate for integration into icing codes. Measurements have also been performed for the case of impacting dendritic drops, i.e. drops in which the dendrites have already formed. Such an impact is pictured in the image sequence of Fig. 16.

Fig. 16

Image sequence of a dendritic frozen drop impacting with a velocity of ${4.2\,\mathrm{ms^{-1}}}$ originating from a drop with an initial supercooling of $\Delta T_0 = {9.2\,\mathrm{\text {K}}}$

4.3 Droplet/-Wall Interactions Under the Presence of Strong Electric Fields

SP-C5 (see Löwe et al. in this volume) involved sessile drops exposed to various high voltage electric fields, including alternating, transient and direct voltage. The specific aims were twofold. For one, the influence of an electric field on the probability of partial discharge was investigated, since partial discharges greatly affect the durability of insulators employed in high voltage power lines [46]. The challenge here is that the influence is indirect: the electric field induces drop oscillations and deformations and these then influence the field concentration around the drop, especially at the three-phase contact line through drop deformation. The second aim was directed at the influence of electric fields on the heterogeneous nucleation rate of supercooled liquid sessile drops. This phenomenon is also of relevance to power transmission, where icing influences the performance of the insulators [7]. Ice accretion alters the shape of the insulators by bridging the space between the weather sheds and may lead to increased creeping currents or to a flash-over. The initial interest in SP-C5 lay in the effect of strong electric fields on sessile drops, whereby not only the strength and nature of the electric field was interesting—alternating, transient or direct voltage, but also the orientation, either tangential or normal. Rather early it became clear, that also the charge on the droplet would play a significant role and one example of this influence is pictured in the image sequences of Fig. 17.

Fig. 17

Reprinted (adapted) figure with permission from [27], Copyright 2020 by the American Physical Society

Comparison of one cycle of an uncharged and a charged drop in resonance mode 1. a Uncharged drop with a volume of 20 $\upmu$l and a voltage frequency of 27 Hz at an electric field strength of 3.81 kV/cm and b charge drop (0.646 nC) with a volume of 20 $\upmu$l and a voltage frequency of 217 Hz at an electric field strength of 4.42 kV/cm.

The drop deformation due to an applied electric field is presumed also to occur on sessile drops residing on insulators of power transmission lines. Especially the deformed drop shape at the three-phase contact line leads to strong electric field concentration and, thus, to a higher probability of partial discharges. Such partial discharges can severely deteriorate the surface of the insulating material and enhance the aging process. In SP-C5 the partial discharge inception field strength could be quantitatively determined and influencing factors identified such as electric field frequency, surface inclination, presence of further droplets, constant or alternating electric field, contact angle, or electric charge on the droplet. A second focus in SP-C5 was the influence of an applied electric field on the nucleation of supercooled sessile drops. Again, solidification of sessile drops on powerline insulators, represent an increased hazard of creeping currents or flash-overs. This was a particular challenging task, since nucleation is a process only captured by statistical models and the question arises whether the influence of an electric field is significant enough to be distinguished above the inherent statistical various of the nucleation? This demanded the development of a dedicated facility to lower the variability introduced in normal experimental procedure. Based on this, experiments with a large ensemble of samples could then be performed, increasing the statistical significance of the results. The conditions under which an electric field can have an influence on heterogeneous nucleation was then quantified in dependence of various boundary conditions.

5 Conclusions

This introductory chapter has outlined the structure of the Collaborative Research Center SFB-TRR 75 and illustrated some selected scientific achievements of this project. This joint initiative involved scientists at the University of Stuttgart, the TU Darmstadt, the TU Berlin, and the German Aerospace Center (DLR) in Lampoldshausen. The project started in January 2010 and ended in June 2022. Within the CRC a strong interaction of the individual projects and involved scientists took place. Great progress can be reported in the area of droplet dynamics related to extreme ambient conditions in this project over the past 12 years. This progress is related to the development of analytical models, for example for phase change problems, sophisticated numerical methods, e.g. for droplet motion in compressible flows, for phase change problems involving droplets or droplet motion in strong electric fields. All analytical and numerical model developments have been supported and validated by highly sophisticated experiments in all investigated areas. After the development of new or improved experimental, theoretical and numerical methods, these methods were increasingly used to study interaction mechanisms and complex fluid systems. The new findings were made publicly available in numerous publications and by selected data sets via the homepage of the CRC: www.sfbtrr75.de.