Our research is focused on Computational Fluid Dynamics (CFD) by means of Direct Numerical Simulation (DNS) including the numerical modeling of interfacial transport processes and physico-chemical phenomena in two- and multiphase systems. For a detailed understanding of elementary transport and transfer processes at fluid interfaces, we actively develop techniques for different numerical simulation methods. For this purpose, we develop taylor-made high-fidelity methods which exhibit distinct advantages for the specific interfacial transport physics under consideration [10, 11, 9, 13].

These fundamentally different numerical approaches are combined under one roof and inside a single numerical framework – OpenFOAM. This open-source C++ based library for Computational Continuum Mechanics and Multiphysics allows us to provide the cutting edge numerical methods for emerging physical models and collaborate, share, and discuss with our partners in science and industry.

OpenFOAM is a versatile and flexible simulation software to address complex interfacial transport process and phenomena involved therein. The modular concept in OpenFOAM (object-oriented and generic programming) is exploited in order to use synergy in method development by means of re-use of solution and discretisation techniques independent of the specific method. For the first time thisparticularly allows the problem-specific choice of suitable methods on one platform, thus, enabling comparisons of different approaches regarding performance and accuracy in an objective manner [12].

The group actively developes both sharp and diffuse interface models:

• Volume-Of-Fluid(VOF) Interface-Capturing (algebraic and geometric) Methods are strictly volume conservative, handle strong deformations of the interface as well as topolgical changes, with one remaining challenge: an accurate and stable curvature approximation. We are developing unstructured unsplit VOF methods [1,2] as well as algebraic VOF methods for challenging viscoelastic flows [3] and species transfer [4].

• ArbitraryLagrangian-Eulerian Interface-Tracking methods represent the liquid gas interface with a part of the mesh and follow its deformation with a movement of the computational mesh. hence, we apply this method, when the main objective is the resolution of physical processes on or close to the interface, such as the adsorption of surfactants and their influence on multiphase system [8], wetting processes [7], or complex mass tranfer processes at the interface [9].

• Front-Tracking methods, although not strictly volume conservative, are highly accurate and applicable to problems that involve strong surface tension forces and interfaces that strongly deform and may change topologically. We are developing hybrid Level Set / Front Tracking methods on unstructured meshes [5,6] with the aim of extending Front Tracking with local dynamic mesh refinement and geometrically complex problems.

• Diffuse interface models involving Allen-Cahn as well as Cahn-Hilliard phase-field methods. The smooth interface representation allows for a less demanding discretization of the interface evolution. Hence it can be applied to larger cases and complex topologies such as structured surfaces with the compromise of a somewhat less accurate localization of interface and thereby an increased complexity to include additional transport processes.

We have a close and longstanding relationship with leading members of the OpenFoam community and educate the next generation of Computational Engineers in using and developing OpenFOAM.

Selected publications:

[1] Marić, T., Marschall, H., & Bothe, D. (2018). An enhanced un-split face-vertex flux-based VoF method. Journal of computational physics, 371, 967-993.

[2 Marić, T., Kothe, D. B., & Bothe, D. (2020). Unstructured un-split geometrical Volume-of-Fluid methods–A review. Journal of Computational Physics, 420, 109695.

[3] Niethammer, M., Brenn, G., Marschall, H., & Bothe, D. (2019). An extended volume of fluid method and its application to single bubbles rising in a viscoelastic liquid. Journal of Computational Physics, 387, 326-355.

[4 ]Deising, D., Marschall, H., & Bothe, D. (2016). A unified single-field model framework for Volume-Of-Fluid simulations of interfacial species transfer applied to bubbly flows. Chemical Engineering Science, 139, 173-195.

[5] Marić, T., Marschall, H., & Bothe, D. (2015). lentFoam–A hybrid Level Set/Front Tracking method on unstructured meshes. Computers & Fluids, 113, 20-31.

[6] Tolle, T., Bothe, D., & Marić, T. (2020). SAAMPLE: A Segregated Accuracy-driven Algorithm for Multiphase Pressure-Linked Equations. Computers & Fluids, 200, 104450.

[7] Gründing, D. (2020). An enhanced model for the capillary rise problem. International Journal of Multiphase Flow, 103210.

[8] Pesci, C., Weiner, A., Marschall, H., & Bothe, D. (2018). Computational analysis of single rising bubbles influenced by soluble surfactant. Journal of Fluid Mechanics, 856, 709-763.

[9] Weber, P. S., & Bothe, D. (2016). Applicability of the linearized theory of the Maxwell–Stefan equations. AIChE Journal, 62(8), 2929-2946.

[10] Bothe, D., & Fleckenstein, S. (2013). A volume-of-fluid-based method for mass transfer processes at fluid particles. Chemical Engineering Science, 101, 283-302.

[11] Gründing, D., Fleckenstein, S., & Bothe, D. (2016). A subgrid-scale model for reactive concentration boundary layers for 3D mass transfer simulations with deformable fluid interfaces. International Journal of Heat and Mass Transfer, 101, 476-487.

[12] Gründing, D., Smuda, M., Antritter, T., Fricke, M., Rettenmaier, D., Kummer, F., … & Bothe, D. (2020). A comparative study of transient capillary rise using direct numerical simulations. Applied Mathematical Modelling.

[13] Weiner, A., & Bothe, D. (2017). Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles. Journal of Computational Physics,347, 261-289.

Part of the simulations in the group of Mathematical Modeling and Analysis are performed with the inhouse Volume-Of-Fluid (VOF) code “Free Surface 3D” (FS3D), originally developed by Rieber and Frohn [1] and Rieber [2] at the Institute of Aerospace Thermodynamics (ITLR), University Stuttgart, Germany. Since then, the code has been massively expanded by the ITLR and the group of Mathematical Modeling and Analysis (MMA) in Darmstadt, Germany.

The research focus of the MMA group has been mainly on the mathematical and numerical modeling of multiphysical and multiscale two-phase flows.

A noticeable difference to original implementation of FS3D relate to substantially improved surface tension treatment within the VOF framework:

• The improved surface tension treatment of the balanced continuum surface force (bCSF) model, with the curvature being calculated from local height functions, was demonstrated for the hydrodynamics of wavy laminar falling films [4]. Simulations using the bCSF model combined with improved curvature calculation allowed to find good agreement with experiments.

• Binary collisions of shear-thinning droplets were investigated. The newly developed lamella stabilization algorithm allows to capture liquid films thinner than one computational cell with the VOF method [3]. The same technique was also used to study the onset of instabilities in high energy head-on droplet collisions [8].

A novel method to initialize the volume fraction field with high accuracy (third- to fourth-order) has been developed in [10]. Moreover, fundamental investigations on the kinematics of interfaces led to the development of novel interface reconstruction methods close to the domain boundary [11]. The latter methods allow for a kinematically consistent transport of the dynamic contact angle and thereby increase the accuracy in the numerical simulation of wetting flows.

Furthermore, a variety of physical models have been developed and applied to research topics of current interest:

• Two-phase flows with thermal Marangoni effect were studied [5], i.e. thermocapillary flows, where local changes of the interfacial temperature cause differences in the surface tension and induce a fluid flow along the interface. An additional energy transport equation is solved to obtain the temperature distribution within the two phases. For the transport of temperature and diluted species in two-phase flows, the two-scalar approach was developed. Furthermore, for the case of droplet migration on a solid wall has been investigated within the VOF framework [6].

• Rising fluid particles have been investigated with a focus on the dynamic behaviour of rising droplets [7] and the effect of surface contamination on mass transfer at rising bubbles. Recent efforts have been on the accurate modeling of physical and reactive mass transfer. Subgrid-scale models allow to capture concentration boundary layers which can be fully embedded in only one computational cell [9].

• The breakup dynamics of a wetting capillary bridge has been investigated in cooperation with the Nano- and Microfluidics group at TU Darmstadt [P1]. The interface reconstruction methods developed in [11] allowed for a detailed numerical simulation of the breakup process in good agreement with experimental data.

The current research focuses on:

• Transfer processes at dynamic contact lines

• Stability analysis of fluid particles

• Droplet collisions and splashing

• Multiphysics of reactive mass transfer

• Higher-order methods for geometry treatment in VOF

Selected external publications:

[1] Rieber, M., Frohn, A. (1999). A numerical study on the mechanism of splashing. International Journal of Heat and Fluid Flow, 20(5), 455–461. https://doi.org/10.1016/S0142-727X(99)00033-8

[2] M. Rieber: Numerische Modellierung der Dynamik freier Grenzflächen in Zweiphasenströmungen. Ph.D. thesis, Universität Stuttgart (2004)

Selected MMA Publications:

[3] Focke, C., Bothe, D. (2011). Computational analysis of binary collisions of shear-thinning droplets. Journal of Non-Newtonian Fluid Mechanics, 166(14–15), 799–810. https://doi.org/10.1016/j.jnnfm.2011.03.011

[4 ] Albert, C., Raach, H., Bothe, D. (2012). Influence of surface tension models on the hydrodynamics of wavy laminar falling films in Volume of Fluid-simulations. International Journal of Multiphase Flow, 43, 66–71. https://doi.org/10.1016/j.ijmultiphaseflow.2012.02.011

[5] Ma, C., Bothe, D. (2013). Numerical modeling of thermocapillary two-phase flows with evaporation using a two-scalar approach for heat transfer. Journal of Computational Physics, 233, 552–573. https://doi.org/10.1016/j.jcp.2012.09.011

[6] Fath, A., Bothe, D. (2015). Direct numerical simulations of thermocapillary migration of a droplet attached to a solid wall. International Journal of Multiphase Flow, 77, 209–221. https://doi.org/10.1016/j.ijmultiphaseflow.2015.08.018

[7] Albert, C., Kromer, J., Robertson, A. M., Bothe, D. (2015). Dynamic behaviour of buoyant high viscosity droplets rising in a quiescent liquid. Journal of Fluid Mechanics, 778, 485–533. https://doi.org/10.1017/jfm.2015.393

[8] Liu, M., Bothe, D. (2016). Numerical study of head-on droplet collisions at high Weber numbers. Journal of Fluid Mechanics, 789, 785–805. https://doi.org/10.1017/jfm.2015.725

[9] Weiner, A., Bothe, D. (2017). Advanced subgrid-scale modeling for convection-dominated species transport at fluid interfaces with application to mass transfer from rising bubbles. Journal of Computational Physics, 347, 261–289. https://doi.org/10.1016/j.jcp.2017.06.040

[10] Kromer, J., Bothe, D. (2019). Highly accurate computation of volume fractions using differential geometry. Journal of Computational Physics, 396, 761–784. https://doi.org/10.1016/j.jcp.2019.07.005

[11] Fricke, M., Marić, T., Bothe, D. (2020). Contact line advection using the geometrical Volume-of-Fluid method. Journal of Computational Physics, 407, 109221. https://doi.org/10.1016/j.jcp.2019.109221

Preprints:

[P1] M. Hartmann, M. Fricke, L. Hauer, D. Gründing, T. Marić, D. Bothe, S. Hardt (2020). Breakup Dynamics of Capillary Bridges on Hydrophobic Stripes, http://arxiv.org/abs/1910.01887

Tomislav Marić is an Athene Young Investigator and a research group leader on Lagrangian / Eulerian Interface Approximation Methods (LEIA). He is developing the unstructured geometrical Volume-of-Fluid method and the hybrid Level Set / Front Tracking method with the goal of simulating multiphase flows driven by surface tension forces in complex geometries.

Holger Marschall leads the research group “Computational Multiphase Flow” (CMF) group, which emerged from MMA and has been established in April 2019 in the Department of Mathematics at TU Darmstadt. The group develops simulation methods for the computer-aided prediction of transport processes in multiphase flows. The scientific focus is on the development of adaptive and hybrid approaches, which are particularly suited to bridge the large spatial and temporal scales typically present in these transport processes.