Subprojects TA-A

# Drop-Gas interaction

The subprojects (SP) of TA-A focus on modelling of evaporation processes, gas kinetic simulations and numerical methods for compressible multi-phase flows.

## SP-A1: Modelling of deformed, multi-component liquid drop evaporation

The evaporation of liquid droplets in gaseous environments, a process that appears in a multitude of industrial applications (spray combustion, fire suppression, spray drying, painting, spray cooling, crystallisation, anti-icing, medical aerosol, etc.), is the object of extensive studies since more than a century. Only sophisticated numerical approaches can accurately analyse the many interacting physical phenomena involved, including phase change coupled with heat and mass transfer, but such approaches are inherently limited by the high computational costs that are needed. Analytical models of heating and evaporation of liquid droplets become necessary when predicting dispersed flows, although the accuracy is limited by many simplifying hypotheses commonly imposed by these models: drop sphericity, quasi-steadiness, constant thermophysical properties, ideal gas behaviour and phase equilibrium at the interface, to just cite the most common ones. In the first phase of DROPIT, the relief of the constraints of drop sphericity and constancy of the properties was handled reaching satisfactory results, while maintaining the quasi-steadiness approximation. However, it was shown that this approximation can introduce non-negligible effects when the drop evaporation is followed along the entire lifetime, and this observation is the motivation of the present sub-project.

The main objective of SP-A1 is the modelling of the effect of time-varying deformation, like shrinking and oscillation, on the heat and mass transfer from liquid drops floating in a gaseous environment. Since the aim is to derive models that can be implemented in CFD codes for spray applications, the computation efficiency is of paramount importance, and an analytical approach will be used. Modelling the heat and mass transfer from shrinking/oscillating drops is inherently a moving boundary (Stefan) problem, and specialised analytic tools must be developed and used. The following tasks are envisioned:

- Heat and mass transfer from spherical shrinking drops
- Evaporation effects on the dynamics of an oscillating drop
- Dynamic effects on the evaporation of an oscillating drop

Since in many practical applications multi-component droplets are found, the species interaction during evaporation should be modelled to hold also when the species concentrations in the gaseous phase are high; to this end, proper constitutive equations for mass diffusion (Maxwell-Stefan) must be used. Two additional activities are planned:

- Species interaction modelling for deformed drop evaporation.
- Modelling of multi-component deformed drops.

Finally, a model validation activity, based on the comparison of the model predictions with available experimental/numerical data is planned.

## SP-A2 : Multi-scale modelling of the evaporation process

A variety of numerical simulations of multi-phase flows are based on the solution of the incompressible flow equations, which is a suitable approximation in many scenarios. However, in case of strong temperature gradients, the influence of pressure waves or compressibility has to be taken into account. The additional difficulty in the compressible case is mainly the strong couplingof hydrodynamics and thermodynamics via the equation of state (EOS). In addition, the computational effort is much higher and there are a lot of open questions in the numerical modelling of phase transition processes. Hence, compressible multi-phase methods are still a topic of numerical research. There are commercial flow solvers that provide approximations based on mixture approaches. But, that is tailored to specific conditions, for which a suitable mixture equation of state is available. Our objective here is to provide an efficient numerical method that allows a detailed investigation of interfacial phenomena in a thermodynamically consistent way.

The objectives of this project are based on the numerical framework and the multi-phase code that has been developed so far in the group of C.-D. Munz. The goal is to simulate evaporating, non-spherical and oscillating droplets with this code. Therefore further development of the code is required, including the validation of single-component evaporation, the extension to multi-component evaporation and the extension to multi-dimensional simulations. A close cooperation to experimental and theoretical results will be used to validate the numerical modeling. The objectives can be formulated as the following sub-goals:

● Improvement of the evaporation kernel for the singe-component case and validation against experimental data, molecular dynamics data as well as numerical results obtained by SFB-TRR 75.

● Extension of the current evaporation model to multi-components and multiple space dimensions to study non-spherical, multi-component droplets and compare the numerical results with analytical investigations in SP-A1 and with numerical investigations in SP-A5.

● Extension of the multiphase code framework by an hp-adaptive numerical scheme, to allow for the accurate and efficient resolution of interfacial phenomena. This enables more accurate and affordable simulations for both SP-A2 and SP-A6 and will thus be implemented cooperatively.

The IRTG gives the chance of the development of numerical evaporation models for multi-component fluids that are based on theoretical, experimental and numerical results and complementary validated.

## SP-A3: Gas-kinetic simulation of microdroplet - gas interaction

Most numerical approaches to handle gas-fluid interactions are based on continuum methods, assuming e.g. local thermodynamic equilibrium or quasi-steady conditions. Therefore, a comparison between simulation results with continuum assumptions and analytical methods is questionable since various non-equilibrium effects are not considered. Especially at the boundary layer between gas and liquid droplets, non-equilibrium effects can play a crucial role in the evaporation and the gas surface interaction processes. Therefore, within this project SP-A3, the interaction between gas flow and liquid droplets was simulated during the first period of DROPIT by using the Direct Simulation Monte Carlo (DSMC) method, which solves the more fundamental, gas kinetic Boltzmann equation. DSMC approximates the gas flow from a microscopic point of view using discrete simulation particles, which allow the treatment of non-equilibrium effects. In the second period of DROPIT, the Vlasov-Enskog equation will be solved by implementing an Enskog Simulation Monte Carlo (ESMC) solver. The Vlasov-Enskog equation is an extension of the Boltzmann equation for high densities, as they occur in liquids. The ESMC technique allows kinetic simulations of such dense phases and can yield molecular dynamics grade results. Thereby, even the interface region between two phases can be simulated with high accuracy, to simulate a droplet and its interaction with a surrounding media in its totality. Consequently, a comparison between DSMC or ESMC and existing or new analytical results is of fundamental interest to analyze the correctness of equilibrium assumptions and flow unsteadiness.

SP-A3 will first focus on the implementation of numerical models for solving the Enskog-Vlasov equation in the in-house code framework PICLas. The information gain and modeling depth will thereby be increased by considering more fundamental influences of microscopic modeling on properties within the droplet and evaporation process, while inherently considering droplet deformation. For this, the following three main work packages are planned:

• Extension of the existing numerical framework by implementation of Enskog collisions. In the case of the Boltzmann equation, which is solved by the DSMC method, the collision integral has a local character. Consequently, particle collisions are computed independently in each computational cell. This collision treatment will be adopted and extended. The Enskog collision integral involves the distribution function at different spatial locations because of the finite extent of a molecule. In order to solve collisions during the collisional step, the particles in a given cell interact with particles located in nearby cells. Thus, the existing DSMC collision procedure must be extended to treat the Enskog equation by additionally considering collisions between particles in neighboring cells. Furthermore, the solution of the Enskog collisions will be considered with a BGK or Fokker-Planck approach, which have the benefit to reduce the computational effort.

• Numerical modeling of attractive forces. In order to achieve phase separation and fluid clustering, the modeling of attractive forces between particles is necessary. Here, different approaches exist, e.g. a direct evaluation of the potential between the particles, a density expansion of the potential or the approximation of the potential using a screened Poisson equation. The different methods will be compared and extended to a 3D framework. Additionally, the screened Poisson equation will be solved by a state-of-the-art hybridizable discontinuous Galerkin approach, which has been already successful utilized with PICLas in high performance computer plasma simulations.

• Numerical simulation of droplet interface using the implemented method. Simulation of gas-droplets interactions with the implemented solution for the Enskog-Vlasov equation will comprise droplet-vapor separation distance, droplet evaporation and temperature (heat flux distribution and information on droplet deformation). Also, interactions of two droplets and coalescence / nucleation shall be investigated. An important part, especially within this work package, is the investigation of parallelization concepts for high performance computing in order to run simulations on cases with appropriate size. Due to the non-local collision procedure, this needs to be addressed in a special way. Thus, current and new parallelization strategies will be tested and compared.

## SP-A4: Numerical methods for compressible multi-phase flows with complex equations of state

This project focuses on development of methods of very high order of accuracy, as well as

low-Mach-compatible semi-implicit schemes for the numerical solution of diffuse interface models of

compressible multi-phase flows. In particular, we concentrate on models of the Baer-Nunziato family,

from the five equation one-pressure, one-velocity model of Kapila, to the full seven-equation

Baer-Nunziato model, with interphase drag and pressure relaxation sources. Additionally, we have at

our disposal a family of promising mathematical models that can describe general continua,

multi-material flows, viscous flows, and surface tension effects in the form of a single unified

first order hyperbolic system.

The advantage of diffuse interface models over sharp interface approaches is that it is not

necessary in the numerical algorithm to resolve the interface explicitly. Instead, it is

automatically captured in a physically based fashion by the governing partial differential equations

(PDE) system. This means that different fluids interact mainly by means of non-conservative terms

within the PDE and at the numerical level such interactions are naturally transferred to the

discrete solution as part of a unified update scheme which does not change across material

interfaces or in their proximity.

Thus the diffuse interface approach has some similarities with the very successful shock capturing

technique developed for shock waves in compressible gas dynamics, where the discontinuous shock

waves can be captured in a stable manner within a few cells on the computational grid at the aid of

a small amount of numerical dissipation. The same is true also for the phase interface in diffuse

interface approaches. However, there is a very important difference: while shock waves are generated

by genuinely nonlinear characteristic fields that automatically tend to steepen the shock front, the

material interface in compressible multiphase flows is governed by a linearly degenerated field,

hence the interface is continuously further diffused and thus more and more spread over the grid.

The main objective of this sub-project is the development of a new family of numerical methods that

is able to resolve compressible multi-phase flows with a diffuse interface approach. The final goal

is to improve the resolution of the material interface within the diffuse interface approach so that

the quality of the interface representation becomes comparable with the one of a sharp interface

method.

To solve the problem of excessive interface smearing that necessarily must be tackled when

adopting diffuse interface models, there are several possible remedies:

1. The use of very accurate and little dissipative Riemann solvers;

2. The use of very high order methods;

3. The use of adaptive mesh refinement (AMR) techniques;

4. The use of Lagrangian and arbitrary Lagrangian-Eulerian methods on moving meshes;

5. When low Mach number flow are considered, the use of ad-hoc semi-implicit schemes that are

stable with smaller numerical dissipation coefficients and that can integrate the governing

equations under a milder timestep restriction, in comparison to explicit schemes.

The research group in Trento will follow all the above lines of research in the present project.

In that sense, we aim at developing a completely new class of diffuse interface algorithms for

compressible multi-phase flows with surface tension and general equation of state. We will also

collaborate with the other members of the IRTG, i.e. with experimental as well as with theoretical

groups, in order to validate our new numerical algorithms. In particular, we consider the following

sub-goals:

1. Extension of our high order DG method with a posteriori subcell limiting to non-conservative

hyperbolic PDE systems, in particular to the Baer-Nunziato model of compressible multiphase

flows with surface tension and viscosity;

2. Development of semi-implicit schemes for two phase flow with surface tension and viscosity, adopting

a first order hyperbolic formulation for such (conventionally second order) fluxes;

3. Application of our new schemes to isolated droplets and to droplet-droplet interactions. Separately,

we will also study wall-droplet interactions. Comparison of our new diffuse interface approach based

on the Baer-Nunziato model with the sharp interface approach based on the level-set method developed

in SP-A2.

The IRTG gives the chance of the development of new high order accurate numerical methods for

compressible multi-phase flows that can then be validated against theoretical and experimental

results obtained within the IRTG, as well as with alternative numerical approaches for similar

problems.

## SP-A5: Modelling of spray evaporation

The evaporation of a spray injected in a gaseous environment is encountered in many industrial (painting, combustion engine, etc.) and civil (fire control, air humidification, etc.) applications. Among others, two effects can be considered to distinguish heat and mass transfer phenomena in a spray from those of single drops: the presence of important convective effects and the neighbouring of a large number of droplets. The first effect is related to the spray generation: a high velocity liquid jet disintegrates by aerodynamic interactions due to the drop-gas relative velocity, which is well-known to enhance heat and mass transfer with respect to the floating drop case. The presence of many droplets relatively close to each other is instead expected to decrease the evaporation rate due to the drop screening effect. Thus, the present project is motivated by the necessity to close the gap between single drop and spray evaporation modelling, and the requirement of CPU efficient predictive tools.

This sub-project has the objective to deepen the knowledge on the heat and mass transfer mechanisms occurring during the evaporation of multi-particle systems, with a particular attention to the effect of convection and droplet neighbouring, developing computationally efficient sub-models to be implemented in more complex numerical codes. The presence of neighbouring droplets is known to have a non-negligible effect on the evaporation rate. Analytic solutions for the simple case of two drops can be found in the literature, also for the case of variable thermophysical properties of the gas. The extension to drop arrays has been studied in the past, mainly relaying on numerical solutions of the energy and species conservation equations. It can be observed that the structure/shape of a drop cloud can influence the overall evaporation characteristics. Thus, a study of the relation between cloud shape and evaporation characteristics is necessary to direct the development of advanced models for heat and mass transport predictions in spray systems. The following tasks are planned:

- Modelling the effect of neighbouring evaporating drops on vapour diffusion characteristics. The point-source method will be used to approach this task in an efficient analytic form for the case of drop clouds. A preliminary evaluation of the inaccuracy due to the non-exact fulfilment of the boundary condition on the drop surface will be performed by comparison with high accuracy numerical solutions for 3D drop arrays (see picture for an example relative to a five drop planar array).
- Modelling the effect of convection by extending the film theory approach to non-spherical drops. The film theory approach used for spherical drops will be extended by defining an appropriate shape of the exterior surface of the film for spheroidal (oblate and prolate) drops, based on the matching with empirical correlations for Nusselt and Sherwood numbers. The approach will be extended to multi-component drops.
- Consistent models will be developed for the convection effects in the presence of multitudes of drops. The method developed in the previous task can be adapted to the case of drop clouds by considering the effect of interaction among the diffusional films of neighbouring drops. The case of twin-drops will be analysed first and the method will possibly be then extended to multi-drop structures by using the point-source approach.

The analytical tools developed in this sub-project can then be implemented in available CFD codes to evaluate the relative weight of the mentioned effects under a wide range of realistic operating conditions.

## SP-A6: Mathematical and numerical modelling of droplet dynamics in weakly compressible multi-component flows

In many situations, the flow fields within a liquid droplet and a surrounding gas are characterized by low Mach numbers, which additionally may differ by orders of magnitude in the different fluids. This suggests the application of a sharp-interface approach because different mathematical models and numerical discretization methods can be used in the domains occupied by the liquid and the gas. For a droplet-gas interaction we will consider several different model pairings based on the Euler or Navier-Stokes equations, precisely compressible/weakly compressible, weakly compressi-ble/weakly compressible and incompressible/weakly compressible pairings. The major modelling challenge for sharp interface models is to find appropriate coupling conditions such that the well-posedness of the overall model can be ensured. While there are results for single-component flow, multi-component and possibly temperature-dependent two-phase flows have rarely been analysed. The focus of this project will be on the analytical and numerical treatment of the multi-component regime including the effects of Maxwell-Stefan diffusion. The project directly links to SP-A1 and SP-A2 dealing with either fully incompressible or fully compressible models as well as to SP-C3 and complements their approaches.

The main goal of this sub-project is the development of reliable sharp interface models for the weakly compressible dynamics of droplet-gas interactions with multiple components and its effi-cient and reliable numerical simulation. The project is structured along the following work packages:

First we will analyse the transition from the compressible to the incompressible regime for multi-component flows starting with the inviscid Euler equations. For this asymptotic analysis via vanishing Mach numbers we can follow. Then we analyse the coupling of two multi-component fluids with different possibly small Mach numbers to cover sharp interface methods using the weakly compressible/compressible or weakly compressible/weakly compressible ap-proach. Essentially, this requires the solution of Riemann-type problems. Finally, the methods will be generalized to an incompressible/weakly compressible model.*Development and analysis of coupling conditions for droplet-gas flow with multiple components.*-
For a fully compressible model with inviscid equations, integration of Maxwell-Stefan diffusion leads still to a first-order hyperbolic system which now contains possibly stiff source terms. The components' equations are strongly coupled by the source terms. Anyhow, the sharp interface coupling as described above should work in this case. However, the technical effort will increase since the source terms change the Riemann solutions qualitatively.*Integration of Maxwell-Stefan type diffusion.* Since the sharp interface approach allows an independent choice of the mathematical model in the bulk phases (incompressible/weakly compressible/compressible), the numerical discretization schemes can be adjusted accordingly. Due to the large difference between the times scales of the pressure disturbances and the convective transport at low Mach numbers, purely explicit time integration schemes for the compressible solvers may become ineffective. Here, an implicit time integration scheme for the pressure in the liquid phase will be investigated. For the numerical approximation in the low Mach number regime a semi-implicit approach is consid-ered based on asymptotic preserving methods. The final numerical algorithm will then be compared at least for simple settings to the fully incompressible or compressible simulations in SP-A1 and SP-A2, respectively.*Development of numerical methods and comparison with alternative approaches.*

### Spokesperson

**Prof. Dr.-Ing.**

### Bernhard Weigand

Spokesman DROPIT, University of Stuttgart

**Prof. Dr.**

### Gianpietro Elvio Cossali

Spokesman for University of Bergamo