Multi-scale three-domain approach for coupling free flow and flow in porous media including droplet-related interface processes
Introduction
Drop formation at the interface between a single-phase gaseous free flow and a two-phase flow in a hydrophobic porous medium occurs in many technical applications. Examples are the water management in fuel cells, thermal insulation of building exteriors or turbine heat exchange processes. With an appropriate pressure gradient, the liquid phase in the porous medium flows towards the interface where it enters the surface pores. For liquid water in a hydrophobic porous medium, drops form on the pore throats and grow into the adjacent free-flow domain. Depending on the surrounding flow conditions, the drops might spread and merge, or be detached by the free flow.
In fuel cells, for example, the drops might block the surface and therefore prevent the reaction of air and oxygen. To describe the processes happening in such applications, two spatial scales are distinguished. On the pore-scale, detailed information about pore sizes, individual drop volumes and drop surfaces is given. The distribution of liquid and gas and their respective interfaces can be described exactly. Averaging all fluid and material properties over representative elementary volumes (REVs) allows a description on the macro-scale, which is usually sufficiently precise for real-life scenarios. On the macro-scale, phase interfaces are no longer resolved.
In order to predict the consequences of drops on coupled free-flow/porous-medium-flow systems, numerical simulators are developed, as for example in [1] and [2]. In these cases, the coupled systems of free flow and flow in porous media are modeled with the help of macro-scale concepts. Even though these models do not take detailed pore-scale information into account, the results are often precise enough for real-world scenarios. In contrast, droplet-related processes are usually studied with pore-scale models which consider properties such as interphasial areas and varying contact angles. Drop dynamics and droplet-related flow processes differ from the multi-phase flow patterns which are assumed and described on the macro-scale. Therefore, the pore-scale drop dynamics need to be combined with the macro-scale model for flow and transport processes in coupled free-flow/porous-medium-flow systems.
The aim of this work is to obtain a model concept on the REV-scale that includes pore-scale droplet-related processes. We therefore develop a multi-scale model which contains droplet-related pore-scale information. The upscaling procedure is designed in such a way that the properties which influence the exchange of mass, momentum and energy are preserved in the macro-scale description.
In the following literature review, we first refer to models for single-phase coupled free-flow/porous-medium-flow systems. Then, approaches for coupled multi-phase systems without and with the influence of drops are presented.
For single-phase systems, commonly either a one-domain or a two-domain approach is applied. In the one-domain approach, one set of balance equations describes the flow and transport processes in the whole system. In the Brinkman equation derived in [3], the Stokes equation is combined with Darcy's law to obtain one momentum balance equation which is valid in the whole domain. Spatial parameters such as porosity or permeability are defined such that they represent the porous medium or the free flow domain respectively, with a smooth transition zone in between.
In the two-domain approach, two different sets of balance equations describe the respective flow regimes. At the sharp interface in between, coupling conditions determine the exchange of mass, momentum and energy between the two domains. Such conditions for single-phase systems are analyzed in [4] and [5].
For multi-phase flows in porous media, modeling the behavior of the liquid phase at the coupling interface is challenging. The approach for compositional non-isothermal systems presented in [6] and [7] is based on the assumption that the normal water flux coming from the porous medium evaporates directly into the gaseous free flow when it reaches the interface. The same is assumed in [8] and [9]. Another possibility is to assume that the liquid phase does not reach the interface and can therefore be neglected in the coupling conditions, as implemented in [10]. Both assumptions neglect the fact that liquid drops might form and move on the porous surface in such a set-up.
A multifluid approach is applied to model the accumulation of liquid water in the gas channel of a fuel cell in [1]. At the interface, the liquid pressure is set as the capillary pressure between the liquid phase pressure in the porous medium and the pore entry pressure which is derived from the Hagen-Poiseuille equation. Another multiphase, multifluid model is presented in [11], where pending drops in the gas channel are investigated with the help of separate transport equations for each phase. In the concept presented in [2], the drops are added to the coupled free-flow/porous-medium-system by applying a mortar method. The additional degrees of freedom allow to store mass and energy in the interface. The drop volume can then be computed as an additional primary variable and is used to predict the drops' influence in fuel cells.
Within this work, we use the model developed in [2] as a base to describe the droplet-related processes. In their approach, only a small number of drops can be computed due to stability issues. In addition, it is impossible to take lateral fluxes into account. Therefore, we implement the principles of [2] in a lower-dimensional domain approach as for example presented in [12], to obtain more flexibility with respect to drop dynamics.
In the next section, we explain the model concepts for free flow, flow in porous media and drops at the interface of these two flow regimes. Section 3 deals with the coupling concept to describe the exchange of mass, momentum and energy. In Section 4, the numerical model is formulated. Then, the results of numerical simulations are presented in Section 5. Finally, a summary and outlook are given in Section 6.
Section snippets
Model concepts
We extend the existing two-domain approach for coupled free-flow/porous-medium-flow systems [6] to a three-domain approach with an additional interface domain as shown in Fig. 1. All droplet-related processes are computed within the interface domain, following the derivations in [2]. First, we take a full-dimensional interface domain into account. Similar to the approach in [12], the respective equations are then upscaled and solved in a lower-dimensional interface domain Γ which is defined
Coupling concept
Modeling coupled flow regimes requires conditions to describe the exchange of mass, momentum and energy. In the following, we present the established simple coupling concept for free-flow/porous-medium-flow systems first. Then, we introduce the three-domain approach which takes interface drops into account.
Numerical model
Discretizing the balance equations in Sections 2.1, 2.2 and 2.3 as well as the coupling conditions in Section 3.2 leads to a global non-linear system where J is the Jacobian matrix, u the vector of unknowns and b the right-hand side. The whole system is solved monolithically with the Newton's method. The structure of the global Jacobian matrix J is shown in Fig. 8. It contains three sub-matrices for the three domains on the diagonal as well as four coupling matrices on the
Results
The coupled model is implemented in DuMux ([27], [10]), an open-source simulator for flow in porous media and free-flow scenarios. The code to reproduce the results of this work is available under https://git.iws.uni-stuttgart.de/dumux-pub/Ackermann2020b.
The general set-up and the outer boundary conditions for all numerical experiments are depicted in Fig. 12. The free-flow domain extends the interface and porous-medium domains to ensure a stable velocity field.
Conclusions
We present a new multi-scale coupling concept based on a three-domain approach to model drops at the interface of a free flow and a flow in a porous medium. Criteria for drop formation, detachment and merging are formulated and evaluated to determine the drop volume evolution. Additionally, the drops' presence at the interface is taken into account when computing the coupling fluxes between free flow and porous medium.
The results presented in the previous section show that drop formation,
CRediT authorship contribution statement
Sina Ackermann: Conceptualization, Methodology, Software, Validation, Writing – original draft. Carina Bringedal: Methodology. Rainer Helmig: Conceptualization, Funding acquisition, Methodology, Writing – review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support for this work in the frame of the International Research Training Group “Droplet Interaction Technologies” (DROPIT) and the SFB 1313, Project Number 327154368.
References (27)
- et al.
Modeling drop dynamics at the interface between free and porous-medium flow using the mortar method
Int. J. Heat Mass Transf.
(2016) - et al.
Coupling compositional liquid gas Darcy and free gas flows at porous and free-flow domains interface
J. Comput. Phys.
(2016) - et al.
A discrete fracture model for two-phase flow in fractured porous media
Adv. Water Resour.
(2017) - et al.
Numerical simulation of non-isothermal multiphase multicomponent processes in porous media.: 1. an efficient solution technique
Adv. Water Resour.
(2002) - et al.
Droplet dynamics in a polymer electrolyte fuel cell gas flow channel: forces, deformation and detachment. I: theoretical and numerical analyses
J. Power Sources
(2012) - et al.
Dumux: dune for multi-phase, component, scale, physics, ... flow and transport in porous media
Adv. Water Resour.
(2011) - et al.
A computational analysis of multiphase flow through pemfc cathode porous media using the multifluid approach
J. Electrochem. Soc.
(2009) A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles
Flow Turbul. Combust.
(1949)- et al.
Navier-Stokes/Darcy coupling: modeling, analysis, and numerical approximation
Rev. Mat. Complut.
(2009) - et al.
Locally conservative coupling of Stokes and Darcy flows
SIAM J. Numer. Anal.
(2005)
A coupling concept for two-phase compositional porous-medium and single-phase compositional free flow
Water Resour. Res.
Numerical scheme for coupling two-phase compositional porous-media flow and one-phase compositional free flow
IMA J. Appl. Math.
A numerical method for a model of two-phase flow in a coupled free flow and porous media system
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