Effect of a new synthetic bubble model on forces in simulations of two-phase flows in tube bundles
Introduction
With the increase of computational power, numerical modelling of two-phase flow has gained importance in the last two decades. On the one hand, some studies focus on accurate numerical modelling of the bubble shape. In that case, it is customary to start from a bubble which is already present inside the domain [1], [2]. On the other hand, some numerical studies start from the assumption that only relatively small bubbles occur in a continuous liquid flow, for which typically Eulerian–Lagrangian modelling [3] is appropriate.
The focus of this research is the inlet modelling of two-phase flow with multiple large bubbles. These flows are important as they cause the largest forces in tube bundles [4]. However, the aforementioned methods are less applicable for these large bubbles. On the one hand, accurate modelling of the interface curvature is not required and therefore encompasses an unnecessary computational cost. On the other hand, in the Eulerian–Lagrangian approach, bubble interaction, impact or shape change should be modelled explicitly, e.g. based on coalescence criteria [5], but this is only done in a limited number of studies. One example is the work by Trapp [6], who successfully applied an Eulerian–Lagrangian framework to calculate bubble growth and changing flow topology, but the pressure drop, bubble velocity and forces on the surrounding or immersed tubes have not been validated. Moreover, a Lagrangian model for the bubbles is closed with semi-empirical models for lift and drag forces [7] which are typically not sufficiently accurate for large air structures such as encountered in slug or intermittent flow. Some attempts have been made to increase the accuracy of such closure models in an Eulerian–Lagrangian framework [8], [9], but these have not been tested on large air structures of arbitrary form at this point, whereas non-spherical bubble shapes appear regularly in axial flow through a tube bundle [10].
This leads to the conclusion that the Eulerian–Lagrangian approach is less suitable than the so-called one-fluid models like the Volume-Of-Fluid (VOF) method when investigating two-phase flows with large bubbles. A one-fluid calculation requires a long precursor domain if steady gas jets are imposed at the inlet. These jets need to break up to form a realistic flow field with large bubbles [11]. In this paper, a new model is proposed which defines a transient inlet boundary condition with bubbles of varying size occurring at a variable location in space and time. This novel approach is coined the “Synthetic Bubble Model” (SBM). To the author’s knowledge, no method described in literature efficiently generates such large bubbles with a (uniform) distribution in size and location. The newly-proposed SBM allows the definition of spherical bubbles of variable size at a varying time instant and at a stochastically determined location on the inlet face. The transient inlet boundary condition is to be applied in a subsequent Computational Fluid Dynamics (CFD) simulation.
The novel model definition is specifically aimed at external flow applications, although it is also applicable to internal flow. However, in internal flow examples, i.e. pipe flow, it is reasonably straightforward to introduce stratified or slug flow at the inlet. The former does not require a transient condition whereas the latter can be modelled with a periodic sequence of air bubbles and water slugs, as was done in previous work [12]. Furthermore, the spatial bubble distribution of bubbles in pipe flow is naturally limited by the tube diameter. By contrast, the definition of a transient inlet boundary condition in external flow applications is more challenging due to the larger available space and the typical lack of temporal periodicity, which is why the SBM is only tested in external flow cases.
Firstly, the algorithm will be discussed in Section 2. After the description of the CFD model in Section 3, the performance of the SBM will be evaluated on a test case containing a square 5-by-3 tube bundle subjected to an axially flowing air/water mixture. The tube diameter and array pitch match the experiments of Ren et al. [13] and Liu et al. [10]. A simulation using the boundary condition defined with the SBM is compared to two precursor domain calculations in Section 4. Finally, the model will be applied to a staggered tube bundle subjected to cross-flow and compared to empirical data of Zhang et al. [14] in Section 5.
Section snippets
Description of the Synthetic Bubble Model (SBM)
In this section, a new model is proposed to generate a transient inlet boundary condition to be used in a subsequent Volume-Of-Fluid simulation. The so-called Synthetic Bubble Model (SBM) introduces spherical bubbles of variable size which are stochastic in space and time. Firstly, the concept behind the model is described. Afterwards, the algorithm is discussed in more detail. Additionally, the complete code is published online on the following website: //github.com/ldmoerlo/InletModelling_VOF
Computational Fluid Dynamics (CFD)
In the following paragraphs, two different cases will be tested in the open-source finite volume software package OpenFOAM®, more specifically using the solver interFoam. This solver assumes incompressible, immiscible and isothermal phases. In the case of externally flowing air–water mixtures at low Mach numbers and at ambient conditions, this assumption is reasonable. The solver has been validated over a range of applications [15], [16], [17], [18]. For the most part, the modelling parameters
5-by-3 tube bundle subjected to axial flow
The SBM which is described in Section 2 will now be applied to a case study comprising of the 5-by-3 tube bundle subjected to axial flow. The tube diameter and array pitch match the experiments of Liu et al. [10] and Ren et al. [13]. The CFD simulation with the transient inlet condition defined by the new model is compared to the flow obtained from calculations using a steady air jet inlet boundary condition. The accuracy of the simulations is evaluated through comparison of the flow profile
Conclusion
An important parameter influencing the performance of Volume-Of-Fluid simulations of two-phase flows is the phase distribution in the region of interest. In this paper, the influence of the inlet condition is quantified by considering the flow-induced forces on tubes in an array. In two separate case studies, steady inlet conditions – consisting of one or multiple air jets surrounded by water – are compared to a transient inlet profile of stochastic nature. The latter is created by application
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
The authors gratefully acknowledge the funding by the Research Foundation-Flanders (FWO), Belgium, through the Ph.D. fellowship of Laurent De Moerloose. The computational resources (Stevin Supercomputer Infrastructure) and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by Ghent University, FWO and the Flemish Government department EWI .
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