Imposing mixed Dirichlet-Neumann-Robin boundary conditions on irregular domains in a level set/ghost fluid based finite difference framework
Introduction
Imposing accurate boundary conditions on potentially evolving, irregular domains is one of the core building blocks in computational fluid dynamics (CFD), and has countless applications in heat and mass transfer, multiphase flows and many other important engineering fields. For instance, the injected liquid in an internal combustion engine deforms drastically from a simple jet to irregular topologies including sheets, ligaments, parcels and droplets [1]. Meanwhile, the liquid evaporates in the hot ambient gas, producing significant mass, momentum and energy transfers on such complex gas-liquid phase interfaces, where multiple boundary conditions are required.
To solve this kind of problem, there is an essential prerequisite to efficiently describe the shape of the irregular domains (denoted as interface in the remainder of the paper). A natural technique is the finite element method (FEM) [2,3], which uses unstructured meshes to comfort to the interface. However, generating high-quality meshes is not a trivial task and is time-consuming especially for complex interface geometries since skewed meshes can limit the accuracy of the method. Furthermore, in evolving interface problems, a re-meshing procedure is required to be performed frequently, which may introduce a considerable work burden [3]. These concerns motivate the development of immersed techniques based on fixed structured meshes. The immersed boundary method (IBM) stands as a typical representative. This numerical strategy [4,5] is simple and easy-to-implement, belonging to the diffuse interface methods that represent the interface with finite width and smear out the singular/discontinuous information, e.g. by using a discrete Dirac-delta function, on the interface. But due to such numerical smearing, unphysical continuities are enforced on the interface that in turn decrease the accuracy in imposing boundary conditions. On the opposite, the sharp interface methods consider the interface as infinitely thin, treat the singular/discontinuous conditions without any smoothening across the interface, and usually solves the governing equations in each sub-domain. To improve the interface description, advanced methods have been proposed including the immersed interface method (IIM) [6,7] and the Voronoi interface method (VIM) [8]. Despite improvements in eliminating the accuracy loss near the interface, these methods usually require complex implementations and cost significant effort to extend to general applications especially in three dimensions (3D). Recently, some articles [9], [10], [11] have coupled the IBM with an additional scalar function whose iso-surface can implicitly indicate the interface location so that the shape and evolution of the interface can be efficiently represented. The core idea of this modification is actually inspired by the level set method (LSM), which is one of the most popular interface-resolved methods in the last three decades [1,[12], [13], [14], [15].
The LSM implicitly represents the interface as the zero level set of the signed distance function, and therefore can automatically handle the topological changes. Once the LSM filed is accurate and available, it is straightforward to calculate the normal and curvature of the interface. Moreover, the LSM proves to be the easiest method for parallelization and extension to 3D configurations [15]. Indeed, the fully parallelized LSM has been developed even on adaptive quadtree/octree Cartesian grids [16], and the fast sweeping method for maintaining the signed-distance property of the level set function has also be been parallelized in [17,18]. Due to its simplicity, the LSM has been widely implemented in modelling multiphase flows [12,19,20]. Despite of inherent advantages, the LSM suffers from unphysical non-conservation of fluid mass as time evolves. To improve this crucial issue, numerous attempts have been made [14,[21], [22], [23], [24], [25], [26]. Recently, Luo et al. [1] have summarized the advantages and disadvantages of the LSM over other popular interface-resolved methods including the front tracking method [27], the volume of fluid method [28] and the phase field method [29], and outlined numerical obstacles with related state-of-the-art strategies. Interested readers could refer to [1] and the references therein.
As aforementioned, the fluid-fluid or fluid-solid interactions may require different types of boundary conditions on the well-captured interface to describe the inherent physical constrains. Specifically, Dirichlet boundary conditions fix the value (for instance the temperature) on the interface while Neumann boundary conditions prescribe the normal derivative (for instance the heat flux). As for more general and complicated Robin boundary conditions, a linear combination of the temperature and heat flux is determined. It is noteworthy that the strategies for imposing boundary conditions are closely related to the adopted interface description method: such as [30], [31], [32], [33], [34] for the FEM, [35], [36], [37] for the IBM, [38,39] for the IIM and [8] for the VIM. In the framework of LSM, which is the primary concern of this paper, efforts have been continuously made to improve the accuracy of boundary enforcements and to simplify the implementations. Chen et al. [40] presented a simple LSM for solving Stefan problems on arbitrarily-shaped domains, which utilizes a quadratic extrapolation based finite difference method (FDM) to implicitly impose the Dirichlet boundary conditions with second-order accuracy. Similar work can be found in Udaykumar et al. [41], where the applications of Dirichlet and Neumann boundary conditions prove to be second-order accurate. In such methods, the boundary conditions are not straightforward to impose due to the implicit definition of the interface, which is a difficulty in the LSM community.
To this end, fictious points are introduced and their ghost values are restricted by boundary conditions. In fact, progresses have been made with the advent of the ghost fluid method (GFM) [42]. The GFM usually places the fictious points along coordinate axes so that the boundary conditions can be treated independently in a dimension-by-dimension pattern. For instance, Gibou et al. proposed a second-order FDM for imposing Dirichlet conditions on regular Cartesian grids [43], and then generalized it to fourth-order accurate [44] and on non-graded adaptive Cartesian grids [45]. However, the GFM neglects the tangential flux terms in order to project the boundary conditions into the coordinate directions separately. Note that this treatment only produces first-order accurate solutions and the gradients do not converge [46], [47], [48]. For this reason, a second-order finite volume method (FVM) has been developed for Neumann [49] and Robin boundary conditions [50], [51], [52], where the area integrals of a diffusive term can be rewritten as the line integrals of a normal derivative by applying the divergence theorem and further simplified by substituting the boundary conditions. Related reviews for numerical obstacles and state-of-art strategies on imposing boundary conditions in the LSM community can be found in [53,54]. Most of these methods, however, cannot be straightforwardly generalized into the special case of mixed boundary conditions [55]. To impose mixed Dirichlet, Neumann and Robin boundary conditions, a hybrid finite difference/finite volume method leveraging on the work of [43,49,50] has been proposed by Helgadóttir et al. [56]. This method preserves the matrix symmetry and achieves second-order accuracy, but it may be hampered in the framework of pure FDM, which represents a popular choice in CFD. Recently, Chai et al. [57] proposed a second-order FDM for Robin boundary conditions, where special care has been devoted to successively extrapolating the normal derivative and ghost value in the normal direction via a linear partial differential equation (PDE) approach, and applied it to heat and mass transfer problems [58]. This method provides a possible solution for imposing mixed boundary conditions in a pure FDM framework.
Challenges of developing such a unified FDM for mixed boundary conditions appear as the contradiction between accuracy and implementation. Specifically, those easy-to-implement methods such as [46], [47], [48] suffer from a loss of accuracy inherent in neglecting the tangential terms. On the other hand, more accurate methods such as [35,55,59] usually consist of complex implementations especially in the case of evolving interface. To address this issue, we further generalize our previous novel FDM in [57] to effectively deal with the mixed boundary conditions on irregular domains in this work. Note that the accuracy of the imposed boundary conditions is closely related to the adopted LSM. Numerous efforts have been made in the literature to develop high-order LSM. For instance, the gradient-augmented level set method [60] addresses the level set function and its gradient within each cell, and thus the sub-grid structure is represented to high-accuracy. However, the implementations for such high-order LSM are usually complex. Interested readers could refer to [1] and the references therein. Since second-order accurate treatments for mixed boundary conditions can meet the demands for most applications, this paper considers the second-order accurate methods. The reminder of this paper is organized as follows. Section 2 presents the governing equations. The details of the proposed method for mixed boundary conditions are then described in Section 3 while numerical validations are performed in Section 4. Finally, some conclusions are drawn in Section 5.
Section snippets
Poisson equation with mixed boundary conditions
The Poisson equation can abstract the core problem and thus is central to PDE based simulations. For instance, when applying the famous projection method [61] to discretize the incompressible Navier–Stokes equations, one should solve the pressure Poisson equation. In this paper, the Poisson equation is given bywhere is the discrete variable stored in the center of Cartesian cells and g is the source term. The mixed boundary conditions work on the interface and can be generally expressed as
Imposing mixed boundary conditions
As introduced previously, difficulties of imposing mixed boundary conditions appear as the contradiction between accuracy and implementation in multidimensional cases where the interface normal direction does not coincide with the coordinate directions. To address this issue, the present paper develops a unified and efficient FDM in the LSM community for mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains leveraging on the work of [57]. The core of this method lies in
Numerical validations
This section validates the proposed method in Poisson problems with mixed boundary conditions. The formulated Poisson problems are from [56,57] and each example poses its own emphasis. The L1 and L∞ norms are used to study the convergence accuracy.
Conclusions
Imposing mixed Dirichlet, Neumann and Robin boundary conditions on irregular domains is one of the core building blocks in CFD. To this end, the present paper has developed a pure FDM, which is a popular choice in CFD, in a level set/ghost fluid framework to efficiently impose mixed boundary conditions on an arbitrarily-shaped interface.
The method uses the LSM to describe the interface implicitly. As a consequence, it can be straightforwardly generalized into 3D evolving-interface problems.
CRediT authorship contribution statement
Min Chai: Conceptualization, Methodology, Investigation, Writing - original draft. Kun Luo: Supervision, Writing - review & editing. Haiou Wang: Validation, Visualization. Shuihua Zheng: Supervision, Writing - review & editing. Jianren Fan: Supervision, Resources.
Declaration of Competing Interest
The authors declare that there is no conflict of interest.
Acknowledgements
This work is financially supported by the National Natural Science Foundation of China (Nos. 91741203 and 51976193).
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