A multi-dimensional finite volume cell-centered direct ALE solver for hydrodynamics
Introduction
Numerical simulations of fluid flow have been extensively performed on a large range of hydrodynamic phenomena over the last 60 years. The governing equations can be implemented using the finite volume approach in a given coordinate system. Several schemes have been developed and these can be divided into three categories: the first is based on a Lagrangian description of the matter for which the computational mesh follows the flow [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13]. Mass conservation is naturally ensured in this approach, which makes Lagrangian schemes the best candidate for preserving and following interfaces in multi-material simulations. However, when the flow is subject to sharp local velocity gradients or local vorticity, mesh distortion can occur and a remeshing/remapping step is inevitable. Unfortunately, this procedure is usually expensive and the remap can be inaccurate and introduce excessive diffusion. The second category adopts an Eulerian point of view for which the computational mesh is fixed, leading to the appearance of convective terms in the governing equations, even in the mass conservation equation [14], [15], [16], [17], [18], [19]. Difficulties arise in multi-material simulations when following material interfaces. Capturing or reconstructing the interfaces can be very complex, time consuming and inaccurate (for example, the order of Young's VOF method is less than one [20]), although significant improvements have been made recently [21], [22], [23].
In the 1970s, the combination of the best properties of the Lagrangian and Eulerian methods led to the creation of a third category, known as arbitrary Lagrangian–Eulerian (ALE) methods [24]. The main feature of ALE is to decorrelate the flow motion and the mesh motion to avoid any mesh tangling which may appear in purely Lagrangian Simulations. ALE methods may be divided into two classes, indirect [25], [26], [27], [28], [29], [30] and direct [31], [32], [33], [34]. Indirect methods are implemented in three steps: (1) a Lagrangian stage in which the solution and the computational mesh are updated; (2) a rezoning stage that regularizes the tangled mesh; (3) a remapping stage in which the Lagrangian solution is transferred to the rezoned mesh. For indirect ALE the decorrelation between mesh and flow is only performed occasionally when the Lagrangian stage becomes critical and might result in simulation failure. In this paper, only direct ALE methods are considered. In these methods, the hydrodynamic equations are solved in an arbitrarily moving coordinate frame, introducing advective terms in the governing equations that depend on the grid velocity. The result is a continuous rezoning capability, with no extra time cost. Overall, ALE methods can manage greater distortions than purely Lagrangian methods but with a better accuracy than purely Eulerian approaches.
In this paper we describe a second order multi-dimensional scheme, belonging to the class of direct Arbitrary Lagrangian–Eulerian (ALE) methods, named DALEC3 since it is a Direct ALE Cell-Centered Code. It is used to solve the well-known Euler equations with the Geometrical Conservation Law for the simulation of compressible hydrodynamic problems. Recall the integral form of these equations in a frame moving at the velocity (see e.g. [34]): where the volume integral is computed over an arbitrary volume moving at velocity . The boundary of is denoted by , and denotes the unitary normal on pointing outwards. Properties characterizing the fluid are the density ρ, velocity , total energy E and pressure p. The relative velocity will be denoted . System (1)–(5) is closed with an equation of state in order to compute the pressure.
The first part of the paper describes the discrete formulation of the system (1)–(5). It is constructed upon a cell-centered Lagrangian solver completed with edge-based upwinded advective fluxes to obtain a full ALE formulation. We fill focus on the two cell-centered schemes GLACE and EUCCLHYD which have demonstrated their properties in references [6], [8], [9], [10]. The treatment of the boundary conditions is presented, as well as the CFL condition used to compute the time step and the second-order extension of the scheme. We also derive the entropy balance and show that our scheme satisfies the 2nd law of thermodynamics.
The second part of the paper presents several numerical results that assess the robustness and the accuracy of the scheme. To do so, most of the results have been obtained by computing the grid velocities as a fraction of the nodal Lagrangian velocities. However, our strategy is to couple the scheme with a method that adaptively computes the grid velocities. Such strategy is a common purpose for the development of direct ALE schemes, and a well-known method (coming from the indirect ALE methods community) is to rely on a rezoning procedure that keeps an acceptable geometrical mesh quality [32], [35], [36]. Recently, a promising technique, the Large Eddy Limitation method, has been developed by Costes and Ghidaglia [1]. The main feature of this technique is to express, from the Hodge decomposition, the nodal Lagrangian velocities as the sum of an irrotational part and a divergence-free part, and to limit each component regarding the local characteristic of the flow to obtain the grid velocities. Our purpose is to couple such method with the DALEC3 scheme. The last part of the paper then proposes a brief theoretical description of the L.E.L. method and preliminary results obtained for the single material triple point problem. Conclusions and perspectives end the paper.
Section snippets
Notation
The computational mesh is assumed to be composed of polygons. We denote the set of nodes, cells and edges defining the mesh respectively , and . In the rest of the paper, specific indices will be used respectively to denote cells, nodes and edges.
In Fig. 1, we represent a 2D polygonal cell ‘j’ whose area will be denoted . The cell is defined by the set of nodes and the set of edges . The length of the edge ‘k’ is denoted , while is the discrete outward pointing
Results
We propose in this section various 2D results to show the robustness of the scheme. Our methodology of validation acts in two phases. First, the grid velocity is computed as a fraction of the Lagrangian velocity to assess the robustness of the ALE scheme. Second, the grid velocity is given by a genuine rezone strategy [1] in order to prove that our approach can be coupled to any artificial mesh motion.
Conclusions and perspectives
In this paper, we have presented in detail a multi-dimensional Direct ALE scheme for the numerical solution to the compressible Euler equations. The scheme has been developed from a Godunov-like cell-centered Lagrangian scheme, like GLACE or EUCCLHYD, which has been supplemented with upwinded advective fluxes. The upwinded values are chosen from the sign of the edge relative velocity . The second-order extension is obtained by computing the numerical fluxes from the MUSCL-Hancock method.
Acknowledgements
This work has been supported by the CMLA located in Cachan and the French Alternative Energies and Atomic Energy Commission, DIF. Authors would like to thank Christophe Fochesato, Antoine Llor, Thibaud Vazquez-Gonzalez and Raphaël Loubère for their valuable comments and suggestions to improve the manuscript.
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