On the Geometric Conservation Law for the Non Linear Frequency Domain and Time-Spectral methods

Abstract

The aim of this paper is to present and validate two new procedures to enforce the Geometric Conservation Law (GCL) on a moving grid for an Arbitrary Lagrangian Eulerian (ALE) formulation of the Euler equations discretized in time for either the Non Linear Frequency Domain (NLFD) or Time-Spectral (TS) methods. The equations are spatially discretized by a structured finite-volume scheme on a hexahedral mesh. The derived methodologies follow a general approach where the positions and the velocities of the grid points are known at each time step. The integrated face mesh velocities are derived either from the Approximation of the Exact Volumetric Increments (AEVI) relative to the undeformed mesh or exactly computed based on a Trilinear Mapping (TRI-MAP) between the physical space and the computational domain. The accuracy of the AEVI method highly depends on the computation of the volumetric increments and limits the temporalorder of accuracy of the deduced integrated face mesh velocities to between one and two. Thus defeating the purpose of the NLFD method which possesses spectral rate of convergence. However, the TRI-MAP method has proven to be more computationally efficient, ensuring the satisfaction of the GCL once the convergence of the time derivative of the cell volume is reached in Fourier space. The methods are validated numerically by verifying the conservation of uniform flow and by comparing the integrated face mesh velocities to the exact values derived from the mapping. a condition limits the inherent spectral in time accuracy of the NLFD approach, an alternative method based on a trilinear mapping between the physical and computational space is introduced.The results are then extent to the Time-Spectral method. These various approaches are presented section 3. The different methodologies are numerically investigated in order to verify that the GCL are enforced through a correct computation of the integrated face mesh velocities in section 4 and 50 the results discussed in section 5. Discretization of the governing equation and mesh deformationThis section presents the formulation of the Euler equations on a moving mesh using the Arbitrary Lagrangian-Eulerian approach, its discretization using the Non-Linear Frequency Domain, and the mesh deformation method through the Radial Basis Functions. 55 Arbitrary Lagrangian-Eulerian formulation of the Euler equationsWhen solving the Euler equations on a moving grid a popular approach is to use an Arbitrary Lagrangian-Eulerian (ALE) formulation [14]. For a control volume Ω enclosed by a boundary ∂Ω and without source terms, the integral form of this formulation is given as follows :by the following corollary 3.6 :