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An Edge Based Finite Volume Approach for the Solution Ofthe Incompressible Navier-stokes Equations Onunstructured Triangular Meshes
| Content Provider | Semantic Scholar |
|---|---|
| Author | Oz, Furkan Sahin, Mehmet Özgür |
| Copyright Year | 2019 |
| Abstract | A faced based unstructured finite volume is presented for the solution of the incompressible Navier-Stokes equation on unstructured triangular meshes. The numerical method is based on the stable side-centered arrangement of primitive variables in which the velocity vector components are located at edge mid-points, meanwhile the pressure term is placed at element centroids. A special attention is given to accurately evaluate the viscous terms at the mid-point of the control volume faces. The convected terms are evaluated using the least square upwind interpolations. The resulting algebraic equations are solved in a monolithic manner. The implementation of the preconditioned Krylov subspace algorithm, matrix-matrix multiplication and the restricted additive Schwarz preconditioner are carried out using the PETSc software package in order to improve the parallel performance. The numerical method is validated for the classical benchmark problem of lid-driven cavity in a square enclosure. The numerical results indicate better accuracy for the viscous fluxes on triangular meshes. INTRODUCTION A face based unstructured finite volume formulation has been developed for the solution of the incompressible Navier-Stokes equation on unstructured triangular meshes in a fully coupled form, where the velocity vector components are defined at the center of edges, meanwhile the pressure term is defined at the element centroid. This face based approach was initially used on boundary-fitted meshes [Maliska and Raithby, 1984] and later, on unstructured triangular meshes [Hwang, 1995; Rida et al., 1997]. The same approach is employed within the finite element framework by using the stable non-conforming Q̃1/Q0 finite element pair [Rannacher and Turek, 1992]. The present face-centered approach has also been initially implemented on quadrilateral and hexagonal meshes within the Arbitrary Lagrangian-Eulerian (ALE) framework for the solution of the moving boundary problems with large displacements and rotations [Erzincanli and Sahin, 2013]. In the current work, the approach has been extended to unstructured triangular meshes in ∗Bsc. in Aeronautical Engineering Department, Email: ozf@itu.edu.tr †Prof in Astronautical Engineering Department, Email: msahin@itu.edu.tr AIAC-2019-053 Oz & Sahin two-dimensions. Although the use of triangular elements leads to increase in both edge numbers and element numbers, which leads a larger algebraic equation compared to quadrilateral elements, it allows further flexibility on mesh modifications for large deformations. In addition, the present formulation on triangular meshes leads to a symmetric mass matrix in contrast to quadrilateral elements. Furthermore, the velocity gradients required for the viscous fluxes are exactly computed at the control volume edge mid-points on highly irregular unstructured triangular meshes, which leads a more accurate numerical algorithm. The present algorithm has been implemented in an object oriented framework in C and it allows us to use the full functionality of the PETSc 3.7.7 library [Balay et al., 2018]. MATHEMATICAL AND NUMERICAL FORMULATION The governing equations for the incompressible fluid flow in the Cartesian coordinate system can be written in the following form: the continuity equation −∇ · u = 0 (1) the momentum equations ρ [ ∂u ∂t + (u · ∇)u ] +∇p = ∇ ·T (2) In these equations, ρ represents the fluid density, u is the velocity vector, p is the pressure and T = μ(∇u + ∇u>) is the viscous stress tensor. Integrating the differential equations (1) and (2) over an unstructured triangular element Ωe with boundary ∂Ωe gives the following nondimensional equations: |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | http://aiac.ae.metu.edu.tr/program/paper.php?Paper=AIAC-2019-053 |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
