Developing a unified FVE-ALE approach to solve unsteady fluid flow with moving boundaries.

*(English)*Zbl 1423.76304Summary: An arbitrary Lagrangian-Eulerian (ALE) approach is incorporated with a mixed finite-volume-element (FVE) method to establish a novel moving boundary method for simulating unsteady incompressible flow on non-stationary meshes. The method collects the advantages of both finite-volume and finite-element (FE) methods as well as the ALE approach in a unified algorithm. In this regard, the convection terms are treated at the cell faces using a physical-influence upwinding scheme, while the diffusion terms are treated using bilinear FE shape functions. On the other hand, the performance of ALE approach is improved by using the Laplace method to improve the hybrid grids, involving triangular and quadrilateral elements, either partially or entirely. The use of hybrid FE grids facilitates this achievement. To show the robustness of the unified algorithm, we examine both the first- and the second-order temporal stencils. The accuracy and performance of the extended method are evaluated via simulating the unsteady flow fields around a fixed cylinder, a transversely oscillating cylinder, and in a channel with an indented wall. The numerical results presented demonstrate significant accuracy benefits for the new hybrid method on coarse meshes and where large time steps are taken. Of importance, the current method yields the second-order temporal accuracy when the second-order stencil is used to discretize the unsteady terms.

##### MSC:

76M12 | Finite volume methods applied to problems in fluid mechanics |

##### Keywords:

finite-volume-element method; arbitrary Lagrangian-Eulerian approach; physical influence upwinding scheme; moving boundary; second-order time accuracy; hybrid mesh
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\textit{A. Naderi} et al., Int. J. Numer. Methods Fluids 63, No. 1, 40--68 (2010; Zbl 1423.76304)

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