A Variational Multiscale method with immersed boundary conditions for incompressible flows
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RECENT ADVANCES IN COMPUTATIONAL MECHANICS AND INNOVATIVE MATERIALS
A Variational Multiscale method with immersed boundary conditions for incompressible flows Soonpil Kang . Arif Masud
Received: 3 March 2020 / Accepted: 10 August 2020 Ó Springer Nature B.V. 2020
Abstract This paper presents a new stabilized form of incompressible Navier–Stokes equations for weak enforcement of Dirichlet boundary conditions at immersed boundaries. The boundary terms are derived via the Variational Multiscale (VMS) method which involves solving the fine-scale variational problem locally within a narrow band along the boundary. The fine-scale model is then variationally embedded into the coarse-scale form that yields a stabilized method which is free of user defined parameters. The derived boundary terms weakly enforce the Dirichlet boundary conditions along the immersed boundaries that may not align with the inter-element edges in the mesh. A unique feature of this rigorous derivation is that the structure of the stabilization tensor which emerges is naturally endowed with the mathematical attributes of area-averaging and stress-averaging. The method is implemented using 4-node quadrilateral and 8-node hexahedral elements. A set of 2D and 3D benchmark problems is presented that investigate the mathematical attributes of the method. These test cases show that the proposed method is mathematically robust as well as computationally stable and accurate for
In honor of Professor J.N. Reddy for his 75th Birthday. S. Kang A. Masud (&) Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA e-mail: [email protected]
modeling boundary layers around immersed objects in the fluid domain. Keywords Weakly imposed essential boundary conditions Immersed boundary Interfacial stabilization Variational Multiscale method Incompressible fluids
1 Introduction In the modeling of flows around submerged objects with involved geometrical configurations, generating body-fitted meshes where nodal points are coincident with the surface profiles has traditionally been considered a bottleneck in Computational Fluid Dynamics (CFD). Surface-fitted meshes not only require advanced mesh generation skills, but also consume substantial time and effort in the modeling and analysis process. This is especially true in applications to biofluid dynamics where patient specific geometries and moving immersed fluid–solid interfaces, namely, modeling of heart valves and ventricle assist devices (VAD), further add to the complexity of the problem. Immersed boundary methods that can help remove these restrictions on the compatibility of the mesh with the immersed objects are poised to make CFD accessible to a wider community. Considerable efforts
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have been devoted to the development of such methods and several techniques have been proposed in the literature that have achieved various levels of success. A common feature in all these methods
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