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A Variational Multiscale method with immersed boundary conditions for incompressible flows

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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 modeling boundary layers around immersed objects in the fluid domain.

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Acknowledgements

This research was partly supported by NIH Grant No. 1R01GN135921-01, and Carle-UIUC Foundation Grant No. SPA-088888. Computing resources were provided by NSF under Grant AC121/DMS 16-20231, for Advanced Computing on the Frontera Supercomputing Platform housed at the Texas Advanced Computing Center (TACC), at The University of Texas at Austin. This support is gratefully acknowledged.

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Correspondence to Arif Masud.

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In honor of Professor J.N. Reddy for his 75th Birthday.

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Appendix: Interfacial edge basis functions

Appendix: Interfacial edge basis functions

In the derivation of the VMIB formulation (31) we have assumed that the interfacial fine scales vanish at the element edges, except for the case when the interface itself is coincident with the element edge. In the former case, interface traverses through the element and therefore a part of the element belongs to the fluid subdomain while the other part belongs to the fictitious subdomain. Consequently, the interface function is non-zero within the element and becomes zero at the element boundaries, and therefore retains the definition of a bubble function. However, when the interface \( \varGamma_{I}^{e} \) is coincident with element edge Γe, the interface function becomes an edge function. For the 1D case, the interface basis functions are shown in Fig. 30.

Fig. 30
figure 30

1D interface basis functions (\( \xi \in [ - 1,1] \) is the natural coordinate and \( \bar{\xi } \) is the location of the interface.)

The design guideline is to modify the regular bubble function by moving its peak in accordance with the location of the interface. In our implementation for 2D quadrilateral and 3D hexahedral elements, the fine-scale basis function \( b_{I}^{e} \) is obtained from the underlying conventional polynomial bubble function be by shifting the position of its peak point from the center of the element to the middle point of the interface segment.The fine-scale basis function \( b_{I}^{e} \) is defined as

$$ b_{I}^{e} (\xi _{1} ,\xi _{2} ,\xi _{3} ) = \left\{ {\begin{array}{*{20}l} {\psi (\xi _{1} )\psi (\xi _{2} )\psi (\xi _{3} )} \hfill & {{\text{if }}{\mathbf{x}}(\xi _{1} ,\xi _{2} ,\xi _{3} ) \in \varOmega _{f} } \hfill \\ 0 \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right. $$
(46)

where

$$ \psi (\xi_{i} ) = \left\{ \begin{aligned} \frac{{\left( {1 - \xi_{i} } \right)\left( {1 - 2\bar{\xi }_{i} + \xi_{i} } \right)}}{{\left( {1 - \bar{\xi }_{i} } \right)^{2} }}{\text{ if }}\xi_{i} \ge \bar{\xi }_{i} \hfill \\ \frac{{\left( {1 + \xi_{i} } \right)\left( {1 + 2\bar{\xi }_{i} - \xi_{i} } \right)}}{{\left( {1 + \bar{\xi }_{i} } \right)^{2} }}{\text{ if }}\xi_{i} < \bar{\xi }_{i} \hfill \\ \end{aligned} \right. $$
(47)
$$ \bar{\xi }_{i} = \frac{{\int_{{\varGamma_{I}^{e} }} {\xi_{i} d\varGamma } }}{{\int_{{\varGamma_{I}^{e} }} {d\varGamma } }} $$
(48)

where \( \xi_{i} { (}i = 1,2,3 ) \) are the natural coordinates in the parent element and \( \bar{\xi }_{i} { (}i = 1,2,3 ) \) are the natural coordinates for the middle point of the interface segment \( \varGamma_{I}^{e} \). The 1D basis functions \( b_{I}^{e} (\xi ) = \psi (\xi ) \) are presented in Fig. 30.

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Kang, S., Masud, A. A Variational Multiscale method with immersed boundary conditions for incompressible flows. Meccanica 56, 1397–1422 (2021). https://doi.org/10.1007/s11012-020-01227-w

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