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Stabilized Finite Element Formulation and High-Performance Solver for Slightly Compressible Navier–Stokes Equations

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Advances in Fluid Mechanics

Part of the book series: Forum for Interdisciplinary Mathematics ((FFIM))

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Abstract

In this study, a high-performance stabilized finite element-based Navier–Stokes formulation for slightly compressible flows is presented for aeroacoustic modeling. Slightly compressible flows are often used to describe waves such as the acoustic waves that have isentropic properties. Here, the slight compressibility is added onto the incompressible form of N–S equations without the full implementation of the coupled energy equation with the continuity and momentum equations. However, the inclusion of the slight compressibility can cause severe numerical instabilities in finite element formulations due to the difficulties that arise in stabilization techniques and parameters. Here, SUPG/PSPG stabilization techniques are adopted for its stabilized, equal-order scheme for finite elements and Newton’s method is used to solve the linearized system. To solve the system matrix efficiently utilizing parallelization processes, a tailored preconditioner based on an incomplete Schur complement is derived and implemented. Two numerical tests are studied to examine the solver’s accuracy in solving aeroacoustic problems. We further evaluate its efficiency performance with different variations of parallel ILU factorization in the preconditioner. The numerical results indicate that the solver has great performance in both accuracy and efficiency with different aeroacoustic problems for various time scales. This study also demonstrates the effectiveness of modeling aeroacoustics using time-based finite element formulation.

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Acknowledgements

Author LTZ would like to acknowledge partial supports from NIH-2R01DC005642-14 and Sandia National Laboratories.

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Authors and Affiliations

  1. Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

    Feimi Yu & Lucy T. Zhang

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  1. Feimi Yu
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  2. Lucy T. Zhang
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Corresponding author

Correspondence to Lucy T. Zhang .

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Editors and Affiliations

  1. School of Basic Sciences and Humanities, German Jordanian University, Amman, Jordan

    Dia Zeidan

  2. Department of Mechanical, Aerospace & Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

    Lucy T. Zhang

  3. ISAE-ENSMA, Institut Pprime, Chasseneuil-du-Poitou, France

    Eric Goncalves Da Silva

  4. MNZ, Leipzig University of Applied Sciences, Leipzig, Germany

    Jochen Merker

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Yu, F., Zhang, L.T. (2022). Stabilized Finite Element Formulation and High-Performance Solver for Slightly Compressible Navier–Stokes Equations. In: Zeidan, D., Zhang, L.T., Da Silva, E.G., Merker, J. (eds) Advances in Fluid Mechanics. Forum for Interdisciplinary Mathematics. Springer, Singapore. https://doi.org/10.1007/978-981-19-1438-6_6

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