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Model Reduction on Approximate Inertial Manifolds for NS Equations through Multilevel Finite Element Method and Hierarchical Basis

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Dynamics and Fault Diagnosis of Nonlinear Rotors and Impellers

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 34))

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Abstract

A numerical method is proposed to approach the Approximate Inertial Manifolds (AIMs) in unsteady incompressible Navier-Stokes equations, using multilevel finite element method with hierarchical basis functions. Following AIMS, the unknown variables, velocity and pressure in the governing equations, are divided into two components, namely low modes and high modes. Then, the couplings between low modes and high modes, which are not accounted by standard Galerkin method, are considered by AIMs, to improve the accuracy of the numerical results. Further, the multilevel finite element method with hierarchical basis functions is introduced to approach low modes and high modes in an efficient way. As an example, the flow around airfoil NACA0012 at different angles of attack has been simulated by the method presented, and the comparisons show that there is a good agreement between the present method and experimental results. In particular, the proposed method takes less computing time than the traditional method. As a conclusion, the present method is efficient in numerical analysis of fluid dynamics, especially in computing time.

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Acknowledgements

The research is supported by the National Basic Research Program of China (973 Program, Grant No. 2012CB026002) and the National Natural Science Foundation of China (Grant No.51305355).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Lahore, Lahore, Punjab, Pakistan

    M. Nauman Aslam

  2. School of Energy and Power Engineering, Xi’an Jiaotong University Xi’an, Xi’an, China

    Jiazhong Zhang & Nannan Dang

  3. Faculty of Science, Yibin University, Yibin, China

    Riaz Ahmad

Authors
  1. M. Nauman Aslam
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  2. Jiazhong Zhang
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  3. Nannan Dang
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  4. Riaz Ahmad
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Editors and Affiliations

  1. School of Energy and Power Engineering, Xi’an Jiaotong University, Shaanxi Province, China

    Jiazhong Zhang

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Aslam, M.N., Zhang, J., Dang, N., Ahmad, R. (2022). Model Reduction on Approximate Inertial Manifolds for NS Equations through Multilevel Finite Element Method and Hierarchical Basis. In: Zhang, J. (eds) Dynamics and Fault Diagnosis of Nonlinear Rotors and Impellers . Nonlinear Systems and Complexity, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-030-94301-1_11

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  • DOI: https://doi.org/10.1007/978-3-030-94301-1_11

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-94300-4

  • Online ISBN: 978-3-030-94301-1

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