Advertisement

Applications of Mathematics

, Volume 64, Issue 2, pp 225–251 | Cite as

On suitable inlet boundary conditions for fluid-structure interaction problems in a channel

  • Jan ValášekEmail author
  • Petr Sváček
  • Jaromír Horáček
Article
  • 15 Downloads

Abstract

We are interested in the numerical solution of a two-dimensional fluid-structure interaction problem. A special attention is paid to the choice of physically relevant inlet boundary conditions for the case of channel closing. Three types of the inlet boundary conditions are considered. Beside the classical Dirichlet and the do-nothing boundary conditions also a generalized boundary condition motivated by the penalization prescription of the Dirichlet boundary condition is applied. The fluid flow is described by the incompressible Navier-Stokes equations in the arbitrary Lagrangian-Eulerian (ALE) form and the elastic body creating a part of the channel wall is modelled with the aid of linear elasticity. Both models are coupled with the boundary conditions prescribed at the common interface.

The elastic and the fluid flow problems are approximated by the finite element method. The detailed derivation of the weak formulation including the boundary conditions is presented. The pseudo-elastic approach for construction of the ALE mapping is used. Results of numerical simulations for three considered inlet boundary conditions are compared. The flutter velocity is determined for a specific model problem and it is shown that the boundary condition with the penalization approach is suitable for the case of the fluid flow in a channel with vibrating walls.

Keywords

flow-induced vibration 2D incompressible Navier-Stokes equations linear elasticity inlet boundary conditions flutter instability 

MSC 2010

76D05 65N30 65N12 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    I. Babuška: The finite element method with penalty. Math. Comput. 27 (1973), 221–228.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    T. Bodnár, G. P. Galdi, Š. Nečasová, (eds.): Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid Mechanics, Birkhäuse/Springer, Basel, 2014.zbMATHGoogle Scholar
  3. [3]
    M. Braack, P. B. Mucha: Directional do-nothing condition for the Navier-Stokes equations. J. Comput. Math. 32 (2014), 507–521.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    A. Curnier: Computational Methods in Solid Mechanics. Solid Mechanics and Its Applications 29, Kluwer Academic Publishers Group, Dordrecht, 1994.Google Scholar
  5. [5]
    D. J. Daily, S. L. Thomson: Acoustically-coupled flow-induced vibration of a computational vocal fold model. Comput. Struct. 116 (2013), 50–58.CrossRefGoogle Scholar
  6. [6]
    T. A. Davis: Direct Methods for Sparse Linear Systems. Fundamentals of Algorithms 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2006.Google Scholar
  7. [7]
    N. G. Diez, S. Belfroid, J. Golliard, (eds.): Flow-Induced Vibration & Noise. Proceedings of 11th International Conference on Flow Induced Vibration & Noise. TNO, Delft, The Hague, The NetherlandsGoogle Scholar
  8. [8]
    E. H. Dowell: A Modern Course in Aeroelasticity. Solid Mechanics and Its Applications 217, Springer, Cham, 2004.Google Scholar
  9. [9]
    M. Feistauer, J. Hasnedlová-Prokopová, J. Horáček, A. Kosík, V. Kučera: DGFEM for dynamical systems describing interaction of compressible fluid and structures. J. Comput. Appl. Math. 254 (2013), 17–30.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    M. Feistauer, P. Sváček, J. Horáček: Numerical simulation of fluid-structure interaction problems with applications to flow in vocal folds. Fluid-Structure Interaction and Biomedical Applications (T. Bodnár et al., eds.). Advances in Mathematical Fluid Mechanics, Birkhäuser/Springer, Basel, 2014, pp. 321–393.Google Scholar
  11. [11]
    L. Formaggia, N. Parolini, M. Pischedda, C. Riccobene: Geometrical multi-scale modeling of liquid packaging system: an example of scientific cross-fertilization. 19th European Conference on Mathematics for Industry (2016), 6 pages.Google Scholar
  12. [12]
    T. Gelhard, G. Lube, M. A. Olshanskii, J.-H. Starcke: Stabilized finite element schemes with LBB-stable elements for incompressible flows. J. Comput. Appl. Math. 177 (2005), 243–267.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Series in Computational Mathematics 5, Springer, Cham, 1986.Google Scholar
  14. [14]
    J. Horáček, V. V. Radolf, V. Bula, J. Košina: Experimental modelling of phonation using artificial models of human vocal folds and vocal tracts. Engineering Mechanics 2017 (V. Fuis, ed.). Brno University of Technology, Faculty of Mechanical Engineering, 2017, pp. 382–385.Google Scholar
  15. [15]
    J. Horáček, P. Šidlof, J. G. Švec: Numerical simulation of self-oscillations of human vocal folds with Hertz model of impact forces. J. Fluids Struct. 20 (2005), 853–869.CrossRefGoogle Scholar
  16. [16]
    J. Horáček, J. G. Švec: Aeroelastic model of vocal-fold-shaped vibrating element for studying the phonation threshold. J. Fluids Struct. 16 (2002), 931–955.CrossRefGoogle Scholar
  17. [17]
    J. Horáček, J. G. Švec: Instability boundaries of a vocal fold modelled as a flexibly supported rigid body vibrating in a channel conveying fluid. ASME 2002 International Mechanical Engineering Congress and Exposition. American Society of Mechanical Engineers, 2002, pp. 1043–1054.Google Scholar
  18. [18]
    C. Johnson: Numerical Solution of Partial Differential Equations by the Finite Element Method. Cambridge University Press, Cambridge, 1987.zbMATHGoogle Scholar
  19. [19]
    M. Kaltenbacher, S. Zörner, A. Hüppe: On the importance of strong fluid-solid coupling with application to human phonation. Prog. Comput. Fluid Dyn. 14 (2014), 2–13.CrossRefzbMATHGoogle Scholar
  20. [20]
    G. Link, M. Kaltenbacher, M. Breuer, M. Döllinger: A 2D finite-element scheme for fluid-solid-acoustic interactions and its application to human phonation. Comput. Methods Appl. Mech. Eng. 198 (2009), 3321–3334.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    H. Sadeghi, S. Kniesburges, M. Kaltenbacher, A. Schützenberger, M. Döllinger: Computational models of laryngeal aerodynamics: Potentials and numerical costs. Journal of Voice (2018).Google Scholar
  22. [22]
    J. H. Seo, R. Mittal: A high-order immersed boundary method for acoustic wave scattering and low-Mach number flow-induced sound in complex geometries. J. Comput. Phys. 230 (2011), 1000–1019.MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Šidlof, J. Kolář, P. Peukert: Flow-induced vibration of a long flexible sheet in tangential flow. Topical Problems of Fluid Mechanics 2018 (D. Šimurda, T. Bodnár, eds.). Institute of Thermomechanics, The Czech Academy of Sciences, Praha, 2018, pp. 251–256Google Scholar
  24. [24]
    W. S. Slaughter: The Linearized Theory of Elasticity. Birkhäuser, Boston, 2002.Google Scholar
  25. [25]
    P. Sváček, J. Horáček: Numerical simulation of glottal flow in interaction with self oscillating vocal folds: comparison of finite element approximation with a simplified model. Commun. Comput. Phys. 12 (2012), 789–806.CrossRefGoogle Scholar
  26. [26]
    P. Sváček, J. Horáček: Finite element approximation of flow induced vibrations of human vocal folds model: effects of inflow boundary conditions and the length of subglottal and supraglottal channel on phonation onset. Appl. Math. Comput. 319 (2018), 178–194.MathSciNetGoogle Scholar
  27. [27]
    N. Takashi, T. J. R. Hughes: An arbitrary Lagrangian-Eulerian finite element method for interaction of fluid and a rigid body. Comput. Methods Appl. Mech. Eng. 95 (1992), 115–138.CrossRefzbMATHGoogle Scholar
  28. [28]
    J. Valášek, M. Kaltenbacher, P. Sváček: On the application of acoustic analogies in the numerical simulation of human phonation process. Flow, Turbul. Combust. (2018), 1–15Google Scholar
  29. [29]
    J. Valášek, P. Sváček, J. Horáček: Numerical solution of fluid-structure interaction represented by human vocal folds in airflow. EPJ Web of Conferences 114 (2016), Article No. 02130, 6 pages.Google Scholar
  30. [30]
    J. Valášek, P. Sváček, J. Horáček: On finite element approximation of flow induced vibration of elastic structure. Programs and Algorithms of Numerical Mathematics 18. Proceedings of the 18th Seminar (PANM), 2016. Institute of Mathematics, Czech Academy of Sciences, Praha, 2017, pp. 144–153.zbMATHGoogle Scholar
  31. [31]
    J. Venkatramani, V. Nair, R. I. Sujith, S. Gupta, S. Sarkar: Multi-fractality in aeroelastic response as a precursor to flutter. J. Sound Vib. 386 (2017), 390–406.CrossRefGoogle Scholar
  32. [32]
    S. Zorner: Numerical Simulation Method for a Precise Calculation of the Human Phonation Under Realistic Conditions. Ph.D. Thesis, Technische Uuniversität Wien, 2013.Google Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Cz 2019

Authors and Affiliations

  • Jan Valášek
    • 1
    • 2
    Email author
  • Petr Sváček
    • 1
    • 2
  • Jaromír Horáček
    • 3
  1. 1.Department of Technical Mathematics, Faculty of Mechanical EngineeringCzech Technical University in PraguePraha 2Czech Republic
  2. 2.Center of Advanced Aerospace Technology, Faculty of Mechanical EngineeringCzech Technical University in PraguePraha 6Czech Republic
  3. 3.Institute of Thermomechanics of the Czech Academy of SciencesPraha 8Czech Republic

Personalised recommendations