Advertisement

Multiple Scale Considerations for Sound Generation in Low Mach Number Flow

  • R. Fortenbach
  • E. Frénod
  • R. Klein
  • C. D. Munz
  • E. Sonnendrücker
Part of the Notes on Numerical Fluid Mechanics and Multidisciplinary Design (NNFM) book series (NNFM, volume 82)

Summary

The calculation of sound generated by fluid flow at low Mach numbers formulates a multiple scale problem due to the very different space and energy scales involved. This leads to a difficult task for the construction of numerical methods which capture sound generation and propagation as well. In this paper, a multiple scale asymptotic analysis in space and in time is used to approach this problem. The analysis results in perturbation equations with source terms describing the sound propagation and generation. Furthermore this has some consequences for direct numerical simulations of flow and acoustics which are captured in numerical results for a co-rotating vortex pair. A multiple time scale asymptotic analysis covering the problem of high frequency acoustic waves is performed for a model equation and convergence results are given in one space dimension.

Keywords

Mach Number Acoustic Wave Euler Equation Sound Generation Incompressible Limit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal. 23 (1992), 1482–1518.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S.C. Crow, Aerodynamic sound emission as a singular perturbation problem, Stud. Appl. Math. 49 (1970), 21–44.Google Scholar
  3. [3]
    A. T. Fedorchenko, On some fundamental flaws in present aeroacoustic theory, Journal of Sound and Vibration 232 (2000), 719–782.CrossRefGoogle Scholar
  4. [4]
    E. Frénod, P. A. Raviart, and E. Sonnendrücker, Two-scale expansion of a singularly perturbed convection equation, J. Pures. App. Math. 80 (2001), 815–843.zbMATHGoogle Scholar
  5. [5]
    E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 (2001), 1227–1247.zbMATHGoogle Scholar
  6. [6]
    E. Grenier, Oscillatory perturbation of the Navier-Stokes equations, J. Maths. Pures Appl. 76 (1997), 477–498.zbMATHMathSciNetGoogle Scholar
  7. [7]
    S. Klainerman and A. Majda, Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math. 34 (1981), 481–524. MR 84d:35089Google Scholar
  8. [8]
    S. Klainerman and A. Majda, Compressible and incompressible fluids, Comm. Pure Appl. Math. 35 (1982), 629–651. MR 84a:35264zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R. Klein, Semi-implicit extension of a Godunov-type scheme based on low Mach number asymptotics I: One dimensional flow, J. of Comp. Phys. 121 (1995), 213–237.zbMATHCrossRefGoogle Scholar
  10. [10]
    M. J. Lighthill, On sound generated aerodynamically I, General theory, Proceedings of Royal Society London 211 (1952), 565–587.MathSciNetGoogle Scholar
  11. [11]
    G. N’Guetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal. 20 (1989), no. 3, 608–623.CrossRefMathSciNetGoogle Scholar
  12. [12]
    S. Roller and C.-D. Munz, The Multiple Pressure Variables Method for low Mach number flows, AIAA 99–0174, Proceedings of the 37th Annual Meeting and Exhibit, January, 11–14, 1999, Reno, NV.Google Scholar
  13. [13]
    S. Roller, C.-D. Munz, R. Klein, and K. J. Geratz, The extension of incompressible flow solvers to the weakly compressible regime, Computers and Fluids, to appear in (2002).Google Scholar
  14. [14]
    D. W. Davis, S. A. Slimon, M. C. Soteriou, Development of computational aeroacoustics equations for subsonic flows using a mach number expansion approach, Journal of Computational Physics 159 (2000), 377–406.zbMATHCrossRefGoogle Scholar
  15. [15]
    S. Schochet, Fast singular limit of hyperbolic PDEs, J. Diff. Equ. 114 (1994), 476–512.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • R. Fortenbach
    • 1
  • E. Frénod
    • 2
  • R. Klein
    • 3
  • C. D. Munz
    • 1
  • E. Sonnendrücker
    • 4
  1. 1.Institut für Aero- und GasdynamikUniversität StuttgartStuttgartGermany
  2. 2.Laboratoire de Mathématiques et Applications des MathématiquesUniversité de Bretagne SudVannes CedexFrance
  3. 3.Fachbereich Mathematik und InformatikFreie Universität BerlinBerlinGermany
  4. 4.Institut de Recherche Mathématique AvancéeUniversité Louis PasteurStrasbourg CedexFrance

Personalised recommendations