Advertisement

Formulation of the problem of sonic boom by a maneuvering aerofoil as a one-parameter family of Cauchy problems

  • S. Baskar
  • Phoolan Prasad
Article

Abstract

For the structure of a sonic boom produced by a simple aerofoil at a large distance from its source we take a physical model which consists of a leading shock (LS), a trailing shock (TS) and a one-parameter family of nonlinear wavefronts in between the two shocks. Then we develop a mathematical model and show that according to this model the LS is governed by a hyperbolic system of equations in conservation form and the system of equations governing the TS has a pair of complex eigenvalues. Similarly, we show that a nonlinear wavefront originating from a point on the front part of the aerofoil is governed by a hyperbolic system of conservation laws and that originating from a point on the rear part is governed by a system of conservation laws, which is elliptic. Consequently, we expect the geometry of the TS to be kink-free and topologically different from the geometry of the LS. In the last section we point out an evidence of kinks on the LS and kink-free TS from the numerical solution of the Euler’s equations by Inoue, Sakai and Nishida [5].

Keywords

Sonic boom shock propagation ray theory elliptic equation conservation laws Cauchy problem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Baskar S, Prasad P, Riemann problem for kinematical conservation laws andgeometric features of a nonlinear wavefront,IMA J. Appl. Maths 69 (2004) 391–420zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    Baskar S and Prasad P, Propagation of curved shock fronts using shock ray theory and comparison with other theories,J. Fluid Mech. 523 (2005) 171–198zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Courant R and Friedrichs K O, Supersonic flow and shock waves (New York: Interscience Publishers) (1948)zbMATHGoogle Scholar
  4. [4]
    Crow S C, Distortion of sonic bangs by atmospheric turbulance,J. Fluid Mech. 37 (1969) 529–563, NPL Aero Report 1260zbMATHCrossRefGoogle Scholar
  5. [5]
    Inoue O, Sakai T and Nishida M, Focusing shock wave generated by an accelerating projectile,Fluid Dynamics Research 21 (1997) 403–416CrossRefGoogle Scholar
  6. [6]
    Kevlahan N K R, The propagation of weak shocks in non-uniform flow,J. Fluid Mech. 327 (1996) 167–197CrossRefMathSciNetGoogle Scholar
  7. [7]
    Morton K W, Prasad P and Ravindran R, Conservation forms of nonlinear ray equations, Tech. Rep., 2, Dept. of Mathematics, Indian Institute of Science (1992)Google Scholar
  8. [8]
    Monica A and Prasad P, Propagation of a curved weak shock,J. Fluid Mech. 434 (2001) 119–151zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Pilon A and Lyrintzis A, A data-parallel TVD method for sonic boom calculations, Preprint 95-003 (Minnesota: University of Minnesota) (1995)Google Scholar
  10. [10]
    Plotkin K J, State of the art of sonic boom modeling,J. Acoust. Soc. Am. 111(1) (2002) 530–536CrossRefGoogle Scholar
  11. [11]
    Prasad P, Nonlinear hyperbolic waves in multi-dimensions, Chapman and Hall/CRC, Monographs and Surveys in Pure and Applied Mathematics 121, (2001)Google Scholar
  12. [12]
    Prasad P and Sangeeta K, Numerical simulation of converging nonlinear wavefronts,J. Fluid Mech. 385 (1999) 1–20zbMATHCrossRefGoogle Scholar
  13. [13]
    Whitham G, Linear and nonlinear waves (New York: John Wiley & Sons) (1974)zbMATHGoogle Scholar

Copyright information

© Indian Academy of Sciences 2006

Authors and Affiliations

  • S. Baskar
    • 1
  • Phoolan Prasad
    • 1
  1. 1.Department of MathematicsIndian Institute of ScienceBangaloreIndia

Personalised recommendations