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Boundary Integral Equations for Potential Flow Past Multiple Aerofoils

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Abstract

Two uniquely solvable boundary integral equations for calculating the incompressible potential flow past multiple aerofoils with smooth boundaries are presented. The kernels of the integral equations are the Neumann kernel and the adjoint Neumann kernel. Numerical examples reveal that the present method offers an effective solution technique for the potential flow problem.

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References

  1. K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.

    Book  MATH  Google Scholar 

  2. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 2000.

    Book  Google Scholar 

  3. A. Carabineanu, A boundary element approach to the 2D potential flow problem around airfoils with cusped trailing edge, Comput. Methods Appl. Mech. Eng. 129 (1996), 213–219.

    Article  MathSciNet  MATH  Google Scholar 

  4. D. G. Crowdy, Analytical solutions for uniform potential flow past multiple cylinders, Eur. J. Mech. B. Fluids. 25 (2006), 459–470.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. G. Crowdy, Calculating the lift on a finite stack of cylindrical aerofoils, Proc. R. Soc. A 462 (2006), 1387–1407.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73(2) (1987), 325–348.

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Helsing and R. Ojala, On the evaluation of layer potentials close to their sources, J. Comput. Phys. 227 (2008), 2899–2921.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Helsing and E. Wadbro, Laplace’s equation and the Dirichlet-Neumann map: a new mode for Mikhlin’s method, J. Comput. Phys. 202 (2005), 391–410.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. I. Hess and A. M. O. Smith, Calculation of potential flow about arbitrary bodies, in: D. Kuchemann (ed.), Progress in Aeronautical Sciences vol. 8, Pergamon, London, 1967, pp. 1–138.

    Google Scholar 

  10. T. Kambe, Elementary Fluid Mechanics, World Scientific, Singapore, 2007.

    Book  MATH  Google Scholar 

  11. J. Katz and A. Plotkin, Low-Speed Aerodynamics, 2nd ed., Cambridge University Press, Cambridge, 2001.

    Book  MATH  Google Scholar 

  12. S. G. Krantz, Geometric Function Theory: Explorations in Complex Analysis, Birkhäuser, Boston, 2006.

    MATH  Google Scholar 

  13. A. R. Krommer and C. W. Ueberhuber, Numerical Integration on Advanced Computer Systems, Springer-Verlag, Berlin Heidelberg, 1994.

    Book  MATH  Google Scholar 

  14. A. I. Markushevich, Theory of Functions of a Complex Variable, Vol. II, Prentice-Hall, Englewood Cliffs, N.J., 1965.

    MATH  Google Scholar 

  15. M. Mokry, Complex Variable Boundary Element Method for External Potential Flows, AIAA 90–0127, 1990.

  16. —, Potential Flow Past Airfoils as a Riemann-Hilbert problem, AIAA 96–2161, 1996.

  17. A. H. M. Murid, M. M. S. Nasser and N. S. Amin, An integral equation for the external potential flow around multi-element obstacles, presented in: Simposium Kebangsaan Sains Matematik Ke-12, IIUM, KL, Malaysia, 2004.

    Google Scholar 

  18. M. M. S. Nasser, A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. Methods Funct. Theory 9 (2009), 127–143.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. M. S. Nasser, Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. Sci. Comput. 31 (2009), 1695–1715.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. M. S. Nasser, The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions, The University Researcher (IBB University Journal) 20 (2009), 47–60.

    Google Scholar 

  21. M. M. S. Nasser, A. Murid and N. Amin, A boundary integral equation for the 2D external potential flow, Int. J. Appl. Mech. Eng. 11(1) (2006), 61–75.

    MATH  Google Scholar 

  22. M. M. S. Nasser, A. H. M. Murid, M. Ismail and E. M. A. Alejaily, A boundary integral equation with the generalized Neumann kernel for Laplace’s equation in multiply connected regions, Appl. Math. Comput. 217 (2011), 4710–4727.

    Article  MathSciNet  MATH  Google Scholar 

  23. V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60(2) (1985), 187–207.

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Saad and M. H. Schultz, GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comp. 7(3) (1986), 856–869.

    Article  MathSciNet  MATH  Google Scholar 

  25. R. Wegmann and M. M. S. Nasser, The Riemann-Hilbert problem and the generalized Neumann kernel on multiply connected regions, J. Comp. Appl. Math. 214 (2008), 36–57.

    Article  MathSciNet  MATH  Google Scholar 

  26. B. R. Williams, An exact test case for the plane potential flow about two adjacent lifting airfoils, Aeronautical research council reports and memoranda, No. 3717, 1971.

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Authors and Affiliations

  1. Faculty of Science, Department of Mathematics, King Khalid University, P. O. Box 9004, Abha, Saudi Arabia

    Mohamed M. S. Nasser

  2. Faculty of Science, Department of Mathematics, Ibb University, P. O. Box 70270, Ibb, Yemen

    Mohamed M. S. Nasser

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  1. Mohamed M. S. Nasser
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Correspondence to Mohamed M. S. Nasser.

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Nasser, M.M.S. Boundary Integral Equations for Potential Flow Past Multiple Aerofoils. Comput. Methods Funct. Theory 11, 375–394 (2012). https://doi.org/10.1007/BF03321868

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  • DOI: https://doi.org/10.1007/BF03321868

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