Transferring the boundary conditions to the middle surface for the numerical solution of a boundary value problem in the linear wing theory

Authors

  • I.V. Pisarev
  • A.V. Setukha

Keywords:

numerical methods
boundary value problems
Laplace equation
integral equations
vortex methods
theory of finite span wings

Abstract

A three-dimensional boundary value problem is considered for the Laplace equation in the framework of an ideal incompressible fluid model in the linear theory of finite span wings. For the numerical solution of this problem, an approach based on the method of potentials and boundary integral equations is used. The thickness of the wing is taken into account in the formulation of the boundary value problem at the middle surface with transferring the boundary conditions to this surface. As a result, the problem is reduced to a system of two-dimensional singular integro-differential equations. A numerical method is proposed for solving these equations on the basis of the vortex-frame method. The efficiency of the proposed method is illustrated by the example of determining the pressure distribution along the surface of the wing.

New computational technologies

Downloads

Published

2014-03-03

Issue

Vol. 15 (2014): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biographies

I.V. Pisarev

Orel State University named after I.S. Turgenev,
Faculty of Physics and Mathematics
• PhD Student

A.V. Setukha

Lomonosov Moscow State University,
Research Computing Center
• Leading Researcher

References

  1. Лойцянский Л.Г. Механика жидкости и газа. М.: Физматгиз, 1959.
  2. Katz J., Plotcin A. Low-speed aerodynamics. New York: Cambridge Univ. Press, 2001.
  3. Белоцерковский С.М., Ништ М.И. Отрывное и безотрывное обтекание тонких крыльев идеальной жидкостью. М.: Наука, 1978.
  4. Лифанов И.К. Метод сингулярных интегральных уравнений и численный эксперимент. М.: Янус, 1995.
  5. Fearn R.L. Airfoil aerodynamics using panel methods // The Mathematica J. 2008. 10, N 4. 725-739.
  6. Clark R.P., Smits A.J. Thrust production and wake structure of a batoid-inspired oscillating fin // J. Fluid Mech. 2006. 562. 415-429.
  7. Persson P.-O., Willis D.J., Peraire J. Numerical simulation of flapping wings using a panel method and a high-order Navier-Stokes solver // Int. J. Numer. Meth. Engng. 2012. 89, N 10. 1296-1316.
  8. Stanford B.K., Beran P.S. Analytical sensitivity analysis of an unsteady vortex-lattice method for flapping-wing optimization // J. Aircraft. 2010. 47, N 2. 647-662.
  9. Uzol O., Yavrucuk I., Sezer-Uzol N. Panel-method-based path planning and collaborative target tracking for swarming micro air vehicles // J. Aircraft. 2010. 47, N 2. 544-550.
  10. Kim J.W., Park S.H., Yu Y.H. Euler and Navier-Stokes simulations of helicopter rotor blade in forward flight using an overlapped grid solver // Proc. 19th AIAA Computational Fluid Dynamics Conf. 2009. AAIA Paper 2009-4268, pp. 1-13.
  11. Seong Y.W., Seongkyu L., Duck J.L. Potential panel and time-marching free-wake coupling analysis for helicopter rotor // J. Aircraft. 2009. 46, N 3. 1030-1041.
  12. Gennaretti M., Bernardini G. Novel boundary integral formulation for blade-vortex interaction aerodynamics of helicopter rotors // AIAA J. 2007. 45, N 6. 1169-1176.
  13. Voutsinas S.G. Vortex methods in aeronautics: how to make things work // Int. J. Comput. Fluid Dyn. 2006. 20, N 1. 3-18.
  14. Willis D.J., Peraire J., White J.K. A combined pFFT-multipole tree code, unsteady panel method with vortex particle wakes // Int. J. Numer. Meth. Fl. 2007. 53, N 8. 1399-1422.
  15. Шипилов С.Д. Применение сингулярных интегральных уравнений второго рода к расчету давления на профиле умеренной толщины // Тр. ВВИА им. Н.Е. Жуковского. 1986. Вып. 1313. 476-487.
  16. Lifanov I.K., Matveev A.F., Molyakov I.M. Flow around permeable and thick airfoils and numerical solution of singular integral equations // Russian J. Numer. Anal. Math. Modelling. 1992. 7, N 2. 109-144.
  17. Валландер С.В. Лекции по гидроаэромеханике. Л.: Изд-во Ленингр. ун-та, 1978.
  18. Вайникко Г.М., Лифанов И.К., Полтавский Л.Н. Численные методы в гиперсингулярных интегральных уравнениях и их приложения. М.: Янус, 2001.
  19. Кочин Н.Е., Кибель И.А., Розе Н.В. Теоретическая гидромеханика. М.: Физматгиз, 1963.
  20. Колтон Д., Кресс Р. Методы интегральных уравнений в теории рассеяния. М.: Мир, 1987.
  21. Гутников В.А, Лифанов И.К., Сетуха А.В. О моделировании аэродинамики зданий и сооружений методом замкнутых вихревых рамок // Изв. РАН. Механ. жидкости и газа. 2006. № 4. 78-92.