DOI QR코드

DOI QR Code

Nonlinear Stability Analysis of Boundary Layers by using Nonlinear Parabolized Stabiltiy Equations

Nonlinear PSE를 이용한 경계층의 비선형 안정성 해석

  • 박동훈 (KAIST 항공우주공학과 대학원) ;
  • 박승오 (KAIST 항공우주공학과)
  • Received : 2011.04.25
  • Accepted : 2011.08.18
  • Published : 2011.09.01

Abstract

Nonlinear Parabolized Stability Equations(NSPE) can be effectively used to study more throughly the transition process. NPSE can efficiently analyze the stability of a nonlinear region in transition process with low computational cost compared to Direct Numerical Simulation(DNS). In this study, NPSE in general coordinate system is formulated and a computer code to solve numerically the equations is developed. Benchmark problems for incompressible and compressible boundary layers over a flat plate are analyzed to validate the present code. It is confirmed that the NPSE methodology constructed in this study is an efficient and effective tool for nonlinear stability analysis.

비선형 포물형 안정성 방정식(Nonlinear Parabolized Stability Equations, NPSE)은 보다 전체적인 천이 과정 연구에 효과적으로 사용될 수 있다. NPSE는 천이 과정에서 비선형 구간의 안정성을 직접 수치 모사(Direct Numerical Simulation, DNS)에 비해 적은 계산 비용을 사용하여 효율적으로 해석 할 수 있다. 본 연구에서는 일반 좌표계에서의 NPSE를 구성하고, 수치 계산을 위한 코드를 개발하였다. 코드의 검증을 위해 비압축성 및 압축성 평판 경계층에서의 벤치마크 문제들을 해석하였다. 본 연구의 NSPE 해석 기법이 비선형 안정성 연구에 효율적이고 효과적인 방법임을 확인하였다.

Keywords

References

  1. E. Reshotko, "Boundary-Layer Stability and Transition", Annu. Rev. Fluid Mech., Vol.9, pp. 311-349, 1976.
  2. A. Fedorov, "Transition and Stability of High-Speed Boundary Layers", Annu. Rev. Fluid Mech., Vol. 43, pp. 79,-95, 2011. https://doi.org/10.1146/annurev-fluid-122109-160750
  3. W. S. Saric, H. L. Reed, and E. J. Kerschen, "Boundary-Layer Receptivity to Freestream Disturbances", Annu. Rev. Fluid Mech., Vol. 34, pp. 291-319, 2002. https://doi.org/10.1146/annurev.fluid.34.082701.161921
  4. L. M. Mack, "Boundary-Layer Linear Stability Theory", AGARD CP-709, NATO, Belgium, 1984.
  5. H. L. Reed and W. S. Saric, "Linear Stability Theory Applied to Boundary Layers", Annu. Rev. Fluid Mech., Vol. 28, pp. 389-428, 1996. https://doi.org/10.1146/annurev.fl.28.010196.002133
  6. J. T. Stuart, "Nonlinear Stability Theory", Annu. Rev. Fluid Mech., Vol.3, pp.347-370, 1971. https://doi.org/10.1146/annurev.fl.03.010171.002023
  7. W. S. Saric, R. B. Carrillo, and M. S. Reibert, "Nonlinear Stability and Transition in 3-D Boundary Layers", Meccanica, Vol.33, pp. 469-487, 1998. https://doi.org/10.1023/A:1004368526215
  8. F. P. Bertolotti and T. Herbert, and P. R. Spalart, "Linear and nonlinear stability of the Blasius boundary layer", Journal of Fluid Mechanics, Vol. 242, pp. 441-474, 1992. https://doi.org/10.1017/S0022112092002453
  9. C. L. Chang, M. R. Malik, G. Erlebacher, and M. Y. Hussaini, "Linear and Nonlinear PSE for Compressible Boundary Layers", NASA CR-191537, ICASE Report No. 93-70, 1993.
  10. R. D. Joslin, C. L. Streett, and C. L. Chang, "Validation of Three-Dimensional Incompressible Spatial Direct Numerical Simulation Code", NASA TP-3205, 1992.
  11. C. D. Pruett, T. A. Zang, C. L. Chang, and M. H. Carpenter, "Spatial Direct Numerical Simulation of High-Speed Boundary Layer Flows Part I: Algorithmic Considerations and Validation", Theoret. Comput. Fluid Dynamics, Vol. 7, pp. 49-76, 1995. https://doi.org/10.1007/BF00312399
  12. L. Jiang, C.-L. Chang, M. Choudhari, and C. Liu, "Instability-wave propagation in boundary-layer flows at subsonic through hypersonic Mach numbers", Mathematics and Computers in Simulation, Vol. 65, pp. 469-487, 2004. https://doi.org/10.1016/j.matcom.2004.01.011
  13. C. S. J. Mayer, H. F. Fasel, M. Choudhari, and C. L. Chang, "Detailed Comparison of DNS with PSE for Oblique Breakdown at Mach 3", AIAA 2010-4596, 2010.
  14. Bertolotti, F.P., "Linear and Nonlinear Stability of Boundary Layers with Streamwise Varying Properties", Ph.D.Dissertation, The Ohio State University, 1990.
  15. N. M. El-Hady, "Nonparallel instability of supersonic and hypersonic boundary layers", Phys. Fluids A. Vol. 3, No. 9, pp. 2164-2178, 1991. https://doi.org/10.1063/1.857898
  16. T. Hebert, "Parabolized Stability Equations", AGARD R-793, 1993.
  17. T. Hebert, "Parabolized Stability Equations", Annu. Rev. Fluid Mech., Vol. 29, pp. 245-283, 1997. https://doi.org/10.1146/annurev.fluid.29.1.245
  18. M. R. Malik, "Finite-Difference Solution of the Compressible Stability Eigenvalue Problem", NASA CR-3584, 1982.
  19. C.-L. Chang and M.R. Malik, "Non-Parallel Stability of Compressible Boundary Layers", AIAA 93-2912, 1993.
  20. C-L. Chang and M.R. Malik, "Oblique-mode breakdown and secondary instability in supersonic boundary layers", J. Fluid Mechanics, Vol. 273, pp. 323-360, 1994. https://doi.org/10.1017/S0022112094001965
  21. M. R. Malik, F. Li, M. M. Choudhari, and C.-L. Chang, "Secondary instability of crossflow vortices and swept-wing boundary-layer transition", J. Fluid Mech, Vol. 399, pp. 85-115, 1999. https://doi.org/10.1017/S0022112099006291
  22. C.-L. Chang, "Langley Stability and Transition Analysis Code(LASTRAC) Version 1.2 User Manual", NASA TM-2004-213233, 2004.
  23. F.Li and M.R.Malik, "On the Nature of PSE Approximation", Theoret. Comput. Fluid Dynamics, Vol. 8, pp. 253-273, 1996. https://doi.org/10.1007/BF00639695
  24. P. Andersson, D. S. Henningson, and A. Hanifi, "On a stabilization procedure for the parabolic stability equations", Journal of Engineering Mathematics, Vol. 33, pp. 311-322, 1998. https://doi.org/10.1023/A:1004367704897
  25. www.netlib.org/lapack
  26. M. R. Malik, "Numerical Methods for Hypersonic Boundary Layer Stability", Journal of Computational Physics, Vol. 86, pp. 376-413, 1990. https://doi.org/10.1016/0021-9991(90)90106-B
  27. C.-L. Chang and M. R. Malik, "Compressible Stability of Growing Boundary Layers Using Parabolized Stability Equation", AIAA 91-1636, 1991.
  28. T. Herbert "Secondary Instability of Boundary Layers", Annu. Rev. Fluid Mech., Vol. 20, pp. 487-526, 1988. https://doi.org/10.1146/annurev.fl.20.010188.002415
  29. N. M. El-Hady, "Secondary Subharmonic Instability of Boundary Layers With Pressure Gradient and Suction", NASA CR-4112, 1988.
  30. Y. S. Kachanov and V. Y. Levchenko, "The resonant interaction of disturbances at laminar-turbulent transition in a boundary layer", Journal of Fluid Mechanics, Vol. 138, pp. 209-247, 1984. https://doi.org/10.1017/S0022112084000100
  31. H. Fasel, A. Thumm, and H. Bestek, "Direct numerical simulation of transition in supersonic boundary layer: oblique breakdown", In Transitional and Turbulent Compressible FLows, FED 151, pp. 77-92, ASME, 1993.
  32. C. S. J. Mayer, D. A. Von Terzi, and H. F. Fasel, "Direct numerical simulation of complete transition to turbulence via obliqe breakdown at Mach 3", Journal of Fluid Mechanics, 2011.
  33. T. Herbert, "Studies of Boundary-Layer Receptivity with Parabolzied Stability Equations", AIAA 93-3053, 1993.
  34. A. M. O. Smith and N. Gamberoni, "Transition, Pressure Gradient, and Stability Theory", Rep. No. ES.26388, Douglas Aircr. Co., Inc.,1956.
  35. G. Schrauf, T. Herbert, and G. Stuckert, "Evaluation of Transition in Flight Tests Using Nonlinear PSE Analysis", AIAA 95-1801, 1995.
  36. C. L. Chang and M. Choudhari, "Boundary-Layer Receptivity and Integrated Transition Prediction", AIAA 2005-0526, 2005.
  37. M. R. Malik and C. L. Chang, "Nonparallel and nonlinear stability of supersonic jet flow", Computers & Fluids, Vol. 29, pp. 327-365, 2000. https://doi.org/10.1016/S0045-7930(99)00013-4
  38. T. Colonius, A. Samanta, and K. Gudmundsson, "Parabolized stability equation models of large-scale jet mixing noise", Procedia Engineering, Vol. 6, pp. 64-73, 2010. https://doi.org/10.1016/j.proeng.2010.09.008

Cited by

  1. Velocity profile measurement of supersonic boundary layer over a flat plate using the PIV technique vol.44, pp.6, 2016, https://doi.org/10.5139/JKSAS.2016.44.6.477