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Compressible Flows with Vortical Disturbances Around a Cascade of Loaded Airfoils

  • Jisheng Fang
  • Hafiz M. Atassi

Abstract

A highly efficient numerical method is developed for three-dimensional, periodic, vortical flows around a cascade of loaded airfoils. The method uses the approximation of the rapid distortion theory and thus fully accounts for the effects of distortion of the vortical disturbances as they propagate through the spatially varying mean flow. The numerical scheme is based on the splitting of the unsteady velocity into a vortical part which is a known function of the upstream flow conditions and the Lagrangian coordinates of the mean flow, and a potential part satisfying a nonconstant-coefficient, inhomogeneous, convective wave equation. A new downstream out-flow boundary condition is derived to complete the unsteady boundary value problem. By writing the downstream condition in terms of the unsteady pressure, the difficulties associated with the singular behavior of the unsteady vortical velocity are avoided. This condition is relatively simple and suitable for numerical computations. By using a body-fitted coordinate system, the unsteady potential is computed in the frequency domain. Results are presented to demonstrate the effects of mean blade loading on the aerodynamic response of a cascade to vortical gust excitations.

Keywords

Mach Number Compressible Flow Blade Surface Unsteady Pressure Blade Loading 
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.

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References

  1. [AA80]
    T. J. Akai and H. M. Atassi. Aerodynamic and aeroelastic characteristics of oscillating loaded cascades at low mach number, part 2: Stability and flutter boundaries. Trans. ASME A: Journal of Engineering for Power, 102: 524–356, 1980.CrossRefGoogle Scholar
  2. [AG89]
    H. M. Atassi and J. Grzedzinski. Unsteady disturbances of streaming motions around bodies. J. Fluid Mech., 209: 385–403, 1989.MathSciNetADSMATHCrossRefGoogle Scholar
  3. [Ata84]
    H. M. Atassi. The sears problem for a lifting airfoil revisited-new results. J. Fluid Mech., 141: 109–122, 1984.ADSMATHCrossRefGoogle Scholar
  4. [Ata86]
    H.M. Atassi. Unsteady vortical disturbances about bodies. In Proceedings of the Tenth U. S. National Congress of Applied Mechanics. J.P. Lamb, ed., ASME, 1986.Google Scholar
  5. [BP54]
    G. K. Batchelor and I. Proudman. The effect of rapid distortion of a fluid in turbulent motion. Quart. J. Mech. Appl. Math., 1: 83–103, 1954.MathSciNetCrossRefGoogle Scholar
  6. [Fan91]
    J. Fang. Compressible Flows with Vortical Disturbances Around A Cascade of Airfoils. PhD thesis, University of Notre Dame, April 1991.Google Scholar
  7. [GA76]
    M. E. Goldstein and H. M. Atassi. A complete second-order theory for the unsteady flow about an airfoils due to a periodic gust. J. Fluid Mech., 74: 741–765, 1976.ADSMATHCrossRefGoogle Scholar
  8. [Go176]
    Marvin E. Goldstein. Aeroacoustics. McGraw-Hill, 1976.Google Scholar
  9. [Go178]
    M. E. Goldstein. Unsteady vortical and entropic distortions of potential flows round arbitrary obstacles. J. Fluid Mech., 89: 433–468, 1978.ADSMATHCrossRefGoogle Scholar
  10. [HV91]
    K.C. Hall and J. M. Verdon. Gust response analysis for cascade operating in nonuniform mean flows. AIAA Journal, 29 (9): 1463–1471, September 1991.ADSCrossRefGoogle Scholar
  11. [Kur74]
    M. Kurosaka. On the unsteady supersonic cascade with a subsonic leading edge–an exact first order theory: Part 1 and 2. Trans. ASME A: Journal of Engineering for Power, 96: 13–31, 1974.CrossRefGoogle Scholar
  12. [McF84]
    E. R. McFarland. A rapid blade-to-blade solution for use in turbomachinery design. Journal of Engineering for Gas Turbines and Power, 106: 376–382, 1984.CrossRefGoogle Scholar
  13. [NW78]
    T. Nagashima and D. S. Whitehead. Linearized supersonic unsteady flows in cascade. Technical Report 3811, A.R.C., 1978.Google Scholar
  14. [Pra33]
    L. Prandtl. Attaining a steady air stream in wind tunnels. Technical Report Tech. Memo., No. 726, N.A.S.A., 1933.Google Scholar
  15. [SA90]
    J. S. Scott and H. M. Atassi. Numerical solutions of the linearized euler equations for unsteady vortical flows around lifting airfoils. In 28th Aerospace Sciences Meeting, Reno. AIAA Paper, 1990.Google Scholar
  16. [Sco90]
    J. S. Scott. Compressible Flows with Periodic Vortical Disturbances around Lift Airfoils. PhD thesis, University of Notre Dame, April 1990.Google Scholar
  17. [VC80]
    J. M. Verdon and J. R. Caspar. Subsonic flow past an oscillating cascade with finite mean flow deflection. AIAA J., 18: 524–356, 1980.CrossRefGoogle Scholar
  18. [VC84]
    J. M. Verdon and J. R. Caspar. A linearized unsteady aerodynamic analysis for transonic cascade. J. Fluid Mech., 149: 403–429, 1984.ADSMATHCrossRefGoogle Scholar
  19. [Ven80]
    C.S. Ventres. A computer program to calculate cascade 2-d kernel. In NASA Technic Memorandum. NASA, 1980.Google Scholar
  20. [Ver77]
    J. M. Verdon. Further development in the aerodynamic analysis of unsteady supersonic cascade, part 1: The unsteady pressure field, and part 2: Aerodynamic response predictions. Trans. ASME A: Journal of Engineering for Power, 99: 509–525, 1977.CrossRefGoogle Scholar
  21. [Ver87]
    J. M. Verdon. Linearized unsteady aerodynamic theory. In AGARD Manual on Aeroelasticity in Axial-Flow Turbomachinery, volume 1. M. F. Platzer and F. O. Carta, Editors, 1987.Google Scholar
  22. [Ver89]
    J. M. Verdon. The unsteady flow in the far field of an isolated blade row. Journal of Fluids and Structures, 3: 123–149, 1989.ADSMATHCrossRefGoogle Scholar
  23. [WG80]
    D. S. Whitehead and R. J. Grant. Force and moment coefficients for high deflection cascade. In 2nt Intl. Symp. on Aeroelasticity in Turbomachines. P. Suter, Editors, 1980.Google Scholar
  24. [Whi60]
    D.S. Whitehead. Force and moment coefficient for vibrating aerofoils in cascade. In A.R.C. 22133. Cambridge Univ. Engineering Department, 1960.Google Scholar
  25. [Whi70]
    D.S. Whitehead. Vibration and sound generation in a cascade of flat plates in subsonic flow. In A.R.C. 32017. Cambridge Univ. Engineering Department, 1970.Google Scholar
  26. [Whi87]
    D. S. Whitehead. Classical two-dimensional method. In AGARD Manual on Aeroelasticity in Axial-Flow Turbomachinery, volume 1. M. F. Platzer and F. O. Carta, Editors, 1987.Google Scholar
  27. [WN85]
    D.S. Whitehead and S.G. Newton. A finite element method for the solution of two-dimensional transonic flows in cascades. International Journal for Numerical Method in Fluids, 5: 115–132, 1985.ADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York, Inc. 1993

Authors and Affiliations

  • Jisheng Fang
    • 1
  • Hafiz M. Atassi
    • 1
  1. 1.Aerospace and Mechanical Engineering DepartmentUniversity of Notre DameNotre DameUSA

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