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Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows

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Abstract

An input/output framework is used to analyze the sensitivity of two- and three-dimensional disturbances in a compressible boundary layer for changes in wall and momentum forcing. The sensitivity is defined as the gradient of the kinetic disturbance energy at a given downstream position with respect to the forcing. The gradients are derived using the parabolized stability equations (PSE) and their adjoint (APSE). The adjoint equations are derived in a consistent way for a quasi-two-dimensional compressible flow in an orthogonal curvilinear coordinate system. The input/output framework provides a basis for optimal control studies. Analysis of two-dimensional boundary layers for Mach numbers between 0 and 1.2 show that wall and momentum forcing close to branch I of the neutral stability curve give the maximum magnitude of the gradient. Forcing at the wall gives the largest magnitude using the wall normal velocity component. In case of incompressible flow, the two-dimensional disturbances are the most sensitive ones to wall inhomogeneity. For compressible flow, the three-dimensional disturbances are the most sensitive ones. Further, it is shown that momentum forcing is most effectively done in the vicinity of the critical layer.

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References

  1. Airiau, C., Non-parallel acoustic receptivity of a Blasius boundary layer using an adjoint approach. Flow,Turbulence and Combustion 65 (2000)347–367.

    Article  MATH  Google Scholar 

  2. Airiau, C., Walther, S. and Bottaro, A., Non-parallel receptivity and the adjoint PSE. In: Fasel, H. and Saric, W. (eds), IUTAM Symposium on Laminar-Turbulent Transition, Sedona, AZ, September 1999. Springer-Verlag, Berlin (2000).

    Google Scholar 

  3. Andersson, Berggren,M. and Henningson, D.S., <nt> </nt>Optimal disturbances and bypass transition in boundary layer. Phys. Fluids 11 (1999)134–150.

    Article  MathSciNet  ADS  Google Scholar 

  4. Andersson, P., Henningson, D.S. and Hanifi, A., On a stabilization procedure for the parabolic stability equations. J. Engrg. Math. 33 (1998)311–332.

    Article  MATH  MathSciNet  Google Scholar 

  5. Bewley, T.R., Temam, R. and Ziane, M., A general framework for robust control in fluid mechanics. Physica D 138 (2000)360–392.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  6. Bertolotti, F.P., Herbert, Th. and Spalart, S.P., Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242 (1992)441–474.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  7. Cathalifaud, P. and Luchini, P., Algebraic growth in a boundary layer: Optimal control by blowing and suction at the wall. Eur. J. Mech. B/Fluids 19(4) (2000)469–490

    Article  MATH  MathSciNet  Google Scholar 

  8. Choudhari, M. and Street, C.L., A finite Reynolds-number approach for the prediction of boundary-layer receptivity in localized regions. Phys. Fluids A 4 (1992)2495–2514.

    Google Scholar 

  9. Corke, T.C., Bar-Sever, A. and Morkovin, M.V., Experiments on transition enhancement by distributed roughness. Phys. Fluids. 29 (1986)3199–3213.

    Article  ADS  Google Scholar 

  10. Crouch, J.D., Localized receptivity of boundary layers. Phys. Fluids A 4 (1992)1408–1414.

    Article  MATH  ADS  Google Scholar 

  11. Crouch, J.D., Non-localized receptivity of boundary layers. J. FluidMech. 244 (1992)567–581.

    MATH  MathSciNet  ADS  Google Scholar 

  12. Goldstein, M.E., The evolution of Tollmien-Schlichting waves near the leading edge. J. Fluid Mech. 127 (1983)59–81.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Goldstein, M.E. and Hultgren, L.S., Boundary-layer receptivity to long-wave free-stream disturbances. Ann. Rev. Fluid Mech. 21 (1989)137–166.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Gunzburger, M., Adjoint equation-based methods for control problems in incompressible, viscous flows. Flow, Turbulence and Combustion 65 (2000)249–272.

    Article  MATH  MathSciNet  Google Scholar 

  15. Hall, M.C.G., Application of adjoint sensitivity theory to an atmospheric general circulation model. J. Atmospheric Sci. 43 (1986)2644–2651.

    Article  ADS  Google Scholar 

  16. Hanifi, A., Henningson, D.S., Hein, S., Bertolotti, F.P. and Simen, M., Linear non-local instability analysis-The linear NOLOT code. FFA TN 1994–54 (1994), see also Hein et al.

  17. Hanifi, A. and Schmid, P.J. and Henningson, D.S., Transient growth in compressible boundary layer flow. Phys. Fluids 8(3) (1996)826–837.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Hein, S., Bertolotti, F.P., Simen, M., Hanifi, A. and Henningson, D.S., Th., Parabolized stability equations. AGARD Report No. 793 (1994)pp.4–1–4–34.

  19. Hill, D.C., Adjoint systems and their role in the receptivity problem for boundary layers. J. Fluid Mech. 292 (1995)183–204.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Hill, D.C., Receptivity in non-parallel boundary layers. In: Proceedings of the 1997 Fluids Engineering Division Summer Meeting, FEDS97–3108, Vancouver, Canada, June 22–26. ASME, New York (1997).

    Google Scholar 

  21. Högberg, M. and Berggren, M., Numerical approaches to optimal control of a model equation for shear flow instabilities. Flow,Turbulence and Combustion (2000) submitted.

  22. Li, F. and Malik, M.R., On the nature of PSE approximation. Theor. Comput. Fluid Dynam. 8 (1996)253–273.

    MATH  ADS  Google Scholar 

  23. Luchini, P. and Bottaro, A., Görtler vortices: A backward-in-time approach to the receptivity problem. J. Fluid Mech. 363 (1998)1–23.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Luchini, P., Reynolds-number-independent instability of the boundarylayer over a flat surface: Optimal perturbations. J. Fluid Mech. 404 (2000)289–309.

    Article  MATH  MathSciNet  ADS  Google Scholar 

  25. Mack, L.M., Boundary-layer stability theory. AGARD Report No. 709 (1986)pp.3–1–3–81.

  26. Nishioka, M. and Morkovin, M.V., Boundary-layer receptivity to unsteady pressure gradients: experiments and overview. J. Fluid Mech. 171 (1986)219–261.

    Article  ADS  Google Scholar 

  27. Pralits, J.O., Hanifi, A. and Henningson, D.S., Adjoint-based suction optimization for 3D boundary layer flows. FFA TN (2000)2000–58.

  28. Saric, W.S., Physical description of boundary-layer transition: Experimental evidence. AGARD Report No. 793 (1993)pp.183–204.

  29. Simen, M., Local and nonlocal stability theory of spatially varying flows. In: Hussaini, H.Y., Kumar, A. and Streett, C.L. (eds), Instability, Transition and Turbulence. Springer-Verlag, Berlin (1992)pp.181–201.

    Google Scholar 

  30. Thomas, A., The control of boundary-layer transition using a wave superposition principle. J. Fluid Mech. 137 (1983)233–250.

    Article  ADS  Google Scholar 

  31. Tumin, A., Receptivity of pipe Poiseuille flow. J. Fluid Mech. 315 (1996)119–137.

    Article  MATH  ADS  Google Scholar 

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Authors and Affiliations

  1. Department of Mechanics, KTH, S-100 44, Stockholm, Sweden

    J.O. Pralits

  2. Institut de Mécanique des Fluides de Toulouse, Allée du professeur Camille Soula, F-31400, Toulouse, France

    C. Airiau

  3. FOI, Aeronautics Division, FFA, Swedish Defence Research Agency, S-172 90, Stockholm, Sweden

    A. Hanifi & D.S. Henningson

Authors
  1. J.O. Pralits
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  2. C. Airiau
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  3. A. Hanifi
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  4. D.S. Henningson
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Pralits, J., Airiau, C., Hanifi, A. et al. Sensitivity Analysis Using Adjoint Parabolized Stability Equations for Compressible Flows. Flow, Turbulence and Combustion 65, 321–346 (2000). https://doi.org/10.1023/A:1011434805046

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