Advertisement

A critical assessment of the parabolized stability equations

  • Aaron TowneEmail author
  • Georgios Rigas
  • Tim Colonius
Original Article
  • 25 Downloads

Abstract

The parabolized stability equations (PSE) are a ubiquitous tool for studying the stability and evolution of disturbances in weakly nonparallel, convectively unstable flows. The PSE method was introduced as an alternative to asymptotic approaches to these problems. More recently, PSE has been applied with mixed results to a more diverse set of problems, often involving flows with multiple relevant instability modes. This paper investigates the limits of validity of PSE via a spectral analysis of the PSE operator. We show that PSE is capable of accurately capturing only disturbances with a single wavelength at each frequency and that other disturbances are not necessarily damped away or properly evolved, as often assumed. This limitation is the result of regularization techniques that are required to suppress instabilities arising from the ill-posedness of treating a boundary value problem as an initial value problem. These findings are valid for both incompressible and compressible formulations of PSE and are particularly relevant for applications involving multiple modes with different wavelengths and growth rates, such as problems involving multiple instability mechanisms, transient growth, and acoustics. Our theoretical results are illustrated using a generic problem from acoustics and a dual-stream jet, and the PSE solutions are compared to both global solutions of the linearized Navier–Stokes equations and a recently developed alternative parabolization.

Keywords

Parabolized stability equations Regularization Error analysis 

Notes

Acknowledgements

A.T. gratefully acknowledges support from NASA Grant No. NNX15AU93A. G.R. and T.C. acknowledge support from ONR Grant N00014-16-1-2445 and The Boeing Company under Strategic Research and Development Relationship Agreement CT-BA-GTA-1.

References

  1. 1.
    Andersson, P., Henningson, D., Hanifi, A.: On a stabilization procedure for the parabolic stability equations. J. Eng. Mech. 33, 311–332 (1998)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Andersson, P., Berggren, M., Henningson, D.S.: Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11(1), 134–150 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Batchelor, G.K., Gill, A.E.: Analysis of the stability of axisymmetric jets. J. Fluid Mech. 14(4), 529–551 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bertolotti, F., Herbert, T.: Analysis of the linear stability of compressible boundary layers using the pse. Theor. Comput. Fluid Dyn. 3(2), 117–124 (1991)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bertolotti, F., Herbert, T., Spalart, P.: Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441–474 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bouthier, M.: Stabilité linéaire des écoulements presque parallèles. J. de Mec. 11, 599–621 (1972)zbMATHGoogle Scholar
  7. 7.
    Brès, G.A., Bose, S., Emory, F. M Ham, Schmidt, O.T., Rigas, G., Colonius, T.: Large-eddy simulations of co-annular turbulent jet using a Voronoi-based mesh generation framework. In: AIAA Paper #2018-3302 (2018a)Google Scholar
  8. 8.
    Brès, G.A., Jordan, P., Le Rallic, M., Jaunet, V., Cavalieri, A.V.G., Towne, A., Lele, S.K., Colonius, T., Schmidt, O.T.: Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. 851, 83–124 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chang, C., Malik, M., Erlebacher, G., Hussaini, M.Y.: Compressible stability of growing boundary layers using parabolized stability equations. In: 22nd Fluid Dynamics, Plasma Dynamics and Lasers Conference, Honolulu, HI, USA (1991)Google Scholar
  10. 10.
    Cheung, L., Lele, S.: Aeroacoustic noise prediction and the dynamics of shear layers and jets using the nonlinear parabolized stability equations. Technical report TF-103 (2007)Google Scholar
  11. 11.
    Cheung, L., Lele, S.: Linear and nonlinear processes in two-dimensional mixing layer dynamics and sound radiation. J. Fluid Mech. 625, 321–351 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Crighton, D.G., Gaster, M.: Stability of slowly diverging jet flow. J. Fluid Mech. 77, 397–413 (1976)CrossRefzbMATHGoogle Scholar
  13. 13.
    Day, M., Mansour, N., Reynolds, W.: Nonlinear stability and structure of compressible reacting mixing layers. J. Fluid Mech. 446, 375–408 (2001)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Fedorov, A.: Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43, 79–95 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gaster, M.: On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66(3), 465–480 (1974)CrossRefzbMATHGoogle Scholar
  16. 16.
    Gudmundsson, K., Colonius, T.: Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128 (2011)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hack, M., Moin, P.: Algebraic disturbance growth by interaction of orr and lift-up mechanisms. J. Fluid Mech. 829, 112–126 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Haj-Hariri, H.: Characteristics analysis of the parabolized stability equations. Stud. Appl. Math. 92(1), 41–53 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Herbert, T.: Parabolized stability equations. In: AGARD-R-793 Special Course on Progress in Transition Modelling (1994)Google Scholar
  20. 20.
    Herbert, T.: Parabolized stability equations. Annu. Rev. Fluid Mech. 29, 245–283 (1997)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Huerre, P., Monkewitz, P.A.: Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473–537 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Jordan, P., Colonius, T.: Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jordan, P., Colonius, T., Bres, G.A., Zhang, M., Towne, A., Lele, S.: Modeling intermittent wavepackets and their radiated sound in a turbulent jet. Technical report. In: Proceedings of the Center for Turbulence Research summer program (2014)Google Scholar
  24. 24.
    Kreiss, H., Lorenz, J.: Initial-Boundary Problems and the Navier-Stokes. Equation Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia (2004)CrossRefGoogle Scholar
  25. 25.
    Li, F., Malik, M.R.: On the nature of PSE approximation. Theoret. Comput. Fluid Dyn. 8, 253–273 (1996)CrossRefzbMATHGoogle Scholar
  26. 26.
    Li, F., Malik, M.R.: Spectral analysis of the parabolized stability equations. Comput. Fluids 26(3), 279–297 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Malik, M., Li, F., Chang, C.L.: Crossflow disturbances in three-dimensional boundary layers: nonlinear development, wave interaction and secondary instability. J. Fluid Mech. 268, 1–36 (1994)CrossRefzbMATHGoogle Scholar
  28. 28.
    Paredes, P., Choudhari, M.M., Li, F.: Transition due to streamwise streaks in a supersonic flat plate boundary layer. Phys. Rev. Fluids 1(8), 083,601 (2016)CrossRefGoogle Scholar
  29. 29.
    Pralits, J.O., Airiau, C., Hanifi, A., Henningson, D.S.: Sensitivity analysis using adjoint parabolized stability equations for compressible flows. Flow Turbul. Combust. 65(3–4), 321–346 (2000)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ran, W., Zare, A., Hack, M., Jovanović, M.: Low-complexity stochastic modeling of spatially-evolving flows. Technical report. In: Proceedings of the Center for Turbulence Research summer program (2016)Google Scholar
  31. 31.
    Rigas, G., Colonius, T., Beyar, M.: Stability of wall-bounded flows using one-way spatial integration of Navier-Stokes equations. In: AIAA Paper #2017-1881 (2017a)Google Scholar
  32. 32.
    Rigas, G., Schmidt, O.T., Colonius, T., Brès, G.A.: One way Navier-Stokes and resolvent analysis for modeling coherent structures in a supersonic turbulent jet. In: AIAA Paper #2017-4046 (2017b)Google Scholar
  33. 33.
    Rodríguez, D., Jotkar, M.R., Gennaro, E.M.: Wavepacket models for subsonic twin jets using 3d parabolized stability equations. Compt. Rend. Mècanique 346(10), 890–902 (2018). (jet noise modelling and control/Modélisation et contrôle du bruit de jet) CrossRefGoogle Scholar
  34. 34.
    Saric, W.S., Reed, H.L., Kerschen, E.J.: Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34(1), 291–319 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows, vol. 142. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  36. 36.
    Schmidt, O.T., Towne, A., Colonius, T., Cavalieri, A.V.G., Jordan, P., Brès, G.A.: Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. J. Fluid Mech. 825, 1153–1181 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Sinha, A., Rodriguez, D., Bres, G., Colonius, T.: Wavepacket models for supersonic jet noise. J. Fluid Mech. 742, 71–95 (2014)CrossRefGoogle Scholar
  38. 38.
    Sinha, A., Gaitonde, D., Sohoni, N.: Parabolized stability analysis of dual-stream jets. In: AIAA Paper #2016-3057 (2016)Google Scholar
  39. 39.
    Tam, C.K.W., Hu, F.Q.: On the three families of instability waves of high-speed jets. J. Fluid Mech. 201, 447–483 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Tempelmann, D., Hanifi, A., Henningson, D.S.: Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech. 646, 5–37 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Towne, A.: Advancements in jet turbulence and noise modeling: accurate one-way solutions and empirical evaluation of the nonlinear forcing of wavepackets. PhD thesis, California Institute of Technology (2016)Google Scholar
  42. 42.
    Towne, A., Colonius, T.: Improved parabolization of the Euler equations. In: AIAA Paper #2013-2171 (2013)Google Scholar
  43. 43.
    Towne, A., Colonius, T.: Continued development of the one-way Euler equations: application to jets. In: AIAA Paper #2014-2903 (2014)Google Scholar
  44. 44.
    Towne, A., Colonius, T.: One-way spatial integration of hyperbolic equations. J. Comput. Phys. 300, 844–861 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Towne, A., Cavalieri, A.V.G., Jordan, P., Colonius, T., Schmidt, O., Jaunet, V., Brès, G.A.: Acoustic resonance in the potential core of subsonic jets. J. Fluid Mech. 825, 1113–1152 (2017)CrossRefzbMATHGoogle Scholar
  46. 46.
    Zhang, X.C., Ran, L.K., Sun, D.J., Wan, Z.H.: Optimal ‘quiet’ inlet perturbation using adjoint-based PSE in supersonic jets. Fluid Dyn. Res. 50(4), 045,504 (2018)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.University of MichiganAnn ArborUSA
  2. 2.California Institute of TechnologyPasadenaUSA

Personalised recommendations