1932

Abstract

This review discusses a recently developed optimization technique for analyzing the nonlinear stability of a flow state. It is based on a nonlinear extension of nonmodal analysis and, in its simplest form, consists of finding the disturbance to the flow state of a given amplitude that experiences the largest energy growth at a certain time later. When coupled with a search over the disturbance amplitude, this can reveal the disturbance of least amplitude—called the minimal seed—for transition to another state such as turbulence. The approach bridges the theoretical gap between (linear) nonmodal theory and the (nonlinear) dynamical systems approach to fluid flows by allowing one to explore phase space at a finite distance from the reference flow state. Various ongoing and potential applications of the technique are discussed.

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2018-01-05
2024-06-17
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Literature Cited

  1. Andersson P, Berggren M, Henningson DS. 1999. Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11:134–50 [Google Scholar]
  2. Arratia C, Caulfield CP, Chomaz J-M. 2013. Transient perturbation growth in time-dependent mixing layers. J. Fluid Mech. 717:90–133 [Google Scholar]
  3. Barkmeijer J. 1996. Constructing fast-growing perturbations for the nonlinear regime. J. Atmos. Sci. 53:2838–51 [Google Scholar]
  4. Berggren M. 1998. Numerical solution of a flow-control problem: vorticity reduction by dynamic boundary action. SIAM J. Sci. Comput. 19:829–60 [Google Scholar]
  5. Bewley TR, Moin P, Temam R. 2001. DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447:179–225 [Google Scholar]
  6. Biau D. 2016. Transient growth of perturbations in Stokes oscillatory flows. J. Fluid Mech. 794:R4 [Google Scholar]
  7. Boberg L, Brosa U. 1988. Onset of turbulence in a pipe. Z. Naturforsch. A 43:697–726 [Google Scholar]
  8. Buizza R, Palmer TN. 1995. The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci. 52:1434–56 [Google Scholar]
  9. Butler KM, Farrell BF. 1992. Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids 4:1637–50 [Google Scholar]
  10. Chen L, Herreman W, Jackson A. 2015. Optimal dynamo action by steady flows confined to a cube. J. Fluid Mech. 783:23–45 [Google Scholar]
  11. Cherubini S, De Palma P. 2013. Nonlinear optimal perturbations in a Couette flow: bursting and transition. J. Fluid Mech. 716:251–79 [Google Scholar]
  12. Cherubini S, De Palma P. 2014. Minimal perturbations approaching the edge of chaos in a Couette flow. Fluid Dyn. Res. 46:041403 [Google Scholar]
  13. Cherubini S, De Palma P. 2015. Minimal-energy perturbations rapidly approaching the edge state in Couette flow. J. Fluid Mech. 764:572–98 [Google Scholar]
  14. Cherubini S, De Palma P, Robinet J-C. 2015. Nonlinear optimals in the asymptotic suction boundary layer: transition thresholds and symmetry breaking. Phys. Fluids 27:034108 [Google Scholar]
  15. Cherubini S, De Palma P, Robinet J-C, Bottaro A. 2010. Rapid path to transition via nonlinear localized optimal perturbations in a boundary layer flow. Phys. Rev. E 82:066302 [Google Scholar]
  16. Cherubini S, De Palma P, Robinet J-C, Bottaro A. 2011. The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689:221–53 [Google Scholar]
  17. Cherubini S, De Palma P, Robinet J-C, Bottaro A. 2012. A purely nonlinear route to transition approaching the edge of chaos in a boundary layer. Fluid Dyn. Res. 44:031404 [Google Scholar]
  18. Cherubini S, de Tullio MD, De Palma P, Pascazio G. 2013a. Optimal perturbations in boundary-layer flows over rough surfaces. J. Fluids Eng. 135:121102 [Google Scholar]
  19. Cherubini S, Robinet J-C, De Palma P. 2013b. Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow. J. Fluid Mech. 737:440–65 [Google Scholar]
  20. Chevalier M, Högberg M, Berggren M, Henningson DS. 2002. Linear and nonlinear optimal control in spatial boundary layers Presented at Theor. Fluid Mech. Meet , 3rd.24–26 June, St. Louis, MO AIAA Pap. 2002-2755 [Google Scholar]
  21. Chomaz J-M. 2005. Global instabilities in spatially developing flow: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37:357–92 [Google Scholar]
  22. Corbett P, Bottaro A. 2000. Optimal perturbations for boundary layers subject to streamwise pressure gradient. Phys. Fluids 12:120–30 [Google Scholar]
  23. Deguchi K, Walton AC. 2013. A swirling spiral wave solution in pipe flow. J. Fluid Mech. 737:R2 [Google Scholar]
  24. Dijkstra H, Viebahn JP. 2015. Sensitivity and resilience of the climate system: a conditional nonlinear optimization approach. Commun. Nonlinear Sci. Numer. Simul. 22:13–22 [Google Scholar]
  25. Duan W, Yu Y, Xu H, Zhao P. 2013. Behaviors of nonlinearities modulating the El Niño events induced by optimal precursory disturbances. Clim. Dyn. 40:1399–413 [Google Scholar]
  26. Duan W, Zhou F. 2013. Non-linear forcing singular vector of a two-dimensional quasi-geostrophic model. Tellus 65:18452 [Google Scholar]
  27. Duguet Y, Brandt L, Larsson BRJ. 2010. Towards minimal perturbations in transitional plane Couette flow. Phys. Rev. E 82:026316 [Google Scholar]
  28. Duguet Y, Monokrousos A, Brandt L, Henningson DS. 2013. Minimal transition thresholds in plane Couette flow. Phys. Fluids 25:084103 [Google Scholar]
  29. Duguet Y, Willis AP, Kerswell RR. 2008. Transition in pipe flow: the saddle structure on the boundary of turbulence. J. Fluid Mech. 613:255–74 [Google Scholar]
  30. Eaves TS, Caulfield CP. 2015. Disruption of SSP/VWI states by a stable stratification. J. Fluid Mech. 784:548–64 [Google Scholar]
  31. Eckhardt B, Schneider TM, Hof B, Westerweel J. 2007. Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39:447–68 [Google Scholar]
  32. Farano M, Cherubini S, Robinet J-C, De Palma P. 2015. Hairpin-like optimal perturbations in plane Poiseuille flow. J. Fluid Mech. 775:R2 [Google Scholar]
  33. Farano M, Cherubini S, Robinet J-C, De Palma P. 2016. Subcritical transition scenarios via linear and nonlinear localized optimal perturbations in plane Poiseuille flow. Fluid Dyn. Res. 46:061409 [Google Scholar]
  34. Farano M, Cherubini S, Robinet J-C, De Palma P. 2017. Optimal bursts in turbulent channel flow. J. Fluid Mech. 817:35–60 [Google Scholar]
  35. Farrell BF. 1988. Optimal excitation of perturbations in viscous shear flow. Phys. Fluids 31:2093–101 [Google Scholar]
  36. Foures DPG, Caulfield CP, Schmid PJ. 2014. Optimal mixing in two-dimensional plane Poiseuille flow at high Péclet number. J. Fluid Mech. 748:241–77 [Google Scholar]
  37. Freidlin MI, Wentzell AD. 1998. Random Perturbations of Dynamical Systems New York: Springer-Verlag [Google Scholar]
  38. Griewank A, Walther A. 2000. Algorithm 799: revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM Trans. Math Softw. 26:19–45 [Google Scholar]
  39. Guégan A, Schmid PJ, Huerre P. 2006. Optimal energy growth and optimal control in swept Hiemenz flow. J. Fluid Mech. 566:11–45 [Google Scholar]
  40. Gunzburger M. 2000. Adjoint equation-based methods for control problems in incompressible, viscous flows. Flow Turbul. Combust. 65:249–72 [Google Scholar]
  41. Gustavsson LH. 1991. Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224:241–60 [Google Scholar]
  42. Henningson DS, Reddy SC. 1994. On the role of linear mechanisms in transition to turbulence. Phys. Fluids 6:1396–98 [Google Scholar]
  43. Hinze M, Walther A, Sternberg J. 2006. An optimal memory-reduced procedure for calculating adjoints of the instationary Navier-Stokes equations. Optim. Control Appl. Methods 27:19–40 [Google Scholar]
  44. Hof B, Juel A, Mullin T. 2003. Scaling of the turbulence transition in pipe flow. Phys. Rev. Lett. 91:244502 [Google Scholar]
  45. Jiang Z, Mu M, Luo D. 2013. A study of the North Atlantic Oscillation using conditional nonlinear optimal perturbation. J. Atmos. Sci. 70:855–75 [Google Scholar]
  46. Jiang Z, Wang D. 2010. A study on precursors to blocking anomalies in climatological flows by using conditional nonlinear optimal perturbations. Q. J. R. Meteorol. Soc. 136:1170–80 [Google Scholar]
  47. Joslin RD, Gunzburger MD, Nicolaides RA, Erlebacher G, Hussaini MY. 1995. Self-contained automated methodology for optimal flow control Tech. Rep. 95-64, Inst. Comput. Appl. Sci. Eng. (ICASE), NASA Langley Res. Cent Hampton, VA: [Google Scholar]
  48. Joslin RD, Gunzburger MD, Nicolaides RA, Erlebacher G, Hussaini MY. 1997. Self-contained automated methodology for optimal flow control validated for transition delay. AIAA J 35:816–24 [Google Scholar]
  49. Juniper MP. 2011a. Transient growth and triggering in the horizontal Rijke tube. Int. J. Spray Combust. Dyn. 3:209–24 [Google Scholar]
  50. Juniper MP. 2011b. Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667:272–308 [Google Scholar]
  51. Juniper MP, Sujith RI. 2018. Sensitivity and nonlinearity of thermoacoustic oscillations. Annu. Rev. Fluid Mech 50661–89 [Google Scholar]
  52. Kawahara G, Uhlmann M, van Veen L. 2012. The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44:203–25 [Google Scholar]
  53. Kerswell RR. 2005. Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18:R17–44 [Google Scholar]
  54. Kerswell RR, Pringle CCP, Willis AP. 2014. An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar. Rep. Prog. Phys. 77:085901 [Google Scholar]
  55. Kim J, Bewley TR. 2007. A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39:383–417 [Google Scholar]
  56. Lecoanet D, Kerswell RR. 2013. Nonlinear optimization of multiple perturbations and stochastic forcing of subcritical ODE systems. Bull. Am. Phys. Soc. 58:L10.4 [Google Scholar]
  57. Levin O, Davidsson EN, Henningson DS. 2005. Transition thresholds in the asymptotic suction boundary layer. Phys. Fluids 17:114104 [Google Scholar]
  58. Levin O, Henningson DS. 2007. Turbulent spots in the asymptotic suction boundary layer. J. Fluid Mech. 584:397–413 [Google Scholar]
  59. Luchini P. 2000. Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404:289–309 [Google Scholar]
  60. Luchini P, Bottaro A. 1998. Görtler vortices: a backward-in-time approach to the receptivity problem. J. Fluid Mech. 363:1–23 [Google Scholar]
  61. Luchini P, Bottaro A. 2014. Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46:493–517 [Google Scholar]
  62. Mellibovsky F, Meseguer A. 2009. Critical threshold in pipe flow transition. Philos. Trans. R. Soc. A 467:545–60 [Google Scholar]
  63. Mellibovsky F, Meseguer A, Schneider TM, Eckhardt B. 2009. Transition in localized pipe flow turbulence. Phys. Rev. Lett. 103:054502 [Google Scholar]
  64. Monokrousos A, Bottaro A, Brandt L, Di Vita A, Henningson DS. 2011. Nonequilibrium thermodynamics and the optimal path to turbulence in shear flows. Phys. Rev. Lett. 106:134502 [Google Scholar]
  65. Mu M. 2000. Nonlinear singular vectors and nonlinear singular values. Sci. China D 43:375–85 [Google Scholar]
  66. Mu M, Duan W, Wang B. 2003. Conditional nonlinear optimal perturbation and its applications. Nonlinear Process. Geophys. 10:493–501 [Google Scholar]
  67. Mu M, Duan W, Wang Q, Zhang R. 2010. An extension of conditional optimal perturbation approach and its applications. Nonlinear Process. Geophys. 17:211–20 [Google Scholar]
  68. Mu M, Jiang Z. 2008a. A method to find perturbations that trigger blocking onset: conditional nonlinear optimal perturbations. Am. Meteorol. Soc. 65:3935–46 [Google Scholar]
  69. Mu M, Jiang Z. 2008b. A new approach to the generation of initial perturbations for ensemble prediction using dynamically conditional perturbations. Chin. Sci. Bull. 53:2062–68 [Google Scholar]
  70. Mu M, Jiang Z. 2011. Similarities between optimal precursors that trigger the onset of blocking events and optimally growing initial errors in onset prediction. J. Atmos. Sci. 68:2860–77 [Google Scholar]
  71. Mu M, Sun L, Dijstra HA. 2004. The sensitivity and stability of the ocean's thermohaline circulation to finite-amplitude perturbations. J. Phys. Oceanogr. 34:2305–15 [Google Scholar]
  72. Mu M, Yu Y, Xu H, Gong T. 2013. Similarities between optimal precursors for ENSO events and optimally growing initial errors in El Niño predictions. Theor. Appl. Climatol. 115:461–69 [Google Scholar]
  73. Mu M, Zhang Z. 2006. Conditional nonlinear optimal perturbations of a two-dimensional quasigeostrophic model. J. Atmos. Sci. 63:1587–604 [Google Scholar]
  74. Nagata M. 1990. Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217:519–27 [Google Scholar]
  75. Olvera D, Kerswell RR. 2017. Optimising energy growth as a tool for finding exact coherent structures. Phys. Rev. Fluids 2:083902 [Google Scholar]
  76. Oortwijn J, Barkmeijer J. 1995. Perturbations that optimally trigger weather regimes. J. Atmos. Sci. 52:3932–44 [Google Scholar]
  77. Orr WM. 1907a. The stability or instability of the steady motions of a perfect fluid and of a viscous fluid. Part I: a perfect liquid. Proc. R. Irish Acad. A 27:9–68 [Google Scholar]
  78. Orr WM. 1907b. The stability or instability of the steady motions of a perfect fluid and of a viscous fluid. Part II: a viscous liquid. Proc. R. Irish Acad. A 27:69–138 [Google Scholar]
  79. Ozdemir CE, Hsu T-J, Balachandar S. 2014. Direct numerical simulations of transition and turbulence in smooth-walled Stokes boundary layer. Phys. Fluids 26:045108 [Google Scholar]
  80. Palmer T, Molteni F, Mureau R, Buizza R, Chapelet P, Tribbia J. 1993. Ensemble prediction. Proc. ECMWF Semin.Valid. Models Eur. 121–66 Shinfield Park, UK: Eur. Cent. Medium-Range Weather Forecast. [Google Scholar]
  81. Passaggia P-Y, Ehrenstein U. 2013. Adjoint based optimization and control of a separated boundary-layer flow. Eur. J. Mech. B 41:169–77 [Google Scholar]
  82. Peixinho J, Mullin T. 2007. Finite amplitude thresholds for transition in pipe flow. J. Fluid Mech. 582:169–78 [Google Scholar]
  83. Pralits JO, Bottaro B, Cherubini S. 2015. Weakly nonlinear optimal perturbations. J. Fluid Mech. 785:135–51 [Google Scholar]
  84. Pralits JO, Hanifi A, Henningson DS. 2002. Adjoint-based optimization of steady suction for disturbance control in incompressible flows. J. Fluid Mech. 467:129–61 [Google Scholar]
  85. Pringle CCT, Kerswell RR. 2010. Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105:154502 [Google Scholar]
  86. Pringle CCT, Willis AP, Kerswell RR. 2012. Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos. J. Fluid Mech. 702:415–43 [Google Scholar]
  87. Pringle CCT, Willis AP, Kerswell RR. 2015. Fully localised nonlinear energy growth optimals in pipe flow. Phys. Fluids 27:064102 [Google Scholar]
  88. Rabin SME. 2013. A variational approach to determining nonlinear optimal perturbations and minimal seeds PhD Thesis, Univ Cambridge: [Google Scholar]
  89. Rabin SME, Caulfield CP, Kerswell RR. 2012. Variational identification of minimal seeds to trigger transition in plane Couette flow. J. Fluid Mech. 712:244–72 [Google Scholar]
  90. Rabin SME, Caulfield CP, Kerswell RR. 2014. Designing a more nonlinearly stable laminar flow via boundary manipulation. J. Fluid Mech. 738:R1 [Google Scholar]
  91. Rayleigh L. 1880. On the stability, or instability, of certain fluid motions. Proc. Lond. Math. Soc. 11:57–70 [Google Scholar]
  92. Rayleigh L. 1916. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philos. Mag. 32:529–46 [Google Scholar]
  93. Reddy SC, Henningson DS. 1993. Energy growth in viscous channel flows. J. Fluid Mech. 252:209–38 [Google Scholar]
  94. Reynolds O. 1883a. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Philos. Trans. R. Soc. A 174:935–82 [Google Scholar]
  95. Reynolds O. 1883b. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and of the law of resistance in parallel channels. Proc. R. Soc. A 35:84–99 [Google Scholar]
  96. Schlichting H. 1955. Boundary Layer Theory New York: McGraw-Hill [Google Scholar]
  97. Schmid PJ. 2007. Nonmodal stability theory. Annu. Rev. Fluid Mech. 39:129–62 [Google Scholar]
  98. Schmid PJ, Henningson DS. 1994. Optimal energy density growth in Hagen–Poiseuille flow. J. Fluid Mech. 277:197–225 [Google Scholar]
  99. Schmid PJ, Henningson DS. 2001. Stability and Transition in Shear Flows New York: Springer-Verlag [Google Scholar]
  100. Schneider TM, Eckhardt B. 2009. Edge states intermediate between laminar and turbulent dynamics in pipe flow. Philos. Trans. R. Soc. A 367:577–87 [Google Scholar]
  101. Schneider TM, Eckhardt B, Yorke JA. 2007. Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99:034502 [Google Scholar]
  102. Schneider TM, Gibson JF, Burke J. 2010. Snakes and ladders: localized solutions in plane Couette flow. Phys. Rev. Lett. 104:104501 [Google Scholar]
  103. Schneider TM, Gibson JF, Lagha M. 2008. Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78:037301 [Google Scholar]
  104. Taylor GI. 1923. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. A 223:289–343 [Google Scholar]
  105. Terwisscha van Scheltinga AD, Dijkstra HA. 2008. Conditional nonlinear optimal perturbations of the double-gyre ocean circulation. Nonlinear Process. Geophys. 15:727–34 [Google Scholar]
  106. Thomson W. 1887. Stability of fluid motion: rectilineal motion of viscous fluid between two parallel plates. Philos. Mag. 24:188–96 [Google Scholar]
  107. Trefethen LN, Trefethen AE, Reddy SC, Driscoll TA. 1993. Hydrodynamic stability without eigenvalues. Science 261:578–84 [Google Scholar]
  108. Wang J, Li Q, E W. 2015. Study of the instability of the Poiseuille flow using a thermodynamic formalism. PNAS 112:9518–23 [Google Scholar]
  109. Wang Q, Mu M, Dijkstra HA. 2012. Application of the conditional nonlinear optimal perturbation method to the predictability study of the Kuroshio large meander. Adv. Atmos. Sci. 29:118–34 [Google Scholar]
  110. Wang Q, Mu M, Dijkstra HA. 2013. Effects of nonlinear physical processes on optimal error growth in predictability experiments of the Kuroshio large meander. J. Geophys. Res. 118:6425–36 [Google Scholar]
  111. Waugh IC, Juniper MP. 2011. Triggering in a thermoacoustic system with stochastic noise. Int. J. Spray Combust. Dyn. 3:225–42 [Google Scholar]
  112. Willis AP. 2012. On the optimised magnetic dynamo. Phys. Rev. Lett. 109:251101 [Google Scholar]
  113. Yu YS, Duan WS, Xu H, Mu M. 2009. Dynamics of nonlinear error growth and season-dependent predictability of El Niño events in the Zebiak–Cane model. Q. J. R. Meteorol. Soc. 135:2146–60 [Google Scholar]
  114. Ziqing ZU, Mu M, Dijkstra HA. 2013. Optimal nonlinear excitation of decadal variability of the North Atlantic thermohaline circulation. Chin. J. Oceanogr. Limnol. 31:1368–74 [Google Scholar]
  115. Zuccher S, Bottaro A, Luchini P. 2006. Algebraic growth in a Blasius boundary layer: nonlinear optimal disturbances. Eur. J. Mech. B 25:1–17 [Google Scholar]
  116. Zuccher S, Luchini P, Bottaro A. 2004. Algebraic growth in a Blasius boundary layer: optimal and robust control by mean suction in the nonlinear regime. J. Fluid Mech. 513:135–60 [Google Scholar]
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