# Laminar-Turbulent Patterning in Transitional Flows

## Abstract

**:**

## 1. Context

`channelflow.org`[15], choosing tab ‘

`movies`’ among the headings. Nontrivial solutions brought about by such inherently nonlinear couplings can then be found away from the base flow in an intermediate R range, $1\ll R\ll {R}_{\mathrm{c}}$ (possibly infinite). Scenarios resting on the presence of a linear instability (infinitesimal disturbances) are, in practice, bypassed by the amplification of finite-amplitude, localized perturbations pushing the flow in the attraction basin of these nontrivial states living on the turbulent solution branch. As previously mentioned, this branch will be stable for $R\ge {R}_{\mathrm{g}}$, but its states are only transient below. Now, on general grounds, a regime of spatially uniform or featureless turbulence [16], is expected at very large R with turbulent fraction or intermittency factor saturating at one [5,17]. On the other hand, just above ${R}_{\mathrm{g}}$, one may expect these quantities to be markedly smaller than one, characterizing the conspicuous laminar-turbulent alternation. How do they approach saturation as R increases, either through a smooth crossover or at a well-defined upper threshold ${R}_{\mathrm{t}}$, and more generally, how do they vary all along the transitional range between ${R}_{\mathrm{g}}$ and the putative ${R}_{\mathrm{t}}$ are the questions of interest.

## 2. Cylindrical Couette Flow

## 3. The Laminar-to-Turbulent and Turbulent-to-Laminar Transition in Planar Flows

#### 3.1. Plane Couette Flow

#### 3.2. Other Planar Configurations

## 4. Decay at R${}_{\mathbf{g}}$ as a Statistical Physics Problem: Directed Percolation

## 5. Emergence of Patterns from the Featureless Regime

## 6. Understanding Laminar-Turbulent Patterning: Theoretical Issues and Modeling Perspectives

## Acknowledgments

## Conflicts of Interest

## Abbreviations

1D/2D/3D | One/two/three-dimensional (depending on 1/2/3 space coordinates) |

ASBL | Asymptotic suction boundary layer (boundary layer along porous wall with through flow) |

CCF | Cylindrical Couette flow (flow between differentially rotated coaxial cylinders) |

CPF | Couette–Poiseuille flow (flow between moving walls under pressure gradient) |

DP | Directed percolation (stochastic competition between decay and contamination) |

FT | Featureless turbulence (uniformly turbulent flow) |

GL | Ginzburg–Landau (formulation accounting for modulated periodic patterns) |

HPF | Hagen Poiseuille flow (flow in a straight cylindrical pipe under pressure gradient) |

MFU | Minimal flow unit (domain size below which no sustained nontrivial flow exist) |

NSE | Navier–Stokes equation, primitive equation governing the flow behavior |

PCF | Plane Couette flow (shear flow between counter-translating walls) |

PPF | Plane Poiseuille flow (channel flow, flow between plane walls under pressure gradient) |

RD | Reaction-diffusion system (field description of reacting chemical mixtures) |

SSP | Self-sustainment process, mechanism for nontrivial nonlinear states |

## References

- Manneville, P. Transition to turbulence in wall-bounded flows: Where do we stand? Mech. Eng. Rev. Bull. JSME
**2016**, 3. [Google Scholar] [CrossRef] - Manneville, P. Instabilities, Chaos and Turbulence; Imperial College Press: London, UK, 2010. [Google Scholar]
- Rayleigh, Lord. On the stability or instability of certain fluid motions. Proc. Lond. Math. Soc.
**1887**, XI, 57–70, and**1887**, XIX, 67–74. [Google Scholar] - Schmid, P.J.; Henningson, D.S. Stability and Transition in Shear Flows; Springer: Berlin, Germany, 2001. [Google Scholar]
- Coles, D. Transition in circular Couette flow. J. Fluid Mech.
**1965**, 21, 385–425. [Google Scholar] [CrossRef] - Grossmann, S. The onset of shear flow turbulence. Rev. Mod. Phys.
**2000**, 72, 603–618. [Google Scholar] [CrossRef] - Mullin, T.; Kerswell, R.R. (Eds.) IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions; Springer: Berlin, Germany, 2005. [Google Scholar]
- Kawahara, G.; Uhlmann, M.; van Veen, L. The Significance of Simple Invariant Solutions in Turbulent Flows. Annu. Rev. Fluid Mech.
**2012**, 44, 203–225. [Google Scholar] [CrossRef] - Gibson, J.F.; Halcrow, J.; Cvitanović, P. Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech.
**2008**, 611, 107–130. [Google Scholar] [CrossRef] - Pomeau, Y. Front motion, metastability and sub-critical bifurcations in hydrodynamics. Physica D
**1986**, 23, 3–11. [CrossRef]The transition to turbulence in parallel flows: A personal view. C. R. Meca**2015**, 343, 210–218. [CrossRef] - Chaté, H.; Manneville, P. Spatio-temporal intermittency. In Turbulence, a Tentative Dictionary; NATO ASI Series, Series B: Physics; Tabeling, P., Cardoso, O., Eds.; Plenum Press: New York, NY, USA, 1994; Volume 341, pp. 111–116. [Google Scholar]
- Barkley, D.; Song, B.; Mukund, V.; Lemoult, G.; Avila, M.; Hof, B. The rise of fully turbulent flow. Nature
**2015**, 526, 550–553. [Google Scholar] [CrossRef] [PubMed] - Hamilton, J.M.; Kim, J.; Waleffe, F. Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech.
**1995**, 287, 317–348. [Google Scholar] [CrossRef] - Waleffe, F. On a self-sustaining process in shear flows. Phys. Fluids
**1997**, 9, 883–900. [Google Scholar] [CrossRef] - Gibson, J.F. ChannelFlow.org. Available online: http://channelflow.org (accessed on 26 June 2017).
- Andereck, C.D.; Liu, S.S.; Swinney, H.L. Flow regimes in a circular Couette flow system with independently rotating cylinders. J. Fluid Mech.
**1986**, 164, 155–183. [Google Scholar] [CrossRef] - Coles, D. Interfaces and intermittency in turbulent shear flow. In Mécanique de la Turbulence; Favre, A., Ed.; CNRS: Paris, France, 1962; pp. 229–248. [Google Scholar]
- Reynolds, O. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous and the law of resistance in parallel channels. Philos. Trans. R. Soc.
**1883**, 174, 935–982. [Google Scholar] [CrossRef] - Eckhardt, B.; Schneider, T.M.; Hof, B.; Westerweel, J. Turbulence transition in pipe flow. Annu. Rev. Fluid Mech.
**2007**, 39, 447–468. [Google Scholar] [CrossRef] - Mullin, T. Experimental studies of transition to turbulence in a pipe. Annu. Rev. Fluid Mech.
**2011**, 43, 1–24. [Google Scholar] [CrossRef] - Barkley, D. Theoretical perspective on the route to turbulence in a pipe. J. Fluid Mech.
**2016**, 803, P1. [Google Scholar] [CrossRef] - Hof, B.; Westerweel, J.; Schneider, T.M.; Eckhardt, B. Finite lifetime of turbulence in shear flows. Nature
**2006**, 443, 59–62. [Google Scholar] [CrossRef] [PubMed] - Avila, K.; Moxey, D.; de Lozar, A.; Avila, M.; Barkley, D.; Hof, B. The onset of turbulence in pipe flow. Science
**2011**, 333, 192–196. [Google Scholar] [CrossRef] [PubMed] - Barkley, D. Simplifying the complexity of pipe flow. Phys. Rev. E
**2011**, 84, 1016309. [Google Scholar] [CrossRef] [PubMed] - Murray, J.D. Mathematical Biology; Springer: Berlin, Germany, 1993. [Google Scholar]
- Ishida, T.; Duguet, Y.; Tsukahara, T. Transitional structures in annular Poiseuille flow depending on radius ratio. J. Fluid Mech.
**2016**, 794, R2. [Google Scholar] [CrossRef] - Ishida, T.; Tsukahara, T. Friction factor of annular Poiseuille flow in a transitional regime. Adv. Mech. Eng.
**2017**, 9, 1–10. [Google Scholar] [CrossRef] - Ishida, T.; Tsukahara, T.; Duguet, Y. Turbulent bifurcations in intermittent shear flows: From puffs to oblique stripes. Phys. Rev. Fluids
**2017**, in press. [Google Scholar] - Kunii, K.; Ishida, T.; Tsukahara, T. Helical turbulence and puff in transitional sliding Couette flow. In Proceedings of the ICTAM 2016, Montréal, QC, Canada, 21–26 August 2016. [Google Scholar]
- Fardin, M.A.; Perge, C.; Taberlet, N. “The Hydrogen atom of fluid dynamics”—Introduction to the Taylor-Couette flow for soft matter scientists. Soft Matter
**2014**, 10, 3523–3535. [Google Scholar] [CrossRef] [PubMed] - Taylor, G.I. Stability of a viscous liquid contained between two rotating cylinders. Philos. Trans. R. Soc. A
**1923**, 223, 289–343. [Google Scholar] [CrossRef] - Manneville, P. Spots and turbulent domains in a model of transitional plane Couette flow. Theor. Comput. Fluid Dyn.
**2004**, 18, 169–181. [Google Scholar] [CrossRef] - Tuckerman, L.S.; Barkley, D. Patterns and dynamics in transitional plane Couette flow. Phys. Fluids
**2011**, 23, 041301. [Google Scholar] [CrossRef] - Barkley, D.; Tuckerman, L.S. Computational study of turbulent laminar patterns in Couette flow. Phys. Rev. Lett.
**2005**, 94, 014502. [Google Scholar] [CrossRef] [PubMed] - Barkley, D.; Tuckerman, L.S. Mean flow of turbulent-laminar patterns in plane Couette flow. J. Fluid Mech.
**2007**, 576, 109–137. [Google Scholar] [CrossRef] - Rayleigh, Lord. On the dynamics of revolving fluids. Proc. R. Soc. A
**1916**, XCIII, 148–154. [Google Scholar] - Grossmann, S.; Lohse, D.; Sun, C. High-Reynolds number Taylor–Couette turbulence. Annu. Rev. Fluid Mech.
**2016**, 48, 53–80. [Google Scholar] [CrossRef] - Di Prima, R.C.; Swinney, H.L. Instabilities and transition in flow between concentric rotating cylinders. In Hydrodynamic Instabilities and the Transition to Turbulence; Swinney, H.L., Gollub, J.P., Eds.; Springer: Berlin, Germany, 1985; pp. 139–180. [Google Scholar]
- Brand, H.R. Phase dynamics—A review and a perspective. In Propagation in Systems Far from Equilibrium; Brand, H.R., Wesfreid, J.E., Brand, H.R., Manneville, P., Albinet, G., Boccara, N., Eds.; Springer: Berlin, Germany, 1988. [Google Scholar]
- Coughlin, K.; Marcus, P.S. Turbulent bursts in Couette–Taylor flow. Phys. Rev. Lett.
**1996**, 77, 2214–2217. [Google Scholar] [CrossRef] [PubMed] - Hegseth, J.J.; Andereck, C.D.; Hayot, F.; Pomeau, Y. Spiral turbulence and phase dynamics. Phys. Rev. Lett.
**1989**, 62, 257–260. [Google Scholar] [CrossRef] [PubMed] - Litschke, H.; Roesner, K.G. New experimental methods for turbulent spots and turbulent spirals in the Taylor–Couette flow. Exp. Fluids
**1998**, 24, 201–209. [Google Scholar] [CrossRef] - Van Atta, C. Exploratory measurements in spiral turbulence. J. Fluid Mech.
**1966**, 25, 495–512. [Google Scholar] [CrossRef] - Borrero-Echeverry, D.; Schatz, M.F.; Tagg, R. Transient turbulence in Taylor–Couette flow. Phys. Rev. E
**2010**, 81, 025301. [Google Scholar] [CrossRef] [PubMed] - Prigent, A.; Dauchot, O. Transition to versus from turbulence in sub-critical Couette flows. In IUTAM Symposium on Laminar-Turbulent Transition and Finite Amplitude Solutions; Springer: Berlin, Germany, 2015; pp. 195–219. [Google Scholar]
- Prigent, A.; Grégoire, G.; Chaté, H.; Dauchot, O. Long-wavelength modulation of turbulent shear flows. Physica D
**2003**, 174, 100–113. [Google Scholar] [CrossRef] - Dong, S. Evidence for internal structures of spiral turbulence. Phys. Rev. E
**2009**, 80, 067301. [Google Scholar] [CrossRef] [PubMed] - Reichardt, H. Über die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couetteströmung. ZAMM
**1956**, 36 (Suppl. S1), S26–S29. [CrossRef]Gezetzmässigkeiten der geradlinigen turbulenten Couetteströmung. Mitt. Max-Planck-Institut für Strömungsforschung Göttingen**1959**, 22, 1–45. - Tillmark, N.; Alfredsson, P.H. Experiments on transition in plane Couette flow. J. Fluid Mech.
**1992**, 235, 89–102. [Google Scholar] [CrossRef] - Daviaud, F.; Hegseth, J.; Bergé, P. Sub-critical transition to turbulence in plane Couette flow. Phys. Rev. Lett.
**1992**, 69, 2511–2514. [Google Scholar] [CrossRef] [PubMed] - Bech, K.H.; Tillmark, N.; Alfredsson, P.H.; Andersson, H.I. An investigation of turbulent plane Couette flow at low Reynolds numbers. J. Fluid Mech.
**1995**, 286, 291–325. [Google Scholar] [CrossRef] - Dauchot, O.; Daviaud, F. Finite amplitude perturbations and spot growth mechanism in plane Couette flow. Phys. Fluids A
**1995**, 7, 335–343. [Google Scholar] [CrossRef] - Bottin, S.; Daviaud, F.; Manneville, P.; Dauchot, O. Discontinuous transition to spatiotemporal intermittency in plane Couette flow. Europhys. Lett.
**1998**, 43, 171–176. [Google Scholar] [CrossRef] - Bottin, S.; Chaté, H. Statistical analysis of the transition to turbulence in plane Couette flow. Eur. Phys. J. B
**1998**, 6, 143–155. [Google Scholar] [CrossRef] - Manneville, P.; Dauchot, O. Patterning and transition in sub-critical systems: The case of plane Couette flow. In Coherent Structures in Complex Systems; Reguera, D., Rubí, J.M., Bonilla, L.L., Eds.; Springer: Berlin, Germany, 2001; pp. 58–79. [Google Scholar]
- Lundbladh, A.; Johansson, A.V. Direct simulations of turbulent spots in plane Couette flow. J. Fluid Mech.
**1989**, 229, 499–516. [Google Scholar] [CrossRef] - Jiménez, J.; Moin, P. The minimal flow unit in near wall turbulence. J. Fluid Mech.
**1991**, 225, 213–240. [Google Scholar] [CrossRef] - Eckhardt, B.; Faisst, H.; Schmiegel, A.; Schneider, T.M. Dynamical systems and the transition to turbulence in linearly stable shear flows. Philos. Trans. R. Soc. A
**2008**, 366, 1297–1315. [Google Scholar] [CrossRef] [PubMed] - Komminaho, J.; Lundbladh, A.; Johansson, A.J. Very large structures in plane turbulent Couette flow. J. Fluid Mech.
**1996**, 320, 259–285. [Google Scholar] [CrossRef] - Duguet, Y.; Schlatter, P.; Henningson, D. Formation of turbulent patterns near the onset of transition in plane Couette flow. J. Fluid Mech.
**2010**, 650, 119–129. [Google Scholar] [CrossRef] - Philip, J.; Manneville, P. From temporal to spatiotemporal dynamics in transitional plane Couette flow. Phys. Rev. E
**2011**, 83, 036308. [Google Scholar] [CrossRef] [PubMed] - Couliou, M.; Monchaux, R. Large scale flows in transitional plane Couette flow: A key ingredient of spot growth mechanism. Phys. Fluids
**2017**, 27, 034101. [CrossRef]Growth dynamics of turbulent spots in plane Couette flow. J. Fluid Mech.**2015**, 819, 1–20. [CrossRef] - Shi, L.; Avila, M.; Hof, B. Scale invariance at the onset of turbulence in Couette flow. Phys. Rev. Lett.
**2013**, 110, 204502. [Google Scholar] [CrossRef] [PubMed] - Tuckerman, L.S.; Barkley, D.; Dauchot, O. Instability of uniform turbulent plane Couette flow: Spectra, probability distribution functions and K–Ω closure model. In Seventh IUTAM Symposium on Laminar-Turbulent Transition; Schlatter, P., Henningson, D., Eds.; Springer: Berlin, Grrmany, 2009; pp. 59–66. [Google Scholar]
- Tuckerman, L.S.; Kreilos, T.; Schrobsdorff, H.; Schneider, T.; Gibson, J.F. Turbulent-laminar patterns in plane Poiseuille flow. Phys. Fluids
**2014**, 26, 114103. [Google Scholar] [CrossRef] - Manneville, P.; Rolland, J. On modelling transitional turbulent flows using under-resolved direct numerical simulations: The case of plane Couette flow. Theor. Comput. Fluid Dyn.
**2011**, 25, 407–420. [Google Scholar] [CrossRef] - Manneville, P. On the decay of turbulence in plane Couette flow. Fluid Dyn. Res.
**2011**, 43, 065501. [CrossRef]On the growth of laminar-turbulent patterns in plane Couette flow. Fluid Dyn. Res.**2012**, 44, 031412. [CrossRef] - Khapko, T.; Schlatter, P.; Duguet, Y.; Henningson, D.S. Turbulence collapse in a suction boundary layer. J. Fluid Mech.
**2016**, 795, 356–379. [Google Scholar] [CrossRef] - Rayleigh, Lord. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philos. Mag.
**1916**, XXXII, 529–546. [Google Scholar] - Chantry, M.; Tuckerman, L.S.; Barkley, D. Turbulent-laminar patterns in shear flows without walls. J. Fluid Mech.
**2016**, 792, R8. [Google Scholar] [CrossRef] - Manneville, P. Turbulent patterns made simple? J. Fluid Mech.
**2016**, 796, 1–4. [Google Scholar] [CrossRef] - Lagha, M.; Manneville, P. Modeling transitional plane Couette flow. Eur. Phys. J. B
**2007**, 58, 433–447. [Google Scholar] [CrossRef] - Seshasayanan, K.; Manneville, P. Laminar-turbulent patterning in wall-bounded shear flow: A Galerkin model. Fluid Dyn. Res.
**2015**, 47, 035512. [Google Scholar] [CrossRef] - Lagha, M.; Manneville, P. Modeling of plane Couette flow. I. Large scale flow around turbulent spots. Phys. Fluids
**2007**, 19, 094105. [Google Scholar] [CrossRef] - Chantry, M.; Tuckerman, L.S.; Barkley, D. Universal continuous transition to turbulence in a planar shear flow. J. Fluid Mech.
**2017**, in press. [Google Scholar] [CrossRef] - Brethouwer, G.; Duguet, Y.; Schlatter, P. Turbulent-laminar coexistence in wall flows with Coriolis, buoyancy or Lorentz forces. J. Fluid Mech.
**2012**, 704, 137–172. [Google Scholar] [CrossRef] - Tsukahara, T.; Tillmark, N.; Alfredsson, P.H. Flow regimes in a plane Couette flow with system rotation. J. Fluid Mech.
**2010**, 648, 5–33. [Google Scholar] [CrossRef] - Klotz, L.; Lemoult, G.; Frontczak, I.; Tuckerman, L.S.; Wesfreid, J.E. New experiment in Couette–Poiseuille flow with zero mean advection velocity: Sub-critical transition to turbulence. Phys. Rev. Fluids
**2017**, 2, 043904. [Google Scholar] [CrossRef] - Carlson, D.R.; Widnall, S.E.; Peeters, M.F. A flow visualization of transition in plane Poiseuille flow. J. Fluid Mech.
**1982**, 121, 487–505. [Google Scholar] [CrossRef] - Lemoult, G.; Gumowski, K.; Aider, J.-L.; Wesfreid, J.E. Turbulent spots in channel flow: An experimental study. Eur. Phys. J. E
**2014**, 37, 25. [Google Scholar] [CrossRef] [PubMed] - Tsukahara, T.; Seki, Y.; Kawamura, H.; Tochio, D. DNS of turbulent channel flow at very low Reynolds numbers. In Turbulence and Shear Flow Phenomena 4; Humphrey, J.A.C., Gatski, T.B., Eds.; Williamsburg: New York, NY, USA, 2005; pp. 935–940. [Google Scholar]
- Hashimoto, S.; Hasobe, A.; Tsukahara, T.; Kawaguchi, Y.; Kawamura, H. An experimental study on turbulent-stripe structure in transitional channel flow. In Proceedings of the Sixth International Symposium on Turbulence, Heat and Mass Transfer, Rome, Italy, 14–18 September 2009. [Google Scholar]
- Tsukahara, T.; Ishida, T. Lower bound of sub-critical transition in plane Poiseuille flow. Nagare
**2015**, 34, 383–386. (In Japanese). Partly reproduced in: EUROMECH Colloquium EC565: Sub-Critical Transition to Turbulence; Corsica, France, 2014. [Google Scholar] - Xiong, X.; Tao, J.; Chen, S.; Brandt, L. Turbulent bands in plane-Poiseuille flow at moderate reynolds numbers. Phys. Fluids
**2015**, 27, 041702. [Google Scholar] [CrossRef] - Kanazawa, T.; Shimizu, M.; Kawahara, G. A two-dimensionally localized turbulence in plane Channel flow. In Proceedings of the Ninth JSME-KSME Thermal and Fluid Engineering Conference, Okinawa, Japan, 27–30 October 2017. [Google Scholar]
- Paranjape, C.; Vasudevan, M.; Hof, B.; IST Austria, Klosterneuburg, Austria; Duguet, Y.; LIMSI, Orsay, France. Private communication, 2017.
- Sano, M.; Tamai, K. A universal transition to turbulence in channel flow. Nat. Phys.
**2016**, 12, 249–253. [Google Scholar] [CrossRef] - Ishida, T.; Tsukahara, T.; Kawaguchi, Y. Effects of spanwise system rotation on turbulent stripes in a plane Poiseuille Flow. J. Turbulence
**2015**, 16, 273–289. [Google Scholar] [CrossRef] - Deusebio, E.; Caulfield, C.P.; Taylor, J.R. The intermittency boundary in stratified plane Couette flow. J. Fluid Mech.
**2015**, 781, 298–329. [Google Scholar] [CrossRef] - Deusebio, E.; Brethouwer, G.; Schlatter, P.; Lindborg, E. A numerical study of the unstratified and stratified Ekman layer. J. Fluid Mech.
**2014**, 755, 672–704. [Google Scholar] [CrossRef] - Launder, B.; Poncet, S.; Serre, E. Laminar, Transitional, and turbulent flow in rotor-stator cavities. Annu. Rev. Fluid Mech.
**2010**, 42, 229–248. [Google Scholar] [CrossRef] - Cros, A.; Le Gal, P. Spatiotemporal intermittency in the torsional Couette flow between a rotating and a stationary disk. Phys. Fluids
**2002**, 14, 3755–3765. [Google Scholar] [CrossRef] - Kreilos, T.; Khapko, T.; Schlatter, P.; Duguet, Y.; Henningson, D.S.; Eckhardt, B. Bypass transition and spot nucleation in boundary layers. Phys. Rev. Fluids
**2016**, 1, 043602. [Google Scholar] [CrossRef] - Antonia, R.A.; Fulachier, L.; Krishnamoorthy, L.V.; Benabid, T.; Anselmet, F. Influence of wall suction on the organized motion in a turbulent boundary layer. J. Fluid Mech.
**1988**, 188, 217–240. [Google Scholar] [CrossRef] - Duguet, Y.; Schlatter, P. Oblique Laminar-Turbulent Interfaces in Plane Shear Flows. Phys. Rev. Lett.
**2013**, 110, 034502. [Google Scholar] - Wikipedia. Phase Transitions. Available online: https://en.wikipedia.org/wiki/Phase_transition (accessed on 26 June 2017).
- Lübeck, S. Universal scaling behavior of non-equilibrium phase transitions. Int. J. Mod. Phys. B
**2004**, 18, 3977–4118. [Google Scholar] [CrossRef] - Skufca, J.D.; Yorke, J.A.; Eckhardt, B. Edge of chaos in a parallel shear flow. Phys. Rev. Lett.
**2006**, 96, 17410. [Google Scholar] [CrossRef] [PubMed] - Duguet, Y.; Schlatter, P.; Henningson, D.S. Localized edge states in plane Couette flow. Phys. Fluids
**2009**, 21, 111701. [Google Scholar] [CrossRef] - Lemoult, G.; Shi, L.; Avila, K.; Jalikop, S.; Avila, M.; Hof, B. Directed percolation phase transition to sustained turbulence in Couette flow. Nat. Phys.
**2016**, 12, 254–258. [Google Scholar] [CrossRef] - Shimizu, M.; Kawahara, G.; Manneville, P. Onset of sustained turbulence in plane Couette flow. In Proceedings of the Ninth JSME-KSME Thermal and Fluid Engineering Conference, Okinawa, Japan, 27–30 October 2017. [Google Scholar]
- Cross, M.; Greenside, H. Pattern Formation and Dynamics in Nonequilibrium Systems; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Manneville, P. On the transition to turbulence of wall-bounded flows in general, and plane Couette flow in particular. Eur. J. Mech. B/Fluids
**2015**, 49, 345–362. [Google Scholar] [CrossRef] - Manneville, P. Turbulent patterns in wall-bounded flows: A Turing instability? Europhys. Lett.
**2012**, 98, 64001. [Google Scholar] [CrossRef] - Hayot, F.; Pomeau, Y. Turbulent domain stabilization in annular flows. Phys. Rev. E
**1994**, 50, 2019–2021. [Google Scholar] [CrossRef] - Graham, R.; Tél, T. Steady-state ensemble for the complex Ginzburg–Landau equation with weak noise. Phys. Rev. A
**1990**, 42, 4661–4677. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**(

**a**) Globally super-critical scenario: more and more modes become progressively active before the flow can be considered turbulent; (

**b**) Globally sub-critical scenario. Qualitatively, sufficiently large perturbations are needed to reach the turbulent branch. Quantitatively, a distance $\mathrm{\Delta}$ to the laminar branch can be defined, but may vary with R discontinuously (

**b1**) or continuously (

**b2**) depending on whether fully-localized coherent structures are long-lived or not, hence whether the turbulent fraction measured in an infinitely-extended system can tend to zero, Case b2 (to be discussed in Section 4).

**Figure 2.**(

**a**) Kelvin–Helmholtz instability of an inflectional velocity profile is mostly responsible for laminar breakdown at low R, here in a mixing layer down a splitter plate. (

**b**–

**d**) Non-inflectional velocity profiles. (

**b**) The Blasius boundary layer velocity profile scales as the square root of the distance to the plate’s leading edge. This downstream evolution can be suppressed by suction through the plate when porous. (

**c**) The plane Couette flow displays a linear velocity profile, with shearing rate $s=({V}_{2}-{V}_{1})/2h$; here ${V}_{1}=-{V}_{2}$, hence no mean advection. (

**d**) The Couette–Poiseuille profile adds a quadratic, pressure driven, component to the Couette contribution, here with non-vanishing mean advection.

**Figure 3.**Laminar-turbulent patterning in PCF modeled via under-resolved direct numerical simulations in a $468h\times 272h$ domain (see [67] for details). Snapshots of typical flow states using color levels of the perturbation energy averaged over the gap $2h$: deep blue is laminar, but different from the base flow due to the large-scale mean flow component featured by the faint yellow lines on the blue background. The streamwise direction is vertical. From left to right: Turbulent spot at $R=281.25$, will decay after a very long transient. Growing oblique turbulent patch for $R=282.50$. Mature, but unsteady pattern with statistically constant turbulent fraction at $R=283.75$. Well-formed pattern with larger turbulent fraction at $R=287.50$. Values of R are shifted downward due to modeling via under-resolution [66], but the pictures displayed give a good idea of experimental findings described at the beginning of the section.

**Figure 4.**(

**Top**) Space-time lattice (

**left**) and contamination rules for different activity configurations as functions of probability p (

**right**) in the 1D case for simplicity. (

**Bottom**) Decay of the banded turbulent regime in the stress-free model of PCF by Chantry et al. [75]. (

**Left**) Typical snapshot of streamwise velocity at mid-gap during decay in a $1280h\times 1280h$ domain; laminar flow in white. (

**Right**) Variation with time of the turbulent fraction (log-log) during decay; red: saturation somewhat above threshold; green: exponential relaxation to zero somewhat below threshold; purple: near critical, power-law decay followed down until finite departure from threshold is felt, hence late exponentially decreasing tail. The dashed line indicates the theoretical expectation for 2D-DP, here valid over about one decade in time (Bottom panels: courtesy Chantry et al.).

**Figure 5.**Emergence of the pattern at ${R}_{\mathrm{t}}$, as seen from color levels of the cross-flow kinetic energy averaged over the gap in well-resolved numerical simulations of NSE [103]. The streamwise direction is vertical and the color scale for the local transverse kinetic energy averaged over the gap is identical for all pictures. From left to right: $R=420$, only short-lived, mostly streamwise-aligned, elongated laminar troughs (deep blue). $R=415$, some laminar troughs become wider, others get inclined. $R=405$, laminar troughs transiently form alleys of both orientations. $R=400$, turbulent fraction decreases significantly owing to bulkier laminar troughs. The illustrations shown are typical of flow patterns around ${R}_{\mathrm{t}}$ [46].

© 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Manneville, P.
Laminar-Turbulent Patterning in Transitional Flows. *Entropy* **2017**, *19*, 316.
https://doi.org/10.3390/e19070316

**AMA Style**

Manneville P.
Laminar-Turbulent Patterning in Transitional Flows. *Entropy*. 2017; 19(7):316.
https://doi.org/10.3390/e19070316

**Chicago/Turabian Style**

Manneville, Paul.
2017. "Laminar-Turbulent Patterning in Transitional Flows" *Entropy* 19, no. 7: 316.
https://doi.org/10.3390/e19070316