Hostname: page-component-6766d58669-fx4k7 Total loading time: 0 Render date: 2026-05-14T20:24:04.200Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Published online by Cambridge University Press:  08 July 2020

Florian Reetz  and
Tobias M. Schneider Open the ORCID record for Tobias M. Schneider [Opens in a new window]
Florian Reetz
Affiliation:
Emergent Complexity in Physical Systems Laboratory (ECPS), Ecole Polytechnique Fédérale de Lausanne, CH-1015, Switzerland
Tobias M. Schneider*
Affiliation:
Emergent Complexity in Physical Systems Laboratory (ECPS), Ecole Polytechnique Fédérale de Lausanne, CH-1015, Switzerland
*
†Email address for correspondence: tobias.schneider@epfl.ch
Get access Rights & Permissions [Opens in a new window]

Abstract

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number $\mathit{Pr}=1.07$. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At a larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck–Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

JFM classification

Nonlinear Dynamical Systems: Pattern formation Instability: Nonlinear instability

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 898 , 10 September 2020 , A22
Copyright
© The Author(s), 2020. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Ahlers, G. & Behringer, R. P. 1978 Evolution of turbulence from the Rayleigh–Bénard instability. Phys. Rev. Lett. 40 (11), 712–716.CrossRefGoogle Scholar
Armbruster, D., Guckenheimer, J. & Holmes, P. 1988 Heteroclinic cycles and modulated travelling waves in systems with O (2) symmetry. Physica D 29 (3), 257–282.Google Scholar
Bergeon, A. & Knobloch, E. 2002 Natural doubly diffusive convection in three-dimensional enclosures. Phys. Fluids 14 (9), 3233–3250.CrossRefGoogle Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32 (1), 709–778.CrossRefGoogle Scholar
Budanur, N. B. & Cvitanović, P. 2017 Unstable manifolds of relative periodic orbits in the symmetry-reduced state space of the Kuramoto–Sivashinsky system. J. Stat. Phys. 167 (3–4), 636–655.CrossRefGoogle Scholar
Busse, F. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41 (12), 1929–1967.CrossRefGoogle Scholar
Busse, F. H. & Clever, R. M. 1979 Instabilities of convection rolls in a fluid of moderate Prandtl number. J. Fluid Mech. 91 (2), 319–335.CrossRefGoogle Scholar
Busse, F. H. & Clever, R. M. 1992 Three-dimensional convection in an inclined layer heated from below. J. Engng Maths 26 (1), 1–19.CrossRefGoogle Scholar
Busse, F. H. & Clever, R. M. 1996 The sequence-of-bifurcations approach towards an understanding of complex flows. In Mathematical Modeling and Simulation in Hydrodynamic Stability (ed. Riahi, D. N.), pp. 15–34. World Scientific.CrossRefGoogle Scholar
Busse, F. H. & Clever, R. M. 2000 Bursts in inclined layer convection. Phys. Fluids 12 (8), 2137–2140.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 2006 Spectral Methods: Fundamentals in Single Domains. Springer.CrossRefGoogle Scholar
Canuto, C. & Landriani, G. S. 1986 Analysis of the Kleiser–Schumann method. Numer. Math. 50 (2), 217–243.CrossRefGoogle Scholar
Chandler, G. J. & Kerswell, R. R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554–595.CrossRefGoogle Scholar
Chen, Y.-M. & Pearlstein, A. J. 1989 Stability of free-convection flows of variable-viscosity fluids in vertical and inclined slots. J. Fluid Mech. 198, 513–541.CrossRefGoogle Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35 (7), 58.Google ScholarPubMed
Clever, R. & Busse, F. 1995 Tertiary and quarternary solutions for convection in a vertical fluid layer heated from the side. Chaos, Solitons Fractals 5 (10), 1795–1803.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1977 Instabilities of longitudinal convection rolls in an inclined layer. J. Fluid Mech. 81 (1), 107–127.CrossRefGoogle Scholar
Clever, R. M. & Busse, F. H. 1992 Three-dimensional convection in a horizontal fluid layer subjected to a constant shear. J. Fluid Mech. 234, 511–527.CrossRefGoogle Scholar
Clever, R. M., Busse, F. H. & Kelly, R. E. 1977 Instabilities of longitudinal convection rolls in Couette flow. Z. Angew. Math. Phys. 28 (5), 771–783.CrossRefGoogle Scholar
Coles, D. 1965 Transition in circular Couette flow. J. Fluid Mech. 21 (3), 385–425.CrossRefGoogle Scholar
Crawford, J. & Knobloch, E. 1991 Symmetry and symmetry-breaking bifurcations in fluid dynamics. Annu. Rev. Fluid Mech. 23 (1), 341–387.CrossRefGoogle Scholar
Cross, M. C. & Hohenberg, P. C. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 (3), 851–1112.CrossRefGoogle Scholar
Cvitanović, P., Artuso, R., Mainieri, R., Tanner, G. & Vattay, G. 2017 Flips, slides and turns. In Chaos: Classical and Quantum, version 15, Pt. I, chap. 10. Niels Bohr Institute.Google Scholar
Daniels, K., Plapp, B. & Bodenschatz, E. 2000 Pattern formation in inclined layer convection. Phys. Rev. Lett. 84 (23), 5320–5323.CrossRefGoogle ScholarPubMed
Daniels, K. E. & Bodenschatz, E. 2002 Defect turbulence in inclined layer convection. Phys. Rev. Lett. 88 (3), 034501.CrossRefGoogle ScholarPubMed
Daniels, K. E., Brausch, O., Pesch, W. & Bodenschatz, E. 2008 Competition and bistability of ordered undulations and undulation chaos in inclined layer convection. J. Fluid Mech. 597, 261–282.CrossRefGoogle Scholar
Daniels, K. E., Wiener, R. J. & Bodenschatz, E. 2003 Localized transverse bursts in inclined layer convection. Phys. Rev. Lett. 91 (11), 114501.CrossRefGoogle ScholarPubMed
Dijkstra, H. A., Wubs, F. W., Cliffe, A. K., Doedel, E., Hazel, A. L., Lucarini, V., Salinger, A. G., Phipps, E. T., Sanchez-Umbria, J., Schuttelaars, H. et al. 2014 Numerical bifurcation methods and their application to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys. 15 (1), 1–45.CrossRefGoogle Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.CrossRefGoogle Scholar
Farano, M., Cherubini, S., Robinet, J.-C., De Palma, P. & Schneider, T. M. 2019 Computing heteroclinic orbits using adjoint-based methods. J. Fluid Mech. 858, R3.CrossRefGoogle Scholar
Frigo, M. & Johnson, S. 2005 The design and implementation of FFTW3. Proc. IEEE 93 (2), 216–231.CrossRefGoogle Scholar
Fujimura, K. & Kelly, R. E. 1993 Mixed mode convection in an inclined slot. J. Fluid Mech. 246, 545–568.CrossRefGoogle Scholar
Gershuni, G. Z. & Zhukhovitskii, E. M. 1969 Stability of plane-parallel convective motion with respect to spatial perturbations. Prikl. Mat. Mekh. 33 (5), 855–860.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.CrossRefGoogle Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2008 Visualizing the geometry of state space in plane Couette flow. J. Fluid Mech. 611, 107–130.CrossRefGoogle Scholar
Gibson, J. F., Reetz, F., Azimi, S., Ferraro, A., Kreilos, T., Schrobsdorff, H., Farano, M., Yesil, A. F., Schütz, S. S., Culpo, M. et al. 2019 Channelflow 2.0 (in preparation). Available at: https://www.channelflow.ch.Google Scholar
de Graaf, J. G. A. & van der Held, E. F. M. 1953 The relation between the heat transfer and the convection phenomena in enclosed plane air layers. Appl. Sci. Res. A 3, 393–409.CrossRefGoogle Scholar
Gray, D. D. & Giorgini, A. 1976 The validity of the Boussinesq approximation for liquids and gases. Intl J. Heat Mass Transfer 19 (5), 545–551.CrossRefGoogle Scholar
Halcrow, J., Gibson, J. F., Cvitanović, P. & Viswanath, D. 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365–376.CrossRefGoogle Scholar
Hart, J. E. 1971a Stability of the flow in a differentially heated inclined box. J. Fluid Mech. 47 (03), 547–576.CrossRefGoogle Scholar
Hart, J. E. 1971b Transition to a wavy vortex régime in convective flow between inclined plates. J. Fluid Mech. 48 (2), 265–271.CrossRefGoogle Scholar
Hof, B. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.CrossRefGoogle ScholarPubMed
Hollands, K. & Konicek, L. 1973 Experimental study of the stability of differentially heated inclined air layers. Intl J. Heat Mass Transfer 16 (7), 1467–1476.CrossRefGoogle Scholar
Kawahara, G., Uhlmann, M. & van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44 (1), 203–225.CrossRefGoogle Scholar
Kelly, R. E. 1994 The onset and development of thermal convection in fully developed shear flows. Adv. Appl. Mech. 31, 35–112.CrossRefGoogle Scholar
Kerr, R. M. 1996 Rayleigh number scaling in numerical convection. J. Fluid Mech. 310, 139–179.CrossRefGoogle Scholar
Kerswell, R. R. 2005 Recent progress in understanding the transition to turbulence in a pipe. Nonlinearity 18 (6), R17–R44.CrossRefGoogle Scholar
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3-D numerical spectral simulations of plane channel flows. In Proceedings of the Third GAMM Conference on Numerical Methods in Fluid Mechanics (ed. Hirschel, E.), pp. 165–173. Vieweg+Teubner Verlag.CrossRefGoogle Scholar
Krupa, M. 1997 Robust heteroclinic cycles. J. Nonlinear Sci. 7 (2), 129–176.CrossRefGoogle Scholar
Krupa, M. & Melbourne, I. 1995 Asymptotic stability of heteroclinic cycles in systems with symmetry. Ergod. Th. & Dyn. Syst. 15 (1), 121–147.CrossRefGoogle Scholar
Lanford, O. E. 1982 The strange attractor theory of turbulence. Annu. Rev. Fluid Mech. 14, 347–364.CrossRefGoogle Scholar
Malkus, W. V. R. 1964 Boussinesq equations. In Notes on the 1964 Summer Study Program in Geophysical Fluid Dynamics at the Woods Hole Oceanographic Institution Vol. 1, pp. 1–12. WHOI.CrossRefGoogle Scholar
Melnikov, K., Kreilos, T. & Eckhardt, B. 2014 Long-wavelength instability of coherent structures in plane Couette flow. Phys. Rev. E 89 (4), 043008.Google ScholarPubMed
Mercader, I., Prat, J. & Knobloch, E. 2002 Robust heteroclinic cycles in two-dimensional Rayleigh–Bénard convection without Boussinesq symmetry. Intl J. Bifurcation Chaos 12 (11), 2501–2522.CrossRefGoogle Scholar
Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. 1993 Spiral defect chaos in large aspect ratio Rayleigh–Bénard convection. Phys. Rev. Lett. 71 (13), 2026–2029.CrossRefGoogle ScholarPubMed
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.CrossRefGoogle Scholar
Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1 : 2 mode interaction in exactly counter-rotating von Kármán swirling flow. J. Fluid Mech. 477 (477), 51–88.CrossRefGoogle Scholar
Nusselt, W.1908 Die Wärmeleitfähigkeit von Wärmeisolierstoffen. PhD thesis, Königl. Technische Hochschule zu München.CrossRefGoogle Scholar
Peyret, R. 2002 Spectral Methods for Incompressible Flows. Springer.CrossRefGoogle Scholar
Prigent, A., Grégoire, G., Chaté, H., Dauchot, O. & van Saarloos, W. 2002 Large-scale finite-wavelength modulation within turbulent shear flows. Phys. Rev. Lett. 89 (1), 014501.CrossRefGoogle ScholarPubMed
Proctor, M. R. E. & Jones, C. A. 1988 The interaction of two spatially resonant patterns in thermal convection. Part 1. Exact 1 : 2 resonance. J. Fluid Mech. 188, 301–335.CrossRefGoogle Scholar
Reetz, F., Subramanian, P. & Schneider, T. M. 2020 Invariant states in inclined layer convection. Part 2. Bifurcations and connections between branches of invariant states. J. Fluid Mech.Google Scholar
Ruth, D. W., Raithby, G. D. & Hollands, K. G. T. 1980 On the secondary instability in inclined air layers. J. Fluid Mech. 96 (3), 481–492.CrossRefGoogle Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. A. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 34502.CrossRefGoogle ScholarPubMed
Schneider, T. M., Gibson, J. F., Lagha, M., De Lillo, F. & Eckhardt, B. 2008 Laminar-turbulent boundary in plane Couette flow. Phys. Rev. E 78 (3), 037301.Google ScholarPubMed
Shadid, J. N. & Goldstein, R. J. 1990 Visualization of longitudinal convection roll instabilities in an inclined enclosure heated from below. J. Fluid Mech. 215, 61–84.CrossRefGoogle Scholar
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.CrossRefGoogle Scholar
Subramanian, P., Brausch, O., Daniels, K. E., Bodenschatz, E., Schneider, T. M. & Pesch, W. 2016 Spatio-temporal patterns in inclined layer convection. J. Fluid Mech. 794, 719–745.CrossRefGoogle Scholar
Suri, B., Tithof, J., Grigoriev, R. O. & Schatz, M. F. 2017 Forecasting fluid flows using the geometry of turbulence. Phys. Rev. Lett. 118 (11), 114501.CrossRefGoogle ScholarPubMed
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.CrossRefGoogle Scholar
Waleffe, F. 1998 Three-dimensional coherent states in plane shear flows. Phys. Rev. Lett. 81, 4140–4143.CrossRefGoogle Scholar