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Transition in the asymptotic suction boundary layer over a heated plate

Published online by Cambridge University Press:  19 August 2016

Stefan Zammert ,
Nicolas Fischer  and
Bruno Eckhardt
Stefan Zammert*
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany Laboratory for Aero and Hydrodynamics, Delft University of Technology, 2628 CD Delft, The Netherlands
Nicolas Fischer
Affiliation:
Fakultät Verkehrswissenschaften “Friedrich List”, Technische Universität Dresden, D-01062 Dresden, Germany
Bruno Eckhardt
Affiliation:
Fachbereich Physik, Philipps-Universität Marburg, D-35032 Marburg, Germany J.M. Burgerscentrum, Delft University of Technology, 2628 CD Delft, The Netherlands
*
†Email address for correspondence: Stefan.Zammert@gmail.com
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Abstract

The asymptotic suction boundary layer (ASBL) is a parallel shear flow that becomes turbulent in a bypass transition in parameter regions where the laminar profile is stable. We here add a temperature gradient perpendicular to the plate and explore the interaction between convection and shear in determining the transition. We find that the laminar state becomes unstable in a subcritical bifurcation and that the critical Rayleigh number and wavenumber depend strongly on the Prandtl number. We also track several secondary bifurcations and identify states that are localized in two directions, showing different symmetries. In the subcritical regime, transient turbulent states which are connected to exact coherent states and follow the same transition scenario as found in linearly stable shear flows are identified and analysed. The study extends the bypass transition scenario from shear flows to thermal boundary layers and highlights the intricate interactions between thermal and shear forces.

JFM classification

Boundary Layers: Boundary Layers Convection: Convection Nonlinear Dynamical Systems: Nonlinear Dynamical Systems
Type
Papers
Information
Journal of Fluid Mechanics , Volume 803 , 25 September 2016 , pp. 175 - 199
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Copyright
© 2016 Cambridge University Press 

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References

Ahlers, G., Grossmann, S. & Lohse, D. 2009 Heat transfer and large scale dynamics in turbulent Rayleigh–Bénard convection. Rev. Mod. Phys. 81 (2), 503–537.Google Scholar
Avila, M., Mellibovsky, F., Roland, N. & Hof, B. 2013 Streamwise-localized solutions at the onset of turbulence in pipe flow. Phys. Rev. Lett. 110, 224502.Google Scholar
Beaume, C., Bergeon, A., Kao, H.-C. & Knobloch, E. 2013 Convectons in a rotating fluid layer. J. Fluid Mech. 717, 417–448.Google Scholar
Blanchflower, S. 1999 Magnetohydrodynamic convectons. Phys. Lett. A 261, 74–81.Google Scholar
Bobke, A., Örlü, R. & Schlatter, P. 2016 Simulations of turbulent asymptotic suction boundary layers. J. Turbul. 17 (2), 157–180.Google Scholar
Brand, E. & Gibson, J. F. 2014 A doubly-localized equilibrium solution of plane Couette flow. J. Fluid Mech. 750, R1.Google Scholar
Busse, F. H. 1978 Non-linear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.Google Scholar
Chantry, M., Willis, A. P. & Kerswell, R. R. 2014 The genesis of streamwise-localized solutions from globally periodic travelling waves in pipe flow. Phys. Rev. Lett. 112, 164501.Google Scholar
Chillà, F. & Schumacher, J. 2012 New perspectives in turbulent Rayleigh–Bénard convection. Eur. Phys. J. E 35, 58.Google 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.Google Scholar
Deguchi, K. & Hall, P. 2014 Free-stream coherent structures in parallel boundary-layer flows. J. Fluid Mech. 752, 602–625.Google Scholar
Dijkstra, H., Wubs, F. W., Cliffe, A. K., Doedel, E., Dragomirescu, I. F., Eckhardt, B., Gelfgat, A. Y., Hazel, A. L., Lucarini, V., Salinger, A. G. et al. 2014 Numerical bifurcation methods and their applicaton to fluid dynamics: analysis beyond simulation. Commun. Comput. Phys. 15 (1), 1–45.Google Scholar
Du Puits, R., Li, L., Resagk, C., Thess, A. & Willert, C. 2013 Turbulent boundary layer in high Rayleigh number convection in air. Phys. Rev. Lett. 112, 124301.Google Scholar
Duguet, Y., Pringle, C. C. T. & Kerswell, R. R. 2008 Relative periodic orbits in transitional pipe flow. Phys. Fluids 20 (11), 114102.Google Scholar
Duguet, Y., Schlatter, P., Henningson, D. S. & Eckhardt, B. 2012 Self-sustained localized structures in a boundary-layer flow. Phys. Rev. Lett. 108, 044501.Google Scholar
Eckhardt, B., Faisst, H., Schmiegel, A. & Schneider, T. M. 2008 Dynamical systems and the transition to turbulence in linearly stable shear flows. Phil. Trans. A 366, 1297–1315.Google Scholar
Eckhardt, B., Schneider, T. M., Hof, B. & Westerweel, J. 2007 Turbulence transition in pipe flow. Annu. Rev. Fluid Mech. 39 (1), 447–468.Google Scholar
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.Google Scholar
Fransson, J. H. M. & Alfredsson, P. H. 2003 On the disturbance growth in an asymptotic suction boundary layer. J. Fluid Mech. 482, 51–90.Google Scholar
Gage, K. S. & Reid, W. H. 1968 The stability of thermally stratified plane Poiseuille flow. J. Fluid Mech. 33 (01), 21–32.Google Scholar
Getling, A. V. 1998 Rayleigh–Bénard Convection – Structures and Dynamics. World Scientific.CrossRefGoogle Scholar
Gibson, J. F.2012 Channelflow: a spectral Navier–Stokes simulator in C++. Tech. Rep., University of New Hampshire.Google Scholar
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 25–61.Google Scholar
Gibson, J. F., Halcrow, J. & Cvitanović, P. 2009 Equilibrium and travelling-wave solutions of plane Couette flow. J. Fluid Mech. 638, 243–266.CrossRefGoogle Scholar
Grossmann, S. 2000 The onset of shear flow turbulence. Rev. Mod. Phys. 72 (2), 603.Google Scholar
Grossmann, S. & Lohse, D. 2000a Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407, 27–56.Google Scholar
Grossmann, S. & Lohse, D. 2000b Thermal convection for large Prandtl numbers. Phys. Rev. Lett. 86, 3316–3319.Google Scholar
Grossmann, S. & Lohse, D. 2002 Prandtl and Rayleigh number dependence of the Reynolds number in turbulent thermal convection. Phys. Rev. E 66, 016305.Google 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
Hocking, L. M. 1975 Non-linear instability of the asymptotic suction velocity profile. Q. J. Mech. Appl. Maths 28 (3), 341–353.CrossRefGoogle Scholar
Hof, B., van Doorne, C. W. H., Westerweel, J., Nieuwstadt, F. T. M., Faisst, H., Eckhardt, B., Wedin, H., Kerswell, R. R. & Waleffe, F. 2004 Experimental observation of nonlinear traveling waves in turbulent pipe flow. Science 305 (5690), 1594–1598.Google Scholar
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291–300.Google Scholar
Khapko, T., Duguet, Y., Kreilos, T., Schlatter, P., Eckhardt, B. & Henningson, D. S. 2014 Complexity of localised coherent structures in a boundary-layer flow. Eur. Phys. J. E 37, 32.Google Scholar
Khapko, T., Kreilos, T., Schlatter, P., Duguet, Y., Eckhardt, B. & Henningson, D. S. 2013 Localized edge states in the asymptotic suction boundary layer. J. Fluid Mech. 717, R6.CrossRefGoogle Scholar
Khapko, T., Schlatter, P., Duguet, Y. & Henningson, D. S. 2016 Turbulence collapse in a suction boundary layer. J. Fluid Mech. 795, 356–379.Google Scholar
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane Couette flow. Chaos 22 (4), 047505.CrossRefGoogle ScholarPubMed
Kreilos, T., Gibson, J. F. & Schneider, T. M. 2016 Localized travelling waves in the asymptotic suction boundary layer. J. Fluid Mech. 795, R3.Google Scholar
Kreilos, T., Veble, G., Schneider, T. M. & Eckhardt, B. 2013 Edge states for the turbulence transition in the asymptotic suction boundary layer. J. Fluid Mech. 726, 100–122.Google Scholar
Kuznetsov, Y. A. 1998 Elements of Applied Bifurcation Theory. Springer.Google Scholar
Lai, Y.-C. & Tél, T. 2011 Transient Chaos – Complex Dynamics on Finite Time Scales. Springer.Google Scholar
Landau, L. D. & Lifschitz, E. M. 1959 Fluid Mechanics. Pergamon Press.Google Scholar
Lo Jacono, D., Bergeon, A. & Knobloch, E. 2011 Magnetohydrodynamic convectons. J. Fluid Mech. 687, 595–605.Google Scholar
Marati, N., Davoudi, J., Casciola, C. M. & Eckhardt, B. 2006 Mean profiles for a passive scalar in wall-bounded flows from symmetry analysis. J. Turbul. 7, N61.Google Scholar
Melnikov, K., Kreilos, T. & Eckhardt, B. 2014 Long-wavelength instability of coherent structures in plane Couette flow. Phys. Rev. E 89, 043008.Google Scholar
Merkin, J. H. 1972 Free convection with blowing and suction. Intl J. Heat Mass Transfer 15 (5), 989–999.Google Scholar
Milinazzo, F. A. & Saffman, P. G. 1985 Finite-amplitude steady waves in plane viscous shear flows. J. Fluid Mech. 160, 281–295.Google Scholar
Moffat, J. & Kays, W. M. 1968 The turbulent boundary layer on a porous plate: experimental heat transfer with uniform blowing and suction. Intl J. Heat Mass Transfer 11, 1547–1566.Google Scholar
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519–527.Google Scholar
Nagata, M. & Deguchi, K. 2013 Mirror-symmetric exact coherent states in plane Poiseuille flow. J. Fluid Mech. 735, R4.CrossRefGoogle Scholar
Nicolas, X. 2002 Bibliographical review on the Poiseuille–Rayleigh–Bénard flows: the mixed convection flows in horizontal rectangular ducts heated from below. Intl J. Therm. Sci. 41, 961–1016.Google Scholar
Parodi, A., von Hardenberg, J., Passoni, G., Provenzale, A. & Spiegel, E. 2004 Clustering of plumes in turbulent convection. Phys. Rev. Lett. 92, 194503.Google Scholar
Pringle, C. C. T. & Kerswell, R. R. 2007 Asymmetric, helical, and mirror-symmetric traveling waves in pipe flow. Phys. Rev. Lett. 99, 074502.Google Scholar
Schlatter, P. & Örlü, R. 2011 Turbulent asymptotic suction boundary layers studied by simulation. J. Phys.: Conf. Ser. 318 (2), 022020.Google Scholar
Schlichting, H. & Gersten, K. 1997 Grenzschicht-Theorie. Springer.Google Scholar
Schmid, P. & Henningson, D. S. 2001 Stability and Transition in Shear Flow. Springer.Google Scholar
Schmiegel, A.1999 Transition to turbulence in linearly stable shear flows. PhD thesis, Philipps-Universitität Marburg, http://archiv.ub.uni-marburg.de/diss/z2000/0062/.Google Scholar
Schneider, T. M., Eckhardt, B. & Yorke, J. 2007 Turbulence transition and the edge of chaos in pipe flow. Phys. Rev. Lett. 99, 034502.Google Scholar
Skufca, J., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.Google Scholar
Sparrow, E. M. & Cess, R. D. 1961 Free convection with blowing and suction. Trans. ASME J. Heat Transfer 83 (3), 387–389.Google Scholar
Trefethen, L. N. 2000 Spectral Methods in MATLAB. Society for Industrial and Applied Mathematics.Google Scholar
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339–358.Google Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.Google Scholar
Wedin, H., Cherubini, S. & Bottaro, A. 2015 Effect of plate permeability on nonlinear stability of the asymptotic suction boundary layer. Phys. Rev. E 92, 013022.Google Scholar
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.Google Scholar
Zammert, S. & Eckhardt, B. 2014a Periodically bursting edge states in plane Poiseuille flow. Fluid Dyn. Res. 46, 041419.Google Scholar
Zammert, S. & Eckhardt, B. 2014b Streamwise and doubly-localised periodic orbits in plane Poiseuille flow. J. Fluid Mech. 761, 348–359.Google Scholar
Zammert, S. & Eckhardt, B. 2015 Crisis bifurcations in plane Poiseuille flow. Phys. Rev. E 91, 041003(R).Google Scholar
Zhou, Q., Sun, C. & Xia, K.-Q. 2007 Morphological evolution of thermal plumes in turbulent Rayleigh–Bénard convection. Phys. Rev. Lett. 98, 074501.Google Scholar
Zhou, Q. & Xia, K. Q. 2010 Physical and geometrical properties of thermal plumes in turbulent Rayleigh–Bénard convection. New J. Phys. 12, 075006.Google Scholar
Zocchi, G., Moses, E. & Libchaber, A. 1990 Coherent structures in turbulent convection, an experimental study. Phys. A 166 (3), 387–407.Google Scholar

Zammert et al. supplementary movie

Dynamics of the temperature field for the periodic orbit $PO_{1}$. Additionally, the temporal evolution of the amplitude $a$ and of the Nusselt number are shown.

Download Zammert et al. supplementary movie(Video)
Video 516.8 KB

Zammert et al. supplementary movie

Dynamics of the temperature field for a trajectory on the chaotic attractor at Ra=9.95. Additionally, the temporal evolution of the amplitude $a$ and of the Nusselt number are shown.

Download Zammert et al. supplementary movie(Video)
Video 681.9 KB

Zammert et al. supplementary movie

Dynamics of the temperature field for a trajectory on the chaotic saddle at Ra=10.2. Additionally, the temporal evolution of the amplitude $a$ and of the Nusselt number are shown.

Download Zammert et al. supplementary movie(Video)
Video 445.7 KB