Skip to main content Accessibility help
×
Home

Vertically localised equilibrium solutions in large-eddy simulations of homogeneous shear flow

  • Atsushi Sekimoto (a1) and Javier Jiménez (a1)

Abstract

Unstable equilibrium solutions in a homogeneous shear flow with sinuous (streamwise-shift-reflection and spanwise-shift-rotation) symmetry are numerically found in large-eddy simulations (LES) with no kinetic viscosity. The small-scale properties are determined by the mixing length scale $l_{S}$ used to define eddy viscosity, and the large-scale motion is induced by the mean shear at the integral scale, which is limited by the spanwise box dimension $L_{z}$ . The fraction $R_{S}=L_{z}/l_{S}$ , which plays the role of a Reynolds number, is used as a numerical continuation parameter. It is shown that equilibrium solutions appear by a saddle-node bifurcation as $R_{S}$ increases, and that the flow structures resemble those in plane Couette flow with the same sinuous symmetry. The vortical structures of both lower- and upper-branch solutions become spontaneously localised in the vertical direction. The lower-branch solution is an edge state at low $R_{S}$ , and takes the form of a thin critical layer as $R_{S}$ increases, as in the asymptotic theory of generic shear flow at high Reynolds numbers. On the other hand, the upper-branch solutions are characterised by a tall velocity streak with multiscale multiple vortical structures. At the higher end of $R_{S}$ , an incipient multiscale structure is found. The LES turbulence occasionally visits vertically localised states whose vortical structure resembles the present vertically localised LES equilibria.

Copyright

Corresponding author

Email address for correspondence: atsushi.sekimoto@monash.edu.au

References

Hide All
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.
Blackburn, H. M., Hall, P. & Sherwin, S. 2013 Lower branch equilibria in Couette flow: the emergence of canonical states for arbitrary shear flows. J. Fluid Mech. 726, R2.
Cambon, C. & Scott, J. F. 1999 Linear and nonlinear models of anisotropic turbulence. Annu. Rev. Fluid Mech. 31, 153.
Champagne, F. H., Harris, V. G. & Corrsin, S. 1970 Experiments on nearly homogeneous turbulent shear flow. J. Fluid Mech. 41, 81139.
Chandler, G. J. & Kerswell, R. R. 2013 Invariant recurrent solutions embedded in a turbulent two-dimensional Kolmogorov flow. J. Fluid Mech. 722, 554595.
Cvitanović, P. 2013 Recurrent flows: the clockwork behind turbulence. J. Fluid Mech. 726, 14.
Deguchi, K. 2015 Self-sustained states at Kolmogorov microscale. J. Fluid Mech. 781, R6.
Deguchi, K. & Hall, P. 2014a Canonical exact coherent structures embedded in high Reynolds number flows. Phil. Trans. R. Soc. Lond. A 372, 20130352.
Deguchi, K. & Hall, P. 2014b The high-Reynolds-number asymptotic development of nonlinear equilibrium states in plane Couette flow. J. Fluid Mech. 750, 99112.
Deguchi, K. & Hall, P. 2016 On the instability of vortex–wave interaction states. J. Fluid Mech. 802, 634666.
Deguchi, K., Hall, P. & Walton, A. 2013 The emergence of localized vortex–wave interaction states in plane Couette flow. J. Fluid Mech. 721, 5885.
Faisst, H. & Eckhardt, B. 2003 Traveling waves in pipe flow. Phys. Rev. Lett. 91, 224502.
Flores, O. & Jiménez, J. 2010 Hierarchy of minimal flow units in the logarithmic layer. Phys. Fluids 22, 071704.
Gerz, T., Schumann, U. & Elghobashi, S. E. 1989 Direct numerical simulation of stratified homogeneous turbulent shear flows. J. Fluid Mech. 200, 563594.
Gibson, J. F. & Brand, E. 2014 Spanwise-localized solutions of planar shear flows. J. Fluid Mech. 745, 2561.
Goto, S. 2008 A physical mechanism of the energy cascade in homogeneous isotropic turbulence. J. Fluid Mech. 605, 355366.
Goto, S. 2012 Coherent structures and energy cascade in homogeneous turbulence. Prog. Theor. Phys. Suppl. 195, 139156.
Gualtieri, P., Casciola, C. M., Benzi, R. & Piva, R. 2007 Preservation of statistical properties in large-eddy simulation of shear turbulence. J. Fluid Mech. 592, 471494.
Hall, P. & Sherwin, S. 2010 Streamwise vortices in shear flows: harbingers of transition and the skeleton of coherent structures. J. Fluid Mech. 661, 178205.
Hall, P. & Smith, F. T. 1991 On strongly nonlinear vortex/wave interactions in boundary-layer transition. J. Fluid Mech. 227, 641666.
Hughes, T. J. R., Oberai, A. A. & Mazzei, L. 2001 Large eddy simulation of turbulent channel flows by the variational multiscale method. Phys. Fluids 13, 6.
Hwang, Y. & Cossu, C. 2010 Self-sustained process at large scales in turbulent channel flow. Phys. Rev. Lett. 105, 044505.
Hwang, Y., Willis, A. P. & Cossu, C. 2016 Invariant solutions of minimal large-scale structures in turbulent channel flow for Re 𝜏 up to 1000. J. Fluid Mech. 802.
Itano, T. & Generalis, S. C. 2009 Hairpin vortex solution in planar Couette flow: a tapestry of knotted vortices. Phys. Rev. Lett. 102 (11), 114501.
Itano, T. & Toh, S. 2001 The dynamics of bursting process in wall turbulence. J. Phys. Soc. Japan 70, 703716.
Jiménez, J.1987 Coherent structures and dynamical systems. In Proceedings of CTR Summer School, pp. 323–324. Stanford University.
Jiménez, J., Kawahara, G., Simens, M. P., Nagata, M. & Shiba, M. 2005 Characterization of near-wall turbulence in terms of equilibrium and ‘bursting’ solutions. Phys. Fluids 17, 015105.
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.
Kawahara, G. 2005 Laminarization of minimal plane Couette flow: going beyond the basin of attraction of turbulence. Phys. Fluids 17, 041702.
Kawahara, G. & Kida, S. 2001 Periodic motion embedded in plane Couette turbulence: regeneration cycle and burst. J. Fluid Mech. 449, 291300.
Kawahara, G., Uhlmann, M. & Van Veen, L. 2012 The significance of simple invariant solutions in turbulent flows. Annu. Rev. Fluid Mech. 44, 203225.
Kerswell, R. R. & Tutty, O. R. 2007 Recurrence of travelling waves in transitional pipe flow. J. Fluid Mech. 584, 69102.
Kim, J., Moin, P. & Moser., R. D. 1987 Turbulent statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133166.
Kreilos, T. & Eckhardt, B. 2012 Periodic orbits near onset of chaos in plane couette flow. Chaos 22 (4), 047505.
Nagata, M. 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.
Park, J. S. & Graham, M. D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430454.
Piomelli, U., Rouhi, A. & Geurts, B. J. 2015 A grid-independent length scale for large-eddy simulations. J. Fluid Mech. 766, 499527.
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.
Pumir, A. 1996 Turbulence in homogeneous shear flows. Phys. Fluids 8, 31123127.
Rawat, S., Cossu, C., Hwang, Y. & Rincon, F. 2015 On the self-sustained nature of large-scale motions in turbulent Couette flow. J. Fluid Mech. 782, 515540.
Rogers, M. M. & Moin, P. 1987 The structure of the vorticity field in homogeneous turbulent flows. J. Fluid Mech. 176, 3366.
Sánchez, J. & Net, M. 2010 On the multiple shooting continuation of periodic orbits by Newton–Krylov methods. Intl J. Bifurcation Chaos 20, 4361.
Sasaki, E., Kawahara, G., Sekimoto, A. & Jiménez, J. 2016 Unstable periodic orbits in plane Couette flow with the Smagorinsky model. J. Phys.: Conf. Ser. 708, 012003.
Schmiegel, A. & Eckhardt, B. 1997 Fractal stability border in plane Couette flow. Phys. Rev. Lett. 277, 197225.
Schneider, T. M., Gibson, J. F. & Burke, J. 2010 Snakes and ladders: localized solutions of plane Couette flow. Phys. Rev. Lett. 104, 104501.
Scovazzi, G., Jiménez, J. & Moin, P. 2001 LES of the very large scales in a Re 𝜏 = 920 channel. In Proc. Div. Fluid Dyn. pp. KF–5, American Physical Society.
Sekimoto, A., Dong, S. & Jiménez, J. 2016 Direct numerical simulation of statistically stationary and homogeneous shear turbulence and its relation to other shear flows. Phys. Fluids 28, 035101.
Skufca, J. D., Yorke, J. A. & Eckhardt, B. 2006 Edge of chaos in a parallel shear flow. Phys. Rev. Lett. 96, 174101.
Smagorinsky, J. 1963 General circulation experiments with the primitive equations. Mon. Weath. Rev. 91, 99164.
Tavoularis, S. & Karnik, U. 1989 Further experiments on the evolution of turbulent stresses and scales in uniformly sheared turbulence. J. Fluid Mech. 204, 457478.
Toh, S. & Itano, T. 2003 A periodic-like solution in channel flow. J. Fluid Mech. 481, 6776.
van Veen, L. & Kawahara, G. 2011 Homoclinic tangle on the edge of shear turbulence. Phys. Rev. Lett. 107, 114501.
van Veen, L., Kawahara, G. & Matsumura, A. 2011 On matrix-free computation of 2D unstable manifolds. SIAM J. Sci. Comput. 33, 2544.
van Veen, L., Kida, S. & Kawahara, G. 2006 Periodic motion representing isotropic turbulence. Fluid Dyn. Res. 38, 1946.
Viswanath, D. 2007 Recurrent motions within plane Couette turbulence. J. Fluid Mech. 580, 339358.
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. Lond. A 367, 561576.
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93102.
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98, 204501.
Wedin, H. & Kerswell, R. 2004 Exact coherent structures in pipe flow: traveling wave solutions. J. Fluid Mech. 435, 333371.
Yasuda, T., Goto, S. & Kawahara, G. 2014 Quasi-cyclic evolution of turbulence driven by a steady force in a periodic cube. Fluid Dyn. Res. 46, 061413.
Zammert, S. & Eckhardt, B. 2015 Crisis bifurcations in plane Poiseuille flow. Phys. Rev. E 91, 041003.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed