# zbMATH — the first resource for mathematics

Edge state modulation by mean viscosity gradients. (English) Zbl 1419.76271
Summary: Motivated by the relevance of edge state solutions as mediators of transition, we use direct numerical simulations to study the effect of spatially non-uniform viscosity on their energy and stability in minimal channel flows. What we seek is a theoretical support rooted in a fully nonlinear framework that explains the modified threshold for transition to turbulence in flows with temperature-dependent viscosity. Consistently over a range of subcritical Reynolds numbers, we find that decreasing viscosity away from the walls weakens the streamwise streaks and the vortical structures responsible for their regeneration. The entire self-sustained cycle of the edge state is maintained on a lower kinetic energy level with a smaller driving force, compared to a flow with constant viscosity. Increasing viscosity away from the walls has the opposite effect. In both cases, the effect is proportional to the strength of the viscosity gradient. The results presented highlight a local shift in the state space of the position of the edge state relative to the laminar attractor with the consequent modulation of its basin of attraction in the proximity of the edge state and of the surrounding manifold. The implication is that the threshold for transition is reduced for perturbations evolving in the neighbourhood of the edge state in the case that viscosity decreases away from the walls, and vice versa.
##### MSC:
 76F06 Transition to turbulence 37N10 Dynamical systems in fluid mechanics, oceanography and meteorology 76E30 Nonlinear effects in hydrodynamic stability
Full Text:
##### References:
 [1] Avila, K.; Moxey, D.; De Lozar, A.; Avila, M.; Barkley, D.; Hof, B., The onset of turbulence in pipe flow, Science, 333, 6039, 192-196, (2011) · Zbl 1411.76035 [2] Avila, M.; Mellibovsky, F.; Roland, N.; Hof, B., Streamwise-localized solutions at the onset of turbulence in pipe flow, Phys. Rev. Lett., 110, 22, (2013) [3] Barker, S. J.; Gile, D., Experiments on heat-stabilized laminar boundary layers in water, J. Fluid Mech., 104, 139-158, (1981) [4] Chantry, M.; Schneider, T. M., Studying edge geometry in transiently turbulent shear flows, J. Fluid Mech., 747, 506-517, (2014) [5] Cherubini, S.; Palma, P. D.; Robinet, J.-C.; Bottaro, A., Edge states in a boundary layer, Phys. Fluids, 23, 5, (2011) · Zbl 1241.76246 [6] Chevalier, M., Schlatter, P., Lundbladh, A. & Henningson, D. S.2007 A pseudo-spectral solver for incompressible boundary layer flows. Tech. Rep. TRITA-MEK 2007:07. KTH Mechanics, Stockholm, Sweden. [7] Chikkadi, V.; Sameen, A.; Govindarajan, R., Preventing transition to turbulence: a viscosity stratification does not always help, Phys. Rev. Lett., 95, 26, (2005) [8] De Lozar, A.; Mellibovsky, F.; Avila, M. D.; Hof, B., Edge state in pipe flow experiments, Phys. Rev. Lett., 108, 21, (2012) [9] Duguet, Y.; Monokrousos, A.; Brandt, L.; Henningson, D. S., Minimal transition thresholds in plane Couette flow, Phys. Fluids, 25, 8, (2013) [10] Duguet, Y.; Pringle, C. C. T.; Kerswell, R. R., Relative periodic orbits in transitional pipe flow, Phys. Fluids, 20, 11, (2008) · Zbl 1182.76222 [11] Duguet, Y.; Schlatter, P.; Henningson, D. S., Localized edge states in plane Couette flow, Phys. Fluids, 21, 11, (2009) · Zbl 1183.76187 [12] Duguet, Y.; Schlatter, P.; Henningson, D. S.; Eckhardt, B., Self-sustained localized structures in a boundary-layer flow, Phys. Rev. Lett., 108, 4, (2012) [13] Duguet, Y.; Willis, A. P.; Kerswell, R. R., Transition in pipe flow: the saddle structure on the boundary of turbulence, J. Fluid Mech., 613, 255-274, (2008) · Zbl 1151.76495 [14] Eckhardt, B.; Schneider, T. M.; Hof, B.; Westerweel, J., Turbulence transition in pipe flow, Annu. Rev. Fluid Mech., 39, 447-468, (2007) · Zbl 1296.76062 [15] Govindarajan, R.; L’Vov, V. S.; Procaccia, I., Retardation of the onset of turbulence by minor viscosity contrasts, Phys. Rev. Lett., 87, 17, (2001) [16] Govindarajan, R.; Sahu, K. C., Instabilities in viscosity-stratified flow, Annu. Rev. Fluid Mech., 46, 331-353, (2014) · Zbl 1297.76067 [17] Hamilton, J. M.; Kim, J.; Waleffe, F., Regeneration mechanisms of near-wall turbulence structures, J. Fluid Mech., 287, 317-348, (1995) · Zbl 0867.76032 [18] Hof, B.; De Lozar, A.; Kuik, D. J.; Westerweel, J., Repeller or attractor? Selecting the dynamical model for the onset of turbulence in pipe flow, Phys. Rev. Lett., 101, 21, (2008) [19] Hof, B.; Westerweel, J.; Schneider, T. M.; Eckhardt, B., Finite lifetime of turbulence in shear flows, Nature, 443, 7107, 59-62, (2006) [20] Jiménez, J.; Moin, P., The minimal flow unit in near-wall turbulence, J. Fluid Mech., 225, 213-240, (1991) · Zbl 0721.76040 [21] Jiménez, J.; Pinelli, A., The autonomous cycle of near-wall turbulence, J. Fluid Mech., 389, 335-359, (1999) · Zbl 0948.76025 [22] Khapko, T.; Duguet, Y.; Kreilos, T.; Schlatter, P.; Eckhardt, B.; Henningson, D. S., Complexity of localised coherent structures in a boundary-layer flow, Eur. Phys. J. E, 37, 32, 1-12, (2014) [23] Khapko, T.; Kreilos, T.; Schlatter, P.; Duguet, Y.; Eckhardt, B.; Henningson, D. S., Localized edge states in the asymptotic suction boundary layer, J. Fluid Mech., 717, R6, (2013) · Zbl 1284.76106 [24] Khapko, T.; Kreilos, T.; Schlatter, P.; Duguet, Y.; Eckhardt, B.; Henningson, D. S., Edge states as mediators of bypass transition in boundary-layer flows, J. Fluid Mech., 801, R2, (2016) [25] Kreilos, T.; Khapko, T.; Schlatter, P.; Duguet, Y.; Henningson, D. S.; Eckhardt, B., Bypass transition and spot nucleation in boundary layers, Phys. Rev. Fluids, 1, (2016) [26] Kreilos, T.; Veble, G.; Schneider, T. M. E.; Eckhardt, B., Edge states for the turbulence transition in the asymptotic suction boundary layer, J. Fluid Mech., 726, 100-122, (2013) · Zbl 1287.76122 [27] Lam, K.; Banerjee, S., On the condition of streak formation in a bounded turbulent flow, Phys. Fluids, 4, 2, 306-320, (1992) · Zbl 0745.76028 [28] Lauchle, G. C.; Gurney, G. B., Laminar boundary layer transition on a heated underwater body, J. Fluid Mech., 144, 79-101, (1984) [29] Liepmann, H. W. & Fila, G. H.1947 Investigations of effects of surface temperature and single roughness elements on boundary-layer transition. NACA Tech. Rep. 890. [30] Nouar, C.; Bottaro, A.; Brancher, J. P., Delaying transition to turbulence in channel flow: revisiting the stability of shear-thinning fluids, J. Fluid Mech., 592, 177-194, (2007) · Zbl 1128.76026 [31] Park, J. S.; Graham, M. D., Exact coherent states and connections to turbulent dynamics in minimal channel flow, J. Fluid Mech., 782, 430-454, (2015) · Zbl 1381.76097 [32] Patel, A.; Boersma, B. J.; Pecnik, R., The influence of near-wall density and viscosity gradients on turbulence in channel flows, J. Fluid Mech., 809, 793-820, (2016) · Zbl 1383.76244 [33] Patel, A.; Peeters, J. W. R.; Boersma, B. J.; Pecnik, R., Semi-local scaling and turbulence modulation in variable property turbulent channel flows, Phys. Fluids, 27, 9, (2015) [34] Pringle, C. C. T.; Kerswell, R. R., Using nonlinear transient growth to construct the minimal seed for shear flow turbulence, Phys. Rev. Lett., 105, 15, (2010) [35] Reddy, S. C.; Henningson, D. S., Energy growth in viscous channel flows, J. Fluid Mech., 252, 209-238, (1993) · Zbl 0789.76026 [36] Roland, N.; Plaut, E.; Nouar, C., Petrov-Galerkin computation of nonlinear waves in pipe flow of shear-thinning fluids: first theoretical evidences for a delayed transition, Comput. Fluids, 39, 9, 1733-1743, (2010) · Zbl 1245.76093 [37] Sameen, A.; Govindarajan, R., The effect of wall heating on instability of channel flow, J. Fluid Mech., 577, 417-442, (2007) · Zbl 1110.76018 [38] Schäfer, P.; Herwig, H., Stability of plane Poiseuille flow with temperature dependent viscosity, Intl J. Heat Mass Transfer, 36, 9, 2441-2448, (1993) · Zbl 0776.76030 [39] Schmid, P. J.; Henningson, D. S., Stability and Transition Shear Flows, (2001), Springer · Zbl 0966.76003 [40] Schneider, T. M.; De Lillo, F.; Buehrle, J.; Eckhardt, B.; Dörnemann, T.; Dörnemann, K.; Freisleben, B., Transient turbulence in plane Couette flow, Phys. Rev. E, 81, 1, (2010) [41] Schneider, T. M.; Eckhardt, B., Edge states intermediate between laminar and turbulent dynamics in pipe flow, Phil. Trans. R. Soc. Lond. A, 367, 1888, 577-587, (2009) · Zbl 1221.76097 [42] Schneider, T. M.; Eckhardt, B.; Yorke, J. A., Turbulence transition and the edge of chaos in pipe flow, Phys. Rev. Lett., 99, 3, (2007) [43] Schneider, T. M.; Gibson, J. F.; Lagha, M.; De Lillo, F.; Eckhardt, B., Laminar-turbulent boundary in plane Couette flow, Phys. Rev. E, 78, 3, (2008) [44] Schneider, T. M.; Marinc, D.; Eckhardt, B., Localized edge states nucleate turbulence in extended plane Couette cells, J. Fluid Mech., 646, 441-451, (2010) · Zbl 1189.76258 [45] Skufca, J. D.; Yorke, J. A.; Eckhardt, B., Edge of chaos in a parallel shear flow, Phys. Rev. Lett., 96, 17, (2006) [46] Strazisar, A.; Reshotko, E., Stability of heated laminar boundary layers in water with nonuniform surface temperature, Phys. Fluids, 21, 5, 727-735, (1978) [47] Strazisar, A. J.; Reshotko, E.; Prahl, J. M., Experimental study of the stability of heated laminar boundary layers in water, J. Fluid Mech., 83, 2, 225-247, (1977) [48] Toh, S.; Itano, T., A periodic-like solution in channel flow, J. Fluid Mech., 481, 67-76, (2003) · Zbl 1034.76014 [49] Waleffe, F., Transition in shear flows. Nonlinear normality versus non-normal linearity, Phys. Fluids, 7, 12, 3060-3066, (1995) · Zbl 1026.76528 [50] Waleffe, F., On a self-sustaining process in shear flows, Phys. Fluids, 9, 4, 883-900, (1997) [51] Wall, D. P.; Wilson, S. K., The linear stability of channel flow of fluid with temperature-dependent viscosity, J. Fluid Mech., 323, 107-132, (1996) · Zbl 0886.76030 [52] Wall, D. P.; Wilson, S. K., The linear stability of flat-plate boundary-layer flow of fluid with temperature-dependent viscosity, Phys. Fluids, 9, 10, 2885-2898, (1997) [53] Xi, L.; Graham, M. D., Dynamics on the laminar-turbulent boundary and the origin of the maximum drag reduction asymptote, Phys. Rev. Lett., 108, 2, (2012) [54] Zammert, S.; Eckhardt, B., Periodically bursting edge states in plane Poiseuille flow, Fluid Dyn. Res., 46, 4, (2014) [55] Zammert, S.; Eckhardt, B., Streamwise and doubly-localised periodic orbits in plane Poiseuille flow, J. Fluid Mech., 761, 348-359, (2014) [56] Zonta, F.; Marchioli, C.; Soldati, A., Modulation of turbulence in forced convection by temperature-dependent viscosity, J. Fluid Mech., 697, 150-174, (2012) · Zbl 1250.76110
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.