# zbMATH — the first resource for mathematics

Stabilisation and drag reduction of pipe flows by flattening the base profile. (English) Zbl 1421.76102
J. Fluid Mech. 863, 850-875 (2019); corrigendum ibid. 888, Paper No. E1, 3 p. (2020).
Summary: Recent experimental observations [J. Kühnen et al., “Destabilizing turbulence in pipe flow”, Nat. Phys. 14, No. 4, 386–390 (2018; doi:10.1038/s41567-017-0018-3)] have shown that flattening a turbulent streamwise velocity profile in pipe flow destabilises the turbulence so that the flow relaminarises. We show that a similar phenomenon exists for laminar pipe flow profiles in the sense that the nonlinear stability of the laminar state is enhanced as the profile becomes more flattened. The flattening of the laminar base profile is produced by an artificial localised body force designed to mimic an obstacle used in the experiments of J. Kühnen et al. [“Relaminarization by steady modification of the streamwise velocity profile in a pipe”, Flow Turbul. Combust. 100, No. 4, 919–943 (2018; doi:10.1007/s10494-018-9896-4)] and the nonlinear stability measured by the size of the energy of the initial perturbations needed to trigger transition. Significant drag reduction is also observed for the turbulent flow when triggered by sufficiently large disturbances. In order to make the nonlinear stability computations more efficient, we examine how indicative the minimal seed – the disturbance of smallest energy for transition – is in measuring transition thresholds. We first show that the minimal seed is relatively robust to base profile changes and spectral filtering. We then compare the (unforced) transition behaviour of the minimal seed with several forms of randomised initial conditions in the range of Reynolds numbers $$Re=2400-10\,000$$ and find that the energy of the minimal seed after the Orr and oblique phases of its evolution is close to that of a critical localised random disturbance. In this sense, the minimal seed at the end of the oblique phase can be regarded as a good proxy for typical disturbances (here taken to be the localised random ones) and is thus used as initial condition in the simulations with the body force. The enhanced nonlinear stability and drag reduction predicted in the present study are an encouraging first step in modelling the experiments of Kühnen et al. [loc. cit.] and should motivate future developments to fully exploit the benefits of this promising direction for flow control.

##### MSC:
 76F06 Transition to turbulence 76F10 Shear flows and turbulence
##### Keywords:
transition to turbulence
Openpipeflow
Full Text:
##### References:
 [1] Auteri, F.; Baron, A.; Belan, M.; Campanardi, G.; Quadrio, M., Experimental assessment of drag reduction by traveling waves in a turbulent pipe flow, Phys. Fluids, 22, 11, (2010) [2] Avila, K.; Moxey, D.; De Lozar, A.; Avila, M.; Barkley, D.; Hof, B., The onset of turbulence in pipe flow, Science, 333, 192-196, (2011) · Zbl 1411.76035 [3] Bewley, T. R., Flow control: new challenges for a new renaissance, Prog. Aerosp. Sci., 37, 1, 21-58, (2001) [4] Blasius, H.1913Das ähnlichkeitsgesetz bei reibungsvorgängen in flüssigkeiten. In Mitteilungen über Forschungsarbeiten auf dem Gebiete des Ingenieurwesens, pp. 1-41. Springer. [5] Brandt, L., The lift-up effect: the linear mechanism behind transition and turbulence in shear flows, Eur. J. Mech. (B/Fluids), 47, 80-96, (2014) · Zbl 1297.76073 [6] Cherubini, S.; De Palma, P.; Robinet, J.-Ch., Nonlinear optimals in the asymptotic suction boundary layer: transition thresholds and symmetry breaking, Phys. Fluids, 27, 3, (2015) [7] Cherubini, S.; De Palma, P.; Robinet, J.-Ch.; Bottaro, A., A purely nonlinear route to transition approaching the edge of chaos in a boundary layer, Fluid Dyn. Res., 44, 3, (2012) · Zbl 1309.76075 [8] Cherubini, S.; Palma, P. D., Minimal perturbations approaching the edge of chaos in a couette flow, Fluid Dyn. Res., 46, 4, (2014) [9] Choi, H.; Moin, P.; Kim, J., Active turbulence control for drag reduction in wall-bounded flows, J. Fluid Mech., 262, 75-110, (1994) · Zbl 0800.76191 [10] Choi, J.-I.; Xu, C.-X.; Sung, H. J., Drag reduction by spanwise wall oscillation in wall-bounded turbulent flows, AIAA, 40, 5, 842-850, (2002) [11] Choi, K.-S.; Graham, M., Drag reduction of turbulent pipe flows by circular-wall oscillation, Phys. Fluids, 10, 1, 7-9, (1998) [12] Darbyshire, A. G.; Mullin, T., Transition to turbulence in constant-mass-flux pipe flow, J. Fluid Mech., 289, 83-114, (1995) [13] Duggleby, A.; Ball, K. S.; Paul, M. R., The effect of spanwise wall oscillation on turbulent pipe flow structures resulting in drag reduction, Phys. Fluids, 19, 12, (2007) · Zbl 1182.76221 [14] Duguet, Y.; Monokrousos, A.; Brandt, L.; Henningson, D. S., Minimal transition thresholds in plane Couette flow, Phys. Fluids, 25, 8, (2013) [15] Eckhardt, B.; Schneider, T. M.; Hof, B.; Westerweel, J., Turbulence transition in pipe flow, Annu. Rev. Fluid Mech., 29, 447-468, (2007) · Zbl 1296.76062 [16] Fadlun, E. A.; Verzicco, R.; Orlandi, P.; Mohd-Yusof, J., Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations, J. Comput. Phys., 161, 1, 35-60, (2000) · Zbl 0972.76073 [17] He, S.; He, K.; Seddighi, M., Laminarisation of flow at low Reynolds number due to streamwise body force, J. Fluid Mech., 809, 31-71, (2016) · Zbl 1383.76228 [18] Hof, B.; De Lozar, A.; Avila, M.; Tu, X.; Schneider, T. M., Eliminating turbulence in spatially intermittent flows, Science, 327, 5972, 1491-1494, (2010) [19] Hof, B.; Juel, A.; Mullin, T., Scaling of the turbulence transition threshold in a pipe, Phys. Rev. Lett., 91, (2003) · Zbl 1049.76511 [20] Högberg, M.; Bewley, T. R.; Henningson, D. S., Linear feedback control and estimation of transition in plane channel flow, J. Fluid Mech., 481, 149-175, (2003) · Zbl 1163.76353 [21] Jovanović, M. R., Turbulence suppression in channel flows by small amplitude transverse wall oscillations, Phys. Fluids, 20, 1, (2008) · Zbl 1182.76359 [22] Kasagi, N.; Suzuki, Y.; Fukagata, K., Microelectromechanical systems-based feedback control of turbulence for skin friction reduction, Annu. Rev. Fluid Mech., 41, 231-251, (2009) · Zbl 1157.76022 [23] Kerswell, R. R., Recent progress in understanding the transition to turbulence in a pipe, Nonlinearity, 18, R17-R44, (2005) · Zbl 1084.76033 [24] Kerswell, R. R., Nonlinear nonmodal stability theory, Annu. Rev. Fluid Mech., 50, 1, 319-345, (2018) · Zbl 1384.76022 [25] Kerswell, R. R.; Pringle, C. C. T.; Willis, A. P., An optimization approach for analysing nonlinear stability with transition to turbulence in fluids as an exemplar, Rep. Prog. Phys., 77, 8, (2014) [26] Kim, J.; Bewley, T. R., A linear systems approach to flow control, Annu. Rev. Fluid Mech., 39, 383-417, (2007) · Zbl 1296.76074 [27] Kim, J.; Moin, P.; Moser, R., Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., 177, 133-166, (1987) · Zbl 0616.76071 [28] Kühnen, J.; Scarselli, D.; Schaner, M.; Hof, B., Relaminarization by steady modification of the streamwise velocity profile in a pipe, Flow Turbul. Combust., 100, 4, 919-943, (2018) [29] Kühnen, J.; Song, B.; Scarselli, D.; Budanur, N. B.; Riedl, M.; Willis, A.; Avila, M.; Hof, B., Destabilizing turbulence in pipe flow, Nat. Phys., 14, 4, 386-390, (2018) [30] Lee, C.; Kim, J.; Choi, H., Suboptimal control of turbulent channel flow for drag reduction, J. Fluid Mech., 358, 245-258, (1998) · Zbl 0907.76039 [31] Lumley, J.; Blossey, P., Control of turbulence, Annu. Rev. Fluid Mech., 30, 1, 311-327, (1998) · Zbl 1398.76083 [32] Mellibovsky, F.; Meseguer, A., Critical threshold in pipe flow transition, Phil. Trans. R. Soc. Lond. A, 367, 1888, 545-560, (2009) · Zbl 1221.76096 [33] Moarref, R.; Jovanović, M. R., Controlling the onset of turbulence by streamwise travelling waves. Part 1. Receptivity analysis, J. Fluid Mech., 663, 70-99, (2010) · Zbl 1205.76129 [34] Mullin, T., Experimental studies of transition to turbulence in a pipe, Annu. Rev. Fluid Mech., 43, 1-24, (2011) · Zbl 1210.76005 [35] Peixinho, J.; Mullin, T., Finite-amplitude thresholds for transition in pipe flow, J. Fluid Mech., 582, 169-178, (2007) · Zbl 1114.76304 [36] Pfenninger, W.1961Transition in the inlet length of tubes at high Reynolds numbers. In Boundary Layer and Flow Control (ed. Lachmann, G. V.), pp. 970-980. Pergamon. [37] Pope, S. B., Turbulent Flows, (2000), Cambridge University Press · Zbl 0966.76002 [38] Pringle, C. C. T.; Kerswell, R. R., Using nonlinear transient growth to construct the minimal seed for shear flow turbulence, Phys. Rev. Lett., 105, (2010) [39] Pringle, C. C. T.; Willis, A. P.; Kerswell, R. R., Minimal seeds for shear flow turbulence: using nonlinear transient growth to touch the edge of chaos, J. Fluid Mech., 702, 415-443, (2012) · Zbl 1248.76075 [40] Quadrio, M., Drag reduction in turbulent boundary layers by in-plane wall motion, Phil. Trans. R. Soc. Lond. A, 369, 1940, 1428-1442, (2011) [41] Quadrio, M.; Sibilla, S., Numerical simulation of turbulent flow in a pipe oscillating around its axis, J. Fluid Mech., 424, 217-241, (2000) · Zbl 0994.76037 [42] Rabin, S. M. E.; Caulfield, C. P.; Kerswell, R. R., Designing a more nonlinearly stable laminar flow via boundary manipulation, J. Fluid Mech., 738, 1-12, (2014) [43] Reddy, S. C.; Schmid, P. J.; Baggett, J. S.; Henningson, D. S., On stability of streamwise streaks and transition thresholds in plane channel flows, J. Fluid Mech., 365, 269-303, (1998) · Zbl 0927.76029 [44] 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, Proc. R. Soc. Lond. A, 174, 935-982, (1883) · JFM 16.0845.02 [45] Schmid, P. J.; Henningson, D. S., Stability and Transition in Shear Flows, vol. 142, (2012), Springer Science & Business Media [46] Schneider, T. M.; Eckhardt, B., Lifetime statistics in transitional pipe flow, Phys. Rev. E, 78, 4, (2008) [47] Schoppa, W.; Hussain, F., Coherent structure generation in near-wall turbulence, J. Fluid Mech., 453, 57-108, (2002) · Zbl 1141.76408 [48] Trefethen, L. N., Chapman, S. J., Henningson, D. S., Meseguer, A., Mullin, T. & Nieuwstadt, F. T.2000 Threshold amplitudes for transition to turbulence in a pipe. Numer. Anal. Rep. 00/17. Oxford University Computer Laboratory. [49] Waleffe, F., On a Self-Sustaining Process in shear flows, Phys. Fluids, 9, 883-900, (1997) [50] Willis, A. P., The Openpipeflow Navier-Stokes solver, SoftwareX, 6, 124-127, (2017) [51] Willis, A. P.; Hwang, Y.; Cossu, C., Optimally amplified large-scale streaks and drag reduction in turbulent pipe flow, Phys. Rev. E, 82, (2010) [52] Willis, A. P.; Peixinho, J.; Kerswell, R. R.; Mullin, T., Experimental and theoretical progress in pipe flow transition, Phil. Trans. R. Soc. Lond. A, 366, 2671-2684, (2008) · Zbl 1153.76313 [53] Wygnanski, I. J.; Champagne, F. H., On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug, J. Fluid Mech., 59, 281-335, (1973) [54] Xu, C.-X.; Choi, J.-I.; Sung, H. J., Suboptimal control for drag reduction in turbulent pipe flow, Fluid Dyn. Res., 30, 4, 217-231, (2002) [55] Yudhistira, I.; Skote, M., Direct numerical simulation of a turbulent boundary layer over an oscillating wall, J. Turbul., 12, N9, (2011) [56] Zhou, D.; Ball, K. S., Turbulent drag reduction by spanwise wall oscillations, Int. J. Eng. Trans., 21, 1, 85, (2008)
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.