Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-12T01:28:47.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Streamwise-constant large-scale structures in Couette and Poiseuille flows

Published online by Cambridge University Press:  21 February 2020

Simon J. Illingworth Open the ORCID record for Simon J. Illingworth [Opens in a new window]
Simon J. Illingworth*
Affiliation:
Mechanical Engineering, University of Melbourne, VIC 3010, Australia
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The linear amplification mechanisms leading to streamwise-constant large-scale structures in laminar and turbulent channel flows are considered. A key feature of the analysis is that the Orr–Sommerfeld and Squire operators are each considered separately. Physically, this corresponds to considering two separate processes: (i) the response of wall-normal velocity fluctuations to external forcing; and (ii) the response of streamwise velocity fluctuations to wall-normal velocity fluctuations. The analysis is performed for both plane Couette flow and plane Poiseuille flow; and for each we consider linear amplification mechanisms about both the laminar and turbulent mean velocity profiles. The analysis reveals two things. First, that the most amplified structures (with a spanwise spacing of approximately $4h$, where $h$ is the channel half-height) are to an important degree encoded in the Orr–Sommerfeld operator alone, thus helping to explain their prevalence. Second – and consistent with numerical and experimental observations – that Couette flow is significantly more efficient than Poiseuille flow in leveraging the mean shear to produce channel-wide streamwise streaks.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Shear layer turbulence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 889 , 25 April 2020 , A13
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.)

References

Adrian, R. J., Meinhart, C. D. & Tomkins, C. D. 2000 Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54.CrossRefGoogle Scholar
del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.CrossRefGoogle Scholar
Avsarkisov, V., Hoyas, S., Oberlack, M. & García-Galache, J. P. 2014 Turbulent plane Couette flow at moderately high Reynolds number. J. Fluid Mech. 751, R1.CrossRefGoogle Scholar
Balakumar, B. J. & Adrian, R. J. 2007 Large-and very-large-scale motions in channel and boundary-layer flows. Phil. Trans. R. Soc. Lond. A 365 (1852), 665–681.CrossRefGoogle ScholarPubMed
Bamieh, B. & Dahleh, M. 2001 Energy amplification in channel flows with stochastic excitation. Phys. Fluids 13 (11), 3258–3269.CrossRefGoogle Scholar
Bark, F. H. 1975 On the wave structure of the wall region of a turbulent boundary layer. J. Fluid Mech. 70 (2), 229–250.CrossRefGoogle Scholar
Bech, K. H. & Andersson, H. I. 1994 Very-large-scale structures in DNS. In Direct and Large-Eddy Simulation I (ed. Voke, P. R. et al. ), pp. 13–24. Springer.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids 4 (8), 1637–1650.CrossRefGoogle Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487–488.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5 (11), 2600–2609.CrossRefGoogle Scholar
Gayme, D. F., McKeon, B. J., Bamieh, B., Papachristodoulou, A. & Doyle, J. C. 2011 Amplification and nonlinear mechanisms in plane Couette flow. Phys. Fluids 23 (6), 065108.CrossRefGoogle Scholar
Gayme, D. F., McKeon, B. J., Papachristodoulou, A., Bamieh, B. & Doyle, J. C. 2010 A streamwise constant model of turbulence in plane Couette flow. J. Fluid Mech. 665, 99–119.CrossRefGoogle Scholar
Guala, M., Hommema, S. E. & Adrian, R. J. 2006 Large-scale and very-large-scale motions in turbulent pipe flow. J. Fluid Mech. 554, 521–542.CrossRefGoogle Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane Poiseuille flow. J. Fluid Mech. 224, 241–260.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2006 Scaling of the velocity fluctuations in turbulent channels up to Re 𝜏 = 2003. Phys. Fluids 18 (1), 011702.Google Scholar
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20 (10), 101511.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010a Amplification of coherent streaks in the turbulent Couette flow: an input–output analysis at low Reynolds number. J. Fluid Mech. 643, 333–348.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010b Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 51–73.CrossRefGoogle Scholar
Illingworth, S. J., Monty, J. P. & Marusic, I. 2018 Estimating large-scale structures in wall turbulence using linear models. J. Fluid Mech. 842, 146–162.CrossRefGoogle Scholar
Jovanovic, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Kim, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 1885–1888.CrossRefGoogle Scholar
Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422.CrossRefGoogle Scholar
Kitoh, O., Nakabyashi, K. & Nishimura, F. 2005 Experimental study on mean velocity and turbulence characteristics of plane Couette flow: low-Reynolds-number effects and large longitudinal vortical structure. J. Fluid Mech. 539, 199–227.CrossRefGoogle Scholar
Kitoh, O. & Umeki, M. 2008 Experimental study on large-scale streak structure in the core region of turbulent plane Couette flow. Phys. Fluids 20 (2), 025107.CrossRefGoogle Scholar
Komminaho, J., Lundbladh, A. & Johansson, A. V. 1996 Very large structures in plane turbulent Couette flow. J. Fluid Mech. 320, 259–285.CrossRefGoogle Scholar
Landahl, M. T. 1967 A wave-guide model for turbulent shear flow. J. Fluid Mech. 29 (3), 441–459.CrossRefGoogle Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98 (2), 243–251.CrossRefGoogle Scholar
Lee, M. & Moser, R. D. 2018 Extreme-scale motions in turbulent plane Couette flows. J. Fluid Mech. 842, 128–145.CrossRefGoogle Scholar
Lee, M. J. & Kim, J. 1991 The structure of turbulence in a simulated plane Couette flow. In 8th Symposium on Turbulent Shear Flows, vol. 1, pp. 5.3.1–5.3.6.Google Scholar
Madhusudanan, A., Illingworth, S. J. & Marusic, I. 2019 Coherent large-scale structures from the linearized Navier–Stokes equations. J. Fluid Mech. 873, 89–109.CrossRefGoogle Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Annu. Rev. Fluid Mech. 18 (1), 405–432.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.CrossRefGoogle Scholar
Monty, J. P., Hutchins, N., Ng, H. C. H., Marusic, I. & Chong, M. S. 2009 A comparison of turbulent pipe, channel and boundary layer flows. J. Fluid Mech. 632, 431–442.CrossRefGoogle Scholar
Monty, J. P., Stewart, J. A., Williams, R. C. & Chong, M. S. 2007 Large-scale features in turbulent pipe and channel flows. J. Fluid Mech. 589, 147–156.CrossRefGoogle Scholar
Papavassiliou, D. V. & Hanratty, T. J. 1997 Interpretation of large-scale structures observed in a turbulent plane Couette flow. Intl J. Heat Fluid Flow 18 (1), 55–69.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M. & Orlandi, P. 2014 Turbulence statistics in Couette flow at high Reynolds number. J. Fluid Mech. 758, 327–343.CrossRefGoogle Scholar
Pirozzoli, S., Bernardini, M., Verzicco, R. & Orlandi, P. 2017 Mixed convection in turbulent channels with unstable stratification. J. Fluid Mech. 821, 482–516.CrossRefGoogle Scholar
Pujals, G., García-Villalba, M., Cossu, C. & Depardon, S. 2009 A note on optimal transient growth in turbulent channel flows. Phys. Fluids 21 (1), 015109.CrossRefGoogle Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in turbulent shear flow. Part 3. Theoretical models and comparisons with experiments. J. Fluid Mech. 54 (02), 263–288.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.CrossRefGoogle Scholar
Tillmark, N. & Alfredsson, P. H. 1994 Structures in turbulent plane Couette flow obtained from correlation measurements. In Advances in Turbulence V (ed. Benzi, R.), pp. 502–507. Springer.Google Scholar
Tillmark, N. & Alfredsson, P. H. 1998 Large scale structures in turbulent plane Couette flow. In Advances in Turbulence VII, pp. 59–62. Springer.CrossRefGoogle Scholar
Townsend, A. A. 1956 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Trefethen, L. N. 2008 Is Gauss quadrature better than Clenshaw–Curtis? SIAM Rev. 50 (1), 67–87.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
Tsukahara, T., Kawamura, H. & Shingai, K. 2006 DNS of turbulent Couette flow with emphasis on the large-scale structure in the core region. J. Turbul. 7 (19), N19.Google Scholar
Vadarevu, S. B., Symon, S., Illingworth, S. J. & Marusic, I. 2019 Coherent structures in the linearized impulse response of turbulent channel flow. J. Fluid Mech. 863, 1190–1203.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85–111.CrossRefGoogle Scholar
Zaki, T. A. & Durbin, P. A. 2006 Continuous mode transition and the effects of pressure gradient. J. Fluid Mech. 563, 357–388.CrossRefGoogle Scholar