Hostname: page-component-546b4f848f-zwmfq Total loading time: 0 Render date: 2023-06-03T19:33:50.681Z Has data issue: false Feature Flags: { "useRatesEcommerce": true } hasContentIssue false

Energy transfer in turbulent channel flows and implications for resolvent modelling

Published online by Cambridge University Press:  25 January 2021

Sean Symon*
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
Simon J. Illingworth
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
Ivan Marusic
Department of Mechanical Engineering, University of Melbourne, Parkville, VIC3010, Australia
Email address for correspondence:


We analyse the inter-scale transfer of energy for two types of plane Poiseuille flow: the P4U exact coherent state of Park & Graham (J. Fluid Mech., vol. 782, 2015, pp. 430–454) and turbulent flow in a minimal channel. For both flows, the dominant energy-producing modes are streamwise-constant streaks with a spanwise spacing of approximately 100 wall units. Since the viscous dissipation for these scales is not sufficient to balance production, the nonlinear terms redistribute the excess energy to other scales. Spanwise-constant scales (that is, Tollmien–Schlichting-like modes with zero spanwise wavenumber), in particular, account for a significant amount of net energy gain from the nonlinear terms. We compare the energy balance to predictions from resolvent analysis, and we show that it does not model energy transfer well. Nevertheless, we find that the energy transferred from the streamwise-constant streaks can be predicted reasonably well by a Cess eddy viscosity profile. As such, eddy viscosity is an effective model for the nonlinear terms in resolvent analysis and explains good predictions for the most energetic streamwise-constant streaks. It also improves resolvent modes as a basis for structures whose streamwise lengths are greater than their spanwise widths by counteracting non-normality of the resolvent operator. This is quantified by computing the inner product between the optimal resolvent forcing and response modes, which is a metric of non-normality. Eddy viscosity does not respect the conservative nature of the nonlinear energy transfer, which must sum to zero over all scales. Since eddy viscosity tends to remove energy, it is less effective in modelling nonlinear transport for scales that receive energy from the nonlinear terms.

JFM Papers
© The Author(s), 2021. 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.)



del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205213.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of the wall boundary layer. J. Fluid Mech. 192, 115173.CrossRefGoogle Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 484504.CrossRefGoogle Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids A 4, 16371650.CrossRefGoogle Scholar
Cess, R. D. 1958 A survey of the literature on heat transfer in turbulent tube flow. Tech. Rep. 8-0529-R24. Westinghouse Research.Google Scholar
Cho, M., Hwang, Y. & Choi, H. 2018 Scale interactions and spectral energy transfer in turbulent channel flow. J. Fluid Mech. 854, 474504.CrossRefGoogle Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357392.CrossRefGoogle Scholar
Chung, D., Monty, J. P. & Ooi, A. 2014 An idealised assessment of Townsend's outer-layer similarity hypothesis for wall turbulence. J. Fluid Mech. 742, R3.CrossRefGoogle Scholar
Dar, G., Verma, M. K. & Eswaran, V. 2001 Energy transfer in two-dimensional magnetohydrodynamic turbulence: formalism and numerical results. Physica D 157, 207225.Google Scholar
Domaradzki, J. A., Liu, W., Härtel, C. & Kleiser, L. 1994 Energy transfer in numerically simulated wall-bounded turbulent flows. Phys. Fluids 6 (4), 15831599.Google Scholar
Hamilton, J. M., Kim, J. & Waleffe, F. 1995 Regeneration mechanisms of near-wall turbulence structures. J. Fluid Mech. 287, 317348.CrossRefGoogle Scholar
Hwang, Y. 2016 Mesolayer of attached eddies in turbulent channel flow. Phys. Rev. Fluids 1, 064401.CrossRefGoogle Scholar
Hwang, Y. & Cossu, C. 2010 Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow. J. Fluid Mech. 664, 5173.Google Scholar
Illingworth, S. J., Monty, J. P. & Marusic, I. 2018 Estimating large-scale structures in wall turbulence using linear models. J. Fluid Mech. 842, 146162.CrossRefGoogle Scholar
Jiménez, J. & Moin, P. 1991 The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213240.CrossRefGoogle Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335359.Google Scholar
Jin, B., Symon, S. & Illingworth, S. J. 2020 Energy transfer mechanisms and resolvent analysis in the cylinder wake. arXiv:2004.14534.CrossRefGoogle Scholar
Jovanović, M. R. & Bamieh, B. 2005 Componentwise energy amplificaton in channel flows. J. Fluid Mech. 534, 145183.CrossRefGoogle Scholar
Karhunen, K. 1946 Uber lineare methoden in der wahrscheinlichkeitsrechnung. Ann. Acad. Sci. Fennicae 37, A1.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 209303.Google Scholar
Landahl, M. T. 1980 A note on algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.Google Scholar
Leclercq, C., Demourant, F., Puossot-Vassal, C. & Sipp, D. 2019 Linear iterative method for closed-loop control of quasiperiodic flows. J. Fluid Mech. 868, 2665.CrossRefGoogle Scholar
Lee, M. & Moser, R. D. 2015 Direct numerical simulation of turbulent channel flow up to ${R}e_{\tau } \approx 5200$. J. Fluid Mech. 774, 395415.CrossRefGoogle Scholar
Lee, M. & Moser, R. D. 2019 Spectral analysis of the budget equation in turbulent channel flows at high Reynolds number. J. Fluid Mech. 860, 886938.CrossRefGoogle Scholar
Loève, M. M. 1955 Probability Theory. Van Nostrand.Google Scholar
Luhar, M., Sharma, A. S. & McKeon, B. J. 2014 Opposition control within the resolvent analysis framework. J. Fluid Mech. 749, 597626.CrossRefGoogle Scholar
Lumley, J. L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Mamatsashvili, G. R., Gogichaishvili, D. Z., Chagelishvili, G. D. & Horton, W. 2014 Nonlinear transverse cascade and two-dimensional magnetohydrodynamic subcritical turbulence in plane shear flows. Phys. Rev. E 89, 043101.CrossRefGoogle ScholarPubMed
Mamatsashvili, G. R., Khujadze, G., Chagelishvili, G. D., Dong, S., Jiménez, J. & Foysi, H. 2016 Dynamics of homogeneous shear turbulence: a key role of the nonlinear transverse cascade in the bypass concept. Phys. Rev. E 94, 023111.CrossRefGoogle ScholarPubMed
Marquet, O., Lombardi, M., Chomaz, J. M., Sipp, D. & Jacquin, L. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 121.CrossRefGoogle Scholar
McKeon, B. J. 2017 The engine behind (wall) turbulence: perspectives on scale interactions. J. Fluid Mech. 817, P1.CrossRefGoogle Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336382.CrossRefGoogle Scholar
McKeon, B. J., Sharma, A. S. & Jacobi, I. 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25 (3), 031301.CrossRefGoogle Scholar
McMullen, R. M., Rosenberg, K. & McKeon, B. J. 2020 Interaction of forced Orr–Sommerfeld and Squire modes in a low-order representation of turbulent channel flow. Phys. Rev. Fluids 5, 084607.CrossRefGoogle Scholar
Mizuno, Y. 2016 Spectra of energy transport in turbulent channel flows for moderate Reynolds numbers. J. Fluid Mech. 805, 171187.CrossRefGoogle Scholar
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling of the streamwise energy denisty in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275316.Google Scholar
Morra, P., Semeraro, O., Henningson, D. S. & Cossu, C. 2019 On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows. J. Fluid Mech. 867, 969984.Google Scholar
Muralidhar, S. D., Podvin, B., Mathelin, L. & Fraigneau, Y. 2019 Spatio-temporal proper orthogonal decomposition of turbulent channel flow. J. Fluid Mech. 864, 614639.CrossRefGoogle Scholar
Orszag, S. A. 1971 Accurate solution of the Orr–Sommerfeld stability equation. J. Fluid Mech. 50, 689703.CrossRefGoogle Scholar
Park, J. S. & Graham, M. D. 2015 Exact coherent states and connections to turbulent dynamics in minimal channel flow. J. Fluid Mech. 782, 430454.CrossRefGoogle Scholar
Picard, C. & Delville, J. 2000 Pressure velocity coupling in a subsonic round jet. Intl J. Heat Fluid Flow 21, 359364.Google Scholar
Pickering, E. M., Towne, A., Jordan, P. & Colonius, T. 2020 Resolvent-based jet noise models: a projection approach. In AIAA SciTech 2020 Forum (AIAA 2020-0999).Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209238.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, 263288.CrossRefGoogle Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow, with application to Malkus's theory. J. Fluid Mech. 27 (2), 253272.CrossRefGoogle Scholar
Rosenberg, K. & McKeon, B. J. 2019 Efficient representation of exact coherent states of the Navier–Stokes equations using resolvent analysis. Fluid Dyn. Res. 51, 011401.CrossRefGoogle Scholar
Rosenberg, K., Symon, S. & McKeon, B. J. 2019 Role of parasitic modes in nonlinear closure via the resolvent feedback loop. Phys. Rev. Fluids 4 (5), 052601.Google Scholar
Schlichting, H. 1933 Berechnung der Anfachung kleiner Störungen bei der plattenströmung. Z. Angew. Math. Mech. 13, 171174.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Sharma, A. S. 2009 Model reduction of turbulent fluid flows using the supply rate. Intl J. Bifurcation Chaos 19, 12671278.CrossRefGoogle Scholar
Sharma, A. S., Moarref, R., McKeon, B. J., Park, J. S., Graham, M. D. & Willis, A. O. 2016 Low-dimensional representations of exact coherent states of the Navier–Stokes equations from the resolvent model of wall turbulence. Phys. Rev. E 93, 021102.CrossRefGoogle ScholarPubMed
Smith, C. R. & Metzler, S. P. 1983 The characteristics of low-speed streaks in the near-wall region of a turbulent boundary layer. J. Fluid Mech. 129, 2754.CrossRefGoogle Scholar
Symon, S., Illingworth, S. J. & Marusic, I. 2020 Large-scale structures predicted by linear models of wall-bounded turbulence. J. Phys.: Conf. Ser. 1522, 012006.Google Scholar
Symon, S., Rosenberg, K., Dawson, S. T. M. & McKeon, B. J. 2018 Non-normality and classification of amplification mechanisms in stability and resolvent analysis. Phys. Rev. Fluids 3 (5), 053902.Google Scholar
Tollmien, W. 1930 Über die entstehung der turbulenz. In Vorträge aus dem Gebiete der Aerodynamik und verwandter Gebiete, pp. 18–21. Springer.CrossRefGoogle Scholar
Towne, A., Lozano-Durán, A. & Yang, X. 2020 Resolvent-based estimation of space-time flow statistics. J. Fluid Mech. 883, A17.CrossRefGoogle Scholar
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition. J. Fluid Mech. 847, 821867.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without the eigenvalues. Science 261, 578584.CrossRefGoogle ScholarPubMed
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9, 883900.CrossRefGoogle Scholar
Welch, P. 1967 The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans. Audio Electroacoust. 15 (2), 7073.CrossRefGoogle Scholar