Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-02T22:03:30.018Z Has data issue: false hasContentIssue false
  • English
  • Français

A critical-layer framework for turbulent pipe flow

Published online by Cambridge University Press:  01 July 2010

B. J. McKEON  and
A. S. SHARMA
B. J. McKEON*
Affiliation:
Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, CA 91125, USA
A. S. SHARMA
Affiliation:
Department of Aeronautics, Imperial College, London SW7 2AZ, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A model-based description of the scaling and radial location of turbulent fluctuations in turbulent pipe flow is presented and used to illuminate the scaling behaviour of the very large scale motions. The model is derived by treating the nonlinearity in the perturbation equation (involving the Reynolds stress) as an unknown forcing, yielding a linear relationship between the velocity field response and this nonlinearity. We do not assume small perturbations. We examine propagating helical velocity response modes that are harmonic in the wall-parallel directions and in time, permitting comparison of our results to experimental data. The steady component of the velocity field that varies only in the wall-normal direction is identified as the turbulent mean profile. A singular value decomposition of the resolvent identifies the forcing shape that will lead to the largest velocity response at a given wavenumber–frequency combination. The hypothesis that these forcing shapes lead to response modes that will be dominant in turbulent pipe flow is tested by using physical arguments to constrain the range of wavenumbers and frequencies to those actually observed in experiments. An investigation of the most amplified velocity response at a given wavenumber–frequency combination reveals critical-layer-like behaviour reminiscent of the neutrally stable solutions of the Orr–Sommerfeld equation in linearly unstable flow. Two distinct regions in the flow where the influence of viscosity becomes important can be identified, namely wall layers that scale with R+1/2 and critical layers where the propagation velocity is equal to the local mean velocity, one of which scales with R+2/3 in pipe flow. This framework appears to be consistent with several scaling results in wall turbulence and reveals a mechanism by which the effects of viscosity can extend well beyond the immediate vicinity of the wall. The model reproduces inner scaling of the small scales near the wall and an approach to outer scaling in the flow interior. We use our analysis to make a first prediction that the appropriate scaling velocity for the very large scale motions is the centreline velocity, and show that this is in agreement with experimental results. Lastly, we interpret the wall modes as the motion required to meet the wall boundary condition, identifying the interaction between the critical and wall modes as a potential origin for an interaction between the large and small scales that has been observed in recent literature as an amplitude modulation of the near-wall turbulence by the very large scales.

JFM classification

Boundary Layers: Boundary layer receptivity Boundary Layers: Boundary layer structure Waves/Free-surface Flows: Critical layers Boundary Layers: Pipe flow boundary layer Turbulent Flows: Turbulence theory Turbulent Flows: Turbulent boundary layers
Type
Papers
Information
Journal of Fluid Mechanics , Volume 658 , 10 September 2010 , pp. 336 - 382
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Adrian, R. J. 2007 Vortex organization in wall turbulence. Phys. Fluids 19 (041301).CrossRefGoogle Scholar
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
del Álamo, J. C., Jiménez, J., Zandonade, P. & Moser, R. D. 2004 Scaling of the energy spectra of turbulent channels. J. Fluid Mech. 500, 135–144.CrossRefGoogle Scholar
Aubry, N., Holmes, P., Lumley, J. L. & Stone, E. 1988 The dynamics of coherent structures in the wall region of a turbulent boundary layer. J. Fluid Mech. 192, 115–173.CrossRefGoogle Scholar
Bailey, S. C. C., Hultmark, M., Smits, A. J. & Schultz, M. P. 2008 Azimuthal structure of turbulence in high Reynolds number pipe flow. J. Fluid Mech. 615, 121–138.CrossRefGoogle Scholar
Bailey, S. C. C. & Smits, A. J. 2009 The structure of large- and very large-scale motions in turbulent pipe flow. Paper 2009-3684. AIAA.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. A 365, 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
Bandyopadhyay, P. R. & Hussain, A. K. M. F. 1984 The coupling between scales in shear flows. Phys. Fluids 27 (9), 2221–2228.CrossRefGoogle Scholar
Blackwelder, R. F. & Kovasznay, L. S. G. 1972 Time scales and correlations in a turbulent boundary layer. J. Fluid Mech. 15, 1545–1554.Google Scholar
Blondel, V. & Megretski, A. (Ed) 2004 Unsolved Problems in Mathematical Systems and Control Theory. Princeton University Press.CrossRefGoogle Scholar
Boberg, L. & Brosa, U. 1988 Onset of turbulence in a pipe. Z. Naturforsch. 43a, 697–726.CrossRefGoogle Scholar
Butler, K. & Farrell, B. 1992 a Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A5, 774–777.Google Scholar
Butler, K. & Farrell, B. 1992 b Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids 4 (8).CrossRefGoogle Scholar
Cess, R. D. 1958 A study of the literature on heat transfer in turbulent tube flow. Tech. Rep. Rep. 8-0529-R24. Westinghouse Research.Google Scholar
Chung, D. & McKeon, B. J. 2010 Large-eddy simulation investigation of large-scale structures in a long channel flow. (submitted).CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.CrossRefGoogle Scholar
Curtain, R. F. & Zwart, H. J. 1995 An Introduction to Infinite-Dimensional Linear Systems Theory. Springer.CrossRefGoogle Scholar
Dean, R. B. & Bradshaw, P. 1976 Measurements of interacting turbulent shear layers in a duct. J. Fluid Mech. 78, 641–676.CrossRefGoogle Scholar
DeGraaff, D. B. & Eaton, J. K. 2000 Reynolds-number scaling of the flat-plate turbulent boundary layer. J. Fluid Mech. 422, 319–346.CrossRefGoogle Scholar
Dennis, D. & Nickels, T. 2008 On the limitations of Taylor's hypothesis in constructing long structures in wall-bounded turbulent flow. J. Fluid Mech. 614, 197–206.CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 2004 Hydrodynamic Stability, 2nd edn.Cambridge University Press.CrossRefGoogle Scholar
Duggleby, A., Ball, K. S., Pail, M. R. & Fischer, P. F. 2007 Dynamical eigenfunction decomposition of turbulent pipe flow. J. Turbul. 8 (43), 1–24.CrossRefGoogle Scholar
Farrell, B. & Ioannou, J. 1993 Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids 5 (11), 2600–2609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 237–250.CrossRefGoogle Scholar
Gayme, D. F., McKeon, B. J., Papachristodolou, A., Bamieh, B. & Doyle, J. C. 2010 Streamwise constant model of turbulence in plane Couette flow. (submitted).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
Guala, M., Metzger, M. & McKeon, B. J. 2010 a Intermittency in the atmospheric surface layer: unresolved or slowly varying? Physica D (in press).CrossRefGoogle Scholar
Guala, M., Metzger, M. J. & McKeon, B. J. 2010 b Interactions within the turbulent boundary layer at high Reynolds number. (submitted).CrossRefGoogle Scholar
Henningson, D. S. & Reddy, S. C. 1994 On the role of linear mechanisms in transition to turbulence. Phys. Fluids 6 (3), 1396–1398.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.CrossRefGoogle Scholar
Hutchins, N. & Marusic, I. 2007 b Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. A 365, 647–664.CrossRefGoogle ScholarPubMed
Jiménez, J., del Álamo, J. C. & Flores, O. 2004 The large-scale dynamics of near-wall turbulence. J. Fluid Mech. 505, 179–199.CrossRefGoogle Scholar
Jovanovic, M. R. & Bamieh, B. 2004 Unstable modes versus non-normal modes in supercritical channel flows. In Proceedings of the 2004 American Control Conference, Boston, MA.Google Scholar
Jovanovic, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–183.CrossRefGoogle Scholar
Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11, 417–422.CrossRefGoogle Scholar
Kim, J. & Lim, J. 1993 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 1885–1888.CrossRefGoogle Scholar
Klewicki, J. C., Fife, P., Wei, T. & McMurtry, P. 2007 A physical model of the turbulent boundary layer consonant with the mean momentum balance structure. Phil. Trans. R. Soc. A 365, 823–839.CrossRefGoogle ScholarPubMed
Kovasznay, L. S. G., Kibens, V. & Blackwelder, R. F. 1970 Large-scale motion in the intermittent region of a turbulent boundary layer. J. Fluid Mech. 41, 283–325.CrossRefGoogle Scholar
Krogstad, P.-A., Kaspersen, J. H. & Rimestad, S. 1998 Convection velocities in a turbulent boundary layer. J. Fluid Mech. 10 (4), 949–957.Google Scholar
Long, R. R. & Chen, T.-C. 1981 Experimental evidence for the existence of the ‘mesolayer’ in turbulent systems. J. Fluid Mech. 105, 19–59.CrossRefGoogle Scholar
Marusic, I. & Heuer, W. D. C. 2007 Reynolds number invariance of the structure inclination angle in wall turbulence. Phys. Rev. Lett. 99, 114504.CrossRefGoogle ScholarPubMed
Marusic, I., McKeon, B. J., Monkewitz, P. A., Nagib, H. M., Smits, A. J. & Sreenivasan, K. R. 2010 Wall-bounded turbulent flows at high Reynolds numbers: recent advances and key issues. Phys. Fluids. (in press).CrossRefGoogle Scholar
Maslowe, S. A. 1981 Shear flow instabilities and transition. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. Swinney, H. L. & Gollub, J. P.), pp. 181–228. Springer.Google Scholar
Maslowe, S. A. 1986 Critical layers in shear flows. Annu. Rev. Fluid Mech. 18, 405–432.CrossRefGoogle Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures of turbulent boundary layers. J. Fluid Mech. 628, 311–337.CrossRefGoogle Scholar
McKeon, B. J. 2008 Scaling in wall turbulence: scale separation and interaction. Paper 2008–4237. AIAA.CrossRefGoogle Scholar
McKeon, B. J., Li, J., Jiang, W., Morrison, J. F. & Smits, A. J. 2004 Further observations on the mean velocity distribution in fully developed pipe flow. J. Fluid Mech. 501, 135–147.CrossRefGoogle Scholar
McKeon, B. J. & Morrison, J. F. 2007 Asymptotic scaling in turbulent pipe flow. Phil. Trans. R. Soc. A 365, 771–787.CrossRefGoogle ScholarPubMed
McKeon, B. J., Zagarola, M. V. & Smits, A. J. 2005 A new friction factor relationship for fully developed pipe flow. J. Fluid Mech. 538, 429–443.CrossRefGoogle Scholar
Meseguer, A. & Trefethen, L. N. 2003 Linearized pipe flow to Reynolds number 107. J. Comput. Phys. 186, 178–197.CrossRefGoogle Scholar
Metzger, M. M. & Klewicki, J. C. 2001 A comparative study of near-wall turbulence in high and low Reynolds number boundary layers. Phys. Fluids 13, 692–701.CrossRefGoogle Scholar
Metzger, M., McKeon, B. J. & Holmes, H. 2007 The near-neutral atmospheric surface layer: turbulence and non-stationarity. Phil. Trans. R. Soc. A 365, 859–876.CrossRefGoogle ScholarPubMed
Monty, J. P., Hafez, S., Jones, M. B. & Chong, M. S. 2001 Turbulent pipe flow: an analysis of mean-flow characteristics. In Australasian Fluid Mechanics Conference, pp. 921–924.Google 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
Morris, S. C., Stolpa, S. R., Slaboch, P. E. & Klewicki, J. 2007 Near-surface particle image velocimetry measurements in a transitionally rough-wall atmospheric boundary layer. J. Fluid Mech. 580, 319–338.CrossRefGoogle Scholar
Morrison, W. R. B., Bullock, K. J. & Kronauer, R. E. 1971 Experimental evidence of waves in the sublayer. J. Fluid Mech. 49 (4), 639–656.CrossRefGoogle Scholar
Morrison, J. F., Jiang, W., McKeon, B. J. & Smits, A. J. 2004 Scaling of the streamwise velocity component in turbulent pipe flow. J. Fluid Mech. 508, 99–131.CrossRefGoogle Scholar
Morrison, W. R. B. & Kronauer, R. E. 1969 Structural similarity for fully developed turbulence in smooth tubes. J. Fluid Mech. 39 (1), 117–141.CrossRefGoogle Scholar
Nagib, H. M. & Chauhan, K. A. 2008 Variations of von Kármán coefficient in canonical flows. Phys. Fluids 20, 101518.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld equation. SIAM J. Appl. Math. 53 (1), 15–47.CrossRefGoogle Scholar
Reynolds, W. C. & Hussain, A. K. M. F. 1972 The mechanics of an organized wave in shear flow. Part 3. Theoretical models and comparisons with experiment. J. Fluid Mech. 54, 263–288.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), 253–272.CrossRefGoogle Scholar
Sahay, A. & Sreenivasan, K. R. 1999 The wall-normal position in pipe and channel flows at which viscous and turbulent shear stresses are equal. Phys. Fluids 11 (10), 3186–3188.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Schoppa, W. & Hussain, F. 2002 Coherent structure generation in near-wall turbulence. J. Fluid Mech. 453, 57–108.CrossRefGoogle Scholar
Sharma, A. S. & McKeon, B. J. 2009 Perturbation energy production in pipe flow over a range of Reynolds numbers using resolvent analysis. In Forty-seventh AIAA Aerospace Sciences Meeting, Paper 2009-1513. AIAA.CrossRefGoogle Scholar
Sharma, A. S., McKeon, B. J., Morrison, J. F. & Limebeer, D. J. N. 2006 Stabilising control laws for the incompressible Navier-Stokes equations using sector stability theory. In Third AIAA Flow Control Conference, Paper 2006-3695. AIAA.CrossRefGoogle Scholar
Shockling, M. A., Allen, J. J. & Smits, A. J. 2006 Effects of machined surface roughness on high-Reynolds-number turbulent pipe flow. J. Fluid Mech. 564, 267–285.CrossRefGoogle Scholar
Sirovich, L., Ball, K. S. & Keefe, L. R. 1990 Plane waves and structures in turbulent channel flow. Phys. Fluids A 2 (12), 2217–2226.CrossRefGoogle Scholar
Sreenivasan, K. R. 1988 A unified view of the origin and morphology of the turbulent boundary layer structure. In Turbulence Management and Relaminarisation; Proceedings of the IUTAM Symposium, Bangalore, India, January 13–23, 1987 (A89-10154 01-34) (ed. Liepmann, H. W. & Narasimha, R.), pp. 37–61. Springer.Google Scholar
Sreenivasan, K. R. & Bershadskii, A. 2006 The mean velocity distribution near the peak of the Reynolds shear stress, extending also to the buffer region. In IUTAM Symposium on One Hundred Years of Boundary Layer Research (ed. Sreenivasan, K. R., Meier, G. E. A. & Heinemann, H.-J.), pp. 241–246. Springer Netherlands.CrossRefGoogle Scholar
Sreenivasan, K. R. & Sahay, A. 1997 The persistence of viscous effects in the overlap region, and the mean velocity in turbulent pipe and channel flows. In Self-Sustaining Mechanisms of Wall Turbulence (ed. Panton, R.), pp. 253–272. Computational Mechanics.Google Scholar
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
Viswanath, D. 2009 The critical layer in pipe flow at high Reynolds number. Phil. Trans. R. Soc. A 367, 561–576.CrossRefGoogle ScholarPubMed
Waleffe, F. 1997 On a self-sustaining process in shear flows. Phys. Fluids 9 (4), 883–900.CrossRefGoogle Scholar
Waleffe, F. 2001 Exact coherent structures in channel flow. J. Fluid Mech. 435, 93–102.CrossRefGoogle Scholar
Waleffe, F. 2003 Homotopy of exact coherent structures in plane shear flows. Phys. Fluids 15 (6), 1517–1534.CrossRefGoogle Scholar
Wang, J., Gibson, J. & Waleffe, F. 2007 Lower branch coherent states in shear flows: transition and control. Phys. Rev. Lett. 98 (204501).CrossRefGoogle ScholarPubMed
Wedin, H. & Kerswell, R. R. 2004 Exact coherent structures in pipe flow: travelling wave solutions. J. Fluid Mech. 508, 333–371.CrossRefGoogle Scholar
Willis, A., Hwang, Y. & Cossu, C. 2009 Drag reduction in pipe flow by optimal forcing. ArXiv 0908.3971.v1.Google Scholar
Wu, X. & Moin, P. 2008 A direct numerical simulation study on the mean velocity characteristics in pipe flow. J. Fluid Mech. 608, 81–112.CrossRefGoogle Scholar
Young, N. 1988 An Introduction to Hilbert Space. Cambridge University Press.CrossRefGoogle Scholar
Zagarola, M. V. & Smits, A. J. 1998 Mean flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.CrossRefGoogle Scholar
Zhao, R. & Smits, A. J. 2007 Scaling of the wall-normal turbulence component in high-Reynolds-number pipe flow. J. Fluid Mech. 576, 457–473.CrossRefGoogle Scholar