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Optimal disturbances and large-scale energetic motions in turbulent boundary layers

Published online by Cambridge University Press:  29 November 2018

Timothy B. Davis Open the ORCID record for Timothy B. Davis [Opens in a new window] ,
Ali Uzun  and
Farrukh S. Alvi
Timothy B. Davis*
Affiliation:
Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida Center for Advanced Aero-Propulsion (FCAAP), Florida State University, Tallahassee, FL 32310, USA
Ali Uzun
Affiliation:
Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida Center for Advanced Aero-Propulsion (FCAAP), Florida State University, Tallahassee, FL 32310, USA
Farrukh S. Alvi
Affiliation:
Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida Center for Advanced Aero-Propulsion (FCAAP), Florida State University, Tallahassee, FL 32310, USA
*
†Email address for correspondence: [email protected]
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Abstract

We examine disturbances leading to optimal energy growth in a spatially developing, zero-pressure-gradient turbulent boundary layer. The slow development of the turbulent mean flow in the streamwise direction is modelled through a parabolized formulation to enable a spatial marching procedure. In the present framework, conventional spatial optimal disturbances arise naturally as the homogeneous solution to the linearized equations subject to a turbulent forcing at particular wavenumber combinations. A wave-like decomposition for the disturbance is considered to incorporate both conventional stationary modes as well as propagating modes formed by non-zero frequency/streamwise wavenumber and representative of convective structures naturally observed in wall turbulence. The optimal streamwise wavenumber, which varies with the spatial development of the turbulent mean flow, is computed locally via an auxiliary optimization constraint. The present approach can then be considered, in part, as an extension of the resolvent-based analyses for slowly developing flows. Optimization results reveal highly amplified disturbances for both stationary and propagating modes. Stationary modes identify peak amplification of structures residing near the centre of the logarithmic layer of the turbulent mean flow. Inner-scaled disturbances reminiscent of near wall streaks, and amplified over short streamwise distances, are identified in the computed streamwise energy spectra. In all cases, however, propagating modes surpass their stationary counterpart in both energy amplification and relative contribution to total fluctuation energy. We identify two classes of large-scale energetic modes associated with the logarithmic and wake layers of the turbulent mean flow. The outer-scaled wake modes agree well with the large-scale motions that populate the wake layer. For high Reynolds numbers, the log modes increasingly dominate the energy spectra with the predicted streamwise and wall-normal scales in agreement with superstructures observed in turbulent boundary layers.

JFM classification

Boundary Layers: Boundary Layers Boundary Layers: Boundary layer stability Turbulent Flows: Turbulent boundary layers
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 860 , 10 February 2019 , pp. 40 - 80
Copyright
© 2018 Cambridge University Press 

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Footnotes

‡

Present address: National Institute of Aerospace, 100 Exploration Way, Hampton, VA 23666, USA

References

Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.Google 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.Google Scholar
del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Alizard, F. 2015 Linear stability of optimal streaks in the log-layer of turbulent channel flows. Phys. Fluids 27 (10), 105103.Google Scholar
Andersson, P., Berggren, M. & Henningsin, D. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.Google 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 (1852), 665–681.Google Scholar
Beneddine, S., Sipp, D., Arnault, A., Dandois, J. & Lesshafft, L. 2016 Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485–504.Google Scholar
Butler, K. & Farrell, B. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids 5, 774–777.Google Scholar
Butler, K. M. & Farrell, B. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4 (8), 1637–1650.Google 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.Google Scholar
Dennis, D. & Nickels, T. 2011a Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 1. Vortex packets. J. Fluid Mech. 673, 180–217.Google Scholar
Dennis, D. & Nickels, T. 2011b Experimental measurement of large-scale three-dimensional structures in a turbulent boundary layer. Part 2. Long structures. J. Fluid Mech. 673, 218–244.Google Scholar
Eitel-Amor, G., Örlü, R. & Schlatter, P. 2014 Simulation and validation of a spatially evolving turbulent boundary layer up to Re 𝜃 = 8300. Intl J. Heat Fluid Flow 47 (C), 57–69.Google Scholar
Eitel-Amor, G., Orlu, R., Schlatter, P. & Flores, O. 2015 Hairpin vortices in turbulent boundary layers. Phys. Fluids 27 (2), 025108.Google Scholar
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18 (4), 487–488.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part II. Nonautonomous operators. J. Atmos. Sci. 53 (14), 2041–2053.Google Scholar
Gustavsson, L. H. 1991 Energy growth of three-dimensional disturbances in plane poiseuille flow. J. Fluid Mech. 224, 241–260.Google Scholar
Hack, M. J. P. & Moin, P. 2017 Algebraic disturbance growth by interaction of Orr and lift-up mechanisms. J. Fluid Mech. 829, 112–126.Google Scholar
Hanifi, A., Schmid, P. & Henningson, D. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826–13.Google Scholar
Head, M. R. & Bandyopadhyay, P. 1981 New aspects of turbulent boundary-layer structure. J. Fluid Mech. 107, 297–338.Google Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245–283.Google Scholar
Hutchins, N., Hambleton, W. T. & Marusic, I. 2005 Inclined cross-stream stereo particle image velocimetry measurements in turbulent boundary layers. J. Fluid Mech. 541, 21–54.Google Scholar
Hutchins, N. & Marusic, I. 2007a Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.Google Scholar
Hutchins, N. & Marusic, I. 2007b Large-scale influences in near-wall turbulence. Phil. Trans. R. Soc. Lond. A 365 (1852), 647–664.Google 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, 51–73.Google Scholar
Jiménez, J. & Pinelli, A. 1999 The autonomous cycle of near-wall turbulence. J. Fluid Mech. 389, 335–359.Google Scholar
Kim, K. C. & Adrian, R. J. 1999 Very large-scale motion in the outer layer. Phys. Fluids 11 (2), 417–422.Google Scholar
Kim, J. & Lim, J. 2000 A linear process in wall-bounded turbulent shear flows. Phys. Fluids 12 (8), 1885–1888.Google Scholar
Kline, S. J., Reynolds, W. C., Schraub, F. A. & Runstadler, P. W. 1967 The structure of turbulent boundary layers. J. Fluid Mech. 30 (04), 741–773.Google Scholar
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243–251.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.Google Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46 (1), 493–517.Google Scholar
Marquet, O., Lombardi, M. & Chomaz, J. M. 2009 Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities. J. Fluid Mech. 622, 1–21.Google Scholar
Marusic, I., Mathis, R. & Hutchins, N. 2010 High Reynolds number effects in wall turbulence. Intl J. Heat Fluid Flow 31 (3), 418–428.Google Scholar
Mathis, R., Hutchins, N. & Marusic, I. 2009 Large-scale amplitude modulation of the small-scale structures in turbulent boundary layers. J. Fluid Mech. 628, 311–337.Google Scholar
Matsubara, M. & Alfredsson, P. H. 2001 Disturbance growth in boundary layers subjected to free-stream turbulence. J. Fluid Mech. 430, 149–168.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382.Google Scholar
McKeon, B. J., Sharma, A. S. & Jacobi, I. 2013 Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25 (3), 031301.Google Scholar
Moarref, R., Sharma, A., Tropp, J. & McKeon, B. 2013 Model-based scaling of the streamwise energy density in high-Reynolds number turbulent channels. J. Fluid Mech. 734, 275–316.Google Scholar
Monkewitz, P., Chauhan, K. & Nagib, H. 2007 Self-consistent high-Reynolds-number asymptotics for zero-pressure-gradient turbulent boundary layers. Phys. Fluids 19 (11), 115101.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.Google Scholar
Österlund, J. M.1999 Experimental studies of zero pressure-gradient turbulent boundary layer flow. PhD thesis, Royal Institute of Technology (KTH), Stockholm, Sweden.Google Scholar
Panton, R. L. 2001 Overview of the self-sustaining mechanisms of wall turbulence. Prog. Aerosp. Sci. 37 (4), 341–383.Google Scholar
Pujals, G., Depardon, S. & Cossu, C. 2010 Drag reduction of a 3D bluff body using coherent streamwise streaks. Exp. Fluids 49 (5), 1085–1094.Google Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google 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, 263–288.Google Scholar
Reynolds, W. C. & Tiederman, W. G. 1967 Stability of turbulent channel flow, with application to Malkus’s theory. J. Fluid Mech. 27, 253–272.Google Scholar
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech. 23 (1), 601–639.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sharma, A. S. & McKeon, B. J. 2013 On coherent structure in wall turbulence. J. Fluid Mech. 728, 196–238.Google Scholar
Smits, A. J., McKeon, B. J. & Marusic, I. 2011 High-Reynolds number wall turbulence. Annu. Rev. Fluid Mech. 43 (1), 353–375.Google Scholar
Sreenivasan, K. R. 1989 The turbulent boundary layer. In Frontiers in Experimental Fluid Mechanics, pp. 159–209. Springer.Google Scholar
Tempelmann, D., Hanifi, A. & Henningson, D. 2010 Spatial optimal growth in three-dimensional boundary layers. J. Fluid Mech. 646, 5–33.Google Scholar
Theodorsen, T. 1952 Mechanism of turbulence. In Proceedings of the Second Midwestern Conference on Fluid Mechanics, pp. 1–18. Ohio State University.Google Scholar
Tomkins, C. D. & Adrian, R. J. 2003 Spanwise structure and scale growth in turbulent boundary layers. J. Fluid Mech. 490, 37–74.Google Scholar
Tomkins, C. D. & Adrian, R. J. 2005 Energetic spanwise modes in the logarithmic layer of a turbulent boundary layer. J. Fluid Mech. 545, 141–162.Google Scholar
Uzun, A., Davis, T. B., Alvi, F. S. & Hussaini, M. Y. 2017 Optimally growing boundary layer disturbances in a convergent nozzle preceded by a circular pipe. Theor. Comput. Fluid Dyn. 770 (1), 124–132.Google Scholar
Vallikivi, M., Ganapathisubramani, B. & Smits, A. J. 2015 Spectral scaling in boundary layers and pipes at very high Reynolds numbers. J. Fluid Mech. 771, 303–326.Google Scholar
Viola, F., Iungo, G. V., Camarri, S., Port-Agel, F. & Gallaire, F. 2014 Prediction of the hub vortex instability in a wind turbine wake: stability analysis with eddy-viscosity models calibrated on wind tunnel data. J. Fluid Mech. 750, R1.Google Scholar
Wu, X. & Moin, P. 2009 Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer. J. Fluid Mech. 630, 5–41.Google Scholar