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Coherent large-scale structures from the linearized Navier–Stokes equations

Published online by Cambridge University Press:  20 June 2019

Anagha Madhusudanan Open the ORCID record for Anagha Madhusudanan [Opens in a new window] ,
Simon. J. Illingworth Open the ORCID record for Simon. J. Illingworth [Opens in a new window]  and
Ivan Marusic Open the ORCID record for Ivan Marusic [Opens in a new window]
Anagha Madhusudanan*
Affiliation:
Department of Mechanical Engineering, University of Melbourne, VIC 3010, Australia
Simon. J. Illingworth
Affiliation:
Department of Mechanical Engineering, University of Melbourne, VIC 3010, Australia
Ivan Marusic
Affiliation:
Department of Mechanical Engineering, University of Melbourne, VIC 3010, Australia
*
†Email address for correspondence: [email protected]
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Abstract

The wall-normal extent of the large-scale structures modelled by the linearized Navier–Stokes equations subject to stochastic forcing is directly compared to direct numerical simulation (DNS) data. A turbulent channel flow at a friction Reynolds number of $Re_{\unicode[STIX]{x1D70F}}=2000$ is considered. We use the two-dimensional (2-D) linear coherence spectrum (LCS) to perform the comparison over a wide range of energy-carrying streamwise and spanwise length scales. The study of the 2-D LCS from DNS indicates the presence of large-scale structures that are coherent over large wall-normal distances and that are self-similar. We find that, with the addition of an eddy viscosity profile, these features of the large-scale structures are captured by the linearized equations, except in the region close to the wall. To further study this coherence, a coherence-based estimation technique, spectral linear stochastic estimation, is used to build linear estimators from the linearized Navier–Stokes equations. The estimator uses the instantaneous streamwise velocity field or the 2-D streamwise energy spectrum at one wall-normal location (obtained from DNS) to predict the same quantity at a different wall-normal location. We find that the addition of an eddy viscosity profile significantly improves the estimation.

JFM classification

Turbulent Flows: Turbulence modelling Turbulent Flows: Turbulence theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 873 , 25 August 2019 , pp. 89 - 109
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Copyright
© 2019 Cambridge University Press 

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References

Adrian, R. J. 1979 Conditional eddies in isotropic turbulence. Phys. Fluids 22 (11), 2065–2070.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
Adrian, R. J. & Moin, P. 1988 Stochastic estimation of organized turbulent structure: homogeneous shear flow. J. Fluid Mech. 190, 531–559.Google Scholar
Baars, W. J., Hutchins, N. & Marusic, I. 2016 Spectral stochastic estimation of high-Reynolds-number wall-bounded turbulence for a refined inner–outer interaction model. Phys. Rev. Fluids 1, 054406.Google Scholar
Baars, W. J., Hutchins, N. & Marusic, I. 2017 Self-similarity of wall-attached turbulence in boundary layers. J. Fluid Mech. 823, R2.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. Lond. 365 (1852), 665–681.Google Scholar
Bonnet, J. P., Delville, J., Glauser, M. N., Antonia, R. A., Bisset, D. K., Cole, D. R., Fiedler, H. E., Garem, J. H., Hilberg, D., Jeong, J. et al. 1998 Collaborative testing of eddy structure identification methods in free turbulent shear flows. Exp. Fluids 25, 197–225.Google Scholar
Brown, G. L. & Thomas, A. S. W. 1977 Large structure in a turbulent boundary layer. Phys. Fluids 20 (10), S243–S252.Google Scholar
Butler, K. M. & Farrell, B. F. 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flow. Phys. Fluids A 5 (3), 774–777.Google 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
Cole, D. R. & Glauser, M. N. 1998 Applications of stochastic estimation in the axisymmetric sudden expansion. Phys. Fluids 10 (11), 2941–2949.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
Del Álamo, J. C. & Jiménez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Dennis, D. J. C. & Nickels, T. B. 2011 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
Encinar, M. P., Vela-Martín, A., García-Gutiérrez, A. & Jiménez, J.2018 A second-order consistent, low-storage method for time-resolved channel flow simulations. arXiv:1808.06461.Google Scholar
Ewing, D. & Citriniti, J. H. 1999 Examination of a LSE/POD complementary technique using single and multi-time information in the axisymmetric shear layer. In IUTAM Symposium on Simulation and Identification of Organized Structures in Flows, pp. 375–384. Springer.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5 (6), 1390–1400.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1998 Perturbation structure and spectra in turbulent channel flow. Theor. Comput. Fluid Dyn. 11, 237–250.Google Scholar
Ganapathisubramani, B., Longmire, E. K. & Marusic, I. 2003 Characteristics of vortex packets in turbulent boundary layers. J. Fluid Mech. 478, 35–46.Google 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.Google Scholar
Hellström, L. H. O., Marusic, I. & Smits, A. J. 2016 Self-similarity of the large-scale motions in turbulent pipe flow. J. Fluid Mech. 792, R1.Google 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
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. 365 (1852), 647–664.Google Scholar
Hwang, Y. 2015 Statistical structure of self-sustaining attached eddies in turbulent channel flow. J. Fluid Mech. 767, 254–289.Google 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.Google 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.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, 146–162.Google Scholar
Jiménez, J. 2012 Cascades in wall-bounded turbulence. Annu. Rev. Fluid Mech. 44, 27–45.Google Scholar
Jiménez, J., Del Álamo, J. C. & Flores, O. 2004 The large-scale dynamics of near-wall turbulence. J. Fluid Mech. 505, 179–199.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
Klewicki, J., Fife, P. & Wei, T. 2009 On the logarithmic mean profile. J. Fluid Mech. 638, 73–93.Google Scholar
Lozano-Durán, A., Flores, O. & Jiménez, J. 2012 The three-dimensional structure of momentum transfer in turbulent channels. J. Fluid Mech. 694, 100–130.Google Scholar
Marusic, I. 2001 On the role of large-scale structures in wall turbulence. Phys. Fluids 13 (3), 735–743.Google Scholar
Marusic, I. & Monty, J. P. 2019 Attached eddy model of wall turbulence. Annu. Rev. Fluid Mech. 51, 49–74.Google Scholar
Marusic, I., Monty, J. P., Hultmark, M. & Smits, A. J. 2013 On the logarithmic region in wall turbulence. J. Fluid Mech. 716, R3.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., Jovanovi, M. R., Tropp, J. A., Sharma, A. S. & McKeon, B. J. 2014 A low-order decomposition of turbulent channel flow via resolvent analysis and convex optimization. Phys. Fluids 26 (5), 051701.Google Scholar
Moarref, R., Sharma, A. S., Tropp, J. A. & McKeon, B. J. 2013 Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316.Google 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.Google 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.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 (2), 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 (2), 253–272.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.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, 353–375.Google Scholar
Tinney, C. E., Coiffet, F., Delville, J., Hall, A. M., Jordan, P. & Glauser, M. N. 2006 On spectral linear stochastic estimation. Exp. Fluids 41 (5), 763–775.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
Townsend, A. A. 1976 The Structure of Turbulent Shear Flow. Cambridge University Press.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar
Waleffe, F., Kim, J. & Hamilton, J. M. 1993 On the origin of streaks in turbulent shear flows. In Turbulent Shear Flows, vol. 8, pp. 37–49. Springer.Google Scholar
Willis, A. P., Hwang, Y. & Cossu, C. 2010 Optimally amplified large-scale streaks and drag reduction in turbulent pipe flow. Phys. Rev. E 82 (3), 036321.Google Scholar
Zhou, J., Adrian, R. J., Balachandar, S. & Kendall, T. M. 1999 Mechanisms for generating coherent packets of hairpin vortices in channel flow. J. Fluid Mech. 387, 353–396.Google Scholar
Zhou, K., Doyle, J. C. & Glover, K. 1996 Robust and Optimal Control, vol. 40. Prentice-Hall.Google Scholar