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Efficient harmonic resolvent analysis via time stepping

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Abstract

We present an extension of the RSVD-\(\Delta t\) algorithm initially developed for resolvent analysis of statistically stationary flows to handle harmonic resolvent analysis of time-periodic flows. The harmonic resolvent operator, as proposed by Padovan et al. (J Fluid Mech 900, 2020), characterizes the linearized dynamics of time-periodic flows in the frequency domain, and its singular value decomposition reveals forcing and response modes with optimal energetic gain. However, computing harmonic resolvent modes poses challenges due to (i) the coupling of all \(N_{\omega }\) retained frequencies into a single harmonic resolvent operator and (ii) the singularity or near-singularity of the operator, making harmonic resolvent analysis considerably more computationally expensive than a standard resolvent analysis. To overcome these challenges, the RSVD-\(\Delta t\) algorithm leverages time stepping of the underlying time-periodic linearized Navier–Stokes operator, which is \(N_{\omega }\) times smaller than the harmonic resolvent operator, to compute the action of the harmonic resolvent operator. We develop strategies to minimize the algorithm’s CPU and memory consumption, and our results demonstrate that these costs scale linearly with the problem dimension. We validate the RSVD-\(\Delta t\) algorithm by computing modes for a periodically varying Ginzburg–Landau equation and demonstrate its performance using the flow over an airfoil.

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References

  1. Padovan, A., Otto, S.E., Rowley, C.W.: Analysis of amplification mechanisms and cross-frequency interactions in nonlinear flows via the harmonic resolvent. J. Fluid Mech. 900, (2020)

  2. Taira, K., Brunton, S.L., Dawson, S.T.M., Rowley, C.W., Colonius, T., McKeon, B.J., Schmidt, O.T., Gordeyev, S., Theofilis, V., Ukeiley, L.S.: Modal analysis of fluid flows: an overview. AIAA J. 55(12), 4013–4041 (2017)

    Article  Google Scholar 

  3. Towne, A., Schmidt, O.T., Colonius, T.: Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867 (2018)

    Article  MathSciNet  Google Scholar 

  4. Dawson, S.T.M., McKeon, B.J.: On the shape of resolvent modes in wall-bounded turbulence. J. Fluid Mech. 877, 682–716 (2019)

    Article  MathSciNet  Google Scholar 

  5. Sipp, D., Marquet, O.: Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theoret. Comput. Fluid Dyn. 27(5), 617–635 (2013)

    Article  Google Scholar 

  6. Pickering, E., Towne, A., Jordan, P., Colonius, T.: Resolvent-based modeling of turbulent jet noise. J. Acoust. Soc. Am. 150(4), 2421–2433 (2021)

    Article  Google Scholar 

  7. Troy, C.D., Koseff, J.R.: The instability and breaking of long internal waves. J. Fluid Mech. 543, 107–136 (2005)

    Article  Google Scholar 

  8. Giannenas, A.E., Laizet, S., Rigas, G.: Harmonic forcing of a laminar bluff body wake with rear pitching flaps. J. Fluid Mech. 945, A5 (2022)

    Article  MathSciNet  Google Scholar 

  9. Farghadan, A., Arzani, A.: The combined effect of wall shear stress topology and magnitude on cardiovascular mass transport. Int. J. Heat Mass Transf. 131, 252–260 (2019)

    Article  Google Scholar 

  10. Wereley, N.M.: Analysis and control of linear periodically time varying systems. PhD thesis, Massachusetts Institute of Technology (1990)

  11. Lin, C.-T., Tsai, M.-L., Tsai, H.-C.: Flow control of a plunging cylinder based on resolvent analysis. J. Fluid Mech. 967, A41 (2023)

    Article  MathSciNet  Google Scholar 

  12. Franceschini, L., Sipp, D., Marquet, O., Moulin, J., Dandois, J.: Identification and reconstruction of high-frequency fluctuations evolving on a low-frequency periodic limit cycle: application to turbulent cylinder flow. J. Fluid Mech. 942, A28 (2022)

    Article  MathSciNet  Google Scholar 

  13. Heidt, L., Colonius, T.: Spectral proper orthogonal decomposition of harmonically forced turbulent flows. J. Fluid Mech. 985, A42 (2024)

    Article  MathSciNet  Google Scholar 

  14. Wu, W., Meneveau, C., Mittal, R., Padovan, A., Rowley, C.W., Cattafesta, L.: Response of a turbulent separation bubble to zero-net-mass-flux jet perturbations. Phys. Rev. Fluids 7(8), 084601 (2022)

    Article  Google Scholar 

  15. Padovan, A., Rowley, C.W.: Analysis of the dynamics of subharmonic flow structures via the harmonic resolvent: application to vortex pairing in an axisymmetric jet. Phys. Rev. Fluids 7(7), 073903 (2022)

    Article  Google Scholar 

  16. Jin, B., Symon, S., Illingworth, S.J.: Energy transfer mechanisms and resolvent analysis in the cylinder wake. Phys. Rev. Fluids 6(2), 024702 (2021)

    Article  Google Scholar 

  17. Herbert, T.: Analysis of the subharmonic route to transition in boundary layers. In: 22nd Aerospace Sciences Meeting, Vol. 9 (1984)

  18. Edgington-Mitchell, D., Wang, T., Nogueira, P., Schmidt, O., Jaunet, V., Duke, D., Jordan, P., Towne, A.: Waves in screeching jets. J. Fluid Mech. 913, A7 (2021)

    Article  MathSciNet  Google Scholar 

  19. Kumar, C., Prakash, A.: Effect of mass injection on secondary instability of hypersonic boundary layer over a blunt cone. Phys. Fluids 34(6), (2022)

  20. Moarref, R., Sharma, A.S., Tropp, J.A., McKeon, B.J.: Model-based scaling of the streamwise energy density in high-Reynolds-number turbulent channels. J. Fluid Mech. 734, 275–316 (2013)

    Article  MathSciNet  Google Scholar 

  21. Ribeiro, J.H.M., Yeh, C.-A., Taira, K.: Randomized resolvent analysis. Phys. Rev. Fluids 5(3), 033902 (2020)

    Article  Google Scholar 

  22. Farghadan, A., Martini, E., Towne, A.: Scalable resolvent analysis for three-dimensional flows (2023). arXiv:2309.04617

  23. Monokrousos, A., Åkervik, E., Brandt, L., Henningson, D.S.: Global three-dimensional optimal disturbances in the Blasius boundary-layer flow using time-steppers. J. Fluid Mech. 650, 181–214 (2010)

    Article  MathSciNet  Google Scholar 

  24. Martini, E., Rodríguez, D., Towne, A., Cavalieri, A.V.G.: Efficient computation of global resolvent modes. J. Fluid Mech. 919 (2021)

  25. McKeon, B.J., Sharma, A.S.: A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382 (2010)

    Article  MathSciNet  Google Scholar 

  26. Leclercq, C., Sipp, D.: Mean resolvent operator of a statistically steady flow. J. Fluid Mech. 968, A13 (2023)

    Article  MathSciNet  Google Scholar 

  27. Halko, N., Martinsson, P., Tropp, J.A.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)

    Article  MathSciNet  Google Scholar 

  28. Nyquist, H.: Certain topics in telegraph transmission theory. Trans. Am. Inst. Electr. Eng. 47(2), 617–644 (1928)

    Article  Google Scholar 

  29. Towne, A., Rigas, G., Kamal, O., Pickering, E., Colonius, T.: Efficient global resolvent analysis via the one-way Navier–Stokes equations. J. Fluid Mech. 948, A9 (2022)

    Article  MathSciNet  Google Scholar 

  30. Rigas, G., Sipp, D., Colonius, T.: Nonlinear input/output analysis: application to boundary layer transition. J. Fluid Mech. 911, A15 (2021)

    Article  MathSciNet  Google Scholar 

  31. Chomaz, J.M., Huerre, P., Redekopp, L.G.: Bifurcations to local and global modes in spatially developing flows. Phys. Rev. Lett. 60(1), 25 (1988)

    Article  Google Scholar 

  32. Hunt, R.E., Crighton, D.G.: Instability of flows in spatially developing media. Proc. R. Soc. Lond. Ser. A: Math. Phys. Sci. 435(1893), 109–128 (1991)

    Google Scholar 

  33. Bagheri, S., Henningson, D.S., Hoepffner, J., Schmid, P.J.: Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. 62(2), (2009)

  34. Chen, K.K., Rowley, C.W.: H2 optimal actuator and sensor placement in the linearised complex Ginzburg–Landau system. J. Fluid Mech. 681, 241–260 (2011)

    Article  MathSciNet  Google Scholar 

  35. Cavalieri, A. V. G., Jordan, P., Lesshafft, L.: Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71(2), (2019)

  36. Jung, J., Bhagwat, R., Towne, A.: Resolvent-based estimation of laminar flow around an airfoil. AIAA Paper \(\#\)2023-0077 (2023)

  37. Chu, B.-T.: On the energy transfer to small disturbances in fluid flow (part i). Acta Mech. 1(3), 215–234 (1965)

    Article  Google Scholar 

  38. Schmidt, O.T., Towne, A., Rigas, G., Colonius, T., Brès, G.A.: Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982 (2018)

    Article  MathSciNet  Google Scholar 

  39. Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Dalcin, L., Dener, A., Eijkhout, V., Gropp, W., et al.: PETSc users manual. Argonne National Laboratory (2019)

  40. Hernandez, V., Roman, J.E., Vidal, V.: Slepc: A scalable and flexible toolkit for the solution of eigenvalue problems. ACM Trans. Math. Software (TOMS) 31(3), 351–362 (2005)

    Article  MathSciNet  Google Scholar 

  41. de Pando, M.F., Sipp, D., Schmid, P.J.: Efficient evaluation of the direct and adjoint linearized dynamics from compressible flow solvers. J. Comput. Phys. 231(23), 7739–7755 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We acknowledge the University of Michigan’s Great Lakes cluster for providing the essential computational resources to conduct all the simulations in this study.

Funding

This work was funded in part by Air Force Office of Scientific Research (AFOSR) grant no. FA9550-20-1-0214 and by a Catalyst Grant from the Michigan Institute for Computational Discovery and Engineering (MICDE).

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI, 48109, USA

    Ali Farghadan, Junoh Jung, Rutvij Bhagwat & Aaron Towne

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  1. Ali Farghadan
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  2. Junoh Jung
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  3. Rutvij Bhagwat
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  4. Aaron Towne
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Contributions

Conceptualization: Aaron Towne, Ali Farghadan; Methodology: Aaron Towne, Ali Farghadan; Formal analysis and investigation: Ali Farghadan; Writing - original draft preparation: Ali Farghadan; Writing - review and editing: All authors; Software: Ali Farghadan, Rutvij Bhagwat; Validation: Junoh Jung; Funding acquisition: Aaron Towne; Supervision: Aaron Towne.

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Correspondence to Aaron Towne.

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Communicated by André Cavalieri.

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RSVD-\(\Delta t\) for the subharmonic resolvent operator

RSVD-\(\Delta t\) for the subharmonic resolvent operator

In Sect. 3.5, we provided an overview of subharmonic resolvent analysis. In this appendix, we briefly outline the application of RSVD-\(\Delta t\) in computing subharmonic resolvent modes and gains. Consider the frequency of interest as \(\gamma \in \Omega _{\gamma }\). This set specifies the perturbation frequency, while the base flow frequency content is confined to \(\Omega _{\bar{q}}\).

To compute the actions of \(\varvec{H}\) and \(\varvec{H}^*\) using time stepping, we must adhere to both the base flow frequency, enforcing a duration of \(T = 2\pi /\omega _f\), and the perturbation frequency, enforcing a duration of \(T_p = 2\pi /\gamma \), in order to obtain \(N_{\omega }\) steady-state solutions to (7). Thus, we need to integrate for \(T_{sub}\) such that \(T_{sub}/T \in \mathbb {N}\) and \(T_{sub}/T_p \in \mathbb {N}\), during which \(N_{\omega }\) equidistant snapshots are saved. For instance, if we consider \(\gamma = \omega _f/5\), requiring \(T_p = 2\pi /(\omega _f/5) = 5T\), integrating for \(T_{sub} = 5T\) is sufficient to obtain the steady-state solutions. All the other steps remain the same as introduced in Algorithm 1.

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Farghadan, A., Jung, J., Bhagwat, R. et al. Efficient harmonic resolvent analysis via time stepping. Theor. Comput. Fluid Dyn. (2024). https://doi.org/10.1007/s00162-024-00694-1

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