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On the low-frequency dynamics of turbulent separation bubbles

Published online by Cambridge University Press:  22 August 2024

C. Cura Open the ORCID record for C. Cura [Opens in a new window] ,
A. Hanifi Open the ORCID record for A. Hanifi [Opens in a new window] ,
A.V.G. Cavalieri Open the ORCID record for A.V.G. Cavalieri [Opens in a new window]  and
J. Weiss Open the ORCID record for J. Weiss [Opens in a new window]
C. Cura*
Affiliation:
Institut für Luft- und Raumfahrt, Technische Universität Berlin, 10587 Berlin, Germany
A. Hanifi
Affiliation:
FLOW, Engineering Mechanics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden
A.V.G. Cavalieri
Affiliation:
Divisão de Engenharia Aeronáutica, Instituto Tecnológico de Aeronáutica, 12228-900 São José dos Campos, SP, Brazil
J. Weiss
Affiliation:
Institut für Luft- und Raumfahrt, Technische Universität Berlin, 10587 Berlin, Germany
*
†Email address for correspondence: c.cura@tu-berlin.de
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Abstract

The low-frequency modal and non-modal linear dynamics of an incompressible, pressure-gradient-induced turbulent separation bubble (TSB) are investigated, with the objective of studying the mechanism responsible for the low-frequency contraction and expansion (breathing) commonly observed in experimental studies. The configuration of interest is a TSB generated on a flat test surface by a succession of adverse and favourable pressure gradients. The base flow selected for the analysis is the average TSB from the direct numerical simulation of Coleman et al. (J. Fluid Mech., vol. 847, 2018, pp. 28–70). Global mode analysis reveals that the eigenmodes of the linear operator are damped for all frequencies and wavenumbers. Furthermore, the least damped eigenmode appears to occur at zero frequency and low, non-zero spanwise wavenumber when scaled with the separation length. Resolvent analysis is then employed to examine the forced dynamics of the flow. At low frequency, a region of low, non-zero spanwise wavenumber is also discernible, where the receptivity appears to be driven by the identified weakly damped global mode. The corresponding optimal energy gain is shown to have the shape of a first-order, low-pass filter with a cut-off frequency consistent with the low-frequency unsteadiness in TSBs. The results from resolvent analysis are compared to the unsteady experimental database of Le Floc'h et al. (J. Fluid Mech., vol. 902, 2020, A13) in a similar TSB flow. The alignment between the optimal response and the first spectral proper orthogonal decomposition mode computed from the experiments is shown to be close to $95\,\%$, while the spanwise wavenumber of the optimal response is consistent with that of the low-frequency breathing motion captured experimentally. This indicates that the fluctuations observed experimentally at low frequency closely match the response computed from resolvent analysis. Based on these results, we propose that the forced dynamics of the flow, driven by the weakly damped global mode, serve as a plausible mechanism for the origin of the low-frequency breathing motion commonly observed in experimental studies of TSBs.

JFM classification

Boundary Layers: Boundary layer separation Wakes/Jets: Separated flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 991 , 25 July 2024 , A11
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© The Author(s), 2024. Published by Cambridge University Press

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References

Abe, H. 2017 Reynolds-number dependence of wall-pressure fluctuations in a pressure-induced turbulent separation bubble. J. Fluid Mech. 833, 563–598.Google Scholar
Abreu, L.I., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R. & Henningson, D.S. 2020 Spectral proper orthogonal decomposition and resolvent analysis of near-wall coherent structures in turbulent pipe flows. J. Fluid Mech. 900, A11.Google Scholar
Abreu, L.I., Tanarro, A., Cavalieri, A.V.G., Schlatter, P., Vinuesa, R., Hanifi, A. & Henningson, D.S. 2021 Spanwise-coherent hydrodynamic waves around flat plates and airfoils. J. Fluid Mech. 927, A1.Google Scholar
Adler, M.C. & Gaitonde, D.V. 2018 Dynamic linear response of a shock/turbulent-boundary-layer interaction using constrained perturbations. J. Fluid Mech. 840, 291–341.Google Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75 (5), 750–756.Google Scholar
Beresh, S.J., Clemens, N.T. & Dolling, D.S. 2002 Relationship between upstream turbulent boundary-layer velocity fluctuations and separation shock unsteadiness. AIAA J. 40 (12), 2412–2422.Google Scholar
Blanco, D.C.P., Martini, E., Sasaki, K. & Cavalieri, A.V.G. 2022 Improved convergence of the spectral proper orthogonal decomposition through time shifting. J. Fluid Mech. 950, A9.Google Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.Google Scholar
Boutilier, M.S.H. & Yarusevych, S. 2013 Sensitivity of linear stability analysis of measured separated shear layers. Eur. J. Mech. (B/Fluids) 37, 129–142.Google Scholar
Bugeat, B., Robinet, J.-C., Chassaing, J.-C. & Sagaut, P. 2022 Low-frequency resolvent analysis of the laminar oblique shock wave/boundary layer interaction. J. Fluid Mech. 942, A43.Google Scholar
Cavalieri, A.V.G., Jordan, P. & Lesshafft, L. 2019 Wave-packet models for jet dynamics and sound radiation. Appl. Mech. Rev. 71 (2), 020802.Google Scholar
Cherry, N.J., Hillier, R. & Latour, M.E.M.P. 1984 Unsteady measurements in a separated and reattaching flow. J. Fluid Mech. 144, 13–46.Google Scholar
Clemens, N.T. & Narayanaswamy, V. 2014 Low-frequency unsteadiness of shock wave/turbulent boundary layer interactions. Annu. Rev. Fluid Mech. 46, 469–492.Google Scholar
Coleman, G.N. 2018 DNS: 2-D turbulent separation bubbles. Available at https://turbmodels.larc.nasa.gov/Other_DNS_Data/separation_bubble_2d.html.Google Scholar
Coleman, G.N., Rumsey, C.L. & Spalart, P.R. 2018 Numerical study of turbulent separation bubbles with varying pressure gradient and Reynolds number. J. Fluid Mech. 847, 28–70.Google ScholarPubMed
Crow, S.C. & Champagne, F.H. 1971 Orderly structure in jet turbulence. J. Fluid Mech. 48 (3), 547–591.Google Scholar
Cura, C., Hanifi, A. & Weiss, J. 2023 Medium-frequency unsteadiness in a turbulent separation bubble. In AIAA SCITECH 2023 Forum, 0095.Google Scholar
Dau, K., Borgmann, D., Little, J.C. & Weiss, J. 2023 Investigation of low frequency unsteadiness in the turbulent separation bubble produced by a wall mounted hump. In AIAA AVIATION 2023 Forum, 4008.Google Scholar
Del Alamo, J.C. & Jimenez, J. 2006 Linear energy amplification in turbulent channels. J. Fluid Mech. 559, 205–213.Google Scholar
Dolling, D.S. 2001 Fifty years of shock-wave/boundary-layer interaction research: what next? AIAA J. 39 (8), 1517–1531.Google Scholar
Dovgal, A.V., Kozlov, V.V. & Michalke, A. 1994 Laminar boundary layer separation: instability and associated phenomena. Prog. Aerosp. Sci. 30 (1), 61–94.Google Scholar
Dussauge, J.-P., Dupont, P. & Debiève, J.-F. 2006 Unsteadiness in shock wave boundary layer interactions with separation. Aerosp. Sci. Technol. 10 (2), 85–91.Google Scholar
Eaton, J.K. & Johnston, J.P. 1981 A review of research on subsonic turbulent flow reattachment. AIAA J. 19 (9), 1093–1100.Google Scholar
Ehrenstein, U. & Gallaire, F. 2008 Two-dimensional global low-frequency oscillations in a separating boundary-layer flow. J. Fluid Mech. 614, 315–327.Google Scholar
Gaitonde, D.V. 2015 Progress in shock wave/boundary layer interactions. Prog. Aerosp. Sci. 72, 80–99.Google Scholar
Ganapathisubramani, B., Clemens, N.T. & Dolling, D.S. 2007 Effects of upstream boundary layer on the unsteadiness of shock-induced separation. J. Fluid Mech. 585, 369–394.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
Görtler, H. 1954 On the three-dimensional instability of laminar boundary layers on concave walls. Tech. Memo. 1375. National Advisory Commitee for Aeronautics.Google Scholar
Hao, J. 2023 On the low-frequency unsteadiness in shock wave–turbulent boundary layer interactions. J. Fluid Mech. 971, A28.Google Scholar
Huerre, P. & Monkewitz, P.A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22 (1), 473–537.Google Scholar
Hutchins, N. & Marusic, I. 2007 Evidence of very long meandering features in the logarithmic region of turbulent boundary layers. J. Fluid Mech. 579, 1–28.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
Kiya, M. & Sasaki, K. 1983 Structure of a turbulent separation bubble. J. Fluid Mech. 137, 83–113.Google Scholar
Larchevêque, L. 2020 Normalizing low-frequency unsteadiness in compressible separated flows. AIAA 2020-0561. In AIAA SciTech 2020 Forum.Google Scholar
Le Floc'h, A., Mohammed-Taifour, A., Weiss, J., Dufresne, L., Vétel, J. & Jondeau, E. 2016 Effect of boundary-layer superstructures on separation bubble unsteadiness. In 24th International Congress of Theoretical and Applied Mechanics, Montreal, QC, Canada.Google Scholar
Le Floc'h, A., Weiss, J., Mohammed-Taifour, A. & Dufresne, L. 2020 Measurements of pressure and velocity fluctuations in a family of turbulent separation bubbles. J. Fluid Mech. 902, A13.Google Scholar
Le Floc'h, A.S., Mohammed-Taifour, A.T., Dufresne, L. & Weiss, J. 2018 Spanwise aspects of unsteadiness in a pressure-induced turbulent separation bubble. In 2018 Fluid Dynamics Conference, 3538.Google Scholar
Lesshafft, L., Semeraro, O., Jaunet, V., Cavalieri, A.V.G. & Jordan, P. 2019 Resolvent-based modeling of coherent wave packets in a turbulent jet. Phys. Rev. Fluids 4 (6), 063901.Google Scholar
Lumley, J.L. 1970 Stochastic Tools in Turbulence. Academic Press.Google Scholar
Mabey, D.G. 1972 Analysis and correlation of data on pressure fluctuations in separated flow. J. Aircr. 9 (9), 642–645.Google Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds-number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.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
Michalke, A. 1984 Survey on jet instability theory. Prog. Aerosp. Sci. 21, 159–199.Google Scholar
Mohammed-Taifour, A. 2017 Instationnarités dans une bulle de décollement turbulente: étude expérimentale. PhD thesis, École de technologie supérieure.Google Scholar
Mohammed-Taifour, A., Schwaab, Q., Pioton, J. & Weiss, J. 2015 A new wind tunnel for the study of pressure-induced separating and reattaching flows. Aeronaut. J. 119 (1211), 91–108.Google Scholar
Mohammed-Taifour, A. & Weiss, J. 2016 Unsteadiness in a large turbulent separation bubble. J. Fluid Mech. 799, 383–412.Google Scholar
Mohammed-Taifour, A. & Weiss, J. 2021 Periodic forcing of a large turbulent separation bubble. J. Fluid Mech. 915, A24.Google Scholar
Morra, P., Nogueira, P.A.S., Cavalieri, A.V.G. & Henningson, D.S. 2021 The colour of forcing statistics in resolvent analyses of turbulent channel flows. J. Fluid Mech. 907, A24.Google Scholar
Morra, P., Semeraro, O., Henningson, D.S. & Cossu, C. 2019 On the relevance of Reynolds stresses in resolvent analyses of turbulent wall-bounded flows. J. Fluid Mech. 867, 969–984.Google Scholar
Na, Y. & Moin, P. 1998 Direct numerical simulation of a separated turbulent boundary layer. J. Fluid Mech. 374, 379–405.Google Scholar
Nichols, J.W., Larsson, J., Bernardini, M. & Pirozzoli, S. 2017 Stability and modal analysis of shock/boundary layer interactions. Theor. Comput. Fluid Dyn. 31, 33–50.Google Scholar
Nishioka, M., Asai, M. & Yoshida, S. 1990 Control of flow separation by acoustic excitation. AIAA J. 28 (11), 1909–1915.Google Scholar
Nogueira, P.A.S., Morra, P., Martini, E., Cavalieri, A.V.G. & Henningson, D.S. 2021 Forcing statistics in resolvent analysis: application in minimal turbulent Couette flow. J. Fluid Mech. 908, A32.Google Scholar
Padovan, A., Otto, S.E. & Rowley, C.W. 2020 Analysis of amplification mechanisms and cross-frequency interactions in nonlinear flows via the harmonic resolvent. J. Fluid Mech. 900, A14.Google Scholar
Pickering, E., Rigas, G., Nogueira, P.A.S., Cavalieri, A.V.G., Schmidt, O.T. & Colonius, T. 2020 Lift-up, Kelvin–Helmholtz and Orr mechanisms in turbulent jets. J. Fluid Mech. 896, A2.Google Scholar
Piponniau, S., Dussauge, J.-P., Debieve, J.-F. & Dupont, P. 2009 A simple model for low-frequency unsteadiness in shock-induced separation. J. Fluid Mech. 629, 87–108.Google Scholar
Pirozzoli, S. & Grasso, F. 2006 Direct numerical simulation of impinging shock wave/turbulent boundary layer interaction at $M= 2.25$. Phys. Fluids 18 (6), 065113.Google Scholar
Plotkin, K.J. 1975 Shock wave oscillation driven by turbulent boundary-layer fluctuations. AIAA J. 13 (8), 1036–1040.Google Scholar
Poggie, J., Bisek, N.J., Kimmel, R.L. & Stanfield, S.A. 2015 Spectral characteristics of separation shock unsteadiness. AIAA J. 53 (1), 200–214.Google Scholar
Porter, K.M. & Poggie, J. 2019 Selective upstream influence on the unsteadiness of a separated turbulent compression ramp flow. Phys. Fluids 31 (1), 016104.Google Scholar
Priebe, S., Tu, J.H., Rowley, C.W. & Martín, M.P. 2016 Low-frequency dynamics in a shock-induced separated flow. J. Fluid Mech. 807, 441–477.Google Scholar
Rajaee, M., Karlsson, S.K.F. & Sirovich, L. 1994 Low-dimensional description of free-shear-flow coherent structures and their dynamical behaviour. J. Fluid Mech. 258, 1–29.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
Richardson, R., Zhang, Y., Cattafesta, L.N. & Wu, W. 2023 Low frequency characteristics of a pressure-gradient induced turbulent separation bubble. In AIAA SCITECH 2023 Forum, 0079.Google Scholar
Robinet, J.-C. 2007 Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach. J. Fluid Mech. 579, 85–112.Google Scholar
Rodríguez, D. & Gennaro, E.M. 2017 Three-dimensional flow stability analysis based on the matrix-forming approach made affordable. In Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2016: Selected Papers from the ICOSAHOM conference, June 27–July 1, 2016, Rio de Janeiro, Brazil, pp. 639–650. Springer International.Google Scholar
Rodríguez, D., Gennaro, E.M. & Juniper, M.P. 2013 The two classes of primary modal instability in laminar separation bubbles. J. Fluid Mech. 734, R4.Google Scholar
Sansica, A., Sandham, N.D. & Hu, Z. 2016 Instability and low-frequency unsteadiness in a shock-induced laminar separation bubble. J. Fluid Mech. 798, 5–26.Google Scholar
Sartor, F., Mettot, C., Bur, R. & Sipp, D. 2015 Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis. J. Fluid Mech. 781, 550–577.Google Scholar
Sasaki, K., Barros, D.C., Cavalieri, A.V.G. & Larchevêque, L. 2021 Causality in the shock wave/turbulent boundary layer interaction. Phys. Rev. Fluids 6 (6), 064609.Google Scholar
Schmidt, O.T., Towne, A., Colonius, T., Cavalieri, A.V.G., Jordan, P. & Brès, G.A. 2017 Wavepackets and trapped acoustic modes in a turbulent jet: coherent structure eduction and global stability. J. Fluid Mech. 825, 1153–1181.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.Google Scholar
Smits, A.J. & Dussauge, J.-P. 2006 Turbulent Shear Layers in Supersonic Flow. Springer Science & Business Media.Google Scholar
Steinfurth, B., Cura, C. & Weiss, J. 2022 Three-dimensional effects associated with the low-frequency breathing motion of a turbulent separation bubble. In 20th International Symposium on Application of Laser and Imaging Techniques to Fluid Mechanics, Lisbon.Google Scholar
Theofilis, V. 2003 Advances in global linear instability analysis of nonparallel and three-dimensional flows. Prog. Aerosp. Sci. 39 (4), 249–315.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar
Theofilis, V., Hein, S. & Dallmann, U. 2000 On the origins of unsteadiness and three-dimensionality in a laminar separation bubble. Phil. Trans. R. Soc. Lond. A 358 (1777), 3229–3246.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
Touber, E. & Sandham, N. 2011 Low-order stochastic modelling of low-frequency motions in reflected shock-wave/boundary-layer interactions. J. Fluid Mech. 671, 417–465.Google Scholar
Touber, E. & Sandham, N.D. 2009 Large-eddy simulation of low-frequency unsteadiness in a turbulent shock-induced separation bubble. Theor. Comput. Fluid Dyn. 23, 79–107.Google Scholar
Towne, A., Cavalieri, A.V.G., Jordan, P., Colonius, T., Schmidt, O., Jaunet, V. & Brès, G.A. 2017 Acoustic resonance in the potential core of subsonic jets. J. Fluid Mech. 825, 1113–1152.Google Scholar
Towne, A., Schmidt, O.T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.Google Scholar
Wang, S. & Ghaemi, S. 2022 Unsteady motions in the turbulent separation bubble of a two-dimensional wing. J. Fluid Mech. 948, A3.Google Scholar
Weiss, J., Little, J., Threadgill, J. & Gross, A. 2021 Low-frequency unsteadiness in pressure-induced separation bubbles. AIAA 2021-1324. In AIAA SciTech 2021 Forum.Google Scholar
Weiss, J., Mohammed-Taifour, A. & Schwaab, Q. 2015 Unsteady behavior of a pressure-induced turbulent separation bubble. AIAA J. 53 (9), 2634–2645.Google Scholar
Weiss, J., Steinfurth, B., Chamard, L., Giani, A. & Combette, P. 2022 Spectral proper orthogonal decomposition of unsteady wall shear stress under a turbulent separation bubble. AIAA J. 60 (4), 2150–2159.Google Scholar
Wu, W., Meneveau, C. & Mittal, R. 2020 Spatio-temporal dynamics of turbulent separation bubbles. J. Fluid Mech. 883, A45.Google Scholar
Wu, W., Meneveau, C., Mittal, R., Padovan, A., Rowley, C.W. & Cattafesta, L. 2022 Response of a turbulent separation bubble to zero-net-mass-flux jet perturbations. Phys. Rev. Fluids 7 (8), 084601.Google Scholar
Yarusevych, S., Sullivan, P.E. & Kawall, J.G. 2006 Coherent structures in an airfoil boundary layer and wake at low Reynolds numbers. Phys. Fluids 18 (4), 044101.Google Scholar
Zare, A., Jovanović, M.R. & Georgiou, T.T. 2017 Colour of turbulence. J. Fluid Mech. 812, 636–680.Google Scholar