Hostname: page-component-848d4c4894-ndmmz Total loading time: 0 Render date: 2024-05-02T05:35:58.372Z Has data issue: false hasContentIssue false
  • English
  • Français

Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis

Published online by Cambridge University Press:  24 September 2015

F. Sartor ,
C. Mettot ,
R. Bur  and
D. Sipp
F. Sartor*
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
C. Mettot
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
R. Bur
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
D. Sipp
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
*
†Email address for correspondence: fulvio.sartor@onera.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

A transonic interaction between a shock wave and a turbulent boundary layer is experimentally and theoretically investigated. The configuration is a transonic channel flow over a bump, where a shock wave causes the separation of the boundary layer in the form of a recirculating bubble downstream of the shock foot. Different experimental techniques allow for the identification of the main unsteadiness features. As recognised in similar shock-wave/boundary-layer interactions, the flow field exhibits two distinct characteristic frequencies, whose origins are still controversial: a low-frequency motion which primarily affects the shock wave; and medium-frequency perturbations localised in the shear layer. A Fourier analysis of a series of Schlieren snapshots is performed to precisely characterise the structure of the perturbations at low- and medium-frequencies. Then, the Reynolds-averaged Navier–Stokes (RANS) equations closed with a Spalart–Allmaras turbulence model are solved to obtain a mean flow, which favourably compares with the experimental results. A global stability analysis based on the linearization of the full RANS equations is then performed. The eigenvalues of the Jacobian operator are all damped, indicating that the interaction dynamic cannot be explained by the existence of unstable global modes. The input/output behaviour of the flow is then analysed by performing a singular-value decomposition of the Resolvent operator; pseudo-resonances of the flow may be identified and optimal forcings/responses determined as a function of frequency. It is found that the flow strongly amplifies both medium-frequency perturbations, generating fluctuations in the mixing layer, and low-frequency perturbations, affecting the shock wave. The structure of the optimal perturbations and the preferred frequencies agree with the experimental observations.

JFM classification

Instability: Absolute/convective instability Compressible Flows: Compressible Flows Compressible Flows: Shock waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 781 , 25 October 2015 , pp. 550 - 577
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdessemed, N., Sharma, A. S., Sherwin, S. J. & Theofilis, V. 2009 Transient growth analysis of the flow past a circular cylinder. Phys. Fluids 21, 044103.Google Scholar
Agostini, L., Larchevêque, L., Dupont, P., Debiève, J. F. & Dussauge, J. P. 2012 Zones of influence and shock motion in a shock/boundary-layer interaction. AIAA J. 50 (6), 1377–1387.Google Scholar
Alizard, F., Cherubini, S. & Robinet, J. C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21 (6), 064108.Google Scholar
Andreopoulos, J. & Muck, K. C. 1987 Some new aspects of the shock-wave/boundary-layer interaction in compression-ramp flows. J. Fluid Mech. 180 (1), 405–428.Google Scholar
Beneddine, S., Mettot, C. & Sipp, D. 2015 Global stability analysis of underexpanded screeching jets. Eur. J. Mech. (B/Fluids) 49, 392–399.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
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.CrossRefGoogle Scholar
Brandt, L., Sipp, D., Pralits, J. O. & Marquet, O. 2011 Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.CrossRefGoogle Scholar
Bur, R., Benay, R., Galli, A. & Berthouze, P. 2006 Experimental and numerical study of forced shock-wave oscillations in a transonic channel. Aerosp. Sci. Technol. 10 (4), 265–278.Google Scholar
Bur, R., Coponet, D. & Carpels, Y. 2009 Separation control by vortex generator devices in a transonic channel flow. Shock Waves 19 (6), 521–530.Google Scholar
Bur, R., Corbel, B. & Délery, J. 1998 Study of passive control in a transonic shock wave/boundary-layer interaction. AIAA J. 36 (3), 394–400.Google Scholar
Cambier, Laurent, Heib, Sébastien & Plot, Sylvie 2013 The onera elsA CFD software: input from research and feedback from industry. Mechanics Industry 14 (03), 159–174.Google Scholar
Cerqueira, S. & Sipp, D. 2014 Eigenvalue sensitivity, singular values and discrete frequency selection mechanism in noise amplifiers: the case of flow induced by radial wall injection. J. Fluid Mech. 757, 770–799.CrossRefGoogle 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
Cossu, C. & Chomaz, J. M. 1997 Global measures of local convective instabilities. Phys. Rev. Lett. 78 (23), 4387.Google Scholar
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comput. Phys. 224 (2), 924–940.Google Scholar
Crouch, J. D., Garbaruk, A., Magidov, D. & Travin, A. 2009 Origin of transonic buffet on aerofoils. J. Fluid Mech. 628 (1), 357–369.Google Scholar
Deck, S. 2005 Numerical simulation of transonic buffet over the OAT15A airfoil. AIAA J. 43 (7), 1556–1566.Google Scholar
Délery, J. 1978 Analysis of the separation due to shock wave-turbulent boundary layer interaction in transonic flow. La Recherche Aerospatiale 305–320.Google Scholar
Délery, Jean 2010 Handbook of Compressible Aerodynamics. ISTE.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
Dolling, D. S. & Brusniak, L. 1989 Separation shock motion in fin, cylinder, and compression ramp-induced turbulent interactions. AIAA J. 27 (6), 734–742.Google Scholar
Dolling, D. S. & Erengil, M. E. 1991 Unsteady wave structure near separation in a Mach 5 compression ramp interaction. AIAA J. 29 (5), 728–735.Google Scholar
Drazin, P. G. & Reid, W. H. 1980 Solutions to the Problems in Hydrodynamic Stability. Cambridge University Press.Google Scholar
Dupont, P., Haddad, C. & Debiève, J. F. 2006 Space and time organization in a shock-induced separated boundary layer. J. Fluid Mech. 559, 255–278.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
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.Google Scholar
Erengil, M. E. & Dolling, D. S. 1991a Correlation of separation shock motion with pressure fluctuations in the incoming boundary layer. AIAA J. 29 (11), 1868–1877.Google Scholar
Erengil, M. E. & Dolling, D. S. 1991b Unsteady wave structure near separation in a Mach 5 compression ramp interaction. AIAA J. 29 (5), 728–735.CrossRefGoogle Scholar
Farrell, B. F & Ioannou, P. J. 1996 Generalized stability theory. Part I: Autonomous operators. J. Atmos. Sci. 53 (14), 2025–2040.Google Scholar
Grilli, M., Schmid, P. J., Hickel, S. & Adams, N. A. 2012 Analysis of unsteady behaviour in shockwave turbulent boundary layer interaction. J. Fluid Mech. 700, 16–28.Google Scholar
Harten, A. & Hyman, J. M. 1983 Self adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50 (2), 235–269.Google Scholar
Huerre, P. 2000 Open shear flow instabilities. In Perspectives in Fluid Dynamics (ed. Batchelor, G., Moffatt, H. & Worster, M.), Perspectives in Fluid Dynamics, pp. 159–229. Cambrige University Press.Google Scholar
Iaccarino, G., Ooi, A., Durbin, P. A. & Behnia, M. 2003 Reynolds averaged simulation of unsteady separated flow. Intl J. Heat Fluid Flow 24 (2), 147–156.CrossRefGoogle Scholar
Jackson, C. P. 1987 A finite-element study of the onset of vortex shedding in flow past variously shaped bodies. J. Fluid Mech. 182 (1), 23–45.CrossRefGoogle Scholar
Jovanovic, M. R. & Bamieh, B. 2005 Componentwise energy amplification in channel flows. J. Fluid Mech. 534, 145–184.CrossRefGoogle Scholar
Knight, D. D. & Degrez, G. 1998 Shock wave boundary layer interactions in high Mach number flows a critical survey of current numerical prediction capabilities. AGARD ADVISORY REPORT AGARD AR 2, 1–1.Google Scholar
Koch, W. 2005 Acoustic resonances in rectangular open cavities. AIAA J. 43 (11), 2342–2349.Google Scholar
Kussoy, M. I., Brown, J. D., Brown, J. L., Lockman, W. K. & Horstman, C. C. 1988 Fluctuations and massive separation in three-dimensional shock-wave/boundary-layer interactions. In Transport Phenomena in Turbulent Flows: Theory, Experiment, and Numerical Simulation (ed. Hirata, M. & Kasagi, N.), pp. 875–887. Hemisphere.Google Scholar
Larchevêque, L., Dupont, P., De Martel, E., Garnier, E. & Debiève, J. F. 2010 Experimental and numerical study of unsteadiness in boundary layer/shock wave interaction. In Turbulence and Interactions, pp. 263–269. Springer.Google Scholar
Lawson, S. J. & Barakos, G. 2011 Review of numerical simulations for high-speed, turbulent cavity flows. Prog. Aerosp. Sci. 47 (3), 186–216.CrossRefGoogle 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.Google Scholar
Mary, I., Sagaut, P. & Deville, M. 2000 An algorithm for unsteady viscous flows at all speeds. Intl J. Numer. Meth. Fluids 34 (5), 371–401.Google Scholar
McKeon, B. J. & Sharma, A. S. 2010 A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658 (1), 336–382.CrossRefGoogle Scholar
Mettot, C., Renac, F. & Sipp, D. 2014a Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: Application to open-loop control. J. Comput. Phys. 269, 234–258.CrossRefGoogle Scholar
Mettot, C., Sipp, D. & Bézard, H. 2014b Quasi-laminar stability and sensitivity analyses for turbulent flows: Prediction of low-frequency unsteadiness and passive control. Phys. Fluids 26 (4), 045112.Google Scholar
Orr, W. M. F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part II: A viscous liquid. In Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, pp. 69–138. Hodges, Figgis, & Co.Google Scholar
Piponniau, S., Dussauge, J. P., Debiève, 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., Larsson, J., Nichols, J. W., Bernardini, M., Morgan, B. E. & Lele, S. K. 2010 Analysis of unsteady effects in shock/boundary layer interactions. Annu. Res. Briefs 153–164.Google Scholar
Plotkin, K. J. 1975 Shock wave oscillation driven by turbulent boundary-layer fluctuations. AIAA J. 13 (8), 1036–1040.Google Scholar
Robinet, J. C. 2007 Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach. J. Fluid Mech. 579 (1), 85–112.Google Scholar
Rodi, W. 1997 Comparison of LES and RANS calculations of the flow around bluff bodies. J. Wind Engng Ind. Aerodyn. 69, 55–75.Google Scholar
Sartor, F., Losfeld, G. & Bur, R. 2012 PIV study on a shock induced separation in a transonic flow. Exp. Fluids 53 (3), 815–827.Google Scholar
Sartor, F., Mettot, C. & Sipp, D. 2015 Stability, receptivity and sensitivity analyses of buffeting transonic flow over a profile. AIAA J. 53 (7), 1980–1993.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.Google Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 30801.Google Scholar
Spalart, P. R. & Allmaras, S. R.1992 A one equation turbulence model for aerodinamic flows. AIAA paper 1992–0439.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
Touber, E. & Sandham, N. D.2008 Oblique shock impinging on a turbulent boundary layer: low-frequency mechanisms. AIAA paper 2008–4170.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 (2), 79–107.Google Scholar
Touber, E. & Sandham, N. D. 2011 Low-order stochastic modelling of low-frequency motions in shock-wave/boundary-layer interactions. J. Fluid Mech. 671, 417–465.Google Scholar
Trefethen, L., Trefethen, A., Reddy, S. & Driscoll, T. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
Wu, M. & Martin, M. P. 2008 Analysis of shock motion in shockwave and turbulent boundary layer interaction using direct numerical simulation data. J. Fluid Mech. 594, 71.Google Scholar