Reduced-order model of a reacting, turbulent supersonic jet based on proper orthogonal decomposition

  • Antoni AlomarEmail author
  • Aurélie Nicole
  • Denis Sipp
  • Valérie Rialland
  • François Vuillot
Original Article


The present article deals with the spatiotemporal reduction of a reacting, supersonic, turbulent jet. A flowfield dataset is first obtained from a LES simulation including chemical reactions. The spatial reduction is accomplished by performing successively a Fourier transform in the azimuthal direction and a proper orthogonal decomposition (POD), while the temporal reduction is obtained through a selection of Fourier modes in the leading temporal POD modes (chronos). A prior low-pass temporal filter has been used to eliminate a significant portion of high-frequency content, the resulting reduced-order model (ROM) focusing only on the low-frequency, large-scale structures of the jet. The leading axisymmetric (\(m=0\)) POD mode describes a very low-frequency pulsation of the shock cells. The second and third axisymmetric POD modes are paired and describe a wave packet amplified along the potential core and with a rapid decay downstream. The two leading helical (\(m=1\)) POD modes, which are complex modes, appear to be mixed and together describe convecting wave packets of a longer wavelength and a lower frequency. They can be associated to the instability of the shear layer downstream of the potential core. Strong global cross-correlations are observed between the mass fractions and the axial velocity and temperature fields, which lead to similar energy decay rates and POD modes when they are included in the POD. The temporal reduction of the leading sets of chronos via an energy-based selection of Fourier modes has been shown to complement efficiently the spatial reduction, providing a complete spatiotemporal ROM of the flowfield.


Reduced-order model Proper orthogonal decomposition Turbulent jet Reacting flows 



This work has been funded by the ONERA internal project PRF SIMBA. The authors would like to thank Aurélien Guy and Sidonie Lefebvre for a number of useful comments and suggestions.


  1. 1.
    Caraballo, E., Samimy, M., Scott, J., Narayanan, S., DeBonis, J.: Application of proper orthogonal decomposition to a supersonic axisymmetric jet. AIAA J. 41(5), 866–877 (2003)CrossRefGoogle Scholar
  2. 2.
    Zhou, X., Hitt, D.L.: Proper orthogonal decomposition analysis of coherent structures in simulated reacting buoyant jets. AIAA J. 49(5), 945–952 (2011)CrossRefGoogle Scholar
  3. 3.
    Nobach, H., Tropea, C., Cordier, L., Bonnet, J.-P., Delville, J., Lewalle, J., Farge, M., Schneider, K., Adrien, R.: Handbook of experimental fluid mechanics. Chapter 22: Review of Some Fundamentals of Data Processing. Springer (2007)Google Scholar
  4. 4.
    Gudmundsson, K., Colonius, T.: Instability wave models for the near-field fluctuations of turbulent jets. J. Fluid Mech. 689, 97–128 (2011)zbMATHCrossRefGoogle Scholar
  5. 5.
    Davoust, S., Jacquin, L., Leclaire, B.: Dynamics of m=0 and m=1 modes of streamwise vortices in a turbulent axisymmetric mixing layer. J. Fluid Mech. 709, 408–444 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Rowley, C.W., Colonius, T., Murray, R.M.: Model reduction for compressible flows using pod and galerkin projection. Phys. D Nonlinear Phenom. 189, 115–129 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Rempfer, D.: On low-dimensional galerkin models for fluid flow. Theor. Comput. Fluid Dyn. 14, 75–88 (2000)zbMATHCrossRefGoogle Scholar
  8. 8.
    Couplet, M., Sagaut, P., Basdevant, C.: Intermodal energy transfers in a proper orthogonal decomposition-galerkin representation of a turbulent separated flow. J. Fluid Mech. 491, 275–284 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    McKeon, B.J., Sharma, A.S.: A critical-layer framework for turbulent pipe flow. J. Fluid Mech. 658, 336–382 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gomez, F., Blackburn, H.M., Rudman, M., Sharma, A.S., McKeon, B.J.: A reduced-order model of three-dimensional unsteady flow in a cavity based on the resolvent operator. J. Fluid Mech. 798, 408–444 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Beneddine, S., Yegavian, R., Sipp, D., Leclaire, B.: Unsteady flow dynamics reconstruction from mean flow and point sensors: an experimental study. J. Fluid Mech. 824, 174–201 (2017)CrossRefGoogle Scholar
  12. 12.
    Beneddine, S., Sipp, D., Arnault, A., Dandois, J., Lesshafft, L.: Conditions for validity of mean flow stability analysis. J. Fluid Mech. 798, 485–504 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rowley, C.W., Mezic, I., Bagheri, S., Schlatter, P.: Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Chen, K.K., Tu, J.H., Rowley, C.K.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci. 22(6), 887–915 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Cammilleri, A., Gueniat, F., Carlier, J., Pastur, L., Memin, E., Lusseyran, F., Artana, G.: POD-spectral decomposition for fluid flow analysis and model reduction. Theor. Comput. Fluid Dyn. 27(6), 787–815 (2013)CrossRefGoogle Scholar
  17. 17.
    Perret, L., Collin, E., Delville, J.: Polynomial identification of pod based low-order dynamical system. J. Turbul. 7, 1–15 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Berkooz, G., Holmes, P., Lumley, J.L.: The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Duwig, C., Iudiciani, P.: Extended proper orthogonal decomposition for analysis of unsteady flames. Flow Turbul. Combust. 84, 25–47 (2010)zbMATHCrossRefGoogle Scholar
  20. 20.
    Rialland, V., Guy, A., Gueyffier, D., Perez, P., Roblin, A., Smithson, T.: Numerical simulation of ionized rocket plumes. J. Phys. Conf. Ser. 676(–), 1–12 (2016)Google Scholar
  21. 21.
    Langenais, A., Vuillot, F., Troyes, J., Bailly, C.: Numerical investigation of the noise generated by a rocket engine at lift-off conditions using a two-way coupled cfd-caa method. In: 23th AIAA/CEAS Aeroacoustics Conference, Denver, Colorado, June 2017. AIAA, pp. 25–47 (2017)Google Scholar
  22. 22.
    Gueyffier, D., Fromentin-Denoziere, B., Simon, J., Merlen, A., Giovangigli, V.: Numerical simulation of ionized rocket plumes. J. Thermophys. Heat Transf. 28(2), 218–225 (2014)CrossRefGoogle Scholar
  23. 23.
    Guy, A., Fromentin-Denoziere, B., Phan, H-K., Cheraly, A., Gueyffier, D., Rialland, V., Erades, C., Elias, P.Q., Labaune, Jarrige, J., Ristori, A., Brossard, C., Rommeluere, S.: Ionized solid propellant rocket exhaust plume: miles simulation and comparison to experiment. In: 7th European Conference for Aeronautics and Space Sciences (EUCA SS), Milan, Italy, July 2017. EUCASS Association, pp. 1–19 (2017)Google Scholar
  24. 24.
    Refloch, A., Courbet, B., Murrone, A., Villedieu, P., Laurent, C., Gilbank, P., Troyes, J., Tessé, L., Chaineray, G., Dargaud, J.B., Quémerais, E., Vuillot, F.: Cedre software. AerospaceLab 1(2), 1–10 (2011)Google Scholar
  25. 25.
    LeTouze, C., Murrone, A., Guillard, H.: Multislope muscl method for general unstructured meshes. J. Comput. Phys. 284, 389–418 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Toro, E.F., Spruce, M., Speares, W.: Restoration of the contact surface in the HLL-Riemann solver. Shock Waves 4, 25–34 (1994)zbMATHCrossRefGoogle Scholar
  27. 27.
    Gao, F., O’Brien, E.E.: A large-eddy simulation scheme for turbulent reacting flows. Phys. Fluids A Fluid Dyn. 5(6), 1282–1284 (1993)zbMATHCrossRefGoogle Scholar
  28. 28.
    Jimenez, J., Linan, A., Rogers, M.M., Higuera, F.J.: A priori testing of subgrid models for chemically reacting non-premixed turbulent shear flows. J. Fluid Mech. 349, 149–171 (1997)zbMATHCrossRefGoogle Scholar
  29. 29.
    DesJardin, P.E., Frankel, S.H.: Large eddy simulation of a non-premixed reacting jet: application and assessment of subgrid-scale combustion models. Phys. Fluids 10(9), 2298–2314 (1998)CrossRefGoogle Scholar
  30. 30.
    Celik, I., Cehreli, Z., Yavuz, I.: Index of resolution quality for large eddy simulations. ASME J. Fluids Eng. 127, 949–958 (2005)CrossRefGoogle Scholar
  31. 31.
    Pack, D.C.: A note on prandtl’s formula for the wave-length of a supersonic gas jet. Q. J. Mech. Appl. Math. 3(2), 173–181 (1950)zbMATHCrossRefGoogle Scholar
  32. 32.
    De Cacqueray, N., Bogey, C., Bailly, C.: Investigation of a high-mach-number over-expanded jet using large-eddy simulation. AIAA J. 49(10), 2171–2182 (2011)CrossRefGoogle Scholar
  33. 33.
    Brès, G.A., Jordan, P., Jaunet, V., Le Rallic, M., Cavalieri, A.V.G., Towne, A., Lele, S.K., Colonius, T., Schmid, O.T.: Importance of the nozzle-exit boundary-layer state in subsonic turbulent jets. J. Fluid Mech. 851, 83–124 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Lorteau, M., Cléro, F., Vuillot, F.: Analysis of noise radiation mechanism in hot subsonic jet from a validated large eddy simulation solution. Phys. Fluids 27, 075108 (2015)CrossRefGoogle Scholar
  35. 35.
    Langenais, A., Vuillot, F., Troyes, J., Bailly, C.: Accurate simulation of the noise generated by a hot supersonic jet including turbulence tripping and nonlinear acoustic propagation. Phys. Fluids 31, 016105 (2018)CrossRefGoogle Scholar
  36. 36.
    Lumley, J.L.: Atmospheric Turbulence and Radio Wave Propagation: The Structure of Inhomogeneous Turbulent Flows. Nauka, Moscow (1967)Google Scholar
  37. 37.
    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)CrossRefGoogle Scholar
  38. 38.
    Poje, A., Lumley, J.L.: Low-dimensional models for flows with density fluctuations. Phys. Fluids 9(7), 2023–2031 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    Deane, A.E., Kevrekidis, I.G., Karniadakis, G.E., Orszag, S.A.: Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders. Phys. Fluids A Fluid Dyn. 3(10), 2337–2354 (1991)zbMATHCrossRefGoogle Scholar
  40. 40.
    Noack, B., Afanasiev, K., Morzyński, M., Tadmor, G., Thiele, F.: A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Crighton, D.G., Gaster, M.: Stability of slowly diverging jet flow. J. Fluid Mech. 77(2), 397–413 (1976)zbMATHCrossRefGoogle Scholar
  42. 42.
    Michalke, A., Fuchs, H.V.: On turbulence and noise of an axisymmetric shear flow. J. Fluid Mech. 70, 179–205 (1975)zbMATHCrossRefGoogle Scholar
  43. 43.
    Luo, K.H., Sandham, N.D.: Instability of vortical and acoustic modes in supersonic round jets. Phys. Fluids 9(4), 1003–1013 (1997)CrossRefGoogle Scholar
  44. 44.
    Jordan, P., Colonius, T.: Wave packets and turbulent jet noise. Annu. Rev. Fluid Mech. 45, 173–195 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Schmidt, O.T., Towne, A., Rigas, G., Colonius, T., Brès, G.A.: Spectral analysis of jet turbulence. J. Fluid Mech. 855, 953–982 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    Towne, A., Schmid, 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)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, Berlin (2003)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.DAAA, ONERAUniversité Paris-SaclayMeudonFrance
  2. 2.DMPE, ONERAUniversité Paris-SaclayPalaiseauFrance
  3. 3.DOTA, ONERAUniversité Paris-SaclayPalaiseauFrance

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