Experiments in Fluids

, Volume 41, Issue 5, pp 763–775 | Cite as

On spectral linear stochastic estimation

  • C. E. Tinney
  • F. Coiffet
  • J. Delville
  • A. M. Hall
  • P. Jordan
  • M. N. Glauser
Research Article


An extension to classical stochastic estimation techniques is presented, following the formulations of Ewing and Citriniti (1999), whereby spectral based estimation coefficients are derived from the cross spectral relationship between unconditional and conditional events. This is essential where accurate modeling using conditional estimation techniques are considered. The necessity for this approach stems from instances where the conditional estimates are generated from unconditional sources that do not share the same grid subset, or possess different spectral behaviors than the conditional events. In order to filter out incoherent noise from coherent sources, the coherence spectra is employed, and the spectral estimation coefficients are only determined when a threshold value is achieved. A demonstration of the technique is performed using surveys of the dynamic pressure field surrounding a Mach 0.30 and 0.60 axisymmetric jet as the unconditional events, to estimate a combination of turbulent velocity and turbulent pressure signatures as the conditional events. The estimation of the turbulent velocity shows the persistence of compact counter-rotating eddies that grow with quasi-periodic spacing as they convect downstream. These events eventually extend radially past the jet axis where the potential core is known to collapse.


Stochastic Estimation Potential Core Conditional Event Laser Doppler Anemometer Marginal Range 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Fluctuating velocity


Estimate of u


Vector field over which the conditional event is investigated


Vector field over which the unconditional event is investigated


Fluctuating pressure






Wave number


Gas density




Time lag


Jet centerline exit velocity


Spatial separation between unconditional and conditional events

R, D

Jet radius and diameter, respectively


Radial coordinate of the jet from the jet axis


Radial coordinate from the center of the jet shear layer


Axial coordinate of the jet from the jet exit


Strouhal number \((St_{{{D}}}={{fD}\over {U_{{{{\rm cl}}}}}})\) based on jet diameter


Reynolds stresses


Time step increment


High-pass frequency


Low-pass frequency



The authors are grateful to Program Manager Dr. John Schmisseur from the Air Force Office of Scientific Research, the Central New York AGEP Program from the National Science Foundation, and NYSTAR, for funding the Syracuse University portions of this study.


  1. Adrian RJ (1977) On the role of conditional averages in turbulence theory. In: Proceedings of the 4th biennal symposium on turbulence in liquidsGoogle Scholar
  2. Adrian RJ (1979) Conditional eddies in isotropic turbulence. Phys Fluids 22(11):2065–2070zbMATHCrossRefGoogle Scholar
  3. Adrian RJ (1996) Stochastic estimation of the structure of turbulent flows. In: Bonnet JP (eds) Eddy structure identification. Springer, Berlin Heidelberg New York, pp 145–195Google Scholar
  4. Arndt REA, Long DF, Glauser MN (1997) The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet. J Fluid Mech 340:1–33CrossRefGoogle Scholar
  5. Bendat JS, Piersol AG (1980) Engineering applications of correlation and spectral analysis. Wiley, New YorkzbMATHGoogle Scholar
  6. Bonnet J-P, Cole DR, Delville J, Glauser MN, Ukeiley LS (1994) Stochastic estimation and proper orthogonal decomposition: complementary techniques for identifying structure. Exp Fluids 17(5):307–314CrossRefGoogle Scholar
  7. Borée J (2003) Extended proper orthogonal decomposition: a tool to analyse correlated events in turbulent flows. Exp Fluids 35:188–192CrossRefGoogle Scholar
  8. Citriniti JH, George WK (1997) The reduction of spatial aliasing by long hot-wire anemometer probes. Exp Fluids 23:217–224CrossRefGoogle Scholar
  9. Citriniti JH, George WK (2000) Reconstruction of the global velocity field in the axisymmetric mixing layer utilizing the proper orthogonal decomposition. J Fluid Mech 418:137–166zbMATHCrossRefGoogle Scholar
  10. Coiffet F, Delville J, Ricaud F, Valiere JC (2004) Nearfield pressure of a subsonic free jet, estimation and separation of hydrodynamic and acoustic components. In: Anderson HI, Krogstad PA (eds) Proceedings of the 10th European turbulence conference, Trondheim, Norway, p 168Google Scholar
  11. Coiffet F, Jordan P, Delville J, Gervais Y, Ricaud F (2006) Coherent structures in subsonic jets: a quasi-irrotational source mechanism? Int J Aeroacoust 5(1):67–89CrossRefGoogle Scholar
  12. Cole DR, Glauser MN (1998) Applications of stochastic estimation in the axisymmetric sudden expansion. Phys Fluids 10(11):2941–2949CrossRefGoogle Scholar
  13. Delville J (1994) La décomposition orthogonale aux valeurs propres et l’analyse de l’organisation tridimensionelle des écoulements turbulents cisaillés libres. Ph.D. Thesis, l’Université de Poitiers, Potiers, FranceGoogle Scholar
  14. Ewing D, Citriniti J (1999) Examination of a LSE/POD complementary technique using single and multi-time information in the axisymmetric shear layer. In: Sorensen, Hopfinger, Aubry (eds) Proceedings of the IUTAM Symposium on simulation and identification of organized structures in flows, Kluwer, Lyngby, Denmark, 25–29 May 1997, pp 375–384Google Scholar
  15. Gamard S, George WK, Jung D, Woodward S (2002) Application of a “slice” proper orthogonal decomposition to the far field of an axisymmetric jet. Phys Fluids 14:6CrossRefGoogle Scholar
  16. Glauser MN, Young MJ, Higuchi H, Tinney CE, Carlson H (2004) POD based experimental flow control on a NACA-4412 airfoil (Invited). In: 41th AIAA aerospace sciences meeting and exhibit, Reno, NV p 0575Google Scholar
  17. Hall AM, Glauser MN, Tinney CE (2005) An experimental investigation of the pressure–velocity cross-correlation in an axisymmetric jet. ASME FEDSM2005-77338, Houston, TX, June 19–23Google Scholar
  18. Hussain AKMF, Clark AR (1981) On the coherent structure of the axisymetric mixing layer: a flow visualization study. J Fluid Mech 104:263–294CrossRefGoogle Scholar
  19. Jordan P, Tinney CE, Delville J, Coiffet F, Glauser M, Hall AM (2005) Low-dimensional signatures of the sound production mechanicsms in subsonic jets: towards their identification and control. In: 35th AIAA fluid dynamics conference and exhibit, June 6–9, Toronto, Canada, Paper pp 2005–4647Google Scholar
  20. Jung D, Gamard S, George WK (2004) 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 514:173–204zbMATHCrossRefGoogle Scholar
  21. Ko NWM, Davies POAL (1971) The near field within the potential cone of subsonic cold jets. J Fluid Mech 50:49–78CrossRefGoogle Scholar
  22. Lighthill MJ (1958) Introduction to Fourier analysis and generalised functions. Cambridge monographs on mechanics. Cambridge University Press, CambridgeGoogle Scholar
  23. Murray N, Ukeiley LS (2003) Estimation of the flow field from surface pressure measurements in an open cavity. AIAA J 41(5):969–972CrossRefGoogle Scholar
  24. Naguib AM, Wark CE, Juckenhöfel O (2001) Stochastic estimation and flow sources associated with surface pressure events in a turbulent boundary layer. Phys Fluids 13(9):2611–2626CrossRefGoogle Scholar
  25. Otnes RK, Enochson L (1978) Applied time series analysis, vol 1. Wiley, New YorkGoogle Scholar
  26. Picard C, Delville J (2000) Pressure velocity coupling in a subsonic round jet. Int J Heat Fluid Flow 21:359–364CrossRefGoogle Scholar
  27. Ricaud F (2003) étude de l’identification des sources acoustiques è partir du couplage de la pression en champ proche et de l’organisation instantanée de la zone de mélange de jet. Ph.D. Thesis, l’Université de Poitiers, Potiers, FranceGoogle Scholar
  28. Tennekes H, Lumley JL (1972) A first course in turbulence, vol 1. MIT Press, MassachusettsGoogle Scholar
  29. Tinney CE (2005) Low-Dimensional techniques for sound source identification in high speed jets. Ph.D. Thesis, Syracuse University, SyracuseGoogle Scholar
  30. Tinney CE, Glauser MN, Ukeiley LS (2005) The evolution of the most energetic modes in a high subsonic Mach number turbulent jet. In: 43rd AIAA aerospace sciences meeting and exhibit, Reno, 2005-0417Google Scholar
  31. Tung TC, Adrian RJ (1980) Higher-order estimates of conditional eddies in isotropic turbulence. Phys Fluids 23:1469CrossRefGoogle Scholar
  32. Ukeiley L, Mann R, Tinney CE, Glauser MN (2004) Spatial correlations in a transonic jet. In: 34th AIAA fluid dynamics conference, Portland, Oregon, pp 2004–2654Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • C. E. Tinney
    • 1
  • F. Coiffet
    • 2
  • J. Delville
    • 1
  • A. M. Hall
    • 3
  • P. Jordan
    • 1
  • M. N. Glauser
    • 3
  1. 1.Laboratoire d’Etudes Aérodynamiques, UMR CNRS 6609Université de PoitiersPoitiers CedexFrance
  2. 2.Department of Mechanical EngineeringFlorida State UniversityTallahasseeUSA
  3. 3.Department of Mechanical & Aerospace EngineeringSyracuse UniversitySyracuseUSA

Personalised recommendations