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Experiments in Fluids

, Volume 17, Issue 5, pp 307–314 | Cite as

Stochastic estimation and proper orthogonal decomposition: Complementary techniques for identifying structure

  • J. P. Bonnet
  • D. R. Cole
  • J. Delville
  • M. N. Glauser
  • L. S. Ukeiley
Originals

Abstract

The Proper Orthogonal Decomposition (POD) as introduced by Lumley and the Linear Stochastic Estimation (LSE) as introduced by Adrian are used to identify structure in the axisymmetric jet shear layer and the 2-D mixing layer. In this paper we will briefly discuss the application of each method, then focus on a novel technique which employs the strengths of each. This complementary technique consists of projecting the estimated velocity field obtained from application of LSE onto the POD eigenfunctions to obtain estimated random coefficients. These estimated random coefficients are then used in conjunction with the POD eigenfunctions to reconstruct the estimated random velocity field. A qualitative comparison between the first POD mode representation of the estimated random velocity field and that obtained utilizing the original measured field indicates that the two are remarkably similar, in both flows. In order to quantitatively assess the technique, the root mean square (RMS) velocities are computed from the estimated and original velocity fields and comparisons made. In both flows the RMS velocities captured using the first POD mode of the estimated field are very close to those obtained from the first POD mode of the unestimated original field. These results show that the complementary technique, which combines LSE and POD, allows one to obtain time dependent information from the POD while greatly reducing the amount of instantaneous data required. Hence, it may not be necessary to measure the instantaneous velocity field at all points in spacesimultaneously to obtain the phase of the structures, but only at a few select spatial positions. Moreover, this type of an approach can possibly be used to verify or check low dimensional dynamical systems models for the POD coefficients (for the first POD mode) which are currently being developed for both of these flows.

Keywords

Root Mean Square Proper Orthogonal Decomposition Instantaneous Velocity Random Coefficient Proper Orthogonal Decomposition Mode 
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.

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References

  1. 1.
    Adrian RJ (1975) On the role of conditional averages in turbulence theory. Turbulence in Liquids. Science Press, Princeton, NJ, pp 323–332Google Scholar
  2. 2.
    Adrian RJ; Moin P (1988) Stochastic estimation of organized turbulent structure: Homogeneous Shear Flow. J Fluid Mech 190: 531–559Google Scholar
  3. 3.
    Aubry N; Holmes P; Lumley JL; Stone E (1988) The dynamics of coherent structure in the wall region of a turbulent boundary layer. J Fluid Mech 192: 115–173Google Scholar
  4. 4.
    Berkooz G; Holmes P; Lumley JL (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Annu Rev Fluid Mech. 25:539–575Google Scholar
  5. 5.
    Cole DR; Glauser MN; Guezennec YG (1992) An application of stochastic estimation to the jet mixing layer. Physics of Fluids A, 4(1): 192–194.Google Scholar
  6. 6.
    Cole DR; Ukeiley LS; Glauser MN (1991) A comparison of coherent structure detection techniques in the axisymmetric jet mixing layer. Bulletin of the American Physical Society, 36(10)Google Scholar
  7. 7.
    Delville J; Bellin S; Bonnet JP (1989) Use of the proper orthogonal decomposition in a plane turbulent mixing layer. in Turbulence and Coherent Structures. (O Metais and M Leiseur eds.) Kluwer Academic Press, The Netherlands, pp 75–90Google Scholar
  8. 8.
    Delville J; Vincendeau E; Ukeiley L; Garem JH; Bonnet JP (1993) Etude Expérimentale de la Structure d'une Couche de Mélange Plane Turbulente en Fluide Incompressible. Application de la Dècomposition Orthogonale aux Valeurs Propres. Convention DRET 90-171 Rapport Final.Google Scholar
  9. 9.
    Delville J (1993) Characterization of the organization in shear layers via the proper orthogonal decomposition. in Eddy Structure Identification in Free Turbulent Shear Flows. (JP Bonnet and MN Glauser eds.), Kluwer Academic Press, The Netherlands, pp. 225–238.Google Scholar
  10. 10.
    Delville J; Ukeiley L (1993) Vectorial proper orthogonal decomposition, including spanwise dependency, in a plane fully turbulent mixing Layer. Proceedings, Ninth Symposium on Turbulent Shear Flow, Kyoto, Japan, pp 10.3.1–10.3.6.Google Scholar
  11. 11.
    Ewing D (1994) Private Communication, Department of Mechanical and Aerospace Engineering, University at Buffalo/SUNYGoogle Scholar
  12. 12.
    Glauser Mark N; George William K (1992) Application of Multipoint Measurements for Flow Characterization. Experimental Thermal and Fluid Science, 5: 617–632Google Scholar
  13. 13.
    Glauser MN; Leib SJ; George WK (1987) Coherent Structures in the Axisymmetric Jet Mixing Layer. Turbulent Shear Flows 5, Springer Verlag, pp 134–145Google Scholar
  14. 14.
    Glauser MN; George WK (1987) An orthogonal decomposition of the Axisymmetric Jet Mixing Layer Utilizing Cross-Wire Measurements. Proceedings, Sixth Symposium on Turbulent Shear Flow, Toulouse, France, pp 10.1.4–10.1.6.Google Scholar
  15. 15.
    Glauser MN; Zheng X; George WK (1992) An analysis of the Turbulent Axisymmetric Jet Mixing Layer. In Review J Fluid MechGoogle Scholar
  16. 16.
    Glauser MN; Zheng X; Doering C (1992) A low dimensional description of the axisymmetric jet mixing layer utilizing the proper orthogonal decomposition. In review Physics of Fluids AGoogle Scholar
  17. 17.
    Guezennec YG (1989) Stochastic estimation of coherent structures in turbulent boundary layers. Phys. Fluids A, 1(6): 1054–1060Google Scholar
  18. 18.
    Lumley JL (1967) The structure of inhomogeneous turbulent flows. Atm. Turb. and Radio Wave Prop. (Yaglom and Tatarsky eds.) Nauka, Moscow, pp 166–178Google Scholar
  19. 19.
    Moin P; Adrian RJ; Kim J (1987) Stochastic estimation of organized structures in turbulent channel flow. Proceedings, Sixth Symposium on Turbulent shear flows, Toulouse, France, pp 16.9.1–16.9.8.Google Scholar
  20. 20.
    Moin P; Moser RD (1989) Characteristic-Eddy decomposition of turbulence in a channel. J Fluid Mech. 200: 471–504Google Scholar
  21. 21.
    Moser RD (1988) Statistical analysis of near-wall structures in turbulent channel flow. Proceedings, Symposium on Near Wall Turbulence, Dubrovnik, Yugoslavia, pp 45–62.Google Scholar
  22. 22.
    Tung TC; Adrian RJ (1980) Higher-Order estimates of conditional eddies in isotropic turbulence. Phys Fluids 23:1469–1470Google Scholar
  23. 23.
    Ukeiley L; Glauser M; Wick D (1993) Downstream evolution of proper orthogonal decomposition eigenfunctions in a lobed mixer. AIAA Journal 31(8) 1392–1397Google Scholar
  24. 24.
    Ukeiley L; Cole D; Glauser M; (1993) An examination of the axisymmetric jet mixing layer coherent structure detection techniques. in Eddy Structure Identification in Free Turbulent Shear Flows. (JP Bonnet and MN Glauser eds.), Kluwer Academic Press, pp 325–336Google Scholar

Copyright information

© Springer-Verlag 1994

Authors and Affiliations

  • J. P. Bonnet
    • 1
  • D. R. Cole
    • 2
  • J. Delville
    • 1
  • M. N. Glauser
    • 2
  • L. S. Ukeiley
    • 2
  1. 1.Centre D'Etudes Aerodynamiques et ThermiquesPoitiersFrance
  2. 2.Department of Mechanical and Aeronautical EngineeringClarkson UniversityPotsdamUSA

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