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Liutex core line and POD analysis on hairpin vortex formation in natural flow transition

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Abstract

In this study, the new method of the vortex core line based on Liutex definition, also known as Liutex core line, is applied to support the hypothesis that the vortex ring is not a part of the Λ - vortex and the formation of the ring-like vortex is formed separately from the Λ - vortex. The proper orthogonal decomposition (POD) is also applied to analyze the Kelvin-VHelmholtz (K-H) instability happening in hairpin ring areas of the flow transition on the flat plate to understand the mechanism of the ring-like vortex formation. The new vortex identification method named modified Liutex-Omega method is efficiently used to visualize and observe the shapes of vortex structures in 3-D. The streamwise vortex structure characteristics can be found in POD mode one as the mean flow. The other POD modes are in stremwise and spanwise structures and have the fluctuation motions, which are induced by K-H instability. Moreover, the result shows that POD modes are in pairs and share the same characteristics such as amplitudes, mode shapes, and time evolutions. The vortex core and POD results confirm that the Λ - vortex is not self-deformed to a hairpin vortex, but the hairpin vortex is formed by the K-H instability during the development of Lambda vortex to hairpin vortex in the boundary layer flow transition.

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References

  1. Yan Y., Chen C., Huankun F. et al. DNS study on Λ-vortex and vortex ring formation in the flow transition at Mach number 0.5 [J]. Journal of Turbulence, 2014, 15(1): 1–21.

    Article  Google Scholar 

  2. Liu C., Yan Y., Lu P. Physics of turbulence generation and sustenance in a boundary layer [J]. Computers and Fluids, 2014, 102: 353–384.

    Article  Google Scholar 

  3. Hama F. R. Boundary-layer transition induced by a vibrating ribbon on a flat plate [C]. Proceedings of the 1960 Heat Transfer and Fluid Mechanics Institute, Palo Alto, CA, USA, 1960, 92–105.

    Google Scholar 

  4. Hama F. R., Nutant J. Detailed flow-field observations in the transition process in a thick boundary layer [J]. Proceedings of the 1963 Heat Transfer and Fluid Mechanics Institute, Palo Alto, CA, USA, 1963, 77–93.

    Google Scholar 

  5. Knapp C. F., Roache P. J. A combined visual and hot-wire anemometer investigation of boundary-layer transition [J]. AIAA Journal, 1968, 6(1): 29–36.

    Article  Google Scholar 

  6. Moin P., Leonard A., Kim J. Evolution of curved vortex filament into a vortex ring [J]. Physics of Fluids, 1986, 29(4): 955–963.

    Article  Google Scholar 

  7. Helmholtz H. Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen [J]. Journal für die reine und angewandte Mathematik, 1858, 55: 25–55.

    MathSciNet  Google Scholar 

  8. Robinson S. K. Coherent motion in the turbulent boundary layer [J]. Annual Review of Fluid Mechanics, 1991, 23: 601–639.

    Article  Google Scholar 

  9. Jeong J., Hussain F. On the identification of a vortex [J]. Journal of Fluid Mechanics, 1995, 285: 69–94.

    Article  MathSciNet  MATH  Google Scholar 

  10. Liu C., Gao Y., Tian S. et al. Rortex—A new vortex vector definition and vorticity tensor and vector decompositions [J]. Physics of Fluids, 2018, 30(3): 035103.

    Article  Google Scholar 

  11. Gao Y., Liu C. Rortex and comparison with eigenvalue-based vortex identification criteria [J]. Physics of Fluids, 2018, 30(8): 085107.

    Article  Google Scholar 

  12. Liu C., Gao Y. S., Dong X. R. et al. Third generation of vortex identification methods: Omega and Liutex/Rortex based systems [J]. Journal of Hydrodynamics, 2019, 31(2): 205–223.

    Article  Google Scholar 

  13. Gao Y., Liu, C. Rortex based velocity gradient tensor decomposition [J]. Physics of Fluids, 2019, 31(1): 011704.

    Article  Google Scholar 

  14. Gao Y. S., Liu J. M., Yu Y. et al. A Liutex based definition of vortex rotation axis line [J]. Journal of Hydrodynamics, 2019, 31(3): 445–454.

    Article  Google Scholar 

  15. Wang Y. Q., Gao Y. S., Liu J. M. et al. Explicit formula for the Liutex vector and physical meaning of vorticity based on the Liutex-Shear decomposition [J]. Journal of Hydrodynamics, 2019, 31(3): 464–474.

    Article  Google Scholar 

  16. Liu J. M., Deng Y., Gao Y. S. et al. Mathematical foundation of turbulence generation-symmetric to asymmetric Liutex/Rortex [J]. Journal of Hydrodynamics, 2019, 31(4): 632–636.

    Article  Google Scholar 

  17. Wang Y. Q., Gao Y. S., Xu H. et al. Liutex theoretical system and six core elements of vortex identification [J]. Journal of Hydrodynamics, 2020, 32(2): 197–211.

    Article  Google Scholar 

  18. Xu W. Q., Wang Y. Q., Gao Y. S. et al. Liutex similarity in the turbulent boundary layer [J]. Journal of Hydrodynamics, 2019, 31(6): 1259–1262.

    Article  Google Scholar 

  19. Liu J., Liu C. Modified normalized Rortex/vortex identification method [J]. Physics of Fluids, 2019, 31(6): 061704.

    Article  Google Scholar 

  20. Gao Y. S., Liu J. M., Yu Y. et al. A Liutex based definition and identification of vortex core center lines [J]. Journal of Hydrodynamics, 2019, 31(3): 445–454.

    Article  Google Scholar 

  21. Lumley J. L. The structure of inhomogeneous turbulent flows (Yaglom A. M., Tartarsky V. I. Atmospheric turbulence and radio wave propagation) [M]. 1967, 166–178.

    Google Scholar 

  22. Sirovich L. Turbulence and the dynamics of coherent structures. Part I: Coherent structures [J]. Quarterly of Applied Mathematics, 1987, 45(3): 561–571.

    Article  MathSciNet  MATH  Google Scholar 

  23. Duggleby A., Ball K. S., Paul M. R. et al. Dynamical eigenfunction decomposition of turbulent pipe flow [J]. Journal of Turbulence, 2007, 8: N43.

    Article  MathSciNet  MATH  Google Scholar 

  24. Duggleby A., Ball K. S., Paul M. R. The effect of spanwise wall oscillation on turbulent pipe flow structures resulting in drag reduction [J]. Physics of Fluids, 2007, 19(12): 107–125.

    Article  MATH  Google Scholar 

  25. Hellstrom L., Ganapathisubramani B., Smits A. J. Coherent structures in transitional pipe flow [J]. Physical Review Fluids, 2016, 1(2): 024403.

    Article  Google Scholar 

  26. Dong X. R., Cai X. S., Dong Y. et al. POD analysis on vortical structures in MVG wake by Liutex core line identification [J]. Journal of Hydrodynamics, 2020, 32(3): 497–509.

    Article  Google Scholar 

  27. Gunes H., Rist U. Proper orthogonal decomposition reconstruction of a transitional boundary layer with and without control [J]. Physics of Fluids, 2004, 16(8): 2763.

    Article  MathSciNet  MATH  Google Scholar 

  28. Yang Y., Tian S., Liu C. POD analyses on vortex structure in late-stage transition [R]. AIAA paper 2018-0821, 2018.

    Book  Google Scholar 

  29. Charkrit S., Dong X., Liu C. POD analysis of losing symmetry in late flow transition [R]. AIAA paper 2019-1870, 2019.

    Book  Google Scholar 

  30. Cavalieri A., Schlatter P., Vinuesa R. et al. SPOD and resolvent analysis of near-wall coherent structures in turbulent pipe flows [J]. Journal of Fluid Mechanics, 2020, 900: A11.

    Article  MATH  Google Scholar 

  31. Chen L., Liu C. Numerical study on mechanisms of second sweep and positive spikes in transitional flow on a flat plate [J]. Computers of Fluids, 2011, 40(1): 28–41.

    Article  MathSciNet  MATH  Google Scholar 

  32. Bake S., Meyer D., Rist U. Turbulence mechanism in Klebanoff transition: A quantitative comparison of experiment and direct numerical simulation [J]. Journal of Fluid Mechanics, 2002, 459: 217–243.

    Article  MATH  Google Scholar 

  33. Lee C., Li R. A dominant structure in turbulent production of boundary layer transition [J]. Journal of Turbulence, 2007, 8: N55.

    Article  Google Scholar 

  34. Liu C., Wang Y. Q., Yang Y. et al. New Omega vortex identification method [J]. Science China Physics, Mechanics and Astronomy, 2016, 59(8): 684711.

    Article  Google Scholar 

  35. Dong X., Gao Y., Liu C. New normalized Rortex/vortex identification method [J]. Physics of Fluids, 2019, 31(1): 011701.

    Article  Google Scholar 

  36. Liu J. M., Gao Y. S., Wang Y. Q. et al. Objective Omega vortex identification method [J]. Journal of Hydrodynamics, 2019, 31(3): 455–463.

    Article  Google Scholar 

  37. Hunt J. C. R., Wray A. A., Moin P. Eddies, stream, and convergence zones in turbulent flows [R]. proceedings of the Summer Program. Center for Turbulent Research Report CTR-S88, 1988, 193–208.

    Google Scholar 

  38. Zhang Y. N., Wang X. Y., Zhang Y. N. et al. Comparisons and analyses of vortex identification between Omega method and Q criterion [J]. Journal of Hydrodynamics, 2019, 31(2): 224–230.

    Article  Google Scholar 

  39. Zhang Y. N., Liu K. H., Li J. W. et al. Analysis of the vortices in the inner flow of reversible pump-turbine with the new omega vortex identification method [J]. Journal of Hydrodynamics, 2018, 30(3): 463–469.

    Article  Google Scholar 

  40. Dong X. R., Wang Y. Q., Chen X. P. et al. Determination of epsilon for Omega vortex identification method [J]. Journal of Hydrodynamics, 2018, 30(4): 541–548.

    Article  Google Scholar 

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Acknowledgments

The authors thank the Department of Mathematics of University of Texas at Arlington and Royal Thai Government for the financial support. The authors are grateful to Texas Advanced Computation Center (TACC) for providing CPU hours to this research project. The computation is performed by using Code DNSUTA which was released by Dr. Chaoqun Liu at University of Texas at Arlington in 2009.

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Authors and Affiliations

  1. Department of Mathematics, Maejo University, Chiang Mai, Thailand

    Sita Charkrit

  2. Department of Mathematics, University of Texas at Arlington, Arlington, Texas, 76019, USA

    Sita Charkrit, Pushpa Shrestha & Chaoqun Liu

Authors
  1. Sita Charkrit
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  2. Pushpa Shrestha
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  3. Chaoqun Liu
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Correspondence to Chaoqun Liu.

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Biography: Sita Charkrit, Female, Ph. D.

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Charkrit, S., Shrestha, P. & Liu, C. Liutex core line and POD analysis on hairpin vortex formation in natural flow transition. J Hydrodyn 32, 1109–1121 (2020). https://doi.org/10.1007/s42241-020-0079-0

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  • DOI: https://doi.org/10.1007/s42241-020-0079-0

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