Coupling temporal and spatial gradient information in high-density unstructured Lagrangian measurements

Abstract

Particle tracking velocimetry (PTV) produces high-quality temporal information that is often neglected when computing spatial gradients. A method is presented here to utilize this temporal information in order to improve the estimation of spatial gradients for spatially unstructured Lagrangian data sets. Starting with an initial guess, this method penalizes any gradient estimate where the substantial derivative of vorticity along a pathline is not equal to the local vortex stretching/tilting. Furthermore, given an initial guess, this method can proceed on an individual pathline without any further reference to neighbouring pathlines. The equivalence of the substantial derivative and vortex stretching/tilting is based on the vorticity transport equation, where viscous diffusion is neglected. By minimizing the residual of the vorticity-transport equation, the proposed method is first tested to reduce error and noise on a synthetic Taylor–Green vortex field dissipating in time. Furthermore, when the proposed method is applied to high-density experimental data collected with ‘Shake-the-Box’ PTV, noise within the spatial gradients is significantly reduced. In the particular test case investigated here of an accelerating circular plate captured during a single run, the method acts to delineate the shear layer and vortex core, as well as resolve the Kelvin-Helmholtz instabilities, which were previously unidentifiable without the use of ensemble averaging. The proposed method shows promise for improving PTV measurements that require robust spatial gradients while retaining the unstructured Lagrangian perspective.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11

References

  1. Brachet ME, Meiron DI, Orszag SA, Nickel BG, Morf RH, Frisch U (1983) Small-scale structure of the TaylorGreen vortex. J Fluid Mech 130:411–452

    Article  MATH  Google Scholar 

  2. Brunton SL, Rowley CW (2010) Fast computation of FTLE fields for unsteady flows: a comparison of methods. Chaos 20:1–12

    Article  Google Scholar 

  3. Bunin GA, François G, Bonvin D (2013) From discrete measurements to bounded gradient estimates: a look at some regularizing structures. Ind Eng Chem Res 52(35):12500–12513

    Article  Google Scholar 

  4. Canuto CG, Hussaini MY, Quarteroni AM, Zang TA (2007) Spectral methods: evolution to complex geometries and applications to fluid dynamics (scientific computation). Springer, New York

    Google Scholar 

  5. Correa CD, Hero R, Ma KL (2011) A comparison of gradient estimation methods for volume rendering on unstructured meshes. Vis Comput Graph IEEE Trans 17(3):305–319

    Article  Google Scholar 

  6. Fernando JN, Rival DE (2016) Reynolds-number scaling of vortex pinch-off on low-aspect-ratio propulsors. J Fluid Mech 799. doi:10.1017/jfm.2016.396

  7. Gesemann S, Huhn F, Schanz D, Schröder A (2016) From noisy particle tracks to velocity, acceleration and pressure fields using b-splines and penalties. In: 18th international symposium on applications of laser and imaging techniques to fluid mechanics, Lisbon, Portugal

  8. Jeon YJ, Chatellier L, David L (2014) Fluid trajectory evaluation based on an ensemble-averaged cross-correlation in time-resolved piv. Exp Fluids 55:1766

    Article  Google Scholar 

  9. Kaehler CJ, Scharnowski S, Cierpka C (2012) On the uncertainty of digital PIV and PTV near walls. Exp Fluids 52:1641–1656

    Article  Google Scholar 

  10. Kim J, Moin P (1985) Application of a fractional-step method to incompressible Navier-Stokes equations. J Comput Phys 59(2):308–323

    MathSciNet  Article  MATH  Google Scholar 

  11. Meyer HTH, Eriksson M, Maggio RC (2001) Gradient estimation from irregularly spaced data sets. Math Geol 33(3):693–717

    Article  MATH  Google Scholar 

  12. Neeteson NJ, Rival DE (2015) Pressure-field extraction on unstructured flow data using a voronoi tessellation-based networking alogrithm: a proof-of-principle study. Exp Fluids 56(44):44

    Article  Google Scholar 

  13. Neeteson NJ, Bhattacharya S, Rival DE, Michaelis D, Schanz D, Schröder A (2016) Pressure-field extraction from lagrangian flow measurements: first experiences with 4d-ptv data. Exp Fluids 57:102

    Article  Google Scholar 

  14. Orszag SA (1969) Numerical methods for the simulation of turbulence. Phys Fluids 12(12):2–250

    Article  MATH  Google Scholar 

  15. Orszag SA (1971) On the elimination of aliasing in finite-difference schemes by filtering high-wavenumber components. J Atmos Sci 28(6):1074

    Article  Google Scholar 

  16. Orszag SA (1972) Comparison of pseudospectral and spectral approximation. Stud Appl Math 51(3):253–259

    MathSciNet  Article  MATH  Google Scholar 

  17. Raben SG, Ross SD, Vlachos PP (2014) Computation of finite-time Lyapunov exponents from time resolved particle image velocimetry data. Exp Fluids 55(1638):1–14

    Google Scholar 

  18. Raffel M, Willert CE, Wereley ST, Kompenhans J (2007) Particle image velocimetry: a practical guide, 2nd edn. Springer, Berlin

    Google Scholar 

  19. Rockwood MP, Taira K, Green MA (2016) Detecting vortex formation and shedding in cylinder wakes using lagrangian coherent structures. AIAA J 55:15–23

    Article  Google Scholar 

  20. Scarano F, Moore P (2012) An advection-based model to increase the temporal resolution of piv time series. Exp Fluids 52:919–933

    Article  Google Scholar 

  21. Schanz D, Gesemann S, Schröder A (2016) Shake-the-box: Lagrangian particle tracking at high particle image densities. Exp Fluids 57:70

    Article  Google Scholar 

  22. Schneiders J, Scarano F (2016) Dense velocity reconstruction from tomographic ptv with material derivatives. Exp Fluids 57:139

    Article  Google Scholar 

  23. Schneiders J, Singh P, Scarano F (2016) Instantaneous flow reconstruction from particle trajectories with vortex-in-cell. In: 18th international symposium on the application of laser and imaging techniques to fluid mechanics, Lisbon, Portugal

  24. Shadden SC, Dabiri JO, Marsden JE (2006) Lagrangian analysis of fluid transport in empirical vortex ring flows. Phys Fluids 18(047105):1–11

    MathSciNet  MATH  Google Scholar 

  25. Taylor GI, Green AE (1937) Mechanism of the production of small eddies from large ones. Proc R Soc Lond Ser A Math Phys Sci 158(895):499–521

    Article  MATH  Google Scholar 

  26. Wolf M, Holzner M, Krug D, Lüthi B, Kinzelbach W, Tsinober A (2013) Effects of mean shear on the local turbulent entrainment process. J Fluid Mech 731:95–116

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jaime G. Wong.

Electronic supplementary material

Below is the link to the electronic supplementary material.

348_2017_2427_MOESM1_ESM.mp4

Supplementary material 1 (mp4 8874 KB)

348_2017_2427_MOESM2_ESM.mp4

Supplementary material 2 (mp4 7690 KB)

348_2017_2427_MOESM3_ESM.mp4

Supplementary material 3 (mp4 7730 KB)

Supplementary material 1 (mp4 8874 KB)

Supplementary material 2 (mp4 7690 KB)

Supplementary material 3 (mp4 7730 KB)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wong, J.G., Rosi, G.A., Rouhi, A. et al. Coupling temporal and spatial gradient information in high-density unstructured Lagrangian measurements. Exp Fluids 58, 140 (2017). https://doi.org/10.1007/s00348-017-2427-6

Download citation