Advertisement

On identification of self-similar characteristics using the Tensor Train decomposition method with application to channel turbulence flow

  • Thomas von LarcherEmail author
  • Rupert Klein
Original Article
First Online:

Abstract

A study on the application of the Tensor Train decomposition method to 3D direct numerical simulation data of channel turbulence flow is presented. The approach is validated with respect to compression rate and storage requirement. In tests with synthetic data, it is found that grid-aligned self-similar patterns are well captured and also the application to non-grid-aligned self-similarity yields satisfying results. It is observed that the shape of the input Tensor significantly affects the compression rate. Applied to data of channel turbulent flow, the Tensor Train format allows for surprisingly high compression rates whilst ensuring low relative errors. However, the results indicate that representation of highly irregular flows at low ranks cannot be expected.

Keywords

Self-similarity Turbulent flows Tensor Train format 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This research has been funded by Deutsche Forschungsgemeinschaft (DFG) through grant CRC 1114 ‘Scaling Cascades in Complex Systems’, Project B04 ‘Multiscale Tensor decomposition methods for partial differential equations’. The authors thank Prof. Illia Horenko (CRC 1114 Mercator Fellow) as well as Prof. Reinhold Schneider and Prof. Harry Yserentant for rich discussions and for steady support. Thomas von Larcher thanks Sebastian Wolf and Benjamin Huber (both at TU Berlin, Germany) very much for developing the Tensor library Xerus which has been used for data analysis, as well as for their round-the-clock support in the project. The data were generated and processed using resources of the North-German Supercomputing Alliance (HLRN), Germany, and of the Department of Mathematics and Computer Science, Freie Universität Berlin, Germany. The authors thank Alexander Kuhn and Christian Hege (both at Zuse Institute Berlin, Germany) for steady support in data processing and data visualization.

References

  1. 1.
    Adams, N.: A stochastic extension of the approximate deconvolution model. Phys. Fluids 23(055103), 1–9 (2011)Google Scholar
  2. 2.
    Adrian, R., Meinhart, C., Tomkins, C.: Vortex organization in the outer region of the turbulent boundary layer. J. Fluid Mech. 422, 1–54 (2000).  https://doi.org/10.1017/S0022112000001580 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bakewell, H., Lumley, J.: Viscous sublayer and adjacent wall region in turbulent pipe flow. Phys. Fluids 10(9), 1880–1889 (1967).  https://doi.org/10.1063/1.1762382 CrossRefGoogle Scholar
  4. 4.
    Ballani, J., Grasedyck, L.: Tree adaptive approximation in the hierarchical tensor format. SIAM J. Sci. Comput. 36, A1415–A1431 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bürger, K., et al.: Vortices within vortices: hierarchical nature of vortex tubes in turbulence. arXiv:1210.3325 [physics.flu-dyn] (2013)
  6. 6.
    Cattani, C.: Wavelet analysis of self similar functions. J. Dyn. Syst. Geom. Theor. 9(1), 75–97 (2011).  https://doi.org/10.1080/1726037X.2011.10698594 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Corino, E., Brodkey, R.: A visual investigation of the wall region in turbulent flow. J. Fluid Mech. 37, 1–30 (1969).  https://doi.org/10.1017/S0022112069000395 CrossRefGoogle Scholar
  8. 8.
    De Stefano, G., Nejadmalayeri, A., Vasilyev, O.: Wall-resolved wavelet-based adaptive large-eddy simulation of bluff-body flows with variable thresholding. J. Fluid Mech. 788, 303–336 (2016).  https://doi.org/10.1017/jfm.2015.708 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    De Stefano, G., Vasilyev, O.: A fully adaptive wavelet-based approach to homogeneous turbulence simulation. J. Fluid Mech. 695, 149–172 (2012).  https://doi.org/10.1017/jfm.2012.6 MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dolgov, S., Khoromskij, B., Savostyanov, D.: Superfast Fourier transform using QTT approximation. J. Fourier Anal. Appl. 18, 915 (2012).  https://doi.org/10.1007/s00041-012-9227-4 MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Farge, M., Rabreau, G.: Transformée en ondelettes pour détecter et analyser les structures cohérentes dans les écoulements turbulents bidimensionnels. C. R. Acad. Sci. Paris 307, 1479–1486 (1988)Google Scholar
  12. 12.
    Farge, M., Schneider, K., Kevlahan, N.: Non-Gaussianity and coherent vortex simulation for two-dimensional turbulence using an adaptive orthogonal wavelet basis. Phys. Fluids 11, 2187 (1999).  https://doi.org/10.1063/1.870080 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Frisch, U., Sulem, P.L., Nelkin, M.: A simple dynamical model of intermittent fully developed turbulence. J. Fluid Mech. 87(4), 719–736 (1978).  https://doi.org/10.1017/S0022112078001846 CrossRefzbMATHGoogle Scholar
  14. 14.
    Fröhlich, J., Schneider, K.: Numerical simulation of decaying turbulence in an adaptive wavelet basis. Appl. Comput. Harmonic Anal. 3, 393–397 (1996).  https://doi.org/10.1006/acha.1996.0033 CrossRefzbMATHGoogle Scholar
  15. 15.
    Fröhlich, J., Uhlmann, M.: Orthonormal polynomial wavelets on the interval and applications to the analysis of turbulent flow fields. SIAM J. Appl. Math. 63(5), 1789–1830 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Germano, M., Piomelli, U., Moin, P., Cabot, W.H.: A dynamic subgrid scale eddy viscosity model. Phys. Fluids A 48, 273–337 (1991)zbMATHGoogle Scholar
  17. 17.
    Goldstein, D., Vasilyev, O.: Stochastic coherent adaptive large eddy simulation method. Phys. Fluids 16, 2497 (2004).  https://doi.org/10.1063/1.1736671 MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Grasedyck, L., Kressner, D., Tobler, C.: A literature survey of low-rank tensor approximation techniques. GAMM Mitteilungen 36, 53–78 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hackbusch, W.: Tensor Spaces and Numerical Tensor Calculus. Springer, Heidelberg (2012)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hackbusch, W., Kühn, S.: A new scheme for the tensor representation. J. Fourier Anal. Appl. 15, 706–722 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hackbusch, W., Schneider, R.: Tensor spaces and hierarchical tensor representations. In: Dahlke, S., Dahmen, W., Griebel, M., Hackbusch, W., Ritter, K., Schneider, R., Schwab, C., Yserentant, H. (eds.) Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol. 102, pp. 237–361. Springer, New York (2014)Google Scholar
  22. 22.
    Hitchcock, F.: The expression of a tensor or a polyadic as a sum of products. J. Math. Phys. 6, 164–189 (1927)CrossRefzbMATHGoogle Scholar
  23. 23.
    Horenko, I.: On clustering of non-stationary meteorological time series. Dyn. Atmos. Ocean 49, 164–187 (2010)CrossRefGoogle Scholar
  24. 24.
    Huber, B., Wolf, S.: Xerus—a general purpose tensor library. https://libxerus.org/ (2014–2015)
  25. 25.
    Hughes, T.: Multiscale phenomena: Greens functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods. Comput. Methods Appl. Mech. Eng. 127, 387–401 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Hunt, J.C.R.: Vorticity and vortex dynamics in complex turbulent flows. In: Canadian Society for Mechanical Engineering, Transactions (ISSN 0315-8977), vol. 11, no. 1, 1987, pp. 21–35 (1987)Google Scholar
  27. 27.
    Hunt, J.C.R., Wray, A.A., Moin, P.: Eddies, streams, and convergence zones in turbulent flows. In: Studying Turbulence Using Numerical Simulation Databases, vol. 2 (1988)Google Scholar
  28. 28.
    Jimenez, J., Moin, P.: The minimal flow unit in near-wall turbulence. J. Fluid Mech. 225, 213–240 (1991)CrossRefzbMATHGoogle Scholar
  29. 29.
    John, V., Kindl, A.: A variational multiscale method for turbulent flow simulation with adaptive large scale space. J. Comput. Phys. 229, 301–312 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kevlahan, N., Alam, J., Vasilyev, O.: Scaling of space-time modes with Reynolds number in two-dimensional turbulence. J. Fluid Mech. 570, 217–226 (2007).  https://doi.org/10.1017/S0022112006003168 MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Khoromskij, B.: O(\(d \log {N}\))-quantics approximation of \({N}-d\) tensors in high-dimensional numerical modeling. Constr. Approx. 34, 257 (2011).  https://doi.org/10.1007/s00365-011-9131-1 MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Khoromskij, B.: Tensor numerical methods for multidimensional PDEs: theoretical analysis and initial applications. ESAIM Proc. 48, 1–28 (2015).  https://doi.org/10.1051/proc/201448001 MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Khoromskij, B., Miao, S.: Superfast wavelet transform using QTT approximation I: Haar wavelets. Comput. Methods Appl. Math. 14, 537–553 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kim, J., Moin, P., Moser, R.: Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166 (1987)CrossRefzbMATHGoogle Scholar
  35. 35.
    Kline, S., Robinson, S.: Quasi-coherent structures in the turbulent boundary layer. i—status report on a community-wide summary of the data. In: Kline, S., Afgan, N. (eds.) Near-Wall Turbulence, pp. 200–217. Routledge (1990)Google Scholar
  36. 36.
    Kolda, T., Bader, B.: Tensor decompositions and applications. SIAM Rev. 51, 455–500 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kolmogorov, A.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. C. R. Acad. Sci. U.S.S.R. 30, 301 (1941)Google Scholar
  38. 38.
    Kolmogorov, A.: A refinement of previous hypothesis concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 62, 82 (1962)CrossRefzbMATHGoogle Scholar
  39. 39.
    Kutz, J.: Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4 (2017)CrossRefzbMATHGoogle Scholar
  40. 40.
    von Larcher, T., Beck, A., Klein, R., Horenko, I., Metzner, P., Waidmann, M., Igdalov, D., Gassner, G., Munz, C.D.: Towards a framework for the stochastic modelling of subgrid scale fluxes for large eddy simulation. Meteorol. Z. 24(3), 313–342 (2015).  https://doi.org/10.1127/metz/2015/0581 CrossRefzbMATHGoogle Scholar
  41. 41.
    Lesieur, M.: Turbulence in Fluids: Stochastic and Numerical Modelling. Nijhoff, The Hague (1987)CrossRefzbMATHGoogle Scholar
  42. 42.
    Ling, J., Kurzawski, A., Templeton, J.: Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Liu, S., Meneveau, C., Katz, J.: On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet. J. Fluid Mech. 275, 83–91 (1994)CrossRefGoogle Scholar
  44. 44.
    Mallat, S.: A Wavelet Tour of Signal Processing, 3rd edn. Academic Press, Boston (2009)zbMATHGoogle Scholar
  45. 45.
    Metzner, P., Putzig, L., Horenko, I.: Analysis of persistent non-stationary time series and applications. CAMCoS 7, 175–229 (2012)CrossRefzbMATHGoogle Scholar
  46. 46.
    Moser, R., Kim, J., Mansour, N.: Direct numerical simulation of turbulent channel flow up to R\(\text{ e }_\tau \)=590. Phys. Fluids 11(4), 943–945 (1999)CrossRefzbMATHGoogle Scholar
  47. 47.
    Nejadmalayeri, A., Vezolainen, A., DeStefano, G., Vasilyev, O.: Fully adaptive turbulence simulations based on lagrangian spatio-temporally varying wavelet thresholding. J. Fluid Mech. 749, 794–817 (2014).  https://doi.org/10.1017/jfm.2014.241 MathSciNetCrossRefGoogle Scholar
  48. 48.
    Nejadmalayeri, A., Vezolainen, A., Vasilyev, O.: Reynolds number scaling of coherent vortex simulation and stochastic coherent adaptive large eddy simulation. Phys. Fluids 25(11), 110–823 (2013).  https://doi.org/10.1063/1.4825260 CrossRefGoogle Scholar
  49. 49.
    Oboukhov, A.: Some specific features of atmospheric turbulence. J. Fluid Mech. 62, 77 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Oseledets, I.: Approximation of \(2^{d}\times 2^{d}\) matrices using tensor decomposition. SIAM J. Matrix Anal. Appl. 31, 2130–2145 (2010)CrossRefzbMATHGoogle Scholar
  51. 51.
    Oseledets, I., Tyrtyshnikov, E.: Breaking the curse of dimensionality, or how to use SVD in many dimensions. SIAM J. Sci. Comput. 31, 3744–3759 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Oseledets, I., Tyrtyshnikov, E.: TT-cross approximation for multidimensional arrays. Linear Algebra Appl. 432, 70–88 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Oseledets, I., Tyrtyshnikov, E.: Algebraic wavelet transform via quantics tensor train decomposition. SIAM J. Sci. Comput. 33, 1315–1328 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    Paladin, G., Vulpiani, A.: Degrees of freedom of turbulence. Phys. Rev. A 35, 1971–1973 (1987).  https://doi.org/10.1103/PhysRevA.35.1971 CrossRefGoogle Scholar
  55. 55.
    Pope, S.: Turbulent Flows. Cambridge University Press, Cambridge (2000)CrossRefzbMATHGoogle Scholar
  56. 56.
    Richardson, L.: Weather Prediction by Numerical Process. Cambridge University Press, Cambridge (1922)zbMATHGoogle Scholar
  57. 57.
    Robinson, S.: Coherent motions in the turbulent boundary layer. Ann. Rev. Fluid Mech. 23, 601–639 (1991)CrossRefGoogle Scholar
  58. 58.
    Sagaut, P.: Large Eddy Simulation for Incompressible Flows, 3rd edn. Springer, Berlin (2006)zbMATHGoogle Scholar
  59. 59.
    Scotti, A., Meneveau, C.: A fractal model for large eddy simulation of turbulent flows. Physica D 127, 198–232 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  60. 60.
    Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Wea. Rev. 93, 99–164 (1963)CrossRefGoogle Scholar
  61. 61.
    Ting, L., Klein, R., Knio, O.M.: Vortex Dominated Flows: Analysis and Computation for Multiple Scales. Series in Applied Mathematical Sciences, vol. 116. Springer, Berlin (2007)zbMATHGoogle Scholar
  62. 62.
    Tucker, L.: Some mathematical notes on three-mode factor analysis. Psychometrika 31, 279–311 (1966)MathSciNetCrossRefGoogle Scholar
  63. 63.
    Uhlmann, M.: Generation of a temporally well-resolved sequence of snapshots of the flow-field in turbulent plane channel flow (2000). http://www-turbul.ifh.uni-karlsruhe.de/uhlmann/reports/produce.pdf. Accessed July 2017
  64. 64.
    Uhlmann, M.: Generation of initial fields for channel flow investigation. Intermediate Report (2000). http://www-turbul.ifh.uni-karlsruhe.de/uhlmann/home/report.html. Accessed July 2017
  65. 65.
    Uhlmann, M.: The need for de-aliasing in a Chebyshev pseudo-spectral method. Technical note no. 60 (2000). http://www-turbul.ifh.uni-karlsruhe.de/uhlmann/reports/dealias.pdf. Accessed July 2017
  66. 66.
    Vergassola, M., Frisch, U.: Wavelet transforms of self-similar processes. Phys. D Nonlinear Phenom. 54(1), 58–64 (1991).  https://doi.org/10.1016/0167-2789(91)90107-K. http://www.sciencedirect.com/science/article/pii/016727899190107K
  67. 67.
    Wallace, J., Eckelmann, H., Brodkey, R.: The wall region in turbulent shear flow. J. Fluid Mech. 54(1), 39–48 (1972).  https://doi.org/10.1017/S0022112072000515 CrossRefGoogle Scholar
  68. 68.
    Wang, J., Wu, J., Ling, J., Iaccarino, G., Xiao, H.: A comprehensive physics-informed machine learning framework for predictive turbulence modeling. arXiv:1701.07102v1 [physics.flu-dyn] (2017)
  69. 69.
    Willmarth, W., Lu, S.: Structure of the Reynolds stress near the wall. J. Fluid Mech. 55(1), 65–92 (1972).  https://doi.org/10.1017/S002211207200165X CrossRefGoogle Scholar
  70. 70.
    Wu, J., Wang, J., Xiao, H., Ling, J.: A priori assessment of prediction confidence for data-driven turbulence modeling. arXiv:1607.04563v2 [physics.comp-ph] (2017)
  71. 71.
    Zhang, Z., Duraisamy, K.: Machine learning methods for data-driven turbulence modeling. AIAA Aviation (2015).  https://doi.org/10.2514/6.2015-2460

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.RotterdamThe Netherlands
  2. 2.Institute of MathematicsFreie Universität BerlinBerlinGermany

Personalised recommendations

Cite article

Buy options