- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Data-driven modeling for drop size distributions

T. Traverso1,2, T. Abadie3,4, O. K. Matar4,1, and L. Magri2,1,*

  • 1The Alan Turing Institute, British Library, London NW1 2DB, United Kingdom
  • 2Department of Aeronautics, Imperial College London, South Kensington Campus, London SW72AZ, United Kingdom
  • 3School of Chemical Engineering, University of Birmingham, Birmingham B15 2TT, United Kingdom
  • 4Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW72AZ, United Kingdom

  • *l.magri@imperial.ac.uk
PDF

Phys. Rev. Fluids 8, 104302 – Published 27 October, 2023

DOI: https://doi.org/10.1103/PhysRevFluids.8.104302

Abstract

The prediction of the drop size distribution (DSD) resulting from liquid atomization is key to the optimization of multiphase flows from gas-turbine propulsion through agriculture to healthcare. Obtaining high-fidelity data of liquid atomization, either experimentally or numerically, is expensive, which makes the exploration of the design space difficult. First, to tackle these challenges, we propose a framework to predict the DSD of a liquid spray based on data as a function of the spray angle, the Reynolds number, and the Weber number. Second, to guide the design of liquid atomizers, the model accurately predicts the volume of fluid contained in drops of specific sizes while providing uncertainty estimation. To do so, we propose a Gaussian process regression (GPR) model, which infers the DSD and its uncertainty form the knowledge of its integrals and of its first moment, i.e., the mean drop diameter. Third, we deploy multiple GPR models to estimate these quantities at arbitrary points of the design space from data obtained from a large number of numerical simulations of a flat fan spray. The kernel used for reconstructing the DSD incorporates prior physical knowledge, which enables the prediction of sharply peaked and heavy-tailed distributions. Fourth, we compare our method with a benchmark approach, which estimates the DSD by interpolating the frequency polygon of the binned drops with a GPR. We show that our integral approach is significantly more accurate, especially in the tail of the distribution (i.e., large, rare drops), and it reduces the bias of the density estimator by up to 10 times. Finally, we discuss physical aspects of the model's predictions and interpret them against experimental results from the literature. This work opens opportunities for modeling drop size distribution in multiphase flows from data.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (59)

  1. S. Li, C. Chen, Y. Wang, F. Kang, and W. Li, Study on the atomization characteristics of flat fan nozzles for pesticide application at low pressures, Agriculture 11, 309 (2021).
  2. A. H. Lefebvre, The role of fuel preparation in low-emission combustion, J. Eng. Gas Turbines Power 117, 617 (1995).
  3. A. H. Lefebvre and V. G. McDonell, Atomization and Sprays (CRC Press, Boca Raton, FL, 2017)
  4. M. C. B. Ellis, C. R. Tuck, and P. C. H. Miller, The effect of some adjuvants on sprays produced by agricultural flat fan nozzles, Crop Protection 16, 41 (1997).
  5. P. A. Hobson, P. C. H. Miller, P. J. Walklate, C. R. Tuck, and N. M. Western, Spray drift from hydraulic spray nozzles: the use of a computer simulation model to examine factors influencing drift, J. Agricult. Eng. Res. 54, 293 (1993).
  6. E. Babinsky and P. E. Sojka, Modeling drop size distributions, Prog. Energy Combust. Sci. 28, 303 (2002).
  7. B. Mohan, W. Yang, and S. C. kiang, Fuel injection strategies for performance improvement and emissions reduction in compression ignition engines—a review, Renew. Sust. Energy Rev. 28, 664 (2013).
  8. R. Schmehl, G. Maier, and S. Wittig, CFD analysis of fuel atomization, secondary droplet breakup and spray dispersion in the premix duct of a LPP combustor, in Proceedings of the 8th International Conference on Liquid Atomization and Spray Systems (ICLASS'00) (ILASS, Pasadena, California, USA, 2000).
  9. L. Andreassi, S. Ubertini, and L. Allocca, Experimental and numerical analysis of high pressure diesel spray-wall interaction, Int. J. Multiphase Flow 33, 742 (2007).
  10. L. Wang, N. Zhang, J. W. Slocombe, G. E. Thierstein, and D. K. Kuhlman, Experimental analysis of spray distribution pattern uniformity for agricultural nozzles, Appl. Eng. Aric. 11, 51 (1995).
  11. E. Villermaux and C. Clanet, Life of a flapping liquid sheet, J. Fluid Mech. 462, 341 (2002).
  12. E. Villermaux, Fragmentation, Annu. Rev. Fluid Mech. 39, 419 (2007).
  13. A. Khopkar, M. D. Cloeter, and Q. Yuan, Industrial sprays: Experimentalcharacterization and numerical modeling, in Applied Computational Fluid Dynamics (IntechOpen, London, UK, 2012).
  14. C. R. Constante-Amores, L. Kahouadji, A. Batchvarov, S. Shin, J. Chergui, D. Juric, and O. K. Matar, Direct numerical simulations of transient turbulent jets: Vortex-interface interactions, J. Fluid Mech. 922, A6 (2021).
  15. Y. Ling, D. Fuster, S. Zaleski, and G. Tryggvason, Spray formation in a quasiplanar gas-liquid mixing layer at moderate density ratios: A numerical closeup, Phys. Rev. Fluids 2, 014005 (2017).
  16. S. Shin, J. Chergui, D. Juric, L. Kahouadji, O. K. Matar, and R. V. Craster, A hybrid interface tracking–level set technique for multiphase flow with soluble surfactant, J. Comput. Phys. 359, 409 (2018).
  17. J. Sacks, W. J. Welch, T. J. Mitchell, and H. P. Wynn, Design and analysis of computer experiments, Stat. Sci. 4, 409 (1989).
  18. K. Chaloner and I. Verdinelli, Bayesian experimental design: A review, Stat. Sci. 10, 273 (1995).
  19. T. P. Sapsis, Output-weighted optimal sampling for Bayesian regression and rare event statistics using few samples, Proc. R. Soc. A 476, 20190834 (2020).
  20. S. Conti and A. O'Hagan, Bayesian emulation of complex multi-output and dynamic computer models, J. Stat. Plan. Inference 140, 640 (2010).
  21. N. Bempedelis and L. Magri, Bayesian optimization of thelayout of wind farms with a high-fidelity surrogate model, arXiv:2302.01071.
  22. K. Campbell, M. D. McKay, and B. J. Williams, Sensitivity analysis when model outputs are functions, Reliab. Eng. Syst. Safe. 91, 1468 (2006).
  23. M. J. Bayarri, D. Walsh, J. O. Berger, J. Cafeo, G. Garcia-Donato, F. Liu, J. Palomo, R. J. Parthasarathy, R. Paulo, and J. Sacks, Computer model validation with functional output, Ann. Stat. 35, 1874 (2007).
  24. K. Longi, C. Rajani, T. Sillanpaa, J. Makinen, T. Rauhala, A. Salmi, E. Haeggström, and A. Klami, Sensor placement for spatial Gaussian processes with integral observations, in Proceedings of the 36th Conference on Uncertainty in Artificial Intelligence (UAI'20), Proceedings of Machine Learning Research, Vol. 124, edited by J. Peters and D. Sontag (PMLR, 2020), pp. 1009–1018.
  25. H. C. Law, D. Sejdinovic, E. Cameron, T. Lucas, S. Flaxman, K. Battle, and K. Fukumizu, Variational learning on aggregate outputs with Gaussian processes, Advances in Neural Information Processing Systems, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associates, Inc., 2018), Vol. 31.
  26. E. Solak, R. Murray-Smith, W. E. Leithead, D. Leith, and C. Rasmussen, Derivative observations in Gaussian process models of dynamic systems, Advances in Neural Information Processing Systems, edited by S. Becker, S. Thrun, and K. Obermayer (MIT Press, 2002), Vol. 15.
  27. Q. Cao, G. Zhang, E. A. Brandes, and T. J. Schuur, Polarimetric radar rain estimation through retrieval of drop size distribution using a Bayesian approach, J. Appl. Meteorol. Climatol. 49, 973 (2010).
  28. S. Tengeleng and N. Armand, Performance of using cascade forward back propagation neural networks for estimating rain parameters with rain drop size distribution, Atmosphere 5, 454 (2014).
  29. S. Kooij, R. Sijs, M. M. Denn, E. Villermaux, and D. Bonn, What determines the drop size in sprays? Phys. Rev. X 8, 031019 (2018).
  30. C. K. Williams and C. E. Rasmussen, Gaussian Processes for Machine Learning, Vol. 2 (MIT Press, Cambridge, MA, 2006).
  31. GPy, GPy: A Gaussian process framework in python [http://github.com/SheffieldML/GPy].
  32. J. Gardner, G. Pleiss, K. Q. Weinberger, D. Bindel, and A. G. Wilson, GPyTorch: Blackbox matrix-matrix Gaussian process inference with GPU acceleration, Advances in Neural Information Processing Systems, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associates, Inc., 2018), Vol. 31.
  33. M. A. Alvarez, L. Rosasco, N. D. Lawrence et al., Kernels for vector-valued functions: A review, Found. Trends Mach. Learn. 4, 195 (2012).
  34. A. O'Hagan, Bayes-hermite quadrature, J. Stat. Plan. Inference 29, 245 (1991).
  35. C. E. Rasmussen and Z. Ghahramani, Bayesian Monte Carlo, Advances in Neural Information Processing Systems, edited by S. Becker, S. Thrun, and K. Obermayer (MIT Press, 2003), Vol. 15.
  36. T. Gunter, M. A. Osborne, R. Garnett, P. Hennig, and S. J. Roberts, Sampling for inference in probabilistic models with fast Bayesian quadrature, Advances in Neural Information Processing Systems, edited by Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Q. Weinberger (Curran Associates, Inc., 2014), Vol. 27.
  37. J. A. Rice, Mathematical Statistics and Data Analysis (Cengage Learning, Independence, KY, 2006).
  38. T. Abadie, J. Aubin, and D. Legendre, On the combined effects of surface tension force calculation and interface advection on spurious currents within volume of fluid and level set frameworks, J. Comput. Phys. 297, 611 (2015).
  39. S. Popinet, Numerical models of surface tension, Annu. Rev. Fluid Mech. 50, 49 (2018).
  40. S. Popinet, A quadtree-adaptive multigrid solver for the Serre–Green–Naghdi equations, J. Comput. Phys. 302, 336 (2015).
  41. S. Popinet, An accurate adaptive solver for surface-tension-driven interfacial flows, J. Comput. Phys. 228, 5838 (2009).
  42. G. Tryggvason, R. Scardovelli, and S. Zaleski, Direct Numerical Simulations of Gas–Liquid Multiphase Flows (Cambridge University Press, Cambridge, UK, 2011).
  43. W. Aniszewski, T. Ménard, and M. Marek, Volume of fluid (VOF) type advection methods in two-phase flow: A comparative study, Comput. Fluids 97, 52 (2014).
  44. S. Afkhami and M. Bussmann, Height functions for applying contact angles to 2D VOF simulations, Int. J. Numer. Methods Fluids 57, 453 (2008).
  45. E. G. Puckett and J. S. Saltzman, A 3D adaptive mesh refinement algorithm for multimaterial gas dynamics, Physica D 60, 84 (1992).
  46. J. L. Nowak, M. Mohammadi, and S. P. Malinowski, Applicability of the VisiSize D30 shadowgraph system for cloud microphysical measurements, Atmos. Meas. Tech. 14, 2615 (2021).
  47. VisiSize Software Operation Manual-VisiSize 6 (Oxford Lasers Inc., 2013).
  48. Y. Jin, F. R. Beck, B. R. Lowery, D. J. Miller, F. Cheung, S. M. Bajorek, K. Tien, and C. L. Hoxie, Experimental study of droplet sizes across a spacer grid location under various reflood conditions, Exp. Therm. Fluid Sci. 94, 246 (2018).
  49. P. Marmottant and E. Villermaux, Fragmentation of stretched liquid ligaments, Phys. Fluids 16, 2732 (2004).
  50. P. Marmottant and E. Villermaux, On spray formation, J. Fluid Mech. 498, 73 (2004).
  51. R. D. Cohen, Shattering of a liquid drop due to impact, Proc. R. Soc. Lond. A 435, 483 (1991).
  52. M. N. Gibbs, Bayesian Gaussian processes for regression and classification, Ph.D. thesis, Citeseer (1998).
  53. A. Conn, N. Gould, and P. Toint, A globally convergent Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds, Math. Comput. 66, 261 (1997).
  54. Z. Geng, F. Yang, X. Chen, and N. Wu, Gaussian process based modeling and experimental design for sensor calibration in drifting environments, Sens. Actuat. B: Chem. 216, 321 (2015).
  55. J. S. Simonoff, Smoothing Methods in Statistics, Springer Series in Statistics (Springer-Verlag, New York, 1996).
  56. T. Traverso, T. Abadie, O. K. Matar, and L. Magri, Drop count and size of 38 numerical simulation of a flat fan spray [Data set], Zenodo (2023).
  57. R. M. Neal, Bayesian learning for neural networks, Ph.D. thesis, University of Toronto, 1994.
  58. W. Li and L. Zhenyu, A method of SVM with normalization in intrusion detection, Proc. Environ. Sci. 11, 256 (2011).
  59. J. A. Weymark, Generalized gini inequality indices, Math. Soc. Sci. 1, 409 (1981).

Outline

Information

Phys. Rev. Fluids 8, 104302– Published 27 October, 2023
  • Received 28 May 2023
  • Accepted 3 October 2023
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation