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Turbulent scalar flux in inclined jets in crossflow: counter gradient transport and deep learning modelling

Published online by Cambridge University Press:  16 November 2020

Pedro M. Milani Open the ORCID record for Pedro M. Milani [Opens in a new window] ,
Julia Ling  and
John K. Eaton
Pedro M. Milani*
Affiliation:
Mechanical Engineering Department, Stanford University, Stanford, CA 94305, USA
Julia Ling
Affiliation:
Citrine Informatics, Redwood City, CA 94061, USA
John K. Eaton
Affiliation:
Mechanical Engineering Department, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: pmmilani@stanford.edu
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Abstract

A cylindrical and inclined jet in crossflow is studied under two distinct velocity ratios, $r=1$ and $r=2$, using highly resolved large eddy simulations. First, an investigation of turbulent scalar mixing sheds light onto the previously observed but unexplained phenomenon of negative turbulent diffusivity. We identify two distinct types of counter gradient transport, prevalent in different regions: the first, throughout the windward shear layer, is caused by cross-gradient transport; the second, close to the wall right after injection, is caused by non-local effects. Then, we propose a deep learning approach for modelling the turbulent scalar flux by adapting the tensor basis neural network previously developed to model Reynolds stresses (Ling et al., J. Fluid Mech., vol. 807, 2016a, pp. 155–166). This approach uses a deep neural network with embedded coordinate frame invariance to predict a tensorial turbulent diffusivity that is not explicitly available in the high-fidelity data used for training. After ensuring analytically that the matrix diffusivity leads to a stable solution for the advection diffusion equation, we apply this approach in the inclined jets in crossflow under study. The results show significant improvement compared to a simple model, particularly where cross-gradient effects play an important role in turbulent mixing. The model proposed herein is not limited to jets in crossflow; it can be used in any turbulent flow where the Reynolds averaged transport of a scalar is considered.

JFM classification

Mixing: Turbulent mixing Turbulent Flows: Turbulence modelling Wakes/Jets: Jets
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 906 , 10 January 2021 , A27
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., Davis, A., Dean, J., Devin, M., et al. 2016 Tensorflow: large-scale machine learning on heterogeneous distributed systems. arXiv:1603.04467.Google Scholar
Abe, K. & Suga, K. 2001 Towards the development of a Reynolds-averaged algebraic turbulent scalar-flux model. Intl J. Heat Fluid Flow 22 (1), 19–29.CrossRefGoogle Scholar
Acharya, S., Tyagi, M. & Hoda, A. 2001 Flow and heat transfer predictions for film cooling. Ann. N. Y. Acad. Sci. 934 (1), 110–125.CrossRefGoogle ScholarPubMed
Batchelor, G. K. 1949 Diffusion in a field of homogeneous turbulence. I. Eulerian analysis. Austral. J. Chem. 2 (4), 437–450.CrossRefGoogle Scholar
Bodart, J., Coletti, F., Bermejo-Moreno, I. & Eaton, J. K. 2013 High-fidelity simulation of a turbulent inclined jet in a crossflow. Center Turbul. Res. Annu. Res. Briefs 19, 263–275.Google Scholar
Bogard, D. G. & Thole, K. A. 2006 Gas turbine film cooling. J. Propul. Power 22 (2), 249–270.CrossRefGoogle Scholar
Combest, D. P., Ramachandran, P. A. & Dudukovic, M. P. 2011 On the gradient diffusion hypothesis and passive scalar transport in turbulent flows. Ind. Engng Chem. Res. 50 (15), 8817–8823.CrossRefGoogle Scholar
Daly, B. J. & Harlow, F. H. 1970 Transport equations in turbulence. Phys. Fluids 13 (11), 2634–2649.CrossRefGoogle Scholar
Duraisamy, K., Iaccarino, G. & Xiao, H. 2019 Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51, 357–377.CrossRefGoogle Scholar
Fric, T. F. & Roshko, A. 1994 Vortical structure in the wake of a transverse jet. J. Fluid Mech. 279, 1–47.CrossRefGoogle Scholar
Gardner, J. E., Burgisser, A. & Stelling, P. 2007 Eruption and deposition of the Fisher Tuff (Alaska): evidence for the evolution of pyroclastic flows. J. Geol. 115 (4), 417–435.CrossRefGoogle Scholar
Goodfellow, I., Bengio, Y. & Courville, A. 2016 Deep Learning. MIT Press.Google Scholar
Ham, F. 2007 An efficient scheme for large eddy simulation of low-Ma combustion in complex configurations. Center Turbul. Res. Annu. Res. Briefs 4, 41–46.Google Scholar
Hoda, A. & Acharya, S. 1999 Predictions of a film coolant jet in crossflow with different turbulence models. J. Turbomach. 122 (3), 558–569.CrossRefGoogle Scholar
Kays, W. M. 1994 Turbulent Prandtl number – where are we? J. Heat Transfer 116 (2), 284–295.CrossRefGoogle Scholar
Kingma, D. P. & Ba, J. 2014 Adam: a method for stochastic optimization. arXiv:1412.6980.Google Scholar
Kohli, A. & Bogard, D. G. 2005 Turbulent transport in film cooling flows. J. Heat Transfer 127 (5), 513–520.CrossRefGoogle Scholar
Ling, J., Hutchinson, M., Antono, E., DeCost, B., Holm, E. A. & Meredig, B. 2017 Building data-driven models with microstructural images: generalization and interpretability. Mater. Discovery 10, 19–28.CrossRefGoogle Scholar
Ling, J., Kurzawski, A. & Templeton, J. 2016 a Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.CrossRefGoogle Scholar
Ling, J., Ryan, K. J., Bodart, J. & Eaton, J. K. 2016 b Analysis of turbulent scalar flux models for a discrete hole film cooling flow. J. Turbomach. 138 (1), 011006.CrossRefGoogle Scholar
Mahesh, K. 2013 The interaction of jets with crossflow. Annu. Rev. Fluid Mech. 45, 379–407.CrossRefGoogle Scholar
Maulik, R., San, O., Jacob, J. D. & Crick, C. 2019 Sub-grid scale model classification and blending through deep learning. J. Fluid Mech. 870, 784–812.CrossRefGoogle Scholar
Milani, P. M. & Eaton, J. K. 2018 Magnetic resonance imaging, optimization, and machine learning to understand and model turbulent mixing. In 21st Australasian Fluid Mechanics Conference (ed. I. Marusic), Australasian Fluid Mechanics Society.Google Scholar
Milani, P. M., Gunady, I. E., Ching, D. S., Banko, A. J., Elkins, C. J. & Eaton, J. K. 2019 a Enriching MRI mean flow data of inclined jets in crossflow with large eddy simulations. Intl J. Heat Fluid Flow 80, 108472.CrossRefGoogle Scholar
Milani, P. M., Ling, J. & Eaton, J. K. 2019 b Physical interpretation of machine learning models applied to film cooling flows. J. Turbomach. 141 (1), 011004.CrossRefGoogle Scholar
Milani, P. M., Ling, J. & Eaton, J. K. 2020 Generalization of machine-learned turbulent heat flux models applied to film cooling flows. J. Turbomach. 142 (1), 011007.CrossRefGoogle Scholar
Milani, P. M., Ling, J., Saez-Mischlich, G., Bodart, J. & Eaton, J. K. 2018 A machine learning approach for determining the turbulent diffusivity in film cooling flows. J. Turbomach. 140 (2), 021006.CrossRefGoogle Scholar
Muppidi, S. & Mahesh, K. 2008 Direct numerical simulation of passive scalar transport in transverse jets. J. Fluid Mech. 598, 335–360.CrossRefGoogle Scholar
Parish, E. J. & Duraisamy, K. 2016 A paradigm for data-driven predictive modeling using field inversion and machine learning. J. Comput. Phys. 305, 758–774.CrossRefGoogle Scholar
Pope, S. B. 1975 A more general effective-viscosity hypothesis. J. Fluid Mech. 72 (2), 331–340.CrossRefGoogle Scholar
Ryan, K. J., Bodart, J., Folkersma, M., Elkins, C. J. & Eaton, J. K. 2017 Turbulent scalar mixing in a skewed jet in crossflow: experiments and modeling. Flow Turbul. Combust. 98 (3), 781–801.CrossRefGoogle Scholar
Salewski, M., Stankovic, D. & Fuchs, L. 2008 Mixing in circular and non-circular jets in crossflow. Flow Turbul. Combust. 80 (2), 255–283.CrossRefGoogle Scholar
Sandberg, R. D., Tan, R., Weatheritt, J., Ooi, A., Haghiri, A., Michelassi, V. & Laskowski, G. 2018 Applying machine learnt explicit algebraic stress and scalar flux models to a fundamental trailing edge slot. J. Turbomach. 140 (10), 101008.CrossRefGoogle Scholar
Schreivogel, P., Abram, C., Fond, B., Straußwald, M., Beyrau, F. & Pfitzner, M. 2016 Simultaneous kHz-rate temperature and velocity field measurements in the flow emanating from angled and trenched film cooling holes. Intl J. Heat Mass Transfer 103, 390–400.CrossRefGoogle Scholar
Shih, T. H., Zhu, J. & Lumley, J. L. 1995 A new Reynolds stress algebraic equation model. Comput. Methods Appl. Mech. Engng 125 (1), 287–302.CrossRefGoogle Scholar
Sotgiu, C., Weigand, B. & Semmler, K. 2018 A turbulent heat flux prediction framework based on tensor representation theory and machine learning. Intl Commun. Heat Mass Transfer 95, 74–79.CrossRefGoogle Scholar
Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I. & Salakhutdinov, R. 2014 Dropout: a simple way to prevent neural networks from overfitting. J. Machine Learning Res. 15 (1), 1929–1958.Google Scholar
Su, L. K. & Mungal, M. G. 2004 Simultaneous measurements of scalar and velocity field evolution in turbulent crossflowing jets. J. Fluid Mech. 513, 1–45.CrossRefGoogle Scholar
Tavoularis, S. & Corrsin, S. 1981 Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1. J. Fluid Mech. 104, 311–347.CrossRefGoogle Scholar
Vreman, A. W. 2004 An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 16 (10), 3670–3681.CrossRefGoogle Scholar
Weatheritt, J., Sandberg, R. D., Ling, J., Saez, G. & Bodart, J. 2017 A comparative study of contrasting machine learning frameworks applied to RANS modeling of jets in crossflow. In ASME Turbo Expo 2017: Turbomachinery Technical Conference and Exposition. American Society of Mechanical Engineers Digital Collection.CrossRefGoogle Scholar
Weatheritt, J., Zhao, Y., Sandberg, R. D., Mizukami, S. & Tanimoto, K. 2020 Data-driven scalar-flux model development with application to jet in cross flow. Intl J. Heat Mass Transfer 147, 118931.CrossRefGoogle Scholar
Wu, J., Xiao, H., Sun, R. & Wang, Q. 2019 Reynolds-averaged Navier–Stokes equations with explicit data-driven Reynolds stress closure can be ill-conditioned. J. Fluid Mech. 869, 553–586.CrossRefGoogle Scholar
Xie, Z. T. & Castro, I. P. 2008 Efficient generation of inflow conditions for large eddy simulation of street-scale flows. Flow Turbul. Combust. 81 (3), 449–470.CrossRefGoogle Scholar
Younis, B. A., Speziale, C. G. & Clark, T. T. 2005 A rational model for the turbulent scalar fluxes. Proc. R. Soc. A 461 (2054), 575–594.CrossRefGoogle Scholar
Yuan, L. L., Street, R. L. & Ferziger, J. H. 1999 Large-eddy simulations of a round jet in crossflow. J. Fluid Mech. 379, 71–104.CrossRefGoogle Scholar
Zheng, Q. S. 1994 Theory of representations for tensor functions – a unified invariant approach to constitutive equations. Appl. Mech. Rev. 47 (11), 545–587.CrossRefGoogle Scholar