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Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations

Published online by Cambridge University Press:  21 November 2016

H. Babaee*
Affiliation:
Department of Mechanical Engineering, MIT Sea Grant College Program, Massachusetts Institute of Technology, Cambridge, MA 02142, USA
P. Perdikaris
Affiliation:
Department of Mechanical Engineering, MIT Sea Grant College Program, Massachusetts Institute of Technology, Cambridge, MA 02142, USA
C. Chryssostomidis
Affiliation:
Department of Mechanical Engineering, MIT Sea Grant College Program, Massachusetts Institute of Technology, Cambridge, MA 02142, USA
G. E. Karniadakis
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
*
Email address for correspondence: babaee@mit.edu

Abstract

For thermal mixed-convection flows, the Nusselt number is a function of Reynolds number, Grashof number and the angle between the forced- and natural-convection directions. We consider flow over a heated cylinder for which there is no universal correlation that accurately predicts Nusselt number as a function of these parameters, especially in opposing-convection flows, where the natural convection is against the forced convection. Here, we revisit this classical problem by employing modern tools from machine learning to develop a general multi-fidelity framework for constructing a stochastic response surface for the Nusselt number. In particular, we combine previously developed experimental correlations (low-fidelity model) with direct numerical simulations (high-fidelity model) using Gaussian process regression and autoregressive stochastic schemes. In this framework the high-fidelity model is sampled only a few times, while the inexpensive empirical correlation is sampled at a very high rate. We obtain the mean Nusselt number directly from the stochastic multi-fidelity response surface, and we also propose an improved correlation. This new correlation seems to be consistent with the physics of this problem as we correct the vectorial addition of forced and natural convection with a pre-factor that weighs differently the forced convection. This, in turn, results in a new definition of the effective Reynolds number, hence accounting for the ‘incomplete similarity’ between mixed convection and forced convection. In addition, due to the probabilistic construction, we can quantify the uncertainty associated with the predictions. This information-fusion framework is useful for elucidating the physics of the flow, especially in cases where anomalous transport or interesting dynamics may be revealed by contrasting the variable fidelity across the models. While in this paper we focus on the thermal mixed convection, the multi-fidelity framework provides a new paradigm that could be used in many different contexts in fluid mechanics including heat and mass transport, but also in combining various levels of fidelity of models of turbulent flows.

Type
Papers
Copyright
© 2016 Cambridge University Press 

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References

Acrivos, A. 1966 On the combined effect of forced and free convection heat transfer in laminar boundary layer flows. Chem. Engng Sci. 21 (4), 343352.Google Scholar
Babaee, H., Acharya, S. & Wan, X. 2013a Optimization of forcing parameters of film cooling effectiveness. Trans. ASME J. Turbomach. 136 (6), 061016.Google Scholar
Babaee, H., Wan, X. & Acharya, S. 2013b Effect of uncertainty in blowing ratio on film cooling effectiveness. Trans. ASME J. Heat Transfer 136 (3), 031701.Google Scholar
Badr, H. M. 1984 Laminar combined convection from a horizontal cylinder – parallel and contra flow regimes. Intl J. Heat Mass Transfer 27 (1), 1527.CrossRefGoogle Scholar
Churchill, S. W. 1977 A comprehensive correlating equation for laminar, assisting, forced and free convection. AIChE J. 23 (1), 1016.CrossRefGoogle Scholar
Churchill, S. W. 2014 Equivalents – a new concept for the prediction and interpretation of thermal convection. Ind. Engng Chem. Res. 53 (10), 41044118.CrossRefGoogle Scholar
Churchill, S. W. & Usagi, R. 1972 A general expression for the correlation of rates of transfer and other phenomena. AIChE J. 18 (6), 11211128.CrossRefGoogle Scholar
Damianou, A. C. & Lawrence, N. D. 2013 Deep gaussian processes. In Proceedings of the Sixteenth International Conference on Artificial Intelligence and Statistics, pp. 207215.Google Scholar
Eckstein, A. & Vlachos, P. P. 2009 Digital particle image velocimetry (dpiv) robust phase correlation. Meas. Sci. Technol. 20 (5), 055401.Google Scholar
Forrester, A., Sobester, A. & Keane, A. 2008 Engineering Design via Surrogate Modelling: A Practical Guide. John Wiley & Sons.Google Scholar
Gratiet, L. L. & Garnier, J. 2014 Recursive co-kriging model for design of computer experiments with multiple levels of fidelity. Intl J. Uncertainty Quant. 4, 365386.CrossRefGoogle Scholar
Hatton, A. P., James, D. D. & Swire, H. W. 1970 Combined forced and natural convection with low-speed air flow over horizontal cylinders. J. Fluid Mech. 42, 1731.Google Scholar
Hensman, J., Fusi, N. & Lawrence, N. D.2013 Gaussian processes for big data. arXiv:1309.6835.Google Scholar
Hu, H. & Koochesfahani, M. M. 2011 Thermal effects on the wake of a heated circular cylinder operating in mixed convection regime. J. Fluid Mech. 685, 235270.Google Scholar
Karniadakis, G. E. & Sherwin, S. J. 2005 Spectral/hp Element Methods for Computational Fluid Dynamics. Oxford University Press.Google Scholar
Kennedy, M. C. & O’Hagan, A. 2000 Predicting the output from a complex computer code when fast approximations are available. Biometrika 87 (1), 113.Google Scholar
Lemlich, R. & Hoke, R. 1956 A common basis for the correlation of forced and natural convection to horizontal cylinders. AIChE J. 2 (2), 249250.Google Scholar
Mattos, C. L. C., Dai, Z., Damianou, A., Forth, J., Barreto, G. A. & Lawrence, N. D.2015 Recurrent gaussian processes. arXiv:1511.06644.Google Scholar
Oosthuizen, P. H. & Madan, S. 1971 The effect of flow direction on combined convective heat transfer from cylinders to air. Trans. ASME J. Heat Transfer 93 (2), 240242.CrossRefGoogle Scholar
Patnaik, B. S. V., Narayana, P. A. A. & Seetharamu, K. N. 1999 Numerical simulation of vortex shedding past a circular cylinder under the influence of buoyancy. Intl J. Heat Mass Transfer 42 (18), 34953507.Google Scholar
Perdikaris, P. & Karniadakis, G. E. 2016 Model inversion via multi-fidelity bayesian optimization. J. R. Soc. Interface 13 (118).Google Scholar
Perdikaris, P., Venturi, D., Royset, J. O. & Karniadakis, G. E. 2015 Multi-fidelity modelling via recursive co-kriging and gaussian–markov random fields. Proc. R. Soc. Lond. A 471, 20150018.Google ScholarPubMed
Pereira, F., Gharib, M., Dabiri, D. & Modarress, D. 2000 Defocusing digital particle image velocimetry: a 3-component 3-dimensional dpiv measurement technique. application to bubbly flows. Exp. Fluids 29 (1), S078S084.Google Scholar
Rasmussen, C. E.2006 Gaussian processes for machine learning.Google Scholar
Ronald, J. A. 1991 Particle-imaging techniques for experimental fluid mechanics. Annu. Rev. Fluid Mech. 23 (1), 261304.Google Scholar
Sharma, A. & Eswaran, V. 2004 Effect of aiding and opposing buoyancy on the heat and fluid flow across a square cylinder at re = 100. Numer. Heat Transfer 45 (6), 601624.Google Scholar
Snelson, E. & Ghahramani, Z. 2006 Sparse gaussian processes using pseudo-inputs. In Advances in Neural Information Processing Systems, pp. 12571264. The MIT Press.Google Scholar
Sparrow, E. M. & Lee, L. 1976 Analysis of mixed convection about a horizontal cylinder. Intl J. Heat Mass Transfer 19 (2), 229232.Google Scholar