Data-driven scalar-flux model development with application to jet in cross flow

Abstract The classical gradient-diffusion hypothesis has known deficiencies when applied to cooling applications. In this paper, the gene-expression programming (GEP) method, a machine learning approach, has been applied to develop scalar-flux models via symbolic regression. The scalar-flux, the unclosed term of the mean passive-scalar transport equation, is treated by considering the polynomial basis and scalar invariants available from computable Reynolds-averaged quantities. This method has been applied to develop and then assess a model for the test case of jet in crossflow. A high-fidelity database was first probed for insight into which of the candidate bases are the most suitable as modelling terms. The high dimensionality of the function space, spanned by the basis, was then reduced by basic statistical techniques. The resulting data-driven model is presented and tested for a range of different jet in crossflow cases. Compared with eddy-diffusivity models, the new model is shown to produce reliably more accurate results. This demonstrates that the current framework can be used for scalar-flux modelling in complex three-dimensional flows and has potential to provide generalized form closures with improved predictive accuracy for the same classes of flows they were trained on.

[1]  Philippe R. Spalart,et al.  Direct Simulation and RANS Modelling of a Vortex Generator Flow , 2015 .

[2]  Parviz Moin,et al.  Transport of Passive Scalars in a Turbulent Channel Flow , 1989 .

[3]  Paul A. Durbin,et al.  A Procedure for Using DNS Databases , 1998 .

[4]  Hrvoje Jasak,et al.  A tensorial approach to computational continuum mechanics using object-oriented techniques , 1998 .

[5]  Ken-ichi Abe,et al.  Towards the development of a Reynolds-averaged algebraic turbulent scalar-flux model , 2001 .

[6]  W. Rodi,et al.  Experimental investigation of jets in a crossflow , 1982, Journal of Fluid Mechanics.

[7]  Karthik Duraisamy,et al.  A paradigm for data-driven predictive modeling using field inversion and machine learning , 2016, J. Comput. Phys..

[8]  Achmed Schulz,et al.  Correlation of Film-Cooling Effectiveness From Thermographic Measurements at Enginelike Conditions , 2002 .

[9]  Stavros Tavoularis,et al.  Experiments in nearly homogenous turbulent shear flow with a uniform mean temperature gradient. Part 1 , 1981, Journal of Fluid Mechanics.

[10]  Richard D. Sandberg,et al.  A Comparative Study of Contrasting Machine Learning Frameworks Applied to RANS Modeling of Jets in Crossflow , 2017 .

[11]  Jinlong Wu,et al.  Physics-informed machine learning approach for reconstructing Reynolds stress modeling discrepancies based on DNS data , 2016, 1606.07987.

[12]  V. Michelassi,et al.  Investigation of the Accuracy of RANS Models to Predict the Flow Through a Low-Pressure Turbine , 2016 .

[13]  Richard Sandberg,et al.  A novel evolutionary algorithm applied to algebraic modifications of the RANS stress-strain relationship , 2016, J. Comput. Phys..

[14]  Yasutaka Nagano,et al.  A Two-Equation Model for Heat Transport in Wall Turbulent Shear Flows , 1988 .

[15]  R. S. Rivlin,et al.  The formulation of constitutive equations in continuum physics. II , 1959 .

[16]  Ayodeji O. Demuren,et al.  Characteristics of three-dimensional turbulent jets in crossflow , 1993 .

[17]  Christopher J. Elkins,et al.  Turbulent transport in an inclined jet in crossflow , 2013 .

[18]  Andrew T. Hsu,et al.  The effect of Schmidt number on turbulent scalar mixing in a jet-in-crossflow , 1999 .

[19]  Julia Ling,et al.  A Machine Learning Approach for Determining the Turbulent Diffusivity in Film Cooling Flows , 2017 .

[20]  Francis H. Harlow,et al.  Transport Equations in Turbulence , 1970 .

[21]  I. Gartshore,et al.  Multiple Jets in a Crossflow: Detailed Measurements and Numerical Simulations , 1995 .

[22]  V. Michelassi,et al.  Applying Machine Learnt Explicit Algebraic Stress and Scalar Flux Models to a Fundamental Trailing Edge Slot , 2018, Journal of Turbomachinery.

[23]  Richard D. Sandberg,et al.  The development of algebraic stress models using a novel evolutionary algorithm , 2017 .

[24]  John R. Koza,et al.  Genetic programming as a means for programming computers by natural selection , 1994 .

[25]  Krishnan Mahesh,et al.  The Interaction of Jets with Crossflow , 2013 .

[26]  Toshihiko Takahashi,et al.  Numerical Study on Effects of Density Ratio on Film Cooling Flow Structure and Film Cooling Effectiveness , 2017 .

[27]  Christopher J. Elkins,et al.  Turbulent Scalar Mixing in a Skewed Jet in Crossflow: Experiments and Modeling , 2017, Proceeding of Ninth International Symposium on Turbulence and Shear Flow Phenomena.

[28]  J. Templeton,et al.  Reynolds averaged turbulence modelling using deep neural networks with embedded invariance , 2016, Journal of Fluid Mechanics.

[29]  Gianluca Iaccarino,et al.  Numerical simulation of scalar dispersion downstream of a square obstacle using gradient-transport type models , 2009 .

[30]  S. Pope.  A more general effective-viscosity hypothesis , 1975, Journal of Fluid Mechanics.

[31]  David E. Goldberg,et al.  Genetic algorithms and Machine Learning , 1988, Machine Learning.

[32]  Sumanta Acharya,et al.  Analysis of k and epsilon Budgets for Film Cooling Using Direct Numerical Simulation , 2006 .

[33]  Cândida Ferreira,et al.  Gene Expression Programming: A New Adaptive Algorithm for Solving Problems , 2001, Complex Syst..

[34]  G. Laskowski,et al.  Machine Learning for Turbulence Model Development Using a High-Fidelity HPT Cascade Simulation , 2017 .

[35]  G. F. Smith.  On isotropic integrity bases , 1965 .

[36]  Manfred Aigner,et al.  A Numerical Study on the Turbulent Schmidt Numbers in a Jet in Crossflow , 2012 .

[37]  R. M. C. So,et al.  A near-wall two-equation model for turbulent heat fluxes , 1992 .

[38]  David G. Bogard,et al.  Adiabatic Effectiveness, Thermal Fields, and Velocity Fields for Film Cooling With Large Angle Injection , 1995 .

[39]  Sumanta Acharya,et al.  Predictions of a Film Coolant Jet in Crossflow With Different Turbulence Models , 1999 .

[40]  Toshihiko Takahashi,et al.  Large-eddy simulation of an inclined round jet issuing into a crossflow , 2014 .

[41]  Karen A. Thole,et al.  Gas Turbine Film Cooling , 2006 .

[42]  Michael A. Leschziner,et al.  Statistical Turbulence Modelling For Fluid Dynamics - Demystified: An Introductory Text For Graduate Engineering Students , 2015 .

[43]  Karthik Duraisamy,et al.  Turbulence Modeling in the Age of Data , 2018, Annual Review of Fluid Mechanics.

[44]  Nagi N. Mansour,et al.  An algebraic model for the turbulent flux of a passive scalar , 1989, Journal of Fluid Mechanics.

[45]  Julia Ling,et al.  Analysis of Turbulent Scalar Flux Models for a Discrete Hole Film Cooling Flow , 2015 .

[46]  Yoshihide Tominaga,et al.  Turbulent Schmidt numbers for CFD analysis with various types of flowfield , 2007 .

[47]  Gaël Varoquaux,et al.  Scikit-learn: Machine Learning in Python , 2011, J. Mach. Learn. Res..

[48]  G. Batchelor.  Diffusion in a Field of Homogeneous Turbulence. I. Eulerian Analysis , 1949 .