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Research Papers: Fundamental Issues and Canonical Flows

A Phenomenological Model for Turbulent Heat Flux in High-Speed Flows With Shock-Induced Flow Separation

[+] Author and Article Information
Utkarsh Pathak, Subhajit Roy

Department of Aerospace Engineering,
Indian Institute of Technology Bombay,
Mumbai 400076, India

Krishnendu Sinha

Professor
Department of Aerospace Engineering,
Indian Institute of Technology Bombay,
Mumbai 400076, India

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received July 15, 2017; final manuscript received November 25, 2017; published online January 9, 2018. Assoc. Editor: Sergio Pirozzoli.

J. Fluids Eng 140(5), 051203 (Jan 09, 2018) (9 pages) Paper No: FE-17-1432; doi: 10.1115/1.4038760 History: Received July 15, 2017; Revised November 25, 2017

High-speed flows with shock waves impinging on turbulent boundary layers pose severe challenge to current computational methods and models. Specifically, the peak wall heat flux is grossly overpredicted by Reynolds-averaged Navier–Stokes (RANS) simulations using conventional turbulence models. This is because of the constant Prandtl number assumption, which fails in the presence of strong adverse pressure gradient (APG) of the shock waves. Experimental data suggest a reduction of the turbulent Prandtl number in boundary layers subjected to APG. We use a phenomenological approach to develop an algebraic model based on the available data and cast it in a form that can be used in high-speed flows with shock-induced flow separation. The shock-unsteadiness (SU) k–ω model is used as the baseline, since it gives good prediction of flow separation and the regions of APG. The new model gives marked improvement in the peak heat flux prediction near the reattachment point. The formulation is applicable to both attached and separated flows. Additionally, the simplicity of the formulation makes it easily implementable in existing numerical codes.

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Figures

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Fig. 1

Experimental configuration of Schulein [24] to study the interaction of an oblique shock with a turbulent boundary layer

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Fig. 2

Distribution of (a) normalized mean dilatation, (b) fAPG function, and (c) turbulent Prandtl number for β=14 deg and M∞=5 using the SU kω and variable PrT models

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Fig. 3

Comparison of (a) surface pressure and (b) wall heat flux for β=14 deg and M∞=5 using standard kω, SU kω and variable PrT models with the experimental data of Schulein [24]

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Fig. 4

Distribution of the turbulent Prandtl number computed for the 10 deg SBLI case using the variable PrT model

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Fig. 5

Comparison of (a) surface pressure and (b) wall heat flux for β=10 deg and M∞=5 using standard kω, SU modified kω and variable PrT models with the experimental data of Schulein [24]

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Fig. 6

Distribution for the turbulent Prandtl number computed for the 6 deg SBLI case using the variable PrT model

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Fig. 7

Comparison of (a) surface pressure and (b) wall heat flux for β=6 deg and M∞=5 using standard kω, SU modified kω and variable PrT models with the experimental data of Schulein [24]

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Fig. 8

Comparison of wall heat flux with different exponent values for (a) β=14 deg and M∞=5 and (b) β=10 deg and M∞=5 using variable PrT models with the experimental data of Schulein [24]

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Fig. 9

Comparison of wall heat flux with different variable PrT models for (a) β=10 deg and M∞=5 and (b) β=14 deg and M∞=5 with the experimental data of Schulein [24]

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Fig. 10

Comparison of wall heat flux between Xiao et al. [12] (reproduced with permission from AIAA. Copyright 2007.) and current variable PrT model for (a) β=10 deg and M∞=5 and (b) β=14 deg and M∞=5 with the experimental data of Schulein [24]. The heat flux result with kζ model is also reported in the plot.

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Fig. 11

Comparison of (a) surface pressure, (b) skin friction coefficient, and (c) wall heat flux for β=7.8 deg and M∞=3.44 using standard kω, SU modified kω and variable PrT models with the experimental data of Back and Cuffel [30] (Reprinted with permission from AIAA. Copyright 1976 by American Institute of Aeronautics and Astronautics, Inc.).

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Fig. 12

Comparison of (a) surface pressure, (b) skin friction coefficient, and (c) wall heat flux for β=36 deg and M∞=11.3 using standard kω, SU modified kω and variable PrT models with the experimental data of Holden et al. [31]

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