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Research Papers: Forced Convection

Prediction of the Turbine Tip Convective Heat Flux Using Discrete Green's Functions

[+] Author and Article Information
Valeria Andreoli

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: vale.andreoli@gmail.com

David G. Cuadrado

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: david.gonzalez.cuadrado@gmail.com

Guillermo Paniagua

School of Mechanical Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: gpaniagua@me.com

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 18, 2017; final manuscript received November 30, 2017; published online March 30, 2018. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 140(7), 071703 (Mar 30, 2018) (11 pages) Paper No: HT-17-1419; doi: 10.1115/1.4039182 History: Received July 18, 2017; Revised November 30, 2017

The heat fluxes across the turbine tip gap are characterized by large unsteady pressure gradients and shear from the viscous effects. The classical Newton heat convection equation, based on the turbine inlet total temperature, is inadequate. Previous research from our team relied on the use of the adiabatic wall temperature. In this paper, we propose an alternative approach to predict the convective heat transfer problem across the turbine rotor tip using discrete Green's functions (DGF). The linearity of the energy equation in the solid domain with constant thermal properties can be applied with a superposition technique to measure the data extracted from flow simulations to determine the Green's function distribution. The DGF is a matrix of coefficients that relate the temperature spatial (GF) distribution with the heat flux. This methodology is first applied to a backward facing step, validated using experimental data. The final aim of this paper is to demonstrate the method in the rotor turbine tip. A turbine stage at engine-like conditions was assessed using cfd software. The heat flux pulses were applied at different locations in the rotor tip geometry, and the increment of temperature in this zone was evaluated for different clearances, with a consequent variation of the DGF coefficients. Ultimately, a detailed uncertainty analysis of the methodology was included based on the magnitude of the heat flux pulses used in the DGF coefficients calculation and the uncertainty in the experimental measurements of the wall temperature.

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Figures

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Fig. 1

(a) Numerical domain and (b) Nusselt number for isothermal wall conditions

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Fig. 2

Values of the DGF coefficients in the main diagonal, subdiagonals, and super-diagonals of the GF matrix for the baseline case

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Fig. 3

Comparison between experimental data of Eaton and Hacker [8] and the numerical calculation of the Stanton number in the recirculating region of the backward facing step

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Fig. 4

(a) Velocity field in the backward-facing step calculated with ANSYS Fluent in the baseline case, (b) heat flux distribution used for the validation of the DGF approach, and (c) comparison between the CFD calculation of the wall temperature, the wall temperature result using the adiabatic heat transfer coefficient, and the GF result of the wall temperature in the baseline case

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Fig. 5

(a) Comparison between CFD temperature results and DGF approach temperature results at 250 K, (b) comparison between CFD temperature results and DGF approach temperature results at 4 bar, and (c) comparison between CFD temperature results and DGF approach temperature results at Mach 0.85

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Fig. 6

(a) Stage mesh and tip gap details and (b) elements used to impose the heat pulses

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Fig. 7

Grid sensitivity study

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Fig. 8

(a) Isentropic Mach number (experiments—blue dots, CFD—continuous black line) at 85% of the blade span and (b) Nusselt number (experiments—blue dots, CFD—continuous black line) at 85% of the blade span

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Fig. 9

(a) Shear stress lines and heat flux on blade tip for 1%, (b) 0.1% clearance, (c) shear stress lines and heat flux on shroud for 1%, and (d) 0.1% clearance

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Fig. 10

(a) Adiabatic wall temperature (with shear stress lines) at 1% clearance, (b) 0.1% clearance, (c) relative total temperature with streamlines at 1%, and (d) 0.1% clearance

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Fig. 11

Nominal clearance validation case: (a) temperature distribution calculated with CFD, (b) temperature distribution predicted with GF approach, (c) heat flux distribution imposed on the tip surface, and (d) relative temperature error between the calculated CFD distribution and the predicted GF approach distribution

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Fig. 12

Tight clearance validation case: (a) temperature distribution calculated with CFD, (b) temperature distribution predicted with GF approach, (c) heat flux distribution imposed on the tip surface, and (d) relative temperature error between the calculated CFD distribution and the predicted GF approach distribution

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Fig. 13

Histogram of the IDGF for (a) 1% clearance and (b) 0.1% clearance

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Fig. 14

Comparison between adiabatic wall temperature computed from RANS with adiabatic walls (solid line), temperature retrieved from RANS with isothermal walls at 402 K (dashed line) and temperature retrieved from validation case (dotted line)

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Fig. 15

Uncertainty in the wall temperature estimation associated to a heat flux pulse of 10,000 W/m2 and an uncertainty in the temperature measurement of 0.5 K

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Fig. 16

(a) Uncertainty associated to the 1% clearance blade tip temperature calculation using sensitivity estimation, (b) uncertainty associated to the 0.1% clearance blade tip temperature calculation using sensitivity estimation, (c) uncertainty associated to the 1% clearance blade tip temperature calculation using Monte Carlo approach, and (d) uncertainty associated to the 0.1% clearance blade tip temperature calculation using Monte Carlo approach

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