The Discrete Green's Function for Convective Heat Transfer—Part 1: Definition and Physical Understanding

2020 ◽  
Vol 142 (10) ◽  
Author(s):  
John K. Eaton
Keyword(s):  
Heat Transfer ◽  
Green’S Function ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Green's Function ◽  
Plate Boundary ◽  
Direct Calculation ◽  
Discrete Green’S Function ◽  
The Matrix ◽  
Flow Situation

Abstract The discrete Green's function (DGF) is a superposition-based descriptor of the relationship between the surface temperature and the convective heat transfer from a surface. The surface is discretized into a finite number of elements and the DGF matrix elements relate the heat transfer out of any element i to the temperature rise on every element j of the surface. For a given flow situation, the DGF is insensitive to the thermal boundary condition so it allows direct calculation of the heat transfer for any temperature distribution and noniterative solution of conjugate heat transfer problems. The diagonal elements of the matrix are determined solely by the local velocity field while the off-diagonals are determined by the spread of the thermal wake downstream of a heated element. An analytical DGF for the laminar flat-plate boundary layers is included as an example.

Convective Heat Transfer From a Desert Surface

1975 ◽  
Vol 97 (1) ◽  
pp. 104-109 ◽  
Author(s):  
R. T. Bailey ◽  
J. W. Mitchell ◽  
W. A. Beckman
Keyword(s):  
Heat Transfer ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Roughness Parameter ◽  
Direct Calculation ◽  
Soil Surface ◽  
Convection Coefficient ◽  
Atmospheric Flow ◽  
Experimental Heat ◽  
Analytical Relations

The convective heat transfer relations for atmospheric flow over sparsely vegetated areas are reviewed and compared to existing relations for flow in rough ducts. Experimental convection coefficients obtained at four desert sites are compared to the analytical relations. The experimental equipment for measuring the heat transfer convection coefficient between air and ground in a desert environment consists of two electrically heated plates positioned flush with the ground. Measurements of power dissipation and surface temperatures allow direct calculation of the convection coefficient. The experimental heat transfer results are correlated with micrometeorological models from which a soil roughness height is calculated. This roughness parameter is shown to characterize air flow near the soil surface, and may be significantly different from the roughness parameter ordinarily determined from velocity profiles. A simplified heat transfer correlation is presented for desert surfaces.

Engine-Scalable Rotor Casing Convective Heat Flux Evaluation Using Inverse Heat Transfer Methods

2018 ◽  
Vol 141 (1) ◽  
Author(s):  
David Gonzalez Cuadrado ◽  
Francisco Lozano ◽  
Valeria Andreoli ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Green’S Function ◽  
Convective Heat ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Filter Method ◽  
Convective Heat Flux ◽  
Inverse Heat Transfer

In this paper, we propose a two-step methodology to evaluate the convective heat flux along the rotor casing using an engine-scalable approach based on discrete Green's functions . The first step consists in the use of an inverse heat transfer technique to retrieve the heat flux distribution on the shroud inner wall by measuring the temperature of the outside wall; the second step is the calculation of the convective heat flux at engine conditions, using the experimental heat flux and the Green functions engine-scalable technique. Inverse methodologies allow the determination of boundary conditions; in this case, the inner casing surface heat flux, based on measurements from outside of the system, which prevents aerothermal distortion caused by routing the instrumentation into the test article. The heat flux, retrieved from the inverse heat transfer methodology, is related to the rotor tip gap. Therefore, for a given geometry and tip gap, the pressure and temperature can also be retrieved. In this work, the digital filter method is applied in order to take advantage of the response of the temperature to heat flux pulses. The discrete Green's function approach employs a matrix to relate an arbitrary temperature distribution to a series of pulses of heat flux. In this procedure, the terms of the Green's function matrix are evaluated with the output of the inverse heat transfer method. Given that key dimensionless numbers are conserved, the Green's functions matrix can be extrapolated to engine-like conditions. A validation of the methodology is performed by imposing different arbitrary heat flux distributions, to finally demonstrate the scalability of the Green's function method to engine conditions. A detailed uncertainty analysis of the two-step routine is included based on the value of the pulse of heat flux, the temperature measurement uncertainty, the thermal properties of the material, and the physical properties of the rotor casing.

Discrete Green’s Function Measurements in Internal Flows

2005 ◽  
Vol 127 (7) ◽  
pp. 692-698 ◽  
Author(s):  
Charles Booten ◽  
John K. Eaton
Keyword(s):  
Heat Transfer ◽  
Green’S Function ◽  
Green's Function ◽  
Thermal Response ◽  
Turbulent Pipe Flow ◽  
Transfer Coefficients ◽  
Flow Solver ◽  
Numerical Predictions ◽  
Discrete Green’S Function ◽  
Transfer Measurement

The discrete Green’s function (DGF) for convective heat transfer was measured in a fully developed, turbulent pipe flow to test a new technique for simple heat transfer measurement. The 10×10 inverse DGF, G−1, was measured with an element length of approximately one pipe diameter and Reynolds numbers from 30,000 to 100,000 and compared to numerical predictions using a parabolic flow solver called STANTUBE. The advantages of using the DGF method over traditional heat transfer coefficients in predicting the thermal response for flows with nonuniform thermal boundary conditions are also demonstrated.

Conjugate Heat Transfer Analysis Using the Discrete Green's Function

2020 ◽  
Author(s):  
Davis Hoffman ◽  
John Eaton
Keyword(s):  
Heat Transfer ◽  
Steady State ◽  
Green’S Function ◽  
Green's Function ◽  
Conjugate Heat Transfer ◽  
Temperature Fields ◽  
Heat Transfer Analysis ◽  
Transient Heating ◽  
Discrete Green’S Function ◽  
Linear Algebraic Equations

Abstract Conjugate heat transfer problems generally require a coupled solution of the temperature fields in the fluid and solid domains. Implementing the boundary condition at the surface of the solid using a discrete Green's function (DGF) decouples the solutions. A DGF is determined first considering only the fluid domain with prescribed thermal boundary conditions, then the temperature distribution in the solid is calculated using standard numerical methods. The only compatibility requirement is that the DGF must be specified with the same discretization as the surface of the solid. The method is demonstrated for both steady-state and transient heating of a thin plate with laminar boundary layers flowing over both sides. The resulting set of linear algebraic equations for the steady-state problem or linear ordinary differential equations for the transient problem are easily solved using conventional scientific programming packages. The method converges with nearly second-order accuracy as the discretization resolution is increased.

Discrete Green’s Function Measurements in a Single Passage Turbine Model

2005 ◽  
Vol 127 (4) ◽  
pp. 366-377 ◽  
Author(s):  
Debjit Mukerji ◽  
John K. Eaton
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Green’S Function ◽  
Green's Function ◽  
Uniform Heat Flux ◽  
Specific Flow ◽  
Full Field ◽  
Single Passage ◽  
Discrete Green’S Function ◽  
Uniform Heat

The superposition-based Discrete Green’s Function (DGF) technique provides a general representation of convective heat transfer that can capture the numerous flow and thermal complexities of the gas turbine environment and provide benchmark data for the validation of computational codes. The main advantages of the DGF technique are that the measurement apparatus is easier to fabricate than a uniform heat flux or uniform temperature surface, and that the results are applicable to any choice of discretized thermal boundary condition. Once determined for a specific flow condition, the DGF results can be used, for example, with measured surface temperature data to estimate the surface heat flux. In this study, the experimental DGF approach was extended to the suction side blade surface of a single passage model of a turbine cascade. Full-field thermal data were acquired using a steady state, liquid crystal-based imaging technique. The objective was to compute a 10×10 one-dimensional DGF matrix in a realistic turbomachinery geometry. The inverse 1-D DGF matrix, G−1, was calculated and its uncertainties estimated. The DGF-based predictions for the temperature rise and Stanton number distributions on a uniform heat flux surface were found to be in good agreement with experimental data. The G matrix obtained by a direct inversion of G−1 provided reasonable heat transfer predictions for standard thermal boundary conditions.

Engine-Scalable Rotor Casing Convective Heat Flux Evaluation Using Inverse Heat Transfer Methods

2018 ◽  
Author(s):  
David G. Cuadrado ◽  
Francisco Lozano ◽  
Valeria Andreoli ◽  
Guillermo Paniagua
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Green’S Function ◽  
Convective Heat ◽  
Green's Function ◽  
Green's Functions ◽  
Green’S Functions ◽  
Filter Method ◽  
Convective Heat Flux ◽  
Inverse Heat Transfer

In this paper, we propose a two-step methodology to evaluate the convective heat flux along the rotor casing using an engine-scalable approach based on Discrete Green’s Functions. The first step consists in the use of an inverse heat transfer technique to retrieve the heat flux distribution on the shroud inner wall by measuring the temperature of the outside wall; the second step is the calculation of the convective heat flux at engine conditions, using the experimental heat flux and the Green Functions engine-scalable technique. Inverse methodologies allow the determination of boundary conditions, in this case the inner casing surface heat flux, based on measurements from outside of the system, which prevents aerothermal distortion caused by routing the instrumentation into the test article. The heat flux, retrieved from the inverse heat transfer methodology, is related to the rotor tip gap. Therefore, for a given geometry and tip gap, the pressure and temperature can also be retrieved. In this work, the Digital Filter Method is applied in order to take advantage of the response of the temperature to heat flux pulses. The Discrete Green’s Function approach employs a matrix to relate an arbitrary temperature distribution to a series of pulses of heat flux. In the present procedure, the terms of the Green’s Function matrix are evaluated with the output of the inverse heat transfer method. Given that key dimensionless numbers are conserved, the Green’s Functions matrix can be extrapolated to engine-like conditions. A validation of the methodology is performed by imposing different arbitrary heat flux distributions, to finally demonstrate the scalability of the Green’s Function Method to engine conditions. A detailed uncertainty analysis of the two-step routine is included based on the value of the pulse of heat flux, the temperature measurement uncertainty, the thermal properties of the material and the physical properties of the rotor casing.

Discrete Green’s Function Measurements in a Serpentine Cooling Passage

2007 ◽  
Vol 129 (12) ◽  
pp. 1686-1696 ◽  
Author(s):  
Charles W. Booten ◽  
John K. Eaton
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Green’S Function ◽  
Green's Function ◽  
Turbine Blade Cooling ◽  
Discrete Green’S Function ◽  
Rib Turbulators ◽  
Four Square ◽  
Cooling Passage

The inverse discrete Green’s function (IDGF) is a heat transfer coefficient that is valid for arbitrarily complex thermal boundary conditions. It was measured using a rapid experimentation technique in a generic serpentine turbine-blade cooling passage with rib turbulators for Reynolds numbers from 15,000 to 55,000. The model was designed to adhere closely to industry design practice. There were four square cross-section passages with ribs on two opposing walls at 45deg to the main flow. The rib pitch-to-height ratio was 8.5:1 and the blockage ratio was 0.1. The IDGF was measured with an element length of one rib pitch and was used to determine Nusselt numbers that were then compared to the literature. An increase in Nusselt number over thermally fully developed pipe flow of 2.5–3.0 is common in the literature and was consistent with the results in this work. The results showed that the heat transfer coefficient in such complex passages is weakly affected by the thermal boundary condition, which simplifies measurement of this quantity.

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