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

Abstract

The complex heat and mass transfer across the turbine tip gap requires a detailed analysis which cannot be expressed using the classical Newton heat convection approach. In this portion of the turbine, characterized by tight moving clearances, pressure gradients are counterbalanced by viscous effects. Hence, non-dimensional analysis, based on the boundary layer, is inadequate and therefore the use of an adiabatic wall temperature is questionable. In this paper, we propose an alternative approach to predict the convective heat transfer problem across the turbine rotor tip using Discrete Green Functions. The linearity of the energy equation can be applied with a superposition technique to measure the data extracted from flow simulations to determine the Green’s function distribution. The Discrete Green Function is a matrix of coefficients that relate the increment of temperature observed in a surface with the heat flux integrated on the same surface. These coefficients are independent on the inlet temperature of the flow and are associated to the geometry. The controlled surface is discretized into cells and each cell is associated to a vector of coefficients. The Discrete Green Function coefficients are calculated using the temperature response of the cell to a heat flux pulse imposed at different locations. The methodology was previously applied to a backward facing step to prove its validity. Several simulations were performed applying a representative pulse of heat flux in different locations in the bottom wall of the backward facing step. From these simulations, the increment of temperature in each node of the geometry was retrieved and the Discrete Green Function coefficients associated to the bottom wall were calculated. A numerical validation was performed imposing a random pattern of heat flux and predicting the increment of temperature on the bottom wall under different inlet flow conditions. 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 a 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 Discrete Green Function coefficients. A validation of the rotor tip heat flux was accomplished by imposing different heat flux distributions in the studied region. Ultimately, a detailed uncertainty analysis of the methodology was included based in the magnitude of the heat flux pulses used in the Discrete Green Function coefficients calculation and the uncertainty in the experimental measurements of the wall temperature.