Spectral Heat Transfer Coefficient for convection

The present work is aimed at developing a methodology to predict convective heat transfer of flow bounded by a non-isothermal wall, which has been a long-standing issue of interest for Newton's law of cooling. A framework is proposed to centre around the Spectral Heat Transfer Coefficients (SHTC). They are effectively global-to-local temperature-heat flux influence coefficients for a small number of low order spectral modes of wall temperature disturbances for a range of physically relevant and numerically resolvable length scales. The framework is implemented with a unified Fourier spectral method applicable to both spatially periodic and non-periodic wall temperature distributions. The Fourier spectral formulation also leads to the zeroth harmonic base mode corresponding to exactly the standard form of Newton's law of cooling, thus the method is directly applicable to both non-isothermal and isothermal cases, as intended. The method formulations in both 2D and 3D settings and the working procedure are presented. All computational case studies for 2D and 3D configurations clearly and consistently demonstrate the validity and effectiveness of the proposed framework methodology and method implementation.