A Graetz-type thermal-response superposition model for turbulent heat transfer in smooth square channels under non-uniform wall conditions

Non-uniform distributions of wall temperature or wall heat flux in internal convection generate streamwise thermal-history effects that conventional local heat-transfer-coefficient (HTC) closures do not explicitly capture. This study develops a Graetz-type streamwise thermal-response superposition (TRS) model for turbulent flow in stationary smooth square channels. A hydraulic-diameter-scaled response kernel retains upstream thermal memory and supports two complementary formulations: the T -route uses T w ( x ) to predict q w ( x ) , whereas the q -route uses q w ( x ) to predict T w ( x ) ; both also provide the bulk temperature T b ( x ) , heat-transfer coefficient h ( x ) , and Nusselt number N u D ( x ) . The model is validated against three-dimensional square-channel CFD for prescribed non-uniform wall histories, one-sided active-wall configurations, and a counterflow coupled-wall case. For prescribed linear wall-temperature and wall-heat-flux histories and one-sided active-wall cases, the downstream ( x / D h > 3.5 ) root-mean-square (RMS) errors in N u D are 0.6%–4.4%; the corresponding full-interval errors are 4.7%–8.6%, with the largest deviations concentrated in the inlet-sensitive thermal-entry region. Relative to a streamwise-invariant local HTC closure, TRS reduces downstream RMS errors by 48%–87% for the prescribed linear histories. Gaussian-history tests further isolate the wall-history contribution: the downstream accuracy gain over an entry-aware local comparator increases from 0.84–0.89 to 5.57–6.43 percentage points as the imposed intensity increases. In the coupled-wall case, both routes yield downstream N u D RMS errors below 2.5% and aggregate CPU-time speedups of 5.30 × 1 0 3 – 7.70 × 1 0 3 relative to the present coupled-CFD implementation. A case-calibrated cool-side variable-property correction, K vp ( T R ) = T R − 0.317 with T R = T w / T b , in , reduces downstream cool-side N u D , 1s , c RMS errors from 21.44%–22.06% to 0.74%–0.94% in the corresponding variable-property counterflow conjugate heat-transfer (CHT) test. The proposed model therefore provides a streamwise-causal reduced closure for repeated evaluations in partitioned smooth-channel CHT analysis.