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We carry out direct numerical simulation (DNS) of flow in a turbulent squareduct by focusing on heat transfer effects, considering the case of unit Prandtlnumber. Reynolds numbers up to $Re_\tau \approx 2000$ are considered which aremuch higher than in previous studies, and which yield clear scale separationbetween inner- and outer-layer dynamics. Close similarity between the behaviorof the temperature and the streamwise velocity fields is confirmed as inprevious studies related to plane channels and pipes. Just like the meanvelocity, the mean temperature is found to exhibit logarithmic layers as afunction of the nearest wall, however with a different slope. The mostimportant practical implication is the validity of the traditional hydraulicdiameter as the correct reference length for reporting heat transfer data, aswe rigorously show here. Temperature and velocity fluctuations also havesimilar behavior, but apparently logarithmic growth of their inner-scaled peakvariances is not observed here unlike in canonical wall-bounded flows. Analysisof the split contributions to the heat transfer coefficient shows that meancross-stream convection associated with secondary motions is responsible forabout $5\%$ of the total. Finally, we use the DNS database to highlightshortcomings of traditional linear closures for the turbulent heat flux, andshow that substantial modeling improvement may be in principle obtained byretaining at least the three terms in the vector polynomial integrity basisexpansion.

Direct numerical simulation of forced thermal convection in square ducts up to $Re_\tau \approx 2000$