Conjugate heat transfer in turbulent pipe flow with azimuthally non-uniform heating

Turbulent conjugate heat transfer (CHT) in pipes with azimuthally non-uniform heating is studied theoretically and numerically with a focus on low-$\mathrm{P}\mathrm{r}$ fluids. While the influence of the solid properties is well documented for uniform heat flux, their effect under azimuthally non-uniform heating has not been explored systematically. We address this gap by combining spectral analysis of the problem – which reduces arbitrary boundary conditions to a discrete set of azimuthal wavenumbers – with well-resolved numerical simulations of thermally and hydraulically fully developed turbulent pipe flow at ${\mathrm{R}\mathrm{e}}_{\tau }\approx 180$ for $\mathrm{P}\mathrm{r}=\{0.025,0.71\}$, representative of liquid metals and air. Two principal results emerge: (i) the global Nusselt number is independent of the azimuthal heat-flux distribution and exhibits only a minor dependence on the solid properties, resolving an open question for low-Prandtl-number fluids; (ii) CHT nevertheless strongly affects the azimuthally resolved mean temperature at low $\mathrm{P}\mathrm{r}$, where azimuthal conduction in the solid homogenises the interface heat flux, and governs the level of temperature fluctuations through the effusivity ratio (thermal activity ratio) $K$. Finally, an analysis of the turbulent Prandtl number ${\mathrm{P}\mathrm{r}}_t$ shows a strong anisotropy, particularly near the wall, and a dependence on the azimuthal wavenumber.