Effects of temperature-dependent material properties in coldwater Rayleigh-B\'enard convection

Abstract

Water exhibits an anomalous nonlinear density equation of state (EOS) as it approaches freezing, along with an increase in viscosity ($\mu$) and a decrease in thermal conductivity ($k$). Here we ask: how do these temperature-dependent material properties affect thermal convection in coldwater? We examine these effects within the canonical Rayleigh-B\'enard convection (RBC) system, performing direct numerical simulations with coldwater between $0^{\circ}$C and the temperature of maximum density $3.98^{\circ}$C. To disentangle the effects of these properties, we consider two main cases: RBC with a quadratic EOS and constant $\mu$ and $k$; and RBC with a quadratic EOS and temperature-dependent $\mu(T)$ and $k(T)$. We show that the most significant consequence is a shift in the mean fluid temperature relative to `standard RBC'. The nonlinear EOS and the temperature-dependent $\mu$ and $k$ drive this shift in opposite directions, with the former dominating and producing a net decrease in the mean temperature. This decrease varies with the Rayleigh number $Ra$, reaching up to $0.08^{\circ}$C at $Ra=10^{8}$. Additionally, the onset of convection is affected, but the influence from the EOS and $\mu(T)$ and $k(T)$ compensate, resulting in a negligible net effect. Despite these changes, the Nusselt and the Reynolds numbers follow classical scalings in all cases. Our results establish how the anomalous material properties of coldwater affect local and global features of convection, evidencing the implications of model selection for studies of cryospheric water bodies.