The present study first describes the turbulence and turbulent scalar fields modified by the temperature-dependent variable viscosity in a water channel flow at Re_τ =300-650 by performing direct numerical simulations (DNSs) and discusses the effects of correlation terms between the sub-grid-scale velocity and viscosity variance. The filtered DNS data suggest that effects of the viscosity-strain correlation term are ignorable, and thus a large eddy simulation (LES) without its modelling can capture the essential characteristics of variable viscosity turbulence. To provide further reference data, a wall resolved LES is then performed at a higher Reynolds number of Re_τ =1300 considering that the viscosity-strain correlation has minor effects. Referring to these data, we propose a non-equilibrium wall-modelled LES method by introducing a temperature-dependent near-wall layer for the viscosity. In the method, with a simple model function for the viscosity the steep near-wall variation of the temperature-dependent viscosity is described. We then integrate the filtered thin boundary layer equations for the momentum and the temperature to construct an algebraic non-equilibrium wall stress model. The simulation results show that the proposed model successfully reproduces the mean velocity and temperature profiles owing to the temperature-dependent variable viscosity.