The transient natural convection of a mass of water confined within a closed cavity with wall temperature decreasing at a steady rate is considered. For situations where a linear density-temperature relationship can be assumed, a quasi-steady state following an initial transient may be reached, provided that the cooling rate applied to the wall is held constant long enough. Steady-state flow characteristics in the case of a specific geometry are functions of a single parameter, the Rayleigh number, in which a dimensionless temperature, based on the cooling rate, is used. For the particular case of water cooled through 4 °C, the temperature at which maximum density occurs, a linear variation of density with respect to temperature is no more acceptable. However, it can be assumed that a linear relationship between the water thermal-expansion coefficient and the temperature is valid in the neighbourhood of 4 °C. With such an assumption it is still possible to characterize the cooling process that follows the initial transient by a single parameter. Detailed numerical results are presented for the particular case of a square cavity. Existing experimental and numerical results for the case of a horizontal circular pipe are also discussed.