Numerical study of supercritical carbon-dioxide flows across the critical point is presented. The present numerical method is based on the preconditioning method developed by Yamamoto and mathematical models of thermophysical properties for carbon dioxide programmed in the program package for thermophysical properties of fluids, developed by Kyushu University. First, the two-dimensional natural convection of carbon dioxide between two parallel plates is calculated while changing the bulk pressure. The calculated thermophysical properties of the carbon-dioxide flow under supercritical pressure are compared with those in a gas condition. Next, the natural convection of carbon dioxide in an O-shaped cyclic channel is calculated, and the effect of the density difference induced by the phase change to the flow is investigated. For application to high-speed flows, supercritical carbon dioxide flows through a nozzle with free-jet expansion (known as the process of rapid expansion of supercritical solutions) process are calculated. The calculated shock distance to the Mach disk generated in the free jet is compared with experiments and the density variations in the nozzle while changing the inlet temperature are numerically predicted. (C) 2010 Elsevier Ltd. All rights reserved.