Numerical analysis has been carried out to investigate transient turbulent convective heat transfer in a vertical tube for supercritical water near the thermodynamic critical point. Heat transfer and fluid flow in the tube we strongly coupled due to the large variations of thermodynamic and transport properties such as density, specific heat, and turbulent viscosity. As pressure in the tube approaches to the critical pressure, the properties variation with time becomes larger. Heat transfer coefficient rapidly decreases along the tube near the pseudocritical temperature at the tube wall for $P_R<1.2$. Stanton number variation with time is largely reduced in the region of gas-like phase in comparison with Nusselt number. Turbulent viscosity ratio close to the wall increases near the pseudocritical temperature and it gradually decreases with time.