The steady natural-convection heat transfer in a tilted cylindrical envelope with constant but different end temperatures (300 and 80K) is investigated numerically with the lateral surface being adiabatic. The inner diameter of the envelope is 27.8mm and its length/diameter ratio is 9. This is supposed to be a simplified model for the pulse tube in a pulse-tube cryocooler when the pulse tube is positioned at different orientations. The problem studied is a typical nonlinear one in that the thermophysical properties of the working fluid (helium) vary significant from the hot end to the cold end. Three-dimensional steadystate governing equations are solved with fully variable thermal properties. The high nonlinearity of the problem leads to many special characters of the convergence process, and a very peculiar convergence process is found. Initial-field dependence is also revealed. After quite a few preliminary computations, a series of convergence criteria are proposed. Grid-independence examination is conducted for inclination angle of 110°. It is found that the grid system of 20(r)×20(ψ)×80(Z) with grids in the z direction being nonuniformly positioned can obtain a grid-independent solution. Preliminary computations are conducted for the horizontal position with 70℃ of end temperature difference. The predicted velocity and temperature distributions are compared with available measured data. Good agreement between the predicted and measured results provides strong support for the physical model and numerical treatments developed in this article.