This paper presents an advection-diffusion model that describes normal grain growth in 'size-sides' space. Ordinary differential equations governing the self-similar distributions at the steady state of normal grain growth are derived. By solving numerically the continuity equations (time-dependent) and the corresponding ordinary differential equations (time-independent), we get the self-similar grain size distributions in time-dependent and time-independent form. The two sets of distributions have nearly the same shape, confirming the self-consistency of the model. Some comparisons with other models and simulations are given.