This study investigates turbulence structures in steady and non-uniform flows. Equations of Reynolds shear stress and turbulent velocity fluctuations are derived and their physical interpretations are explained. The theoretical results show that, different from previous studies, the variation of water surface can generate the wall-normal velocity, resulting in deviations of Reynolds shear stress and turbulence intensities from those in uniform flows. A self-similarity relationship is found between the Reynolds shear stress and turbulence intensities in non-uniform flows. The existence of self-similarity indicates that the effect of non-uniformity does not influence the mixing length. An empirical equation has been proposed to express the relationship based on experimental data available in the literature. Good agreement is achieved between the measured and predicted turbulence intensities by applying the self-similarity relationship.