The problem of the joint flow of a turbulent gas stream and a vertically falling wavy liquid film is considered. Tangential and normal stresses on the interfaces are calculated. The components of the Reynolds stress tensor are determined within the framework of the Boussinesq hypothesis. For the case of small Reynolds numbers of a liquid, the problem is reduced to a nonlinear integro-differential equation for the deviation of the layer thickness from the unperturbed level. A numerical study of the evolution of periodic perturbations is carried out. Several typical scenarios of their development are presented.