Modern methods of mathematical modeling of the development of hydrodynamic instabilities and turbulent mixing

The study of the development of perturbations under the influence of various hydrodynamic instabilities, as well as the transition to turbulent mixing and turbulence, has been a subject of considerable interest over the past decades. This is primarily due to the importance of these phenomena for various fields of science and technology. It should be noted, that the final results of the turbulent flows characteristics study have not yet been obtained. This fact stimulates a great interest in this topic, both in sense of physical theory and in sense of approaches to mathematical modeling and numerical methods development. The capabilities of modern computer technology make it possible to carry out numerical experiments in two-dimensional and three-dimensional settings, and to analyze the features of new numerical methods. To date, there are a huge number of methods and their modifications are applied in practice. This review is devoted to the most promising, according to the authors, of them.