Symmetry analysis of renormalization group approach for analysis of unsteady turbulence

This article presents a mathematical model of hydrodynamic turbulence developed based on renormalization group analysis, which allows for the consideration of the unsteadiness of the flow itself. The renormalization group technique is based on the properties of symmetry transformations. Equations for the kinetic energy of turbulence and the rate of energy dissipation have been derived. Two problems are considered as applications: oscillating flow in a flat channel and accelerating flow in a flat channel under the condition of a variation in the longitudinal pressure gradient according to the Heaviside function. The influence of the renormalization group correction on the characteristics of the turbulent flow is shown. The renormalization group correction caused by the unsteadiness of the turbulence has a significant impact on the flow characteristics at high-frequency regimes. In the case of low pulsation frequencies and in monotonically developing turbulence, the influence of the mentioned correction is practically negligible, and the traditional mathematical model of turbulence can be used for the calculations of such flow. Therefore, the unsteadiness correction should be taken into account during abrupt temporal changes, such as the occurrence of hydrodynamic instabilities of various nature.