Modeling of turbulent fluid flow based on the solution of the Mathieu equation

Abstract

Objective. This work examines the problem of fluid flow simulation in a turbulent regime. The main reasons why turbulent flow occurs are due to the existence of high velocities of movement in fluids; besides that, there may be obstacles or changes in the shapes of flows. Method. To determine the characteristics of a flow, it is proposed to use Mathieu’s equation. The main steps of the solution algorithm are presented for Mathieu functions that were used over the course of the computer program execution. Result. Upon evaluating eigenvalues, it is shown that it is necessary to rely on the corresponding transcendental equations. It is illustrated how modified Mathieu functions are calculated. It is indicated how calculations should be carried out in the case of a small parameter value in Mathieu functions. A block diagram of the executed turbulence modeling algorithm is presented. In order to conduct turbulence simulation, a GUI shell based on the Qt library was created in the Qt Creator environment. During the simulation process, the value of the Smagorinsky constant was chosen to be equal to 0.01. For the time step, a length value of 0.05 was selected. During the execution of the simulation, 2000 time steps were used. The simulation results were recorded every 10 steps. The results of the simulation are presented visually. Conclusion. A mathematical model has been created, on the basis of which there are possibilities for modeling turbulent media. The mathematical model has been constructed for various parameters during the flow of solid bodies by turbulent flows.