An extended and improved particle‐spectral method for analysis of unsteady inviscid incompressible flows through a channel of finite length

Abstract

AbstractThis paper addresses the development of a method to simulate and analyze unsteady flows of incompressible and inviscid fluids in rectangular channels. Mathematically, the flow problem formulates as Euler equations in terms of the stream function and vorticity. The normal velocity and the vorticity are given at the inlet of the channel, and only the normal velocity is specified at the outlet. The particle‐spectral method describes the vorticity field change in a Lagrangian sense by dynamics of marker particles and an Eulerian description of the velocity field using the global Galerkin approximation. The method involves the subdivision of flow domain on rectangular cells; the approximation of vorticity distribution in cells by least‐squares method; Galerkin coefficients calculation using analytical integration on space variables. For time integration, the five‐stage Runge‐Kutta method of the third accuracy order and the sixth one of symplecticity with adaptive step size control uses. The extension of the particle‐spectral approach comprises algorithms for fluid flows qualitative analysis jointly with the time simulation of unsteady flows. This includes the trajectories of marker particles calculation, computing the particle's residence time in the channel, kinetic energy dynamics, and algorithms for analyzing the dynamics and mixing of vortex patches. Test calculations based on reference solutions showed the effectiveness and good accuracy of the particle‐spectral approach. Using the developed method the dynamics of one and two initial vortex patches in the channel were investigated. We found the final state of their long‐time dynamics is time‐periodic or quasi‐periodic movements. The application of qualitative analysis algorithms demonstrates these final states produce a chaotic scattering of fluid particles in the flow‐through zone of the channel.