Non-linear instability analysis of the three-dimensional Navier-Stokes equations: Taylor-Green vortex problem

摘要

The three-dimensional (3D) Taylor-Green vortex (TGV) flow is one of the simplest systems to study the generation of different scales of vortices due to the growth of disturbances via effects of different physical mechanisms, including vortex-stretching as an additional source for instability, showing not only the creation of turbulence but also turbulent decay. The strong anisotropic and well-organized flow becomes unstable at early time due to transfer of energy to small scales. The analysis of instability of the periodic 3D TGV flow for the Reynolds number of Re = 2000 is reported here. The direct numerical simulation of the periodic 3D TGV flow is carried out using high accuracy numerical methods for the (vector-potential, vorticity)-formulation, which exactly satisfy the solenoidality condition for vector-potential and vorticity in the computational domain. The evolution of disturbances is examined using the instability theories of the disturbance mechanical energy of the Navier-Stokes equation and the role of rotationality by the disturbance enstrophy transport equation (DETE), which is derived from the enstrophy transport equation. The 3D TGV flow exhibits a tornado-type structure at the center of the domain at intermediate stages of transition to turbulence, which is analyzed using the vortex-identification method of lambda (2)-criteria and the DETE method, as described by Sengupta et al. ["Tracking disturbances in transitional and turbulent flows: Coherent structures," Phys. Fluids 31(12), 124106 (2019)]. Here, it is observed that the coherent structure is diffused in the lambda (2)-contours. Third generation vortex-identification methods are analyzed for capturing the transient, rotating vortex. The combination of new Omega- and the Liutex/Rortex-methods, as reviewed by C. Liu et al. ["Third generation of vortex identification methods: Omega and Liutex/Rortex based systems," J. Hydrodyn. 31(2), 205-223 (2019)], captures the evolution of the transient vortex, but the structure identified by these methods appears to be diffused, while the DETE method clearly captures the vortex geometry and highlights the formation of the vortex at early times to aid in predicting the flow evolution.