Chaos and turbulence are two important topics in nonlinear dynamics. In this study, two problems related to chaos and turbulence modelling are presented. They are the chaotic vibration phenomenon in high-dimensional partial differential equations and the emergence of the Navier-Stokes-alpha model for channel flows. The study of the chaotic vibration phenomenon in high-dimensional partial differential equations is explained from both the numerical and theoretical aspects. In the numerical perspective, we have studied the chaotic vibration phenomenon of the 2D wave equation through numerical simulations. Based on the finite-volume method, we have built our own solver "img2Foam" in the Computational Fluid Dynamics software OpenFOAM (Open source Field Operation and Manipulation). We have implemented several numerical simulations containing both chaotic and non-chaotic cases. As for the theoretical perspective, we give a rigorous proof for the chaotic vibration phenomenon of the 2D non-strictly hyperbolic equation. After introducing two linear operators, the initial system of the 2D non-strictly hyperbolic equation is converted into a system of two coupled first order equations. By using the method of characteristics, we have found the explicit solution formulas of the new system. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs by applying the period-doubling bifurcation theorem. Numerical simulations are presented to validate the theoretical results. Inspired by the concept of the regular part of the weak attractor of the 3D Navier- Stokes equations, we concentrate on a restricted class of fluid flows to explore the transition from the Navier-Stokes equations to the Navier-Stokes-alpha model for channel flows. The Navier-Stokes equations have been widely used to describe the motion of viscous incompressible fluid flows. As an averaged version of the Navier- Stokes equations, the Navier-Stokes-alpha model has solid mathematical properties as well as reliable experimental matches. Therefore, the Navier-Stokes-alpha model is taken as an approximation for the dynamics of appropriately averaged turbulent fluid flows. We are interested in finding a connection between Navier-Stokes equations and the Navier-Stokes-alpha model in terms of the physical properties of the fluid flow. Given the hypothesis that the turbulence described by the Navier-Stokes-alpha model was partly due to the roughness of the walls, the transition from the Navier-Stokes equations into the Navier-Stokes-alpha model is presented by introducing a Reynolds type averaging.
Chaos and turbulence are two important topics in nonlinear dynamics. In this study, two problems related to chaos and turbulence modelling are presented. They are the chaotic vibration phenomenon in high-dimensional partial differential equations and the emergence of the Navier-Stokes-alpha model for channel flows.
The study of the chaotic vibration phenomenon in high-dimensional partial differential equations is explained from both the numerical and theoretical aspects. In the numerical perspective, we have studied the chaotic vibration phenomenon of the 2D wave equation through numerical simulations. Based on the finite-volume method, we have built our own solver "img2Foam" in the Computational Fluid Dynamics software OpenFOAM (Open source Field Operation and Manipulation). We have implemented several numerical simulations containing both chaotic and non-chaotic cases. As for the theoretical perspective, we give a rigorous proof for the chaotic vibration phenomenon of the 2D non-strictly hyperbolic equation. After introducing two linear operators, the initial system of the 2D non-strictly hyperbolic equation is converted into a system of two coupled first order equations. By using the method of characteristics, we have found the explicit solution formulas of the new system. We have also found a regime of the parameters when the chaotic vibration phenomenon occurs by applying the period-doubling bifurcation theorem. Numerical simulations are presented to validate the theoretical results.
Inspired by the concept of the regular part of the weak attractor of the 3D Navier- Stokes equations, we concentrate on a restricted class of fluid flows to explore the transition from the Navier-Stokes equations to the Navier-Stokes-alpha model for channel flows. The Navier-Stokes equations have been widely used to describe the motion of viscous incompressible fluid flows. As an averaged version of the Navier- Stokes equations, the Navier-Stokes-alpha model has solid mathematical properties as well as reliable experimental matches. Therefore, the Navier-Stokes-alpha model is taken as an approximation for the dynamics of appropriately averaged turbulent fluid flows. We are interested in finding a connection between Navier-Stokes equations and the Navier-Stokes-alpha model in terms of the physical properties of the fluid flow. Given the hypothesis that the turbulence described by the Navier-Stokes-alpha model was partly due to the roughness of the walls, the transition from the Navier-Stokes equations into the Navier-Stokes-alpha model is presented by introducing a Reynolds type averaging.