Modeling And Dynamics Analysis Of Optimal Dynamical Systems Of Incompressible Navier-Stokes Equations

Based on the optimal dynamical system modeling theory,this paper has established an optimal dynamical system modeling theory adapted to arbitrary speed boundary conditions and incompressible conditions.The main results are as follows:1.By satisfying the boundary conditions in physical space,a Galerkin spectral method adapted to arbitrary velocity boundary conditions was constructed;furthermore,by introducing orthogonality objective functionals and incompressible objective functionals,the optimal dynamical system modeling theory for 3D incompressible Navier-Stokes equations was established,which is adapted to arbitrary velocity boundary conditions and velocity incompressible conditions.In the optimization process,a multi-scale global optimization method was used to search for approximate global optimal basis functions.Based on this modeling theory,an optimal dynamical system model for the flow around a single square column was established,and the flow field and dynamics characteristics of the flow were obtained,including phase space orbit,Poincaré section,Lyapunov exponent set,bifurcation diagram,power spectrum,etc.It is found that the long-term dynamics behavior of that optimal dynamical system is a limit cycle.2.For more complicated flows,such as the flow around two square columns,some important dynamics characteristics are contained in the pulsating flow field.Therefore,the above modeling theory was extended to the fluctuation velocity equations of the Navier-Stokes equations,and the optimal dynamical system modeling theory of the fluctuation velocity equations of the Navier-Stokes equations was established.Based on this modeling theory,by modeling the flow around two square columns and analysing its dynamics characteristics,it is found that with the increase of the Reynolds number,the pulsating optimal dynamical system has an intermittent period-like bifurcation characteristic.3.For extremely complicated flow problems,the treatment of velocity incompressibility condition is very tricky.It is necessary to use the velocity basis function and the pressure basis function to make the velocity incompressibility condition satisfied.Therefore,based on the aforementioned modeling theory,by introducing the pressure basis function,the optimal dynamical system modeling theory of the Navier-Stokes equations with pressure basis was obtained.In this modeling theory,the continuity equation was projected on the pressure basis function and the momentum equations were projected on the velocity basis function.These two kind of projection equations are combined to obtain the optimal dynamical system model of the NavierStokes equations.Modeling and dynamics analysis of the flow around three square columns were conducted by using this modeling theory.It is found that the long-term dynamics behavior of the optimal dynamical system is chaotic,indicating that the dynamics characteristics of the flow around three square columns is extremely complicated.It can be seen that the complexity of wake flow can be increased by multi-column flow,thereby promoting fluid mixing.In summary,the modeling theory of optimal dynamical system adapted to arbitrary velocity boundary conditions and incompressible conditions is of great significance to deeply explore the complex dynamics characteristics inherent in the Navier-Stokes equations and reveal the physical nature of complex flows.