In this paper,we study the nonlinear stability problem for the two-dimensional Boussinesq system around the Couette flow in a finite channel with Navier-slip boundary condition for the velocity and Dirichlet boundary ...In this paper,we study the nonlinear stability problem for the two-dimensional Boussinesq system around the Couette flow in a finite channel with Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature with small viscosityνand small thermal diffusionμ.We establish that if the initial perturbation velocity and initial perturbation temperature satisfy ||u_(0)||H^(2)≤ε_(0) min{μ,ν}1/2, and ||θ_(0)||H^(1)+|||D_(x)|^(1/3)θ_(0)||H^(1)+|||D_(x)|^(1/3)θ_(0)||_(H^(1))≤εi min{μ,ν}^(5/6),for some smallε0 andε1 independent ofμ,ν,then the solution of the two-dimensional NavierStokes Boussinesq system does not transition away from the Couette flow for any time.展开更多
In this paper,we investigate the linear stability/instability of the planar Couette flow to the two-dimensional compressible Euler-Euler system for(ρ,u)and(n,v)with the sound speeds c_(1)and c_(2)respectively,coupled...In this paper,we investigate the linear stability/instability of the planar Couette flow to the two-dimensional compressible Euler-Euler system for(ρ,u)and(n,v)with the sound speeds c_(1)and c_(2)respectively,coupled each other through the drag force on T×R.It is shown in general for the different sound speeds c_(1)≠c_(2)that the perturbations of the densities(ρ,n)and the velocities(u,v)demonstrate the stability in any fixed finite time interval(0,T],besides,excluding the zero mode,the densities(ρ,n)and the irrotational components of the velocities(u,v)obey the algebraic time-growth rates,while the rotational components of the velocities(u,v)exhibit the algebraic time-decay rates due to the inviscid damping.For the case c_(1)=c_(2)(same sound speeds),it is proved that the relative velocity u-v decays faster than those of the velocities u,v caused by the inviscid damping,but the linear instability of the densities(ρ,n)and the irrotational components of the velocities(u,v)is also shown for some large time in spite of the inviscid damping.展开更多
Turbulent spots play a key role in the formation of the turbulence and the transition. The generation and evolution of turbulent spots using the wall impulse model in the plane Couette flow are studied by direct numer...Turbulent spots play a key role in the formation of the turbulence and the transition. The generation and evolution of turbulent spots using the wall impulse model in the plane Couette flow are studied by direct numerical simulation of Navier-Stokes equations. A group of three-dimensional coupling compact difference schemes with high accuracy and high resolution is used in the numerical calculation. The important characteristics of turbulent spots based on the results of examples are analyzed, including the formation of random pulse, the generation of Reynolds stress, the growth of disturbance amplitude, and the continuous change of spot shape, especially the complex evolution process of the streamwise vortices. Computational results confirm that basic properties of turbulent spots in the laminar flow are similar to those in the turbulent flow.展开更多
文摘In this paper,we study the nonlinear stability problem for the two-dimensional Boussinesq system around the Couette flow in a finite channel with Navier-slip boundary condition for the velocity and Dirichlet boundary condition for the temperature with small viscosityνand small thermal diffusionμ.We establish that if the initial perturbation velocity and initial perturbation temperature satisfy ||u_(0)||H^(2)≤ε_(0) min{μ,ν}1/2, and ||θ_(0)||H^(1)+|||D_(x)|^(1/3)θ_(0)||H^(1)+|||D_(x)|^(1/3)θ_(0)||_(H^(1))≤εi min{μ,ν}^(5/6),for some smallε0 andε1 independent ofμ,ν,then the solution of the two-dimensional NavierStokes Boussinesq system does not transition away from the Couette flow for any time.
基金supported by the National Natural Science Foundation of China(11931010,12326613,12331007)the Beijing Scholar Foundation of Beijing Municipal Committeethe key research project of Academy for Multidisciplinary Studies,Capital Normal University。
文摘In this paper,we investigate the linear stability/instability of the planar Couette flow to the two-dimensional compressible Euler-Euler system for(ρ,u)and(n,v)with the sound speeds c_(1)and c_(2)respectively,coupled each other through the drag force on T×R.It is shown in general for the different sound speeds c_(1)≠c_(2)that the perturbations of the densities(ρ,n)and the velocities(u,v)demonstrate the stability in any fixed finite time interval(0,T],besides,excluding the zero mode,the densities(ρ,n)and the irrotational components of the velocities(u,v)obey the algebraic time-growth rates,while the rotational components of the velocities(u,v)exhibit the algebraic time-decay rates due to the inviscid damping.For the case c_(1)=c_(2)(same sound speeds),it is proved that the relative velocity u-v decays faster than those of the velocities u,v caused by the inviscid damping,but the linear instability of the densities(ρ,n)and the irrotational components of the velocities(u,v)is also shown for some large time in spite of the inviscid damping.
文摘Turbulent spots play a key role in the formation of the turbulence and the transition. The generation and evolution of turbulent spots using the wall impulse model in the plane Couette flow are studied by direct numerical simulation of Navier-Stokes equations. A group of three-dimensional coupling compact difference schemes with high accuracy and high resolution is used in the numerical calculation. The important characteristics of turbulent spots based on the results of examples are analyzed, including the formation of random pulse, the generation of Reynolds stress, the growth of disturbance amplitude, and the continuous change of spot shape, especially the complex evolution process of the streamwise vortices. Computational results confirm that basic properties of turbulent spots in the laminar flow are similar to those in the turbulent flow.
