Study On Nonlinear Stability Of Two Classical Flow Problems In Fluid Mechanics

The stability of fluid flow is an important research topic in fluid mechanics.The study of flow stability is not only of great significance in weather forecast,hydraulic engineering and aeronautical engineering,but also helps us understand some stability phenomena in life,such as river flow and thermal convection etc.Linear stability method and energy method are two important methods in the study of flow stability.The research on the stability of a system generally starts with the linearized stability method,but this method can only provide necessary conditions for stability of the system.The energy method can provide sufficient conditions for system stability by defining an energy functional.The stability of the plane parallel shear flows is considered in this paper,the nonlinear stability of the system is mainly discussed in two cases: The perturbation is tilted;the fluid is heated and salted from blow.In this paper the nonlinear stability of the plane parallel shear flows with respect to tilted perturbations is investigated by energy method.Tilted perturbations refer to the perturbations that form an angle θ∈(0,π/2))with the direction of the basic flow.By defining appropriate energy functional it is proved that the plane parallel shear flows are unconditionally nonlinearly exponentially stable for tilted streamwise perturbations,when the Reynolds number is below a certain critical value and the boundary conditions are either rigid or stress-free.Further,in the case of stress-free boundaries,by taking the advantage of the poloidal-toroidal decomposition of a solenoidal field to define energy functionals it can be even shown that the plane parallel shear flows are unconditionally nonlinearly exponentiall-y stable for all Reynolds numbers,where the tilted perturbations can be either spanwise or streamwise.Moreover,the nonlinear stability of the plane parallel shear flows in a conduction-diffusion system is also consid-ered.By defining a generalized energy functional it is proved that the plane parallel shear flows are global nonlinearly exponentially stable,moreover,the concentration diffusion has a stabilizing effect on the basic flow.To obtain an accurate solution of the marginal Rayleigh number for global stability we restrict our research to streamwise perturbations.The critical Rayleigh number in this paper have improved the result in reference,and the stability bounds of Reynolds numbers can be explicitly determined for the given values of Rayleigh number.