Research On Well-posedness And Numerical Simulation Of Anisotropic Navier-Stokes Equations

Navier-stokes equations are the general law of viscous fluid and are of great significance in fluid mechanics.The global existence of smooth solutions to 3D classical Navier-Stokes equations are listed as one of the seven "Millennium Conundrum" problem and are still an open question.In this thesis,we research 3D anisotropic Navier-Stokes equations in theoretical simulation and numerical simulation.In theory,we consider the existence and uniqueness of global solutions of 3D Navier-Stokes equations with horizontal dissipation.By means of energy method,we prove that if the initial value u₀ satisfies ?u₀?H²??₀²,where ?₀ is a sufficiently small positive number,the equation which we considered in this thesis has a unique global solution.Due to lack of the vertical direction in the equation,it is necessary to express the nonlinear in terms vertical direction with the terms in horizontal direction by ?=?×u and ?·u=0.First of all,we prove ?u?L² are small enough,then we obtain a priori estimate of ???L² and ????L² by using the vorticity equation of 3D Navier-Stokes equations with horizontal dissipation.Thus the existence of the small solution is proved.Finally we first assume that there are two small solutions u¹and u²which satisfy the theorem condition.And using the energy method prove that the difference (?)=u¹-u²is zero,that is,the uniqueness of the small solution is proved.In terms of numerical simulation,the first part is to restore the macro equation.Firstly,Taylor expansion and Chapman-Enskog multiscale method are used to restore the evolution equation by adding correction function to the macroscopic 3D anisotropic Navier-Stokes equations,and the relationship between mesoscopic scale and macro scale is established.In this process,the correction function expression which can restore the macro equation is obtained.Secondly,four working conditions are set up,and the 3D anisotropic Navier-Stokes equations and 3D classical Navier-Stokes equations are simulated by D3Q15 LBGK model.We obtain the fluid velocity diagram,and analyze the influence of Reynolds number and the dissipation of that vertical direction is less than horizontal direction.At last,we simulate the 3D anisotropic Navier-Stokes equations in a 3D flume and analyze the fluid flow.