Three-dimensional Full Euler Flows With Nontrivial Swirl In Axisymmetric Nozzles Through A Symmetric Body

Steady Euler equations are the basic equations of aerodynamics,which mainly describe the motion of ideal fluid in equilibrium state,and the Euler flows in axisymmetric nozzles through obstacle is one of the important types of problem with the significant physical background.It is noticeable that the equations are the elliptic-hyperbolic composite-mixed type,including the linear degenerate characteristics and the fully nonlinear elliptic-hyperbolic characteristics,which could not apply the classical partial differential equation theories directly.Up to now,most of the researches focus on the irrotational flow cases or the two-dimensional symmetric rotational flow case.Comparing with the two-dimensional case,in the three-dimensional case,the structure of streamline and vorticities are more complicated.This thesis studies the three-dimensional axisymmetric airfoil nozzles problem which models the jet engine and the wind tunnel test,and proves the well-posedness and the far field convergence rates.First,by introducing the stream function Ψ,for the nondegenerate streamlines,the streamline conservations:B,S and rW are represented by stream function,which solves the linear degenerate characteristics.When(?)less than the sonic speed,the fully nonlinear characteristics are elliptic.It is worth to point out:in the two-dimensional case,the fully nonlinear characteristics are elliptic if and only if the flow is subsonic,while in our case the fully nonlinear characteristics may still be elliptic even when the flow is supersonic.By the relationship between the vorticities and streamline conservations,we transform the problem of steady compressible Euler equations into the boundary value problem of the second-order quasi-linear equation of stream function Ψ.Then,we consider the well-posedness of elliptic equation and nondegenerate of streamlines.Due the domain in unbounded along the x-axis,we introduce the finite length cut-off and show the existence of ψ.Since at r=0,the equation may degenerate,height lifting close to r=0 is employed,which leads the existence in the finite domain.By the uniform estimate in the homogeneous H?lder space,the non-decency of the problem is shown.For the non-backwardness,we start from the irrotational case,and hire the continuity method to show nondegenerate for the rotational case.For corner points at boundaries,a set of smooth boundary is used to approximate the region near the corner.By constructing barrier function,it is proved that the approximate solution is uniformly bounded,and the well-posedness of the solution with corner region is obtained.And the uniqueness is proved by the energy method.Finally,the Bers technique leads to the well-posedness up to the critical mass flux.Base on the above well-posedness theorem,the far field convergence rates of the solutions are studied.If the infinite nozzles are the flat boundary outside the finite length,the solution of the equation converges to an asymptotic state at the exponential rate.If the infinite nozzles converge to the flat boundary with the polynomial rates,the solutions converge to the asymptotic states at the same polynomial rates.Finally,the critical mass limit is shown by the compensated compactness method.And,the above research is extended to the corresponding steady incompressible Euler equation.By constructing a sequence of adiabatic indexes,it is proved that the weak solution of the compressible Euler equation converges to the weak solution of the incompressible Euler equation.The main innovations of this thesis are as follows.First,it is the first result on the three-dimensional compressible Euler flow with more than one nonzero and large vorticity through obstacle.Secondly,we improved conditions at inlet for solutions with large vorticity.Finally,the convergence rate of Euler flows is obtained under more general conditions,even including the supersonic flows.This thesis extends the research on the well-posedness and convergence rates of the steady Euler equation on the airfoil surface problem.By introducing innovative methods and techniques,the fine properties of the solution of the Euler equation are explored.This study improves theoretical support for subsequent theoretical exploration and solving problems in practical applications.