Subsonic Euler Flows Research Articles

Subsonic Euler flows in a three-dimensional finitely long cylinder with arbitrary cross section

Subsonic Euler flows in a three-dimensional finitely long cylinder with arbitrary cross section

Structural stability and optimal convergence rates of subsonic Euler flows with large vorticity in infinitely long nozzles

In this paper, we establish the structural stability and optimal convergence rates of non-isentropic subsonic Euler flows with large vorticity in two-dimensional infinitely long nozzles. By applying the stream function formulation for compressible Euler equations, Euler equations are equivalent to a quasilinear second order elliptic equation of the stream function for subsonic flow. Then, the key points to prove the structural stability of subsonic flows with large vorticity under small perturbations of nozzle boundaries are the standard estimates of elliptic equations. Furthermore, using the maximum principle and the choice of compared functions, we obtain the optimal convergence rates of subsonic flows at far fields.

Existence of global steady subsonic Euler flows with collision through 2D infinitely long nozzles

In this paper, we study the global existence of steady subsonic flows with collision, where the collision is caused by a confluence of two semi‐infinitely incoming flows that are nonmiscible steady subsonic irrotational Euler flows come from two different infinitely nozzles. First, we prove that when the total flux of two incoming flows is less that the critical mass flux, there exists a unique global smooth subsonic flow with collision. Meanwhile, the interface between two flows is a smooth free interface, which is determined uniquely by the mass fluxes of incoming flows. Second, by using the blowup argument, we establish the asymptotic behaviors for the stream function. Finally, we prove that the position of free interface can be determined uniquely by the mass fluxes of incoming flows. Moreover, we establish the monotonicity of the relation between the position of free interface and the mass fluxes of incoming flows.

The optimal convergence rates of non‐isentropic subsonic Euler flows through the infinitely long three‐dimensional axisymmetric nozzles

This paper is concerned about the optimal convergence rates of non‐isentropic subsonic flows at far fields in three‐dimensional infinitely long axisymmetric nozzles. By using the stream function formulation for the compressible Euler equations, the subsonic Euler flows are equivalent to a quasilinear elliptic equation of the stream function. The key points to prove the convergence rates of subsonic flows at far fields are the choice of compared functions and the maximum principles.

Global uniqueness of steady subsonic Euler flows in three-dimensional ducts

ABSTRACTWe show for certain boundary conditions, under suitable assumptions on the velocity field, the steady compressible Euler flows in three-dimensional straight ducts must be irrotational, and hence, prove global uniqueness of uniform subsonic flows in these ducts. The proof depends on careful analysis of transport equations of vorticity and theory of second-order elliptic equations in bounded or unbounded domains.

Subsonic Euler Flows with Large Vorticity Through an Infinitely Long Axisymmetric Nozzle

This paper is a sequel to the earlier work Du and Duan (J Diff Equ 250:813–847, 2011) on well-posedness of steady subsonic Euler flows through infinitely long three-dimensional axisymmetric nozzles. In Du and Duan (J Diff Equ 250:813–847, 2011), the authors showed the existence and uniqueness of the global subsonic Euler flows through an infinitely long axisymmetric nozzle, when the variation of Bernoulli’s function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The smallness of the variation of Bernoulli’s function in the upstream prevents the attendance of the possible singularity in the nozzles, however, at the same time it also leads that the vorticity of the ideal flow is sufficiently small in the whole nozzle and the flows are indeed adjacent to axisymmetric potential flows. The purpose of this paper is to investigate the effects of the vorticity for the smooth subsonic ideal flows in infinitely long axisymmetric nozzles. We modify the formulation of the problem in the previous work Du and Duan (J Diff Equ 250:813–847, 2011) and the existence and uniqueness results on the smooth subsonic ideal polytropic flows in infinitely long axisymmetric nozzles without the restriction on the smallness of the vorticity are shown in this paper.

Two Dimensional Subsonic Euler Flows Past a Wall or a Symmetric Body

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal velocity in the upstream, there exists a critical value \({\rho_{\rm cr}}\) such that if the incoming density in the upstream is larger than \({\rho_{\rm cr}}\), then there exists a subsonic flow past a wall. Furthermore, \({\rho_{\rm cr}}\) is critical in the sense that there is no such subsonic flow if the density of the incoming flow is less than \({\rho_{\rm cr}}\). The subsonic flows possess large vorticity and positive horizontal velocity above the wall except at the corner points on the boundary. Moreover, the existence and uniqueness of a two dimensional subsonic Euler flow past a symmetric body are also obtained when the incoming velocity field is a general small perturbation of a constant velocity field and the density of the incoming flow is larger than a critical value. The asymptotic behavior of the flows is obtained with the aid of some integral estimates for the difference between the velocity field and its far field states.

A new formulation for the 3-D Euler equations with an application to subsonic flows in a cylinder

In this paper, a new formulation for the three dimensional Euler equations is derived. Since the Euler system is hyperbolic-elliptic coupled in a subsonic region, so an effective decoupling of the hyperbolic and elliptic modes is essential for any development of the theory. The key idea in our formulation is to use the Bernoulli’s law to reduce the dimension of the velocity field by defining new variables (1,β2 = u 2 u1 ,β3 = u3 u1 ) and replacing u1 by the Bernoulli’s function B through u 2 = 2(B h(�)) 1+�2+�2 . We find a conserved quantity for flows with a constant Bernoulli’s function, which behaves like the scaled vorticity in the 2-D case. More surprisingly, a system of new conservation laws can be derived, which is new even in the two dimensional case. We use this new formulation to construct a smooth subsonic Euler flow in a rectangular cylinder, which is also required to be adjacent to some special subsonic states. The same idea can be applied to obtain similar information for the 3-D incompressible Euler equations, the self-similar Euler equations, the steady Euler equations with damping, the steady Euler-Poisson equations and the steady Euler-Maxwell equations. 2010 Mathematics Subject Classification:35Q31; 35Q35;76G25.

Three dimensional non-isentropic subsonic Euler flows in rectangular nozzles

In this paper, we study the existence and uniqueness of three dimensional non-isentropic subsonic Euler flows in rectangular nozzles. This work is an extension of Chen and Xie’s work on isentropic subsonic Euler flows. If, besides small normal component of vorticity, Bernoulli’s function and entropy function with small variations are given on the entrance, the existence and uniqueness of non-isentropic subsonic Euler flows are established.

Three dimensional steady subsonic Euler flows in bounded nozzles

The existence and uniqueness of three dimensional steady subsonic Euler flows in rectangular nozzles were obtained when prescribing normal component of momentum at both the entrance and exit. If, in addition, the normal component of the voriticity and the variation of Bernoulli's function at the entrance are both zero, then there exists a unique subsonic potential flow when the magnitude of the normal component of the momentum is less than a critical number. As the magnitude of the normal component of the momentum approaches the critical number, the associated flows converge to a subsonic–sonic flow. Furthermore, when the normal component of vorticity and the variation of Bernoulli function are both small, the existence and uniqueness of subsonic Euler flows with non-zero vorticity are established. The proof of these results is based on a new formulation for the Euler system, a priori estimate for nonlinear elliptic equations with nonlinear boundary conditions, detailed study for a linear div–curl system, and delicate estimate for the transport equations.

Steady Subsonic Ideal Flows Through an Infinitely Long Nozzle with Large Vorticity

In this paper, the existence, uniqueness, and far field behavior of a class of subsonic flows with large vorticity for the steady Euler equations in infinitely long nozzles are established. More precisely, for any given convex horizontal velocity of incoming flow in the upstream, there exists a critical value mcr, if the mass flux is larger than mcr, then there exists a unique smooth subsonic Euler flow through the infinitely long nozzle. This well-posedness result is proved by a new observation for the method developed in Xie and Xin (SIAM J Math Anal 42:751–784, 2010) to deal with the Euler system. Furthermore, the optimal convergence rates of the subsonic flows at far fields are obtained via the maximum principle and an elaborate choice of the comparison functions.

On subsonic Euler flows with stagnation points in two-dimensional nozzles

On subsonic Euler flows with stagnation points in two-dimensional nozzles

Subsonic Euler flows in a divergent nozzle

We characterize a class of physical boundary conditions that guarantee the existence and uniqueness of the subsonic Euler flow in a general finitely long nozzle. More precisely, by prescribing the incoming flow angle and the Bernoulli’s function at the inlet and the end pressure at the exit of the nozzle, we establish an existence and uniqueness theorem for subsonic Euler flows in a 2-D nozzle, which is also required to be adjacent to some special background solutions. Such a result can also be extended to the 3-D asymmetric case.

Existence of steady subsonic Euler flows through infinitely long periodic nozzles

In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles which are periodic in x1-direction with the period L. It is shown that when the variation of Bernoulli function at some given section is small and mass flux is in a suitable regime, there exists a unique global subsonic flow in the nozzle. Furthermore, the flow is also periodic in x1-direction with the period L. If, in particular, the Bernoulli function is a constant, we also get the existence of subsonic-sonic flows when the mass flux takes the critical value.

Note on the uniqueness of subsonic Euler flows in an axisymmetric nozzle

In this paper, we consider the uniqueness of globally subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and satisfies no-flow boundary conditions on the nozzle walls. We will show that for given mass flux and Bernoulli’s function in the upstream, the subsonic flow is unique in the class of all axisymmetric solutions, which possess the asymptotic behaviors at the far fields. This result extends the uniqueness of solutions in the previous paper Du and Duan (2011) [1].

Uniqueness of symmetric steady subsonic flows in infinitely long divergent nozzles

We prove that the spherically symmetric subsonic flows in an infinitely long straight divergent nozzle with arbitrary smooth cross-section are unique for the three-dimensional steady potential flow equation. The proof depends on an extreme principle for elliptic equations in an unbounded conical domain, under the assumption that the gradient of the solution is of order \({O\left(\frac{1}{|x|}\right)}\) as \({|x|\rightarrow\infty}\) . Similar result holds for steady subsonic Euler flows in two-dimensional infinitely long straight divergent nozzles.

Global subsonic Euler flows in an infinitely long axisymmetric nozzle

In this paper, we consider global subsonic compressible flows through an infinitely long axisymmetric nozzle. The flow is governed by the steady Euler equations and has boundary conditions on the nozzle walls. Existence and uniqueness of global subsonic solution are established for an infinitely long axisymmetric nozzle, when the variation of Bernoulli's function in the upstream is sufficiently small and the mass flux of the incoming flow is less than some critical value. The results give a strictly mathematical proof to the assertion in Bers (1958) [2]: there exists a critical value of the incoming mass flux such that a global subsonic flow exists uniquely in a nozzle, provided that the incoming mass flux is less than the critical value. The existence of subsonic flow is obtained by the precisely a priori estimates for the elliptic equation of two variables. With the assumptions on the nozzle in the far fields, the asymptotic behavior can be derived by a blow-up argument for the infinitely long nozzle. Finally, we obtain the uniqueness of uniformly subsonic flow by energy estimate and derive the existence of the critical value of incoming mass flux.

Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles

In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value, then there exists a unique global subsonic solution in a suitable class for a general variable nozzle. One of the main difficulties for the general steady Euler flows is that the governing equations are a mixed elliptic-hyperbolic system even for uniformly subsonic flows. A key point in our theory is to use a stream function formulation for compressible Euler equations. By this formulation, Euler equations are equivalent to a quasilinear second order equation for a stream function so that the hyperbolicity of the particle path is already involved. The existence of a solution to the boundary value problem for stream function is obtained with the help of the estimate for an elliptic equation of t...

Adjoint and defect error bounding and correction for functional estimates

We present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag. Adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation. Defect methods can be used to bound or reduce the error in the entire solution, with corresponding improvements to functional estimates. Both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds. The adjoint theory is presented for both smooth and shocked problems; numerical experiments confirm fourth-order error estimates for a pressure integral of shocked quasi-1D Euler flow. By employing defect and adjoint methods together and accounting for errors in approximating the geometry, it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach. Superconvergent drag estimates are obtained for subsonic Euler flow over a lifting airfoil.

Spurious, zeroth‐order entropy generation along a kinked wall

AbstractNumerical entropy generation is studied in the case of steady, subsonic Euler flow along a kinked solid wall. For a standard upwind finite volume discretization the numerical entropy error, a component of the global discretization error, appears to be zeroth‐order in mesh size. Two possible causes of the zeroth‐order entropy error are studied. First an investigation is made of the local truncation error on a kinked grid. Although this error also appears to be zeroth‐order in the neighbourhood of the kink, it probably does not cause the zeroth‐order entropy error. Next a study is made of the existence of a singularity in the exact solution. Probably, the Euler flow solution is singular at the kink in the wall. The form of this likely singularity is unknown. Therefore the construction of a computational method which uses a priori knowledge about the singularity is not possible. Finally it is shown by numerical experiments that the subsonic Euler flow along a kinked wall still can be computed with vanishing entropy errors by using an appropriate sequence of continuously curved walls which converge to the kinked wall in the limit of zero mesh width.