Subsonic-sonic Flows Research Articles

On Subsonic and Subsonic-Sonic Flows with General Conservatives Force in Exterior Domains

In this paper, we study the irrotational subsonic and subsonic-sonic flows with general conservative forces in the exterior domains. The conservative forces indicate the new Bernoulli law naturally. For the subsonic case, we introduce a modified cut-off system depending on the conservative forces which needs the varied Bers skill, and construct the solution by the new variational formula. Moreover, comparing with previous results, our result extends the pressure-density relation to the general case. Afterwards we obtain the subsonic-sonic limit solution by taking the extract subsonic solutions as the approximate sequences.

Regular subsonic-sonic flows in general nozzles

This paper concerns subsonic-sonic potential flows in general two dimensional nozzles. For finitely long symmetric nozzles, we formulate the subsonic-sonic flow problem by prescribing the flow angle at the inlet and the outlet. It is shown that this problem admits a unique Lipschitz continuous subsonic-sonic flow, and the sonic points of the flow must occur at the wall or the throat. This is the first result on the well-posedness for general subsonic-sonic flow problems. More importantly, the location of sonic points is classified completely. Indeed, it is shown that there exists a critical value depending only on the length and the geometry of the nozzle such that the flow is sonic on the whole throat if the height of the nozzle is not greater than this critical value, while the sonic points must be located at the wall if the height is greater than this value. Furthermore, the critical height is positive iff the nozzle is suitably flat near the throat. As a direct application of this theory, we can obtain conditions on whether there is a smooth transonic flow of Meyer type whose sonic points are all exceptional in de Laval nozzles.

A degenerate elliptic problem from subsonic-sonic flows in convergent nozzles

<p style="text-indent:20px;">This paper concerns continuous subsonic-sonic potential flows in a two dimensional convergent nozzle, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. It is shown that for a given nozzle which is a perturbation of an straight one, and a given mass flux, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only <inline-formula><tex-math id="M1">\begin{document}$ C^{1/2} $\end{document}</tex-math></inline-formula> Hölder continuous and the acceleration blows up at the sonic state.</p>

Continuous subsonic-sonic flows in a two-dimensional semi-infinitely long nozzle

<p style='text-indent:20px;'>This paper focuses on two-dimensional continuous subsonic-sonic potential flows in a semi-infinitely long nozzle with a straight lower wall and an upper wall which is convergent at the outlet while straight at the far fields. It is proved that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique continuous subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the sonic curve. Furthermore, the sonic curve is free, where the flow is singular in the sense that the flow speed is only Hölder continuous and the flow acceleration blows up. Additionally, the asymptotic behaviors of the flow speed at the far fields is shown.</p>

A degenerate elliptic problem from subsonic-sonic flows in general nozzles

This paper deals with two-dimensional subsonic-sonic potential flows in general nozzles, which is governed by a free boundary problem of a quasilinear degenerate elliptic equation. For a large class of nozzles, it is shown that if the variation rate of the cross section of the nozzle is suitably small, there exists a unique subsonic-sonic flows in the nozzle such that the sonic curve intersects the upper wall at a fixed point and the velocity of the flow is along the normal direction at the inlet and the sonic curve. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only Hölder continuous and the acceleration blows up at the sonic state.

Smooth Transonic Flows of Meyer Type in De Laval Nozzles

A smooth transonic flow problem is formulated as follows: for a de Laval nozzle, one looks for a smooth transonic flow of Meyer type whose sonic points are all exceptional and whose flow angle at the inlet is prescribed. If such a flow exists, its sonic curve must be located at the throat of the nozzle and the nozzle should be suitably flat at its throat. The flow is governed by a quasilinear elliptic–hyperbolic mixed type equation and it is strongly degenerate at the sonic curve in the sense that all characteristics from the sonic points coincide with the sonic curve and never approach the supersonic region. For a suitably flat de Laval nozzle, the existence of a local subsonic–sonic flow in the convergent part and a local sonic–supersonic flow in the divergent part is proved by some elaborate elliptic and hyperbolic estimates. The precise asymptotic behavior of these two flows near the sonic state is shown and they can be connected to a smooth transonic flow whose acceleration is Lipschitz continuous. The flow is also shown to be unique by an elaborate energy estimate. Moreover, we give a set of infinitely long de Laval nozzles, such that each nozzle admits uniquely a global smooth transonic flow of Meyer type whose sonic points are all exceptional, while the same result does not hold for smooth transonic flows of Meyer type with nonexceptional points.

A Free Boundary Problem of a Degenerate Elliptic Equation and Subsonic-Sonic Flows with General Sonic Curves

This paper concerns a free boundary problem of a quasilinear degenerate elliptic equation, which arises from the studying of subsonic-sonic flows with both exceptional and nonexceptional sonic poin...

Continuous Subsonic-Sonic Flows in a Convergent Nozzle

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle. It is shown that for a given nozzle which is a perturbation of a straight one, a given point on its wall where the curvature is zero, and a given inlet which is a perturbation of an arc centered at the vertex, there exists uniquely a continuous subsonic-sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the given point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C1/2 Holder continuous and the acceleration blows up. The perturbation problem is solved in the potential plane, where the flow is governed by a free boundary problem of a degenerate elliptic equation with two free boundaries and two nonlocal boundary conditions, and the equation is degenerate at one free boundary.

On subsonic and subsonic-sonic flows in the infinity long nozzle with general conservatives force

In this article, we study irrotational subsonic and subsonic-sonic flows with general conservative forces in the infinity long nozzle. For the subsonic case, the varified Bernoulli law leads a modified cut-off system. Because of the local average estimate, conservative forces do not need any decay condition. Afterwards, the subsonic-sonic limit solutions are constructed by taking the extract subsonic solutions as the approximate sequences.

Continuous subsonic-sonic flows in convergent nozzles with straight solid walls

This paper concerns continuous subsonic-sonic potential flows in a two-dimensional convergent nozzle with straight solid walls. It is shown that for the given inlet, which is a perturbation of an arc centered at the vertex of the nozzle, and the given incoming flow angle which is a perturbation of the angle of the inner normal of the inlet, and the given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic-sonic flow from the given inlet with the given incoming flow angle and the given incoming mass flux. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only Hölder continuous and the acceleration blows up at the sonic state. The perturbation problem is solved in the potential plane, where the flow is governed by a degenerate elliptic problem with two free boundaries, on one of which the equation is degenerate and on the other of which the boundary condition is nonlocal.

Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations

A compactness framework is established for approximate solutions to subsonic-sonic flows governed by the steady full Euler equations for compressible fluids in arbitrary dimension. The existing compactness frameworks for the two-dimensional irrotational case do not directly apply for the steady full Euler equations in higher dimensions. The new compactness framework we develop applies for both non-homentropic and rotational flows. One of our main observations is that the compactness can be achieved by using only natural weak estimates for the mass balance and the vorticity, along with the Bernoulli law and the entropy relation, through a more delicate analysis on the phase space. As direct applications, we establish two existence theorems for multidimensional subsonic-sonic full Euler flows through infinitely long nozzles.

Continuous subsonic–sonic flows in a general nozzle

This paper concerns continuous subsonic–sonic potential flows in a two dimensional finite nozzle with a general upper wall and a straight lower wall. We give a class of nozzles where continuous subsonic–sonic flows may exist. Consider a continuous subsonic–sonic flow in such a nozzle after rescaling the upper wall in a small scale. It is shown that for a given inlet and a fixed point at the upper wall, there exists uniquely a continuous subsonic–sonic flow whose velocity vector is along the normal direction at the inlet and the sonic curve, which satisfies the slip conditions on the nozzle walls and whose sonic curve intersects the upper wall at the fixed point. Furthermore, the sonic curve of this flow is a free boundary, where the flow is singular in the sense that the speed is only C1/2 Hölder continuous and the acceleration blows up at the sonic state. As the scale tends to zero, the precise convergent rate of the continuous subsonic–sonic flow converging to the sonic state is also determined.

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.

On a Degenerate Free Boundary Problem and Continuous Subsonic–Sonic Flows in a Convergent Nozzle

This paper concerns the well-posedness of a boundary value problem for a quasilinear second order elliptic equation which is degenerate on a free boundary. Such problems arise when studying continuous subsonic–sonic flows in a convergent nozzle with straight solid walls. It is shown that for a given inlet being a perturbation of an arc centered at the vertex of the nozzle and a given incoming mass flux belonging to an open interval depending only on the adiabatic exponent and the length of the arc, there is a unique continuous subsonic–sonic flow from the given inlet with the angle of the velocity orthogonal to the inlet and the given incoming mass flux. Furthermore, the sonic curve of this continuous subsonic–sonic flow is a free boundary, where the flow is singular in the sense that while the speed is C1/2 Holder continuous at the sonic state, the acceleration blows up at the sonic state.

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.

Optimal Hölder Continuity for a Class of Degenerate Elliptic Problems with an Application to Subsonic-Sonic Flows

In this paper, we first study a class of elliptic equations with anisotropic boundary degeneracy. Besides establishing the existence, uniqueness and comparison principle, we obtain the optimal Hölder estimates for weak solutions by the estimates in the Campanato space. Based on such Hölder estimates, we then investigate subsonic-sonic flows with singularities at the sonic curves in a symmetric convergent nozzle with straight wall for an approximate model of the potential flow equation. It is proved that the perturbation problem of the symmetric subsonic-sonic flow is solvable and the symmetric subsonic-sonic flow is stable.

Global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method. More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π / 2 ; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.

Examples of steady subsonic flows in a convergent–divergent approximate nozzle

We construct special solutions of the full Euler system for steady compressible flows in a convergent–divergent approximate nozzle and study the stability of the purely subsonic flows. For a given pressure p 0 prescribed at the entry of the nozzle, as the pressure p 1 at the exit decreases, the flow patterns in the nozzle change continuously: there appear subsonic flow, subsonic–sonic flow, transonic flow and transonic shocks. Our results indicate that, to determine a subsonic flow in a two-dimensional nozzle, if the Bernoulli constant is uniform in the flow field, then this constant should not be prescribed if the pressure, density at the entry and the pressure at the exit of the nozzle are given; if the Bernoulli constant and both the pressures at the entrance and the exit are given, the average of the density at the entrance is then totally determined.

Global subsonic and subsonic-sonic flows through infinitely long nozzles

In this paper, we study global subsonic and subsonicsonic flows through a general infinitely long nozzle. First, it is proved that there exists a critical value for the incoming mass flux so that a global uniformly subsonic flow exists in the nozzle as long as the incoming mass flux is less than the critical value. More importantly, we establish some uniform estimates for the deflection angles and the minimum speed of the subsonic flows by combining hodograph transformation and the comparison principle for elliptic equations. With the help of these properties and a compensated compactness framework, we get the existence of a global subsonic-sonic flow solution in the case of the critical incoming mass flux.