Elsevier

Journal of Differential Equations

Volume 286, 15 June 2021, Pages 433-454
Journal of Differential Equations

Steady compressible radially symmetric flows with nonzero angular velocity in an annulus

https://doi.org/10.1016/j.jde.2021.03.028Get rights and content

Abstract

In this paper, we investigate steady inviscid compressible flows with radial symmetry in an annulus. The major concerns are transonic flows with or without shocks. One of the main motivations is to elucidate the role played by the angular velocity in the structure of steady inviscid compressible flows. We give a complete classification of flow patterns in terms of boundary conditions at the inner and outer circle. Due to the nonzero angular velocity, many new flow patterns will appear. There exists accelerating or decelerating smooth transonic flows in an annulus satisfying one side boundary conditions at the inner or outer circle with all sonic points being nonexceptional and noncharacteristically degenerate. More importantly, it is found that besides the well-known supersonic-subsonic shock in a divergent nozzle as in the case without angular velocity, there exists a supersonic-supersonic shock solution, where the downstream state may change smoothly from supersonic to subsonic. Furthermore, there exists a supersonic-sonic shock solution where the shock circle and the sonic circle coincide, which is new and interesting.

Introduction

In this paper, we consider two-dimensional steady compressible Euler flows in an annulus Ω={(x1,x2):r0<r=x12+x22<r1}, which are governed by the following system{x1(ρu1)+x2(ρu2)=0,x1(ρu12)+x2(ρu1u2)+x1p=0,x1(ρu1u2)+x2(ρu22)+x2p=0,x1(ρu1E+u1p)+x2(ρu2E+u2p)=0, where u=(u1,u2)t is the velocity, ρ is the density, p is the pressure, E is the energy. Here we consider only the polytropic gas, therefore p=A(S)ργ, where A is a smooth function of the entropy A(S)=ReS with R>0,γ(1,+) are positive constants. Denote the Bernoulli function by B=12|U|2+γp(γ1)ρ.

The study on the steady compressible Euler system is not only of fundamental importance in developing the mathematical theory of partial differential equations arising from fluid dynamics, but also has been making a great contribution to the design of projectiles, rockets, aircrafts etc. Courant and Friedrichs [11, Section 104] used the Hodograph transformation to rewrite the Euler system (1.1) into a linear second order PDE on the plane of the flow speed and angle, and obtained some special flows without concerning the boundary conditions. These special flows include circulatory flows and purely radial flows, which are symmetric flows with only angular and radial velocity respectively. Their superpositions are called spiral flows. It had been proved in [11, Section 104] that spiral flows can take place only outside a limiting circle where the Jacobian of the hodograph transformation is zero and may change smoothly from subsonic to supersonic or vice verse. However, few results are available concerning properties of transonic spiral flows on the physical plane. This motivates us strongly to investigation of the effects of the angular velocity on general radially symmetric transonic flows with/or without shocks on the physical plane in this paper. The focus will be to find some new transonic flow patterns in the presence of shocks compared with the case without angular velocity.

We will study radially symmetric transonic spiral flows in an annulus with suitable boundary conditions on the inner and outer circle. We will classify all possible transonic radially symmetric flow patterns in terms of physical boundary conditions and study their detailed properties. To this end, we introduce the polar coordinate (r,θ) asx=rcosθ,y=rsinθ, and decompose the velocity as u=U1er+U2eθ wither=(cosθsinθ),eθ=(sinθcosθ). Therefore the system (1.1) can be rewritten as{r(ρU1)+1rθ(ρU2)+1rρU1=0,(U1r+U2rθ)U1+1ρrpU22r=0,(U1r+U2rθ)U2+1rρθp+U1U2r=0,(U1r+U2rθ)A=0.

For the radially symmetric solutions to (1.4) of the form u=U1(r)er+U2(r)eθ, ρ=ρ(r), A=A(r) in Ω, the steady Euler system (1.4) reduces to{ddr(ρU1)+1rρU1=0,r0<r<r1,U1U1+1ρddrpU22r=0,r0<r<r1,U1U2+U1U2r=0,r0<r<r1,U1A=0,r0<r<r1. We start with smooth solutions to (1.5) with one side boundary conditions prescribed on the inner or outer circle and give a classification of possible flow patterns in an annulus including purely smooth supersonic, subsonic flows, and smooth decelerating or accelerating transonic flows. We further analyze the detailed properties of these solutions and their dependence on the boundary data. If the boundary data is prescribed on the outer circle and the fluid moves from outer circle to the inner one, there exists a limiting circle such that the acceleration will blow up when the fluid moves to the limiting circle. The sonic circle and the limiting circle are also determined by boundary data. Although these smooth transonic spiral flows are essentially same as the ones obtained by Courant-Friedrichs [11] on the hodograph plane, the detailed behaviors of these solutions would be very helpful for the investigation of structural stability of symmetric transonic flows under suitable perturbations on the boundaries [27].

An important fact is on the degeneracy type of the sonic points in the smooth radially symmetric transonic spiral flows. Recall the definition by Bers [2], a sonic point in a C2 smooth transonic flow is exceptional if and only if the velocity is orthogonal to the sonic curve at this point. Then all sonic points of smooth transonic spiral flows here are nonexceptional and noncharacteristically degenerate due to the nonzero angular velocity. This is different from smooth transonic flows constructed by Wang and Xin [22], [23], [24], [25] in De Laval nozzles. In particular, the authors in [24] proved that if the nozzle is suitably flat at its throat, then there exists a unique smooth transonic irrotational flow of Meyer type with all sonic points being exceptional in De Laval nozzles. The sonic points must be located at the throat of the nozzle and are strongly degenerate in the sense that all the characteristics from sonic points coincide with the sonic line and can not approach the supersonic region. It should be mentioned that Kuzmin [15] had studied the structural stability of accelerating smooth transonic flows with some artificial boundary conditions on the potential and stream function plane. The existence of subsonic-sonic weak solutions to the 2-D steady potential equation were proved in [6], [28] by utilizing the compensated compactness and later on the subsonic-sonic limit for multidimensional potential flows and steady Euler flows were examined in [7], [14]. However, the solutions obtained by the subsonic-sonic limit only satisfied the equations in the sense of distribution and the regularity of the solutions and the properties of sonic points are not clear at all.

More importantly, we study transonic solutions with shocks to (1.4) by prescribing suitable boundary conditions on the inner and out circle. Recall that, a piecewise smooth solution (U1±,U2±,ρ±,A±)C1(Ω±) with a jump on the curve r=f(θ) is a shock solution to the system (1.4), if (U1±,U2±,ρ±,A±) satisfy the system (1.4) in Ω± respectively, and the entropy condition [p]>0 and the following Rankine-Hugoniot jump conditions hold on r=f(θ):{[ρU1]f(θ)f(θ)[ρU2]=0,[ρU12+p]f(θ)f(θ)[ρU1U2]=0,[ρU1U2]f(θ)f(θ)[ρU22+p]=0,[B]=0, where [v]=v+(f(θ),θ)v(f(θ),θ). For radially symmetric transonic solutions with shocks, we would consider the system (1.5) supplemented with suitable boundary conditions on both inner and outer circle and the shock curve will be a circle r=rb and the Rankine-Hugoniot jump conditions reduce to{[ρU1](rb)=[ρU12+p](rb)=0,[ρU1U2](rb)=[B](rb)=0.

The existence and uniqueness of radially symmetric supersonic-subsonic shock solutions without angular velocity in a divergent nozzle have been proved in [11], [31]. The dynamical and structural stability of these symmetric transonic shock solutions have been an important and difficult research topic in the mathematical studies of gas dynamics. Symmetric transonic shocks are shown in [31] to be dynamically stable in divergent nozzles and are dynamically unstable in convergent nozzles. The transonic shock problem with given exit pressure was shown to be ill-posed for the potential flow model in [29], [30]. The authors in [17], [20] proved that, for the two-dimensional steady compressible Euler system, the symmetric transonic shock solutions in [11], [31] are structurally stable under the perturbation of the exit pressure and the nozzle wall. They further investigated the existence and monotonicity of the axi-symmetric transonic shock by perturbing axi-symmetrically exit pressure in [18], [19]. The authors in [26] examined the structural stability under axi-symmetric perturbations of the nozzle wall by introducing a modified invertible Lagrangian transformation. There have been many interesting results on transonic shock problems in flat or divergent nozzles for different models with various exit boundary conditions, such as the non-isentropic potential model, the exit boundary condition for the normal velocity, the spherical symmetric flows without solid boundary, etc. One may refer to [1], [3], [4], [5], [8], [9], [10], [12], [13], [16], [21] and the references therein.

However, in the presence of the nonzero angular velocity, many new wave patterns appear in the annulus. It follows from (1.7) that the radial velocity will jump from supersonic to subsonic and the angular velocity experiences no jump across the shock. Therefore the total velocity after the shock may be supersonic, sonic and subsonic. We will give a complete classification of flow patterns according to the exit pressure and the position of the outer circle. Besides the well-known supersonic-subsonic shock solutions in a divergent nozzle as in the case without the angular velocity, we find that there are supersonic-supersonic shock solutions and the flow at downstream may change smoothly from supersonic to subsonic. Furthermore, there exists also a supersonic-sonic shock solution where the shock front and the sonic circle coincide, which is new and surprising.

The structure of the paper will be organized as follows. In Section 2, we will consider the one side boundary value problem to (1.5) and show that there are accelerating or decelerating smooth transonic flows. In Section 3, the precise description of transonic flows with shocks and new wave patterns will be presented by studying (1.5) with two sides boundary values.

Section snippets

Smooth radially symmetric flows with nonzero angular velocity

In this section, we will construct a radially symmetric smooth solution to (1.1) in an annulus Ω={(x1,x2):r0<r=x12+x22<r1}, where the fluid flows from the inner circle to the outer one. Thus we consider the following problem:

Problem I. Find smooth functions u(x)=U1(r)er+U2(r)eθ,ρ(x)=ρ(r) and p(x)=p(r),A(x)=A(r), which solve the system (1.5) with the boundary conditions on r=r0:ρ(r0)=ρ0,U1(r0)=U10>0,U2(r0)=U20,A(r0)=A0.

It is easy to see that the problem (1.5) with (2.1) is equivalent to{ρU1=κ1r,κ

Transonic shock flows in an annulus

In this section, we will prove the existence of radially symmetric transonic solutions with shocks and nonzero angular velocity to (1.5) satisfying suitable boundary conditions both at the inner and outer circle. Especially, motivated by previous studies due to Courant-Friedrichs [11], we prescribe the pressure at the exit. Therefore the problem to be solved can be formulated precisely as

Problem III. Construct a piecewise smooth radially symmetric flow on an annulus Ω={(x1,x2):r0<r=x12+x22<r1}

Acknowledgement

Part of this work was done when Weng visited The Institute of Mathematical Sciences of The Chinese University of Hong Kong. He is grateful to the institute for providing nice research environment. Weng is partially supported by National Natural Science Foundation of China 11701431, 11971307, 12071359. Xin is supported in part by the Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants CUHK-14305315, CUHK-14300917, CUHK-14302819 and CUHK-14302917, and by Guangdong Basic and Applied

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