Analysis on the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes

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Abstract

This paper concerns the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes. One of the key points is to quantitatively determine the position of the shock front as the gravity is small. To this end, a free boundary problem for the linearized Euler system with small gravity will be proposed. Then one can obtain the information of the approximating position of the shock front. Once the initial approximation of the shock solution is obtained, a nonlinear iteration scheme can be constructed to show that the position of the shock front is a small perturbation of the initial approximation, which establishes a quantitative expression for the shock solution. Such an expression shows the asymptotic behavior as the gravity goes to zero.

Introduction

In this paper, we are concerned with the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes(see Fig. 1.1). In particular, we are trying to quantitatively determine the position of the shock front as the acceleration of gravity is small, and to analyze its asymptotic behavior as the gravity vanishes. In order to determine the position of the shock front, the pressure condition will be imposed on the exit of the nozzle, as suggested by R. Courant and K.O. Friedrichs in [1]. For 1-D steady flows, it can be qualitatively shown for given value of the acceleration of gravity by applying shooting method that the position of the shock front can be uniquely determined as the receiver pressure at the exit lies in a certain interval. While as the gravity vanishes, it is well-known that the position of the shock front can be arbitrary and the pressure at the exit is uniquely determined by the incoming flow. Thus, the asymptotic behavior of the position of the shock front as the gravity vanishes would be an interesting problem that is worth of further investigation. To study this problem, in this paper, we are going to quantitatively determine the position of the shock front for given gravity as its value is sufficiently small, and the expression helps to show the asymptotic behavior as the gravity vanishes.

In this paper, the direction of the gravity is assumed to be parallel to the nozzle walls, and the flow in the nozzle is governed by the compressible isentropic steady Euler equations with gravity: x(ρu)+y(ρv)=0,x(ρu2+p)+y(ρuv)=ρg,x(ρuv)+y(ρv2+p)=0, where ρ is the density, p is the pressure, (u,v) is the velocity field, the non-zero constant g is the acceleration of gravity. When g>0, the direction of the gravity is to the left, as Fig. 1.1 shows. When g<0, the gravity has the opposite direction. In the following arguments, we may assume that g>0, the case of g<0 can be treated similarly. For polytropic gases, its state equation is assumed to be p(ρ)=ργ, where γ>1 is the adiabatic exponent.

Let D{(x,y)R2:0<x<L,0<y<1} be a flat nozzle (see Fig. 1.1). Define U(p,θ,q), where θ=arctanvu is the flow angle and q=u2+v2 is the magnitude of the flow velocity. In addition, let c=ρp be the sonic speed and M=qc the Mach number. For the supersonic flow, M>1 and for the subsonic flow, M<1. Assume that there is a shock front Σs{x=φ(y)}, then the following Rankine–Hugoniot (R–H) conditions on Σs should be satisfied [ρu]φ[ρv]=0,[ρu2+p]φ[ρuv]=0,[ρuv]φ[ρv2+p]=0.When g=0, for the given uniform supersonic state ŪD(p̄,0,q̄) in D, there exists a unique uniform subsonic state ŪD+(p̄+,0,q̄+) in D+ such that [ρ̄q̄]=ρ̄+q̄+ρ̄q̄=0,[p̄+ρ̄q̄2]=(p̄++ρ̄+q̄+2)(p̄+ρ̄q̄2)=0,which can be connected to ŪD through a normal transonic shock front x=x̄s with x̄s(0,L) being arbitrary. (In this paper, the subscript “” will represent the parameters of the flow in the supersonic domain and the subscript “+” in the subsonic domain.) In addition, without loss of generality, we may assume that ρ̄+q̄+=ρ̄q̄=1 in the following arguments.

Problem ABP: The position of the shock front and its asymptotic behavior as g0.

Let g>0 be a given constant and PEC2,α(R̄+) be a given function with α(0,1). Try to determine the position of shock front φ(y;g,PE) for the given g and PE, and to analyze the asymptotic behavior of transonic shock solution (U+,U;φ) as g0, where the transonic shock solution (U+,U;φ) satisfies the following properties:

  • (i)

    U satisfies Eqs. (1.1)–(1.3) in the domain D with the initial–boundary data: U(0,y)=ŪD=(p̄,0,q̄),onΣ1,θ=0,on(Σ2Σ4)D¯.

  • (ii)

    U+ satisfies Eqs. (1.1)–(1.3) in the domain D+ with the boundary conditions: p(L,y)=p̄++gPE(y),onΣ3,θ+=0,on(Σ2Σ4)D+¯.

  • (iii)

    On the shock front Σs, U and U+ connected by the R–H conditions (1.4). In addition, the following entropy condition holds (see [1]): p+>p,onΣs.

This paper will deal with the problem ABP. The main idea is to quantitatively determine the position of the shock front for sufficiently small gravity g, which helps to show its asymptotic behavior as the gravity vanishes. We are going to show that the following theorem holds.

Theorem 1.1

Assume that the given function PE satisfies: ρ̄+M̄+21L<01PE(y)dy<ρ̄M̄+21L.Then there exists a sufficiently small positive constant g0, only depends on Ū±,L and PE, such that for any 0<gg0, there exists a transonic shock solution (U,U+,φ) to the problem ABP. In particular, the position of the shock front is φ(y;g,PE)=ξ0+δξy1φ(τ;g,PE)dτ,where ξ0=ρ̄+ρ̄+ρ̄LM̄+21ρ̄+ρ̄01PE(y)dy,|δξ|Cg,φC0(Σs)Csg, and the constants C and Cs depend on Ū±,L and PE. Applying (1.12)(1.13), it is obvious that when g goes to zero, one has limg0φ(y;g,PE)=ξ0.

Remark 1.2

ξ0 is the initial approximating position of the shock front and it will be determined by the proposed free boundary problem of the linearized Euler system (see Section 3.1 for more details). In addition, the estimates (1.12)–(1.13) will be established in Section 3.2.

Remark 1.3

For the given uniform supersonic state at the entrance of the nozzle and the receiver pressure at the exit, (1.10) implies that the position of the shock front can be determined uniquely for any given sufficiently small gravity g. Note that the approximating position ξ0 also depends on PE satisfying (1.9). It indicates that the limit of the position of the shock front depends on PE when the gravity g goes to zero. That is, the position in the flat nozzle is not unique as the gravity g vanishes. Moreover, the receiver pressure condition at the exit can be replaced by p(L,y)Pe(y,g), where PeC2,α(R̄+2) with Pe(y,0)=p̄+ and Pe(y,0)PE(y) satisfying the condition (1.9).

To analyze the asymptotic behavior of transonic shock solutions for the problem ABP, one of the key points is to quantitatively determine the position of the shock front as the acceleration of gravity g is small. To this end, motivated by the ideas of Fang–Xin in [2], a free boundary problem for the linearized Euler system with small gravity will be proposed. Applying the solvability condition of the linearized equations (3.4)–(3.5) for the subsonic flow, one can obtain the information of the approximating position of the shock front. Once the initial approximation of the shock solution is obtained, a nonlinear iteration scheme can be constructed and one can prove that the position of the shock front is a small perturbation of the initial approximation. With these quantitative analyses for the position of the shock front, one can observe the asymptotic behavior as the gravity g goes to zero.

For the transonic shock problems in nozzles, there have been extensive literatures on it from different viewpoints, for instance, see [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and reference therein. In particular, for steady transonic shocks in flat nozzles, Chen–Feldman in [6] proved the existence of transonic shock solutions in a flat nozzle with given potential value at the exit and an assumption that the shock front passes through a given point. Later in [5], Chen–Chen–Song established the existence of the shock solutions for 2-D steady Euler flows with given vertical component of the velocity at the exit. In [16], Xin–Yan–Yin showed that the transonic shock solution was ill-posed under the small perturbations of the supersonic incoming flows and the receiver pressure given at the exit of the nozzle. Recently, Fang–Xin in [2] developed a method to determine the position of the shock front in an almost flat nozzle with the pressure condition at the exit. In this paper, we try to analyze the asymptotic behavior of the transonic shock solutions as the gravity g0.

In Section 2, we reformulate the transonic shock problem ABP via the Lagrange transformation and state the main theorem. In Section 3, a free boundary problem for the linearized Euler system will be proposed, which helps to determine the initial approximating transonic shock solution. And based on this initial approximation, a nonlinear iteration scheme can be constructed and one can prove that the position of the shock front is a small perturbation of the approximating initial position. Furthermore, the limit of the position of the shock front can be obtained as the gravity g vanishes.

Section snippets

Reformulation of the problem ABP and main theorem

First, the Lagrange transformation will be introduced to straighten the streamline. See [7], [13] and references therein for more details. Let (ξ,η)=(x,(0,0)(x,y)ρu(s,t)dtρv(s,t)ds).Under this transformation, Eqs. (1.1)–(1.3) become ξ(1ρu)η(vu)=0,ξ(u+pρu)η(pvu)+gu=0,ξv+ηp=0.Then further computations yield that Eqs. (2.1) can be rewritten as: ηpsinθρqξp+qcosθξθgsinθq=0,ηθsinθρqξθcosθρq1M2ρq2ξpgcosθρq3=0,qξq+1ρξp+g=0. The domain D becomes Ω{(ξ,η)R2:0<ξ<L,0<η<1}. In

The initial approximation problem and the position of the shock front for the problem ABPL

In this section, motivated by the idea introduced in [2], a free boundary problem for the linearized Euler system will be proposed, whose solution gives an initial approximation of the transonic shock solution. Then one can obtain the existence of the solution to the problem ABPL similar as [2], which proves Theorem 2.1.

Acknowledgments

The research was supported by Natural Science Foundation of China under Grant Nos. 11971308 and 11631008.

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