Analysis on the asymptotic behavior of transonic shocks for steady Euler flows with gravity in a flat nozzle as the gravity vanishes
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: where is the density, is the pressure, is the velocity field, the non-zero constant is the acceleration of gravity. When , the direction of the gravity is to the left, as Fig. 1.1 shows. When , the gravity has the opposite direction. In the following arguments, we may assume that , the case of can be treated similarly. For polytropic gases, its state equation is assumed to be , where is the adiabatic exponent.
Let be a flat nozzle (see Fig. 1.1). Define , where is the flow angle and is the magnitude of the flow velocity. In addition, let be the sonic speed and the Mach number. For the supersonic flow, and for the subsonic flow, . Assume that there is a shock front , then the following Rankine–Hugoniot (R–H) conditions on should be satisfied When , for the given uniform supersonic state in , there exists a unique uniform subsonic state in such that which can be connected to through a normal transonic shock front with 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 in the following arguments.
Problem : The position of the shock front and its asymptotic behavior as .
Let be a given constant and be a given function with . Try to determine the position of shock front for the given and , and to analyze the asymptotic behavior of transonic shock solution as , where the transonic shock solution satisfies the following properties:
- (i)
satisfies Eqs. (1.1)–(1.3) in the domain with the initial–boundary data:
- (ii)
satisfies Eqs. (1.1)–(1.3) in the domain with the boundary conditions:
- (iii)
On the shock front , and connected by the R–H conditions (1.4). In addition, the following entropy condition holds (see [1]):
This paper will deal with the problem . The main idea is to quantitatively determine the position of the shock front for sufficiently small gravity , 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 satisfies: Then there exists a sufficiently small positive constant , only depends on and , such that for any , there exists a transonic shock solution to the problem . In particular, the position of the shock front is where and the constants and depend on and . Applying (1.12)–(1.13), it is obvious that when goes to zero, one has
Remark 1.2 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 . Note that the approximating position also depends on satisfying (1.9). It indicates that the limit of the position of the shock front depends on when the gravity goes to zero. That is, the position in the flat nozzle is not unique as the gravity vanishes. Moreover, the receiver pressure condition at the exit can be replaced by , where with and satisfying the condition (1.9).
To analyze the asymptotic behavior of transonic shock solutions for the problem , one of the key points is to quantitatively determine the position of the shock front as the acceleration of gravity 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 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 .
In Section 2, we reformulate the transonic shock problem 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 vanishes.
Section snippets
Reformulation of the problem and main theorem
First, the Lagrange transformation will be introduced to straighten the streamline. See [7], [13] and references therein for more details. Let Under this transformation, Eqs. (1.1)–(1.3) become Then further computations yield that Eqs. (2.1) can be rewritten as: The domain becomes . In
The initial approximation problem and the position of the shock front for the problem
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 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.
References (18)
- et al.
Transonic shocks and free boundary problems for the full Euler equations in infinite nozzles
J. Math. Pures Appl. (9)
(2007) - et al.
Transonic nozzle flows and free boundary problems for the full Euler equation
J. Differential Equations
(2006) - et al.
On admissible positions of transonic shocks for steady Euler flows in a 3-D axisymmetric cylindrical nozzle
J. Differential Equations
(2021) - et al.
Supersonic Flow and Shock Waves
(1948) - et al.
On admissible locations of transonic shock fronts for steady Euler flows in an almost flat finite nozzle with prescribed receiver pressure
Comm. Pure Appl. Math.
(2021) - et al.
Transonic shocks in multidimensional divergent nozzles
Arch. Ration. Mech. Anal.
(2011) - et al.
Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type
J. Amer. Math. Soc.
(2003) Stability of transonic shock fronts in two-dimensional Euler systems
Trans. Amer. Math. Soc.
(2005)Compressible flow and transonic shock in a diverging nozzle
Comm. Math. Phys.
(2009)
Cited by (1)
Asymptotic analysis of transonic shocks in divergent nozzles with respect to the expanding angle
2024, Journal of Differential Equations