Regular subsonic-sonic flows in general nozzles
Introduction
In this paper we study subsonic-sonic flows in two dimensional nozzles. Such problems arise naturally in physical experiments and engineering designs ([2], [16]). For steady irrotational subsonic-sonic flows, their well-posedness and properties, especially the regularity and the location of sonic points, are so important that they have received much attention ([2, Chapter 3]). There are many efforts in the past decades, such as the existence of the critical Mach number and the critical mass flux (it is unknown whether there is a subsonic or subsonic-sonic flow or not if the Mach number or the mass flux is greater than the critical value), the location of sonic points for a given smooth flow (the existence is unknown), the existence of weak solutions (the uniqueness and the location of sonic points are unknown). However, their well-posedness, regularity and location of sonic points are far from complete. The purpose of this paper is to give a complete answer for the subsonic-sonic flow problem in general nozzles, including the well-posedness, the precise regularity and the location of sonic points.
Two kinds of subsonic-sonic flow patterns have been concerned for a long time: one is flows past a profile and the other is flows in a nozzle. Shiffman [18] and Bers [1] showed that there is a unique subsonic flow past a given two dimensional profile if the freestream Mach number is less than a critical value; furthermore, the maximum flow speed tends to the sound speed as the freestream Mach number tends to the critical value. Finn and Gilbarg [12] studied the asymptotic behavior of these subsonic flows at the far field. Later, [13] and [8] proved the same results for multidimentional cases. Yet, the theory and methods in these works fail to deal with subsonic-sonic flows with the critical freestream Mach number. Based on the compensated compactness method, it was shown in [4] that the two dimensional flows with sonic points past a profile may be realized as weak limits of sequences of strictly subsonic flows. However, their regularity and uniqueness are unknown yet. If a two dimensional subsonic-sonic flow past a profile is smooth, Gilbarg and Shiffman [14] proved that the sonic points must occur at the profile. The similar situation occurs for subsonic-sonic flows in a nozzle. For a two dimensional infinitely long nozzle, it was proved in [23] that there exists a critical value such that a strictly subsonic flow exists uniquely as long as the incoming mass flux is less than the critical value, and subsonic-sonic flows exist as weak limits of strictly subsonic flows. The multidimentional cases were dealt with in [11]. In the recent decade, there are many works on subsonic flow problems for steady irrotational and rotational Euler system ([3], [5], [9], [10], [17], [24] and the references therein). But strong degeneracy at sonic points prevents substantial progress on subsonic-sonic flows ([6], [19]). The well-posedness, precise regularity and location of sonic points for general subsonic-sonic flow problems are still open.
A steady isentropic inviscid compressible flow is governed by the following Euler system where , P and ρ represent the velocity, pressure and density of the flow, respectively, and is a smooth function. In particular, for a polytropic gas with the adiabatic exponent , is the normalized pressure. Assume further that the flow is irrotational. Then the density ρ is expressed in terms of the speed q by the Bernoulli law ([7]) The sound speed c is defined as . At the sonic state, the speed is , which is critical in the sense that the flow is subsonic when , sonic when and supersonic when . It is well known that the above system can be transformed into the full potential equation where φ is a velocity potential with , and ρ is given by (1.1). It is well known that (1.2) is elliptic in the subsonic region, hyperbolic in the supersonic region, and degenerate at the sonic points. For a subsonic-sonic flow, (1.2) is a degenerate elliptic equation, which is degenerate at the sonic state. Furthermore, the location of sonic points is unknown in general. Hence it is difficult to investigate the well-posedness and the regularity of a subsonic-sonic flow problem. Note that the regularity of the subsonic-sonic flows past a profile and the ones in a nozzle obtained in [4], [23] is unknown, and the uniqueness is still open.
In the present paper, we are interested in not only the well-posedness and the precise regularity of subsonic-sonic flows in general nozzles, but also the location of the sonic points. Another important motivation for such problems lies in the study on smooth transonic flows of Meyer type in de Laval nozzles in [21], [22], where we sought smooth transonic flows of Meyer type whose sonic points are all exceptional in de Laval nozzles. For such a flow, its sonic curve is located at the throat of the nozzle (where the cross section is smallest) and the potential on its sonic curve equals identically to a constant. As usual, a de Laval nozzle, whose upper wall is given by , is assumed to be symmetric with respect to the x-axis and its throat lies in the y-axis. In [22] we formulated a smooth transonic problem by prescribing the flow angle at the inlet and proved its local well-posedness if and are suitably small. The asymptotic behavior of is almost necessary. Indeed, if there is such a smooth transonic flow in the nozzle, then . It was shown in [20] that the flow obtained in [22] can be extended into a global smooth transonic flow in an infinite expanding nozzle whose upper wall is convex. Hence a question is whether small is necessary and how to get a global smooth transonic flow in a general de Laval nozzle. It is noted that if a symmetric nozzle admits a smooth subsonic-sonic flow whose sonic points are a curve from one wall to another, then it follows from Gilbarg and Shiffman [14] that the nozzle must has a smallest cross section where the two walls are parallel. Moreover, a continuous subsonic-sonic flow problem in a straight convergent nozzle was studied in [19], where the set of the sonic points is a free curve from one wall to another, and the flow was shown to be singular in the sense that the speed is only Hölder continuous and the acceleration blows up at the sonic curve.
In this paper, it is assumed that the nozzle is symmetric with respect to the x-axis, and its inlet and outlet lie in and (), respectively. The flow satisfies the slip condition on the wall, and the velocity is horizontal at the inlet and the outlet. Due to the compatibility condition for continuous flows, the wall of the nozzle should be horizontal at the two endpoints. Hence there is a smallest cross section for the nozzle, which is assumed to be unique and located at the y-axis without loss of generality (the case that there are several smallest cross sections can be considered similarly). Precisely, the wall of the nozzle is given by satisfying where . We consider subsonic-sonic flows in the nozzle . The subsonic-sonic flow problem is formulated as where are free constants, and is normalized to be zero. Thanks to (1.4)–(1.6), holds for some constant m, called the mass flux of the flow, where S is any transversal section of the nozzle and is its unit normal in the positive x-axis.
To solve the subsonic-sonic flow problem (1.4)–(1.10), we replace (1.10) with (1.11), and reformulate such a subsonic or subsonic-sonic flow problem in terms of the stream function in the physical plane. Using the elliptic regularization method, together with some delicate estimates both in the physical plane and in the potential plane, it is shown that there is a critical value such that a smooth subsonic flow exists uniquely as long as , while there is a unique Lipschitz continuous subsonic-sonic flow if . In particular, we get the Lipschitz estimates of the speed by using the special structure of the Chaplygin equations in the physical plane. As for the a priori estimate about the location of sonic points for subsonic-sonic flows, it is noted that Gilbarg and Shiffman proved in [14] that the sonic points of a smooth subsonic-sonic flow to the problem (1.4)–(1.10) must occur at the upper wall or the throat; and if a point at the interior of the throat is sonic, the flow is sonic on the whole throat. In this paper, by establishing a comparison principle for a degenerate elliptic equation in the potential plane, we are able to prove that this result still holds for a continuous subsonic-sonic flow. Furthermore, we obtain the geometrical property of the upper wall at sonic points: If a Lipschitz continuous subsonic-sonic flow to the problem (1.4)–(1.10) is sonic at a point on the upper wall (the flow is subsonic in the nozzle if this point is located at the throat), then the curvature of the wall at this point is positive. By means of these a priori results on the location of sonic points and some elaborate arguments, one can prove that there is not a Lipschitz continuous subsonic or subsonic-sonic flow to the problem (1.4)–(1.9), (1.11) in the case . Summing up, it is proved that there exists uniquely a Lipschitz continuous subsonic-sonic flow to the problem (1.4)–(1.10), and its sonic points must occur at the throat or the upper wall . Furthermore, if a point belonging to is sonic, then the flow is sonic on the whole ; if the flow is sonic at for some (the flow is subsonic at if ), then .
More importantly, we can classify the location of sonic points more precisely for the subsonic-sonic flow to the problem (1.4)–(1.10) in terms of the height of the throat of the nozzle provided that the length and the geometry of the nozzle are given. Indeed, it is shown that there exist two constants depending only on γ, and such that the subsonic-sonic flow to the problem (1.4)–(1.10) is sonic on the whole throat if , while its sonic points must be located at the wall if ; furthermore, the set of sonic points is the whole throat if , while there is also another sonic point at the wall if . It is the asymptotic behavior of at that determines whether is positive or not. Indeed, an almost sufficient and necessary condition for is . Precisely, if while if Since the acceleration of the subsonic-sonic flow is bounded, this paper answers completely the subsonic extension of the local smooth transonic flows obtained in [22]. It is shown that for a de Laval nozzle whose length and geometry are given, its height determines whether there is a Lipschitz continuous transonic flows of Meyer type whose sonic points are all exceptional. The same results hold true if the smallest cross section of the nozzle is located at the inlet or outlet, or there are several smallest cross sections. Furthermore, for subsonic-sonic flows in infinitely long symmetric nozzles or past symmetric profiles, one can get the similar results by using the similar methods as in this paper, together with some a priori estimates for degenerate elliptic equations in unbounded domains, which will be dealt with in our forthcoming studies.
The rest of the paper is arranged as follows. In § 2 we state the main results for the problem (1.4)–(1.10) and introduce the formulations of the subsonic or subsonic-sonic flow problem both in terms of the stream function in the physical plane and in terms of the speed in the potential plane. The regularized problem is studied in § 3. Subsequently, in § 4 we solve the subsonic-sonic flow problem and study the precise regularity of subsonic-sonic flows. Finally, we classify the location of sonic points for the subsonic-sonic flows completely in § 5.
Section snippets
Main results for the subsonic-sonic flow problem and its other formulations
The main results for the subsonic-sonic flow problem (1.4)–(1.10) are the following well-posedness, regularity and classification theorems.
Theorem 2.1 Assume that satisfies (1.3). For , the problem (1.4)–(1.10) admits a unique solution . Furthermore, the sonic points must occur at or , and
Theorem 2.2 Assume that satisfies (1.3). Let be the solution
Regularized problem
It is noted that (2.1) is singular and (2.5) is degenerate at sonic points. We regularize them in the following way. For , choose , which exists due to (2.6) and (2.7), such that where , is a positive constant depending only on γ, while and is the inverse
Well-posedness and regularity of subsonic-sonic flows
In this section, we solve the subsonic-sonic problem and study the regularity of subsonic-sonic flows. As mentioned in , it will be shown that there exists a critical value such that the problem (2.9)–(2.13) admits a unique subsonic flow if , a unique subsonic-sonic flow if , while not a subsonic or subsonic-sonic flow if ; furthermore, the speed of the subsonic-sonic flow is Lipschitz continuous.
Classification of the location of sonic points for subsonic-sonic flows
In this section, we classify the location of sonic points for the subsonic-sonic flows completely. Let be the solution to the problem (2.9)–(2.13) with for . It follows from Proposition 4.1 that the sonic points of must occur at or . Moreover, if is sonic at a point on , then is sonic on the whole .
Lemma 5.1 Let satisfy (2.8), and be the solution to the problem (2.9)–(2.13) with for . Assume
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Supported by Zheng Ge Ru Foundation, Hong Kong RGC Earmarked Research Grants: CUHK-14305315, CUHK-14300917, CUHK-14302819 and CUHK-14302917, NSFC/RGC Joint Research Scheme Grant N/CUHK443/14, and by Guangdong Basic and Applied Basic Research Foundation 2020B1515310002.