On the steady compressible flows in a nozzle Zhouping Xin The Institute of Mathematical Sciences, The Chinese University of Hong Kong 2008, Xiangtan
Contents §1 Introduction • Compressible Euler system and transonic flows • Global subsonic flows • Subsonic-Sonic flows • Transonic flows with shocks ** A Problem due to Bers ** A problem due to Courant-Friedrich on transonic-shocks in a nozzle
§2 Global Subsonic and Subsonic-Sonic Potential Flows in Infinite Long Axially Symmetric Nozzles • Main Results • Ideas of Analysis §3 Global Subsonic Flows in a 2-D Infinite Long Nozzles • Main Results
§4 Transonic Shocks In A Finite Nozzle • Uniqueness • Non-Existence • Well-posedness for a class of nozzles
§1 Introduction The ideal steady compressible fluids are governed by the following Euler system: where
Key Features: • nonlinearities ( shocks in general) • mixed-type system for many interesting wave patterns (change of types, degeneracies, etc.) It seems difficult to develop a general theory for such a system. However, there have been huge literatures studying some of important physical wave patterns, such as • Flows past a solid body; • Flows in a nozzle; • Wave reflections, etc.
Even for such special flow patterns, there are still great difficulties due to the change type of the system, free boundaries, internal and corner singularities etc.. Some simplified models: Potential Flows: Assume that In terms of velocity potential , Then (0.1) can be replaced by the following Potential Flow Equation.
with and the Bernoulli’s law which can be solved to yield here is the enthalpy given by .
Remark 1: The potential equation (0.4) is a 2nd order quasilinear PDE which is Remark 2: (0.4) also appears in geometric analysis such as mean curvature flows. 2-D Isentropic Euler Flows Assume that S = constant. Then the 2-D compressible flow equations are
The characteristic polynomial of (0.7) has three roots given as Thus, (0.7) is hyperbolic for supersonic flows (0.7) is coupled elliptic-hyperbolic for subsonic flow (0.7) is degenerate for sonic flow CHALLENGE: Transonic Flow patterns
M Huge literatures on the studies of the potential equation (0.4). In particular for subsonic flows. The most significant work is due to L. Bers (CPAM, Vol. 7, 1954, 441-504):
Fact: For 2-D flow past a profile, if the Mach number of the freestream is small enough, then the flow field is subsonic outside the profile. Furthermore, as the freestream Mach number increases, the maximum of the speed will tend to the sound speed. (See also Finn-Gilbarg CPAM (1957) Vol. 10, 23-63). These results were later generalized to 3-D by Gilbarg and then G. Dong, they obtained similar theory. And recently, a weak subsonic-sonic around a 2-D body has been established by Chen-Dafermos-Slemrod-Wang.
A lot of the rich wave phenomena in M-D compressible fluids appear in steady flows in a nozzle, which are important in fluid dynamics and aeronautic. In his famous survey (1958), Bers proposed the following problem: For a given infinite long 2-D or 3-D axially symmetric solid nozzle, show that there is a global subsonic flow through the nozzle for an appropriately given incoming mass flux One would expect a similar theory as for the airfoil would hold for the nozzle problem. Question: How the flow changes by varying m0?
s However, this problem has not been solved dispite many studieson subsonic flows is a finite nozzle. One of Keys: To understand sonic state
Our main strategy to studying compressible flows in a nozzle is: • Step 1 Existence of subsonic flow in a nozzle for suitably small incoming mass flux • It is expected that if the incoming mass flux is small, then global uniform subsonic flow in a nozzle exists. • Some of the difficulties are: • Global problem with different far fields, so the compactification through Kelvin-type transmation becomes impossible;
Possibility of appearance of sonic points • For rotational flows, it is unclear how to formulate a global subsonic problem. Step 2 Transition to subsonic-sonic flow We study the dependence of the maximum flow speed on the incoming flux and to investigation whether there exists a critical incoming flux such that if the incoming mass flux m increases to , then the corresponding maximum flow speed approaches the sound speed.
Step 3 Obtain a subsonic-sonic flow in a nozzle as a limit of a sequence of subsonic flows. Assume that Step 1 and Step 2 have been done. Let Let be the corresponding subsonic flow velocity field in the nozzle. Questions: 1. 2. Can solve (0.4)?
If both questions can be answered positively, then will yield a subsonic-sonic flow in a general nozzle!! • Remark: Due to the strong degeneracy at sonic state, it is a long standing open problem how to obtain smooth flows containing sonic states, exceptions: • accelerating transonic flows (Kutsumin, M. Feistauer) (for special nozzles and special B.C.) • subsonic flow which becomes sonic at the exit of a straight expanding nozzle (Wang-Xin, 2007)
shock wave Finally, we deal with transonic flows with shocks. When , in general, transonic flows must appear. However, it can be shown that smooth transonic flows must be unstable (C. Morawetz).
Thus, SHOCK WAVES must appear in general, and the flows patterns can become extremely complicated. Then the analysis of such flow patterns becomes a challenge for the field due to: • complicated wave reflections, • degeneracies, • free boundaries, • change type of equations, • mixed-type equations, etc.
Thus, Morawetz proposed to study the general weak solution by the framework of Compensated-Compactness for the 2-D potential flows. Yet, this approach has not been successful so far. The quasi-1D model has been successfully analyzed by many people, Embid-Majda-Goodam, Gamba, Liu, etc. Some special steady multi-dimensional transonic wave patterns with shock have been investigated recently by Chan-Feldman, Xin-Yin, S. Chen, Fang etc.
pe Motivated by engineering studies, Courant-Friedrichs proposed the following problem on transonic shock phenomena in a de Laval nozzle: 0, (q0, 0, 0)
Consider an uniform supersonic flow entering a de Laval nozzle. Given an appropriately large receiver pressure pe at the exit of the nozzle, if the supersonic flow extends passing through the throat of the nozzle, then at the certain place of the divergent part of the nozzle, a shock wave must intervene and the flow is compressed and slowed down to a subsonic speed, and the location and strength of the shock are adjusted automatically so that the pressure at the exit becomes the given pressure pe.
Experimentally and physically, it seems to be a very reasonable conjecture. Indeed, there are cases, such as quasi-one-dimensional model, the conjecture is definitely true. As we will show later, it also holds for symmetric flows. Unfortunately, this seems to be a very tricky question in general as we will show later. Some surprising facts appear!!! • general uniqueness results • non-existence • well-posedness for a class of nozzles
§2 Global Subsonic and Subsonic-Sonic Potential Flows in Infinite Long Axially-Symmetric Nozzles We first give a complete positive answer to the problem of Bers on global subsonic flows a general infinite nozzle. Furthermore, we will obtain a subsonic-sonic flow in the nozzle also as mentioned in the introduction. §2.1 Formulation of the problem Consider 3-D potential equation (0.4) with
Set and assume that Bernoulli’s law, (0.5), becomes with being the maximal speed.
Normalize the flow by the critical speed Then (2.2) can be rewritten as For example, for polytrophic gases, , (2.4) is
Some facts: 1. Subsonic 2. is a two-valued function of and subsonic branch corresponds to 1
Now G = G (q2) such that then Then the potential equation can be rewritten as
Assume that the nozzle is axi-symmetric and given by where is assumed to be smooth such that for some Assume also that the nozzle wall is impermeable, so that the boundary condition is
Note that for any smooth solution to (2.9) satisfying the boundary condition (2.12), the mass flux through any section of the nozzle transversal to the x-axis is constant, the nozzle problem can be formulated as: Find a solution to (2.9) and (2.12) such that where s is a section of the nozzle transversal to x-axis, and is the normal of s forming an accurate angle with x-axis.
§2.2 The Main Results Then the following existence results on the global uniform subsonic flow in the nozzle hold: Theorem 2.1 (Xie-Xin) Assume that nozzle is given by (2.10) satisfying (2.11). Then a positive constant , which depends only on f, such that if , the boundary value problem (2.9), (2.12) and (2.13) has a smooth solution , such that and the flow is axi-symmetric in the sense that where , (U, V) (x, r) are smooth, and V (x, 0) = 0.
To study some important properties of the subsonic flows in a nozzle, in particular, the dependence of the flows on the incoming mass flux m0, we assume that the wall of the nozzle tends to be flat at far fields, say (rescaling if necessary) Then we have following sharper results. Theorem 2.2 (Xie-Xin) Let the nozzle satisfy (2.11) and (2.16). Then a positive constant with the following properties:
(1) axially-symmetric uniformly subsonic solution to the problem (2.19), (2.12), and (2.13) with the properties and uniformly in r, where G is given in (2.8).
(2) is critical in the sense that ranges over [0,1) as m0 varies in [0, ). (3) For , the axial velocity is always positive in , i.e., (4) (Flow angle estimates): For , the flow angle satisfies where
(5) (Flow speed estimates) For any , In particular, (No stagnation uniformly). Finally, we show the asymptotic behavior of these subsonic solutions when the incoming mass flux m0 approaches the critical value . Based on Theorem 2.2 and a framework of compensated-compactness, we can obtain the existence of a global subsonic-sonic weak solution to (2.9), (2.12) and (2.13).
Theorem 2.3 (Xie-Xin) Assume that (i) The nozzle given by (2.10) satisfies (2.11) and (2.16). (ii) The fluids satisfy (iii) Let mn be any sequence such that Denote by the global uniformly subsonic flow corresponding to mn . Then subsequence of mn, still labeled as mn, such that
with almost every where convergence. Moreover, the limit yields a 3-D flow with density and velocity satisfying in the sense of distribution, and for any .
Remark 1 (2.26) implies that the boundary condition (2.12) is satisfied by the limiting velocity field as the normal trace of the divergence free field on the boundary. Remark 2 Similar theory holds for the 2-D flows (of Xie-Xin).
Remark 3 Compared with 3-D airfoil problem, the main difficulty is how to obtain the uniform ellipticity of (2.9). Remark 4 Key ideas of analysis: - Cut-off and desigularization; - Hodograph transformation part-hodograph transformation; - Rescaling and blow-up estimates for uniformly elliptic equations of two variables; - Compensated-compactness.
§3 Global Isentropic Subsonic Euler Flow in a Nozzle In this section, we present some results on the existence of global subsonic isentropic flows through a general 2-D infinite long nozzle. Formulation of the problem Note that the steady, isentropic compressible flow is governed by (0.7), which is a coupled elliptic-hyperbolic system.
Let the 2-D nozzle be with boundaries: Assumptions on si:
Impermeable Solid Wall Condition: Incoming Mass Flux: Let l be any smooth curve transversal to the x1-direction, and is the normal of l in the positive x1-axis direction, l
Set which is a constant independent of l. Due to the hyperbolic mode, one needs to impose one boundary condition at infinity. Set where is the anthalpy normalized so that h(0) = 0.
Then we propose the following boundary condition on B where B(x2) is smooth given function defined on [0,1]. Problem (*): Find a global subsonic solution to (0.7) on satisfying (3.3), (3.4), and (3.6).
Main Results Theorem 3.1 (Xie-Xin) Assume that 1. (3.2) holds, 2. Then such that if then with the property that for all , the problem (*) has a solution such that the following properties hold true:
uniformly on any sets k1 cc (0, 1), and k2 cc (a, b). 4. The solution to the problem (*) is unique under the additional assumptions (3.10)- (3.11). Furthermore, is the upper critical mass flux for the existence of subsonic flow in the following sense, it holds that either
or such that for all the problem (*) has a solution with the properties (3.9)-(3.11) and Remark 1: Similar results hold for the full non-isentropic Euler system if, in addition, the entropy is specified at the upstream.