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### 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.

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

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?

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)

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.

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

Normalize the flow by the critical speed

Then (2.2) can be rewritten as

For example, for polytrophic gases, , (2.4) is

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.

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.

Remark 2: One of main steps in the proof of the main results is to reduce a non-local boundary value problem for a coupled-hyperbolic-elliptic system (0.7) to a standard boundary value problem for a 2nd quasilinear equations on a unbounded domain. The key to this is a stream-function formulation of the problem. Assume u > 0. Then

(0.7)

(3.14)

(3.15)

Remark 3 Open problems:

- Uniqueness of Subsonic flows
- Regularity of the Subsonic-Sonic flows and Geometry of the degeneracies
- General 3-D Nozzle
- Existence of smooth subsonic-sonic flows

§4 Transonic Shock in a nozzle

In this section, we will present some recent progress on transonic flows with shock in a nozzle due to Courant-Friedrich’s. For simplicity in presentation, we will concentrate on 2D, steady, isentropic Euler equations.

§4.1 Formulation of The Problem

Consider a uniform supersonic flow (q0,0) with constant density 0 > 0 which enters a nozzle with slowly-varying sections

Let be the shock surface we are looking for, which is assumed to go through a fixed point on the wall sometimes, i.e.,

The boundary conditions can be described as:

where the given large density at the exit satisfies

with the constant state satisfying

Thus the problem is to find a piecewise smooth solution solving (0.7) with conditions (4.3), and (4.5)-(4.9).

Then we have the following uniqueness results.

Theorem 4.1 (Xin-Yan-Yin) a positive constant such

that if and (4.2) and (4.10) hold, then the transonic shock problem(0.7), (4.3), and (4.5)-(4.9) has no more than one solution such that satisfy the following estimates with :

Remark 4.1 It should be emphasized that although one of the key issues to solve some mixed boundary value problem with corners, and thus may be a reasonable class for well-posedness, yet the regularity assumptions in Theorem 3.1 are plausible. Indeed,

implies that R-H condition (4.5) is compatible with solid-wall B.C. (4.8), while yields the compatibility of (4.8) and (4.9) at the fixed corners. Then regularity of a special class of 2nd order elliptic equations can be improved.

Remark 4.2 Similar uniqueness holds if the end pressure pe is prescribed on a c3-smooth curve which is a small perturbation of x1=1.

Remark 4.3 For general nozzle, the condition (4.3) is required for uniqueness, due to the example of flat nozzles.

Remark 4.4 The condition (4.2) is necessary for the transonic shock wave patterns conjectured by Courant-Friedrich’s in general. Since, otherwise, there might be supersonic shocks in the supersonic region and supersonic bulbs in the subsonic flows.

Remark 4.5 Similar results holds for non-isentropic flows & in 3-D.

§4.2 Non-Existence

Although the formulation of the transonic shock problem, (0.7), (4.3), and (4.5)–(4.9) looks reasonable physically, this problem HAS NO SOLUTION in general, indeed, we can show that for a class of nozzles, there exists no such transonic solutions for general given supersonic incoming flow and end pressure.

Our first example is 2-D nozzles with flat walls.

Theorem 4.2 (Xin-Yan-Yin) Assume that the nozzle is flat, i.e.

Then for the constant supersonic incoming flow

with , and the end pressure , the Euler equations (0.7), with boundary condition (4.5)-(4.9) has no transonic solutions so that

satisfies the following requirements with some :

where is a suitable small constant which depends only on the

Remark 4.6 It should be emphasized that Theorem 4.2 does not require that the transonic shock wave goes through a fixed point, i.e. we do not assume (4.3).

Remark 4.7 For flat nozzles, similar non-existence results hold true for the non-isentropic Euler system with a similar analysis.

§4.3 Well-posedness

We now solve the conjecture of Courant-Friedrich for a class of nozzle.

We consider a class of non-flat nozzles which c5-regular, whose wall consist of two parts on [-1,1].

Set

Let satisfy

so that for a symmetric shock for an angular section nozzle.

Furthermore, assume the incoming supersonic flow is symmetric on in the sense that

and is a small perturbation of . Indeed of (4.3), the shock is assumed to go through (0,0).

and instead of (4.9), one imposes the B.C. at the exit as

Theorem 4.3 (Xin-Yan-Yin) Let the nozzle be given as above and

be suitably small. Then the transonic shock problem (0.7) with boundary condition (4.3)’, (4.5)-(4.8), and (4.9)’ is ill-posed for large | |. More precisely, supersonic incoming flows, which are small perturbations of , such that the problem (0.7), (4.3)’, (4.5)-(4.8), and (4.9)’ has no transonic shock solution with satisfying the following properties for some .

where

are the intersection points of the shock wave curve

with the solid wall respectively.

Remark 4.8 Similar results hold for non-isentropic flow and 3D fluids.

Despite the non-existence results in above, it is possible to have the transonic shock wave pattern conjecture by

Courant-Friedrich’s for some interesting class of nozzle and special exit boundary condition. Instead, consider the nozzle give in (4.14). If one gives up the requirement (4.3)’, that is, the shock positive is completely free, then it is possible to have a solution. Indeed, one has

Theorem 4.4 (Xin-Yan-Yin) Let the nozzle be given in (4.14) and the incoming supersonic flow be described as in Theorem 4.3. Then

(1) positive constants p1 and p2, p1 < p2, which are determined by the incoming flow and the shape of the nozzle,

such that for a given constant pressure , symmetric transonic shock solution to the problem (0.7), (4.5)-(4.8), (4.9)’ with the shock location at which depends on pe monotonically. Furthermore, in the subsonic region, the solution is denoted by

(2) Let . Then the above symmetric transonic shock are unique in the class

for suitably small .

(3) Such a transonic-shock is dynamically stable!

Finally, we consider the general case that the exit and pressure is a variable

with suitably small,

Then we have the following general results:

Theorem 4.5 (Li-Xin-Yin)

Let the nozzle be given as in (4.14) and the incoming supersonic flow be described as in Theorem 4.3 such that

Then constant such that for all

the transonic shock problem (0.7), (4.5) – (4.8), (4.17) (here (4.7) becomes )

Remark 4.9 Same results hold for non-isentropic flow.

Remark 4.10 In fact, the shock position depends on the exit and pressure monotonically, this is the key for the proof of the existence in Theorem 4.5. The proof of this depends crucially on the properties of incoming supersonic flow.

Remark 4.11 Similar results have been obtained by Li-Xin-Yin for 3-D case.

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