Transcript Multi-Dimensional Shock Waves

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 transonicshocks 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
 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):
M
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
studies on subsonic flows is a finite nozzle.
s
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
subsonic-sonic flow in a general nozzle!!
will yield a
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).
shock wave
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 ChanFeldman, 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)
pe
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 quasione-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
corresponds to
1
and subsonic branch
3. Let H be the specific volume, i.e.,
Then
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
varies in [0, ).
(3) For
, the axial velocity is always positive in
(4) (Flow angle estimates): For
satisfies
where
ranges over [0,1) as m0
, the flow angle
, i.e.,
(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 compensatedcompactness, 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
corresponding to mn .
the global uniformly subsonic flow
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:
,
uniquely determined by m, B(x2), and b – a such that
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 coupledhyperbolic-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)
 in terms of stream function

where
and
J (M, S) can be derived from the equation of state.
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 CourantFriedrich’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
x2
..
x1
x2 = f2 (x1)
x1 =  (x2)
-
+
x2 = f1 (x1)
pe = p(e)
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 across the shock surface, we require that
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
on the
is a suitable small constant which depends only
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].
x2
-½ -¼
(1,0)
x1
Set
Let
satisfy
so that
angular section nozzle.
for a symmetric shock for an
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
)
has a unique solution
following properties:
(i)
(ii)
with
being the subsonic region
with the
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-XinYin for 3-D case.
Thank You!