On the steady compressible f lows 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
The Institute of Mathematical Sciences, The Chinese University of Hong Kong
** A Problem due to Bers
** A problem due to Courant-Friedrich on transonic-shocks in a nozzle
§3 Global Subsonic Flows in a 2-D Infinite Long Nozzles
The ideal steady compressible fluids are governed by the following Euler system:
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
Some simplified models:
Potential Flows: Assume that
In terms of velocity potential ,
Then (0.1) can be replaced by the following Potential Flow Equation.
and the Bernoulli’s law
which can be solved to yield
here is the enthalpy given by .
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
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):
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.
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
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.
Let be the corresponding subsonic flow velocity field in the nozzle.
2. Can solve (0.4)?
If both questions can be answered positively, then will yield a subsonic-sonic flow in a general nozzle!!
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).
Motivated by engineering studies, Courant-Friedrichs proposed the following problem on transonic shock phenomena in a de Laval nozzle:
0, (q0, 0, 0)
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
and assume that
Bernoulli’s law, (0.5), becomes
with being the maximal speed.
Then (2.2) can be rewritten as
For example, for polytrophic gases, , (2.4) is
2. is a two-valued function of and subsonic branch corresponds to
Then the potential equation can be rewritten as
where is assumed to be smooth such that
Assume also that the nozzle wall is impermeable, so that the boundary condition is
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.
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.
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:
uniformly in r, where G is given in (2.8).
(3) For , the axial velocity is always positive in , i.e.,
(4) (Flow angle estimates): For , the flow angle
(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).
(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
in the sense of distribution, and
for any .
Remark 2 Similar theory holds for the 2-D flows (of Xie-Xin).
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;
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.
Assumptions on si:
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,
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.
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).
Theorem 3.1 (Xie-Xin) Assume that
1. (3.2) holds,
Then such that if
then with the property that for all ,
the problem (*) has a solution such that the following properties hold true:
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
Remark 1: Similar results hold for the full non-isentropic Euler system if, in addition, the entropy is specified at the upstream.
J (M, S) can be derived from the equation of state.
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
x2 = f2 (x1)
x1 = (x2)
pe = p(e)
x2 = f1 (x1)
where the given large density at the exit satisfies
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.
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 :
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 4.3 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.
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.
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 :
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.
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].
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
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 .
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
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,
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,
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 )
with being the subsonic region
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.