Shuxing Chen(Fudan University)

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Shuxing Chen(Fudan University)

• Mathematical Analysis of Supersonic flow
  past bodies

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• Since the twenty century the flight
  technology developed rapidly. Today different
  aircrafts with speed more than ten times of the
  sonic one have been designed.

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• When a supersonic aircraft flies in the air
  there will be a shock ahead of the aircraft. The
  appearance let the resistance increase greatly.
  To clearly understand the location of the shock,
  as well as the flow field between the shock and
  the body is very important.

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• Many experiments show that in the supersonic flow
  field the shock ahead of a sharp body is
  generally attached, while the shock ahead of a
  blunt body is detached.
• Next we first consider the supersonic flow past
  sharp bodies.

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Supersonic flow past wing
shock
wing
Supersonic flow
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Supersonic flow past conical body
Conical body
shock
Supersonic flow
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• Our task is to explain the rule and the
  character for such supersonic motion. From the
  mathematical viewpoint it is to prove the
  existence and the stability of the corresponding
  solutions.

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For the importance of the proof
R.Courant gave the following writing in his
famous book Supersonic flow and shock waves
The confidence of the engineer and physicist in
the result of mathematical analysis should
ultimately rest on a proof that the solution
obtained is singled out by the data of the
problem. A great effort will be necessary to
develop the theories presented in this book to a
stage where they satisfy both the needs of
applications and the basic requirements of
natural philosophy.
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Euler System
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For the potential flow the system can be written
as a second order equation
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Main difficulties

• Nonlinearity

• Multidimensional

• Free boundaries

• Strong singularity
• Mixed type equations

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• Many mathematicians paid much efforts to the
  study on the mathematical analysis of supersonic
  flow past bodies

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• Courant-Friedrichs in 1948 put forward this
  problem, and give solutions for the case, when
  the body is a wedge or a cone.

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• When the body is a wedge, the problem can be
  solved by solving a set of algebraic equations
  (R-H conditions)
• When the body is a cone, the problem is reduced
  to solve a b.v.p. of an ordinary system.

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Gu Chaohao, Li Daqian and others (1960s)

• They studied the case when the body is a curved
  wedge.
• They first applied the theory of partial
  differential equations to solve this problem.

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• The contributions of other mathematicians ( for
  instance, Peter Lax,David Shaeffer )

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Chen Shuxing (3-d wings)

• Existence of local solution to supersonic flow
  around a three dimensional wing (Advances in
  Appl. Math. 1992)

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Main points of the method

• The blow up of the edge.
• Near the surface of the body the existence of
  the solution to the initial boundary value
  problems of hyperbolic systems
• Near the shock the existence of the initial
  value problem with discontinuous initial data for
  the nonlinear hyperbolic system of conservation
  laws.

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Chen Shuxing (Curved cone)

• Existence of stationary supersonic flow past
  a pointed body (Archive Rat. Mech. Anal. 2001)

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Mathematical formulation
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The theory on characteristic lines is not
available.
The usual technique for treating free
boundary problem is not available.

• The domain could not be reduced to a normal
  domain without singularities by a single blow-up.

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• First consider the supersonic flow past a cone
  with straight generators
• (This amounts to determine the main part of the
  potential)

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Make the following approximate expansion

• satisfies
• 
  
• 
  on the surface of body
• 
  on
  the shock

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Introduce the partial hodograph transformation
It changes the position of the fix boundary and
the shock, so that it fix the location of the
shock.
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Shock (fixed)
Surface of the body (free)
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To avoid the new free boundary we make a domain
decomposition, so that a series of boundary value
problems are introduced
The equation in (r,q) coordinate. It takes
original boundary condition on the surface of the
body.
The equation in (p,s) coordinate. It takes
original Boundary condition on shock.
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• Via alternative iteration we established a series
  of solutions of these sub-roblems, then like a
  manifold we glue up all of them to get a solution
• By solving all components and then
  construct an approximate solution with higher
  order

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• Looking a general curved cone as a perturbation
  of a cone with straight generators

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Again apply blow-up and a
generalized transformation to lead a new boundary
value problem with fixed boundary

• By solving the new boundary value problem and
  applying the inverse transformation we finally
  establish the existence of the solution of the
  original problem.

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Theorem supersonic

sharp vertex angle
small perturbation of a cone
where e is a sufficiently small number, then
there exists a stable piecewise smooth solution
for the assigned problem satisfying all boundary
conditions.
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• The rigorough proof of the existence and
  stability of the solution with an attached shock
  for the supersonic flow past a sharp conical body
  clarifies the role and the character of such a
  motion from the mathematical point of view, so
  that offers a solid foundation for all related
  experiments and computations.

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• Many further study
• On the existence and asymptotic behaviour of
  global solution W.Lian, T.P.Liu
• S.X.Chen, Z.P.Xin, Y.C.Yin,
• Jun Li, Dachen Cui,
• On the uniqueness
• Hairong Yuan, Li Liu

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• Supersonic flow past abodies with nonsmooth
  boundary
• Y.Q.Zhang, G.Q.Chen,
• Supersonic flow with combustionY.Q.Zhang,
• Application to piston problems
• S.X.Chen, Z.J.Wang,

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How to determine the appearance of the weak shock
or the strong shock?

• S.X.Chen, B.X.Fang, G.Q.Chen, Y.C.Yin, Gang Xu
• V.Elling, T.P.Liu

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• Many aircrafts with hypersonic speed are often
  globally designed as a triangle

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Supersonic flow past a triangle
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Chen Shuxing Yi Chao (2014)

• Theorem When the vertex angle of the triangle
  wing is near to p, the incoming flow is
  supersonic and with a small attack angle, then
  there exists an solution of the problem with a
  shock attached on the two edges of the wing.

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Open Problem 1

• Supersonic flow past a conical body in the case
  with big disturbance

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Open Problem 2

• For the supersonic flow past a wedge in the case
  when the attack angle is negative, or change its
  sign along the edge of the wing

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Open Problem 3

• Supersonic flow past a general thick triangle
  wing

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Open Problem 4

• Study the problem of supersonic flow past bodies
  under the scheme of Euler system

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Open Problem 5Supersonic flow past a blunt body
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Thank you !
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