Mathematical Analysis of
Supersonic flow past bodies
Shuxing Chen（Fudan University）
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.
核能利用 燃烧与爆破
飞行器设计，航空航天技术
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.
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.
Supersonic flow past wing
shock
wing
Supersonic flow
Supersonic flow past conical body
Conical body
shock
Supersonic flow
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.
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.
Euler System

 div(m)  0
t
m
mm
 div(
)  p  0
t

E
E p
 div(m(  ))  0
t
 
For the potential flow the system
can be written as a second order
equation
N
(  ( )t   ( xi  ( )) xi  0
i 1
Main difficulties
 Nonlinearity
 Multidimensional
 Free boundaries
 Strong singularity
 Mixed type equations
Many mathematicians
paid much efforts to the
study on the mathematical
analysis of supersonic flow
past bodies
Courant-Friedrichs in
1948 put forward this
problem, and give solutions
for the case, when the body
is a wedge or a cone.
• 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.
Gu Chaohao, Li Daqian
and others （1960’s)
• They studied the case when the
body is a curved wedge.
• They first applied the theory of
partial differential equations to
solve this problem.
The contributions of other
mathematicians ( for
instance, Peter Lax，David
Shaeffer )
Chen Shuxing (3-d wings）
Existence of local solution to
supersonic flow around a three
dimensional wing (Advances in
Appl. Math. 1992)
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.
Chen Shuxing （Curved cone）
Existence of stationary
supersonic flow past a
pointed body (Archive Rat.
Mech. Anal. 2001)
Mathematical formulation
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.
First consider the
supersonic flow past a
cone with straight
generators
(This amounts to determine
the main part of the potential)
Make the following approximate expansion
N
 ( z, r , )   z n 1 n (r , )  O( z N  2 )
n 0
N
s( z, )   sn ( ) z n  O( z N 1 )
n 0
   0 ( r , )
satisfies
a11 rr  a22   2a12 r  A( , r ,  ) n  0,
1
on the surface of body
b  2 b   (1  b 2 ) r  0,
b
1
( r2  2  2  (  r r )r r )  0  r r q   ,   q. on the shock
r
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.
u2
((1  u )  2  upu p   0
u
2
u  b( )
b '( )
(1  b ( ))  2
u  b( ) pu p  0
b ( )
2
Shock (fixed)
Surface of the body
(free)
To avoid the new free boundary we make a
domain decomposition, so that a series of
boundary value problems are introduced:
( NL)(i ) : The equation in (r,) coordinate. It takes
original boundary condition on the surface of the
body.
( NL)( e ) : The equation in (p,) coordinate. It takes original
Boundary condition on shock. (1   k  1)
• 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   0 (r, )
• By solving all components n (r, )
and then construct an approximate
solution with higher order （z, r , )
Looking a general
curved cone as a
perturbation of a cone
with straight generators
Again apply blow-upz  e
and
a generalized transformation to
lead a new boundary value
problem with fixed boundary
t
By solving the new boundary value
problem and applying the inverse
transformation we finally establish
the existence of the solution of the
original problem.
Theorem: supersonic（q  a）
sharp vertex angle （max b( z, )  b*）
small perturbation of a cone
 kz b(0, )  0
b(0, )  b0
C k2
(1  k  k1 )
e
where e is a sufficiently small number,
then there exists a stable piecewise smooth
solution for the assigned problem satisfying
all boundary conditions.
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.
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
• Supersonic flow past abodies with
nonsmooth boundary
Y.Q.Zhang, G.Q.Chen, …
• Supersonic flow with combustion
Y.Q.Zhang, …
• Application to piston problems
S.X.Chen, Z.J.Wang, …
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
Many aircrafts with
hypersonic speed are
often globally designed
as a triangle
Supersonic flow past a triangle
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.
Open Problem 1
Supersonic flow past a
conical body in the case with
big disturbance
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
Open Problem 3
Supersonic flow past a
general thick triangle wing
Open Problem 4
Study the problem of
supersonic flow past bodies
under the scheme of Euler
system
Open Problem 5
Supersonic flow past a blunt body