Shuxing Chen （ Fudan University ）

Mathematical Analysis of Supersonic flow past bodies

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.

飞行器设计，航空航天技术

核能利用 燃烧与爆破

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

Supersonic flow

wing

Supersonic flow past conical body

Supersonic flow

Conical body

shock

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

( ) 0

( ) 0

( ( )) 0

div mt

m m mdiv p

t

E E pdiv m

t

��������������

������������������������������������������

��������������

For the potential flow the system can be written as a second order

equation

1

( ( ) ( ( )) 0i i

N

t x xi

Nonlinearity Multidimensional Free boundaries Strong singularity Mixed type equations

Main difficulties

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

The usual technique for treating free boundary problem is not available.

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

satisfies

on the surface of body

on the shock

1 2

0

( , , ) ( , ) ( )N

n Nn

n

z r z r O z

1

0

( , ) ( ) ( )N

n Nn

n

s z s z O z

0 ( , )r

11 22 12

22

2 20 .2

2 ( , , ) 0,

1(1 ) 0,

1( ( ) ) ,

rr r r n

r

r r r r

a a a A

b b bb

r r r q qr

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.

22

2((1 ) 0p

uu upu

u Shock (fixed)

22

( )

'( )(1 ( )) ( ) 0

( ) p

u b

bb u b pu

b

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:

( )( ) :iNL The equation in (r,coordinate. It takes original boundary condition on the surface of the body.

( )( ) :eNL

The equation in (p,) coordinate. It takes original Boundary condition on shock.(1 1)k

• 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

0 ( , )r

( , )n r

, , )z r （

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

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.

tz e

where e is a sufficiently small number, then there exists a stable piecewise smooth solution for the assigned problem satisfying all boundary conditions.

Theorem: supersonicsharp vertex angle

small perturbation of a cone

q a （ ）*max ( , )b z b （ ）

2

1

0

(0, ) 0 (1 )

(0, ) k

kz

C

b k k

b b e

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 combustionY.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 5Supersonic flow past a blunt body