Shuxing Chen （ Fudan University ）

1. Shuxing Chen（Fudan University） Mathematical Analysis of Supersonic flow past bodies

2. 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.

3. 核能利用 燃烧与爆破 飞行器设计，航空航天技术

5. 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.

6. 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.

7. Supersonic flow past wing shock wing Supersonic flow

8. Supersonic flow past conical body Conical body shock Supersonic flow

9. 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.

10. 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.

11. Euler System

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

13. Main difficulties • Nonlinearity • Multidimensional • Free boundaries • Strong singularity • Mixed type equations

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

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

16. 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.

17. 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.

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

19. Chen Shuxing (3-d wings） Existence of local solution to supersonic flow around a three dimensional wing (Advances in Appl. Math. 1992)

21. 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.

22. Chen Shuxing （Curved cone） Existence of stationary supersonic flow past a pointed body (Archive Rat.Mech. Anal. 2001)

24. Mathematical formulation

25. 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.

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

27. Make the following approximate expansion satisfies on the surface of body on the shock

28. 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.

29. Shock (fixed) Surface of the body (free)

30. 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.

31. 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 componentsand then construct an approximate solution with higher order

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

33. 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.

34. Theorem: supersonic sharp vertex angle small perturbation of a cone wheree is a sufficiently small number, then there exists a stable piecewise smooth solution for the assigned problem satisfying all boundary conditions.

35. 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.

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

37. 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, …

38. 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

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

42. Supersonic flow past a triangle

43. 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.

44. Open Problem 1 Supersonic flow past a conical body in the case with big disturbance

45. 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

46. Open Problem 3 Supersonic flow past a general thick triangle wing

47. Open Problem 4 Study the problem of supersonic flow past bodies under the scheme of Euler system

48. Open Problem 5Supersonic flow past a blunt body

49. Thank you ！