1. Free Streamline Theory Separated Flows Wakes and Cavities

2. Flow approximation Viscosity is necessary to provoke separation, but if we introduce the separation "by hand", viscosity is not relevant anymore. Solves the D'Alambert Paradoxe : Drag on bodies with zero viscosity

3. 3.1 Flow over a plate The pressure (and then the velocity modulus) is constant along the separation streamline = The separation streamline is a free streamline   is the cavity parameter

4. 3.1 Flow over a plate

5. Separation has to be smooth otherwise U=0 at separation is not consistent with the velocity on the free stream line Form of the potential near separation

6. 3.1 Flow over a plate Cases study with k

7. 3.1 Flow over a plate Villat condition U S =U  : the cavity pressure is the lowest Subcritical flow Supercritical flow 1. Separation angle deduced from Villat condition (k= 0 at separation) 2. Pressure cavity is prescribed to p 

8. 3.1 Flow over a plate Subcritical flow Supercritical flow 1. Separation angle is prescribed and k>0 2. Pressure cavity is prescribed to p 

9. 3.1 Flow over a plate

10. Flow boundaries in the z-plane (physical space) Represent the flow in the  -plane and then apply the SC theorem (W=0)

11. 3.1 Flow over a plate Show that +1

12. 3.1 Flow over a plate Represent the flow in the W-plane and then in the W 1/2 plane (W=0)

13. 3.1 Flow over a plate Show that : +1

14. 3.1 Flow over a plate Correspondance between two half planes gives : Extract and show that :

15. 3.1 Flow over a plate Compute z 0 and k = d/(4+  ) and the shape of the free streamline

16. 3.1 Flow over a plate From the pressure distribution around the plate, the drag is: In experiments, C D  2

17. Similar problem with circular cylinder : C D0 =0.5 while in experiments C D  1.2 The pressure in the cavity is not p , but lower !

18. 1. Separation angle is prescribed and k>0 2. Pressure cavity is prescribed to p b It is a fit of the experimental data ! Improvment of the theory

19. 3.1 Flow over a plate Work only if the separation position is similar to that of the theory at p c =p  ( i.e.  C =0, is called the Helmholtz flow that gives C D0 )

20. 3.1 Flow over a plate A cavity cannot close freely in the fluid (if no gravity effect)  Closure models L/d ~ (-C pb ) -n

21. Limiting of the stationary NS solution as Re  ∞ Academic case L ~ d Re Imagine the flow stays stationary as Re  ∞ free streamline theory solution (b) and (c) Stationary simulation of NS (a) Theoretical sketch A candidate solution of NS as Re  ∞ ? Cpb  0 Cx  0.5 L = O(Re) : infinite length Kirchoff helmholtz flow :

22. Limiting stationary solution as Re  ∞ Academic case Cpb>0 !!! C D  0 ? Numerical simulation

23. Limiting stationary solution as Re  ∞ Academic case (b) and (c) Stationary simulation of NS (a) Theoretical sketch A possibility :Non uniqueness of the Solution as Re  

24. Super cavitating wakes Kirchoff helmholtz flow ? : vapor liquid

25. Super cavitating wakes