1. Separation in B.L.T. context
Separated Flows Wakes and Cavities
Separation in B.L.T. context

2. 2.1 Boundary Layer Theory (Prandtl)
Simplification of the Navier-Stokes equation for large Reynolds flows
inviscid solution (Euler)
thin boundary layer
pressure uniform accross the B.L.

3. 2.1 Boundary Layer Theory (Prandtl)
using the dimensionless quantities in the NS eq.
In the limit of large Re:

4. 2.1 Boundary Layer Theory (Prandtl)
The pressure gradient is considered to be given by the inviscid flow, approximated by Bernoulli's theorem :

5. 2.1 Boundary Layer Theory (Prandtl)
Problem to solve : flow around a body
Compute the inviscid flow around the body given by the potential flow theory to have the pressure gradient.
Use this pressure gradient to compute the history of the BL on the body.
Is it easy ?
Inviscid solution is not unique.
Lack of coupling: the inviscid flow cannot react to the BL dynamics.
The BLT needs an initial velocity profile: what happens if back flow occurs for a given inviscid solution that do not have any back flow ?
It is not so easy !, and even very limited but gives the theoretical description for the understanding of a large Re flow at separation.

6. 2.2 Attached boundary layers
In the external stream
Const
no pressure gradient y
External stream
adverse
the lower the velocity the greatest the rate of change of the velocity
favourable
thinning
thickening BL 1
u/Um(x)

7. 2.2 Attached boundary layers
In the BL, close to the wall and at the wall
The pressure gradient is observable in the curvature of the velocity profile
thinning
thichening
negative zero positive

8. 2.2 Attached boundary layers
Both the BL and the inviscid flow behaves in the same way :
Acceleration = thinning of the BL (favourable pressure gradient)
Deceleration = thickening of the BL (adverse pressure gradient)

9. 2.2 Attached boundary layers
adverse
favourable
Potential flow theory for inviscid part of the flow:

10. 2.2 Attached boundary layers
Result of the BLT (matched asymptotic theory):
m<0, non-uniqueness
m< back-flow : BLT not physical

11. 2.2 Attached boundary layers
Result of the BLT:
Show that the displacement thickness is :
and that the wall shear stress is :
Discussion:
Which is the flow with a constant boundary layer thickness ? why ?
Describe the flow with the constant wall shear stress

12. 2.2 Attached boundary layers
The BL starts to grow at the point when the dividing streamline coming from far upstream intersects the body.
Two cases for initial velocity profile :
Smooth edge : m=1, i.e. the BL of a stagnation point
Sharp edge : one member of the family m>=0

13. 2.3 Separated boundary layers
BLT limitations
Goldstein singularity :
BLT Integration for BL S
The solution blows when pressure gradient becomes adverse
Something is missing in the BLT !
Starting point
zone of adverse pressure gradient
Pressure gradient from real pressure field

14. 2.3 Separated boundary layers
Origin of the Goldstein's singularity
BLT = prime quantities are O(1) as Re  
near separation, due to fluid ejection : 
There is no possibility to balance this ejection (neither pressure gradient nor viscous diffusion in y-direction in BLT) '
The singularity develops in a sublayer :

15. 2.3 Separated boundary layers
The triple deck theory
upper deck
main deck
lower deck
The flow ejection (lower deck) at separation introduces a pressure gradient in the external flow

16. 2.4 Unsteady flow separation
Initial condition such that at t=0 the flow is potential, i.e. inviscid everywhere :
Dynamics :
No steady solution, the BL will grow and saturate due to the action of pressure gradient and viscous diffusion
Initial condition for a circular cylinder:
After ?

17. 2.4 Unsteady flow separation
Circular cylinder impulsively started from rest at constant velocity.
Dynamics :

18. 2.4 Unsteady flow separation
at t ~ 0, the equation can be approximated by u1 with :

19. 2.4 Unsteady flow separation

20. 2.4 Unsteady flow separation
f/erf is max at =0 (worth 1/0.7 at the wall).
the first back flow (u<0) occurs, if dU/dx < 0 at =0
The time that makes the wall shear stress zero :

21. 2.4 Unsteady flow separation
Application to the circular cylinder

22. 2.4 Unsteady flow separation
High Reynolds : secondary separations

23. 2.8 Conclusion
In the BLT context, the necessary condition for separation with the adverse pressure gradient corresponds to a deceleration of the potential flow (Bernoulli theorem).
For instance, if strong enough, it gives a "location" for which the friction at the wall is zero that will announce a separation. However the BLT of Prandtl is not complete and unable to give a description of the BL at separation.
At a separation the BL develops a triple layer structure (improvment of the Prandtl BLT)
As far as the potential flow solution is not much modified by the separation, the boundary layer theory remains relevant to predict the onset of separation
It is still relevant if the potential flow of the separated flow (chapter Free Streamline Theory) is used to compute the BL solution.