Separated Flows
Wakes and Cavities
Separation in B.L.T.
context
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.
2.1 Boundary Layer Theory (Prandtl)
using the dimensionless quantities in the NS eq.
In the limit of large Re:
2.1 Boundary Layer Theory (Prandtl)
The pressure gradient is considered to be given by the inviscid flow,
approximated by Bernoulli's theorem :
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.
2.2 Attached boundary layers
In the external stream
Const
no pressure
gradient
y
the lower the velocity the greatest
the rate of change of the velocity
External stream
adverse
favourable
BL
thinning
thickening
1
u/Um(x)
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
thichening
thinning
negative
zero
positive
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)
2.2 Attached boundary layers
adverse
Potential flow theory for
inviscid part of the flow:
favourable
2.2 Attached boundary layers
Result of the BLT (matched asymptotic theory):
m<0, non-uniqueness
m<-0.091 back-flow : BLT not physical
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
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
2.3 Separated boundary layers
BLT limitations
Goldstein singularity :
BLT
Integration
for BL
S
Starting
point
zone of adverse
pressure gradient
Pressure gradient from real pressure field
The solution blows
when pressure
gradient becomes
adverse
Something is
missing in the BLT !
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 :
'
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
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 ?
2.4 Unsteady flow separation
Circular cylinder impulsively
started from rest at constant
velocity.
Dynamics :
2.4 Unsteady flow separation
at t ~ 0, the equation can be approximated
by u1 with :
2.4 Unsteady flow separation
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 :
2.4 Unsteady flow separation
Application to the circular cylinder
t
U0
a
2.4 Unsteady flow separation
High Reynolds : secondary separations
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.