### Generalities

```Separated Flows
Wakes and Cavities
Generalities
1.1 What is separation ?
A streamline leaves the body and turns into the interior of the fluid
2D separation
3D separation
1.1 What is separation ?
Separation is intimitaley related to the no-slip condition
for instance: stagnation point flow
y=0
1.2 The mechanism of smooth 2D separation
Vorticity is an intrinsic local ingredient of the flow dynamics.
Vorticity at the wall :
Separation = reversed vorticity flow region
1.2 The mechanism of smooth 2D separation
The key to understanding when separation may occur is :
How is reversed vorticity introduced in the flow ?
Before separation
After separation
1.2 The mechanism of smooth 2D separation
Steady 2D flow - x-momentum equation
At the wall (exact)
the wall responsible for vorticity transport (diffusion) in the flow.
and this is the mechanism for the reversed vorticity introduction
1.2 The mechanism of smooth 2D separation
U
vorticity transport by viscous
diffusion
+
Only negative vorticity at the wall
Need to introduce positive vorticity by
viscous diffusion
1.2 The mechanism of smooth 2D separation
since
can be realized if :
>0
U
need to have a positive or
wall
1.2 The mechanism of smooth 2D separation
>0
If
If
>0
is strong enough :
is not strong enough the vorticity magnitude is
reduced but the vorticity not reversed = no separation
1.2 The mechanism of smooth 2D separation
At the wall: relationship between slope of vorticity, curvature of velocity and pressure gradient
=
negative
zero (Blasius)
positive

1.2 The mechanism of smooth 2D separation
Separation may occur as long as the flow develops a strong
viscous diffusion in the flow)
separation, but not sufficient.
Nothing has been said so far about the flow Reynolds number !
Actually, the mechanism for separation applies whatever the
Reynolds number is.
1.3 local criteria : on-wall signature
shear at the wall (or skin friction)
1.3 local criteria : on-wall signature
For 2D flows, the shear is a scalar and :
(with boundary convention)
At S, the wall shear stress
It is zero at S.
(or wall vorticity) changes sign,
Prandtl criteria
1.3 local criteria : on-wall signature
For 3D flows, it is more complicated ...
skin friction lines
roll-up into an
eddy
h
streamlines surface
• Skin friction lines
convergence
• Zero skin friction
On the separation line S, the skin
friction is generally different from zero
(shear along the line) Prandtl criteria
not applicable
Lighthill criteria
1.4 Low Re separation
an example at (Re=0.01)...
Very low Re: no convection : upstream-downstream symmetry
1.5 Intermediate Re separation - cylinder
A bit of convection : upstream-downstream symmetry is broken
1.5 Intermediate Re separation - cylinder
if Re= Ud/ > 4
eddies
recirculation region L
reattachment
separation angle S
1.5 Intermediate Re separation - cylinder
L ~ d Re where Re= Ud/
Re = 10
S
Streamlines
Vorticity
Re = 40
S
Viscous diffusion + advection : S ~ Cte +Re -1/2
1.5 Intermediate Re separation - step flow
L
L ~ d Re
are the result of :
• vertical diffusion by viscosity
1.5 Intermediate Re separation - step flow
Re = 100
Re = 230
Rc = 350
threshold
Re = 400
Re = 500
6h
1.5 Intermediate Re separation - step flow
Re = 630
Re = 850
Re = 1050
Re = 1200
6h
1.5 Intermediate Re separation - step flow
Fixed separation points (separation at edge)
L varies as :
L/h
L  h Re
14
12
10
8
6
4
2
0
0
500
1000
1500
2000
2500
3000
3500
Re
1.5 Large Re separation
Boundary layer separation and reattachment
THEORETICAL FRAME
Vorticity is only confined to the
solid boundary in a layer <<d.

 ~dRe-1/2
d
• Inviscid motion outside the layer
• Boundary layer equation inside the
layer (Boundary Layer Theory, BLT)
• Matched asymptotic theory
y/h
1.6 Large Re separation
The separated boundary layer
Re = 60; 100; 160; 210; 270; 2600
Re = 200
5
Laminar
Turbulent
Mixing layer profile
4,5
4
3,5
3
2,5
2
1,5
1
0,5
0
0
0,2
0,4
0,6
0,8
1
x/xR
1,2
1,4
1,6
1,8
Re = 1000
2
1.6 Large Re separation
Sketch of a separated boundary layer
Laminar
Turbulent
1.6 Large Re separation
Stability of the laminar separated boundary layer
increases downstream  inertial instability
x>xt Kelvin-Helmholtz instability
xt -xS: transition point moves upstream as Re
increases
xt
S
xt-xS ~d Re -1/2
xt
1.6 Large Re separation
Stability of the separated boundary layer
Re=100
Re=10000
1.6 Large Re separation
Stability of the separated boundary layer
Re=10000
1.7 Conclusion
An adverse pressure gradient at the wall is a necessary condition for
separation, but not sufficient.