generalities separated flows wakes and cavities. 1.1 what is separation ? a streamline leaves the...
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Generalities
Separated FlowsWakes and Cavities
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
How is reversed vorticity introduced in the flow ?
After separationBefore separation
The key to understanding when separation may occur is :
1.2 The mechanism of smooth 2D separation
Steady 2D flow - x-momentum equation
At the wall (exact)
The pressure gradient at the wall creates a vorticity gradient at 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
-
+
Only negative vorticity at the wallNeed to introduce positive vorticity by viscous diffusion
vorticity gradient
vorticity transport by viscous diffusion
1.2 The mechanism of smooth 2D separation
can be realized if : >0
U
-
+
need to have a positive or adverse pressure gradient at the wall
since
1.2 The mechanism of smooth 2D separation
If is strong enough :
If is not strong enough the vorticity magnitude is
reduced but the vorticity not reversed = no separation
>0
>0
1.2 The mechanism of smooth 2D separation
negative zero (Blasius) positive
=
1.2 The mechanism of smooth 2D separation
At the wall: relationship between slope of vorticity, curvature of velocity and pressure gradient
Adverse pressure gradient at the wall is a necessary condition for 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.
Separation may occur as long as the flow develops a strong adverse pressure gradient (introducing reversed vorticity by viscous diffusion in the flow)
1.2 The mechanism of smooth 2D separation
1.3 local criteria : on-wall signature
shear at the wall (or skin friction)
1.3 local criteria : on-wall signature
At S, the wall shear stress (or wall vorticity) changes sign, It is zero at S.
(with boundary convention)
Prandtl criteria
For 2D flows, the shear is a scalar and :
1.3 local criteria : on-wall signature
Lighthill criteria
For 3D flows, it is more complicated ...
streamlines surface
roll-up into an eddy
On the separation line S, the skin friction is generally different from zero (shear along the line) Prandtl criteria not applicable
skin friction lines
• Skin friction lines convergence
• Zero skin friction
h
1.4 Low Re separation
Very low Re: no convection : upstream-downstream symmetry
an example at (Re=0.01)...
Where does the adverse pressure gradient come from ?
1.5 Intermediate Re separation - cylinder
A bit of convection : upstream-downstream symmetry is broken
1.5 Intermediate Re separation - cylinder
eddies
recirculation region L
reattachment
separation angle S
if Re= Ud/ > 4
1.5 Intermediate Re separation - cylinder
L ~ d Re where Re= Ud/
Re = 10
Re = 40
S
S
Viscous diffusion + advection : S ~ Cte +Re -1/2
Streamlines Vorticity
1.5 Intermediate Re separation - step flow
L ~ d Re
are the result of :
• horizontal advection by U
• vertical diffusion by viscosity
L
1.5 Intermediate Re separation - step flow
Re = 100
Re = 230
Re = 400
Re = 500
Steady
Unsteady
Rc = 350 threshold
6h
1.5 Intermediate Re separation - step flow
Re = 630
Re = 850
Re = 1050
Re = 1200
6h
Unsteady
1.5 Intermediate Re separation - step flow
Fixed separation points (separation at edge)
L varies as :
0
2
4
6
8
10
12
14
0 500 1000 1500 2000 2500 3000 3500Re
L/h
RehL st
eady
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
• Inviscid motion outside the layer
• Boundary layer equation inside the layer (Boundary Layer Theory, BLT)
• Matched asymptotic theory
d
1.6 Large Re separation
Re = 60; 100; 160; 210; 270; 2600
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8 2
x/xR
y/h
Re = 200
Re = 1000
The separated boundary layer
Mixing layer profile
LaminarTurbulent
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
xt -xS: transition point moves upstream as Re increases
xt-xS ~d Re -1/2
x>xt Kelvin-Helmholtz instability
S
xt
xt
increases downstream inertial instability
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.
The adverse pressure gradient can be either created by friction (creeping flows) or of inertia (Euler flows)
The separated boundary layer is similar to a mixing layer which entrains the flow from low speed region toward the ML.
How strong the adverse pressure gradient should be ?
We are going to study the case of large Reynolds number flows.