Generalities Separated Flows Wakes and Cavities. 1.1 What is separation ? A streamline leaves the...

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.