Separation in B.L.T. context

Separated FlowsWakes and Cavities

2.1 Boundary Layer Theory (Prandtl)

pressure uniform accross the B.L.

Simplification of the Navier-Stokes equation for large Reynolds flows

inviscid solution (Euler)thin boundary layer

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

the lower the velocity the greatest the rate of change of the velocity

u/Um(x)

y

no pressure gradient

adverse

favourable

BL

External stream

Const

1

thinningthickening

In the BL, close to the wall and at the wall

The pressure gradient is observable in the curvature of the velocity profile

negative zero positive

2.2 Attached boundary layers

thicheningthinning

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

Potential flow theory for inviscid part of the flow:

adverse 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

2.2 Attached boundary layers

2.3 Separated boundary layers

BLT Integration for BL

Starting point

S

Goldstein singularity :

Pressure gradient from real pressure field

BLT limitations

zone of adverse pressure gradient

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

The flow ejection (lower deck) at separation introduces a pressure gradient in the external flow

upper deck

main deck

lower deck

2.4 Unsteady flow separation

Initial condition such that at t=0 the flow is potential, i.e. inviscid everywhere :

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:

Dynamics :

After ?

Circular cylinder impulsively started from rest at constant velocity.

2.4 Unsteady flow separation

Dynamics :

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

2.4 Unsteady flow separation

Application to the circular cylinder

a

Ut 0

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.