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 Wen Long August 20, 2002 22:35

I'm reading Stephen J. Cowley's review :"Laminar Boundary-layer theory: a 20th century paradox?"

It surely makes sense to me, and I'd like to share some quick thoughts with everybody.

Classical boundar layer theory employes the assumption

delta/L << 1

where delta is the boundary layer thickness, L is the streamwise length scale. And this assumption gives rise to a critical point:

Re=rho*u*L/mu >>1

The above assumption is used to reduce the horizontal momentun equation to be:

u_t + uu_x + v u+y =-p_x/rho+ mu * u_yy

and vertical momentun equation to be :

0=-p_y

The upper Boundary layer equation successfully explained the phenomena of steady flow past flat plate which is famous Blasius(1908) solution. But it came out to be a disaster for flow past a circular cylinder. Why? Separation occurs at the lee side for Re>>1 flow. If RE ~1 is creeping flow not going to have thin boundary layer. And for the thin boundary layer to occure, Re has to >>1 and turns out to be turbulent and separation flow.

So there is NO Laminar Boundary Layer flow past a Circular Cylinder or a backstep. This is the paradox.

The important point is at lee side of the cylinder, the assumption delta << L is not true. Because there a large L doesn't exist and a small delta doesn't exist either. So boundary layer theory fails there drastically.

In fact, at the lee side of the cylinder, there is an adverse pressure gradient, i.e. p_x >0 , and there is an inflection point of u(y), a flow profile with an inflection point is apt to be unstable, and will take the transition to turbluence fastly.

What I want to say is that, boundary layer theory will fail not only for cylinder/blunt body case, but also for flat plate/wall case when there is an adverse pressure gradient at play. Such as surface waves on flat bottom. Here outer flow is oscillatory, and when the waves are big enough, H/h ~ O(1), H is height and h is depth. The bottom boundary layer thickness delta is not small at all hence can't call it a "boundary layer" and separation of flow will occure. I guess the separation is closely related to the motion of bottom sediments in harbor/coastal engineering. I hope that someone can give a clear understanding about it in the near future,

wen

 Axel Rohde August 21, 2002 04:00

Classical B.L. theory is only valid up to the point of separation. Then it no longer applies.

 Steve August 21, 2002 16:30

The circular cylinder does not experience transition until Re= 400000. Between 1 < Re < 400000 the boundary layer is so sluggish that as soon as it hits the adverse pressure gradient it leaves the surface as laminar separation.

 Tom September 24, 2002 08:47

This result has been known for a long time - the singularity is known as the Goldstein singularity (Goldstein; 1948) and involves the skin friction tending to zero as sqrt(x_s - x) where x_s is the separation point and x is a point upstream of x_s. This square root behaviour then induces an infinite, like 1/sqrt(x_s - x), vertical velocity which violates the assumption that that the vertical velocity is small.

A similar phenomenon occurs in the unsteady boundary layer equations (van Dommelen and Shen - don't have the reference to hand) but now instead of the singularity occurring at the wall it occurs at the edge of the boundary layer.

Basically the singularity tells you that the assumption that you can solve the inviscid outer flow independently of the boundary layer is incorrect and when the flow is near separation account must be taking of the displacement of the boundary layer upon the inviscid outer flow - this in turn forces the boundary layer in an feedback loop. This is the essence of Stewartson's triple-deck theory.

In practice, on a bluff body such as a circular cylinder, the boundary layer actually separates on the windward side of the body where inviscid potential theory says the pressure gradient is favourable! This fact was used in the seminal paper of Sychev (1971?) to show how, using tripledeck theory, the separation singularity could be removed if the inviscid flow was of Kirchoff type with the freestream line leaving the body smoothly. It's this final condition which forces the separation point to the windward side of the cylinder!

As a final point the it takes about a 10% increase in the pressure for separation to occur within the boundary layer - this is one of the reasons that thin aerofoils (5% thickness) don't separate. However once the suction peak has been passed the streamwise velocity develops an inflection point and is unstable to waves whose length is of the order of the boundary-layer thickness. These Rayleigh waves are the early stages of transition within the boundary layer.

I think Stephen's paradox is simply the fact that, at the Reynolds numbers where boundary layer theory is valid, the flow should be fully turbulent and so is not applicable while at low(ish) Reynolds numbers it can be quite accurate.

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