Separation of Boundary Layers

The conditions at the extrema of the Boundary Layer are:-

Outer edge ( $y = \Delta$): $u = u_{o} , \ \frac{\partial u}{\partial y} = 0 , \ \frac{\partial^{2} u}{\partial y^{2}} = 0$

Wall (y = 0): $u = 0 , \ v = 0 , \ \tau_{o} = \mu \left(\frac{\partial u}{\partial y} \right)_{o}$

The Navier Stokes Equation for the Boundary Layer,

\begin{displaymath}u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial ...
...tial P}{\partial x} + \nu\frac{\partial^{2}
u}{\partial y^{2}}
\end{displaymath} (1)

(negligible terms already eliminated) reduces, at y = 0, to

\begin{displaymath}\frac{1}{\rho}\frac{\partial P}{\partial x} = \nu\frac{\partial^{2}
u}{\partial y^{2}}
\end{displaymath} (2)

or allowing for boundary layer suction

\begin{displaymath}v\frac{\partial u}{\partial y} + \frac{1}{\rho}\frac{\partial P}{\partial x}
= \nu\frac{\partial^{2} u}{\partial y^{2}}
\end{displaymath} (3)

(Boundary layer suction, ie v < 0, makes the first term negative, since $\frac{\partial u}{\partial y}$ is positive, thus reducing the destabilising effect of a positive value of $\frac{\partial P}{\partial x}$).

Returning to the case with no suction at the wall, v = 0 and equation (2) applies.

If the external pressure gradient, $\frac{\partial P}{\partial x} = 0$, then, $\frac{\partial^{2} u}{\partial y^{2}} = 0$ at the wall and hence $\frac{\partial u}{\partial y}$ is at a maximum there and falls away steadily.

If, however, $\frac{\partial P}{\partial x} > 0$, we have an adverse pressure gradient, and $\frac{\partial^{2} u}{\partial y^{2}} > 0$ at the wall. Hence, $\frac{\partial u}{\partial y}$ must first increase and then decrease as y increases.

As the severity of the adverse pressure gradient increases, the velocity profiles become increasingly distorted until $\left(\frac{\partial u}{\partial y} \right)_{o} = 0$and then even becomes negative.

The flow pattern then looks like this, and the effect is known as `separation'.

This phenomenon is particularly well marked in the case of flow round a cylinder, when bound vortices form in the wake at Re > 1.

Note that the pressure gradient is of the form

The flow patterns obtained on increasing Re are as follows

For a sphere (similar to cylinder), the drag coefficient chart then looks something like this:-

The onset of the turbulent boundary layer may be promoted by roughening the sphere.

Indeed, laminar boundary layer flow is unstable. It will, in any case, become turbulent if $\frac{u_{o} \delta^{*}}{\nu} > 600$, but may do so at lower Re values if, for example, the wall is rough.

Effect of Separation

From the above discussion we see that separation occurs at the point where $\left(\frac{\partial u}{\partial y} \right)_{o} = 0$.

Whilst separation occurs in both laminar and turbulent flows, it has been studied to a greater extent in turbulent flows. This is because
a) Turbulent flows are more commonly encountered than laminar flows.
b) Separation is more likely to occur when the flow is turbulent.
c) Due to inertial effects, separation has a much greater influence in turbulent flows. It accounts for the majority of the drag on a bluff body. Delaying separation until near the end of a bluff body can greatly reduce drag. This is borne out by the fact that the drag on a streamlined body is around $\frac{1}{15}$ of that on a cylinder of the same frontal area.

The effect of separation is to carry vorticity (normally constrained to the boundary layer) into the bulk flow. The presence of separation therefore modifies the inviscid flow solution, breaking down the basis of the assumption that the (inviscid) flow solution outside the boundary layer is independent of the boundary layer.

Prevention of Separation

1) Move the boundary with the stream. eg for a rotating cylinder there may be no boundary layer on the side of the cylinder rotating with the flow.
2) Suction of fluid through the wall (eg porous diffuser). This removes decelerated fluid from the wall region.
3) Acceleration of the boundary layer (blowing), eg slotted wing. Gives a high kinetic energy to fluid in the boundary layer to overcome the adverse $\frac{\partial P}{\partial x}$.
4) Vanes to direct flow (eg before bend in diffuser).

Attachment/Reattachment (Opposite of Separation)

Coanda Effect

A jet of fluid normally entrains surrounding fluid as its momentum diffuses outwards. If the jet is close to a wall then the jet cannot draw fluid through the wall and the jet itself is attracted towards the wall, attaching itself to the wall.

A similar phenomenon can occur with boundary layers. The laminar boundary layer separates more easily than a turbulent boundary layer under an adverse pressure gradient due to lack of inertia of the fluid within the boundary layer. This fluid on entering the bulk flow will usually become turbulent and, acting like a turbulent jet, may reattach itself to the wall as a turbulent boundary layer unless the pressure gradient is great enough to prevent its reattachemnt.

David Balmer
Last modified: Wed Dec 2 16:09:02 GMT