Subject: Fluid Mechanics 2

Topic: Boundary layer theory

Difficulty: Low

BOUNDARY LAYER SEPARATION

• Consider the flow of fluid past a solid surface. let P represent the static pressure at any point and x denote the distance measured along the flow direction, then$\frac{dp}{dx}$ is called the pressure gradient

• For the flow of a fluid over a smooth thin plate which is flat and placed parallel to the direction for free stream of fluid, $\frac{dp}{dx}$ is zero.

• If the pressure gradient is zero, the boundary layer thickness increases continuously with increase in distance from the leading edge of the plate along the flow.

• But for flow over a curved surface the pressure gradient is not zero and hence its effect on boundary layer growth is to be considered.

• Consider the flow over a curved surface as shown in the figure.

• In the region ABC of the curved surface, the flow area decreases and therefore the velocity increases. This means that the flow is accelerated. Due to increase in velocity, the pressure decreases in the direction of flow.

• Therefore the pressure gradient $\frac{dp}{dx}$ is negative in the region of ABC.

• In this case the pressure force acts in the direction of flow, against the frictional resistance. That gives the fluid an additional push in the direction of flow.

• Negative pressure gradient is called favourable pressure gradient.
• The boundary layer moves forward in the region ABC
• In the region CSD of curved surface, the pressure is minimum at point C.

• Along the path CSD of the curved surface, the area of flow increases and hence the velocity decreases and the pressure increases in the direction of flow. Therefore the pressure gradient $\frac{dp}{dx}$ is positive in the region of CSD.

• Here the pressure force acts opposite to the direction of flow. The positive pressure gradient is called adverse pressure gradient.

• This positive pressure gradient is unfavourable because some kinetic energy of fluid particles is utilized to increase the pressure. Also some part of kinetic energy is used to overcome the surface friction of the solid body.

• Thus the combined effect of adverse pressure gradient and surface frictional resistance reduce the momentum of the fluid.
• A stage comes when the momentum of the fluid is unable to overcome the surface resistance and the boundary layer starts separating from the solid boundary at point S.
• After the point S the flow direction in the boundary layer is reversed and the velocity gradient becomes negative.
• The position of the point of separation depends on the roughness of surface, Reynolds number, the geometry of curved surface, and nature of boundary layer.
• Turbulent boundary layer resists separation better than laminar boundary layer.
• This is because turbulent boundary layer possess more energy than laminar boundary layer and are capable of moving against a great pressure gradient without separation.

Location of Separation Point

At Separation point S,

$$ \left(\frac{d u}{d y}\right)_{y=0}=0 $$

• For a given velocity profile, it can be determined whether the boundary layer is separated, or on the verge of separation or will not separate using the following conditions:

1. If $\left(\frac{d u}{d y}\right)_{y=0}$ is negative, the flow is separated

2. If $\left(\frac{d u}{d y}\right)_{y=0}$ the flow is on the verge of separation

3. If $\left(\frac{d u}{d y}\right)_{y=0}$is positive, the flow will not separate

METHODS TO PREVENT BOUNDARY LAYER SEPARATION

• When the boundary layer separates from the solid surface, a certain portion adjacent to the surface has a back flow or reverse flow and eddies are continuously formed in this region and hence continuous loss of energy takes place.
• Thus separation of boundary layer is undesirable and attempts should be made to avoid separation by various methods.

Following are the methods for preventing separation of boundary layer

a. Suction of slow moving fluid by a suction slot

b. Supplying additional energy from a blower

c. Providing a bypass in the slotted wing

d. Providing rotating cylinder near leading edge

e. Providing small divergence in a diffuser

f. Providing guide blades in a bend

g. Providing a trip-wire in the laminar region for the flow over a sphere