The displacement thickness (δ*) in the boundary layer is given by the following formula:
δ* = ∫0h (1 - u/Ue) dy
Where:
- δ* is the displacement thickness.
- u is the velocity within the boundary layer at a height y.
- Ue is the free stream velocity (velocity outside the boundary layer).
- y is the distance normal to the wall.
- h is a distance sufficiently large to be outside the boundary layer (theoretically, this is often represented as infinity, but practically, a finite distance beyond the edge of the boundary layer is used).
Explanation
The displacement thickness represents the distance by which the outer potential flow is displaced away from the wall due to the reduced velocity within the boundary layer. In other words, the boundary layer effectively makes the wall appear thicker to the external flow. The formula calculates this effective thickening.
The term (1 - u/Ue) represents the velocity deficit within the boundary layer, compared to the free stream velocity. The integral sums this velocity deficit over the height of the boundary layer to determine the displacement thickness.
Derivation (brief)
The formula can be derived by considering the conservation of mass. The mass flow rate within the boundary layer is less than what it would be if the velocity were uniformly equal to the free stream velocity Ue. The displacement thickness is the distance required to reduce the mass flow rate of a uniform stream with velocity Ue by an amount equal to the deficit in mass flow rate caused by the presence of the boundary layer.
Significance
The displacement thickness is crucial in various fluid mechanics applications, including:
- Calculating pressure gradients: It affects the pressure distribution around an object in a flow.
- Analyzing aerodynamic performance: It influences the lift and drag characteristics of airfoils and other aerodynamic shapes.
- Designing flow control devices: It's used in the design of devices aimed at manipulating the boundary layer.
