Short Answer:

Displacement thickness is defined as the distance by which the external or free-stream flow is displaced outward due to the formation of the boundary layer on a solid surface. It represents the reduction in mass flow rate caused by the presence of the slower-moving fluid within the boundary layer.

In simpler terms, displacement thickness shows how much the actual flow streamlines are pushed away from the surface compared to an ideal inviscid flow (without viscosity). It helps engineers calculate the additional drag and flow resistance caused by the boundary layer on bodies such as wings, plates, or pipes.

Detailed Explanation:

Displacement Thickness

When a viscous fluid flows over a solid surface, such as a flat plate, the velocity of the fluid at the surface becomes zero due to the no-slip condition. As we move away from the surface, the velocity increases gradually until it reaches the free-stream velocity . This gradual velocity variation creates the boundary layer, within which the flow is slowed down due to viscous effects.

Because of this velocity reduction near the surface, the actual mass flow rate in the boundary layer region is less than what would occur if the fluid moved at the free-stream velocity throughout. To maintain the same total mass flow rate, the outer streamlines of the flow must shift outward.

The measure of this outward shift or the imaginary displacement of the external flow due to the boundary layer formation is known as the displacement thickness, denoted by .

Definition

Mathematically, the displacement thickness is defined as:

Where,

•  = Displacement thickness (m)
•  = Local velocity of the fluid at a distance  from the surface (m/s)
•  = Free-stream velocity outside the boundary layer (m/s)
•  = Distance normal to the surface (m)

This equation expresses the loss of mass flow rate due to the slower velocities near the wall as an equivalent thickness, showing how much the outer flow is displaced.

Physical Meaning

The displacement thickness does not represent a real physical layer or boundary; instead, it is a theoretical concept used to account for the effect of the boundary layer on the outer flow. It indicates how much the boundary layer “pushes” the external flow away from the surface.

If there were no viscosity, the flow would move freely along the surface without any velocity loss, and the displacement thickness would be zero. However, in real flows, viscosity reduces the velocity near the wall, reducing the effective flow area and causing an apparent displacement of the outer streamlines.

In simpler terms:

• The larger the displacement thickness, the stronger the impact of viscous forces.
• A thicker boundary layer leads to a larger displacement thickness.

Displacement Thickness for Laminar Flow over a Flat Plate

For a laminar boundary layer over a flat plate, the velocity distribution can be described approximately by empirical relations. Based on such relations, the displacement thickness is given by:

Where,

•  = Distance from the leading edge (m)
•  = Local Reynolds number
•  = Kinematic viscosity (m²/s)

This shows that the displacement thickness increases along the length of the plate and decreases as the Reynolds number increases (higher velocity or lower viscosity).

Displacement Thickness for Turbulent Flow

In the case of turbulent flow, due to strong mixing and higher momentum transfer, the velocity gradient near the surface is steeper, and the displacement thickness is smaller relative to the total boundary layer thickness. It can be approximately expressed as:

This equation shows that even though turbulent flow has a thicker boundary layer, the displacement effect relative to its overall thickness is smaller than in laminar flow because of higher momentum exchange.

Importance of Displacement Thickness

1. Determines Flow Displacement Effect:
   It quantifies how much the external flow is shifted due to viscosity.
2. Useful in Aerodynamics:
   In aircraft design, displacement thickness helps in predicting drag and lift characteristics caused by the boundary layer.
3. Boundary Layer Analysis:
   It is an essential parameter for understanding how momentum and mass transfer occur within the boundary layer.
4. Flow Separation Prediction:
   A large displacement thickness can lead to early flow separation, reducing aerodynamic performance.
5. Energy Loss Estimation:
   It helps engineers estimate friction losses and design surfaces that minimize viscous effects.

Relation with Other Thicknesses

In boundary layer theory, two other important thicknesses are related to displacement thickness:

1. Momentum Thickness (θ):
   It represents the loss of momentum due to viscous effects within the boundary layer and is defined as:

1. Energy Thickness (δe):
   It represents the loss of kinetic energy due to the velocity gradient in the boundary layer:

Although these three parameters are related, displacement thickness specifically focuses on the mass flow reduction and streamline displacement effect.

Applications in Engineering

• Aircraft Wing Design: Used to calculate aerodynamic drag and streamline displacement around wings.
• Turbine and Compressor Blades: Helps in reducing losses caused by viscous effects.
• Ship and Submarine Design: Important for studying fluid resistance on hulls.
• Heat Exchangers: Helps understand thermal boundary layer effects.
• Pipelines: Used to predict flow losses and optimize internal flow surfaces.

Conclusion

The displacement thickness represents the imaginary distance by which the external flow is displaced outward because of the boundary layer formed near a surface. It indicates the reduction in mass flow due to viscous effects and provides a measure of how much the boundary layer affects the main flow. Mathematically, it is the integral of the velocity defect within the boundary layer. Displacement thickness plays a key role in aerodynamic design, drag estimation, and boundary layer control, helping engineers design efficient and streamlined systems.