Short Answer:

Momentum thickness is defined as the distance by which the boundary layer should be displaced from the solid surface so that the loss of momentum in the boundary layer is equivalent to that of the actual flow. It represents the reduction in momentum flow due to viscous effects within the boundary layer.

In simple terms, momentum thickness measures how much the momentum of the real (viscous) fluid is reduced compared to an ideal flow without viscosity. It is an important parameter in analyzing drag, shear stress, and energy losses in fluid systems.

Detailed Explanation:

Momentum Thickness

When a viscous fluid flows over a solid surface, such as a flat plate, the fluid particles in contact with the surface experience friction, causing their velocity to reduce to zero due to the no-slip condition. As we move away from the surface, the velocity of the fluid increases gradually until it reaches the free-stream velocity .

This region, known as the boundary layer, has a velocity profile that varies from zero at the surface to nearly  at the outer edge. Because of the reduced velocity near the wall, the momentum of the fluid particles in this region is lower than that in the free stream. The total momentum of the flow through the boundary layer is therefore less than that of an ideal, inviscid flow where all particles move at .

The momentum thickness, denoted by , is a theoretical distance that represents this loss of momentum as an equivalent uniform displacement of the free-stream flow. It helps engineers calculate drag forces and analyze the performance of surfaces in contact with moving fluids, such as airplane wings, ship hulls, and turbine blades.

Definition of Momentum Thickness

The momentum thickness is mathematically defined as:

Where,

•  = Momentum thickness (m)
•  = Local velocity of fluid at a distance  from the wall (m/s)
•  = Free-stream velocity (m/s)
•  = Distance measured normal to the surface (m)

This equation expresses the momentum deficit caused by viscous effects in terms of an equivalent thickness. The term  represents the ratio of local velocity to free-stream velocity, and the integral accounts for the entire boundary layer region.

Physical Meaning

The concept of momentum thickness can be better understood by comparing the real and ideal flows:

• In an ideal, non-viscous flow, all fluid particles would move at , so there would be no momentum loss.
• In a real viscous flow, however, the velocity near the wall is much lower due to friction, causing a loss in momentum.

The momentum thickness  represents the imaginary shift of the free stream that would produce the same momentum loss as the real flow. Essentially, it quantifies the momentum defect within the boundary layer.

A larger momentum thickness means greater momentum loss, indicating higher drag and friction on the surface.

Momentum Thickness for Laminar Boundary Layer

For laminar flow over a flat plate, the velocity distribution within the boundary layer can be approximated by empirical relations. Using these relations, the momentum thickness is given by:

Where,

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

This equation shows that for laminar flow, the momentum thickness increases with the square root of the distance from the leading edge and decreases with higher Reynolds number.

Momentum Thickness for Turbulent Boundary Layer

For turbulent flow over a flat plate, where fluid motion is irregular and highly mixed, the velocity distribution is flatter and fuller compared to laminar flow. The approximate empirical relation for turbulent flow is:

This indicates that the momentum thickness grows faster with distance along the surface in turbulent flow than in laminar flow, due to greater mixing and higher momentum transfer.

Importance of Momentum Thickness

1. Determines Skin Friction Drag:
   Momentum thickness directly relates to the shear stress on the surface and helps estimate drag forces on objects moving through fluids.
2. Analyzes Boundary Layer Growth:
   It helps in understanding how the boundary layer develops and changes along the surface of a body.
3. Flow Separation Prediction:
   By studying the variation of momentum thickness, engineers can predict points where the flow may separate from the surface, which is critical in aerodynamic design.
4. Energy Loss Evaluation:
   It helps calculate losses due to friction and viscous effects in flow systems.
5. Used in Momentum Integral Equation:
   Momentum thickness is an important part of the momentum integral equation, which simplifies boundary layer analysis for engineering applications.

Relationship with Other Thicknesses

The boundary layer is characterized by three different thicknesses, each representing a different physical property:

1. Displacement Thickness () – Represents the reduction in mass flow rate due to the boundary layer.

1. Momentum Thickness () – Represents the reduction in momentum flow due to the boundary layer.

1. Energy Thickness () – Represents the reduction in kinetic energy due to the boundary layer.

Among these, momentum thickness provides the most direct link to drag and shear stress calculations.

Applications in Engineering

1. Aerodynamics:
   Used to estimate drag on aircraft wings, car bodies, and other aerodynamic surfaces.
2. Hydraulic Systems:
   Helps in predicting pressure drop and energy loss due to viscous effects.
3. Turbomachinery:
   Used in the design of turbine blades and compressors to minimize frictional losses.
4. Naval Architecture:
   Assists in calculating the resistance of ships and submarines moving through water.
5. Flow Control Devices:
   Helps design boundary layer control mechanisms to delay separation and improve performance.

Conclusion

The momentum thickness represents the distance by which the external flow must be displaced to account for the momentum loss caused by viscous effects in the boundary layer. It quantifies the momentum defect of the real flow compared to the ideal one and is vital in determining drag, shear stress, and energy losses. The concept is central to boundary layer theory and is used in analyzing both laminar and turbulent flow behavior in various engineering systems.