Short Answer:

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

In simple terms, energy thickness indicates how much the energy of the fluid is reduced because of the slower-moving layers inside the boundary layer. It is denoted by  and is an important parameter in analyzing drag, energy losses, and flow efficiency in boundary layer theory.

Detailed Explanation:

Energy Thickness

When a viscous fluid flows over a surface such as a plate, wall, or wing, the velocity of the fluid particles near the surface becomes zero because of the no-slip condition. The velocity then gradually increases with the distance from the surface until it reaches the free-stream velocity () outside the boundary layer.

This variation in velocity across the boundary layer causes a reduction in the kinetic energy of the flow compared to the ideal case where viscosity is absent. The energy thickness is introduced as a measure of this loss in kinetic energy. It represents a hypothetical thickness by which the external flow must be displaced so that the energy loss due to viscous effects inside the boundary layer is equivalent to the energy loss in the actual flow.

It is an important part of boundary layer theory, as it helps engineers estimate friction losses, calculate drag, and study heat and mass transfer in engineering systems such as turbines, pipes, and aircraft surfaces.

Definition

The energy thickness is denoted by  and mathematically defined as:

Where,

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

This equation considers the loss of kinetic energy across the boundary layer as compared to the free-stream region. The term  represents the ratio of local velocity to the free-stream velocity, while the expression inside the brackets  corresponds to the kinetic energy deficit.

Physical Meaning

The concept of energy thickness helps to quantify the loss of kinetic energy due to viscosity in a real fluid flow.

• In an ideal flow (inviscid flow), all fluid particles move at , so no energy is lost, and .
• In a real viscous flow, however, the velocity of fluid particles near the wall is lower than , resulting in reduced kinetic energy.

The energy thickness expresses this energy loss as an equivalent imaginary layer of displaced fluid. It essentially measures the extent to which the boundary layer has decreased the total kinetic energy of the moving fluid compared to an ideal, frictionless flow.

The thicker the boundary layer, the larger the energy thickness, indicating higher viscous losses and greater drag.

Relation with Other Thicknesses

In boundary layer theory, there are three primary types of thicknesses used to describe different effects of viscosity:

1. Displacement Thickness ()
   • It represents the reduction in mass flow rate due to the boundary layer.

1. Momentum Thickness ()
   • It represents the reduction in momentum flow due to viscosity.

1. Energy Thickness ()
   • It represents the reduction in kinetic energy within the boundary layer.

From these equations, it is clear that:

This means that the energy thickness is the largest because it accounts for the loss of kinetic energy, which is more affected by velocity variations than mass or momentum losses.

Energy Thickness for Laminar Flow

For a laminar boundary layer over a flat plate, experimental and theoretical analyses give:

Where,

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

This shows that energy thickness increases with distance from the leading edge and decreases as Reynolds number increases.

Energy Thickness for Turbulent Flow

For turbulent flow, due to the higher degree of mixing and momentum transfer, the boundary layer becomes thicker. The corresponding energy thickness is approximately given by:

Turbulent flows have larger energy losses because of the random motion of fluid particles, which increases the shear stress and the rate of energy dissipation.

Importance of Energy Thickness

1. Indicates Energy Loss:
   It helps quantify the kinetic energy reduction caused by viscous effects in the boundary layer.
2. Drag Force Calculation:
   Used to estimate frictional resistance and drag on bodies moving through fluids.
3. Flow Separation Prediction:
   A large energy thickness may indicate regions where the flow is losing significant momentum, leading to flow separation.
4. Design of Aerodynamic Bodies:
   Engineers use energy thickness to design surfaces (like aircraft wings and turbine blades) for reduced energy loss and improved performance.
5. Boundary Layer Control:
   Helps in developing flow control techniques such as suction, blowing, or surface shaping to minimize energy loss.

Applications

• Aerodynamic Design: Used in predicting drag and optimizing lift-to-drag ratio.
• Turbomachinery: Helps in analyzing friction and energy losses in compressors, turbines, and pumps.
• Ship and Submarine Design: Used to estimate viscous resistance due to water flow.
• Heat Exchangers: Helps understand energy loss in the thermal boundary layer.
• Automobile Engineering: Useful in studying air resistance on vehicle bodies.

Conclusion

The energy thickness is a measure of the loss of kinetic energy in a fluid due to viscous effects within the boundary layer. It is defined as the distance by which the outer flow should be displaced to account for the reduction in kinetic energy compared to an ideal, inviscid flow. The value of energy thickness depends on flow conditions—being smaller for laminar flow and larger for turbulent flow. Understanding energy thickness is vital in predicting drag, flow separation, and energy efficiency in various mechanical and aerodynamic systems.