What is displacement thickness in boundary layer theory?

Short Answer:

In boundary layer theory, displacement thickness is the imaginary distance by which the solid surface would need to be moved outward to account for the reduction in flow rate due to the presence of the boundary layer. It measures the effect of the slowed-down fluid near the surface on the overall flow.

Because the fluid velocity near the surface is less than the free-stream velocity, the mass flow rate is reduced. Displacement thickness helps engineers understand this loss and is important in designing shapes for pipes, airfoils, and hydraulic surfaces to minimize flow resistance.

Detailed Explanation:

Displacement thickness in boundary layer theory

In fluid mechanics, especially in boundary layer analysis, the fluid near a solid surface does not flow at full speed because of viscous effects. The fluid particles stick to the surface (no-slip condition) and gradually accelerate away from it. This creates a velocity profile in the boundary layer where some of the fluid moves slower than the free-stream velocity.

Due to this slowing down, there is a reduction in the actual mass flow rate through the section compared to an ideal case with no boundary layer. To account for this difference, engineers use the concept of displacement thickness, denoted as δ∗\delta^*δ∗ (delta star).

Meaning and Formula

Displacement thickness is defined as the distance by which the outer inviscid flow must be shifted outward so that the reduced mass flow due to the boundary layer matches the full free-stream flow outside the boundary layer.

Mathematically, it is given by:

δ∗=∫0∞(1−u(y)U)dy\delta^* = \int_0^{\infty} \left(1 – \frac{u(y)}{U}\right) dyδ∗=∫0∞​(1−Uu(y)​)dy

Where:

  • δ∗\delta^*δ∗ = displacement thickness
  • u(y)u(y)u(y) = velocity at height yyy above the surface
  • UUU = free-stream velocity
  • dydydy = small thickness of fluid layer

This equation shows that displacement thickness depends on how much the velocity at each point in the boundary layer differs from the full-speed flow.

Physical Interpretation

Imagine a flat plate in a flow. If there were no boundary layer, the full free-stream velocity would exist at every point above the surface, and mass flow would be maximum. However, the boundary layer slows the fluid near the surface. This loss of velocity behaves as if the solid plate was a little thicker, reducing the space available for flow. This “imaginary increase” in thickness is the displacement thickness.

Importance in Civil Engineering

  1. Flow Prediction
    It helps in predicting how the boundary layer affects the flow around surfaces like bridge decks, canal linings, or spillways.
  2. Shape Design
    Engineers use displacement thickness in streamlining designs to reduce flow resistance and improve performance in water-carrying systems or structures exposed to wind or water.
  3. Pressure Distribution
    It helps in calculating how flow changes affect the pressure on surfaces, which is critical in structural analysis.
  4. Computational Fluid Dynamics (CFD)
    Displacement thickness is often used in CFD simulations for setting boundary conditions or comparing flow behavior with real-world observations.
  5. Control of Flow Separation
    Knowing how the boundary layer displaces the main flow can help in delaying or avoiding flow separation, which is essential in civil and hydraulic systems.
Conclusion:

Displacement thickness in boundary layer theory is the distance by which the flow would need to shift outward to compensate for the slower velocity near the surface. It represents the flow rate reduction due to the boundary layer and is important for predicting flow behavior, designing smooth surfaces, and minimizing energy losses in civil engineering systems.