There are several definitions of boundary layer thickness, each quantifying different aspects of the velocity profile within the boundary layer. These include displacement thickness, momentum thickness, and energy thickness, in addition to the nominal boundary layer thickness.
Types of Boundary Layer Thickness
Here's a breakdown of the different types:
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Nominal Boundary Layer Thickness (δ): This is the most straightforward definition. It's the distance y from the wall where the flow velocity u reaches 99% of the free stream velocity U (i.e., u = 0.99U). While intuitive, its definition is somewhat arbitrary.
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Displacement Thickness (δ*): This represents the distance by which the solid surface would have to be displaced outward into the free stream to account for the reduction in mass flow rate caused by the boundary layer. Mathematically, it's defined as:
δ* = ∫0∞ (1 - u/U) dy
It essentially quantifies the "missing" flow due to the velocity deficit in the boundary layer.
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Momentum Thickness (θ): This represents the loss of momentum flux due to the presence of the boundary layer. It's the thickness of a hypothetical layer of fluid with velocity U that has the same reduction in momentum as the actual boundary layer. Mathematically:
θ = ∫0∞ (u/U)(1 - u/U) dy
Momentum thickness is crucial for calculating drag forces.
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Energy Thickness (δe): This represents the reduction in kinetic energy flux due to the boundary layer. It's defined as:
δe = ∫0∞ (u/U)(1 - (u/U)2) dy
Energy thickness is relevant in situations involving heat transfer and compressible flows.
Summary Table
| Thickness Type | Definition | Physical Significance |
|---|---|---|
| Nominal (δ) | Distance where u = 0.99U | Simple measure of boundary layer extent. |
| Displacement (δ*) | ∫0∞ (1 - u/U) dy | Displacement of the surface due to the boundary layer. |
| Momentum (θ) | ∫0∞ (u/U)(1 - u/U) dy | Loss of momentum due to the boundary layer. Relates to drag. |
| Energy (δe) | ∫0∞ (u/U)(1 - (u/U)2) dy | Reduction in kinetic energy flux. Relevant for heat transfer and compressible flow. |
Importance
Understanding the different boundary layer thicknesses is crucial for accurately modeling and predicting fluid flow behavior in various engineering applications, including aerodynamics, hydrodynamics, and heat transfer. Each thickness parameter provides a different perspective on how the boundary layer affects the overall flow field.
