Air passes through a square plenum of an air-handling system...

Air passes through a square plenum of an air-handling system with a velocity of $5 \mathrm{~m} / \mathrm{s}$ and a temperature of $20^{\circ} \mathrm{C}$. Determine the displacement thickness of the boundary layer at $x=1.2 \mathrm{~m}$. Also, what is thè uniform speè of thè air at this location?

Key Concepts

Boundary Layer

The boundary layer is a thin region adjacent to a solid surface where fluid velocity changes from zero (due to the no?slip condition) to nearly the free-stream value. Within this layer, viscous effects dominate and the velocity profile develops in a way that influences key properties such as the momentum and energy transfer between the surface and the flow.

Displacement Thickness

Displacement thickness is a measure of how the presence of the boundary layer effectively reduces the cross-sectional area available for the inviscid flow. It quantifies the net loss in mass flow rate compared to an ideal flow with a uniform velocity profile and arises because the velocity in the boundary layer is lower than the free-stream velocity.

Free-Stream Velocity

The free-stream velocity refers to the uniform velocity of the fluid in regions unaffected by viscosity and far from the influence of the boundary layer. This constant speed is critical as it serves as the reference value when evaluating how properties like the boundary layer profile and displacement thickness develop along a surface.

Blasius Boundary Layer Solution

The Blasius solution is a classical similarity solution for the laminar boundary layer that develops along a flat plate. It provides analytical expressions for the boundary layer profile and associated parameters like displacement thickness, highlighting how these quantities evolve spatially along the plate based on the free-stream velocity and fluid properties.

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