A laboratory wind tunnel has a square test section with...

A laboratory wind tunnel has a square test section with sides of width $W=1$ ft and length $L=2$ ft. When the freestream air speed at the test section entrance is $U_{1}=80 \mathrm{ft} / \mathrm{s}$, the head loss from the atmosphere is 0.3 in. $\mathrm{H}_{2} \mathrm{O}$. Turbulent boundary layers form on the top, bottom, and side walls of the test section. Measurements show the boundary-layer thicknesses are $\delta_{1}=0.8$ in at the entrance and $\delta_{2}=1$ in at the outlet of the test section. The velocity profiles are of $\frac{1}{7}$ -power form. Evaluate the freestream air speed at the outlet from the test section. Determine the static pressures at the test section inlet and outlet.

Key Concepts

Boundary Layer Development

Boundary layers form along solid surfaces where viscous effects cause the fluid velocity to change from zero (at the wall) to the free?stream value. In wind tunnel test sections, these layers can grow along the walls, thereby reducing the effective cross-sectional area available for the higher-velocity core flow. Understanding boundary layer development is key in predicting how the flow profile evolves from the inlet to the outlet of a test section.

Turbulent Velocity Profiles and the 1/7-Power Law

In turbulent flows, the velocity profile in the boundary layer is often approximated by power-law distributions such as the 1/7-power law. This empirical profile helps describe how velocity increases with distance from the wall in a turbulent boundary layer. It is crucial for estimating the reduction in effective flow area and the associated changes in freestream velocity in confined test sections.

Conservation of Mass and Flow Area Reduction

When boundary layers develop along the walls of a wind tunnel, the effective cross-sectional area for the free stream is diminished. Conservation of mass requires that the product of the effective area and the local velocity remains constant (assuming incompressible flow). This principle is used to determine changes in freestream velocity as the boundary layers grow from the inlet to the outlet.

Bernoulli’s Equation and Head Loss

Bernoulli’s equation relates the pressure and velocity in a flowing fluid along a streamline, assuming inviscid and incompressible flows. In practical applications, energy losses (head losses) due to friction, flow separation, or other factors must be accounted for. Converting measured head losses into pressure differences allows one to evaluate the static pressures at various locations in the test section, providing insight into the interplay between velocity changes and pressure changes in the flow.

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