Flow of air develops in a horizontal cylindrical duct, of...

Flow of air develops in a horizontal cylindrical duct, of diameter $D=15$ in., following a well-rounded entrance. A turbulent boundary grows on the duct wall, but the flow is not yet fully developed. Assume that the velocity profile in the boundary layer is $u / U=(y / \delta)^{1 / 7} .$ The inlet flow is $U=50 \mathrm{ft} / \mathrm{s}$ at section ?. At section ? , the boundary-layer thickness is $\delta_{2}=4$ in. Evaluate the static gage pressure at section ? located at $L=20$ ft. Find the average wall shear stress.

Key Concepts

Turbulent Boundary Layer

In internal flows, such as in ducts or pipes, a boundary layer develops along the walls where viscous effects are significant. In turbulent flow, this boundary layer grows more rapidly than in laminar flow due to the mixing and momentum exchange caused by eddies. The characteristics of this layer, including its thickness and velocity gradient at the wall, are critical for determining the frictional forces that act on the walls and contribute to the overall pressure drop along the duct.

1/7th Power Law Velocity Profile

The 1/7th power law velocity profile is an empirical relation commonly used to describe the mean velocity in the turbulent boundary layer. This profile, where the local velocity scales as the seventh root of the distance from the wall relative to the boundary-layer thickness, reflects the enhanced mixing in turbulent flows and is often employed as a simplifying assumption for estimating shear stresses and velocity distributions in engineering calculations.

Wall Shear Stress

The average wall shear stress is a measure of the frictional force per unit area exerted by the fluid on the duct wall, resulting from the velocity gradient at the wall. It is calculated based on the derivative of the velocity profile in the boundary layer. In turbulent flows, determining the wall shear stress is essential for predicting momentum loss and is closely connected to the friction factor used in pressure drop calculations for internal flows.

Pressure Drop in Duct Flow

The static pressure drop along a duct is largely influenced by friction losses, which are directly related to the development and characteristics of the boundary layer and the associated wall shear stress. In engineering applications, the energy losses due to friction are computed by integrating the shear stress along the wall over the length of the duct. Accurate estimation of the pressure drop is essential for designing efficient fluid transport systems in various industrial applications.

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