Air flows in a horizontal cylindrical duct of diameter D=100 mm . At a section a few meters from

step 1 In this question we have to find the velocity U2 in the inverted core and the pressure top. So f

Air flows in a horizontal cylindrical duct of diameter D=100...

Air flows in a horizontal cylindrical duct of diameter $D=100 \mathrm{mm} .$ At a section a few meters from the entrance, the turbulent boundary layer is of thickness $\delta_{1}=5.25 \mathrm{mm}$ and the velocity in the inviscid central core is $U_{1}=12.5 \mathrm{m} / \mathrm{s}$ Farther downstream the boundary layer is of thickness $\delta_{2}=$ $24 \mathrm{mm} .$ The velocity profile in the boundary layer is approximated well by the $\frac{1}{7}$ -power expression. Find the velocity, $U_{2},$ in the inviscid central core at the second section, and the pressure drop between the two sections.

Key Concepts

Pressure Drop in Duct Flows

Pressure drop in internal flows is associated with frictional losses and the momentum exchange between different regions of the flow, such as the inviscid core and the turbulent boundary layer. As the flow develops and the boundary layer thickens, the increasing losses due to shear stress over the duct walls manifest as a pressure drop along the length of the duct. Quantifying this drop requires integrating the effects of the velocity profile and frictional forces acting on the flow.

1/7-Power Law Velocity Profile

This concept describes a common empirical approximation for the velocity distribution in turbulent boundary layers, where the velocity at any point in the boundary layer is expressed as a fraction of the centerline (or free-stream) velocity raised to the 1/7 power relative to its distance from the wall. It captures the gradual acceleration of the flow away from the wall and provides a simple yet effective means of describing turbulent shear flows in ducts and over flat plates.

Conservation of Mass in Internal Flows

In incompressible flows through ducts or pipes, the mass flow rate must remain constant along the length of the flow. As the turbulent boundary layer grows, the effective flow area available for the high-speed core decreases, and the core velocity must change accordingly to maintain the overall mass flow rate. This principle is fundamental when relating changes in the velocity profile due to the development of the boundary layer.

Transcript

Please give Ace some feedback