The wind passes through a tunnel at a temperature of 30^∘ C with a free-stream velocity of 50 m

step 1 Given U is 50 meters per second, X is 5 meters, the core is 1 .2 meters. And if we use air at 30

The wind passes through a tunnel at a temperature of 30^∘ C...

The wind passes through a tunnel at a temperature of $30^{\circ} \mathrm{C}$ with a free-stream velocity of $50 \mathrm{~m} / \mathrm{s}$. If the velocity of the wind is to be maintained at the central $1.2-\mathrm{m}$ core throughout the tunnel, determine the dimension $a$ at the exit in order to accommodate the growing boundary layer. Show that the boundary layer is turbulent, and use $\delta^{*}=$ $0.0463 x /\left(\operatorname{Re}_{x}\right)^{1 / 5}$ to calculate the displacement thickness.

Key Concepts

Boundary Layer

This concept describes the thin region adjacent to a solid surface where the fluid velocity gradient is significant due to viscous effects. In the context of flow in a tunnel, the boundary layer grows along the surface, gradually reducing the velocity near the walls compared to the free-stream value, which is crucial when designing channels to ensure a uniform core velocity.

Turbulent Flow

Turbulence in a boundary layer is characterized by chaotic and eddy-driven motion, leading to enhanced mixing and momentum transfer compared to laminar flow. Identifying the flow as turbulent usually involves checking the Reynolds number, with turbulent boundary layers having thinner viscous sublayers and different scaling laws, which directly impacts how the boundary layer grows along the wall.

Displacement Thickness

Displacement thickness quantifies the effective thickness of a boundary layer by representing the distance by which the external inviscid flow is displaced due to the gradual reduction in velocity near a wall. It is an important parameter in boundary layer analyses as it directly influences the geometry of the duct or tunnel required to maintain a uniform flow region.

Reynolds Number

The Reynolds number is a dimensionless parameter that compares inertial forces to viscous forces in a fluid flow. It is essential for predicting the flow regime (laminar or turbulent) in the boundary layer, and higher Reynolds numbers typically indicate turbulent flow. In this problem, the Reynolds number is used to validate the turbulence assumption and to apply appropriate correlations for boundary layer characteristics.

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