Determine the ratios of displacement and momentum thickness to the boundary layer thickness when the velocity profile is represented by \(u / u_{\mathrm{m}}=\sin (\pi \eta / 2)\) where \(\eta=y / \delta\).

Understanding the velocity profile is essential for solving boundary layer problems. The velocity profile describes how the fluid velocity changes as you move away from the wall. In our given problem, the velocity profile is represented by \( \frac{u}{u_{m}} = \sin \left( \frac{\pi \eta}{2} \right) \(, where \ \eta = \frac{y}{\delta} \. Here, \ u \ is the local velocity at a distance \ y \ from the wall, \ u_{m} \ is the maximum velocity, and \ \delta \ is the boundary layer thickness.

Simply put: as you move away from the wall (increasing \ y \), the velocity profile tells you how the speed of the fluid changes until it reaches the maximum velocity (\ u_{m} \). In the problem, the sine function describes this change. By understanding this profile, we can analyze further aspects like displacement and momentum thickness.

Displacement thickness \( \delta^* \) is an imaginary concept that helps quantify the effect of the boundary layer on the overall flow. It is defined as the distance by which the outer potential flow would be displaced due to the presence of the boundary layer. Mathematically, it is given as:

\[ \ \delta^* = \int_{0}^{\delta} \left( 1 - \frac{u}{u_{m}} \right) dy \ \]

In our problem, substitution of the given velocity profile into the integral gives us:

\[ \ \delta^* = \int_{0}^{\delta} \left( 1 - \sin \left( \frac{\pi y}{2 \delta} \right) \right) dy \ \]
We solve this integral to find that: \ \delta^* = \delta \left( 1 - \frac{4}{\pi} \right) \

This result tells us the amount by which the streamlines of the outer flow are shifted because of the boundary layer. By comparing \ \delta^* \ to \ \delta \, we find the ratio \ \frac{\delta^*}{\delta} = 1 - \frac{4}{\pi} \.

Momentum thickness \( \theta \) is another important boundary layer parameter that represents the loss of momentum flux due to the boundary layer. It is given by:

\[ \ \theta = \int_{0}^{\delta} \frac{u}{u_{m}} \left( 1 - \frac{u}{u_{m}} \right) dy \ \]

For our velocity profile, substitution gives us:

\[ \ \theta = \int_{0}^{\delta} \sin \left( \frac{\pi y}{2 \delta} \right) \left( 1 - \sin \left( \frac{\pi y}{2 \delta} \right) \right) dy \ \]

This can be simplified using trigonometric identities, resulting in:

\[ \ \theta = \delta \left( 1 - \frac{8}{3\pi} \right) \ \]

The ratio of momentum thickness to boundary layer thickness is then: \[ \ \frac{\theta}{\delta} = 1 - \frac{8}{3\pi} \ \]

This ratio helps us understand how much the momentum is reduced within the boundary layer compared to an ideal flow without the boundary layer.

Boundary layer integrals help simplify and solve various boundary layer problems by converting differential equations into integral equations. For displacement and momentum thickness, we use integrals to compute these values as it involves summing up effects across the boundary layer.

For example:

• Displacement thickness integrates the difference between the local and maximum velocities.
• Momentum thickness integrates the momentum loss due to the boundary layer.

By understanding and using these boundary layer integrals, we can easily determine how the velocity profile affects the flow properties such as displacement and momentum thickness. These integrals are often based on simplifying assumptions but provide a clear picture of the boundary layer characteristics.