1.1. Prof. Mohammad Saud Afzal

Displacement thickness is the distance by which a streamline, just outside the boundary layer, is displaced away from the wall due to viscous effects on the plate, as we have seen in the last slide and answer to the question, which we started the lecture with.

Displacement thickness is a critical concept in fluid mechanics that represents the deviation of flow due to the presence of a boundary layer. When a fluid flows over a surface, the velocity at the boundary (the surface of the plate) is essentially zero due to the no-slip condition. As you move away from the surface into the fluid, the velocity increases to a maximum free-stream value. Displacement thickness quantifies how much the actual flow is pushed away from the wall, effectively increasing the distance that a streamline would travel compared to a case without the boundary layer. This thickness affects the overall flow characteristics and momentum of the fluid.

Consider a river flowing over a shallow sandbank. The water right against the sandbank moves slowly due to friction (like the plate's surface), while the water a bit farther away flows faster. This slower water creates a sort of 'buffer' zone that pushes the faster flowing water away from the bank. That buffer zone's width can be likened to the displacement thickness.

This is the boundary layer. Again, look at the definition, distance by which a streamline, just outside the boundary layer is, displaced away from the wall due to the viscous effects on the plate. What exactly is that thickness? Delta dash x.

To derive the displacement thickness, we need to integrate the velocity profile across the thickness of the boundary layer. The displacement thickness is represented as ak is defined mathematically as: δ* = ∫(1 - (u/U)) dy from y=0 to y=δ, where U is the velocity of the fluid outside the boundary layer and u is the velocity at a distance y from the plate. This integral calculates the total area between the velocity profile and the free stream velocity, giving us the effective displacement of streamline due to viscous effects.

Think of a crowded elevator where the elevator is moving upwards. The people next to the walls can't move at all while the people in the center can move freely. The boundary where people start to feel less crowded and can move more freely represents the boundary layer. The ‘push’ that those people in the center have to make is analogous to how the streamline gets displaced away from the wall. The displacement thickness quantifies how far those people need to effectively move away from the wall to maintain their speed compared to those in the center.

Now, proceeding, what is the momentum thickness theta? So, momentum thickness theta is the loss of momentum flux in the boundary layer, as compared to that of the potential flow.

Momentum thickness (θ) describes how much momentum is lost due to the presence of the boundary layer. It is defined as the thickness of a hypothetical layer of fluid moving at uniform velocity U that would carry the same momentum as the actual fluid flow. It helps engineers understand how the boundary layer results in energy losses in flow systems, as viscous forces cause a drop in momentum flux compared to an ideal flow situation where no boundary layer exists.

Imagine a bunch of children on a playground swing. If most of the kids are swinging uniformly, that's like an ideal flow with consistent momentum. Now, if one child is heavier and is slightly out of sync, it disrupts the rhythm, which is akin to the disruptions caused by the boundary layer in fluid flow. The momentum thickness tells us how much the disruption (or the slower swing caused by friction) affects the overall play.

Now, there is something called energy thickness, delta double dash. We are not going to derive it. But some authors define, another thickness of the boundary layer, based on the reduction of the kinetic energy of the fluid flow due to the velocity defect.

Energy thickness (Δ) is another important metric similar to displacement and momentum thicknesses. It quantifies the loss in kinetic energy due to the velocity defect caused by the boundary layer. This thickness is defined based on the energy loss per unit length and helps engineers evaluate the efficiency of energy usage in flow systems. The greater the energy thickness, the more kinetic energy is lost to viscosity, indicating a less efficient flow regime.

Consider a water slide. If the water flows smoothly, it helps kids slide down quickly. However, if there is friction or resistance (like hands or feet scraping the slide), it slows them down. The difference in sliding speed due to this resistance corresponds to the energy thickness—effectively representing how much potential energy is lost as kinetic energy is dissipated.

Now we have to make note of some important points, that is, the boundary layer theory is based on the fact that the boundary layer is thin. So, at any location, at any location x, this x must be very much greater than delta.

One of the fundamental assumptions of boundary layer theory is that the boundary layer is thin relative to the dimensions of the object it is flowing past. This assumption is crucial because many of the equations and models rely on the condition that the flow behaves uniformly outside the boundary layer. In practice, this means that the length scale of the flow domain should be significantly larger than the boundary layer thickness, which helps simplify calculations and improve the accuracy of predictions based on these models.

Think of a thin film of icing on a cake. If the cake itself is huge but the icing layer is thin, you can easily estimate how much icing is on the surface without worrying too much about the detailed structure underneath—this mirrors how boundaries in fluid flow can often be treated in a simplified, more manageable way.

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