4.2. Comparison of Thickness Types

The 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.

Displacement thickness is a measure that quantifies how much the flow is affected by the presence of a boundary, such as a plate. In fluid mechanics, when a fluid flows past a solid surface, the layer of fluid closest to the surface moves slower due to frictional forces between the fluid and the surface. The displacement thickness reflects this effect by defining a distance that alters the flow pattern just outside the boundary layer, thereby adjusting the velocity profile of the surrounding fluid. Essentially, it shows how much the flow 'thinks' the solid surface has moved outward due to the resistance the fluid feels as it travels past the smooth plate.

Consider a river flowing near a riverbank. Imagine placing a fence (representing the plate) at the edge of the riverbank. The water closest to the bank slows down due to friction with the fence. This slowing effect can be likened to the displacement thickness; it shows how far the 'influence' of the fence disrupts the flow of the river, pushing the main flow slightly outward.

Momentum thickness θ is defined as the loss of momentum flux in the boundary layer, compared to that of the potential flow.

Momentum thickness measures the reduction in the momentum flux due to the boundary layer's presence. When the flow passes a surface, fluid velocity decreases near the surface, which leads to less momentum being carried away by the fluid compared to an ideal situation where the fluid flows without any obstructions. This reduction in momentum is expressed mathematically using momentum thickness, with the formula incorporating the velocity profile across the boundary layer. The less the momentum flux, the thicker the boundary layer becomes, highlighting how viscosity affects flow.

Imagine a car moving through the air. If there were no air resistance (akin to the absence of a boundary layer), the car would maintain a high speed easily. However, as the car moves, it creates turbulence and a 'wake' behind it, slowing down the flow of air. The momentum thickness can be compared to how much slower the air moves behind the car as per the changes in momentum due to its shape and speed.

Energy thickness δ'' is defined based on the reduction of kinetic energy of fluid flow due to velocity defects.

Energy thickness is an additional measure in boundary layer flow that accounts for energy losses. Unlike displacement thickness, which focuses on the mass of fluid moved, and momentum thickness, which examines the momentum flow, energy thickness focuses on how the energy of the moving fluid decreases due to viscosity as flow approaches the boundary. The formula for energy thickness integrates the influences of velocity profile and kinetic energy. It helps in understanding how energy loss varies within the boundary layer, represented as an effective loss in energy transport.

Think of a water slide. When a person slides down, they have a certain amount of kinetic energy. If the slide has bumps or friction (similar to viscosity), the person slows down at each bump, losing energy along the way. The energy thickness can be visualized as the distance needed to account for that lost energy due to the interaction with the slide's surface.

The displacement thickness reflects the distance of the displaced streamline, momentum thickness considers momentum loss, and energy thickness illustrates energy reduction due to velocity defects.

In summary: displacement thickness shows how much the streamline shifts from its original path due to viscosity; momentum thickness highlights the impact of flow deceleration on momentum transport; energy thickness captures energy losses in the flow. These three thicknesses serve distinct yet interconnected roles in analyzing the effects of the boundary layer, allowing engineers and scientists to design better systems by understanding flow characteristics.

Picture students moving through a narrow hallway (the boundary layer) while trying to rush to class. Some students (representing the faster-moving fluid) can easily navigate through, while others (slower fluid) are pushed to one side. The way the crowd shifts can be compared to displacement thickness; their hurry (momentum) being affected compares to momentum thickness; and the overall frustration felt by the shift in flow can be related to energy thickness, as it causes delays and inefficiencies.

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