2.1. Definition of Displacement Thickness

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 (B4*) refers to how much the flow streamlines are shifted from their original position outside the boundary layer because of viscous forces acting on the fluid as it flows past a solid surface (like a plate). This shifting occurs because the fluid velocity near the wall (in the boundary layer) is reduced due to friction, causing the outer streamlines to adjust position to maintain flow continuity.

Imagine a river flowing smoothly over a flat ground. Now, consider a smooth rock placed in the middle of the river. As the water flows around the rock, the flow just above the rock is slower compared to the outer regions of the river. The slower motion close to the rock 'pushes' the faster water further away from the rock. The 'displacement thickness' is similar to how much the surface of the water is shifted downstream as it flows around the rock.

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.

The term 'boundary layer' refers to the thin region near the surface where the effects of viscosity are significant. The displacement thickness quantifies how far the theoretical streamline (the ideal flow line in the absence of viscous effects) is pushed away due to these viscosity-induced changes. The displacement thickness ensures that the mass flow rate remains constant across the section of the flow.

Consider a person walking in a crowded room. As they walk, they push people aside slightly. The path they take now deviates from a straight line since they can't walk perfectly straight due to the presence of others. The distance they had to push people aside corresponds to what we can metaphorically refer to as the displacement thickness in the flow, which shows how real-life obstacles alter straightforward movements.

Now, we consider the flow over a smooth flat plate, like this. So, there is a flow which is coming with a speed U, there is a flat plate and at any distance x, there is a section 1 - 1.

When analyzing fluid flow over a flat plate, we consider a constant velocity U of the fluid approaching the plate. At some distance x from the leading edge of the plate, the fluid will have created a boundary layer. The displacement thickness characterizes how this boundary layer affects the overall flow dynamics, as it accounts for the reduction of mass flow due to velocity gradients within the fluid’s boundary layer.

Think of riding a bicycle next to a smooth wall. As you pedal at a constant speed, if you were to measure the speed of the air next to your face compared to further away from the wall, you'd find that the air closer to the wall moves slower due to the wall's influence. The displacement thickness in this bicycle analogy helps understand how that slower air changes the overall airflow around you.

Therefore, the reduction in the mass flux through the elemental strip, compared to the uniform velocity profile will be the difference of the mass fluxes.

In fluid dynamics, mass flux represents the amount of mass flowing through a unit area per time. When the flow is affected by the displacement thickness, the mass flux through an elemental strip becomes different from that in a scenario without-the plate or boundary layer effects. This reduction can be quantified by comparing the actual mass flux through the boundary layer and the theoretical mass flux that would exist in a uniform flow without the plate's interference.

Consider a highway where cars travel at a constant speed. If a roadblock causes a slowdown, the total number of cars passing through a given point drops, seen in the decreased traffic flow. The difference in car numbers can be likened to the reduction in mass flux affected by the 'blockade' of the boundary layer. Just as vehicles that are disrupted change the flow of traffic, the viscous forces near a plate alter how mass flows in fluids.

Therefore, the total reduction in mass flux through BC will be...we are going to simply integrate over this length.

To calculate the total reduction in mass flux due to the displacement thickness, we mathematically integrate the differences in mass flux across the boundary layer. This process of integration allows us to quantify the overall impact of the boundary layer, providing an equation that offers insight into the displacement thickness based on flow conditions.

Consider measuring the total space occupied by a bunch of balloons in a room. If each balloon does not fill the entire space due to being partially squished against each other, you would need to add up the volumes they individually occupy. Similarly, we integrate the impact of each part of the boundary layer to understand the total effect on the flow.

Thus, for an incompressible fluid, we obtain the equation for the displacement thickness.

After integrating and simplifying the terms that reflect the relationships between flow velocities and the boundary layer effects, we arrive at a mathematical expression for displacement thickness. This equation serves as a tool for engineers and scientists to calculate how much the streamline moves away from the surface due to viscous effects. It’s crucial for predicting how fluids behave in various engineering applications.

Think of completing a puzzle where each piece perfectly fits together. The final formulation is like the completed picture that shows the entire process you went through to understand how displacement thickness operates, highlighting how mathematical descriptions demystify complex physical behaviors.

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