6.1. First Problem: Displacement, Momentum and Energy Thickness

Displacement thickness is defined as the distance by which a streamline, just outside the boundary layer, is displaced away from the wall due to viscous effects on the plate. This can be represented as delta dash x.

Displacement thickness quantifies how much the flow is affected by a boundary layer at a wall. As fluid flows past a solid boundary (like a plate), it experiences drag due to viscosity, which is the thickness of the boundary layer. Delta dash x represents this thickness where the outer streamline is pushed away from the wall compared to what it would be without the plate. Essentially, if there were no viscous effects, the fluid would flow directly along the wall, but the presence of the wall causes a displacement in the flow lines.

Imagine trying to walk through a tight hallway filled with people (the plate). You have to adjust your path to avoid them, which represents the displacement. If the hallway was empty, you would walk straight; the more people there are, the more your route is altered. Similarly, the displacement thickness indicates how much the fluid's path changes due to viscous forces near the boundary.

Considering the flow over a smooth flat plate, with incoming flow speed U, and analyzing an elemental strip at a distance y from the plate.

When analyzing the flow over a flat plate, we consider small strips (dy) within the boundary layer at various heights (y). These strips have a width (b) and together they contribute to the total mass flux of the flow over the plate. The speed of fluid in the outer layer is U, while within the boundary layer, the speed varies (denoted as u). This sets the stage for calculating reductions in mass flux due to viscosity and helps in deriving equations for displacement thickness.

Think of the flat plate like a water slide and the water running down it. As you slide down (representing the outer flow), if there were rough patches or bumps, it would slow you down (representing viscous effects). The varying speeds of water at different heights illustrate how fluid velocity changes with distance from the surface.

The mass flux through the elemental strip is given by ρu * dA, where dA = b dy, leading to a formula ρUbdY.

Mass flux quantifies how much mass of fluid is passing through a certain area per unit time. We derive this by multiplying the density (ρ) of the fluid, its speed (u) within the boundary layer, and the area of the elemental strip (dA = b dy). This yields the expression for mass flux. On comparison with a section where the velocity is constant, it allows for understanding how the boundary layer influences overall flow characteristics.

Imagine measuring how much water flows through a hose. If you know the water speed and the area of the hose’s opening, you can calculate how much water comes out in a second. The same principle applies with fluid flow around the plate; you’re just calculating how much fluid 'flows through' our elemental strip area.

The total reduction in mass flux due to the presence of the boundary layer is evaluated by comparing the flow in the boundary to that without it. The equation used is ∫(0 to δ) (ρ(U - u)bdY).

This integral calculates the total mass flux reduction in the presence of the boundary layer. We compare the mass flux in the boundary layer (where speed is less due to viscous forces) with a scenario where the flow is uniform. This is crucial for understanding the dynamic impact of boundary layers on flow and helps in quantifying displacement and momentum thickness.

Consider a situation where you are trying to fill a swimming pool with a hose. If you only have the end of the hose submerged (like a boundary layer effect), less water flows out compared to when the hose is fully submerged in water (uniform flow). Here, the integral gives you a total measure of how much flow is 'lost' due to the boundary layer effects.

Momentum thickness θ is defined as the loss of momentum flux in the boundary layer compared to that in the potential flow, mathematically represented as the relationship between ∫(0 to δ) ρU²θ = ∫(0 to δ) ρU(u)dY.

Momentum thickness quantifies how much momentum is lost due to viscous effects in the fluid as it flows along a surface. We arrive at this by finding the difference in momentum flux between flow with a boundary layer and that with ideal flow (potential flow). The more significant the momentum thickness, the greater the impact of viscosity, and this helps in the design and analysis of fluid systems around surfaces.

Think about driving a car on a smooth road versus a bumpy one. The smoother ride represents the potential flow with no loss of momentum, while the bumpy ride simulates the boundary layer's effect, leading to energy loss and requiring extra effort (force) to maintain speed. The momentum thickness helps quantify this effect.

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

Energy thickness considers how much kinetic energy is reduced in a fluid due to the influence of viscosity and velocity variations near a boundary. It's another layer of understanding fluid dynamics, emphasizing how energy is consumed in overcoming viscous forces. While momentum and displacement thickness primarily deal with mass flow, energy thickness provides insight into energy losses, which are critical in engineering applications like pipe flow design.

Imagine riding a bike up a hill. You start with a certain kinetic energy, but as you ride up against the incline (analogous to viscous forces), that energy is reduced. Energy thickness shows how much 'kinetic energy' is lost due to this 'hill' of resistance, giving a measure of how your energy is being consumed in confronting friction.

The lecture concludes by encouraging practical problems regarding displacement, momentum, and energy thickness, ensuring these concepts are applied in real-world scenarios.

Understanding these thicknesses is foundational in fluid mechanics, affecting designs in civil engineering, aerospace, and many other fields. Problems often arise in various scenarios, such as flow over wings, pipes, or other structures where viscous effects change how fluids flow. Practicing these calculations prepares students to analyze real-world engineering problems more effectively.

Consider engineers designing an airplane wing. They need to know how the air flows over the wing surface (boundary layers' effects) to ensure the plane is streamlined for efficient flight. Practical applications like this bring thickness concepts into real-world solutions, ultimately leading to improved designs that save fuel and enhance performance.

--