4.1. Definition of Energy Thickness

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 is a concept within boundary layer theory that refers to the reduction of kinetic energy in the flow of a fluid due to a defect in velocity. In simple terms, when a fluid flows over a surface, its speed can be affected by the nature of that surface and the fluid's viscosity. This thickness measures how much the fluid's energy (which is related to its velocity) is reduced because of these defects.

Imagine a racetrack where cars (representing fluid) are moving. If the surface of the track is smooth, the cars can maintain high speeds easily. However, if there are obstacles or barriers (like a rough surface), the cars will slow down, resulting in less speed and therefore less kinetic energy. The energy thickness gives us a way to quantify how much energy is 'lost' due to such disruptions.

So, the momentum thickness was due to the loss of the momentum flux, displacement was due to the loss of the mass flux and the energy thickness reduction of the kinetic energy of the fluid due to the velocity deficit. And we will just write down the equation here. It is given as, .

The energy thickness equation relates to how we measure the loss of kinetic energy in a boundary layer due to the differences in fluid velocity across that layer. While the specific equation is not provided here, its formulation generally involves integrating the velocity profile to quantify the loss in kinetic energy. This integration helps to understand how much energy is 'missing' compared to the ideal scenario where the flow would remain uninterrupted.

Think of a water slide where water flows smoothly down the slide. If the slide is free of obstacles and friction, the water retains most of its energy and speed when it exits. However, if there are bumps or friction from the slide's material, the water will lose speed and energy. The energy thickness is akin to calculating how much speed the water has lost due to those obstacles compared to a smooth ride.

Now, we have to make note of some important point, 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, boundary layer thickness and also the displacement thickness. This x must also be greater than momentum thickness and also the energy thickness.

Boundary layer theory revolves around the assumption that the effects of the boundary layer (the layer of fluid in contact with the surface) are confined to a relatively small region. For the theory to be valid, the distance from the leading edge of the object (location x) should be significantly greater than the various thicknesses including boundary layer thickness, displacement thickness, momentum thickness, and energy thickness. This ensures that we consider a proper representation of the flow without interference from the boundary layer effects.

Consider a swimmer pushing through the water. The swimmer creates a wake that alters the flow of water behind them. If you are a certain distance away from the swimmer, their influence on the water flow diminishes. This analogy illustrates that to understand the swimmer's movement and the resulting water dynamics correctly, you need to be significantly far from the boundaries of their wake (representing the boundary layer).

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