2.1. Effects of Velocity Gradient

And then there is a transition from laminar to turbulent boundary layer. So, this is the transitional zone here. So, this short length over which the laminar boundary layer changes to turbulent is called the transition zone, indicated by this distance here. Now, the downstream of the transition zone, the boundary layer becomes turbulent because x keeps on increasing and therefore, Reynolds number increases leading to fully turbulent region.

In fluid dynamics, the boundary layer is the region of fluid in the immediate vicinity of a bounding surface where the effects of viscosity are significant. Initially, the fluid flow is laminar, meaning it's smooth and orderly. However, as the flow moves downstream (designated by 'x'), the velocity increases which changes the flow from laminar to turbulent. This transition zone is where the characteristics of flow begin to change due to increasing Reynolds number, which signifies a more chaotic state of flow.

Think of a river flowing from a calm pool (laminar flow) to white water rapids (turbulent flow). The pool represents laminar flow with smooth, even water movement. As the water flows over rocks and obstacles, it becomes turbulent, much like the transition from a laminar to a turbulent boundary layer.

Now, as you see in this diagram, there is something called laminar sub-layer. And what is that laminar sub-layer? This is a region where the turbulent boundary layer zone and it is very close to the solid boundary. So, basically it is a region in the turbulent boundary layer zone. So, this does not happen here, but it happens in the turbulent boundary layer and it occurs very close to the solid boundary and here, because viscosity will play an important role. Therefore, the viscous effects are dominant; they are much more than the other type of forces. Since, the thickness of this layer, as we can see, this is very, very small compared to this, the variation of the velocity can be assumed to be linear.

Within the turbulent boundary layer, we find a thin region called the laminar sub-layer, which is adjacent to the solid boundary. In this layer, viscosity plays a critical role, and because the thickness is very small, we can assume the velocity changes linearly with respect to distance from the solid boundary. This means that the flow near the boundary moves more slowly than the flow farther away, resulting in a constant velocity gradient.

Imagine a tall glass of water. If you slowly swirl the water in the middle, you'll notice the water near the glass's surface doesn't move as fast as the water in the middle of the glass. This slow-moving water layer represents the laminar sub-layer, where viscous forces dominate.

Now, we will talk about another phenomenon, that is, distortion of a fluid particle within the boundary layer. What happens? So, this figure has been taken from Munson Young and Okiishi’s Fundamentals of Fluid Mechanics published by Wiley and Sons. So, let me just, so, what it says is that the fluid particle retains its original shape in the uniform flow outside the boundary layer. That is very true, because outside the boundary layer there are no effects and the fluid particle, this is the fluid particle above the boundary layer, this is the fluid particle that is going to be in the boundary layer. So, when it moves in this direction, there is no problem at all because all the points will have equal velocities. However, this particle here, after entering the boundary layer the particles begin to distort, as you can see.

When fluid particles flow within the boundary layer, they experience different velocities due to the velocity gradient. The particles near the boundary move slower compared to those farther away from the boundary. This difference causes fluid particles to become distorted as they flow through this region. The flow becomes more chaotic, especially in turbulent conditions, leading to significant distortion of the particles.

Consider blowing up a balloon gently. The air inside the balloon remains uniform and round as it swells, maintaining its shape. However, if you squeeze one side of the balloon, the other side distorts, similar to how fluid particles inside the boundary layer are deformed due to varying velocity.

Now, we are going to see, what the boundary layer thickness is. In real sense, physically, there is no sharp edge to the boundary layer. Now, the boundary layer thickness is the distance from the plate at which the fluid velocity is within some arbitrary value of the free stream velocity. This is the boundary layer thickness delta. We assume, when the velocity reaches almost 99% of free stream velocity, the boundary layers cease to exist above that. And that thickness is called the boundary layer thickness.

Boundary layer thickness is defined as the distance from a stationary surface (like a flat plate) where the fluid velocity transitions to a value very close to the free stream velocity. Typically, this is taken to be about 99% of the free stream velocity, indicating where the effects of viscosity become negligible.

Imagine standing in a swimming pool. When you swim near the edge, you can feel the resistance of the water (similar to viscosity). As you move away from the side, you feel less resistance. The area close to the pool wall where you feel that resistance is analogous to the boundary layer. Once you are far enough away (like 99% of the pool’s width), you feel the full effect of the water's movement, metaphorically representing the free stream velocity.

To remove this confusion, we will now look at some definitions. Some of the definitions is displacement thickness, given by, delta star, very important term, in this particular module of hydraulic engineering. This is another thing called momentum thickness that is called theta. And then there is something called energy thickness which is given by, delta double star. So, these 3 are very important terms in boundary layer analysis.

In boundary layer analysis, several definitions are critical for understanding fluid mechanics. Displacement thickness refers to the distance fluid is displaced due to the presence of the boundary layer, momentum thickness relates to the momentum deficit in the boundary layer, and energy thickness measures the energy loss due to viscosity. These concepts help engineers analyze flow effectively around surfaces.

Imagine a train moving down a track. The train’s body pushes some air aside (displacement thickness), causing a reduced airflow behind it (momentum thickness), and as the train moves, it also uses energy, affecting how far it travels (energy thickness). Understanding these concepts helps us gauge the train's efficiency and speed in the air.

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