5.1. Laminar and Turbulent Boundary Layers

So, what happens is, far away from the boundary, the velocity of the fluid is actually higher. And the velocity increases from zero value on the stationary surface to free-stream of the velocity, free stream velocity of the fluid in the direction normal to the boundary. So, if this is the case that happens when the surface is stationary. If it is moving then the velocity at the boundary is going to be the velocity of the surface that it is on.

In fluid dynamics, when a fluid flows over a solid surface, the fluid particles close to the surface experience a reduction in velocity due to the 'no-slip' boundary condition. This means that the fluid velocity at the solid boundary is zero if the boundary is stationary. As we move away from the boundary, the velocity of the fluid increases to its free-stream value. This transition from zero velocity at the boundary to the free-stream velocity creates a velocity gradient, which is crucial in the formation of boundary layers.

Imagine slathering butter on a piece of toast. The butter (representing the fluid) sticks to the surface of the toast (the solid boundary) and doesn't move until you lift the knife away from the toast. Just like the butter, fluid particles at the solid surface can't slip away freely, leading to a buildup of velocity change from the stationary surface to the faster-moving fluid above.

So, actually this is the most commonly used no slip boundary condition that we use in a viscous fluid flow. This velocity variation occurs in a very thin region of flow near the solid surface. So, this whole phenomenon occurs in a very thin region and this layer, this thin region is called the boundary layer.

The boundary layer is a thin region adjacent to the surface where the flow velocity changes significantly. The no-slip boundary condition signifies that the fluid velocity at the boundary itself is zero, while at a distance (thickness of the boundary layer) above this, the velocity increases to free stream conditions. This velocity variation creates a gradient, referred to as the velocity gradient, which can be expressed mathematically by the relation du/dy, where 'u' is the fluid velocity, and 'y' is the distance from the boundary.

Think of the boundary layer like the syrup poured over a stack of pancakes. The syrup, while touching the pancakes, stays still (zero velocity), but as you move away from the pancakes into the air, it flows freely. The layer of syrup that sticks to the pancakes is similar to the boundary layer; it has a gradient of thickness where it slowly transitions from being still to fluid motion.

So, Prandtl actually divided the flow of the fluid in the neighbourhood of the solid boundary into 2 regions, to have a more simplified look. So, one is called the boundary layer. So, it is in the immediate vicinity of the solid boundary where the viscous forces and rotationality cannot be ignored. The second region is called the outer flow region. The velocity is constant here and is equal to the free stream velocity.

Prandtl's model simplifies the understanding of fluid flow by separating it into two distinct regions: the boundary layer and the outer flow region. The boundary layer is where significant viscous forces and rotational effects occur, necessitating more complex calculations. In contrast, the outer flow region is characterized by constant velocity, effectively treated as inviscid flow, where viscosity is negligible and standard potential flow equations can be applied.

Imagine swimming in a pool. The water close to your body moves slowly compared to the water a little distance away, which flows quickly and smoothly. The region near your body where the water doesn't flow freely because of your presence is akin to the boundary layer while the faster-flowing water around you represents the outer flow region.

Now, after this brief introduction, we will talk about how the boundary layer grows over a flat plate. This is the most simplified object that you think of is a flat plate above when the flow will passed over it how the boundary layer will grow above that. So, the first few units over the flat plate is going to be laminar, that is, the laminar zone. After which this the flow in this region is going to be a transitional zone and after that, this is a turbulent zone where the entire flow will become turbulent.

As fluid flows over a flat plate, the boundary layer develops in stages. Close to the leading edge, the flow remains laminar, characterized by smooth and orderly fluid motion. As we move downstream along the plate, the dynamics begin to change. Once the Reynolds number reaches a critical point (around 5 x 10^5), the flow becomes unstable and transitions to a turbulent state, characterized by chaotic and irregular fluid motion. This transition is essential for understanding the behavior of fluids in various systems.

Consider flowing water in a garden hose. Initially, when you first start the flow, the water moves in a smooth stream (laminar flow). As you increase the flow rate, at a certain point, the water starts to swirl and splatter everywhere (turbulent flow). This transformation illustrates how fluids can shift from a calm state to a chaotic state based on speed and boundary interactions.

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