2.2. Non-zero Vorticity

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.

The transition from laminar to turbulent flow occurs over a specific distance known as the transition zone. In fluid mechanics, laminar flow is smooth and orderly, while turbulent flow is chaotic and irregular. As we move downstream in a pipe or over a surface, the Reynolds number increases, which is a dimensionless quantity that predicts flow patterns in different fluid flow situations. When this number crosses a certain threshold, the flow transitions from laminar to turbulent.

Think of water flowing smoothly through a straw (laminar flow). If you increase the water flow or change the straw’s width, at some point, the water starts to swirl and mix chaotically, similar to the transition zone. The swirling water represents turbulent flow.

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. So, in laminar sub-layer velocity profile is assumed linear. Linear with respect to what? With the distance increasing distance linear, that means, with increasing y. And we also assume that there is has a constant velocity gradient.

The laminar sub-layer is a thin layer at the very edge of the turbulent boundary layer, located close to a solid boundary. Here, the effects of viscosity are significant, resulting in a linear velocity profile. This linearity means that as you move away from the boundary into the turbulent flow, the velocity increases consistently. The velocity gradient, which indicates how quickly velocity changes with distance, remains constant in this sub-layer.

Imagine a book resting on a table with a steady layer of syrup underneath. The syrup closest to the book (the solid boundary) moves very slowly due to friction (viscous effects), while the syrup farther away is mixed and moving faster (turbulent flow). The gradual increase in syrup speed from the book to the faster-moving layer above represents the linear velocity profile of the laminar sub-layer.

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. This is where it starts distorting and when it goes it distorts more, it distorts more, depending upon what the velocity gradient but there is definitely is a distortion from this point to this point, as soon as it enters the boundary layer.

Fluid particles behave differently when they are in uniform flow compared to when they enter the boundary layer. In uniform flow outside the boundary layer, fluid particles maintain their shape because they experience uniform velocities. However, as they enter the boundary layer, they encounter different velocities due to the fluid's interaction with the solid boundary. This variation in velocity leads to a distortion of the fluid particle as the top part moves faster than the bottom part, creating a rotational motion.

Imagine a group of people holding hands in a straight line (uniform flow). As they move into a narrow hallway (boundary layer), the people near the walls move slower due to tight space, while those in the middle keep moving at their original pace. This creates a distortion in the formation, similar to how fluid particles change shape when entering the boundary layer.

So, the why this distortion occurs? This distortion occurs due to the velocity gradient inside the boundary layer. How? Because the top of the particle has a larger velocity than its bottom. So, this point here, the bottom will have lesser velocity than at the top because this is more closer to the solid surface the lower, the surface which is closer to the solid surface here. Therefore, the flow inside the boundary layer has a non-zero vorticity. Because what does vorticity causes? It causes rotation and this is what we see, the fluid particle will rotate. And this happens because of the differential velocity at the top and the bottom surface. And what we can see is, in the turbulent boundary layer the particle becomes greatly distorted. So, in the boundary layer the flow is rotational.

The distortion and rotation of fluid particles is explained by the concept of vorticity, which is a measure of rotation in fluid dynamics. Inside the boundary layer, there is a velocity gradient, meaning that the fluid's speed varies at different heights. The fluid particle's top moves faster than its bottom, causing the particle to rotate and create vorticity. When we talk about non-zero vorticity, we indicate that the flow within the boundary layer is rotational, unlike the flow outside where particles do not experience such forces.

Think of a spinning top. When you spin it, the top's center rotates, creating a vortex. Similarly, in the boundary layer, the difference in speeds causes the fluid particles to spin and create a rotational flow pattern, just like the spinning top.

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. So, this is an important term, boundary layer thickness delta. We have seen similar term in some slides before. And what is this boundary layer thickness? It is distance from the plate, the flat plate over which we have considered the flow. So, distance from the plate at which the fluid velocity is within some arbitrary value of the free stream velocity.

The boundary layer thickness is a key concept in fluid dynamics, defined as the distance from the surface of a plate where fluid velocity approaches a certain percentage of the free-stream velocity. It’s important to note that the boundary layer does not have a distinct edge; instead, it's a gradual transition area. Typically, this thickness is measured where the velocity reaches about 99% of the velocity in the free stream. This percentage indicates that the boundary layer's influence diminishes significantly beyond this point.

Imagine a cake being frosted. The frosting close to the edge takes a while to blend with the cake layers underneath. However, as you move away from the edge, the frosting's effect is diminished. Likewise, fluid velocity changes gradually within the boundary layer until it closely matches the flow speed far away from the surface.

Ideally, at the top of this boundary layer, the velocity should be equal to the free stream velocity, normally, at the top. So, this is the boundary layer. So, because of this is the velocity is going to be zero and at one point we have to consider where the boundary layer. And so 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.

The choice of the 0.99 velocity ratio is significant because it effectively indicates where the influence of the boundary layer ends and the free stream begins. By definition, when the flow velocity reaches 99% of its free stream velocity, the effects of the boundary layer are negligible, and we can consider the flow above this point as behaving differently, essentially as free stream flow.

This can be compared to climbing a hill – as you approach the summit, the incline becomes less steep. At a certain height, the landscape flattens out, demonstrating that you are no longer affected by the hill’s slope, similarly to how flow characteristics change at the boundary layer's upper limit.

To remove this confusion, we will now look at some of the 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, there are several important definitions: displacement thickness (5), momentum thickness (�03), and energy thickness (�03�02). Each of these terms plays a critical role in understanding how the boundary layer affects the flow of fluids. Displacement thickness refers to the virtual increase in the height of the flow due to the presence of the boundary layer, momentum thickness accounts for the momentum loss due to the boundary layer, and energy thickness represents the energy loss in the flow.

Think of a crowded subway train. The displacement thickness can be compared to the extra space needed for passengers to move around the crowd (displacement), momentum thickness is akin to how much slower the train travels due to all the stops and starts (momentum loss), and energy thickness reflects the extra effort needed to push through the crowd (energy loss in the flow).

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