5. Displacement Thickness

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. 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. 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 boundary layer thickness refers to the distance from a solid surface (like a plate) to a point in the fluid where the flow velocity is sufficiently close to the free stream velocity (the speed of the fluid far from the surface). This is not a sharp transition but rather a gradual change. Typically, we consider the boundary layer to be well-defined at points where the velocity is about 99% of the free stream velocity, meaning the fluid is almost moving as fast as the fluid unaffected by the surface.

Imagine swimming in a pool. Close to the side of the pool (the surface), the water feels different because you’re pushing against it. As you swim further away from the wall, the water flows more freely, and your speed increases. The area where your speed begins to match the speed of the freely flowing water represents the boundary layer thickness.

So, what is so special about 0.99? Why not for 0.96 or 0.98? 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.

Displacement thickness is a critical concept in fluid mechanics and hydraulic engineering. It is denoted by delta star (δ*) and helps describe the effect of a boundary layer on the flow of fluid. Unlike boundary layer thickness which specifies physical distance, displacement thickness takes into account the reduction in flow rate due to the presence of the boundary layer. It can be thought of as an 'equivalent thickness' that would displace the flow to maintain the same mass flow rate without the boundary layer.

Consider moving a large group of people congested in a narrow hallway. Even though the hallway can fit more people, the presence of the crowd (the boundary layer) reduces the flow of individuals. If you were to imagine shifting the rear line forward just enough so that fewer people crowded the exit, that 'shift' would equate to the displacement thickness. It shows how far you would need to adjust the crowd to maintain the same escape rate as if they weren’t congested.

So, we consider 2 velocity profiles for flow past a flat plate. So, this is flat plate here. This is the 1 velocity profile, this is 2 and both has equal areas. This figure has been taken from Munson, Young and Okiishi Fundamentals of Fluid Mechanics. So, as you can see, I will explain these terms, as it comes. So, this is a uniform profile, where mu is zero. So, there is no viscosity and there is going to be slip at the wall, this place. It is not, I mean, here, there will be no slip condition. The second is the boundary layer profile, this was a uniform profile and this is the boundary layer profile. Here, there will be some viscosity and mu is not equal to zero and there is going to be no slip at the wall. No slip at the wall means, the fluid velocity just above the plate is going to have the same velocity as the plate. In this case, since, the plate is stationary, the fluid particle just above this will have a zero velocity.

In fluid flow over a flat plate, we can compare two profiles: a uniform flow profile (where fluid moves with consistent velocity and there’s no viscosity) and a boundary layer profile (where viscosity matters). In the uniform profile, fluid has the same speed at all points, while in the boundary layer, the fluid closest to the plate (near the surface) moves slower because of friction with the plate. There’s a velocity deficit in this layer, meaning fluid in the boundary layer has a speed lower than that of the unaffected flow.

Think of a highway with a high constant speed limit. In the center lane, vehicles can travel at that limit without obstruction (uniform flow). However, on the edges where vehicles are merging or slowing for exits, the speed drops. The slower-moving vehicles on the edge represent the boundary layer where flow is impacted due to interaction with the surface (the road). The difference in speed here illustrates the velocity deficit.

So, the flow rate across the section b - b is less than the flow rate across section a – a, that is correct. Because the velocity here is limited, I mean, less than this U / u. So, the velocity in the boundary layer is lesser than the uniform velocity profile by U minus u. Suppose what happens, if the plate, my question to you is, what happens if the plate is displaced at section a - a by an amount delta dash? So, here, this delta dash is called the displacement thickness.

When analyzing flow rates in the boundary layer, we notice that the flow rate at section 'b-b' (in the boundary layer) is less than at section 'a-a' (outside the boundary layer). This is due to the slower velocities in the boundary layer. When we pose the question about displacing the plate by the amount known as 'displacement thickness,' we acknowledge how the actual flow rate would change and how that impacts overall flow.

Imagine a water slide where the water flow is generally fast. If you were to put an inflatable bumper (plate) halfway down, the flow would slow down just in that area due to the bumper's presence creating a barrier. If we were to consider the bumper's impact on water flow and shift its position slightly downstream (displacement thickness), this would change how quickly the water flows past that point, just like displacement thickness affects fluid movement.

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