4.2. Velocity Deficit

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 transition from laminar flow to turbulent flow occurs in a specific region called the transition zone. Laminar flow is characterized by smooth and orderly fluid motion, while turbulent flow is chaotic and irregular. As the distance along the flow increases (denoted by 'x'), the Reynolds number, which measures the ratio of inertial forces to viscous forces, increases. This increase leads to a fully turbulent boundary layer downstream. The transition is significant as it affects the behavior and properties of the flow.

Think of fluid flow as a crowded dance floor. Initially, dancers (fluid particles) move smoothly and in sync (laminar flow). As more people enter and the space gets tighter (increasing x), the movements become chaotic and tangled (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.

The laminar sub-layer exists within a turbulent boundary layer and is located just adjacent to the solid boundary (like a wall or plate). In this thin layer, flow behaves more like laminar flow, where viscosity significantly influences the motion of the fluid. The effects of turbulence are less pronounced in this region due to its proximity to the solid surface.

Imagine a river with calm water at its edges (the laminar sub-layer) and strong currents in the middle (the turbulent boundary layer). The calm water near the shoreline allows for smoother movement compared to the chaotic flow further out.

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 increasing distance linear, that means, with increasing y. And we also assume that there is has a constant velocity gradient. So, the velocity gradient du / dy is constant.

In the laminar sub-layer, the small thickness allows us to assume that the velocity changes linearly as you move away from the solid surface (denoted as 'y'). This linear velocity profile implies that the change in velocity with respect to distance is constant (referred to as the velocity gradient). Mathematically, this is expressed as 'du/dy', indicating a uniform change in velocity over this thin layer.

Think of a gentle slope on a hill where the altitude (height) changes steadily as you walk up. Each step you take increases your height in a consistent manner—that's similar to how velocity gradients behave in the laminar sub-layer.

Therefore, for linear variation of velocity, we can write. Now, the shear stress in this layer is constant and is equal to the boundary shear stress given by taonot, as we have already been using not for tau not, for the shear stress near the wall this is also is it is like a sort of a wall only this is the boundary.

In the laminar sub-layer, the constant velocity gradient indicates that shear stress—force per unit area exerted parallel to the surface—remains consistent. This shear stress can be denoted as 'taonot' (τ₀), which is critical for understanding the interaction between the fluid and the solid surface.

Consider spreading butter on bread. The smooth, consistent pressure you apply creates an even layer of butter. Similarly, shear stress maintains a constant influence over the fluid within the laminar sub-layer.

Now, we will talk about another phenomenon, that is, distortion of a fluid particle within the boundary layer... This distortion occurs due to the velocity gradient inside the boundary layer.

When a fluid particle enters the boundary layer, it begins to distort due to differences in velocity across its height. The top part of the particle moves faster than the bottom because it is further from the solid surface. This differential motion creates a rotation in the fluid particle, signifying the complex interactions within the boundary layer.

Imagine a person trying to wade through a pool where the water moves at different speeds at various heights. They would feel the top half moving quickly, while their feet may feel the slower-moving water, causing them to twist and turn as they try to stabilize.

Now, we are going to see, what 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.

Boundary layer thickness is defined as the distance from the solid surface to the point where the fluid's velocity reaches approximately 99% of the free stream velocity (the velocity far from the surface). This defines where the effects of viscosity diminish and the flow behaves more uniformly like the free stream.

Picture a hot air balloon lifting off the ground. The air immediately surrounding the balloon is affected by its presence. The boundary layer is like the air closest to the balloon—the air may not be rising as fast as the air higher up, indicating a decrease in velocity until it reaches free stream speeds.

To remove this confusion, we will now look at some of the definitions... displacement thickness, given by, delta star, very important term, in this particular module of hydraulic engineering.

In boundary layer analysis, three important thicknesses are defined: displacement thickness (δ*), momentum thickness (θ), and energy thickness (δ**). Each of these provides unique insights into how the boundary layer affects the flow characteristics, energy, and momentum transfer within the fluid.

Think of a sponge absorbing water. The displacement thickness would represent how much water is taken up by the sponge (displacing volume), momentum thickness would reflect how the sponge pushes water as it absorbs (momentum effect), and energy thickness would signify how much energy is used to push the water through the sponge.

So, we consider 2 velocity profiles for flow past a flat plate... and there is going to be no slip at the wall.

Two types of velocity profiles can be studied when analyzing flow past a flat plate: a uniform profile (without viscosity) and a boundary layer profile (with viscosity). The key difference is the no-slip condition at the wall, meaning that fluid particles adhere to the surface of the flat plate and have a velocity of zero right at the surface.

Envision a racecar zooming on a straight track. The air moving around the car is fast (free stream), but the air immediately against the car's surface is stationary. This is similar to how fluid behaves in the boundary layer—fast moving away from the surface, but stationary at the point of contact.

Within the boundary layer there is a velocity deficit... So, the velocity in the boundary layer is lesser than the uniform velocity profile by U minus u.

In the boundary layer, the velocity of the fluid is less than the free stream velocity, creating what is known as velocity deficit (U - u). This deficit means that the flow rate across a section of the boundary layer is lower than that across a section of free stream flow. This understanding is crucial for calculations that involve forced flow such as in hydraulic engineering.

Imagine a traffic jam on a highway; the vehicles in the jam are moving slower (boundary layer) compared to those speeding past on clear lanes (free stream). The slower traffic flow represents velocity deficit, reducing overall transportation efficiency.

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