4. Velocity Profiles

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.

1. The flow of fluid can be categorized based on its velocity characteristics, such as 'laminar' when the flow is smooth and layered, and 'turbulent' when it becomes chaotic and mixed.
2. The section where laminar flow changes to turbulent is referred to as the 'transition zone.'
3. As the flow moves along a surface, the distance (x) from the leading edge of the surface increases, affecting the Reynolds number, a dimensionless quantity used to predict flow patterns.
4. When the Reynolds number becomes sufficiently high, the flow transitions from laminar to turbulent behavior.

Imagine a calm river flowing steadily (laminar flow). As the river meets boulders and gets steeper, the water begins to crash and swirl (turbulent flow). The point where this change happens is like the transition zone.

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. 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.

1. Inside a turbulent boundary layer, there exists a smaller region called the 'laminar sub-layer.'
2. This sub-layer is located very close to the solid surface and exhibits laminar characteristics even though the overall flow is turbulent.
3. Viscosity, a measure of a fluid's resistance to flow, plays a crucial role in this sub-layer, as the effects of viscosity are more pronounced than in the turbulent flow above it.

Think of a bustling crowd at a concert where everyone is moving chaotically (turbulent flow), but if you move in quickly near the stage (the solid boundary), it becomes more orderly and calm, almost like a smooth flow (laminar sub-layer).

Since, the thickness of this layer 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 a constant velocity gradient.

1. The thickness of the laminar sub-layer is much smaller compared to the entire boundary layer, allowing us to simplify calculations.
2. In this sub-layer, the velocity changes uniformly with distance (i.e., it is linear with respect to the vertical distance 'y' from the solid boundary).
3. A constant velocity gradient means that the rate of change of velocity (du/dy) is steady throughout this small layer.

Imagine a flat ramp where cars accelerate slowly. The speed of cars at ground level increases in a straight line as they reach the first few meters up. This is similar to how velocity increases in a linear manner near the surface of 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 tao not, as we have already been using not for the 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.

1. The linear variation of velocity in the laminar sub-layer allows us to establish mathematical relationships for fluid flow.
2. The shear stress, which represents the force per unit area due to the fluid's viscosity, remains constant across this layer and is attributed to the boundary shear stress.
3. The term 'tao not' refers to this shear stress right next to the wall, indicating that it influences the fluid very close to solid surfaces.

Consider butter on a warm piece of bread. The top layer of butter remains smooth and spreads evenly across the surface (constant shear stress), while the butter at the base, contacted by the bread, applies a strong force to stay in place because of viscosity.

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.

1. When a fluid particle flows outside the boundary layer in steady, uniform flow, it maintains its characteristic shape.
2. Once it enters the boundary layer, the varying velocities across the particle cause it to distort, particularly because of differences in velocity from the top of the particle to the bottom.
3. This distortion is a direct result of the velocity gradient within the boundary layer.

Imagine a pencil floating on calm water (uniform flow), appearing straight and intact. If you were to push one end of the pencil under a moving stream of water (entering the boundary layer), the pencil would bend and contort due to the water's varied motion, representing how fluid particles behave in a boundary layer.

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.

1. The boundary layer is a gradual region, not a sudden drop-off—hence no sharp edge.
2. Boundary layer thickness defines where the fluid velocity approaches a certain threshold of the free stream velocity, typically noted as nearly reaching 99%.
3. Below this threshold (at the wall), the fluid velocity is nearly zero due to viscous effects, and at the outside edge of the boundary layer, it nearly matches the free stream velocity.

Envision a deep dish of soup: the top is calm and uniform, while the bottom is sticky against the bowl. As you scoop soup, the smoothest layer is like the boundary layer where the texture visibly changes. The thickness where smoothness slips back into stickiness 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.

1. The choice of 0.99 as a threshold is somewhat arbitrary but provides a clear and recognizable standard for when the boundary layer can be considered to have ended.
2. Related to boundary layer concepts are terms such as displacement thickness (delta star), momentum thickness (theta), and energy thickness (delta double star), which are significant for hydraulic engineering and fluid dynamics.

Consider an artist shading on paper. The shaded area should reach a certain depth to create the intended effect (like reaching 99% of the smooth, flowing area), with this boundary indicating where the detailed portion ends. Similarly, 0.99 indicates where the influence of the boundary layer fades.

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. And this figure has been taken from Munson, Young and Okiishi Fundamentals of Fluid Mechanics. 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.

1. We can compare two different velocity profiles for fluid flow around a flat plate to observe the effects of viscosity.
2. The first profile represents a scenario with zero viscosity, allowing fluid to maintain a constant velocity and exhibiting what is termed as 'slip' at the wall.
3. The second profile includes the effects of viscosity, which leads to no slip at the wall—indicating the fluid's velocity is zero directly at the surface of the plate.

Think of a smooth ice skating rink versus a much rougher surface. On the smooth surface, a skater can glide with minimal resistance (zero viscosity), while on a rough surface, they will slow down and stop more quickly due to friction (viscosity), illustrating the concept of slip versus no-slip.

So, within the boundary layer there is a velocity deficit. So, this is the boundary layer. So, U capital U is the uniform velocity profile, here also the free stream velocity is same. But say, at this distance if the velocity is u, then the deficit of the velocity that is, happening is U minus u.

1. Inside the boundary layer, there exists a phenomenon known as the 'velocity deficit,' which means the fluid does not maintain the same speed as the free stream velocity (U).
2. At any point within the boundary layer, the velocity (u) is lower than the free stream, resulting in a difference (U - u), termed the velocity deficit.
3. This difference leads to reduced flow rates across cross-sections within the boundary layer, indicating a shift in how the fluid moves compared to free flow.

Imagine traffic flowing smoothly down a highway (free stream velocity). As you enter a parking lot with many cars (the boundary layer), the cars slow down significantly (velocity deficit) compared to the highway speed. This experience mirrors how flow changes when entering a boundary layer.

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. With this we will see in the upcoming lecture.

1. The concept of displacement thickness relates to the idea of measuring how much the fluid flow would be affected if the plate were pushed or displaced a small distance (delta dash).
2. Displacement thickness accounts for the velocity deficit in the flow, indicating how this shift impacts overall flow behavior.
3. This concept connects to broader fluid mechanics and will be explored in more detail in forthcoming discussions.

Consider how pushing a boat through water alters its path (similar to displacing the plate). The distance pushed (delta dash) can change how smoothly the boat travels, just like how displacement thickness affects flow patterns.

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