3.1. Comparison with Laminar Flow Profile

So, if we use equation 14 in equation 15, what was equation 14? l m was kappa into y, this was what we said in equation number 14. So, we can simply write or we can simply write and this is equation number 16, very simple. So, for small values of y it can be assumed. So, if the y is very small we can assume that tau is equal to tau not, where tau not is the shear stress at the pipe wall and can be assumed to be a constant. So, at the wall the shear stress is assumed to be constant and equal to tau not.

In this chunk, we are exploring the relationship between shear stress and turbulent flow using equations from fluid dynamics. The basic equation being referenced involves the term 'kappa,' which relates the turbulent flow characteristics (like shear stress) to the distance from the wall of the pipe (y). It emphasizes that when y (the distance from the wall) is very small, the shear stress (tau) can be approximated as a constant value (tau not), which is specifically the value of shear stress right at the wall. This simplification is crucial in analyzing turbulent flow because it allows us to use a simpler equation for calculations.

Imagine a smooth slide at a playground. When a kid is very close to the slide's edge (just at the top), they experience maximum friction, similar to the shear stress at the wall of the pipe. Once they slide down further, the level of 'stickiness' or contact reduces, just as the shear stress changes further away from the wall.

And therefore, what we can say, if we substitute tau is equal to tau not in equation 16, we can obtain or du / dy, this quantity actually tau not can be written as rho. So, but the catch here is, what is the catch? We have considered small value of y. So, du / dy can be written as, 1 / kappa y and under root tau / rho is rho u *. So, it becomes and this u * under root tau not / rho is the sheer velocity and this has the dimension of velocity. And if you integrate the equation number 17, so, what we can get is, simple integration, it will get.

Here, the text discusses how substituting the constant shear stress (tau not) into the turbulent flow equation leads to a new expression (du/dy), which describes how velocity changes with distance from the wall. This integration and the relation to shear velocity (u*) helps simplify the understanding of velocity profiles within the turbulent flow. The key takeaway is that the equations can simplify the complexity of turbulent flow into more manageable forms, allowing for predictions about fluid behavior to be made more easily.

Think of mixing a smoothie. At the bottom of the blender (the 'wall'), the smoothie ingredients get mixed differently than at the top. The closer ingredients are to the wall (or blades), the more they experience 'friction' – or in this case, mixing. As you integrate these differences, you can predict the overall consistency of the smoothie based on how well everything is mixed together.

Now, the equation 20 this equation 20 can be expressed as, u max, so, what we do is, we bring u on the other side. So, we bring u on this side and we take this whole side component this side, then what the result is, u max minus u because u max will always be larger than u is equal to 2.5 u * ln R / y. y will always be less than R or we bring u frictional velocity down, then we get u max, you bring it down here by dividing then you get u max minus u / u * equals to 2.5 ln R / y. And, so, this is ln.

The focus of this chunk is on how to manipulate the derived equation to arrive at the velocity defect law, which describes how the maximum velocity (u max) at the center of the pipe contrasts with the velocity at any point y within the pipe. This law shows that there’s a logarithmic relationship between the distance from the wall and the velocities of fluid particles, a significant feature in turbulent flow due to increased friction and mixing.

Consider a busy road during rush hour. The cars move slower near the sidewalks than in the middle of the street (akin to u and u max). As you calculate out how much slower they are, it reflects the velocity defect law; drivers in the middle enjoy faster speeds while those near the edges have to contend with more 'friction' (in this case, pedestrians, traffic signals, etc.).

So, now, the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. Actually this is the flow, this is the true picture, this is a laminar flow that we have seen before. But below is, this is the V average and the velocity fluctuates or deviates from these depending upon the flow condition. So, this is the V average line. There are several other layers, viscous sublayer, buffer layer, overlap layer and turbulent layer.

This part describes how the velocity profile in turbulent flow differs significantly from laminar flow. Unlike the smooth, parabolic shape of laminar flow, turbulent flow presents a more complex profile characterized by layers: the viscous sublayer (which is closest to the wall), the buffer layer, overlap layer, and the turbulent layer. Each of these layers behaves differently, influencing how the flow behavior changes from laminar to turbulent conditions.

Think of it like layers of a cake. The icing (turbulent layer) might be thick and uneven, while the cake’s core (viscous sublayer) is soft and stable. Just as the cake’s texture varies from the layers to the frosting, the velocity profiles change from smooth flow at the wall to chaotic flow further away.

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