4.2.1. Equation 18

From the above equation, the velocity at the wall u at y is equal to 0 will be minus infinity, correct. If we put y is equal to 0. So, if we put ln 0, it will be minus infinity. u is positive at some distance far away from the wall.

This chunk explains that when we try to calculate the velocity (u) at the wall (y = 0), we encounter a mathematical issue. The natural logarithm of zero (ln(0)) approaches negative infinity. This means that directly at the wall, our velocity calculation is not defined. However, as we move away from the wall to some distance, the velocity becomes a positive value, which indicates fluid movement at a distance from the wall.

Think of a water faucet. When it's off (similar to y=0), no water flows (u = minus infinity in our analogy), but as you slowly open it (moving away from the wall), water starts to flow at increasing speeds. Hence, we need to consider some distance away from the wall for realistic calculations.

Hence, u is 0 at some finite distance y prime from the wall, that is, y at y prime is equal to 0. Hence, from equation 18, we can say that at distance y prime C will be minus u star by Kappa ln y prime.

At some finite distance from the wall (referred to as y prime), we assume the velocity (u) becomes zero. We can derive a constant (C) in equation 18 that relates to the parameters of the flow. Here, C is determined to be negative u star divided by Kappa multiplied by the natural logarithm of y prime. This equation helps us understand how the velocity profile changes as we move away from the wall.

Imagine standing at a beach where the waves gradually dissipate near the shore (y=0). Farther out in the water (at y prime), you feel the water starting to push against your legs, resembling how velocity approaches zero at a point closer to the wall. This gradual change is modeled in this logarithmic equation.

Now, this as you see, the velocity distribution for turbulent flow in smooth pipe. This is the velocity distribution for turbulent flow in a smooth pipe, where there is no irregularity.

In this part, we acknowledge that the derived equation represents the velocity distribution of turbulent flow in smooth pipes. This is important because it lays out how turbulent flow behaves without any surface roughness, giving us a mathematical understanding of flow characteristics in ideal conditions.

Think of a well-maintained, smooth road that allows a car to drive quickly without any bumps or obstacles. Here, the smooth surface helps maintain a steady speed, akin to how turbulent flow behaves in a smooth pipe.

Now, we have to see equation 23. So, actually this is valid for rough surface as well because all the approximation that we did was on this y dash.

This piece indicates that the earlier derived equations are also applicable when considering rough surfaces, not just smooth ones. Surface roughness significantly influences fluid flow, and the equations allow us to approximate flow behavior by adjusting for surface characteristics.

Consider riding a bike on a sidewalk versus a gravel path. While you can ride smoothly on the sidewalk, the rough gravel path makes it harder to maintain speed. Similarly, our equations account for these changes in surface conditions when predicting turbulent flow.

Now, it is a good idea to solve one problem here, on this particular concept, and the solving the problem will give more understanding now.

In this section, the lecture introduces a practical problem related to turbulent flow. Solving such problems helps reinforce the concepts learned theoretically by applying them to real-world scenarios. The problem relates to measuring fluid velocities at different distances from the wall in a rough pipe.

Imagine you are trying to measure how fast a river flows at different points, like near the bank versus further in the middle. By solving similar problems mathematically, we understand flow behavior better and how different factors influence that flow.

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