3. References and Conclusion

Now, about the power law velocity profile, a little bit on that. So, the power law velocity profile for smooth pipes can be expressed as, it is, u by u max can be written as y by R to the power 1 by n. This is the power law velocity that was given or we can always write in terms of R because y is, y is R minus small r and that you substitute here, you will end up in this equation. So, this 1/n depends upon the Reynolds number, putting n is equal to 7 in above equation gives one-seventh power law velocity profile. This is one of the very famous velocity profiles.

This chunk introduces the power law velocity profile, which describes how fluid velocity varies with distance from the wall in a smooth pipe. The relationship is defined by the equation u/u_max = (y/R)^(1/n), where 'u' is the fluid velocity at radial position 'y', 'R' is the pipe's radius, and 'n' is an exponent that is influenced by the flow's Reynolds number. Setting n to 7 gives a specific and commonly used profile, revealing how complex fluid dynamics can be quantified using mathematical models.

Think of the power law velocity profile like a ramp that gradually slopes up. As you move from the bottom to the top of the ramp (representing distance from the pipe wall), you gain height (representing fluid velocity). Different ramps can be steeper or more gradual, much like how different values of 'n' alter the fluid's behavior in the pipe.

The value of n will increase with increasing Reynolds number. Power law velocity profiles cannot give 0 slope at the pipe center. So, power law of velocity profiles also cannot calculate the wall shear stress. Why? Because the power law profiles gives a velocity gradient of infinity at the walls. That you can try, by substituting r is equal to, small r is equal to capital R.

In this chunk, we learn that as the Reynolds number (a measure of flow turbulence) rises, so does 'n', affecting how the velocity profile behaves. A noteworthy characteristic of power law profiles is that they cannot produce a zero slope at the center of the pipe, which is a theoretical ideal for steady laminar flow. Additionally, power law profiles don't account for wall shear stress accurately since they suggest an infinite velocity gradient at the pipe's walls—meaning that there's an unrealistic change in velocity right at the surface where the fluid touches the pipe.

Imagine swimming in a pool. When you swim close to the edge (pipe wall), you may feel a strong pull from the wall, indicating how the water rushes rapidly against it. However, as you move toward the center of the pool, the water's movement is less influenced by the walls and more influenced by the overall pool current. This analogy reflects how power law profiles struggle to accurately depict this variability in velocity gradient at the walls.

Now, we are going to solve one of the problems. So, the velocity profile for incompressible turbulent fluid in a pipe of radius R is given by u of r as u max into 1 minus r by R to the power 1 by 7. Obtain an expression for the average velocity in the pipe.

This chunk sets the stage for a practical example where we calculate the average velocity for a turbulent fluid represented by the equation: u(r) = u_max(1 - (r/R)^(1/7)). The next steps would include integration to find the average velocity across the entire cross-section of the pipe using calculus. The detailed steps involve substituting values, integrating within the specified limits, and simplifying the equations, which ultimately leads to a calculated average velocity.

Consider filling a round swimming pool with water that flows out of a hose. While water at the surface flows faster than at the depths, calculating the average speed of water exiting the hose can be a challenge. Just like finding the average speed of water in the pool using the formula we discussed, you'll consider the varying speeds at different depths before finding the overall average.

So, these are the references, as I have already shown you in the book, I mean, also in the introduction slides. And so, this week’s lecture on the laminar and turbulent flow is finished. I will see you next week with another set of lectures on hydraulic engineering. Thank you so much. Have a nice weekend.

In this concluding chunk, we summarize the key concepts discussed during the lecture on laminar and turbulent flow. The speaker expresses gratitude and informs students about the upcoming topics in the next class, reinforcing the continuity of learning in the field of hydraulic engineering.

Imagine finishing a chapter of a book where each section has built upon the last. The conclusion serves to recap the learning journey and prepare you for the next adventure in the story—much like how engineers build upon principles learned to tackle increasingly complex challenges in fluid dynamics.

--