3.2. Lecture Wrap-up

So, what is this? This is the average velocity divided by the frictional velocity for the turbulent pipe flow case. Now, the difference of the velocity at any point and the average velocity for smooth pipes. For smooth pipes, we have seen that the u by u star we had, we came, we derived this equation, correct. And we also just now we saw that V average by u star is going is this equation and therefore, from the above equation we can simply subtract these two equations and we are able to find u minus V average by u star. So, we do this equation, this minus this, this minus this. So, it will be u star is common, so, it will be u minus V average by u star is equal to 5.75 log to the base 10 u star y by nu u star R by nu.

In this chunk, we are discussing the concept of velocity difference in turbulent pipe flow. The average velocity and frictional velocity are compared, specifically within smooth pipes. The equation presented shows how the difference between point velocity (u) and average velocity (V average) can be expressed in terms of logarithmic functions. This relationship helps in understanding how velocity behaves across different sections of the pipe.

Imagine you're driving on a highway where the average speed is 60 mph. However, your speed at a specific moment could be different due to intersections, traffic, or other factors. The difference between how fast you drive at that moment (point velocity) and the average speed on the highway illustrates how the concept of average and instantaneous velocity works in turbulent pipe flow.

Now, we did it for smooth pipes. Now, for rough pipes, we have an equation and we also obtained V average by u star. We do the same procedure, we subtract this equation, we subtract this equation from this equation. So, this minus this and utilizing the above equation, we get u minus V average by u star.

Here, the content illustrates how the difference between velocity at a point and the average velocity is found for rough pipes as well. The same methodology used for smooth pipes is applied to rough pipes. This emphasizes that the fundamental concepts and procedures in fluid dynamics remain consistent across different types of pipes, whether they are smooth or rough.

Think about how different surfaces affect a skateboard's speed. If you’re on a smooth surface, you gain speed easily; on a rough surface, you might find resistance. Similarly, the equations reflect how the nature of the pipe's surface can impact the average velocity compared to the velocity at specific points.

Now, about the power law velocity profile, a little bit on that. 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.

This chunk introduces the power law velocity profile, which provides a mathematical model to describe how velocity varies across the diameter of the pipe. It points out that this profile is useful for smooth pipes, and the exponent 'n' varies based on the Reynolds number, which indicates the flow regime—laminar or turbulent. These relationships allow for the prediction of velocities under different flow conditions.

Consider how water flows more quickly at the center of a garden hose than at the edges. The power law velocity profile essentially captures this difference mathematically, showcasing how flow speeds change across the cross-section of a 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.

In this section, we explore how the exponent 'n' affects the velocity profile, particularly in relation to the flow type. The statement about the inability to provide zero slope at the center illustrates the limitations of the power law profile in accurately representing real flow conditions, particularly when calculating wall shear stress, which is critical for understanding flow resistance.

Envision a crowded event where people flow to the entrance. At the center, there might be a constant flow of attendees, but the edges might have bottlenecks. Just like this scenario, the formula struggles to accurately represent behaviors under certain conditions, particularly the frictional interactions at the pipe wall.

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.

In this chunk, a practical problem is posed to calculate average velocity based on a given velocity profile formula. The step-by-step approach illustrates how to derive the average velocity mathematically. Through integration, we can find the average velocity, demonstrating how theoretical concepts translate into practical applications in fluid dynamics.

Imagine pouring different sizes of marbles into a jar. Each marble represents fluid at various points along the pipe’s cross-section. By calculating how many marbles fit within the jar's height, we observe how much water volume (velocity) travels through. Similar calculations help determine average velocity in fluid flow.

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.

This concluding chunk wraps up the lecture content, indicating the importance of the topics discussed, and transitioning to further studies in hydraulic engineering. This closure reinforces the relevance of the concepts learned and sets the stage for future learning opportunities.

Ending a chapter in a storybook leaves the reader eager for the next installment. Similarly, wrapping up the lecture ignites anticipation for future discussions in hydraulic engineering, where previously learned concepts will continue to evolve and apply.

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