3.1. References

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 equation 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.

So, nu and nu can get cancelled, it will be y by R or we can say, u minus V average by u star is equal to 5.75 log y by R plus 3.75. So, this is an important equation again. So, all these you either you remember or you can actually derive it. It is very simple, starting from the, starting from the basic logarithmic velocity profile.

In this chunk, we're discussing how to calculate the difference between the velocity at any point in a smooth pipe and the average velocity. We start by defining the relevant terms — average velocity and frictional velocity, specifically for turbulent flow. The relation between the velocity at any point, denoted as 'u,' and the average velocity, V_average, can be formulated through an equation derived from a logarithmic profile. The significant discovery is that the difference is expressed as a function of log terms that relate the properties of the fluid and pipe.

From the derived equation, we recognize that the term 'nu' (kinematic viscosity) appears but cancels out, simplifying our expression to focus on the geometric ratio of the height above the pipe surface (y) to the pipe radius (R). This cascading understanding leads us to a manageable equation to work with, emphasizing the utility of logarithmic relationships in fluid dynamics.

Imagine water flowing through a garden hose. The average speed of water flowing through the hose is like V_average. When you position your finger on the hose's opening, the water velocity right at your fingertip can be higher than the average speed due to the pressure from the hose — this difference corresponds to our concepts of 'u' and 'V_average.' The equation derived helps us understand exactly how much faster the water flows at that point compared to the average throughout the hose.

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. So, we have got y by k divided by R by k plus 8.5 minus 4.75.

Or we can simply get u minus V average by u star is equal to 5.75 log to the base 10 y by R plus 3.75. Is there any observation? If you see for either for smooth or for rough this difference came out to be the same. So, the observation is the difference of velocity at any point and average velocity is same for both smooth and rough pipes.

This chunk expands our understanding by applying the same principles to rough pipes. After establishing the previous calculations for smooth pipes, we repeat the procedure for rough pipes to derive the difference in velocity. The key observation is that the fundamental equation remains unchanged regardless of whether the pipe is smooth or rough. This means the relationship we derived for smooth pipes is equally applicable to rough pipes, which is an important insight in fluid dynamics, indicating broader applicability of our formula across different conditions.

Think of a water slide — the smooth part gives you speed easily, much like a smooth pipe. However, the rough patches (like the ridges or bumps on a slide) don’t drastically change how fast you are going at any given moment. The principle remains that you might speed up slightly in certain spots, but overall, the reports of fast flows at specific points still hold true, regardless of the slide’s texture!

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 concept of the power law velocity profile, which is a specific way to describe how velocity varies within a pipe. Particularly for smooth pipes, we can express the velocity at a height 'y' relative to the maximum velocity, 'u_max.' The relationship shown here is based on the ratio of 'y' to the radius 'R' raised to the power determined by 'n,' which is a function of the Reynolds number. The common selection for 'n' is 7, yielding the one-seventh power law profile, which provides a widely used approximation in fluid dynamics for turbulent flow in pipes.

Consider how a fountain sprays water; the water close to the fountain's base is moving much faster than the droplets further away. The power law helps describe this scenario where velocities vary with height. Using power laws is similar to adjusting the speed settings on a fan based on the distance from it — the closer you are, the stronger the wind you feel, much like how water speed changes in a pipe.

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. Hear by chance, or by, you know, you get 1 by 7 power law. So, how to attack this type of problems and actually you can be given any such profile and you should follow the same procedure as I am going to do now, so that, you are able to calculate the average velocity in the pipe.

In this part, we’re introduced to a practical exercise involving a specific velocity profile for a turbulent fluid within a pipe. The function is defined as u(r) in terms of a known maximum velocity and the radius of the pipe. The objective is to derive the average velocity from this profile using integration. By establishing the method in which to approach this problem — starting from the given function and applying calculus to determine the average across the pipe's cross-section — students learn a systematic way to solve such fluid dynamics problems, regardless of the initial complexity.

Picture trying to find the average speed of cars on a highway. If you know the maximum speed limit (u_max) and the speeds change due to various conditions, like merging lanes, you calculate the average over a certain stretch (like using integration over the area of the highway). Here it's similar where you're taking the average speed of fluid flowing through a pipe by considering how its velocity changes along the radius of the pipe.

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