1.1. Smooth Pipes Equation

So, what is this? This is the average velocity divided by the frictional velocity for the turbulent pipe flow case.

In turbulent pipe flow, we examine the relationship between the average velocity of the fluid and the frictional velocity, which is a measure of the velocity fluctuations due to turbulence. This ratio helps to characterize the flow regime within the pipe.

Think of a river during a flood. The average speed of the water in the river is like the average velocity in a pipe, while pockets of fast and slow-moving water represent the frictional velocities due to turbulence and obstacles in the river.

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.

The focus here is on how the velocity at any point in the pipe (u) relates to the average velocity (V average). The equation derived relates these two velocities in a way that helps us understand the flow behavior within smooth pipes.

Imagine driving on a highway. The average speed of cars is like the average velocity in a pipe, while individual cars may speed up or slow down, reflecting the difference between the velocity at any point (u) and the average velocity (V average).

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.

In this step, we manipulate the derived equations to express the difference between point velocity and average velocity as a function of turbulence characteristics, leading to a specific formula involving logarithmic functions.

Think of it as putting together pieces of a puzzle. By understanding how each piece (or in this case, equation) fits together, you can create a complete picture (the overall flow behavior in the pipe).

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.

This equation represents an important relationship in smooth pipe flow, connecting the local velocity, average velocity, and viscosity. By isolating 'u' and 'V average' terms, we can further explore how these velocities interact within smooth pipe conditions.

Consider a smooth escalator. The average speed of movement is constant, but individual movements can vary (like with turbulence). This equation helps us understand how these variations relate to the smooth operation of the escalator.

Now, we did it for smooth pipes. Now, for rough pipes, we have an equation and we also obtained V average by u star.

This section introduces the concept of rough pipes, indicating that similar equations can be derived for these types of pipes as well, albeit with modified parameters to account for increased friction due to surface roughness.

Imagine a road that's been chipped and cracked versus a newly paved road. The roughness of the surface affects how smoothly vehicles can pass, similar to how rough pipes influence fluid flow.

Is there any observation? If you see for either for smooth or for rough this difference came out to be the same.

An important observation is that the difference between local velocity and average velocity remains consistent regardless of whether the pipe surface is smooth or rough. This consistency is significant for understanding flow behavior.

It's like how the gusts of wind feel similar on both a calm and windy day. The overall effect (difference in weather) can feel the same despite the conditions changing.

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 introduces the power law velocity profile, which is an alternative way to express how velocity changes in smooth pipes with respect to the distance from the center. The exponent 'n' relates to the Reynolds number and changes for different flow conditions.

Think of a balloon gradually deflating. The power law describes how the speed of air exiting changes based on how much pressure is left, similar to how fluid velocity changes with distance in a pipe.

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.

While power law profiles provide useful information, they have limitations such as not being able to represent a flat profile at the center of the pipe. This poses a challenge in accurately calculating shear stress at the wall, which is critical for understanding fluid behavior.

Think of trying to measure how smooth a surface is with inaccurate tools. If the tools (like power law profiles) can’t give you precise measurements, understanding key aspects like shear stress becomes difficult.

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.

This section introduces a practical problem where we use a specific velocity profile to derive an expression for average velocity in a pipe. This demonstrates how theoretical concepts apply in real-world situations.

It’s like following a recipe while cooking. Each step you take (calculation) must be followed to create the final dish (average velocity) successfully.

So, V bar is 2 u max into, just copying the same thing from the last line, 7/15 and V bar is going to be 0.816 u max.

After going through a series of calculations, we arrive at a final expression for average velocity (V bar), showing a clear relationship to the maximum velocity. This is a crucial outcome for practical applications in fluid dynamics.

Just like in sports, where you might end up with a final score after many plays, these calculations lead us to a 'score' for average velocity based on our initial conditions.

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