1.4. Power Law Velocity Profile

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.

In this section, we start by discussing the relationship between average velocity and frictional velocity in turbulent pipe flow. Specifically, we are considering how the velocity at any point in the pipe relates to the average velocity. The concept is crucial for understanding how fluid behaves as it flows through pipes, particularly in terms of friction and turbulence.

Imagine a group of runners on a track. The average speed of the group can be compared to the frictional velocity, which is influenced by how quickly each runner can go. Just like each runner might have a different speed, each point in the pipe has a different velocity compared to the average flow.

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.

We refer to previously derived equations that relate average velocity to frictional velocity. For smooth pipes, we can derive a specific equation regarding how the velocity at a point differs from the average velocity. By subtracting the equations, we simplify our analysis, leading to a more comprehensive understanding of the flow dynamics.

Think of a car traveling on a smooth highway. The car's speed can be compared to the velocity of fluid at a point in the pipe, while the average speed of all cars on that highway represents the average velocity of the fluid. The difference in how fast each car is going versus the average speed can be likened to our calculations.

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.

Next, we apply the same analytical process to rough pipes as we did to smooth pipes. Reference to another equation specific to rough pipes allows us to derive similar expressions for their velocity profiles. The goal remains the same: to understand how velocity at specific points relates to the average effectively, regardless of the pipe's texture.

Imagine now traversing a bumpy dirt road. The speed of your car, analogous to fluid velocity, changes more drastically as your tires hit the rough patches. Just as we calculate the average speed of the vehicle over both smooth and rough terrains, we find similar comparisons in fluid dynamics.

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, u by u max can be written as y by R to the power 1 by n.

This chunk introduces the power law velocity profile, particularly for smooth pipes. The profile describes how velocity varies radial from the center of the pipe. It is defined mathematically where the ratio of the velocity at a point to the maximum velocity follows a specific power relationship with the radial position in the pipe.

Think of a fountain where water jets up at varying heights depending on how far from the center you measure. The closer you are to the jet, the more powerfully it shoots up, similar to the velocity profile in a pipe where the relationship varies based on how far you are from the center.

The value of n will increase with increasing Reynolds number. Power law velocity profiles cannot give 0 slope at the pipe center.

As the Reynolds number, which indicates flow regime, increases, the value of 'n' in the power law profile also increases. This influences the velocity profile and its characteristics significantly. Additionally, it's important to note that the power law velocity profile cannot produce a zero slope at the center of the pipe, which has implications for how we interpret wall shear stress.

Imagine a roller coaster speeding up as it climbs higher hills. As it gains speed (akin to an increasing Reynolds number), the dynamics change in ways that affect how you experience the ride. Similarly, our formula changes as fluid flows become turbulent.

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.

Finally, we move towards practical applications by solving a problem that utilizes the power law velocity profile for a turbulent fluid. A specific velocity profile equation is given, and we'll go through the derivation to find the average velocity in the pipe, emphasizing the steps and calculations involved.

Picture measuring the amount of syrup flowing through different sections of a pipe. By knowing the profile of syrup flow, we can calculate how much syrup we will have on average flowing through the entire pipe, much like how we derive average velocity in our example.

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