2.1. Integral Calculation Steps

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.

In this chunk, we are discussing how to find the difference between the velocity at a specific point in a smooth pipe and the average velocity across that pipe. We start with two derived equations involving the mean velocity (V average) and a reference velocity (u star). By subtracting one equivalent from the other, we can simplify the equation to express the difference in terms of u minus V average, with u star as a common factor.

Imagine a busy amusement park line where some people are walking (the specific point velocity) faster than the average pace of the line (the average velocity). By measuring the difference in speeds, you can understand how much faster certain individuals are compared to the group's pace.

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 deriving a crucial equation, it is shown that the common factor of u star can be used to express the relationship between u minus V average and the logarithmic function involving other parameters. This illustrates how the characteristics of flow can be simplified into a singular, usable equation that factors in the radius (R), the distance from the wall (y), and the kinematic viscosity (nu).

Think of this equation like a recipe where different ingredients (u star, y, R, nu) are combined in just the right proportions to create a final dish (the relationship) that describes the flow in the pipe.

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.

This chunk explains how the same approach used for smooth pipes applies to rough pipes as well. Although rough pipes have a different set of equations, the principle remains the same. By subtracting the derived equations for rough pipes, we arrive at the same form relating u minus V average, indicating a consistent behavior across different pipe types.

Consider driving a car over a smooth road versus a bumpy road. While the surface affects your speed at different points (rough vs. smooth), the method to calculate your average travel time remains the same, highlighting the importance of the overall driving experience relative to the surface.

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.

In this section, we introduce the power law velocity profile, expressing how the velocity at a point in a smooth pipe (u) relates to the maximum velocity (u max) using a power function of the distance from the center (y) and the radius of the pipe (R). The exponent (1/n) is significant as it varies depending on flow conditions characterized by the Reynolds number.

Imagine a river where the speed of the current decreases as you move away from the center. The power law can be thought of as a way to describe how that speed changes based on how far you are from the fastest current in the middle of the river.

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.

This chunk highlights limitations of power law velocity profiles, indicating that they cannot accurately predict conditions (like a zero slope) at the pipe's center. This becomes problematic, particularly when calculating wall shear stress, which is crucial for understanding forces at pipe walls in fluid dynamics.

Think of a water slide where the speed of the water gets very fast in the middle but slows down at the edges. If we can't measure (or calculate) that center speed accurately, it means we might not understand how much water pressure we are feeling at different spots – akin to calculating shear stress 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.

Here, we are introduced to a specific problem where a given velocity profile is provided for a turbulent fluid. The task is to derive the average velocity based on this defined relationship. By integrating the velocity profile across a given area of the pipe, we substitute the velocities and solve the integrals step-by-step.

Imagine trying to find out the average speed of a car on a round trip journey. Knowing different speeds at various sections of the road (like traffic conditions) allows you to calculate an overall average speed by analyzing the entire trip rather than looking at isolated segments.

So, using the, we write V bar is equal to 2 u max integral x to the power 1 by 7 one minus x minus dx. So, that is from 1 to 0.

This chunk provides an equation for calculating the average velocity based on the specific profile. Here, we switch variables and formulate the integral appropriately. The overall goal is to assess the flow characteristics over the entire cross-section of the pipe, taking into account the variables at play.

If you divide a cake (the integral area of the pipe) into manageable slices (small segments of the flow), calculating how much of the cake you have already eaten becomes easier by summing up those small pieces rather than trying to swallow the entire cake at once.

So, V bar will be written by 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 performing the integral calculations, we arrive at a final numerical result for average velocity (V bar). The relationship derived showcases how even with a complex velocity profile, the result can simplify to a practical expression that reflects real-world flow characteristics.

Think of cleaning a large window with a cloth. You systematically clean sections (like performing integrals) to ensure you get the whole window spotless (calculate average velocity). In the end, you can see the clear outcome despite the complexity of the task.

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