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Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Average Velocity in Smooth vs. Rough Pipes
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Today, we will continue our discussion on turbulent pipe flow, focusing on the critical concept of average velocity. Can anyone remind me how we define the average velocity in this context?
Isn't it the total flow divided by the cross-sectional area of the pipe?
Exactly! Now, we derived a relationship for smooth pipes; can anyone share the key equation we discussed?
It's the equation that shows the difference between point velocity and average velocity.
Correct! We noted that this relationship can be expressed with a logarithmic equation, right? Now, what did we find when comparing smooth and rough pipes regarding their average velocities?
The difference in velocity is the same for both, wasn't it?
That's right! This observation is important because it indicates a consistent behavior in turbulent flow characteristic. Always remember that these principles are fundamental across different types of pipes.
Power Law Velocity Profile
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Let's shift gears and talk about the power law velocity profile. Who can explain what it represents?
It describes how velocity varies with distance from the pipe center, based on a specific exponent.
That's accurate! Remember, the exponent n changes based on the Reynolds number. What happens if we set n equal to 7?
We get the one-seventh power law velocity profile!
Correct! This profile is significant because it provides a useful model, even if it cannot give us accurate shear stress values. Why do we think that is?
Because it results in infinite velocity gradients at the wall.
Exactly! Now let's summarize the power law velocity - remember this concept when dealing with turbulent flows.
Calculating Average Velocity
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Now, let’s apply what we’ve discussed with a practical example. Given a velocity profile for incompressible turbulent fluid, how do we begin calculating the average velocity?
We should start by rewriting the profile equation.
Good start! What does the equation look like when we express it in terms of the area?
It's 1 over pi times the integral from 0 to R of the given function, multiplied by 2 pi r.
Correct! Remember to pull constants out of the integral. Can someone summarize how we arrived at the average velocity formula?
After substituting and solving the integral, we found V bar equals 0.816 times u max.
Exactly! This systematic approach can be applied to any similar problem. Practice deriving such equations!
Introduction & Overview
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Quick Overview
Standard
The section focuses on the equations governing the average velocity in turbulent pipe flow, specifically highlighting the differences between smooth and rough pipes, and introducing the power law velocity profile. It culminates with practical examples of calculating the average velocity using a given velocity profile.
Detailed
Detailed Summary of Lecture Wrap-up
This section serves as a conclusion to a discussion on turbulent pipe flow, emphasizing key equations that relate to average velocity. It begins by reviewing the ratio of average velocity to frictional velocity, noting important differences between smooth and rough pipes.
- Equation Derivation: The average velocity in smooth pipes is related to the turbulent velocity profiles through a logarithmic equation. The section highlights how average velocity can be expressed as a function of variables like frictional velocity and radius of the pipe.
- Comparison of Smooth and Rough Pipes: Both types of pipes yield similar differences between point velocities and average velocities, indicating fundamental flow characteristics in turbulent conditions. The derivation provides insight into the universal aspects of fluid mechanics in these contexts.
- Power Law Velocity Profile: The section also introduces the power law velocity profile, indicating how the maximum velocity is expressed through a specific power relationship. Notably, it states that the value of the exponent n depends on the Reynolds number, offering one instance where n equals 7.
- Practical Example: A practical problem is presented to calculate the average velocity based on a defined velocity profile. Through systematic integration of the flow equations, the section shows how to derive the significant average velocity formula of 0.816 times the maximum velocity in the flow, illustrating the application of theoretical knowledge.
The section concludes by reinforcing the importance of understanding both turbulent flow characteristics and velocity profiles in hydraulic engineering.
Audio Book
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Understanding Velocity Difference in Pipe Flow
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Equation for Rough and Smooth Pipes
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Power Law Velocity Profile
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Application of the Power Law Profile
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Calculating Average Velocity
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Final Thoughts and Future Directions
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Smooth and Rough Pipes: Differences in behavior in turbulent flow.
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Average Velocity Equation: Mathematical representation of average velocity.
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Power Law Profile: Describes velocity distribution in a pipe.
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Reynolds Number Impact: How it affects the velocity profile.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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An example of calculating average velocity using a power law velocity profile in a turbulent flow situation.
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Demonstrating how average velocity equations differ for smooth versus rough internal surfaces.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In a pipe so round and wide, average flows do often slide.
📖 Fascinating Stories
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Imagine a river flowing with different currents; its average flow tells us how fast it moves overall, despite the chaos of small ripples.
🧠 Other Memory Gems
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PAV (Power Average Velocity) to remember how flow presents itself.
🎯 Super Acronyms
RAVE (Reynolds Average Velocity Equation) can help recall the concepts under turbulent flow.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Average Velocity
Definition:
The mean velocity of a fluid flow, calculated as the total flow rate divided by the cross-sectional area.
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Term: Frictional Velocity
Definition:
A characteristic velocity scale used in turbulent flow, related to shear stress.
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Term: Power Law Velocity Profile
Definition:
A mathematical expression that describes how the velocity of fluid varies across the radius of a pipe, defined by y/R raised to the power 1/n.
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Term: Turbulent Flow
Definition:
A type of fluid flow characterized by chaotic changes in pressure and velocity.
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Term: Reynolds Number
Definition:
A dimensionless number used to predict flow patterns in different fluid flow situations.