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Interactive Audio Lesson
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Understanding Turbulent Flow in Smooth Pipes
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Today, we're diving into turbulent flow in smooth pipes. Can anyone recall what the Reynolds number indicates about flow types?
The Reynolds number helps determine if the flow is laminar or turbulent based on its value.
Correct! Once we understand that turbulent flow exists, we can explore how it behaves in smooth pipes. The velocity at the wall is actually negative infinity if we follow equation 18. Does anyone understand why?
Because as we calculate it, we substitute y with 0, which leads us to the negative infinity result.
Exactly! So, we denote the distance from the wall where the velocity is zero as 'y prime.' What's the next step in our equation?
We substitute C to find the velocity distribution, using natural logs.
Great point! Using C helps us define the velocity distribution. Remember, smooth pipes mean a particular type of behavior in these fluids.
What happens if the pipe surface is rough?
Excellent question! For rough surfaces, we see adaptations in our equations like those from Nikuradse’s studies. Let’s keep this in mind as we progress.
In summary, the way turbulent flow is expressed through equations takes into account the pipe's surface texture and Reynolds number.
Velocity Distribution Equations
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Now, let's talk about the specific equations that govern velocity distribution in turbulent flows. Who can tell me the significance of equation number 23?
It shows how we can express velocity distribution for turbulent flow in smooth pipes.
Correct! And for rough surfaces, we also derived a similar equation. What did we learn from Nikuradse’s experiments?
He determined that the thickness of the viscous sublayer changed depending on the surface roughness.
Right! The thickness of the viscous sublayer indeed has a significant impact. Now, let's solve a practical problem together regarding rough pipes.
I remember the example you mentioned in the lecture about the pipe diameter and velocity variations.
Exactly! Let’s analyze how we can arrive at the average height of roughness using the given data about diameters and velocities.
As a summary, the complexity of the equations is dictated by the flow type, and we need to adjust our parameters based on pipe roughness.
Practical Application of Turbulent Flow Concepts
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Let’s take a practical approach and apply what we’ve learned to a real-world scenario, specifically a problem regarding pipe flow. What was the initial data we had?
We know the diameter of the pipe and the velocities at different distances from the wall.
Correct! And what’s the relationship we established between those velocities?
The velocity at 4 cm is 40% more than at 1 cm from the wall.
Exactly! Now, how do we incorporate this into our equations?
We can express the velocities in terms of u star and set up our equations accordingly.
Yes! The logarithmic relationships help us backtrack to find constants like k for roughness. What about the implications?
Understanding these distributions means we can better design for efficient fluid flow in real applications.
Correct! Let’s summarize: we applied core concepts to find solutions that impact design choices in engineering applications.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In Lecture 16, we explore the dynamics of turbulent flow in smooth pipes, specifically the velocity distribution and how it is affected by parameters such as roughness. We will derive important equations, examine the implications of Nikuradse’s experiment, and solve practical problems to enhance understanding.
Detailed
In this lecture, we delve into the complexities of turbulent flow within smooth pipes, building on previous discussions about laminar and turbulent flow. By examining equation 18, we find that the velocity at the wall is negative infinity, leading to the conclusion that u is zero at a finite distance from the wall, which we denote as y prime. We substitute values and constants to derive the velocity profile for turbulent flow, which is characterized by its dependence on the distance from the wall and the roughness of the pipe. Notably, we reference Nikuradse's work to highlight the thickness of the viscous sublayer and derive additional equations relevant to rough surfaces. Problem-solving is emphasized through an example that not only reinforces theoretical understanding but also provides practical applications in hydraulic engineering. The transition from smooth to rough pipe considerations showcases the shift in turbulence behavior and velocity distributions, ensuring a comprehensive grasp of the subject.
Audio Book
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Introduction to Turbulent Flow in Smooth Pipes
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Welcome back to this last lecture of turbulent flow and laminar flows where we are going to talk about turbulent flow in smooth pipes. Last time in the last lecture we had seen what a smooth and rough bed is based on the Reynolds particle Reynolds number Re*.
Detailed Explanation
In this introduction, the speaker sets the stage by summarizing the previous lecture's focus on Reynolds numbers, which categorize flows into laminar and turbulent types. Now, the attention shifts specifically to turbulent flow in smooth pipes, a key area of study in hydraulic engineering.
Examples & Analogies
Think of driving on a smooth road versus a bumpy one. The smooth road allows for a constant speed and flow, much like turbulent flow in smooth pipes, where the fluid moves uniformly with less resistance compared to rough surfaces.
Velocity at the Wall and its Implications
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From the above equation, the velocity at the wall u at y is equal to 0 will be minus infinity, correct. If we put y is equal to 0. So, if we put ln 0, it will be minus infinity. u is positive at some distance far away from the wall.
Detailed Explanation
This statement discusses the behavior of fluid velocity in relation to the position near the pipe wall. At the wall, the fluid velocity is theoretically zero. However, as we move away from the wall (to a point 'y'), the velocity increases. The logarithmic relationship is emphasized because it reveals how fluid velocity changes based on distance from the wall.
Examples & Analogies
Imagine being at the edge of a swimming pool. When you are right at the edge (the wall), you aren’t moving at all. But as you swim away from the wall into the deeper part of the pool, your speed increases significantly.
Logarithmic Velocity Profile
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Hence, from equation 18, we can say that at distance y prime C will be minus u star by Kappa ln y prime. Because if we put y is equal to 0 it is going to be infinity. So, this logarithmic profile does not fit this. Therefore, we assume that u is 0 at some finite distance y dash y prime from the wall.
Detailed Explanation
The speaker explains a mathematical model that describes how velocity behaves in the turbulent flow near pipe walls. The value of 'C' is adjusted to accommodate the logarithmic profile of velocity distribution, indicating that there’s a distance from the wall (not just at the wall) where the velocity can effectively be considered zero.
Examples & Analogies
It's like how the wind behaves around a building. Next to the wall of the building (pipe wall), there’s little to no wind (zero velocity), but a few meters away, the wind speeds up significantly. The logarithmic nature of this relationship helps engineers predict how fluid behaves in various situations.
Nikuradse's Experiment and Implications
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From the Nikuradse’s experiment, y prime actually has been found out to be delta prime, where this is a thickness of the laminar sublayer viscous sublayer by hundred and seven.
Detailed Explanation
Nikuradse's experiments led to the discovery of the thickness of the laminar sublayer, which is crucial for determining how fluid flows over surfaces. Understanding this thickness helps engineers design more efficient piping systems by enabling better predictions of resistance and turbulence.
Examples & Analogies
Think of a river. Just like the water flows fast in the center but slows down as it brushes the riverbed, the findings from Nikuradse's experiments help us understand and predict how a fluid interacts with surfaces in pipes.
Velocity Profile for Rough Pipes
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Now, if you take this 30 out, it is . Now, you see, this is the velocity distribution of turbulent flow in rough pipes.
Detailed Explanation
The content discusses the velocity distribution for turbulent flow in rough pipes, which differs from smooth pipes. This difference can significantly affect calculations in hydraulic engineering applications as roughness impacts the friction and resistance experienced by the fluid.
Examples & Analogies
Consider a scenario where you are riding a bike on a smooth road versus a gravel path. The smooth surface allows you to gain speed easily, while the rough surface slows you down. Similarly, the type of surface (rough or smooth) greatly influences the velocity of the fluid in pipes.
Solving a Practical Problem
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Now, we are going to talk about turbulent velocity distribution in terms of average velocity. So, there is a flow and there is an elementary circular ring here, as you can see, this is of radius R.
Detailed Explanation
In this part, a practical example is introduced, focusing on how to determine average velocity in a pipe with turbulent flow. The speaker encourages applying learned equations to real-world problems, which is a critical skill in engineering. This encourages students to engage with the material hands-on.
Examples & Analogies
Imagine measuring the flow of water in a hose. Just like you would calculate how much water passes through a certain length of the hose over time, engineers compute average flow velocity to understand and design piping systems efficiently.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Turbulent Flow: A flow regime characterized by chaotic and irregular currents.
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Velocity Profile: The distribution of velocity across a cross-section of flow.
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Reynolds Number: A critical number determining whether flow is laminar or turbulent.
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Roughness in Pipes: Surface texture that affects how fluid flows and can lead to greater turbulence.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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When analyzing the velocity distribution in rough pipes, engineers use empirically derived relationships to understand fluid behavior.
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In applications like water supply networks, understanding turbulent flow is vital for predicting energy losses and managing flow rates.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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For Reynolds to flow type, just remember the number, below two thousand is calm, but higher makes the flow thunder!
📖 Fascinating Stories
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Imagine a group of ducks swimming in a pond (laminar). They swim smoothly, but when a storm hits, they frantically scatter (turbulent) in different directions.
🧠 Other Memory Gems
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Rough Pipes Rule: Roughness affects flow, causing chaos and woe!
🎯 Super Acronyms
TURB
- Turbulence Unleashes Random Behavior.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Turbulent Flow
Definition:
A type of flow characterized by chaotic changes in pressure and flow velocity.
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Term: Reynolds Number
Definition:
A dimensionless number used to predict flow patterns in different fluid flow situations.
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Term: Velocity Distribution
Definition:
The variation of velocity across different perpendicular distances from the wall of a pipe.
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Term: Nikuradse’s Experiments
Definition:
Research that studied the behavior of flow in pipes and established relationships for roughness.