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Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Introduction to Shear Stress
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Today we will explore shear stress in turbulent flow. Can anyone tell me what shear stress is in the context of fluid dynamics?
Shear stress is the stress component parallel to the material cross section.
Exactly! It measures how much force per unit area acts parallel to a surface. Now, in turbulent flow, how do we express shear stress at the wall?
Isn’t it represented as tau not (τ₀)?
Correct! τ₀ is the constant shear stress at the wall. We will use it in implementing our equations later.
Velocity Profiles in Turbulent Flow
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Let’s discuss velocity profiles. How does the turbulent velocity profile differ from that in laminar flow?
In laminar flow, the velocity profile is parabolic, but in turbulent flow, it has a more logarithmic shape.
Great observation! The turbulent profile is indeed fuller. This will be important when we calculate shear stresses.
What are the implications of this profile shape?
The shape affects how we calculate velocities at different points in the pipe, which leads us to determine τ₀ accurately.
Calculating Shear Stress
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Let’s move on to a problem. We're given the diameters and velocities in a pipe. How would we proceed to find τ₀?
We need to convert all measurements to SI units first.
Exactly! What will be our first step in the calculations?
Substituting the values into the velocity defect equation!
Right! Make sure to integrate and use boundary conditions properly.
Velocity Defect Law
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Does anyone remember what the velocity defect law states?
It shows the difference between maximum velocity and actual velocity in turbulent flow.
Perfect! That difference helps us understand how much energy is lost due to frictional forces. Let's summarize that before we conclude.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, students learn to calculate shear stress using the turbulent flow theory. The example provided involves determining wall shear stress based on velocity measurements at specific points in a pipe, integrating necessary equations to arrive at the final result.
Detailed
In this section, we explore the determination of shear stress in turbulent flow through pipes. By employing the relationship derived from the Prandtl mixing length theory, we express shear stress as proportional to the velocities at different points within the pipe. The conceptual progression begins with defining shear stress at the wall, represented by tau not (τ₀). The section guides students through the integration of key equations, culminating in the application of boundary conditions. An illustrative problem concerning water flow in a pipe is solved using specific diameters and velocities, ultimately leading to the calculation of τ₀. Through this detailed procedure, we demonstrate the significance of understanding turbulent velocity profiles and their divergence from laminar flow profiles. As a result, students gain a comprehensive view of the methods used to analyze shear stress in flow dynamics.
Audio Book
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Introduction to the Problem
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So, now we are going to solve one of the problems, problem number 7. And what it says is, the velocity of water. So, what we have learned in this particular lecture is about the turbulent flow and this problem 7 will help you in solving any problem that is based on this particular concept. So, it says the velocities of water through a pipe of diameter 10 centimeter are 4 meters per second and 3.5 meters per second at the center of the pipe and 2 centimeters from the pipe center, respectively. Considering turbulent flow in pipe, determine the sheer stress at the wall.
Detailed Explanation
In this section, we are introduced to Problem 7, which focuses on calculating the shear stress in turbulent flow through a pipe. We are provided with specific data regarding the velocities of water at two points in the pipe: at the center (4 m/s) and at a point 2 cm from the center (3.5 m/s). The problem specifically requires us to determine the shear stress at the wall's surface of the pipe.
Examples & Analogies
Imagine a garden hose: when you turn on the water, it flows quickly in the center but may flow slower near the edges where it touches the hose. The different speeds represent the varying velocities of water in the pipe, and we can calculate the force (shear stress) acting at the hose's surface.
Data Preparation
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As always what we do we solve, we write given, diameter is given as 10 centimeter, try to always write down in SI units. So, we write 0.1 meter. So, diameter is 10. So, radius is going to be 0.05 meter. u max is given, is given as 4 meters per second, that is, at y is equal to R.
Detailed Explanation
Here, we convert the given diameter (10 cm) into SI units for easier calculations. The diameter is converted into meters (0.1 m), and we derive the radius as half of the diameter, which is 0.05 m. We also note that the maximum velocity (u max) is at the maximum radius of the pipe and equals 4 m/s, as determined from the problem.
Examples & Analogies
Think of measuring ingredients for a recipe. Just as you might convert cups to liters for clarity, we are converting the pipe's measurements into meters to make our calculations easier and standard.
Calculating Shear Velocity
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So, now u max we are using the minus u / u * was 5.75 log R / y. So, substituting the values here, this from here, this equation, 4 - 3.5 divided by u * is equal to 5.75 log base 10 5 / 3. This will give us, u * as 0.392 meters per second.
Detailed Explanation
In this chunk, we apply a logarithmic velocity defect equation to find the shear velocity (u). By substituting the known values, we rearrange the terms to isolate u. We find that u* equals 0.392 m/s, which is critical for determining the shear stress.
Examples & Analogies
Imagine you are adjusting the speed on a treadmill. Just like trying to find the right settings when you reduce or increase speed, we are tweaking our variables to get the right velocity we need for further calculations.
Calculating Shear Stress
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We also know, u is under root tau not / rho or tau not is rho u whole square. Therefore, tau not rho is 1000 and u * we already got, 0.392 whole square. So, tau not is coming out to be 153.6 Newton per meter square.
Detailed Explanation
At this point, we use the relationship between shear velocity and shear stress. We square the previously determined shear velocity and multiply it by the fluid's density (1000 kg/m³ for water). This calculation yields a shear stress (tau_not) of approximately 153.6 N/m² at the pipe wall.
Examples & Analogies
Think of pressing your hand flat against a surface. The harder you push (analogous to shear stress), the more resistance you feel. In our calculations, the shear stress indicates how strongly the water is pushing against the wall of the pipe.
Summary of Results
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So, going back again to the slide, so, what we got was approximately 153 Newton per meter square the sheer stress at the wall.
Detailed Explanation
In summary, after performing the necessary calculations, we conclude that the shear stress at the wall of the pipe is approximately 153 N/m². Thus, our computations successfully followed from the initial problem statement through various steps to arrive at the required result.
Examples & Analogies
Imagine measuring how hard you're pressing against a wall; by understanding this force (shear stress), we can infer how fluid flow interacts with surfaces, much like how we experience friction when sliding our hand against a rough surface.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Shear Stress (τ₀): It is a constant at the pipe wall during turbulent flow.
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Turbulent Flow Characteristics: Chaotic and non-linear compared to laminar flow.
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Velocity Profile: The relationship between maximum and actual velocities across different points in the pipe.
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Integration Methodology: Essential for deriving shear stress from initial equations and boundary conditions.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Calculating shear stress for a turbulent water flow in a pipe with given velocities at specific radial distances.
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Comparing velocity profiles of laminar and turbulent flows to demonstrate their differences.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When the flow is loud and fast, turbulent moments can't be outclassed.
📖 Fascinating Stories
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Imagine a turbulent river crashing over rocks; it shows how energy is dissipated while flowing.
🧠 Other Memory Gems
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Remember 'TSU' for Turbulent Shear at the Wall: T for Turbulent, S for Shear, U for at the Wall (τ₀).
🎯 Super Acronyms
‘PVC’ helps to remember
- Pipe Velocity Constant related to τ₀.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Shear Stress (τ)
Definition:
A measure of the force per unit area exerted parallel to a surface.
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Term: Turbulent Flow
Definition:
A type of flow characterized by chaotic changes in pressure and velocity.
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Term: Velocity Defect Law
Definition:
A law that relates the difference between maximum and actual velocities in turbulent flow.
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Term: Prandtl Mixing Length Theory
Definition:
A theory that describes the velocity distribution in turbulent flow based on turbulent mixing.
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Term: Boundary Conditions
Definition:
Conditions that must be satisfied at the boundaries of a physical system.