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Interactive Audio Lesson
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Introduction to Shear Stress
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Today, we’re discussing biophysical concepts, particularly the shear stress in turbulent flows. Can anyone tell me how laminar and turbulent shear stress differ?
I think laminar shear stress is only due to viscosity, right?
Exactly, Student_1! In laminar flow, shear stress indeed is only due to viscosity. However, in turbulent flow, there's an extra component due to turbulence. We can call it 'eddy viscosity'.
How does eddy viscosity differ from regular viscosity?
Eddy viscosity varies with flow conditions, unlike standard viscosity. It becomes zero at the wall! Remember: `EDDY` can help you remember 'Eddy viscosity varies'.
So, the shear stress in turbulent flow is higher than in laminar flow?
Correct! Now, what would be the mathematical form of turbulent shear stress?
Isn't it related to velocity difference?
Yes! It’s represented as τ = -ρ(u'v'). Great summary, everyone!
Reynolds Shear Stress
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Moving on, let’s discuss Reynolds shear stress. Who remembers his expression for turbulent shear?
I remember it involves fluctuating velocity!
Exactly! It can be written as τ = -ρ(u'v'). This equation shows how turbulent shear stress is derived using velocity fluctuations. Would you like to know why it’s a negative quantity?
Yes, please!
Good! It’s negative because it represents opposing forces in different fluid layers. Remember: 'Negative reflects opposition!' Now, let's think of practical instances where this concept applies.
How about in design of hydraulic systems?
Yes! That's an excellent example. Keep thinking about these applications as we continue.
Prandtl’s Mixing Length Theory
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Now let’s delve into the mixing length theory proposed by Prandtl. Who can summarize what mixing length means?
Isn’t it the distance fluid particles travel to mix momentum?
Exactly! It's a critical concept for estimating turbulent shear stress. Could anyone write out how it relates to velocity?
It’s l_m = κy, right?
Great! And what does κ represent?
It's the von Karman constant, about 0.4!
Excellent! This relationship simplifies calculating turbulent shear stress. Remember: `MIX` for `Mixing Length`.
Applications in Turbulent Flows
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Let’s wrap up by applying these concepts. How might turbulent shear stress affect pipe design?
It could influence how we calculate pressure drops.
Yes! Understanding these stresses is crucial. Additionally, how does knowing ε help in flow predictions?
Eddy viscosity affects flow behavior close to walls!
Perfect! Now, let’s summarize what we learned today. What are three key aspects of Boussinesq's model?
Turbulent shear stress is more complex than laminar, influenced by eddy viscosity and mixing length theory.
Exactly! Keep those concepts in mind for future applications.
Introduction & Overview
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Quick Overview
Standard
The focus of Boussinesq's model is on the differences in shear stress between laminar and turbulent flows, emphasizing the additional component of shear stress caused by turbulence. The model introduces eddy viscosity as a critical factor, which varies with flow conditions and becomes essential in calculating turbulent shear stress, leading to important insights in fluid mechanics.
Detailed
Boussinesq’s Model
Boussinesq’s model is a significant concept in hydraulic engineering, specifically when analyzing turbulent flow. Unlike laminar flow, where shear stress is due solely to viscosity, turbulent flow involves an additional shear stress component due to the chaotic nature of turbulence. This distinction leads to larger total shear stress values in turbulent flows compared to laminar ones.
In turbulent flow modeling, the shear stress contribution due to turbulence can be expressed similarly to that of laminar flow but utilizes a new coefficient known as ‘eddy viscosity’ (η). The relationship can be denoted as:
Shear Stress Formula
- Turbulent shear stress (τ): τ = η (du/dy), where η represents eddy viscosity, and this leads to the dynamic representation of the shear stress caused by turbulence.
The kinematic eddy viscosity (ε) is defined as:
- Kinematic eddy viscosity (ε): ε = η/ρ
Where ρ is the fluid density. Unlike dynamic and kinematic viscosities in laminar flow (represented by μ and ν, respectively), eddy viscosity values depend on changing flow conditions, decreasing near the wall and becoming zero at the wall.
Reynolds Shear Stress
Developed by Reynolds in 1886, the formula for turbulent shear stress includes fluctuating velocity components in different directions (u' and v'). The Reynolds shear stress is defined as:
- Reynolds shear stress: τ = -ρ (u'v')
Prandtl's Mixing Length Theory
Prandtl introduced the mixing length concept in 1925 to quantify turbulent shear stress. He proposed that the mixing length (l_m) is defined as:
- Mixing length (l_m): the distance between two fluid layers in the vertical direction that allows fluid particles to transfer momentum effectively.
The correlations derived from mixing length theory provide insights into the relationship between turbulent shear stress and measurable parameters. Prandtl assumed that:
- l_m is a linear function of the distance (y) from the wall, leading to:
- Prandtl's relation: l_m = κy, where κ (kappa) is the von Karman constant, approximately 0.4.
Conclusion
In turbulent flowing pipes, the shear stress predominantly arises from turbulent shear stress, whereas viscous shear stress is near the boundary only. Hence it can be simplified into a more manageable equation. This modeling insight is crucial for designing efficient hydraulic systems.
Audio Book
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Understanding Shear Stress in Turbulent Flow
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So, shear stress in turbulent flow. We are going to talk about a model that is called Boussinesq’s model, where the total shear stress, in case of laminar flow it was due to the viscosity viscous. Sorry. Yeah, that was only due to the viscous. But in a turbulent flow, there is an additional component of shear stress that happens because of the turbulence in the flow. So, therefore, the shear stress in total is much, much larger than the viscous flow.
Detailed Explanation
In turbulent flow, the shear stress is different from that in laminar flow. Unlike laminar flow where shear stress is solely due to viscosity, turbulent flow has additional shear stress components arising from velocity fluctuations within the fluid. This means that the total shear stress in turbulent flow is significantly higher than in laminar flow, highlighting the impact that turbulence has on fluid movement.
Examples & Analogies
Think of laminar flow as a smooth, orderly dance where every dancer follows their path without deviation. In contrast, turbulent flow is like a lively, chaotic party where people are constantly bumping into one another, creating more excitement and energy — this chaos contributes to greater shear stresses.
Eddy Viscosity and its Measurement
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Boussinesq’s says as for laminar flow, therefore, the shear stress due to the turbulence component is. Here, you see, this is similar. So, instead of µ there is something called ɳ a new coefficient of viscosity, and this is called eddy viscosity. Therefore, if we want to write a kinematic eddy viscosity then we write it by epsilon, for example.
Detailed Explanation
In the Boussinesq model, turbulent shear stress can be expressed using a new parameter called eddy viscosity (η). This is analogous to the dynamic viscosity (µ) used in laminar flow. Eddy viscosity accounts for the chaotic and fluctuating motion of turbulent flows and allows us to quantify the resulting shear stress. Additionally, kinematic eddy viscosity (ε) can be defined, which relates to the density of the fluid, further facilitating calculations in turbulent scenarios.
Examples & Analogies
Consider eddy viscosity as how much warmer a room feels when people are moving around inside compared to a quiet, still room. Just like the mixing of warmer and cooler air creates a more dynamic and rich environment, the turbulent motions of fluid particles create a higher eddy viscosity.
Reynolds Shear Stress Overview
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Now, coming to what is Reynolds shear stress. So, Reynolds in 1886 gave expressions for turbulent shear stress between two fluid layers separated by a small distance. And he said that the shear stress due to turbulence can be written as, minus rho u prime v prime whole bar.
Detailed Explanation
Reynolds shear stress is a fundamental concept in the study of turbulence. It quantifies the additional stress from fluctuations in velocity between different layers of fluid. The formula provided by Reynolds illustrates that this stress can be described as a product of average density (ρ) and the correlation of the fluctuating velocity components (u' and v'). This means that Reynolds shear stress incorporates the effects of turbulence on flow characteristics.
Examples & Analogies
Imagine two layers of water in a lake, with the top layer being affected by wind and the bottom layer being calm. The interaction between these two layers can create stress at their boundary, similar to how Reynolds shear stress quantifies the turbulence between layers. Picture a tug-of-war where one side pulls more erratically — their combined strength has a significant impact on the overall outcome.
Prandtl's Mixing Length Theory
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Now, there is a concept of Prandtl's mixing length theory. So, turbulence shear stress can be calculated if this thing is known, u prime v prime whole bar is known. In 1925 Prandtls introduced the concept of mixing length, which can be utilized to express the shear stress here, in terms of some measurable quantity.
Detailed Explanation
Prandtl's mixing length theory provides a method for calculating turbulent shear stress by introducing the idea of mixing length (l_m). This mixing length represents the distance over which fluid particles can mix and exchange momentum. By determining this mixing length, we can derive a formula for turbulent shear stress that utilizes measurable quantities like average velocity gradients, thus making the calculations more practical.
Examples & Analogies
Consider mixing two different colors of paint. The distance between the two colors where they blend creates a 'mixing length'. Similarly, in fluid dynamics, the mixing length is where the motion of one layer of fluid impacts another. The smoother the transition (just like a smooth blend of paint), the more accurately we can predict how the fluids will behave.
Linear Relationship of Mixing Length
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Prandtl assumed that the mixing length lm is a linear function of distance y from the wall or any solid boundary. Therefore, he said lm can be written as ky, where kappa is known as von Karman constant and it has been found to be equal to 0.4.
Detailed Explanation
Prandtl's assumption that mixing length (l_m) is linearly related to the distance from a boundary (y) simplifies the calculation of shear stress in turbulent flow. By expressing l_m as a product of a constant (kappa, approximately 0.4) and the distance from the wall, we can easily plug values into equations to find shear stress values based on measurable distances from a boundary.
Examples & Analogies
Think of measuring the height of a tree. If we know how long the trunk is (the distance from the ground), we can estimate how high the branches reach by using a proportion or coefficient (like kappa). In turbulence theory, this relationship helps us predict how mixing length varies based on proximity to surfaces.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Boussinesq's Model: A model explaining shear stress in turbulent flow with the introduction of eddy viscosity.
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Eddy Viscosity: A variable coefficient representing turbulence's contribution to shear stress.
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Reynolds Shear Stress: Defined by the fluctuating components of velocity, representing turbulent shear stress.
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Mixing Length Theory: A theory relating turbulent shear stress to measurable quantities through mixing length concepts.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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In the engineering design of pipelines, understanding turbulent shear stress helps in predicting flow resistance.
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Using Prandtl’s mixing length theory, engineers can estimate the shear stress in turbulent flows in confined spaces.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To measure flow, the shear shows, An eddy here, as turbulence grows!
📖 Fascinating Stories
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Imagine a river flowing smoothly, representing laminar flow. When it encounters rocks, it swirls and roils—this chaos shows turbulent flow, thus leading us to understand shear stresses caused by such movement.
🧠 Other Memory Gems
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Remember
TREMBLE: Turbulent Reynolds Eddy Mixing Blends Laminar Energy for flow.
🎯 Super Acronyms
MIX for Mixing Length to recall how momentum mixes across layers.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Shear Stress
Definition:
A measure of how much force is applied parallel to a surface.
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Term: Eddy Viscosity
Definition:
A coefficient that reflects the additional viscosity due to turbulence in a fluid.
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Term: Reynolds Shear Stress
Definition:
The shear stress due to turbulent fluctuations in fluid velocity.
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Term: Mixing Length
Definition:
The distance over which momentum mixing occurs between fluid layers.
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Term: Von Karman Constant (κ)
Definition:
A dimensionless constant, approximately 0.4, used in the relationship between mixing length and distance from surfaces.