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Interactive Audio Lesson
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Introduction to Shear Stress in Turbulent Flow
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Today, we'll delve into Reynolds shear stress, an essential component of turbulent flow. Can anyone tell me what shear stress signifies?
Isn’t it the force per unit area acting parallel to a surface?
Great! That's correct. In turbulent flow, we have to consider not only the viscous shear stress but also the additional shear stress due to turbulence. This is where Boussinesq's eddy viscosity concept comes into play.
What do you mean by eddy viscosity?
Eddy viscosity measures the turbulence in the flow. It enhances total shear stress significantly beyond what is seen in laminar flow.
So, does that mean the total shear stress in turbulent flow is always larger?
Exactly! It usually is, due to the turbulent contributions.
Understanding Reynolds Shear Stress
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Reynolds introduced a formula for calculating shear stress in turbulent flow back in 1886. Can anyone recall how it's expressed?
It's noted as -ρ⟨u'v'⟩, right?
Exactly! This expression illustrates that the shear stress is based on the average of fluctuating velocities. Why might this average be a negative value?
Maybe because of the correlation between the flow directions?
You're on point! The negative correlation leads to positive shear stress once evaluated.
Prandtl's Mixing Length Theory
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Now, let’s transition to Prandtl's mixing length theory, which is crucial in determining turbulent shear stress. Who can summarize what mixing length is?
It’s the distance between two fluid layers that can interact and exchange momentum?
Correct! It’s essential for quantifying shear stress. Prandtl simplified it by stating that mixing length is proportional to the distance from the wall, described mathematically as lm = k * y. What’s the constant k known as?
It's called the von Karman constant, and it’s approximately 0.4.
Exactly! This relationship helps us link shear stress with measurable quantities.
Viscous Shear Stress in Turbulent Flow
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Finally, we must address how viscous shear stress operates within turbulent flow. Where does viscous shear stress largely apply?
It mainly exists near the wall, right?
Exactly! Most turbulent shear stress dominates away from the wall. So, how can we express the total shear stress in turbulent conditions?
It would be mixed with turbulent shear stress, significantly outweighing viscous stress?
That's right! As you can see, the turbulence in flow changes our approach significantly.
Introduction & Overview
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Quick Overview
Standard
Reynolds shear stress is introduced as a critical parameter in turbulent flow theory, distinguished by the unique role of turbulent viscosity. The section discusses the essential concepts of Boussinesq's model, Prandtl's mixing length theory, and how shear stress components in turbulent flow differ from those in laminar flow.
Detailed
Detailed Summary
In turbulent flow, shear stress is notably influenced by turbulence on top of the viscous effects already present in laminar flow. Boussinesq's model introduces the concept of eddy viscosity, which contributes significantly to total shear stress, leading to much larger values than those found in purely laminar conditions.
Reynolds shear stress, articulated in 1886, captures this turbulence effect and can be mathematically expressed as -ρ⟨u'v'⟩, which is a negative correlation indicative of the interaction between fluctuating velocity components (u' and v').
Prandtl's mixing length theory posits that the turbulent shear stress can be calculated using a mixing length (lm) that represents the vertical distance over which fluid layers can interact efficiently. This mixing length is defined as proportional to the distance from the wall, leading to the conclusion that turbulent shear stress can largely be attributed to the non-viscous components dominating the flow away from the wall.
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Introduction to Reynolds Shear Stress
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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 refers to the stress that occurs in a turbulent flow between two different fluid layers. In 1886, scientist Osborne Reynolds introduced a mathematical expression describing this shear stress. It is formulated as the negative product of the fluid density (rho) and the average product of the fluctuating velocity components in two different directions (u' and v'). This means that Reynolds found a way to quantify the chaos of turbulent flow mathematically.
Examples & Analogies
Imagine two layers of honey being stirred in a pot. The top layer moves quickly, while the bottom layer is more stationary. The interaction between these layers creates stresses that can be felt as resistance while stirring. Reynolds' expression helps us understand the nature and the magnitude of that resistance in a turbulent flow scenario.
Components of Reynolds Shear Stress
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So, you have to understand, Reynolds shear stress is given by and it does not have only one component, it has minus u dash w dash, it will have minus v dash w dash. So, there are different, there are some normal shear stress, but this is one of the shear stress components.
Detailed Explanation
Reynolds shear stress consists not only of the interaction between fluctuating velocities in the x and y directions (u' and v') but also includes combinations with another fluctuating component (w'). This means that Reynolds shear stress is multi-faceted; various directional forces contribute to it. The various components can influence how turbulent flows engage with one another as well as their surrounding structures.
Examples & Analogies
Think about a dance floor where dancers are moving in different directions. Each dancer's movements affect others around them, creating a complex interaction of forces. Similarly, the fluctuation of different velocity components can create a complex interplay in turbulent flows, contributing to the overall shear stress experienced by the fluid.
Negative Quantity of Fluctuating Velocities
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Experiments show that u prime v prime is usually a negative quantity. Therefore, the tau turbulence or minus rho u prime v prime whole bar is total positive quantity, it has negative correlation that we will see.
Detailed Explanation
In turbulent flow, the product of the fluctuating velocity components (u' and v') generally yields a negative value. Despite this negative nature, when combined with the negative density factor in Reynolds' equation, the final Reynolds shear stress becomes positive. Understanding this is crucial as it indicates that turbulence contributes positively to overall shear stresses, even if the individual fluctuating components may have a negative correlation.
Examples & Analogies
Consider a football game where each player tries to evade opponents while maintaining possession of the ball. While a player's action (like moving away) might create a negative impact on how close they can get to the opponent, it ultimately leads to positive outcomes, like scoring. In flow dynamics, seemingly negative interactions can also lead to positive results in terms of turbulence and shear stress.
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. Because as we see, the tau turbulence by Reynolds was given by minus rho u prime v prime whole bar. So, what a nice thing it would be if we can calculate u prime v prime whole bar because that is unknown until now.
Detailed Explanation
Prandtl's mixing length theory provides a framework for calculating turbulent shear stress. The theory posits that if we can determine the average of the product of the fluctuating velocities (u' and v'), we can then precisely quantify shear stress in a turbulent flow. The goal here is to bridge the gap between the challenging task of measuring the shear stress directly and finding an elegant way to calculate it using measurable parameters.
Examples & Analogies
Imagine measuring the flow of traffic on a busy highway. Instead of counting every car, you might take periodic snapshots to gauge average speed and behavior. Similarly, in fluid dynamics, Prandtl's theory allows scientists to indirectly measure turbulent stresses through averages rather than direct observations.
Definition of Mixing Length
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So, mixing length l m, he said it can be described in terms of mixing length l m. He said mixing length l m is the distance between 2 fluid layers in the vertical direction, in the y direction, such that, the bundles of fluid particles from one layer could reach the other layer and mix in the new layer in such a way that the momentum of the particle along the flow direction is the same.
Detailed Explanation
The mixing length (l m) is defined as the vertical distance between two fluid layers in which fluid particles can mix while maintaining the same momentum in the flow direction. This definition is fundamental as it allows us to relate the turbulent flow and momentum transfer between different layers. The mixing length essentially reflects how turbulence aids in distributing energy within a fluid.
Examples & Analogies
Think of two layers of colored water in a glass. If you stir them, the mixing length would be akin to the space through which the color particles transition smoothly from one layer to another, effectively blending without losing their individual momentum. This mixing is key to understanding how momentum behaves in turbulent flow.
Relationship Between Mixing Length and Velocity Gradient
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Prandtl related u prime to mixing length lm. He said that this u prime, as you can see in the figure here, he said proportional to u prime but I am going to write it in the next slide. He related u prime to the mixing length lm as, he said this u prime can be written as, mixing length lm multiplied by the gradient of the average velocity.
Detailed Explanation
Prandtl established a proportional relationship between the fluctuating velocity component (u') and the mixing length (l m) combined with the velocity gradient (/changed velocity). This means that the intensity of turbulence can be quantified and directly related to how steeply the average velocity changes across a distance. Understanding this relationship is crucial for predicting shear stresses in turbulent flows.
Examples & Analogies
Imagine measuring how steep a hill is while biking. The steeper the hill (the greater the gradient), the harder it is to pedal. Similarly, in fluid dynamics, a steeper velocity gradient means more turbulence and, correspondingly, greater momentum transfer, impacting overall shear stress.
Final Expression for Turbulent Shear Stress
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And if you substitute equation 11 and equation 12 in Reynolds stress model, which was tau turbulent is equal to minus u prime v prime whole bar you get, so, minus and minus will become positive it will become tau turbulence is .
Detailed Explanation
By substituting the relationships derived from mixing length theory into Reynolds' shear stress equation, we can simplify and express turbulent shear stress in terms of known quantities like the average velocity gradient and mixing length. This process illustrates how to calculate turbulent shear stress with more manageable variables, which is essential for engineers and fluid dynamicists.
Examples & Analogies
Consider a chef refining a recipe. By systematically adjusting ingredients (substituting known factors), the chef can create a delicious new dish. Similarly, fluid dynamicists can refine complex turbulent flow equations into practical formulas for predicting shear stress in real-world applications.
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: Introduces eddy viscosity affecting turbulent shear stress.
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Reynolds Shear Stress: Defined mathematically to capture the effect of turbulence.
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Prandtl's Mixing Length Theory: Connects shear stress to measurable quantities and the flow 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 Reynolds shear stress calculation in a turbulent pipe flow scenario, showing how fluctuating velocities influence the total stress experienced.
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In a river, the turbulent flow of water over rocks can result in increased shear stress due to both turbulence and viscosity combined, illustrating the concepts discussed.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In turbulent flow, stresses do grow, shear and eddy, steal the show.
📖 Fascinating Stories
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Once in a fluid realm, turbulence danced wildly. It swirled so much that it confused the walls. That's when eddy viscosity came in to quantify the chaos!
🧠 Other Memory Gems
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Remember 'REM' - Reynolds, Eddy viscosity, Mixing length for easy recall.
🎯 Super Acronyms
S.E.R.V.E - Shear stress, Eddy viscosity, Reynolds shear stress, Viscous flow, and Effect of turbulence.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Shear Stress
Definition:
The force acting parallel to the surface per unit area.
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Term: Eddy Viscosity
Definition:
A value representing turbulent flow characteristics affecting shear stress.
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Term: Reynolds Shear Stress
Definition:
The average shear stress resulting from fluctuations in velocity in turbulent flow.
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Term: Mixing Length
Definition:
The vertical distance over which fluid particles can mix between layers.
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Term: Prandtl's Mixing Length Theory
Definition:
A model that relates turbulent shear stress to measurable quantities through mixing length.