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Interactive Audio Lesson
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Introduction to Shear Stress in Turbulent Flow
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Welcome, everyone! Today, we will continue our discussion on shear stresses in turbulent flow. Did anyone notice how turbulent flow differs from laminar flow?
Yes, I think turbulent flow has more chaotic movements?
Exactly! In turbulent flow, there's an added component of shear stress from turbulence itself, unlike laminar flow, which is purely viscous.
So, how do we quantify that?
Great question! We use a model called Boussinesq’s model which introduces eddy viscosity. Recall, eddy viscosity is crucial for turbulent flow and varies based on flow conditions.
How does eddy viscosity relate to regular viscosity?
Eddy viscosity is not a fixed property; it changes with flow conditions, particularly decreasing toward the wall and becoming zero at the wall. Remember this: it’s frequently denoted by eta (η).
So, do we still consider traditional viscosity?
Yes, but in turbulent flow, the focus shifts more to the turbulence-induced shear stress.
To recapitulate, the key components we're looking at are shear stress due to eddy viscosity and the necessity to account for both turbulence and viscosity in analysis.
Reynolds Shear Stress
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Next, let's talk about Reynolds shear stress, developed by Osborne Reynolds in 1886. Can anyone summarize what this is?
Isn’t it the average turbulence component that affects shear stress between layers?
Exactly! It’s defined as τ = -ρ (u'v')̅. Here, u' and v' are fluctuating velocity components. Why do you think Reynolds stressed the negative value?
Because it’s conceptualized between different fluid layers that are moving in different velocities?
Correct! There’s a negative correlation between fluctuating velocities, which makes total shear positive despite the negative signs in the equation.
So how do we derive that completely?
That involves more advanced derivations we’ll cover later, but for now, keep in mind the importance of this concept in calculating turbulent shear stress.
Summing up, Reynolds shear stress is crucial for understanding how shear behaves in turbulent flows and sets the stage for further analysis.
Prandtl's Mixing Length Theory
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Continuing our discussion, let’s dive into Prandtl’s mixing length theory introduced in 1925. How do you think this theory makes analyzing turbulence easier?
By allowing us to relate shear stress to measurable quantities?
Exactly! The theory describes mixing length (lm) as the distance that fluid particles can travel from their layer allowing for mixing.
How is that practically used in fluid dynamics?
We express shear stress in terms of lm and the velocity gradient. For instance, we can write τ = ρ lm² (du/dy)², where 'du/dy' is the velocity gradient.
And how does one determine the mixing length?
Prandtl assumed that lm is linear with respect to the distance from the wall, so lm = κy, where κ is von Karman’s constant, roughly 0.4.
This simplifies the whole turbulent shear stress equation!
Precisely, this theory allows engineers to predict shear stress effectively. In summary, Prandtl’s mixing length provides a key link in analyzing turbulent flows.
Shear Stress in Turbulent Flow
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Finally, let’s summarize what we've learned about shear stress in turbulent flow. Why is this different from laminar flow?
The dominance of turbulent shear stress rather than viscous shear stress, especially away from the wall.
Right! Most shear stress in turbulent flow is attributable to turbulence itself, allowing for approximations in calculations.
So we can focus primarily on Reynolds shear stress and shortcuts from Prandtl’s theory?
Exactly! Thus, in practice, you’ll often neglect viscous shear stress when analyzing turbulent flow far from walls.
And that helps simplify fluid dynamics considerably?
Absolutely! To wrap up, we focused on shear stress in turbulent flow, understanding key models and theories that simplify these analyses.
Introduction & Overview
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Quick Overview
Standard
The section elaborates on the difference in shear stress calculations between laminar and turbulent flow, emphasizing the turbulence component's importance through Boussinesq's model and detailing the application of Prandtl's mixing length theory to predict turbulent shear stress in flows through pipes.
Detailed
Turbulent Flow in Pipes
The section explores the characteristics of turbulent flow in pipes, particularly focusing on shear stress calculations. Unlike laminar flow, where shear stress relates only to viscosity, turbulent flow introduces additional shear stress components due to turbulence. This difference necessitates the introduction of a new coefficient of viscosity, termed eddy viscosity, represented by η. Eddy viscosity behaves differently under flow conditions and varies with distance from the pipe wall, where it approaches zero.
The concept of Reynolds shear stress is introduced, explaining how the shear between fluid layers is derived from fluctuating velocities. Prandtl’s mixing length theory offers a practical means of estimating turbulent shear stress using measurable quantities. The mixing length (lm) is described as the distance over which fluid particles can mix, linking these dynamics to the gradient of average velocity.
In turbulent flow modeling, the majority of shear stress in a pipe can be attributed to turbulence, allowing for the simplification of shear stress equations by neglecting the viscous component except near the boundary. Overall, these principles provide significant insight into the complex behaviors of fluid dynamics within engineering applications.
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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. But in a turbulent flow, there is an additional component of shear stress that happens because of the turbulence in the flow. Therefore, the shear stress in total is much larger than the viscous flow because there is shear stress that is associated with turbulence too. So, Boussinesq’s says, the shear stress due to turbulence is given by eta du/dy, where eta is a new coefficient of viscosity called eddy viscosity.
Detailed Explanation
In turbulent flow, the shear stress is influenced by both traditional viscous forces and additional turbulent forces. This is captured in Boussinesq’s model, which introduces a new parameter known as 'eddy viscosity'. Eddy viscosity accounts for the chaotic and irregular movements within the fluid that lead to increased momentum transfer. As a result, the total shear stress is significantly higher in turbulent flows compared to laminar flows, where shear stress is solely a factor of viscosity.
Examples & Analogies
Imagine a crowded dance floor where dancers (fluid particles) are moving smoothly in pairs (laminar flow). When the music tempo increases and everyone starts dancing in a chaotic way, they collide with each other frequently, leading to faster movements across the dance floor. This change in behavior is similar to how turbulence increases shear stress in fluids.
Eddy Viscosity and Flow Conditions
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Unlike the dynamic viscosity mu and kinematic viscosity nu, eta and epsilon are not fluid properties. The values of eta and epsilon are dependent on the flow conditions. So, epsilon decreases towards the wall becoming 0 at the wall.
Detailed Explanation
Eddy viscosity (eta) and kinematic eddy viscosity (epsilon) differ from traditional viscosity measures, as they are not constant properties of the fluid; rather, they vary with the flow conditions. Specifically, kinematic eddy viscosity approaches zero as you move closer to a surface (like a pipe wall) because the turbulent fluctuations diminish in those layers due to frictional effects. This implies that the turbulent behavior is most pronounced away from solid boundaries.
Examples & Analogies
Think of a river flowing past a rocky riverbed. Near the rocks, the water doesn't flow as vigorously due to friction (zero turbulence), but in the middle of the river, the water swirls and moves faster. Just like the turbulent velocity increases in the middle of the river, the eddy viscosity is also higher away from the walls of a pipe.
Reynolds Shear Stress
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Reynolds in 1886 gave expressions for turbulent shear stress between two fluid layers separated by a small distance. Reynolds shear stress is given by minus rho u' v' whole bar, where u' is the fluctuating velocity component in x direction, and v' is the fluctuating component of velocity in y direction.
Detailed Explanation
Reynolds shear stress represents the effect of turbulence on the momentum transfer between fluid layers. The formula includes fluctuating components of velocity (u' and v') to account for the disordered movements that occur in turbulent flows. This results in a shear stress that is influenced not only by average velocity but also by the variations in velocity. An important characteristic is that experiments show the product of these fluctuations is often negative, suggesting a correlation between the two directions.
Examples & Analogies
Imagine a busy highway where some cars are speeding while others are slow. The 'fluctuating velocities' mimic how traffic behaves: some cars move fast while others slow down or stop. The shear stress here represents the pressure exerted among cars, akin to how Reynolds shear stress is derived from the differences in velocities.
Prandtl's Mixing Length Theory
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In 1925, Prandtl introduced the concept of mixing length, which can be utilized to express the shear stress. Prandtl assumed that the mixing length lm is a linear function of distance y from the wall.
Detailed Explanation
Prandtl's mixing length theory simplifies the calculation of turbulent shear stress by introducing a variable called mixing length (lm). Mixing length is defined as the average distance over which fluid particles can interact with one another and transfer momentum. Prandtl proposed that this mixing length increases linearly with the distance from the wall, suggesting a predictable relationship between the flow near the wall and the turbulent flow in the bulk of the fluid.
Examples & Analogies
Think of a classroom full of students where they move around to collaborate. Near the walls, students are constrained and cannot move (similar to low mixing length), while in the center, they can move freely and organize (higher mixing length). The concept of mixing length reflects how far students can influence one another's movement based on their position in the room.
Using Mixing Length to Calculate Shear Stress
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If we substitute the expressions related to the mixing length into the Reynolds stress model, we arrive at an equation relating shear stress to average velocity gradients and mixing length.
Detailed Explanation
By substituting Prandtl's expressions for u' related to the mixing length into the Reynolds shear stress equation, we can derive a clear relationship between turbulent shear stress, fluid density, and the velocity gradient of the flow. This relationship allows engineers to calculate shear stress in turbulent flow scenarios more readily, using measurable parameters.
Examples & Analogies
Think of an air conditioner in a room that affects air flow throughout the space. Just as the placement of the air conditioning unit dictates the movement and mixing of air in the room, understanding mixing length helps determine how forces are distributed in turbulent flows.
Concluding Shear Stress in Turbulent Flow
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In turbulent flow, the viscous shear stresses exist only near the boundary, and most of the region is dominated by turbulence. Therefore, we can neglect the viscous shear stress for practical calculations, allowing us to focus on the turbulent component.
Detailed Explanation
In turbulent flow situations, the impact of viscous shear stress is minimal except near boundaries, meaning that the turbulent shear stress is the dominant factor affecting fluid behavior. This allows for simplifications in calculations, where engineers primarily consider turbulent effects in their analyses rather than the relatively minor viscous stresses.
Examples & Analogies
Picture moving through a busy crowd at a concert. The individuals near the edges (viscous stresses) might be barely moving, while the chaotic center (turbulent stresses) is where the majority of the action occurs. For engineers, it's easier to focus on this turbulent middle rather than the still edges.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Turbulence: Refers to the chaotic, stochastic fluid motion that differs from orderly laminar flow.
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Eddy Viscosity: A measure of turbulence that relates to the shear stresses resulting from chaotic eddies in the fluid.
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Reynolds Shear Stress: A component of total shear stress in turbulent flow, calculated using fluctuating velocity components.
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Prandtl's Mixing Length Theory: A theory allowing the expression of turbulent shear stress in terms of measurable quantities.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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A fluid flowing through a rough pipe will experience turbulent flow patterns and significantly differing shear stress compared to a smooth pipe.
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In an experiment, measuring the velocity gradient and boundary layer thickness can help determine the mixing length for a specific fluid.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In turbulent flow where chaos reigns, Eddy viscosity strengthens the strain.
📖 Fascinating Stories
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Imagine a river where calm meets storm. In the steadiness of laminar, friction keeps uniform. As turbulence brews, shear stress they find, introducing eddies that twist and unwind.
🧠 Other Memory Gems
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To remember Reynolds shear stress: 'Uplifting Viscosity Rises (τ = -ρ (u'v'))'
🎯 Super Acronyms
REMEMBER
- EDRY (Eddy Viscosity
- Reynolds Stress
- Mixing Length
- Eddy Shear
- and Distance)
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Eddy Viscosity (η)
Definition:
A turbulence-related viscosity that adjusts based on flow conditions, crucial for accurately describing turbulent shear stresses.
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Term: Reynolds Shear Stress (τ)
Definition:
The average shear stress arising from turbulent flow between fluid layers, expressed as τ = -ρ (u'v')̅.
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Term: Prandtl's Mixing Length (lm)
Definition:
The distance vital for mixing fluid layers, critical in estimating turbulent shear stress.
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Term: Von Karman Constant (κ)
Definition:
A dimensionless constant approximately equal to 0.4, used in the Prandtl mixing length equation.