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Interactive Audio Lesson
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Shear Stress in Turbulent Flow
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Today, we're focusing on shear stress in turbulent flow. Recall that in laminar flow, shear stress is only due to viscosity. In turbulent flow, we have an additional component due to turbulence. Can anyone explain what this means?
Does this mean that turbulent flow has more shear stress compared to laminar flow?
Exactly, that's a great point. The total shear stress in turbulent flow is increased because of this turbulence-related shear. We use Boussinesq's model to define it. Can anyone tell me what eddy viscosity refers to?
Isn’t eddy viscosity a coefficient that represents the effect of turbulence on viscosity?
Correct! Eddy viscosity describes how turbulence increases momentum transfer, and it differs from the fluid's dynamic viscosity. Remember this distinction; it's vital for our further discussions.
Reynolds Shear Stress
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Let's discuss Reynolds shear stress, established by Osborne Reynolds in 1886. Who can remind us how it's defined?
Reynolds shear stress relates to the fluctuating velocities between two fluid layers, right? Isn’t its expression negative due to how it accounts for turbulence?
Exactly! The expression includes the fluctuating velocities, u' and v'. The negative sign reflects the correlation between these components. What do we call the average value of this stress?
Is it tau turbulence?
Yes, good job! Remember this term because it ties back into how we calculate shear stress and influences many engineering applications.
Prandtl’s Mixing Length Theory
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Now, let's turn our attention to Prandtl's mixing length theory. Can someone explain what mixing length (l_m) is?
The mixing length is the distance over which fluid layers mix; it helps in understanding momentum transfer.
Right! Prandtl proposed that we could relate this mixing length to the average velocity gradient. How might we apply this in real-world scenarios?
So we can calculate shear stress using average velocity, which simplifies our engineering calculations a lot!
Excellent! And when we define l_m as a linear function of the distance from the wall, we get a common relation that uses the von Karman constant.
Applications in Turbulent Flow
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Finally, let’s discuss how these concepts apply to turbulent flow in pipes. What do we typically find regarding viscous and turbulent shear stress?
Viscous shear stress mostly occurs near the boundaries, while turbulent shear stress dominates in the majority of the flow.
Correct! That understanding lets us approximate total shear stress mainly through turbulent components. Any insights on how this affects real systems?
It helps engineers design more efficient pipes by understanding where to focus on turbulent flow dynamics.
Great summary! Always remember how fluid dynamics integrates fluid properties and flow regimes.
Introduction & Overview
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Quick Overview
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The lecture continues from the previous discussion by explaining shear stresses in turbulent flow using Boussinesq’s model. It covers the concept of eddy viscosity, Reynolds shear stress, and Prandtl’s mixing length theory, highlighting their practical implications in hydraulic engineering.
Detailed
Detailed Summary
In hydraulic engineering, the understanding of flow behavior is crucial. This section delves into shear stress in turbulent flow, contrasting it with laminar flow.
- Boussinesq's Model: In this model, the total shear stress in turbulent flow incorporates not just viscous effects but also an additional component due to turbulence, represented by eddy viscosity. The shear stress due to turbulence is calculated as an equivalent of viscous shear but with a different coefficient.
- Eddy Viscosity: Unlike conventional dynamic and kinematic viscosities, eddy viscosity (B7) and kinematic eddy viscosity (B5) are flow condition-dependent. B5 decreases towards the wall, reaching zero at the boundary.
- Reynolds Shear Stress: Established by Osborne Reynolds, this quantity describes the turbulent shear stress, taking into account fluctuating velocity components (u’ and v’). It results in expressions that relate to the momentum transfer in fluid layers.
- Prandtl’s Mixing Length Theory: In 1925, Prandtl proposed a method to quantify turbulence through a parameter known as mixing length (l_m), defined as the distance over which fluid bundles mix. This concept allows shear stress to be expressed in terms of measurable average velocity gradients. Through various equations, the mixing length is related to the distance from the wall, facilitating calculations in turbulent flow.
- Application to Turbulent Flow in Pipes: The section recognizes that in turbulent flow, viscous shear stress dominates near the boundaries while turbulent stresses are more significant in the core regions of flow. Thus, shear stress can be generally approximated focusing on turbulent contributions.
Audio Book
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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 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, shear stress is influenced by two factors: the fluid's viscosity and the turbulence itself. In laminar flow, shear stress is caused solely by viscous forces, while in turbulent flow, turbulence adds an extra layer of complexity. This means the total shear stress encountered in turbulent flow is significantly greater than that in laminar flow. Boussinesq’s model helps explain this by incorporating both viscosity and an additional component due to turbulence.
Examples & Analogies
Imagine water flowing smoothly through a pipe (laminar flow) versus water churning violently in a river (turbulent flow). In the smooth flow, the forces acting on the water are predictable and uniform. However, in turbulent flow, eddies and swirls create chaotic motion, adding extra force to the fluid's movement.
Eddy Viscosity
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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. The shear stress due to turbulence is eddy viscosity du / dy, very similar to the shear stress in the laminar flow.
Detailed Explanation
Eddy viscosity is a key concept in understanding turbulent flow. It is a coefficient that represents the additional shear stress caused by turbulence in the fluid. This concept parallelly mimics laminar flow, where shear stress is a function of the fluid's viscosity. Eddy viscosity, however, varies based on flow conditions, unlike the standard dynamic viscosity, which is an intrinsic property of the fluid.
Examples & Analogies
Think of a crowded dance floor where people move chaotically. They can bump into one another, creating a sort of 'friction' or resistance to movement. In this analogy, the 'crowding' corresponds to turbulence, and the influence of this crowding is similar to how eddy viscosity affects the flow of fluid, indicating how much additional resistance is present due to chaotic movement.
Reynolds Shear Stress
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Now, coming to what is Reynolds shear stress. Reynolds in 1886 gave expressions for turbulent shear stress between two fluid layers separated by a small distance. Shear stress due to turbulence can be written as, minus rho u prime v prime whole bar.
Detailed Explanation
Reynolds shear stress is a crucial concept for understanding how turbulence affects shear stress within a fluid. It describes the average shear stress resulting from velocity fluctuations (u prime and v prime) in the flow. This shear stress is often negative due to the nature of turbulence, indicating that the stresses act in opposing directions. Reynolds provided a mathematical expression for this shear stress, enabling engineers to analyze and predict flow behavior.
Examples & Analogies
Imagine a group of people passing a ball among themselves while blindfolded. The unpredictability in passing the ball introduces random fluctuations in the flow of the game, similar to how turbulence introduces fluctuations in a fluid's velocity. The average effect of these chaotic interactions—like the cumulative stress on the ball during game play—can be understood through Reynolds' formula.
Prandtl's Mixing Length Theory
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Now, there is a concept of Prandtls mixing length theory. So, turbulence shear stress can be calculated if this thing is known, u prime v prime whole bar is known. Consequently, in 1925 Prandlts introduced the concept of mixing length, which can be utilized to express the shear stress in terms of some measurable quantity.
Detailed Explanation
Prandtl's mixing length theory simplifies the calculation of turbulent shear stress by introducing the idea of mixing length, denoted as lm. This length represents the distance over which fluid particles from one layer can interact with those in another layer, effectively mixing and conveying momentum. By relating shear stress to this mixing length, engineers can apply measurable quantities to compute complex turbulence phenomena.
Examples & Analogies
Consider a classroom where students in the front row pass notes to those in the back. The 'mixing length' corresponds to the distance across the rows; if students in the front can convey a message effectively to those further away, there's a mixing dimension involved. Similarly, in fluid dynamics, mixing length allows for understanding how momentum is transferred across different layers of the fluid.
Application of Mixing Length Theory
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Prandtl also 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.
Detailed Explanation
In his work, Prandtl established that the mixing length increases linearly with distance from a boundary, represented as lm = kappa * y. Here, kappa is a constant (approximately 0.4), known as the von Karman constant. This assumption greatly simplifies calculations for turbulent flows, especially near wall boundaries; the only unknowns become more manageable.
Examples & Analogies
Imagine measuring how far you can throw a paper airplane—if you step further away from a wall and each step allows you to throw it farther, then the distance (y) you've moved directly affects your airplane's performance. In fluid dynamics, the further a particle is from a surface, the more room there is for momentum transfer (mixing).
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Shear Stress: The stress parallel to fluid layers due to applied forces.
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Eddy Viscosity: Additional viscosity due to turbulence impacting momentum.
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Reynolds Shear Stress: Shear stress influenced by fluctuating velocities within turbulent flow.
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Mixing Length: Distance at which fluid layers interact and momentum is exchanged.
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Von Karman Constant: A proportionality factor in mixing length theory.
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 turbulent flow can be seen in rivers where the water speed varies significantly at different points.
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In engineering, understanding turbulent shear allows for better pipe design to minimize energy losses.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In laminar flows, the stress is clear, but turbulence adds to what we fear.
📖 Fascinating Stories
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Imagine a busy river, layers rushing to mix; the turbulence creates chaos where the stillness used to fix.
🧠 Other Memory Gems
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Remember 'Eddy's Viscous Roller Coaster' – it represents how turbulence accelerates momentum transfer.
🎯 Super Acronyms
M.E.R.C – Mixing Length, Eddy Viscosity, Reynolds Stress, and Core Turbulence.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Shear Stress
Definition:
The stress that arises from the force applied parallel to a surface or section.
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Term: Eddy Viscosity
Definition:
An additional viscosity component in turbulent flow accounting for momentum transfer due to turbulence.
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Term: Reynolds Shear Stress
Definition:
Turbulent shear stress expressed in terms of fluctuating velocities between fluid layers.
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Term: Mixing Length (l_m)
Definition:
The average distance over which momentum mixing occurs between adjacent fluid layers.
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Term: Von Karman Constant
Definition:
A dimensionless constant (approximately 0.4) relating mixing length and distance from the wall.