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Interactive Audio Lesson
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Introduction to Shear Stress in Turbulent Flow
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Let's begin our discussion by recalling the concept of shear stress, τ, at the wall of a pipe. We incorporate this into our equations, particularly focusing on how τ is constant at the wall in turbulent flow.
Why do we consider τ to be constant at the wall?
Great question! In turbulent flow, especially close to the wall where the distance y is small, we can simplify our calculations by assuming τ can be treated as a constant value, τ₀. This aids in creating our velocity profiles.
So, when y is small, τ is constant, does this apply to laminar flow too?
That’s correct, but the overall profiles differ! In laminar flow, τ varies, leading to a simpler parabolic profile.
Velocity Profiles: Laminar vs Turbulent
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Let's delve deeper into velocity profiles. The flow profile in laminar flow is parabolic, while in turbulent flow, we have a logarithmic profile which is much fuller. This fundamental difference affects many calculations in fluid dynamics.
Can you explain why turbulent flow has a fuller profile?
Certainly! In turbulent flow, the mixing of eddies leads to a more uniform velocity distribution across the pipe's cross-section, contributing to that fuller profile.
How do we express this mathematically?
We can use equations derived from the Prandtl mixing length theory to describe this log profile mathematically.
What’s the significance of the velocity defect law mentioned?
The velocity defect law helps us understand the loss in velocity from the maximum flow speed, crucial for calculating shear stress in design applications.
Layers of Turbulent Flow
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In turbulent flow, we recognize different layers: the viscous sublayer, buffer layer, overlap layer, and turbulent layer. Each plays a critical role in the overall dynamics.
What distinguishes the viscous sublayer from the turbulent layer?
Excellent query! The viscous sublayer is closest to the wall, where flow is dominated by viscosity, while the turbulent layer is where mixing dominates. They interact but maintain distinct characteristics.
How can surface roughness influence these layers?
Surface roughness certainly matters! If the irregularities on a surface are significant compared to the viscous sublayer's thickness, they can disturb flow and enhance turbulence.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section describes how shear stress is calculated in cylindrical pipes under turbulent flow conditions, contrasting it with laminar flow's parabolic profile. It introduces the velocity defect law and discusses various flow layers, which include the viscous sublayer and the turbulent layer, while also touching on boundary types relevant to flow characteristics.
Detailed
Comparison with Laminar Flow Profile
In this section, we analyze the differences between turbulent and laminar flow profiles, starting with the derivation of key equations from the Prandtl mixing length theory. Specifically, we relate shear stress () to the distance from the wall ( ) and the velocity profile within the flow. We explore how shear stress at the wall ( ) can be viewed as a constant value under conditions when the distance is small. Upon substitution into the equations, we derive expressions leading to the velocity defect law, which is a significant characteristic of turbulent flow in pipes.
Additionally, we compare these profiles: laminar flow exhibits a parabolic shape while turbulent flow has a fuller profile that diverges from this simplicity. The section also explains the various layers present in turbulent flow, such as the viscous sublayer, the buffer layer, the overlap layer, and the turbulent layer. Finally, we introduce the concepts of rough and smooth boundaries, linking them to how surface irregularities affect flow characteristics.
The definitions pertaining to these concepts (e.g., the height of surface irregularities, k) and their significance in determining flow type are critical for understanding practical fluid dynamics applications.
Audio Book
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Introduction to Shear Stress and Turbulent Flow Equation
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So, if we use equation 14 in equation 15, what was equation 14? l m was kappa into y, this was what we said in equation number 14. So, we can simply write or we can simply write and this is equation number 16, very simple. So, for small values of y it can be assumed. So, if the y is very small we can assume that tau is equal to tau not, where tau not is the shear stress at the pipe wall and can be assumed to be a constant. So, at the wall the shear stress is assumed to be constant and equal to tau not.
Detailed Explanation
In this chunk, we are exploring the relationship between shear stress and turbulent flow using equations from fluid dynamics. The basic equation being referenced involves the term 'kappa,' which relates the turbulent flow characteristics (like shear stress) to the distance from the wall of the pipe (y). It emphasizes that when y (the distance from the wall) is very small, the shear stress (tau) can be approximated as a constant value (tau not), which is specifically the value of shear stress right at the wall. This simplification is crucial in analyzing turbulent flow because it allows us to use a simpler equation for calculations.
Examples & Analogies
Imagine a smooth slide at a playground. When a kid is very close to the slide's edge (just at the top), they experience maximum friction, similar to the shear stress at the wall of the pipe. Once they slide down further, the level of 'stickiness' or contact reduces, just as the shear stress changes further away from the wall.
Integrating the Turbulent Flow Equation
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And therefore, what we can say, if we substitute tau is equal to tau not in equation 16, we can obtain or du / dy, this quantity actually tau not can be written as rho. So, but the catch here is, what is the catch? We have considered small value of y. So, du / dy can be written as, 1 / kappa y and under root tau / rho is rho u *. So, it becomes and this u * under root tau not / rho is the sheer velocity and this has the dimension of velocity. And if you integrate the equation number 17, so, what we can get is, simple integration, it will get.
Detailed Explanation
Here, the text discusses how substituting the constant shear stress (tau not) into the turbulent flow equation leads to a new expression (du/dy), which describes how velocity changes with distance from the wall. This integration and the relation to shear velocity (u*) helps simplify the understanding of velocity profiles within the turbulent flow. The key takeaway is that the equations can simplify the complexity of turbulent flow into more manageable forms, allowing for predictions about fluid behavior to be made more easily.
Examples & Analogies
Think of mixing a smoothie. At the bottom of the blender (the 'wall'), the smoothie ingredients get mixed differently than at the top. The closer ingredients are to the wall (or blades), the more they experience 'friction' – or in this case, mixing. As you integrate these differences, you can predict the overall consistency of the smoothie based on how well everything is mixed together.
Defining Velocity Profile in Turbulent Flow
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Now, the equation 20 this equation 20 can be expressed as, u max, so, what we do is, we bring u on the other side. So, we bring u on this side and we take this whole side component this side, then what the result is, u max minus u because u max will always be larger than u is equal to 2.5 u * ln R / y. y will always be less than R or we bring u frictional velocity down, then we get u max, you bring it down here by dividing then you get u max minus u / u * equals to 2.5 ln R / y. And, so, this is ln.
Detailed Explanation
The focus of this chunk is on how to manipulate the derived equation to arrive at the velocity defect law, which describes how the maximum velocity (u max) at the center of the pipe contrasts with the velocity at any point y within the pipe. This law shows that there’s a logarithmic relationship between the distance from the wall and the velocities of fluid particles, a significant feature in turbulent flow due to increased friction and mixing.
Examples & Analogies
Consider a busy road during rush hour. The cars move slower near the sidewalks than in the middle of the street (akin to u and u max). As you calculate out how much slower they are, it reflects the velocity defect law; drivers in the middle enjoy faster speeds while those near the edges have to contend with more 'friction' (in this case, pedestrians, traffic signals, etc.).
Explaining Turbulent Flow Regions
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So, now, the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. Actually this is the flow, this is the true picture, this is a laminar flow that we have seen before. But below is, this is the V average and the velocity fluctuates or deviates from these depending upon the flow condition. So, this is the V average line. There are several other layers, viscous sublayer, buffer layer, overlap layer and turbulent layer.
Detailed Explanation
This part describes how the velocity profile in turbulent flow differs significantly from laminar flow. Unlike the smooth, parabolic shape of laminar flow, turbulent flow presents a more complex profile characterized by layers: the viscous sublayer (which is closest to the wall), the buffer layer, overlap layer, and the turbulent layer. Each of these layers behaves differently, influencing how the flow behavior changes from laminar to turbulent conditions.
Examples & Analogies
Think of it like layers of a cake. The icing (turbulent layer) might be thick and uneven, while the cake’s core (viscous sublayer) is soft and stable. Just as the cake’s texture varies from the layers to the frosting, the velocity profiles change from smooth flow at the wall to chaotic flow further away.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Shear Stress: Represents the internal friction within fluids, influencing flow behavior.
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Laminar Flow Profile: Characterized by smooth, parabolic velocity distribution.
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Turbulent Flow Profile: Exhibits a fuller shape due to chaotic mixing, deviating from simplicity.
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Velocity Defect Law: Quantifies loss of flow velocity from maximum flow speed.
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Viscous Sublayer: Region where the effects of viscosity are most pronounced.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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In a pipe with a diameter of 10 cm, the velocity profile in turbulent flow can be described using the log profile derived from the Prandtl mixing length theory.
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In laminar flow, if the velocity at the center of the pipe is 4 m/s, at the wall it decreases following a parabolic distribution.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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In pipes so wide, flows don't collide; Laminar's smooth, while turbulence is a wild ride!
📖 Fascinating Stories
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Imagine a quiet stream (laminar flow), where every drop flows smoothly. Then imagine a wild river (turbulent flow), where rocks create whirlpools and chaos rules the rapids.
🧠 Other Memory Gems
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To remember flow types: 'L for Laminar's Light curve, T for Turbulent's Thick sway'.
🎯 Super Acronyms
P-Profile for Parabolic (laminar), L-Logarithmic for turbulent.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Shear Stress (τ)
Definition:
The force per unit area acting parallel to the surface, significant in determining fluid flow characteristics.
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Term: Velocity Profile
Definition:
The variation of velocity across a cross-section of a flow, differing significantly between laminar and turbulent flows.
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Term: Viscous Sublayer
Definition:
The thin layer of fluid next to the wall of a pipe where viscous effects dominate, leading to a linear velocity profile.
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Term: Velocity Defect Law
Definition:
A law defining the difference in velocity from the maximum flow rate to the actual flow rate in turbulent flow.
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Term: Laminar Flow
Definition:
A type of flow characterized by smooth and orderly motion, resulting in a parabolic velocity profile.
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Term: Turbulent Flow
Definition:
A type of flow characterized by chaotic and irregular motion, leading to a fuller and logarithmic velocity profile.
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Term: Roughness Reynolds Number
Definition:
A dimensionless number used to characterize the flow regime over a rough surface.
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Term: Smooth Boundary
Definition:
A boundary condition where surface irregularities are smaller than the thickness of the viscous sublayer.