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Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Fundamentals of Shear Stress
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Let's start by discussing shear stress in turbulent flow. Can anyone tell me what shear stress at the wall is typically assumed to be?
Isn't it constant at the wall, represented as tau_naught?
Exactly! We assume it’s constant, especially for small distances from the wall, or 'y'. This assumption helps us simplify our equations.
How does that relate to the velocity profiles we study?
Great question! It’s part of deriving our velocity profiles, especially the logarithmic ones we use for turbulent flows. This is a crucial aspect - remember that!
In fact, let's remember 'tau_naught at the wall, stability for the fluid's call.' This rhyme helps you keep in mind that shear stress is stable at the wall.
Understanding Velocity Profiles
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Now, let's discuss velocity profiles. Student_3, can you describe how a laminar flow profile appears compared to a turbulent flow profile?
Uh, I think laminar flow has a parabolic profile, while turbulent flow has a sort of logarithmic shape?
Precisely! The laminar flow is smooth and parabolic, whereas turbulent flow creates a fuller velocity profile. It reflects the chaotic nature due to eddies.
What do we call the difference between the maximum velocity and actual velocity?
That's called the velocity defect! Great catch, Student_4. Remember, 'Velocity defect: the flow’s suspect.' It's a crucial point when analyzing turbulent flows.
Prandtl Mixing Length Theory
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Let's dive into Prandtl's mixing length theory. How does it help us?
Isn't it supposed to help derive those logarithmic profiles we just talked about?
Exactly! By considering factors like turbulent intensity, we can derive key velocity equations from it.
What about the boundary conditions used in the equations? How important are they?
Boundary conditions are critical! They help us set our equations properly to derive relationships such as shear stress from turbulence theory. 'Boundary here, simplify near!'
Calculating Shear Stress
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Now, let's apply what we've learned to calculate shear stress. Can someone remind us how to calculate wall shear stress?
We use tau_naught equals rho times u_star squared, right?
Correct! And let’s recall that u_star is the frictional velocity. Are there any other parameters we need to consider?
The density of the fluid too, right? Like for water, it’s around 1000 kg/m³.
Very good! Remember to consider fluid density. 'Shear stress, density's a must!' Let’s try a calculation together next.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we explore the mathematical relationships governing turbulent flow, particularly the shear stress at the pipe wall and the velocity profiles derived from Prandtl's mixing length theory. Key equations are manipulated to reveal insights into flow behavior and shear stress calculations.
Detailed
Use of Equations in Turbulent Flow Analysis
Overview
This section delves into the fundamental equations used to analyze turbulent flow, focusing primarily on shear stress and velocity profiles derived from fluid mechanics principles. Equations such as equation 14 and equation 15 are referenced to illustrate how shear stress () behaves under turbulent conditions.
Key Concepts
- Shear Stress at Wall:
- It is often assumed to be constant and equal to in small values of y (distance from the wall).
- Velocity Profiles:
- The flow velocity is described by a logarithmic profile in contrast to the parabolic profile observed in laminar flow.
- Integrating derived equations leads to the establishment of the famous velocity defect law.
- Boundary Conditions:
- Boundary conditions are crucial for integrating fluid motion equations and determining shear stress.
- Examples include setting velocity conditions to derive constants and thus solving for turbulent properties.
- Characteristics of Turbulent Flow:
- Turbulent flow consists of multiple regions with varying characteristics, including the viscous sublayer and turbulent layer.
This section provides a comprehensive framework for understanding how mathematical equations can model turbulent flow and predict key parameters such as shear stress at the pipe wall.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introducing the Equations
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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.
Detailed Explanation
In this part, the speaker refers to two equations, where Equation 14 involves a relationship represented as l m = kappa × y. Here, 'kappa' is a constant that relates to the turbulent flow characteristics, and 'y' is a variable that can denote the distance from the pipe wall. By substituting Equation 14 into Equation 15, the speaker derives a new equation, Equation 16. This demonstrates the process of substituting known equations to simplify the math related to turbulent flow analysis.
Examples & Analogies
Think of it like cooking: you have a recipe (Equation 14) that requires a specific ingredient (kappa). You then find another recipe (Equation 15) that can use that ingredient—by adding it in, you can create a whole new dish (Equation 16).
Assumptions for Shear Stress
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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.
Detailed Explanation
The statement highlights an important assumption in turbulent flow analysis: when 'y' (distance from the wall) is very small, the shear stress (represented as tau) at that point can effectively be considered equal to the shear stress at the wall, denoted as tau not. This simplification allows for easier calculations and is particularly useful when working close to the pipe wall.
Examples & Analogies
Imagine you're at the edge of a swimming pool. The water right at the edge (or wall) has less disturbance, hence the pressure (like shear stress) is more constant compared to the center of the pool where waves can form. This helps us simplify the pressure calculations at the edge.
Understanding Velocity Profile Derivation
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Then using the boundary conditions, what are the boundary conditions? So, u at y is equal to R, where R is the radius of the pipe. We will get, u is equal to u max. ... C will be u max minus u * / kappa ln R and if we substitute this as C, then we can get equation or or if when we substitute kappa as 0.4, we can get and this is equation number 20.
Detailed Explanation
The analysis involves boundary conditions essential for determining velocity profiles in turbulent flows. It establishes that at the pipe's wall ('y' equals R), the velocity reaches its maximum (u max). By substituting this condition into the equations and manipulating the terms, a constant 'C' is derived, further leading to a logarithmic velocity profile described in Equation 20. This profile showcases how velocity changes from the center to the wall of the pipe.
Examples & Analogies
Think of it like measuring a slope. If you take a height measurement (like u max) at the top of a hill (the center of the pipe), you can apply that to understand the slope down to the base (the wall), calculating various points along the way to see how steep it gets (the velocity profile).
Velocity Defect Law
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This is ln. So, we can put it in form of log. This is simple manipulation, we can get u max minus u / u * is equal to 5.75 log to the base 10 R / y. u max minus u is called the velocity defect or velocity defect law...
Detailed Explanation
Here, the equation is manipulated to express the relationship between the maximum velocity, actual velocity, and shear velocity in a logarithmic form. The term 'velocity defect' refers to the difference between the maximum velocity (u max) and the actual velocity (u) at a certain distance from the wall. This law helps predict how turbulent flow behaves in terms of velocity at different points in the pipe.
Examples & Analogies
Think of a car speeding down a road (u max) but slowing down due to traffic (u). The difference in speed represents the 'velocity defect.' It helps us understand how effectively the car can travel through congested areas (the turbulent flow in pipes).
Layers in Turbulent Flow
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So, now the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. ... Turbulent flow along a wall consists of 4 regions: viscous sublayer, buffer layer, overlap layer, and turbulent layer.
Detailed Explanation
The turbulent flow profile differs significantly from laminar flow, which typically shows a parabolic curve. In turbulent flow, the velocity profile is fuller and complex due to interactions at various layers. These layers include the viscous sublayer (where viscous effects dominate), the buffer layer, the overlap layer (where turbulent effects begin to take precedence), and the turbulent layer (where turbulence dominates). Understanding these layers is crucial for analyzing flow behavior.
Examples & Analogies
Imagine a crowd at a concert. At the very front (viscous sublayer), everyone is orderly and calm. As you move back, the crowd begins to mix and sway (buffer, overlap layers), until at the back, it's chaos with people moving in many directions (turbulent layer). Understanding these distinctions helps manage crowd safety and flow.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Shear Stress at Wall:
-
It is often assumed to be constant and equal to in small values of y (distance from the wall).
-
Velocity Profiles:
-
The flow velocity is described by a logarithmic profile in contrast to the parabolic profile observed in laminar flow.
-
Integrating derived equations leads to the establishment of the famous velocity defect law.
-
Boundary Conditions:
-
Boundary conditions are crucial for integrating fluid motion equations and determining shear stress.
-
Examples include setting velocity conditions to derive constants and thus solving for turbulent properties.
-
Characteristics of Turbulent Flow:
-
Turbulent flow consists of multiple regions with varying characteristics, including the viscous sublayer and turbulent layer.
-
This section provides a comprehensive framework for understanding how mathematical equations can model turbulent flow and predict key parameters such as shear stress at the pipe wall.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
If water has a velocity of 4 m/s at the pipe center and 3.5 m/s at 2 cm from the center, the wall shear stress can be calculated using derived equations.
-
During the discussion of turbulent flow, if one considers a pipe with a rough surface, the wall shear stress would reflect greater turbulent effects compared to a smooth pipe.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When shear appears, at the wall it steers; constant it may stay, guiding the flow’s way.
📖 Fascinating Stories
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Imagine a pipe where water flows freely, at the surface it's calm, but inside it swirls chaotically—a contrast between the outside and turbulent currents.
🧠 Other Memory Gems
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PVT: Prandtl’s Velocity Theory helps remember that turbulent flows have varied velocity behavior.
🎯 Super Acronyms
FLOW
- F: = Friction
- L: = Logarithmic shapes
- O: = Overlap in layers
- W: = Wall shear stress.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Shear Stress (τ)
Definition:
The stress that acts parallel to the surface of a material, particularly in a fluid, often assumed constant at the wall in turbulent flow.
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Term: Frictional Velocity (u*)
Definition:
A measure of the velocity used to characterize the effect of turbulence near the wall in flow.
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Term: Velocity Defect Law
Definition:
A law stating the difference between maximum velocity and actual velocity in turbulent flow.
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Term: Logarithmic Velocity Profile
Definition:
A representation of velocity distribution in turbulent flow which shows how velocity increases logarithmically with distance from the wall.