Professional Courses
Industry-relevant training in Business, Technology, and Design to help professionals and graduates upskill for real-world careers.
Categories
Interactive Games
Fun, engaging games to boost memory, math fluency, typing speed, and English skills—perfect for learners of all ages.
Typing
Memory
Math
English Adventures
Knowledge
Enroll to start learning
You’ve not yet enrolled in this course. Please enroll for free to listen to audio lessons, classroom podcasts and take practice test.
Interactive Audio Lesson
Listen to a student-teacher conversation explaining the topic in a relatable way.
Understanding Shear Stress in Turbulent Flow
Unlock Audio Lesson
Welcome everyone! Today, we’ll dive deeper into the nature of shear stress, especially focusing on turbulent flow. Can anyone remind me what distinguishes laminar from turbulent flow?
Isn't laminar flow smooth while turbulent flow is chaotic?
Exactly! In laminar flow, shear stress is primarily from viscosity. However, in turbulent flow, we need to consider additional shear stress caused by turbulence. This brings us to Boussinesq’s model which introduces the concept of eddy viscosity.
What’s eddy viscosity, though?
Great question! Eddy viscosity quantifies the turbulent component of shear stress. It’s not just about fluid viscosity anymore, but how well the particles mix. And as turbulence increases, this effect gets more prominent.
Does that mean turbulent shear stress is always higher than laminar shear stress?
That's correct! In turbulent flows, shear stress is usually greater due to the additional turbulence factor. Remember, shear stress in turbulent flow can be defined as τ_turbulent = -ρ u' v'.
Oh, does that mean we're depending on u' and v' for calculations?
Yes, but these are fluctuating velocities which can be tricky to measure directly, leading us to the mixing length theory. Let’s summarize our key takeaways: we differentiate turbulent and laminar flow by their shear stress contributions and understand eddy viscosity plays a crucial role.
Excellent discussions team! Remember the acronym ET for 'Eddy Turbulence' to recall the additional shear from turbulence in your studies.
Introducing Mixing Length
Unlock Audio Lesson
Now, let’s shift gears. What is the mixing length, and why is it essential in our discussions?
It seems to be the distance for particles to transfer momentum, right?
Exactly! The mixing length (lm) is the vertical distance between fluid layers that allows for mixing. Prandtl proposed that this length should be directly influenced by how far you are from a wall.
How does this help with turbulent shear stress?
By relating the fluctuating velocities, u', to mixing length, we can express shear stress in terms of a measurable quantity. Specifically, Prandtl defined u' as lm * (du_bar/dy).
And then what about the kappa constant?
Good catch! Kappa, or the von Karman constant, approximated at 0.4, allows us to linearize the mixing length, making it a function of distance from the wall. This helps simplify our calculations significantly!
Can we summarize what we've learned so far?
Certainly! We’ve established that mixing length is crucial for understanding turbulent flow and calculating shear stress. Remember, lm = kappa * y, where y is the distance from the wall. Keep this in mind as you work through calculations. What's our memory aid again?
ET for 'Eddy Turbulence' and K for 'kappa'!
Perfect! Let’s transition into how these concepts apply to turbulent flow in pipes.
Applications of Mixing Length Theory
Unlock Audio Lesson
To finish our discussion, let’s look at how this theory applies practically, especially concerning turbulent flows in pipes. What do you think the primary considerations are?
Is it about calculating the total shear stress?
Exactly! In turbulent flows within pipes, the viscous shear stress only exists near the boundary. We mainly consider turbulent shear stress there.
So, can we say that turbulent shear stress is dominant?
Yes, that’s correct! We can simplify our expression for total shear stress as τ = ρ lm² (du_bar / dy)². We primarily focus on the turbulent contributions in the core of the flow.
What's the takeaway about measuring shear stress in practical applications?
The importance lies in understanding averages and how they relate to fluctuating components. Once we comprehend this, we substantially ease our calculations.
Can we summarize what we've learned in this session?
Absolutely! Key points include understanding mixing length importance for shear stress calculation, the dominance of turbulent shear stress over viscous stress, and the reminder that turbulent flows primarily contribute to our calculations. Use the acronym T for 'Turbulent Dominance' to hold this concept in mind. Remember, practice is key!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
This section explores Prandtl's Mixing Length Theory, which describes how shear stress in turbulent flow can be expressed in terms of a mixing length defined by the distance between fluid layers. The relationship between fluctuating velocities and shear stress is outlined, as well as the assumptions made to simplify calculations.
Detailed
Prandtl's Mixing Length Theory
Prandtl's Mixing Length Theory, introduced in 1925, revolutionized the understanding of turbulent shear stress in fluid dynamics. It asserts that turbulent shear stress can be computed if the product of fluctuating velocities, represented as minus rho u' v' , is established. The challenge is determining u' v', as it is not easily measurable.
Prandtl introduced the concept of a mixing length (lm), which is defined as the vertical distance necessary for fluid particles from one layer to mix into another layer while preserving momentum. This relationship is crucial as it helps express shear stress in terms of a measurable quantity.
Additionally, Prandtl theorized that the mixing length is a linear function of the distance from a solid boundary (y), expressed as lm = kappa * y, where kappa is the von Karman constant (~0.4). This leads to a simplification where the turbulent shear stress can be calculated using the average velocity gradients, significantly aiding the analysis of turbulent flows.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introduction to Reynolds Shear Stress
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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. The shear stress due to turbulence can be written as, minus rho u prime v prime whole bar. This can actually be derived, which we will do at some point in this hydraulic engineering course.
Detailed Explanation
Reynolds shear stress is a critical concept in fluid mechanics that quantifies the effects of turbulence on fluid motion. It is expressed as the product of the density of the fluid (rho) and the average of the product of fluctuating velocity components in different directions (u' and v'). This mathematical expression helps in understanding how turbulence contributes to shear stress in flowing fluids. Reynolds formulated this relationship in the late 19th century, providing a foundation for modern turbulent flow analysis.
Examples & Analogies
Think of a busy highway at rush hour where cars are moving at different speeds and changing lanes unpredictably. The average speed of cars in each lane represents a component of velocity, while the fluctuations in speed as cars weave in and out can be likened to the fluctuating velocity components in Reynolds shear stress. Just as in traffic flow, where the interactions between vehicles cause additional forces (stress), turbulence in a fluid generates complex shear stresses.
Prandtl's Introduction of Mixing Length
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Now, there is a concept of Prandtl's mixing length theory. In 1925, Prandtl introduced the concept of mixing length to express shear stress in a measurable way. The mixing length, l_m, is the distance between two vertical fluid layers such that fluid particles can mix together while preserving their momentum along the direction of flow.
Detailed Explanation
Prandtl proposed the mixing length theory to bridge the gap in measuring shear stress due to turbulence. According to this theory, mixing length (l_m) describes how far fluid particles can travel vertically before they mix and distribute their momentum. It serves as a conceptual tool to quantify the contribution of turbulent motion to shear stress. This intuitive approach allows engineers to calculate turbulent shear stress using measurable variables, which is essential in practical applications such as pipe flow.
Examples & Analogies
Imagine a blender mixing ingredients for a smoothie. The distance the blades beat through the mixture can be thought of as the mixing length. Just like in fluid dynamics, where particles mix together to maintain momentum while being blended, the mixing length in a fluid describes how layers of fluid interact and transfer momentum in turbulent flow.
Relationship Between Shear Stress and Mixing Length
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Prandtl related u prime to mixing length lm stating that u prime can be written as mixing length lm multiplied by the gradient of the average velocity: u prime = l_m * (du bar/dy). By substituting this into Reynolds' stress model, it becomes tau turbulence = rho l_m^2 * (du bar/dy)^2.
Detailed Explanation
In this relationship, Prandtl formulated an equation that connects the fluctuating velocity (u') with the mixing length and the average velocity gradient. By substituting this relationship into the expression for turbulent shear stress, it simplifies the problem to a form that emphasizes the importance of the mixing length in calculating shear stress in turbulent flow. The derived expression shows how shear stress grows with the square of the mixing length and the velocity gradient, highlighting the influence of these factors in turbulent conditions.
Examples & Analogies
Consider a basketball player dribbling while running. The rate at which they bounce the ball (rate of change of average velocity) represents the gradient of their motion, while the distance they travel while dribbling could be likened to the mixing length. Just as a player maintains control and transfers energy to the ball within a certain distance while running, the mixing length describes how fluid particles interact within a defined space, affecting the overall flow's shear stress.
Linear Function of Mixing Length
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Prandtl assumed that the mixing length lm is a linear function of distance y from the wall or any solid boundary: l_m = kappa * y. Here, kappa is known as the von Karman constant and is approximately equal to 0.4.
Detailed Explanation
In this model, Prandtl simplifies the relationship between mixing length and distance from a boundary. By treating mixing length as a linear function of distance y, where kappa is a constant, it allows for easier calculation of mixing length across various flows. This assumption helps in practical applications, where predicting shear stress becomes more manageable because it relies on measurable quantities, significantly aiding in the design of systems involving turbulent flow.
Examples & Analogies
Think of a tree growing taller as it moves away from a fence. The height of the tree (mixing length) increases linearly as the distance from the fence (the boundary) increases. Just like this relationship helps us visualize the tree's growth dynamics, Prandtl's model allows engineers to understand how mixing length varies with distance from the wall in fluid dynamics.
Application in Turbulent Flow
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
In turbulent flow, viscous shear stresses exist only near the boundary, while most regions are dominated by turbulent shear stress. Thus, the total shear stress can be approximated, neglecting viscous shear stress.
Detailed Explanation
During turbulent flow, the effect of viscosity is primarily felt near the solid boundaries where the fluid can adhere and slow down. However, in the majority of the flowing body, turbulence dominates, leading to a different equation for shear stress that emphasizes the role of turbulence. This simplification is crucial in engineering scenarios where turbulent flow conditions are prevalent, making analysis and predictions more feasible.
Examples & Analogies
Imagine a river flowing swiftly past a rocky riverbank. Near the bank, the water's surface is slower due to friction (viscous shear stress), whereas in the middle of the river, the flow is rapid and chaotic (turbulent shear stress). Just as most of the river's movement is dictated by turbulence away from the banks, most turbulent flows in engineering contexts are dominated by turbulent shear stress rather than viscous effects.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Shear Stress: The force per unit area exerted by fluid layers, crucial in determining flow behavior.
-
Eddy Viscosity: Represents turbulence's effect on shear stress calculations.
-
Mixing Length Theory: A model that simplifies turbulent shear stress calculations through the concept of a mixing length.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
In a turbulent pipe flow, the mixing length can be used to calculate how much the velocity varies across different layers of the fluid, impacting shear stress predictions.
-
When analyzing natural rivers, incorporating the mixing-length model greatly improves predictions on sediment transport due to turbulent shear.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
When fluid layers meet and mix, momentum's preserved, that's the trick.
📖 Fascinating Stories
-
Imagine a bustling kitchen where chefs are hustling. Each has their own area but needs to mix ingredients. The distance they have to stir and blend the flavors represents the mixing length in fluid dynamics.
🧠 Other Memory Gems
-
KUD for 'Kappa, Upward, distance' helps recall mixing length in terms of distance from the wall.
🎯 Super Acronyms
MIX helps to remember
- Mixing length
- influencing shear stress
- and 'mixing' brings velocities together.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Eddy Viscosity
Definition:
A coefficient that quantifies the turbulent shear stress in a fluid.
-
Term: Reynolds Shear Stress
Definition:
The stress due to turbulent fluctuations, calculated as minus rho u' v'.
-
Term: Mixing Length (lm)
Definition:
The vertical distance between fluid layers necessary for mixing to occur.
-
Term: Kappa (κ)
Definition:
The von Karman constant, approximately 0.4, used in mixing length theory.