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
Today, we will explore how shear stress operates in turbulent flow. Let's start with the basic equation τ = ρu*², where τ is the shear stress.
Can you explain what u* is?
Great question! u* is known as the friction velocity, which helps bridge the relationship between shear stress and velocity. It's often calculated as u* = √(τ₀/ρ).
And what does the ρ stand for?
ρ is the fluid density, and understanding this helps in determining how shear stress is proportional to the velocity squares. This will aid us in our later calculations.
Velocity Profile in Pipe Flow
Unlock Audio Lesson
Let’s discuss how we derive the velocity profile using boundary conditions. We'll use the fact that at the wall (y=R), the velocity is at its maximum.
So, if I understand correctly, u at the wall is equal to u_max?
Exactly! As we set u at y = R equal to u_max, we can write our equations to express C, which allows us to derive the velocity profile.
What's the form of the velocity profile?
It leads us to a logarithmic profile: u = u_max - 2.5u* ln(R/y). This is crucial in determining how flow behaves near a boundary.
Practical Applications: Determining Shear Stress
Unlock Audio Lesson
Now we'll tackle a practical problem on shear stress using water flow through a pipe. Can anyone tell me the first step?
We need to identify our given values, like the diameter and velocity at specific points.
Correct! For this problem, we have a diameter of 0.1m and velocities at specified points. Write these down as it helps set the stage for our calculations.
Once we have those, how do we find the shear stress at the wall?
We use the equation for velocity defect and the shear stress formula τ₀ = ρu*² to get our result. Let’s work through the calculation for clarity.
Understanding Boundary Layer Properties
Unlock Audio Lesson
Now let's look at the types of boundary layers—viscous, buffer, overlap, and turbulent layers. Who can explain one of them?
The viscous sublayer is where the velocity profile is almost linear, right?
Exactly! As we move away from the wall, turbulent effects become more pronounced in the layers like the overlap layer.
And how does this relate to rough and smooth boundaries?
Great query! If the irregularities on the surface are larger than the viscous sublayer, we consider the boundary to be rough. We'll dive deeper into that next.
Determining Boundary Characteristics
Unlock Audio Lesson
Let’s finalize our understanding by differentiating between rough and smooth boundaries. Who can recall the metrics to evaluate them?
Nikuradse’s criteria for k/delta as less than 0.25 means it's smooth.
Excellent! And when it exceeds 6, we know it’s rough. Understanding these characteristics is crucial for application in fluid mechanics.
What about the transitional boundaries?
Transitional boundaries occur between 0.25 and 6. Remembering this helps in categorizing flows correctly.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
The section explains the derivation of the logarithmic velocity profile in turbulent flow using shear stress equations and boundary conditions. It covers important fluid dynamics concepts including the presentation of laminar and turbulent flows, calculation of shear stress at pipe walls, and discussions of boundary layer properties.
Detailed
In this section, we begin with the integration of shear stress equations within the context of turbulent fluid flow. Specifically, it discusses the assumption that for small values of y, shear stress at the wall, denoted as τ₀, can be taken as constant. By substituting τ as τ₀ in the derived equations, we can express the velocity gradient and derive useful equations that relate flow characteristics. A key focus is the derivation of the logarithmic velocity profile based on the Prandtl mixing length theory. The different characteristics of laminar versus turbulent flow are highlighted, and we explore the implications of boundary conditions on these profiles. The section also addresses practical applications, such as determining shear stress with worked examples that contextualize theoretical concepts in real-world scenarios. Ultimately, this understanding aids in both the theoretical formulation and practical calculations associated with turbulent flow and boundary layers.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Shear Stress at the Wall
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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're discussing conditions where the shear stress (denoted as tau) can be equated to a constant value at the wall of the pipe, referred to as tau not. This assumption holds true particularly when the vertical distance (y) from the wall is very small. The shear stress is a measure of the internal friction in the fluid, which affects how it flows against the walls of the pipe.
Examples & Analogies
Imagine trying to slide a book across a table. The friction you feel represents shear stress. Now, if you gently press the book against the table (like being very close to the wall), the friction becomes a constant force until you push harder. So, near the wall of a pipe, the shear stress remains steady, making calculations easier for engineers.
Deriving Shear Velocity
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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.
Detailed Explanation
In this part, the relationship between the shear rate (du/dy) and the shear velocity (u*) is established. The equation indicates that when we replace tau with tau not, we can derive an expression involving kappa (a constant) and the distance y. The shear velocity represents how fast the fluid layers slip past one another, crucial for understanding turbulent flow in pipes.
Examples & Analogies
Consider a river flowing over a series of rocks. The top layer of water moves more quickly than the deeper layers. The difference in speed between these layers is like shear velocity. As the top layer moves swiftly, if we picture measuring how that layer interacts with the rocks (the tau), we can understand how smoothly or turbulently the water flows.
Using Boundary Conditions for Integration
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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. That is what we have seen at the center line of the pipe the velocity is going to be the maximum.
Detailed Explanation
This chunk explains how boundary conditions are used in deriving formulas for fluid flow. The boundary condition states that at the radius of the pipe (R), the fluid velocity (u) reaches its maximum value (u max). This is crucial for accurately modeling how the fluid behaves within the pipe and allows for solving the equations derived earlier.
Examples & Analogies
Imagine a water slide. When someone sits at the top, that’s the point of maximum speed (u max) right before they go down. The moment you reach the slide (the edge), you begin at maximum speed compared to being near the ground. Similar to how water reaches its fastest flow at the radius of the pipe.
Logarithmic Velocity Profile
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, laminar flow was something like this, a parabolic profile. Here, a profile is little different u is u maximum plus a logarithmic profile.
Detailed Explanation
This chunk contrasts laminar flow with turbulent flow conditions. While laminar flow has a smooth, parabolic velocity profile, turbulent flow results in a logarithmic velocity profile, indicating that flow is irregular and chaotic. Understanding this difference is vital in fluid mechanics for predicting how fluids behave under various conditions.
Examples & Analogies
Think of a calm pond (laminar flow) where you see smooth ripples. Now consider a river during a heavy storm (turbulent flow) where the water splashes in all directions. The transition from smooth to chaotic can be visually represented by changing from a nice curve to a jagged line, like how we see velocity profiles differ.
Velocity Defect Law
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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, this is velocity defect law.
Detailed Explanation
The velocity defect law describes the difference between the maximum fluid velocity and the actual velocity at various points within the pipe. The equation derived here further illustrates how the logarithmic nature of the flow contributes to the differences in velocity, emphasized by the logarithmic scale. This is essential for engineers to predict and manage fluid behavior in pipelines.
Examples & Analogies
Imagine a racetrack. The fastest car (u max) is at the front, while slower cars (u) are behind. The difference in speed is the velocity defect. If we plotted their positions, the differences would paint a visually sharp picture of how speed varies, much like plotting fluid speeds in a pipe.
Summary of Turbulent Velocity Profile
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
So, now the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. ... So, as I told you in the last slide, there are different layers, different layers in turbulent flow.
Detailed Explanation
The turbulent velocity profile is more complex than that of laminar flow. The explanation introduces various layers within turbulent flow, highlighting the structure of how turbulence affects velocity at different distances from the wall. Understanding these layers helps in predicting flow characteristics and improving pipe design.
Examples & Analogies
Think about layers in a cake. The frosting on top (the turbulent layer) is thick and varied unlike the smooth cake underneath (the laminar flow). Each layer plays a unique role in overall texture and flavor, just as each layer of fluid flow influences the movement through a pipe.
Understanding Flow Regions
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
Turbulent flow along a wall consists of 4 regions. Viscous sublayer, this layer is thin layer next to the wall...
Detailed Explanation
In turbulent flow, four distinct regions are identified: the viscous sublayer (where viscous effects dominate), the buffer layer (both turbulent and viscous effects are present), the overlap layer (turbulent mostly, with some viscous influence), and the turbulent layer (where turbulent effects are predominant). These regions are important for understanding how flow behaves near surfaces.
Examples & Analogies
Consider a crowd of people at a concert. The people closest to the stage (viscous sublayer) are more controlled and orderly, while those farther away (turbulent layer) are moving chaotically with lots of pushing. Just like in flow, different regions yield differing behaviors depending on the crowd's proximity to the stage.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Shear Stress: A measure of the force exerted parallel to a surface per unit area.
-
Friction Velocity: A key scale used in fluid dynamics that relates to shear stress and flow characteristics.
-
Viscous Sublayer: The proximity zone adjacent to a wall where viscous effects dominate.
-
Turbulent Layer: The farthest flow area from a wall where turbulence presents distinct flow attributes.
-
Logarithmic Profile: A description of how velocity varies logarithmically with height in turbulent flow situations.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Calculating shear stress at the wall using the friction velocity for a pipe carrying water.
-
Deriving a logarithmic velocity profile using boundary conditions, demonstrating the practical implications of theory.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
When fluid flows with force so grand, shear stress gives us a helping hand.
📖 Fascinating Stories
-
Imagine a river with smooth stones; the water flows easily. But on a rocky bed, the flow is chaos. This story illustrates how smooth versus rough boundaries affect flow.
🧠 Other Memory Gems
-
Remember the word 'SLAP' for shear, laminar, acceleration, and profile.
🎯 Super Acronyms
Use 'BOLT' to remember Boundaries, Overlap, Laminar, and Turbulent flow characteristics.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Shear Stress (τ)
Definition:
The force per unit area exerted parallel to the surface of a material.
-
Term: Friction Velocity (u*)
Definition:
A velocity scale in boundary layer theory that relates shear stress and velocity.
-
Term: Viscous Sublayer
Definition:
The thin layer closest to the wall where the flow is dominated by viscous effects, and the velocity profile is nearly linear.
-
Term: Turbulent Layer
Definition:
The layer farthest from the wall where turbulence dominates the flow characteristics.
-
Term: Boundary Layer
Definition:
The region next to a surface where the effects of the surface affect the flow of the fluid.
-
Term: Logarithmic Velocity Profile
Definition:
A velocity profile characterized by a logarithmic dependence on distance from the wall, commonly seen in turbulent flow.
-
Term: Transitional Boundary
Definition:
A boundary regime where the surface roughness height is between smooth and rough limits.
-
Term: Nikuradse’s Roughness
Definition:
A criterion for defining surface roughness based on the ratio of surface irregularity height to the thickness of the viscous boundary layer.