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.
Introduction to Turbulent Flow
Unlock Audio Lesson
Let's start with turbulent flow. Can anyone tell me how we differentiate between laminar and turbulent flow?
I think laminar flow is smooth, while turbulent flow has a more chaotic pattern.
Exactly! In turbulent flow, the velocity profile is not smooth but rather complex. We can use the Prandtl mixing length theory to describe this. Does anyone remember what shear stress signifies in this context?
Shear stress relates to the force per unit area acting parallel to the flow.
That's correct! And at the wall of the pipe, we denote this shear stress as tau not. Why do you think it's constant there?
Because the flow velocity near the wall is significantly lower due to friction.
Exactly! Great job summarizing key concepts. Now, let's also consider how we derive the turbulent velocity profile using logarithmic equations.
Derivation of the Turbulent Velocity Profile
Unlock Audio Lesson
Now, we can derive the velocity profile based on our discussions. What happens when we apply the boundary conditions at the radius of the pipe?
We can find u at R, and that it equals u max— the maximum velocity at the center!
Yes! Following that, we derive the expression for velocity defect law. Anyone remember how we write it?
It's formulated as u max minus u over u * equals to 5.75 log to the base 10 R over y.
Great recall! And what does this law help us understand? Can someone provide a context for its application?
It can help in determining the frictional forces in turbulent systems!
Exactly! You're making meaningful connections here.
Layers in Turbulent Flow
Unlock Audio Lesson
Let's now discuss the different layers in turbulent flow. What are the different regions we have?
We have the viscous sublayer, buffer layer, overlap layer, and turbulent layer.
Right! Can anyone explain what happens in the viscous sublayer?
In the viscous sublayer, the velocity profile is almost linear due to significant viscous effects.
Correct! As we move away from the wall, we enter the buffer layer. What characterizes this layer?
Here, turbulence effects become significant but viscous effects still dominate.
Well articulated! And why is it crucial to understand these layers in turbulent flow?
It helps in predicting how fluids behave in practical applications like pipe flow.
Perfectly put! Understanding these layers can help optimize designs in engineering applications.
Identifying Boundary Conditions
Unlock Audio Lesson
Now that we've discussed layers, let's dive into boundary conditions. How do we define smooth versus rough boundaries?
Smooth boundaries have irregularities smaller than the viscous sublayer thickness.
Exactly right! And how does Nikuradse's analysis help in classifying these boundaries?
He established that if k/delta dash is less than 0.25, the boundary is smooth. If it's more than 6, it's rough!
Correct! These criteria really help in engineering applications. Can you explain what happens between those ranges?
If it's between 0.25 and 6, the boundary is transitional, meaning it's neither entirely rough nor smooth.
Excellent! You're all grasping these concepts well!
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, we discuss the derivation of the turbulent velocity profile from the Prandtl mixing length theory, including the logarithmic velocity profile and the concepts of velocity defect law. Additionally, the section explains the layers in turbulent flow, including the viscous sublayer, buffer layer, overlap layer, and turbulent layer, including boundary types.
Detailed
Turbulent Velocity Profile
The turbulent velocity profile is a crucial aspect of fluid mechanics that describes how velocity varies in turbulent flow conditions. This section outlines how to derive the turbulent flow velocity, starting from the foundational concepts of shear stress at the wall and the use of boundary conditions. Using the Prandtl mixing length theory, we arrive at a logarithmic velocity profile representing turbulent flow, contrasting it with the parabolic profile observed in laminar flow. Furthermore, we explore the significant role of various layers in turbulent flow -- the viscous sublayer, buffer layer, overlap layer, and turbulent layer -- and how they interact with surface irregularities, which fundamentally shape the characteristics of the flow. The definitions of smooth and rough boundaries are also presented through Nikuradse’s experiments, which help classify flow conditions.
Audio Book
Dive deep into the subject with an immersive audiobook experience.
Introduction to Turbulent Flow and Shear Stress
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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.
Detailed Explanation
In this chunk, we are introduced to the equations that define the turbulent velocity profile. Equation 14 sets up the relationship between the mixing length (l) and the distance from the wall (y) through the constant kappa. We then see that for small values of y, the shear stress (tau) can be assumed to equal the constant shear stress at the pipe wall (tau not). This is crucial because it allows us to simplify the equations involved in the analysis of turbulent flow.
Examples & Analogies
Imagine you are observing traffic on a highway. Near the walls of the highway (the walls of our pipe), you see cars moving slowly due to friction with the ground, similar to how the fluid interacts with the wall. As you move towards the center of the highway (increasing y), the traffic flows more freely and quickly, analogous to how the shear stress changes from the wall (tau not) to the center of the flow.
Deriving Shear Velocity
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
If we substitute tau is equal to tau not in equation 16, we can obtain 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.
Detailed Explanation
Here we substitute tau not into equation 16 to derive another important aspect, du/dy, which describes how velocity changes with respect to distance from the wall. We express tau not in terms of density (rho), transforming our equation. The term u star (u*) is introduced here, representing the shear velocity, which measures how rapidly the velocity of the fluid changes. This portion of the analysis is key in understanding the mechanics of turbulent flows.
Examples & Analogies
Think of a crowded dance floor. Near the edges (where the dancers slow down due to the walls), the velocities of those dancing are relatively low. As they move towards the center where the dancing is lively, the speed increases. The concept of shear velocity (u*) is similar; it captures how rapidly the speed increases as you move away from the wall.
Integration and Boundary Conditions
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
If you integrate the equation number 17, so, what we can get is, simple integration, it will get. 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.
Detailed Explanation
In this step, we perform a simple integration of the derived equation to find a more complete expression for velocity. This results in establishing boundary conditions: particularly, the velocity at the wall (y = R, the radius of the pipe) where it reaches its maximum (u_max). Setting the boundary conditions is essential for getting meaningful solutions in fluid dynamics.
Examples & Analogies
Imagine throwing a ball vertically into the air. As the ball moves upward, it has a maximum height it can reach. Similarly, as fluid flows through the pipe, it has a maximum velocity at the center of the pipe. Establishing where this maximum occurs helps predict how the fluid flows within the pipe.
Logarithmic Velocity Profile Derivation
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
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
In this part, we determine the constant C using our previously established boundary conditions, relating u_max, u*, and kappa (a constant related to roughness). Upon substituting kappa with a specific value (0.4), we can finally arrive at the logarithmic velocity profile equation (equation number 20). This logarithmic relationship signifies how turbulence affects flow characteristics, contrasting with laminar flow profiles.
Examples & Analogies
Consider how a river flows over rocks. Close to the riverbed (like near the walls of the pipe), the water flows slower due to friction while it speeds up in the center. The logarithmic profile is like a map describing these speed changes as you move horizontally across the river at different depths.
Velocity Defect Law
Unlock Audio Book
Signup and Enroll to the course for listening the Audio Book
u max minus u is called the velocity defect or velocity defect law, this is velocity defect law. This is just simple, you know, manipulation of these terms here.
Detailed Explanation
The concept of velocity defect law arises from the earlier equations, where we note the difference between the maximum velocity and the actual velocity at any point u. This difference (u_max - u) signifies how much the velocity is 'defective' or reduced as fluid flows through a pipe due to turbulence and friction. Understanding this law is critical when analyzing fluid flow dynamics.
Examples & Analogies
Think of going downhill on a bike. Ideally, you can reach a certain top speed without obstacles (u_max). However, if you encounter bumps (similar to turbulence in a fluid), your speed decreases (u), creating a 'defect' or difference that we can measure.
Summary of Turbulent Flow Characteristics
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.
Detailed Explanation
This final chunk summarizes the key differences between turbulent and laminar flow. It emphasizes how turbulent flow creates a fuller velocity profile compared to the parabolic shape seen in laminar conditions. This understanding of the velocity profile differences is crucial for engineers and scientists dealing with fluid dynamics.
Examples & Analogies
Just as in a busy city intersection, where cars flow in complex patterns that build momentum and energy (turbulent), versus cars moving smoothly in a straight line down a quiet road (laminar), this analogy illustrates how turbulence can enhance mixing and energy transfer in a fluid.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
-
Prandtl Mixing Length Theory: A model for predicting turbulence in flowing fluids.
-
Logarithmic Velocity Profile: The velocity distribution of turbulent flow derived from boundary conditions.
-
Velocity Defect Law: Describes the relationship between maximum and actual velocity in turbulent flow.
-
Surface Irregularities: Features of boundary surfaces that influence the flow rate and turbulence.
-
Boundary Classification: Identifying smooth, rough, and transitional states based on surface characteristics.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
-
Example of turbulent flow in a river where the surface may be uneven causing chaotic fluid motion.
-
Applying the velocity defect law to assess the performance of fluid flow in a pipe with specific diameter and shear stress conditions.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
-
In turbulent flow, things twist and twirl, / Shear stress and layers make the water swirl.
📖 Fascinating Stories
-
Imagine a river with twists - some parts are calm and smooth, while others rush wildly! This mirrors turbulent flow, where different layers create chaos near the bank.
🧠 Other Memory Gems
-
Remember 'V-BOT' for the layers: Viscous, Buffer, Overlap, Turbulent.
🎯 Super Acronyms
Use the acronym 'TRUST' to remember types of boundaries
- Transitional
- Rough
- and Smooth boundaries.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
-
Term: Turbulent Flow
Definition:
A flow regime characterized by chaotic changes in pressure and velocity.
-
Term: Shear Stress (τ)
Definition:
The force per unit area acting parallel to the flow direction.
-
Term: Logarithmic Velocity Profile
Definition:
A mathematical representation of how velocity varies in turbulent flow conditions.
-
Term: Velocity Defect Law
Definition:
An expression relating the velocity profile in turbulent flow to the wall shear stress.
-
Term: Viscous Sublayer
Definition:
The layer closest to the wall where viscous effects dominate.
-
Term: Buffer Layer
Definition:
The layer where both turbulent and viscous effects are significant.
-
Term: Rough Boundary
Definition:
A boundary where surface irregularities are significant and affect flow characteristics.
-
Term: Smooth Boundary
Definition:
A boundary where surface irregularities are negligible compared to the viscous sublayer thickness.