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A short class with Sarah and her students. The other ways are under More.
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Today, we will discuss how we treat shear stress, specifically τ₀, when dealing with small distances 'y' from the pipe wall. Can anyone tell me what shear stress represents?
I think it measures the force per unit area acting parallel to the surface.
Exactly! And for small values of 'y', we assume τ is constant and equal to τ₀. Why do you think this assumption is useful?
It simplifies our equations and helps in predicting flow behavior without complicated variables.
Good point! This assumption allows us to derive further equations which will lead us to the velocity profiles in turbulent flow. Let's move forward.
Now that we know τ is constant, we can use it in our equations. Can someone remind me how we express the relationship involving du/dy?
It relates the change in velocity with respect to distance y, right?
Exactly! And by substituting τ₀ in our equations, we find this simplifies our integration somewhat. Who remembers the form of the velocity profile we get?
Is it a logarithmic profile that shows how velocity increases with distance from the wall?
Yes! A logarithmic velocity profile shows that the velocity at the centerline is higher, confirming our initial assumption.
Let's apply what we've learned. Given a pipe with an average velocity of water at the center and a specific distance from the center, how do we begin?
We need to use the velocity defect law to relate velocities at different distances.
That's correct! We'll use the equation u_max - u / u* = 5.75 log(R/y) to calculate u*. Can someone explain what R and y represent?
R is the radius of the pipe and y is the distance from the wall.
Absolutely! By plugging in our values, we can derive tau₀ and deepen our understanding through actual data.
Finally, let’s discuss the four layers of turbulent flow: the viscous sublayer, buffer layer, overlap layer, and turbulent layer. What is the significance of these layers?
They each have distinct properties and effects on how the fluid behaves near the wall.
Correct! The viscous sublayer is the thin area closest to the wall. What happens to the velocity profile there?
It's almost linear due to the dominance of viscous effects.
That's right! As we move away from the wall, the behavior shifts to turbulent effects, where the velocity is more complex. Good job today!
For small 'y', shear stress at the wall (τ₀) is assumed constant.
Derived from turbulent flow assumptions, shows a non-linear relationship between velocity and distance from the wall.
The closest layer to the wall where viscous forces dominate, leading to linear velocity profiles.
When you're near the wall, τ is at call, constant at 'y', where no turbulence can ply.
Imagine a tiny fish swimming close to the wall of a pipe. It swims steadily because that's where the water flows smoothly — that's the viscous sublayer, while further away the current starts to swirl and dance, representing the turbulence.
V-T-B-T for remembering layers: Viscous, Transitional, Buffer, Turbulent.
STU for remembering: Shear Stress (τ₀), Turbulent flow, Unchanging at small 'y'.
Term
What does τ₀ represent?
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Term
What characterizes the logarithmic velocity profile?
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What is the viscous sublayer?
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The force per unit area acting parallel to the surface.
The shear stress at the wall, assumed to be constant for small values of 'y'.
A velocity profile that describes how velocity increases logarithmically with distance from the wall.
The layer closest to the wall where viscous effects dominate, creating a nearly linear velocity profile.
The outermost layer in turbulent flow where turbulent effects dominate over viscous ones.
This section discusses the assumptions applicable to small values of 'y' in turbulent fluid flow, particularly shear stress at the pipe wall.
The section elaborates on the shear stress behavior in turbulent flow with small 'y' values, emphasizing that the shear stress at the pipe wall is considered constant. It also introduces the logarithmic velocity profile derived from Prandtl's mixing length theory and illustrates these concepts with example problems.
In this section, we analyze the assumptions for turbulent flow behaviors at small values of 'y,' where 'y' is the distance from the pipe wall. We start by defining the shear stress at the wall, known as tau not (τ₀), which is assumed to be constant at very small 'y' values. By substituting τ = τ₀ into the governing equations, we derive a relationship between du/dy and other parameters, revealing that the velocity profile of turbulent flow differs significantly from laminar flow. The section progresses to introduce a logarithmic velocity profile, derived from integrating the relationships established. Moreover, detailed examples help illustrate the application of these theoretical concepts in determining shear stress effectively. The section also underscores the differences in velocity profiles and delves into the four layers of turbulent flow immediately adjacent to the wall, each possessing unique characteristics and effects on flow dynamics.