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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!
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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.
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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.
In fluid dynamics, particularly when dealing with pipe flows, shear stress (denoted as tau) can change based on the distance from the wall, represented as y. However, for small values of y (very near the wall), we simplify our assumptions by saying that the shear stress remains constant (tau = tau not). Here, tau not represents the shear stress at the very wall of the pipe. This simplification allows us to treat tau not as a constant value for calculations, making our analysis easier.
Imagine you are stirring a thick syrup. Right near the walls of the container where the syrup touches, the syrup has a slow movement (viscous effect). If you measure the speed of the syrup at a point very close to the wall, you can approximately consider its flow speed the same as at the wall itself because there isn't much room for it to move. Thus, we treat this 'wall speed' as a constant for those calculations.
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And therefore, what we can say, 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 *.
When we substitute tau with tau not in our equations (specifically in equation 16), we derive a relationship for the gradient of velocity (du/dy). Due to our assumption of small values of y, we can express this gradient as 1/(kappa * y). Here, kappa is a constant related to the geometry and flow properties. Additionally, we can express tau not using the density (rho) of the fluid and a velocity term (u*), linking the shear stress back to the dynamic behavior of the flow.
Think of the flow of a stream of water in a narrow channel. As you get closer to the banks (representing small y values), the speed of water changes, but if we assume certain conditions (like laminar flow), we can predict this change more easily using simpler relationships. By writing equations that involve these constants (like kappa and rho), we find patterns in how the speed of the water changes near the banks.
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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.
Once we have our equations established, we often need to integrate them to find velocity profiles across the flow. As we integrate equation 17, we then apply boundary conditions—certain known values at specific locations. For example, at the radius of the pipe (y = R), we assume that the velocity (u) reaches its maximum value (u max). This provides a solid point from which we can fully solve for the velocity distribution along the radius of the pipe.
Imagine you're trying to predict how the flow of honey in a pipe varies from the center to the edge. At the very edge (y equals radius), you know it moves the slowest, while at the center, where it bulges out, it moves the fastest. By integrating and applying these known positions, you can piece together an entire flow pattern just like connecting the dots on a graph to predict how quickly the honey flows.
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So, if we use this boundary condition u at y is equal to R is u max, we can get u is equal to... if we substitute kappa as 0.4, we can get... and this is just simple manipulation...
By applying our boundary conditions, we can solve for unknown constants in our equations. Substituting values like kappa (0.4 here) allows us to derive a specific form of the velocity profile across the pipe. This results in a logarithmic relationship as opposed to the parabolic profile seen in laminar flow. This aspect is crucial in understanding how turbulent flows differ significantly from laminar ones.
Think of a water slide that is steep in the middle and shallow at the edges. The way water speeds up and slows down as it travels down the slide can be modeled with similar equations. When conditions change (like the slope), the way we describe how quickly the water travels (the velocity profile) also changes from smooth (like laminar flow) to more chaotic (like turbulent flow).
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u max minus u is called the velocity defect or velocity defect law, this is velocity defect law.
The velocity defect law describes the difference between the maximum velocity in a flow (u max) and the actual fluid velocity (u) at any point within the flow. This difference is crucial for understanding how much fluid velocity 'defects' or falls behind the maximum due to turbulence and wall friction, impacting flow performance in various applications.
Imagine a runner on a track. The fastest speed recorded on a track is like u max, while each runner will run at their own speed, causing some to finish slower than the best time. The difference in their speeds (the defect) gives insight into how factors like fitness level (like wall friction) impact their performance—the same way flow conditions affect fluid behavior in a pipe.
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
Constant Shear Stress: For small 'y', shear stress at the wall (τ₀) is assumed constant.
Logarithmic Velocity Profile: Derived from turbulent flow assumptions, shows a non-linear relationship between velocity and distance from the wall.
Viscous Sublayer: The closest layer to the wall where viscous forces dominate, leading to linear velocity profiles.
See how the concepts apply in real-world scenarios to understand their practical implications.
For a pipe of radius R with a centerline velocity of u_max, calculate τ₀ using given velocities at various distances.
Given R = 0.05m and y = 0.005m, derive the velocity from the logarithmic profile equation.
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
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.
Review key concepts with flashcards.
Review the Definitions for terms.
Term: Shear Stress (τ)
Definition:
The force per unit area acting parallel to the surface.
Term: τ₀ (tau naught)
Definition:
The shear stress at the wall, assumed to be constant for small values of 'y'.
Term: Logarithmic Velocity Profile
Definition:
A velocity profile that describes how velocity increases logarithmically with distance from the wall.
Term: Viscous Sublayer
Definition:
The layer closest to the wall where viscous effects dominate, creating a nearly linear velocity profile.
Term: Turbulent Layer
Definition:
The outermost layer in turbulent flow where turbulent effects dominate over viscous ones.