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A short class with Sarah and her students. The other ways are under More.
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Today, let's explore the power law velocity profile. Does anyone remember the formula for it?
Isn't it u/u_max = (y/R)^(1/n)?
Great! Now, can anyone tell me what the variable n represents?
I think n depends on the Reynolds number, right?
Exactly! And as the Reynolds number increases, n also increases. Remember, higher n gives a steeper profile. But, unfortunately, there are limitations. What do you think happens at the center of the pipe?
It doesn't give a zero slope there.
Correct! So keep in mind, power law profiles cannot predict wall shear stress either as it leads to infinite velocity gradients at the wall. That's crucial!
Let's talk about how we apply these power law profiles in real-world situations. What do we need to consider when using them?
We have to account for the pipe surface texture, whether it's smooth or rough.
Good point! And remember, despite those differences, the velocity profile at any point minus the average remains the same for both types of pipes. Why might that be significant?
It shows that the fundamental behavior of fluid flow doesn't change that much.
Exactly! Lastly, can anyone summarize why we can't use power law profiles to calculate wall shear stress?
Because they give us an infinite velocity gradient at the wall.
Spot on! Let's summarize the key limitations we discussed today.
Now, let’s derive the average velocity from our power law profile. Can anyone remind us how we express the average velocity mathematically?
I think we integrate the velocity profile over the cross-sectional area.
Correct, we use A = πR² for the area and integrate. What’s the expression for the average velocity from u_max?
It's V_bar = 2u_max times the integral of the power law expression.
Exactly! And remember, after integrating and applying boundaries, we find the average velocity. Who remembers the final expression for average velocity using n=7?
That would be V_bar = 0.816u_max.
Perfect! Remember this value, as it can be critical in practical applications.
The difference between a point velocity and average velocity in pipes is consistent for smooth and rough surfaces.
Power law profiles cannot predict zero slope at the pipe center or calculate wall shear stress accurately due to infinite gradients.
In a turbulent water flow in a pipeline, the velocity at the center might reach maximum levels, while the average would fall short, illustrating the practical application of velocity profiles.
Considering Reynolds number changes with flow rates, the behavior of the power law profile can influence the design of piping systems significantly.
Power laws don't show zero slope, at centers, they can't help us cope.
Imagine a smooth and a rough pipe side by side, both carrying water. They both have differences in their average and point velocities, but their velocity behavior is surprisingly similar, meaning we have distinct design considerations!
Remember 'Smoothness Saves Stress' - smooth pipes have a predictable behavior while rough pipes can lead to unpredictable shear stress.
RAPID - Recognize Average Point Interactions Derive - remember the average velocity interaction when considering turbulent flows.
Term
What is a power law velocity profile?
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Term
What happens to wall shear stress using power law profiles?
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A mathematical model representing velocity distribution in a turbulent flow characterized by a governing exponent dependent on flow conditions.
A dimensionless number used to predict flow patterns in different fluid flow situations.
The force per unit area exerted by a fluid at the interface with a solid surface.
This section discusses the limitations of power law velocity profiles in turbulent pipe flow, specifically their inability to predict wall shear stress and zero slope at the pipe center.
The limitations of power law profiles in turbulent flow through pipes are highlighted, especially regarding the calculations for wall shear stress and the behavior of velocity gradients. The section explores how both smooth and rough pipes exhibit similar differences between local and average velocities but face challenges in accurately modeling shear conditions.
In this section, we explore the limitations of power law profiles for turbulent flow in pipes. The key focus is on how the average velocity divided by the frictional velocity reveals insights into local flow conditions. The difference between any point velocity and the average velocity remains consistent across smooth and rough pipes, which implies similarities in fluid behavior despite the surface texture differences.
Furthermore, the power law velocity profile, expressed as u/u_max = (y/R)^(1/n), lacks the capability to predict zero slope at the pipe center and fails to calculate wall shear stress accurately. This is primarily due to the indication that the velocity gradient approaches infinity at the wall (r = R), highlighting a fundamental limitation in the analytical frameworks used. Moreover, calculations involving the average velocity demonstrate a systematic approach to address these velocity profiles, culminating in a practical example that showcases the method of integrating the power law expression to derive average velocities.