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4.5.2. Integration for average velocity
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Today, we are going to understand the turbulent flow in smooth pipes. Can anyone tell me what we mean by turbulent flow?
Isn’t it the type of flow where the fluid moves chaotically?
Exactly, great job! Turbulent flow is characterized by irregular fluctuations, or mixing, in the fluid. It’s crucial to understand how this flow behaves in a pipe. Let’s recall our logarithmic velocity distribution equation.
Is that the one with the natural log function?
Yes! The equation looks something like this: u = (u* / kappa) ln(y prime) + C. This equation is vital in determining how velocity behaves at different distances from the wall.
What does 'C' represent in that equation?
Great question! 'C' is a constant that we determine based on boundary conditions. Can anyone think of how boundary conditions might affect our calculations?
They would define the velocity at the wall, right?
Correct! The velocity at the wall is critical for our calculations.
Let’s summarize. We explored turbulent flow and the impact of boundary conditions. Great discussion, everyone!
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Now that we’ve mastered smooth pipe flow, let’s tackle rough pipe flow. Why do you think this is different?
I think the roughness would influence how the fluid travels over the pipe's surface.
Spot on! Rough surfaces create disturbances that change the flow characteristics. For rough pipes, we use Nikuradse’s equations which define the roughness height 'k'.
How do we calculate the average velocity here?
We integrate across the pipe’s radius. Would anyone like to explain how we set up this integral?
We’d need to set the limits for integration from 0 to the radius of the pipe!
Exactly! This setup leads us to an average velocity equation that reflects the turbulent nature of flow in rough pipes.
To recap, we discussed the transition from smooth to rough flows and how each affects our calculations and understanding. Well done!
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Let’s solve a problem involving the average velocity in turbulent flow in a rough pipe. Are we ready?
Yes! What’s the problem?
You need to find the average height of roughness when the velocity at different points is given. Let’s outline our knowns and unknowns.
Alright, we have a diameter of 10 cm and velocity differences at 1 cm and 4 cm from the wall.
Perfect! Knowing this, we need to apply the equations we’ve discussed. Who can recall which equation we use for rough pipes?
We use the one derived from Nikuradse's experiments!
Absolutely! Now, let’s integrate this and find 'k'.
This challenge really reinforces the application of our theory.
Exactly, and it shows how theoretical concepts translate into real-world challenges.
In summary, we engaged in a detailed application of the principles we've learned through problem-solving. Great teamwork, everyone!
Overview
Short Summary
This section discusses the integration processes to obtain the average velocity for turbulent flow in smooth and rough pipes.
Medium Summary
An exploration of how average velocity can be derived from turbulent velocity distributions using integration techniques is presented. The section provides equations for both smooth and rough pipes, emphasizing significant results obtained from experimental data, such as Nikuradse's findings.
Detailed Summary
Integration for Average Velocity
In this section, we delve into the methodology used to compute average velocities in turbulent flow within smooth and rough pipes. The lecture begins by recalling fundamental equations for turbulent velocity distributions in smooth pipes, showcasing the significance of the distance from the wall in influencing velocity profiles. Specifically, we utilize the logarithmic velocity profile equations which relate flow velocities at varied distances from the wall surface and substitute appropriate conditions informed by empirical data from experiments, such as Nikuradse's findings. Subsequent equations express this in terms of average velocities, leading to clear formulations for calculating average flow rates through integration across specific cross-sectional areas. We conclude this section by solving real-world problems to highlight practical applications and deepen conceptual understanding of turbulent flows.
Audio Book
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Create a free accountNow, we are going to talk about turbulent velocity distribution in terms of average velocity. So, there is a flow and there is an elementary circular ring here, as you can see. This is of radius R, so we have an elementary circular ring of radius r and thickness dr which we have considered.
Detailed Explanation
In turbulent flow, the velocity at any point is not uniform. We need to average out these velocities over a section of flow to find the average velocity. We can consider a cross-section of a pipe and break it down into smaller circular rings. Each of these circular rings has a radius 'r' and a small thickness 'dr'. Understanding this helps us calculate the total discharge or average velocity flowing through these rings.
Examples & Analogies
Think of the flow in a pipe like water flowing through several layers of a cake. Each layer represents a circular ring that contributes to how much water flows out (discharge) of the cake. Just like how you average the thickness of the cake layers to understand the overall thickness, we average the flow velocities in these layers to find the average velocity.
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Create a free accountDischarge Q is given by. Now, can we calculate it for smooth pipes? Yes, since y is equal to R minus capital R minus small r, we can write, equation 24, you know, if you do not remember the equation 24.
Detailed Explanation
Discharge (Q) in fluid flow is the volume of fluid that passes through a given surface per unit time. The relationship involves integrating the velocity profiles of these small rings over the entire area of the pipe. We can use the known equations (like equation 24) to express velocity in terms of the distance from the wall and the average flow properties.
Examples & Analogies
Consider a garden hose: the flow of water can be considered as the discharge. The smoother the inside of the hose (like a smooth pipe), the quicker the water flows. To calculate how much water flows through, we use the same idea of summing up (integrating) the contributions from small segments of the hose's interior.
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Create a free accountOr we can also write, u is equal to 5.75 log base 10 u star R minus r by nu plus 5.55 y into u star.
Detailed Explanation
We can express the velocity 'u' in relation to the parameters such as 'R' (the radius of the pipe), 'u star' (the friction velocity), and 'nu' (the kinematic viscosity). Once we substitute these values into our discharge formula, we can compute the average velocity of the fluid across the flow section by integrating these relationships.
Examples & Analogies
Think of using a variable-speed blender to mix a drink. Depending on how fast you set it (friction velocity) and how viscous the ingredients are (kinematic viscosity), different parts of the drink get blended at different rates. We can use the same mathematical approach to find out how well mixed our drink becomes (average velocity) by accounting for all the speeds throughout the blending process.
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Create a free accountIf we integrate this equation number 28 and using the results in equation 29, that is, dividing by pi R square for obtaining V average, we can get.
Detailed Explanation
To calculate the average velocity, we must integrate the variable velocity function over the area of the pipe and then divide by the total cross-sectional area (πR²). This process smooths out the variations in velocity across the flow profile, allowing us to understand the average behavior of the fluid as it travels through the pipe.
Examples & Analogies
Imagine stirring a pot of soup: as you stir, some parts are mixed more thoroughly than others. To know how mixed the soup is overall, you would take a spoonful from different areas, averaging out the flavors to represent the entire pot's flavor. The integration in fluid dynamics is how we 'taste' different points in the flow to find a consistent 'flavor' (the average velocity).
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Key concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
- Turbulent flow:
Irregular fluid motion that typically occurs in pipes at high velocities.
- Logarithmic velocity profile:
A mathematical formulation that describes how velocity varies with distance from the wall in pipe flow.
- Roughness height (k):
Represents the surface roughness affecting flow characteristics in rough pipes.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
In a smooth pipe of diameter 10 cm, if the velocity 1 cm from the wall is known, we can calculate velocities at greater distances using the logarithmic velocity profile.
A practical application of turbulent flow equations can be seen when determining the friction factor in rough pipes, which affects the average velocity.
Memory aids
Imagine a river with smooth stones; the flow is gentle and slow. Now, picture a rocky path, where the water dances wildly, resembling turbulent flow in a pipe.
Remember T-R-A for turbulent flow: T for turbulent, R for rough surface, A for average velocity.
Flash Cards
Glossary
Turbulent Flow
A type of fluid flow characterized by chaotic changes in pressure and flow velocity.
Logarithmic Velocity Profile
A mathematical representation of the velocity distribution of fluid in a pipe, which often follows a logarithmic pattern.
Nikuradse's Experiment
Experiments in fluid mechanics that studied the effects of surface roughness on pipe flow.
Roughness Height (k)
A measure of the roughness of a pipe's interior surface, influencing the flow characteristics.