1.2. Rough Pipes Equation

So, what is this? This is the average velocity divided by the frictional velocity for the turbulent pipe flow case.

The Rough Pipes Equation is primarily concerned with understanding how flow behaves in a turbulent state within a rough pipe. Specifically, it examines the relationship between the average flow velocity (V average) and the frictional velocity (u star). This ratio helps in analyzing how roughness of the pipe surface impacts flow attributes, particularly in turbulent flow conditions.

Think of a rough pipe like a rocky riverbed where water flows. The more rocks there are, the more the water has to 'fight' against them to move smoothly. Just like how the water flow varies with the number and size of rocks, the velocity in a rough pipe is affected by its interior surface condition.

Now, the difference of the velocity at any point and the average velocity for smooth pipes. For smooth pipes, we have seen that the u by u star we had, we came, we derived this equation, correct.

This segment discusses how we can derive the equation that describes the velocity difference between a point in the pipe and the average velocity. When dealing with smooth pipes, previous equations can be referenced, allowing us to find a simplified relationship through subtraction. The expression lays the groundwork for understanding how to derive behaviors in rough pipes using similar steps.

Imagine measuring the speed of a car (representing fluid at a point) as it drives along a perfectly smooth road (the smooth pipe). Comparing its speed to the overall speed limit (average velocity) shows how quickly that specific car is moving relative to the entire flow of traffic.

So, we do this equation, this minus this, this minus this. So, it will be u star is common, so, it will be u minus V average by u star is equal to 5.75 log to the base 10 y by R plus 3.75.

At this point, we derive the main equation relating the velocity difference to the logarithm of the distances and parameters involved in the flow through the pipe. This equation is significant because it encapsulates how differences in velocity at various points within the turbulent flow are calculated in terms of logarithmic relationships, which are essential for engineering applications.

Consider a marathon race. Just like how the distance from the starting line influences how fast certain runners can go, this equation shows how the position inside the pipe influences the differences in velocity, with each position relating back to a logarithmically-based calculation.

Now, we did it for smooth pipes. Now, for rough pipes, we have an equation and we also obtained V average by u star.

After establishing the fundamentals for smooth pipes, we shift focus to rough pipes, which also requires understanding how V average relates to the frictional velocity. The derivation process follows a similar logic as before, which underlines the consistency in approach when tackling different types of pipe surfaces.

When transitioning from a smooth road to a gravel path, the way a car accelerates changes, much like how we change the analysis of flow from smooth to rough pipes. The principles are the same, but the context shifts based on surface conditions.

Is there any observation? If you see for either for smooth or for rough this difference came out to be the same. So, the observation is the difference of velocity at any point and average velocity is same for both smooth and rough pipes. This is important to know that.

An interesting finding is that despite the differences between smooth and rough pipes, the velocity difference remains consistent. This consistency is crucial for engineers, as it suggests that the same modeling and equations can broadly apply to both scenarios; it's a fundamental principle that simplifies design and analysis.

Consider how students score on exams; whether they sit for a difficult test or an easy one, the difference between their score and the average might still hold constant due to their preparation. Similarly, even with varying pipe conditions, the pattern of velocity difference remains a fundamental constant.

Now, about the power law velocity profile, a little bit on that. So, the power law velocity profile for smooth pipes can be expressed as, it is, u by u max can be written as y by R to the power 1 by n.

In this part, we introduce the power law velocity profile, which provides another way to describe the velocity distribution in smooth pipes. The equation shows that the velocity at any radius can be expressed with respect to the maximum velocity and depends on the parameter n, which varies with the Reynolds number. This relationship is critical for understanding how velocity profiles behave in different flow conditions.

Think about water flowing through a funnel; the speed of the water changes based on its position in the funnel. Similarly, the power law velocity profile predicts how quickly fluid moves from the center of the pipe to its edges based on the surface conditions.

Power law velocity profiles cannot give 0 slope at the pipe center. So, power law of velocity profiles also cannot calculate the wall shear stress. Why? Because the power law profiles gives a velocity gradient of infinity at the walls.

This section indicates limitations of the power law velocity profile, including its inability to predict certain behaviors accurately, such as the wall shear stress. The profiled rates of change at the pipe's edge approach infinity, leading to mathematical inconsistencies that must be factored into practical applications.

Using the funnel example again, if you were to look at how fast the water flows directly at the edge, it might go too quickly to measure accurately. The challenges faced in exact calculations with the power law profiles reflect similar difficulties in this context.

Now, we are going to solve one of the problems. So, the velocity profile for incompressible turbulent fluid in a pipe of radius R is given by u of r as u max into 1 minus r by R to the power 1 by 7.

In this final section, we apply our understanding of flow profiles to a specific problem that involves calculating average velocity in a pipe. By using the given velocity profile equation, students are walked through the steps of how to derive the average velocity using integral calculus, reinforcing the practical application of theoretical concepts learned earlier.

Imagine a group of friends running through a park, with their speed varying based on where they are in relation to the path. By using the provided speed profile equation, we can find out how fast they, on average, are moving throughout the park, much in the way engineers use similar equations to find average flow in turbines.

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