Logarithmic profile and derivations

Today, we'll continue exploring turbulent flow, starting with the velocity profile in smooth pipes. Can anyone tell me what we mean by a 'smooth pipe'?

Isn't it a pipe with a very fine surface texture that minimizes resistance?

Exactly! In smooth pipes, flow velocity can be described using the logarithmic profile. The equation we refer to is directly related to the Reynolds number, which helps determine whether the flow is laminar or turbulent.

What's the main equation we’re focusing on?

We have u = (u_star / Kappa) * ln(y_prime), where u is the velocity at a distance y_prime from the wall. Remember, Kappa is approximately 0.4 for turbulent flow.

What happens when y is equal to zero in this equation?

Good question! If y is zero, the equation suggests that velocity becomes infinitely negative, which shows us that we cannot approach the wall directly. Instead, we introduce a finite distance where we measure y. Now, can anyone tell me the significance of C in our equation?

Is it the constant that helps to adjust values in our logarithmic profile?

Precisely! C adjusts our logarithmic equation to fit the conditions. Understanding this adjustment is vital.

Moving on, let's discuss rough pipes. How do you think the turbulent flow profile changes when we consider a rough surface?

I assume the roughness would create more turbulence, changing the velocities?

Exactly! Nikuradse’s experiments tell us that we can express the velocity distribution in a similar logarithmic form, but we must use different parameters. Who remembers the roughness height in this context?

Is it represented as k for roughness?

Yes, k represents the average height of roughness. The equation derived then for rough pipes generally looks similar but is adjusted due to this k factor.

How does Nikuradse's experiment influence our calculations?

His work allows us to determine the equivalent roughness from velocity profiles, which is crucial for practical engineering applications. If roughness increases, the flow characteristics also change—leading to higher energy losses.

Now let’s solve a problem! Given a pipe with a diameter of 10 centimeters, if the velocity at 4 centimeters from the wall is 40% greater than at 1 centimeter, how can we find the average height of roughness?

We should use the equations for both positions and set them up based on their velocities.

Exactly! Start by expressing the velocities based on our established equations. What do we derive?

We can establish a relationship using u values at the two distances. Wouldn’t we also consider k for roughness?

Yes, and after substituting our values, set the equations equal, simplifying until we can isolate k. This method provides a robust way to find roughness height.

I see! So, manipulating logarithmic expressions is key to solving for k.

Absolutely! Just a reminder, practice this more for better retention, and make sure to refer back to the equations often. Now, let’s summarize today's learnings.

Can you recap the importance of these equations?

Sure! The logarithmic equations provide essential insights into the flow characteristics in both smooth and rough pipes, and understanding how to manipulate them is key in engineering applications.