Today, we're diving into turbulent flow in pipes. Let's start by discussing the relationship between the average velocity and local velocity. Can someone tell me why this relationship is important?

Exactly! In turbulent pipe flow, we use the equation: $$ u - V_{average} = 5.75 \log_{10}\frac{y}{R} + 3.75 $$ where $u$ is the local velocity, $V_{average}$ is the average velocity, $y$ is the distance from the wall, and $R$ is the radius of the pipe. Remember this formula; you can derive it using basic logarithmic profiles. Any thoughts on what this means?

Right! And it's significant that you'll observe the same relationship for both smooth and rough pipes. Let's remember this with the acronym **VARY**: V- velocity at any point, A- average velocity, R- radius, and Y- distance from the wall.

Glad you think so! For our summary: we derive that $ u - V_{average} $ can help predict the flow characteristics in turbulent conditions. Let's move on to rough pipes in our next session.

Moving on to rough pipes, the equation we previously discussed can be modified. Can anyone tell me what that might involve?

Exactly! It's essentially similar in format: $$ u - V_{average} = 5.75 \log_{10} \frac{y}{R} + 3.75 $$ still holds, showing that the local-to-average velocity relationship remains consistent across different pipe types. Isn't that fascinating?

That's a common misconception! It mainly affects the total resistance but not the relationship we derived. We often see this as a principle of turbulence. A mnemonic for this could be **SAME**: S- smooth, A- average, M- multiple types, E- equation remains.

Great! Let's summarize: the differences in velocities for smooth and rough pipes yield similar fundamental equations, a key insight in turbulent flow dynamics.

Next, we discuss the power law velocity profile. Someone take a guess on what that might represent?

Correct! It is given by $ \frac{u}{u_{max}} = \left(\frac{y}{R}\right)^{\frac{1}{n}} $, where $u_{max}$ is the maximum velocity. The power value $n$ depends on the Reynolds number. For $n=7$, we get a famous one-seventh power law profile. Why do we use $n$?

Exactly! It's crucial for evaluating how velocity changes within the pipe. As a memory aid, think of **PULSE**: P- Power, U- Useful in flows, L- Level of turbulence, S- Slopes for calculations, E- Expressing ratios.

To summarize: the power law model is applicable in varying turbulence, aiding analytical predictions but has limitations with wall shear stress due to infinite gradients. It's an essential concept in fluid mechanics.

Let's apply what we've learned by calculating average velocity using a specific turbulent profile provided: $$ u(r) = u_{max}(1 - \frac{r}{R})^{\frac{1}{7}} $$. How should we approach this?

Yes! The integral will be over the entire area of the pipe. Can someone help set up that integral?

Correct. Remember the representation: average velocity $V_{bar} = \frac{1}{\pi R^2} \int_0^R u_{max} \left(1 - \frac{r}{R}\right)^{\frac{1}{7}} 2\pi r \, dr$. What values will cancel out in our calculations?

Yes! Great job! After simplifications, we can find that $V_{bar} = 0.816 u_{max}$. Keep this as a reference for calculating average velocity in other turbulent flows.

Excellent! So to recap: we set up integration for velocity profiles demonstrating how to derive average flow velocity. This method will be vital in real-world applications!