2. Department of Civil Engineering

Welcome back to this last lecture of turbulent flow and laminar flows where we are going to talk about turbulent flow in smooth pipes.

This introduction sets the stage for discussing turbulent flow in smooth pipes. It indicates that the lecture will build upon previous topics and concepts related to fluid dynamics, specifically focusing on turbulent flow, which is characterized by chaotic changes in pressure and flow velocity.

Think of a river; when the water flows smoothly, it resembles laminar flow. But when it hits rocks, creating rapids, it turns into turbulent flow. This lecture helps us understand the physics behind how fluid behaves in both calm and chaotic states.

From the above equation, the velocity at the wall u at y is equal to 0 will be minus infinity, correct. If we put y is equal to 0. So, if we put ln 0, it will be minus infinity.

This section emphasizes a critical point in fluid mechanics: the velocity of fluid at the very wall of a pipe (where y = 0) is theoretically zero, as consequences from Logarithmic Velocity Profile equations. It shows that as you move closer to a wall, the velocity decreases until it hits zero due to molecular adhesion, known as the 'no-slip condition'.

Imagine two cars driving on a highway; one stops right at the edge of the road (like fluid at the wall), while the other is further back on the road. The closer a car gets to the road's edge, the slower it can go due to congestion at the barrier, similar to how fluid slows at the wall.

Hence, from equation 18, we say that at distance y prime C will be minus u star by Kappa ln y prime. Therefore, we obtain, if we use this in equation number 18 and substitute Kappa is equal to 0.4.

The derivation discusses how the velocity distribution of turbulent flow can be expressed mathematically. Substituting constants into the logarithmic equations provides a more concrete understanding of how velocity varies within the fluid. The 'C' comes from boundary conditions reflecting the effects of viscosity and distance from the wall on velocity.

Think of this equation as a temperature chart on the road. At the center of the road (far from edges), the temperature might be a constant high, while along the edges, it gradually gets cooler. Similarly, fluid speed is fastest at the center of the flow and slows down as it approaches the boundaries.

For rough pipes Nikuradse obtained the value of y prime as k/30. This is obtained by Nikuradse.

The discussion shifts focus to turbulent flow in rough pipes, where the roughness affects the velocity distribution. Nikuradse's research gives us a comparative model to analyze how variation in pipe surface influences fluid dynamics. The introduced variables help in deriving equations applicable specifically to rough surfaces.

Imagine riding a bike on a smooth road compared to a gravel path. On the smooth road, you maintain speed easily; however, on the rough path, your speed drops due to increased friction. This analogy shows how rough pipes can impede fluid flow, resembling how rough terrain affects a cyclist.

Now, it is a good idea to solve one problem here, on this particular concept...

The framed problem engages students to actively apply the learned concepts. Students are asked to calculate the roughness height of a pipe based on given conditions, reinforcing the theoretical knowledge presented in previous sections by transitioning it to practical applications.

Solving this problem can be compared to cooking: following a recipe (analogous to the problem statement) helps you create a dish while understanding the science behind it. Just like precision in measurements can result in the perfect cake, accuracy in applying these formulas will refine our understanding in fluid mechanics.

--