3. Indian Institute of Technology – Kharagpur

Welcome back to this last lecture of turbulent flow and laminar flows where we are going to talk about turbulent flow in smooth pipes. Last time in the last lecture we had seen what a smooth and rough bed is based on the Reynolds particle Reynolds number Re*. Now we are going to continue over the turbulent flow in smooth pipes.

This chunk introduces the topic of turbulent flow in smooth pipes, a key concept in hydraulic engineering. Turbulent flow occurs when the fluid moves chaotically with eddies and vortices, as opposed to laminar flow, where the fluid moves in parallel layers. The discussion references the Reynolds number, which is a dimensionless value used to predict flow patterns; values below 2000 typically indicate laminar flow, while values above indicate turbulent flow.

Imagine a river. When the water flows smoothly in a straight line (laminar), you can see the clear surface of the water. Now think about the same river after a storm: the water swirls and churns, creating turbulence and making the surface frothy. This is like turbulent flow, which can affect how we design pipes for carrying water.

So, if we refer to equation 18. 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. u is positive at some distance far away from the wall. Hence, u is 0 at some finite distance y prime from the wall.

This section discusses the behavior of fluid velocity at the boundaries (the wall) of the pipe. The velocity profile is often expressed with logarithmic functions, which help in understanding how the velocity changes from the wall to the center of the pipe. At the pipe wall (y=0), the theoretical velocity would be zero, but as you move away from the wall, the velocity increases until it reaches a maximum in the center.

Think of a sponge submerged in water. The water near the sponge's surface moves slowly due to friction (like the wall in the pipe), but the water further from the sponge can move swiftly in the center. The sponge's surface is like the wall of the pipe, and the water flow represents the velocity profile.

This is the velocity distribution for turbulent flow in a smooth pipe, where there is no irregularity. What about the turbulent flow? So, it is better to note down this equation.

The smooth pipe allows the turbulent flow to have a more predictable distribution of velocity compared to a rough surface. The velocity distribution can significantly affect how efficiently fluids are transported through pipelines. Understanding these concepts allows engineers to design more effective systems for fluid transport, ensuring both speed and efficiency.

Consider a smooth slide at a water park. The water slides down with ease and speed due to the smooth surface. In contrast, a rough surface would slow the water down, just like how rough pipes increase resistance and decrease efficiency in fluid flow.

Now, it is a good idea to solve one problem here, on this particular concept, and the solving the problem will give more understanding now.

This chunk introduces the importance of applying theoretical concepts to practical problems. The equations discussed help predict how quick and effective the flow will be under different conditions. By solving real-world problems, students can grasp how turbulent flow principles apply to actual engineering scenarios, such as pipeline design, environmental management, and fluid dynamics.

Think of it like exercising. You learn about your body’s limits through running, but truly understanding how your body operates during a marathon comes from experiencing it. Similarly, solving fluid flow problems helps students see how theory translates to practical engineering challenges.

So, this was what was asked. So, k is coming out to be 0.9399 centimeter. This is the answer and how to solve? We have already done that in the sheet.

In this conclusion, there is a practical result derived from the equations and methods discussed earlier. The k value represents a coefficient relevant to the roughness of the pipe, which is crucial for calculating flow rates. This process reinforces the concept that theoretical mathematics and physics can yield concrete, measurable outcomes.

Consider baking a cake: the recipe (the theory) gives you expected outcomes based on ingredients' measurements. When you bake using the measurements and follow the recipe, you produce a cake (the actual result). Similarly, applying fluid dynamics equations gives engineers the information they need to design effective systems.

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