4.5.1. Discharge Q and equation derivations

Hence, u is 0 at some finite distance y prime from the wall, that is, y at y prime is equal to 0. Hence, from equation 18, we can say that at distance y prime C will be minus u star by Kappa ln y prime. Because if we put y is equal to 0 it is going to be infinity. So, this logarithmic profile does not fit this. Therefore, we assume that u is 0 at some finite distance y dash y prime from the wall.

In fluid mechanics, particularly when studying flow near a boundary (like a pipe wall), we find that the speed of fluid (velocity) is not uniform. At the wall, the velocity is 0 due to the no-slip condition, where the fluid 'sticks' to the surface. As we move away from the wall (to a distance y prime), the velocity increases. The equation indicates that this pattern resembles a logarithmic curve, showing that at a specific point, the velocity becomes a function of the distance from the wall and a constant that scales with velocity, known as the von Kármán constant (Kappa). However, at an infinitely close distance to the wall (y=0), the logarithm function approaches negative infinity, indicating we cannot just plug in 0 into our equations.

Think of a thick blanket lying on a table. At the point where it touches the table (the wall), it doesn’t move; this represents the no-slip condition (velocity is zero). However, if you lift the blanket just a little off the table, it begins to ripple and creates motion. The distance at which it starts to ripple (the velocity begins to increase) is similar to our y prime, showing how the velocity changes with distance from the boundary.

Now, this equation number 22, can be expressed in terms of common logarithm as, so, what we have done? We have done, converted ln into log, because the previous equation was in terms of natural log.

In fluid dynamics, equations can be expressed using different types of logarithms depending on the context. Here, the natural logarithm (ln) used in earlier equations has been converted into a base-10 logarithm. This is often done for ease of calculation or to conform to standardized equations that some engineers may prefer to use. This step ensures that our turbulence equations will be widely applicable and easier to handle in calculations.

Imagine you are trying to explain how tall a tree is, but you have a ruler that measures in inches and your friend uses a meter stick. To make your communication clearer, you convert your measurements into meters, which your friend understands better. Similarly, converting from natural log to common logarithm makes our calculations and communication about fluid flow much more straightforward and collaborative.

For rough pipes Nikuradse obtained the value of y prime as k/30. This is obtained by Nikuradse and if you substitute this y prime in equation number 23 this one.

The roughness of a pipe surface affects how the fluid flows through it. Nikuradse's work established that the distance at which the velocity profile behaves in a specific manner (y prime) for rough surfaces can be approximated as k divided by 30. Here, 'k' represents the roughness height of the pipe surface. This finding implies that the velocity profile will vary depending on the texture of the pipe surface, emphasizing the importance of accounting for surface roughness in calculations related to turbulent flow.

Consider sliding down a slide at a playground. A smooth slide allows you to go down quickly, while a slide made of rough material will slow you down significantly. The roughness of the slide is akin to the roughness of a pipe; it impacts how smoothly and quickly the fluid flows through.

Given is, D is equal to 10 centimeters or 10 into 10 to the power minus 2 meters or 0.1 meter. It is given, u at y is equal to 4 centimeters or 4 is equal to 1.4 times u at y is equal to 1 and the surface is rough.

In this example, we are provided with specific values, such as the diameter of the pipe and the velocity of fluid at various points from the wall. The concept here is to determine the average height of surface roughness through calculations based on fluid velocities at specific distances from the wall. This example walks through the necessary steps and substitute values into established equations to get to the result, showcasing the practicality of the previously derived equations in real-world scenarios.

Imagine you're trying to assess how steep a hill is by measuring the slope at different points with a protractor. Here, each measurement you take at a different distance is like measuring the fluid's velocity at those specified distances in the pipe. By comparing these measurements through calculations, you can understand the overall steepness (or roughness) of the hill (or pipe).

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