1. Introduction to Turbulent Flow

So, if we use equation 14 in equation 15, what was equation 14? l m was kappa into y, this was what we said in equation number 14. So, we can simply write...can be assumed to be a constant. So, at the wall the shear stress is assumed to be constant and equal to tau not.

To understand turbulent flow, it's important to recognize how shear stress behaves in these conditions. In turbulent flow, shear stress at the wall (tau not) is considered constant, which allows simplifications in calculations. Here, the symbol 'kappa' relates to the mixing length, which quantifies how much mixing occurs at different distances from the pipe wall. This is crucial in deriving other equations relevant to turbulent flow dynamics.

Think of a river with varying flow speeds. Near the riverbed (like the wall of a pipe), the water moves much slower due to friction (analogous to shear stress), while the top layers of water flow more freely. The idea of constant shear stress helps simplify how we understand the interaction between these layers.

And therefore, what we can say, if we substitute tau is equal to tau not in equation 16, we can obtain...under root tau / rho is rho u *.

By substituting shear stress (tau) with its constant value at the wall (tau not) in our equations, we derive a relationship that connects the velocity gradient (du/dy) to shear velocity (u*). This relationship is fundamental in fluid dynamics as it allows us to relate shear stress to flow behavior. The term 'under root tau / rho' relates to the concept of shear velocity, which helps in determining the flow characteristics of turbulent fluid.

Imagine a sliding box on a surface. The speed at which the box accelerates depends on the force applied (analogous to shear stress) and the weight of the box (analogous to density). By understanding this relationship, we can predict how quickly the box will slide, just like shear velocity helps us predict the flow rate in turbulent conditions.

If you integrate the equation number 17, so, what we can get is...C will be u max minus u * / kappa ln R.

After deriving the equations, the next step is to integrate them to find the velocity profile of the turbulent flow. A key boundary condition used is that at the maximum radius (R) of the pipe, the velocity is at its maximum (u max). By substituting this boundary condition into our equations, we arrive at a formula that allows us to express the velocity at any point within the pipe's radius based on its distance from the wall.

Think of this as mapping a hill. At the peak of the hill (u max), you can determine the slope (rate of change) by observing how steep it is down to the base (integration). The boundary conditions help us establish the highest point, making it easier to analyze the rest of the terrain.

So, we can put it in form of log...this is just simple, you know, manipulation of these terms here.

The result of our calculations leads us to a logarithmic equation that describes the velocity defect law. This law helps us understand how the velocity of turbulent flow decreases as we move away from the wall of a pipe. The logarithmic function portrays a more accurate representation of flow compared to the simpler parabolic models used for laminar flow.

Think about how the wind speed drops as you move closer to a building. Just like the air slows down because of the building's surface, the velocity defect law explains how the flow slows down near the pipe walls.

So, now the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case...So, this is the true picture.

Comparing turbulent flow to laminar flow, we observe that turbulent flow exhibits a fuller velocity profile. In laminar flow, the velocity distribution is parabolic, meaning flow is smooth and well-organized. In contrast, turbulent flow has varying speeds and produces a more complex pattern, resulting in enhanced mixing and higher energy losses.

Consider a calm lake (laminar flow) where you can see the smooth surface, versus a fast-flowing river (turbulent flow) where waves and currents create a chaotic scene. This illustrates how turbulent flow leads to a richer variety of flow patterns.

So, now we are going to talk about that. Turbulent flow along a wall consists of 4 regions...in the turbulent layer, the turbulent effects dominate over these viscous effects.

Turbulent flow can be broken down into four distinct regions: the viscous sublayer, buffer layer, overlap layer, and turbulent layer. Each zone has a different behavior and significance. For instance, in the viscous sublayer, viscosity effects dominate, presenting almost linear velocity profiles. As we move toward the turbulent layer, turbulence plays a more significant role in how fluid moves.

Think of layers in a cake. The bottom layers (viscous sublayer) are denser and more structured, while the top layers (turbulent layer) are whippy and can contain more air, representing the chaotic nature of turbulence.

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