1.1. Use of Equations in Turbulent Flow Analysis

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 or we can simply write and this is equation number 16, very simple.

In this part, the speaker refers to two equations, where Equation 14 involves a relationship represented as l m = kappa × y. Here, 'kappa' is a constant that relates to the turbulent flow characteristics, and 'y' is a variable that can denote the distance from the pipe wall. By substituting Equation 14 into Equation 15, the speaker derives a new equation, Equation 16. This demonstrates the process of substituting known equations to simplify the math related to turbulent flow analysis.

Think of it like cooking: you have a recipe (Equation 14) that requires a specific ingredient (kappa). You then find another recipe (Equation 15) that can use that ingredient—by adding it in, you can create a whole new dish (Equation 16).

So, for small values of y it can be assumed. So, if the y is very small we can assume that tau is equal to tau not, where tau not is the shear stress at the pipe wall and can be assumed to be a constant.

The statement highlights an important assumption in turbulent flow analysis: when 'y' (distance from the wall) is very small, the shear stress (represented as tau) at that point can effectively be considered equal to the shear stress at the wall, denoted as tau not. This simplification allows for easier calculations and is particularly useful when working close to the pipe wall.

Imagine you're at the edge of a swimming pool. The water right at the edge (or wall) has less disturbance, hence the pressure (like shear stress) is more constant compared to the center of the pool where waves can form. This helps us simplify the pressure calculations at the edge.

Then using the boundary conditions, what are the boundary conditions? So, u at y is equal to R, where R is the radius of the pipe. We will get, u is equal to u max. ... C will be u max minus u * / kappa ln R and if we substitute this as C, then we can get equation or or if when we substitute kappa as 0.4, we can get and this is equation number 20.

The analysis involves boundary conditions essential for determining velocity profiles in turbulent flows. It establishes that at the pipe's wall ('y' equals R), the velocity reaches its maximum (u max). By substituting this condition into the equations and manipulating the terms, a constant 'C' is derived, further leading to a logarithmic velocity profile described in Equation 20. This profile showcases how velocity changes from the center to the wall of the pipe.

Think of it like measuring a slope. If you take a height measurement (like u max) at the top of a hill (the center of the pipe), you can apply that to understand the slope down to the base (the wall), calculating various points along the way to see how steep it gets (the velocity profile).

This is ln. So, we can put it in form of log. This is simple manipulation, we can get u max minus u / u * is equal to 5.75 log to the base 10 R / y. u max minus u is called the velocity defect or velocity defect law...

Here, the equation is manipulated to express the relationship between the maximum velocity, actual velocity, and shear velocity in a logarithmic form. The term 'velocity defect' refers to the difference between the maximum velocity (u max) and the actual velocity (u) at a certain distance from the wall. This law helps predict how turbulent flow behaves in terms of velocity at different points in the pipe.

Think of a car speeding down a road (u max) but slowing down due to traffic (u). The difference in speed represents the 'velocity defect.' It helps us understand how effectively the car can travel through congested areas (the turbulent flow in pipes).

So, now the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. ... Turbulent flow along a wall consists of 4 regions: viscous sublayer, buffer layer, overlap layer, and turbulent layer.

The turbulent flow profile differs significantly from laminar flow, which typically shows a parabolic curve. In turbulent flow, the velocity profile is fuller and complex due to interactions at various layers. These layers include the viscous sublayer (where viscous effects dominate), the buffer layer, the overlap layer (where turbulent effects begin to take precedence), and the turbulent layer (where turbulence dominates). Understanding these layers is crucial for analyzing flow behavior.

Imagine a crowd at a concert. At the very front (viscous sublayer), everyone is orderly and calm. As you move back, the crowd begins to mix and sway (buffer, overlap layers), until at the back, it's chaos with people moving in many directions (turbulent layer). Understanding these distinctions helps manage crowd safety and flow.

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