3. Turbulent Velocity Profile

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. 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.

In this chunk, we are introduced to the equations that define the turbulent velocity profile. Equation 14 sets up the relationship between the mixing length (l) and the distance from the wall (y) through the constant kappa. We then see that for small values of y, the shear stress (tau) can be assumed to equal the constant shear stress at the pipe wall (tau not). This is crucial because it allows us to simplify the equations involved in the analysis of turbulent flow.

Imagine you are observing traffic on a highway. Near the walls of the highway (the walls of our pipe), you see cars moving slowly due to friction with the ground, similar to how the fluid interacts with the wall. As you move towards the center of the highway (increasing y), the traffic flows more freely and quickly, analogous to how the shear stress changes from the wall (tau not) to the center of the flow.

If we substitute tau is equal to tau not in equation 16, we can obtain du / dy, this quantity actually tau not can be written as rho. So, but the catch here is, what is the catch? We have considered small value of y. So, du / dy can be written as, 1 / kappa y and under root tau / rho is rho u *. So, it becomes and this u * under root tau not / rho is the sheer velocity and this has the dimension of velocity.

Here we substitute tau not into equation 16 to derive another important aspect, du/dy, which describes how velocity changes with respect to distance from the wall. We express tau not in terms of density (rho), transforming our equation. The term u star (u*) is introduced here, representing the shear velocity, which measures how rapidly the velocity of the fluid changes. This portion of the analysis is key in understanding the mechanics of turbulent flows.

Think of a crowded dance floor. Near the edges (where the dancers slow down due to the walls), the velocities of those dancing are relatively low. As they move towards the center where the dancing is lively, the speed increases. The concept of shear velocity (u*) is similar; it captures how rapidly the speed increases as you move away from the wall.

If you integrate the equation number 17, so, what we can get is, simple integration, it will get. 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.

In this step, we perform a simple integration of the derived equation to find a more complete expression for velocity. This results in establishing boundary conditions: particularly, the velocity at the wall (y = R, the radius of the pipe) where it reaches its maximum (u_max). Setting the boundary conditions is essential for getting meaningful solutions in fluid dynamics.

Imagine throwing a ball vertically into the air. As the ball moves upward, it has a maximum height it can reach. Similarly, as fluid flows through the pipe, it has a maximum velocity at the center of the pipe. Establishing where this maximum occurs helps predict how the fluid flows within the pipe.

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.

In this part, we determine the constant C using our previously established boundary conditions, relating u_max, u*, and kappa (a constant related to roughness). Upon substituting kappa with a specific value (0.4), we can finally arrive at the logarithmic velocity profile equation (equation number 20). This logarithmic relationship signifies how turbulence affects flow characteristics, contrasting with laminar flow profiles.

Consider how a river flows over rocks. Close to the riverbed (like near the walls of the pipe), the water flows slower due to friction while it speeds up in the center. The logarithmic profile is like a map describing these speed changes as you move horizontally across the river at different depths.

u max minus u is called the velocity defect or velocity defect law, this is velocity defect law. This is just simple, you know, manipulation of these terms here.

The concept of velocity defect law arises from the earlier equations, where we note the difference between the maximum velocity and the actual velocity at any point u. This difference (u_max - u) signifies how much the velocity is 'defective' or reduced as fluid flows through a pipe due to turbulence and friction. Understanding this law is critical when analyzing fluid flow dynamics.

Think of going downhill on a bike. Ideally, you can reach a certain top speed without obstacles (u_max). However, if you encounter bumps (similar to turbulence in a fluid), your speed decreases (u), creating a 'defect' or difference that we can measure.

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

This final chunk summarizes the key differences between turbulent and laminar flow. It emphasizes how turbulent flow creates a fuller velocity profile compared to the parabolic shape seen in laminar conditions. This understanding of the velocity profile differences is crucial for engineers and scientists dealing with fluid dynamics.

Just as in a busy city intersection, where cars flow in complex patterns that build momentum and energy (turbulent), versus cars moving smoothly in a straight line down a quiet road (laminar), this analogy illustrates how turbulence can enhance mixing and energy transfer in a fluid.

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