1.3. Integration with Boundary Conditions

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. So, at the wall the shear stress is assumed to be constant and equal to tau not.

In this chunk, we're discussing conditions where the shear stress (denoted as tau) can be equated to a constant value at the wall of the pipe, referred to as tau not. This assumption holds true particularly when the vertical distance (y) from the wall is very small. The shear stress is a measure of the internal friction in the fluid, which affects how it flows against the walls of the pipe.

Imagine trying to slide a book across a table. The friction you feel represents shear stress. Now, if you gently press the book against the table (like being very close to the wall), the friction becomes a constant force until you push harder. So, near the wall of a pipe, the shear stress remains steady, making calculations easier for engineers.

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.

In this part, the relationship between the shear rate (du/dy) and the shear velocity (u*) is established. The equation indicates that when we replace tau with tau not, we can derive an expression involving kappa (a constant) and the distance y. The shear velocity represents how fast the fluid layers slip past one another, crucial for understanding turbulent flow in pipes.

Consider a river flowing over a series of rocks. The top layer of water moves more quickly than the deeper layers. The difference in speed between these layers is like shear velocity. As the top layer moves swiftly, if we picture measuring how that layer interacts with the rocks (the tau), we can understand how smoothly or turbulently the water flows.

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. That is what we have seen at the center line of the pipe the velocity is going to be the maximum.

This chunk explains how boundary conditions are used in deriving formulas for fluid flow. The boundary condition states that at the radius of the pipe (R), the fluid velocity (u) reaches its maximum value (u max). This is crucial for accurately modeling how the fluid behaves within the pipe and allows for solving the equations derived earlier.

Imagine a water slide. When someone sits at the top, that’s the point of maximum speed (u max) right before they go down. The moment you reach the slide (the edge), you begin at maximum speed compared to being near the ground. Similar to how water reaches its fastest flow at the radius of the pipe.

So, laminar flow was something like this, a parabolic profile. Here, a profile is little different u is u maximum plus a logarithmic profile.

This chunk contrasts laminar flow with turbulent flow conditions. While laminar flow has a smooth, parabolic velocity profile, turbulent flow results in a logarithmic velocity profile, indicating that flow is irregular and chaotic. Understanding this difference is vital in fluid mechanics for predicting how fluids behave under various conditions.

Think of a calm pond (laminar flow) where you see smooth ripples. Now consider a river during a heavy storm (turbulent flow) where the water splashes in all directions. The transition from smooth to chaotic can be visually represented by changing from a nice curve to a jagged line, like how we see velocity profiles differ.

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, this is velocity defect law.

The velocity defect law describes the difference between the maximum fluid velocity and the actual velocity at various points within the pipe. The equation derived here further illustrates how the logarithmic nature of the flow contributes to the differences in velocity, emphasized by the logarithmic scale. This is essential for engineers to predict and manage fluid behavior in pipelines.

Imagine a racetrack. The fastest car (u max) is at the front, while slower cars (u) are behind. The difference in speed is the velocity defect. If we plotted their positions, the differences would paint a visually sharp picture of how speed varies, much like plotting fluid speeds in a pipe.

So, now the turbulent velocity profile is much fuller compared to the parabolic profile of laminar flow case. ... So, as I told you in the last slide, there are different layers, different layers in turbulent flow.

The turbulent velocity profile is more complex than that of laminar flow. The explanation introduces various layers within turbulent flow, highlighting the structure of how turbulence affects velocity at different distances from the wall. Understanding these layers helps in predicting flow characteristics and improving pipe design.

Think about layers in a cake. The frosting on top (the turbulent layer) is thick and varied unlike the smooth cake underneath (the laminar flow). Each layer plays a unique role in overall texture and flavor, just as each layer of fluid flow influences the movement through a pipe.

Turbulent flow along a wall consists of 4 regions. Viscous sublayer, this layer is thin layer next to the wall...

In turbulent flow, four distinct regions are identified: the viscous sublayer (where viscous effects dominate), the buffer layer (both turbulent and viscous effects are present), the overlap layer (turbulent mostly, with some viscous influence), and the turbulent layer (where turbulent effects are predominant). These regions are important for understanding how flow behaves near surfaces.

Consider a crowd of people at a concert. The people closest to the stage (viscous sublayer) are more controlled and orderly, while those farther away (turbulent layer) are moving chaotically with lots of pushing. Just like in flow, different regions yield differing behaviors depending on the crowd's proximity to the stage.

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