11.3.2. Velocity Distributions in Laminar and Turbulent Flows

Today we are looking at what will be the wall stress, shear stress at this point, also this point, stream function, vorticity, velocity potential and the average velocity if known the velocity field.

In this chunk, we're introduced to several important concepts in fluid mechanics related to how fluids behave when they flow. Specifically, we focus on wall stress (force exerted by the fluid on the wall), shear stress (resistance of the fluid to flow), stream functions (which help describe flow patterns), and vorticity (the measure of rotation in the fluid). Understanding these concepts is crucial for analyzing both laminar (smooth) and turbulent (chaotic) flow.

Think of a river as it flows past a bank. The interaction of the water with the bank involves wall stress, and depending on how fast the water flows (like a gentle stream vs. a rapid current), the characteristics such as shear stress and vorticity change. This helps us understand how rivers shape their banks over time.

In the last class, we estimated this velocity field from Navier-Stokes equations that we assume it vw is a 0 okay, neglecting the gravity force components. We get this u component...

This segment explains how the velocity field is computed using the Navier-Stokes equations, which describe how fluid substances move. By neglecting certain forces (like gravity) and focusing on specific components of velocity (like 'u'), we can derive an expression for velocity that takes into account changes in pressure.

Imagine water flowing through a narrow pipe: by applying the rules of flow (like the Navier-Stokes equations), you can predict how fast the water will go at different points, even if at first glance it seems chaotic.

Now if you look it I am just substituting wall shear stress using the Newton's laws of viscosity...

In this section, the speaker discusses calculating wall shear stress using Newton's laws of viscosity. This involves determining how much the fluid's velocity changes as it approaches the wall, which helps understand the frictional forces at work in fluid flow.

Consider spreading butter on bread: the way the butter moves and slightly changes shape due to the pressure of the knife is similar to how fluid layers slide past each other due to shear stress.

Since it is a 2-dimensional flow, we can get the stream functions...

Stream functions simplify the analysis of flow by providing a way to visualize and calculate fluid flow patterns, especially in 2-dimensional scenarios. Vorticity pertains to the local spinning motion of the fluid, and understanding both these concepts is key to grasping fluid behavior in various flows.

Imagine looking at a spinning top; the speed at which it spins represents vorticity. Likewise, stream functions can be seen as pathways that show us the direction the top would be thrown if we pushed it in various directions.

We cannot get the velocity potential functions because to get a velocity potential function flow should be irrotational.

Velocity potential functions can only be defined for irrotational flows, meaning that there is no rotation in the fluid's movement. This section explains that, because the flows being discussed are not irrotational (due to vorticity), we cannot derive a velocity potential function.

Think of calm water in a pond. If you stir the water, it becomes rotational (swirling). The calm state represents irrotational flow, where a velocity potential could exist. Once stirred, that potential state is lost.

You have to compute the average velocity... I will get it integrations over this.

Here, the average velocity is computed by integrating flow velocities over a cross-sectional area, which provides a comprehensive view of flow behavior across different sections of the fluid body.

Think about averaging your speed over a long road trip. Some parts of the trip you may drive faster, while in others, you might slow down. Calculating your average speed gives a better idea of your overall journey than simply looking at one aspect.

We try to get it what will be the average velocity, what will be wall shear stress, what will be the stream functions...

In this concluding part, the speaker summarizes how all the discussed concepts come together to provide a holistic view of fluid motion. Each principle plays a role in understanding the complete flow dynamics, whether it is shear stress, stream functions, or average velocity.

Imagine being a chef making a sauce. Each ingredient (like flavors from spices, thickness from butter, and tang from tomatoes) impacts the final dish. Similarly, understanding each fluid mechanic concept helps us see the complete 'dish' of fluid dynamics.

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