11.3.3. Jet Flow Dynamics

We are looking at what will be the wall stress, shear stress at this point, stream function, vorticity, velocity potential and the average velocity if known the velocity field. So in the last class we estimate this velocity field from Navier-Stokes equations that we assume it vw is a 0 okay, neglecting the gravity force components.

In this chunk, we focus on the concepts of wall stress and shear stress within fluid dynamics. Wall stress is the force exerted by the fluid on the boundary (or wall), while shear stress relates to how the fluid moves parallel to that boundary. The Navier-Stokes equations help us model these phenomena using assumptions, like zero gravity influence. These equations allow us to estimate the velocity field, which is crucial for determining various stresses, vorticity, and flow patterns in the fluid.

Imagine a swimmer pushing through water. The water exerts a force against the swimmer's hands and body, which is similar to wall stress. The way the swimmer's hands slice through the water creates a shear force as the water moves past, illustrating shear stress in action.

We get this u component that is what we did it last class minus dp by dx okay that is what dp by dx is equal to h square by 2 mu types of 1 minus y square by h square.

This segment explains how we calculate the velocity field in a jet flow scenario. The u component relates to the velocity at a given position within the fluid, considering changes in pressure gradient (dp/dx). The equation provided shows how pressure differential is influenced by fluid properties like viscosity (mu) and boundary layer thickness (h). By analyzing this equation, we can predict how fluid velocity behaves near the boundaries and in free stream areas.

Think about a water slide, where the height of the slide (h) impacts the speed of water flowing down. As the water moves, its pressure changes due to height, similar to how our velocity field changes with pressure gradient in fluid dynamics.

Substituting wall shear stress using the Newton's laws of viscosity we can get tau xy wall fluid kinematics we have done this part mu is equal to del u by del y del v by del x at wall y equal to plus minus h.

This chunk introduces Newton's law of viscosity, which relates shear stress (tau) to the rate of change of velocity (u) with respect to distance from the wall (y). By applying this law, we can compute wall shear stress in the context of fluid kinematics. This is crucial in characterizing how the fluid flows and interacts with the wall.

Picture spreading butter on toast; the force you apply (similar to shear stress) determines how smoothly it spreads. In fluid dynamics, the viscosity reflects how easily the fluid flows under shear stress, just as the softness of butter impacts its spreadability.

We are looking for curl v okay. So we are looking a curl v in z direction so that what is dv by dx I just writing u dy. So v is 0 so only you know this partial derivative.

In this chunk, the focus is on calculating vorticity, which is a measure of rotation in fluid flow. We use the curl of the velocity field (v) to assess how fluid elements rotate as they move. Since fluid flow can have different components, we identify the z-direction vorticity using partial derivatives. This understanding is essential for characterizing the flow behavior and predicting its patterns.

Think of a whirlpool in a river; the water spins and rotates around a central point. The vorticity here is high, indicating a significant rotation. Analyzing fluids in a similar way lets us understand how and why water swirls around bends and obstacles.

Now you have to compute the average velocity if I considering this discharge which is u dA velocity in area integrations from minus h to plus h divide by this area.

This section discusses how to calculate average velocity in the jet flow, which is crucial for understanding the overall fluid dynamics. By integrating the velocity across a certain area, ranging from the lower boundary (-h) to upper boundary (+h), and then normalizing that by the total area, we can find an average velocity. This approach gives insight into how quickly the fluid is moving as a whole, rather than at a single point.

Imagine filling a bathtub with water from a faucet. The average flow rate of water filling the tub tells you how fast it will fill up, even if the water speed varies at different points from the faucet's output. Likewise, calculating average velocity helps us contextualize the flow across a whole area.

The boundary layer formations will be presenting in that which is because of presence of these two walls okay. As the fluid particles are coming it not the all the fluid particles here will be active for the flow is going through this. There are the fluid will commit a regions very close to this will have a flow will come it flow will go like this.

In this chunk, we learn about boundary layers, which are thin layers of fluid near the walls where viscosity and shear stress dominate. The boundary layer forms because not all fluid particles move equally; some are slowed by friction against the wall. This unique flow behavior influences drag and flow patterns, making understanding boundary layers essential for designs like aircraft wings and piping systems.

Consider how when you stir a cup of coffee, the liquid near the edge hardly moves as compared to the center. This slower-moving region is analogous to the boundary layer, where the friction with the cup slows fluid motion, affecting overall flow.

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