11.1.3. Stream Function and Vorticity Analysis

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. 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.

This chunk describes the objectives of today's lesson, focusing on key fluid dynamics concepts such as wall stress and shear stress, as well as the calculations of stream function, vorticity, velocity potential, and average velocity. It also recalls the previous lesson where a velocity field was estimated using Navier-Stokes equations with some assumptions, notably neglecting gravitational effects.

Imagine a water hose spraying water out. As you observe, some water hits the wall nearby and forms a thin film that illustrates wall stress and shear stress as the water flows. In fluid dynamics, just as you can analyze how water moves through a hose and its impact on nearby surfaces, you can apply Navier-Stokes equations to understand velocities in various scenarios.

Now if you look it I am just 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 is what. So plus h this is minus h okay. So del v is 0 we know it. So you only have a u functions that is what the functions you just substitute here. do the partial derivatives finally you get it this part.

Here, the calculation of wall shear stress is based on Newton's laws of viscosity. The notation tau_xy represents shear stress and involves calculating the gradient of velocity u with respect to distance y. At the wall (given by positions y = ±h), the component v is zero, simplifying the derivatives needed for the calculation.

Think of the way your hand feels when it’s pushed along a smooth surface versus a rough one. The tangential force you experience is similar to shear stress in fluids. The mathematical approach taken here to express shear stress at solid boundaries helps us determine how fluids interact with surfaces in situations like pipe flow or air flowing over an aircraft wing.

Now if I look it as the plane okay it is a 2 dimensional plane flow that is what so it can have the stream flow okay. Stream functions we can get it because it is a 2 dimensional flow okay as the x and y flow is steady and incompressible. So we know it u equal to definitions okay. So just integrate the u then you will get it the functions u max.

This section discusses the application of stream functions in a two-dimensional flow, emphasizing that the flow is steady and incompressible. The stream function, psi, is an essential concept in fluid dynamics, helping visualize flow patterns and calculate velocities. By integrating the velocity function u, we can express maximum velocities, u_max, further aiding analysis.

Imagine a river flowing steadily. The stream function helps visualize the current in various parts of the river without having to measure every flow of water directly. By knowing the current at specific points, we can derive patterns of flow and identify where the water flows fastest—similar to how stream functions help in understanding fluid flow in engineering applications.

So, if I look at the vorticity okay. So vorticity in the zth plane that is what we you have a x and y. So you are looking the vorticity which is perpendicular to this plane.

This chunk introduces the concept of vorticity, which is a measure of the local rotation of a fluid. Vorticity is essential in fluid dynamics as it represents the tendency of fluid particles to spin about an axis. In this case, it specifically considers vorticity in a 2D plane, where it is calculated as a curl of the velocity vector.

Consider a region of water swirling around a drain as it empties. The swirling motion is an illustration of vorticity in action—how parts of the water flow rotate is linked to the idea of rotational motion in fluid mechanics, allowing us to analyze flow behaviors in rivers, air around wings, or blood in arteries.

Then you have to compute the average velocity it is a very easy that for a if I considering this discharge which is u dA velocity in area integrations from minus h to plus h divide by this area I will get it integrations over this we will go at it.

This section outlines a straightforward method for calculating average velocity in a channel. By integrating velocity across a defined area (from -h to +h) and dividing by the area, one can determine the average velocity, representing the overall flow in the fluid dynamics context.

Think of measuring the average speed of a car over a long drive. You might record the speed at different intervals but combine all observations to find a single average speed for the entire journey. Similarly, in fluid dynamics, average velocity gives a simplified view of how fast fluid is moving throughout a system, important for design and analysis.

--