2.10. Practice Problem: Calculating Stream Function and Flow Rate

Now, we will see a practice problem, for the following flows, determine the components of rotation about various axes. So, we have been given u is equal to x y cube z, V is equal to- y square z square, w is yz square - y cube z square / 2.

In this section, we introduce the concept of a stream function in the context of 2D flow. The stream function is a mathematical tool used to describe the flow of incompressible fluids. Here, we begin by stating the problem where we have specific equations for the velocity components (u, v, w) of a fluid flow. The goal is to analyze these components and derive relevant properties from them. The variables u, v, and w represent the velocity components in the x, y, and z directions, respectively.

Think of water flowing through a garden hose. The speed of water at any given point in the hose corresponds to these velocity components. Just as you can measure how fast water flows, we can use mathematical functions to describe that flow systematically.

So, based on that a stream function psi is defined that it is constant along the stream line and the difference of these stream line for the 2 streamline is equal to the flow rate between them.

The stream function, denoted as psi (Ψ), is a key concept in fluid mechanics, particularly for incompressible flows. It simplifies the analysis of fluid movement, as it remains constant along a streamline, meaning it does not change as you follow the flow. The difference in the stream function values between two streamlines is directly related to the flow rate between those streamlines. This relationship helps in understanding how fluid particles move in relation to these specific paths in the flow.

Imagine riding a bike along a riverbank. Each of your bike paths can be represented by streamlines. The flow rate between two points along these paths can be thought of as how quickly you ride from one point to another. The stream function helps us determine this 'speed' between similar paths without having to measure every twist and turn in the actual river.

Thus, phi A - phi B is equal to flow rate between S 1 and S 2. The flow from left to right is taken as positive, in the sign convention. So, this is what we assumed, that if the flow is from left to right it is positive.

In this step, we focus on calculating the flow rate based on the difference of the stream function values at two points, A and B. This difference (Ψ_A - Ψ_B) provides insight into the flow rate across two streamlines (S1 and S2). The positive flow convention indicates the direction of flow; by defining left to right as positive, it helps standardize calculations and interpretations in fluid dynamics.

Consider a set of traffic lights on a road. The difference in road traffic at two specific lights (A and B) can be thought of as the flow rate between the two locations. If more cars are passing from A to B, we consider that flow as 'positive' traffic. So similar to cars traveling along a road, fluid motion can be visualized and calculated using these principles.

The velocity is u and v in x and s directions are given by, this is important. So, if there is a stream function psi. So, u is given by del psi / del y and v is given as del psi / del x.

This chunk elaborates on how the velocity components (u and v) are derived from the stream function (Ψ). The formulas describe that the velocity in the x-direction (u) is the partial derivative of the stream function with respect to y, and the velocity in the y-direction (v) is the negative partial derivative of the stream function with respect to x. This relationship is critical because it allows us to translate the flow representation (using Ψ) into actual fluid velocities.

Imagine a spinning pinwheel in wind. The wind (stream function) makes the pinwheel spin, representing the motion of air. The faster it spins in a particular direction is akin to the speed of the wind (velocity components) in those directions. Just like how we can measure wind speed based on the spinning speed of the pinwheel, we can derive fluid velocities from the stream function.

Therefore, this vorticity along the axis is equal to 0, in simple words, this concept will be used in numericals that you are going to solve.

In summary, the discussion concerning stream functions leads to the understanding of vorticity, a measure of rotation in a fluid. For an irrotational flow, the components of rotation or vorticity are zero (indicating no swirl or rotation), which is a crucial concept in fluid mechanics. This element simplifies many problems in fluid dynamics, allowing us to apply this concept in practical numerical problems and real-world applications.

Think of a calm pond that doesn’t have any waves. If you drop a pebble in it, you can see ripples, but before that, the water is irrotational with no movement. Just like this calm water, many flows can be assumed to be irrotational for easier calculations in engineering scenarios, simplifying our understanding of how they behave.

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