17.3.1. Experimental Setup

Now, let me come in to the, if the flow is incompressible. Again, we have a lot of simplifications. Again, I can repeat it. The flow systems when you have mac number less than 0.3, okay, whether it is gas, whether it is a liquid or any flow system, if you think that the within the flow system the flow becomes less than the mac number less than 0.3, then there will be density variation, but that variation of density is much much negligible comparing to other components. So, we can assume the flow is incompressible nature. Again, I am going to summarise that. When you have any flow systems, mac number is less than 0.3, so we can use flow as incomprehensible flow, density does not vary significantly. So, density becomes constant, as density becomes constant, as you know it, it is very simplified problem what we are going to solve. So, density varies negligible, as the density variation is not significant and beta equal to 1, so only this equ ation is left for us. Simple thing.

In this chunk, the concept of incompressible flow is introduced. An incompressible flow is characterized by a small Mach number (less than 0.3). When the Mach number is low, density variations due to pressure changes are negligible, allowing us to simplify the analysis by assuming a constant density. This simplification makes fluid dynamics problems easier to solve. The connection made is that if density is constant, we simplify equations, particularly in fluid mechanics, which can lead to a more straightforward solution.

Consider a child's balloon. As you blow air into it slowly, the density of the air inside does not change significantly, and the balloon expands easily. This is like an incompressible flow where the changes in pressure do not substantially affect the density.

So, please do not confuse, this is a different equation. If the inlet and outlet are one-dimensional where, Q = A V. Only that we are not showing the density multiplication. If you multiply with a Q, the V into A is Q is discharge. So, Q = V α, is the discharge. So, most of the conservation of mass you write it, since density is a constant, you make it come out from that equation. So, it looks like volumetric level we are comparing but all are mass conservation equations.

In this chunk, the distinction between mass flux (related to mass flow per unit area) and volumetric flux (which deals with the volume of fluid per unit time passing through a section) is clarified. When considering flow through a one-dimensional inlet or outlet, the flow discharge (Q) can be simply represented as the product of the area (A) and the average velocity (V). Since density is constant in incompressible flow scenarios, it can be factored out, simplifying the calculations while still adhering to mass conservation principles.

Think about water flowing through a garden hose. The speed of the water (V) combined with the cross-sectional area of the hose (A) determines how much water (Q) comes out of the hose per second. You can calculate this without worrying about the density of the water changing because it stays the same through the hose.

The flow coming and going out. If this is simple in and out system, you can know this velocity distribution area, find out the discharge from inflow and outflow, equate it, then you can solve the problem. So, what I am looking at is again summarised here. To solve this mass conservation equation I should have knowledge on velocity field. I should know how the velocity varies or I should know whether the velocity is a constant or the velocity varies.

This chunk discusses the importance of understanding how velocity varies across a flow field when applying the mass conservation equation. In fluid dynamics, flow is rarely uniform; it can have different velocities at different points, particularly in complex systems like pipes or over surfaces. Knowing how the velocity changes helps in accurately measuring inflow and outflow rates within control volumes, which is key to resolving problems in fluid mechanics.

Imagine a river flowing faster in the center than at the edges due to friction with the riverbed and banks. If you were to measure the average speed of the water, you’d need to take into account that it isn’t uniform across the river’s width. Just like in fluid dynamics, recognizing this change helps provide a more accurate picture of how much water flows downstream.

So, whenever you take fluid mechanics problems, first you think what could be approximate velocity field, what could be the velocity direction. If I assume the flow is one dimensional, is it enough for me that one dimensional flow is okay for us or not, for that problem or not, or you need to have two-dimensional velocity fields, or what could be the direction, whether it is a direction with respect to the control surface normal vectors.

In this chunk, there is an emphasis on the critical thought process required when tackling fluid dynamics problems. Students should first evaluate the system for approximate velocity fields and determine whether a one-dimensional or more complex two-dimensional flow model is required. Recognizing the flow directions relative to control surfaces is essential for applying the conservation laws appropriately to ensure accurate calculations.

Consider how a simple water slide can have a constant flow if you assume only one path for the water. However, if the slide has curves or levels, it’s crucial to account for how water moves in different directions at different points; this complexity is similar when working with fluid flow in various engineering scenarios.

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