1.5. Force Ratio

Now, the discharge ratio Qr will be the ratios of velocity into area. So, I will just. So, velocity is given by Vr and area is the length, I mean, the length whole square. So, Vr we have already found out that Vr was Lr to the power half, in the previous slide, multiplied by Lr square so it becomes Lr to the power 5 / 2, as indicated here.

The discharge ratio, denoted as Qr, is determined by multiplying the velocity ratio (Vr) by the area through which the fluid is flowing. The area is influenced by the length dimensions of the model (Lr). We have previously established the velocity ratio as the square root of the length ratio (Lr), now when we calculate the discharge, we must account for the area which involves squaring the length ratio. This results in the overall discharge ratio being raised to the power of 5/2, as it combines the effects of both velocity and area.

Think of a garden hose. If you have a shorter hose (the model) and you want to see how fast water flows through it compared to a longer hose (the prototype), the water speed and the cross-sectional area where it flows both matter. When you measure how much water comes out in a second in both hoses, you are looking at both how fast the water moves and how wide the hose is, giving you the total discharge.

Force ratio will be the ratio of densities multiplied by Lr square into velocity square. So, ratio of densities we said was ρr into Lr square and Vr was under root Lr whole square, so it will be ρr Lr cube. But anyways I will take this down myself. So, force ratio is going to be ρr Lr cube, as we just derived.

The force ratio takes into account the properties of the fluid, specifically the density, as well as the dimensions of the model and prototype. The equation states that the force ratio is proportional to the density ratio multiplied by the square of the length ratio and the square of velocity. Thus, the square root velocity contributes to raising the entire term to the cube in total for the force ratio. This helps understand how various factors play into the forces acting on both model and prototype structures.

Imagine a small balloon filled with air versus a large weather balloon. Even though both balloons are made of the same material (same density), the larger balloon will experience greater forces due to its larger volume, akin to how fluid properties and dimensions affect forces in hydraulic engineering.

Now, power is force into velocity. So, it is force ratio into Vr. So, it is force is ρr Lr cube into Vr is under root Lr. So, it becomes ρr Lr to the power 7 / 2. So, same thing, ρr into Lr to the power 7 / 2.

In hydraulic systems, power is calculated as the product of force and velocity. Here, we established the force in terms of the density and the length ratio cubed, then combined it with the velocity ratio (which is the square root of the model length ratio). When combined for power, this results in a term that reflects how energy is affected in a model versus a prototype, outlined as ρr times Lr to the power 7/2. This signifies how energy transfers depend on both force and speed.

Think about sprinting: the faster you run (velocity), the more energy (power) you exert, even if you're carrying the same weight (force). In hydraulic terms, adjusting either the force or speed drastically affects overall energy and performance.

Now, there are something called distorted models. Those were an example with, you know, proper modelling, where all the laws were taken into account. There are something called distorted models. So, idea behind similitude is that, we simply equate all pi terms. But in reality, it is not always possible to satisfy all the known requirements so as to be able to equate all pi terms.

Distorted models arise when it is impractical to ensure that all similitude conditions (or pi terms) are satisfied precisely. In real-world applications, factors such as space and material availability compel engineers to adjust dimensions or simulate only specific elements of flow behavior. This flexibility allows engineers to still gain insights from their models, despite potential discrepancies in certain aspects.

Consider crafting a small model of a grand building using LEGOs; you may not have the exact pieces or space to replicate every detail. However, by focusing on the structure’s overall appearance and stability, you still convey the building's essence. This is similar to how engineers create distorted models to understand a system without duplicating it perfectly.

So, in practical, ideally, if we can have all the type of similarities, that is the best case solution. But in reality, it is it might not be always, you know, I mean, we might not be able to find it all the time.

In summary, achieving full similarity between model and prototype is the ideal scenario, but practical constraints frequently prevent this from happening. Engineers must be resourceful and apply fundamental principles of fluid dynamics to create models that, while imperfect, still yield valuable insights into expected behaviors in real situations.

Consider trying to bake a cake from scratch using a recipe you found online; it may not call for all the exact ingredients you have at hand. However, by substituting some items and relying on your understanding of baking, you can still create a tasty dessert. Similarly, hydraulic engineers adapt their models creatively to ensure they effectively represent real-world conditions.

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