1.8. Distorted Models

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. In reality, it becomes very difficult to equate all these terms.

Distorted models are used in fluid dynamics when it's not feasible to satisfy all similarity conditions while modeling. The concept of similitude states that when creating models (typically in hydraulic engineering), one attempts to equate all relevant non-dimensional parameters (pi terms). However, due to practical limitations, achieving this ideal state is often impossible. Designers must acknowledge that certain parameters may not fully align, hence leading to the use of distorted models to simplify the portrayal of physical phenomena in a constrained environment.

Imagine you are trying to create a scale model of a city. You might have to exaggerate certain features, like the height of buildings, while reducing the size of streets to fit everything in a small scale. Just like in a city model, sometimes engineers have to distort certain dimensions to make them work within the constraints of a laboratory or modeling environment.

Example; Study of open channel or free surface flows, we will see through an example. So, in this case, the Froude number similarity will give V m / under route g m l m is equal to V p under root g p l p. And if we have the same gravitational field, just considering, this will give Vm / Vp. So, the ratio of the velocity will be lambda. So, under root of length ratio, that we have seen in the previous example so, under root of length ratio.

In the context of fluid modeling, especially for open channel flows, it is essential to consider both Froude and Reynolds number similarities to ensure the dynamics observed in the model closely replicate those in the prototype. The Froude number evaluates the influence of gravitational force on a fluid's motion, while the Reynolds number assesses the flow regime—whether it is laminar or turbulent. These numbers must maintain their ratios in both the model and the prototype to ensure accurate representation of flow behavior.

Consider comparing a small model boat floating on a pond (the model) to a giant ship (the prototype) sailing in the ocean. Both must be analyzed under similar conditions: the waves caused by the wind (analogous to gravity) must be proportionally similar for the model to mimic the ship's movement accurately. If the boat's size and weight aren't aligned with the laws of fluid dynamics, the results won't be reliable.

So, to overcome difficulties like before modelling, I mean, the one that we did, we use distorted scales or distorted models. So, what is done here? In open channel the vertical dimension is used to simulate Froude laws while the other 2 dimensions are scaled to suit the available space. So, in Froude law, in Froude number the vertical dimension so basically, for Froude law we use only the vertical dimensional scaling and for the other 2 Dimension X and Y, we use the geometric scaling, for example.

In practice, when creating distorted models for simulation, engineers often scale only the vertical dimension in accordance with Froude's Law to properly represent gravitational effects. The horizontal dimensions are then scaled independently to fit the constraints of the modeling environment. This selective scaling allows for a simplified but effective simulation of fluid behavior, overcoming real-world limitations such as space and material availability.

Think of a sandcastle that needs to resemble a large castle. Instead of accurately proportioning every building to fit the sand size, you might decide to make the towers taller and the bases slimmer to look spectacular visually, while ensuring it can be built within the limited area of sand. Similarly, engineers prioritize the most critical dimensions that influence flow behavior while adapting others for practical reasons.

Froude number ratio is one. So, this gives us, F m / F p is equal to 1 and this is V m. So, that is what it means, F r is equal to 1. So, through this, so implies V m / V p is equal to under root y m / y p. And what is the ratio of y m / y p? Is h r, so, it is under root h r, but anyways, we will see. So, V m / V p whole square / y m / y p is equal to V r square / h r, another way of solving and we get the same result.

When analyzing distorted models, the Froude number ratio is typically set to unity to ensure similarity between the model and prototype velocities. This means that the relationship between the model velocity (VM) and the prototype velocity (VP) can be expressed as the square root of the ratio of their corresponding vertical dimensions (yM/yP). As a result, fluid dynamics can be predicted effectively, allowing engineers to derive meaningful insights from scale model experiments.

Imagine you are timing two different types of moving objects: a toy car and a real car. If you measure their speeds accurately only under similar intended conditions (flat surface, same slope), you can reliably infer the toy car's performance based on the real car's data, given that they are proportionately scaled. Similarly, in fluid dynamics, understanding the velocity ratios allows us to predict how behavior changes between models and real-life scenarios.

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