1.12. Distorted Scale 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 scenarios where ideal conditions (like perfect similitude) cannot be achieved. In engineering fluid models, the principle of similitude relies on ensuring all relevant dimensionless parameters (pi terms) are equal between the model and the prototype. However, in practice, it is challenging to meet all these conditions, often leading to the need for distorted models that simplify some aspects while maintaining necessary relationships.

Imagine trying to replicate a large painting on a smaller canvas. While you can keep the basic subjects similar, you might have to omit some fine details or change proportions to fit the smaller size. Similarly, in fluid dynamics, distorted models allow engineers to create simplified representations of larger systems that may not be able to replicate every detail accurately but still provide useful information.

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 distorted models, the relationships established by the Froude number become crucial. For open channel flows, the similarity condition based on the Froude number ensures that the velocity ratios between the model (Vm) and the prototype (Vp) are linked to their respective lengths. Given a common gravitational field, this relationship helps maintain a semblance of accuracy in how the model simulates the prototype's behavior.

Consider a small water channel model used to test how a dam would behave in a river. The model must maintain a specific width-to-depth ratio that simulates the actual river conditions to ensure that the resulting flow patterns are similar to the real-world scenario. Although the small model can’t perfectly mimic everything, proper scaling allows for meaningful experiments.

To overcome difficulties like before modelling, I mean, the one that we did, we have 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.

In practical applications of distorted models, the vertical scaling of dimensions is generally prioritized to adhere to Froude laws, while the horizontal dimensions may be adjusted geometrically to accommodate constraints of the test environment. This specialized scaling allows for relevant behavior to be accurately represented without the need for absolute similarity in all dimensions.

Think of a child playing with a toy model of a skyscraper. The child might accurately replicate the building's height but not the width precisely because of limited space on their play table. Despite this distortion, the model still represents the basic attributes of the skyscraper, thereby allowing the child to creatively engage with the concept of height and scale without needing an exact replica.

Froude number ratio is one. So, this give 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.

The relationships established in distorted models allow for simplified calculations. For instance, if the Froude number is equal for both model and prototype, it implies that the velocity corresponding to the model directly relates to the depth dimensions scaled by h_r (the vertical scaling ratio). By maintaining these relationships, engineers can derive meaningful ratios without needing perfect scaling across all dimensions.

Consider a chef adjusting a recipe for a cake. If scaling the recipe down, the chef knows how to adjust the amounts of each ingredient while keeping the proportions the same. Similarly, when calculating velocities and depths in distorted models, the relationships allow engineers to maintain functional properties based on scaled ratios, even if the absolute sizes differ.

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