Dimensional analysis (fluid mechanics context)

[Kenny Lee]

This follows from Buckingham's Pi theorem and is more of a conceptual problem... I'm doing fluid mechanics 101, so everything's kinda new to me.

They say that one reason dimensional analysis is so useful - I'm referring to grouping n variables into n-m dimensionless parameters, where m is the number of fundamental units etc. etc - is because it allows the experimenter to limit the scope of his investigation to those dimensionless parameters only.
So for example, one could simply plot Reynold's number against the drag coefficient, instead of varying and holding constant consecutively, each density, velocity, viscosity, diameter and area.

What I'm wondering is, what's to stop us from randomly selecting variables and experimenting on them. Why must they be dimensionless?

If in the context of similarity (models), then I understand that dimensionless groups have their uses. But I have problem accepting the former.

Can someone please clarify?

 

[Astronuc]

For one thing, it is the cost involved in fabricating a full size prototype - with all the materials, tooling, parts fabrication, possibly new fabrication techniques. It's much less expensive to build a scale model.

The idea of dimensional analysis allows one to build a scale model of something - car, truck, aircraft, ship, nuclear fuel assembly, rocket - and then test it over a range of thermohydraulic parameters, including flow sweep tests.

Suppose one to test at range of temperature and pressure conditions - e.g. temperatures 200 - 350°C and pressure 7 bar - 18 bar. Testing a large product requires a large (full size) testing rig. So a scale model of the prototype means a smaller testing system.

Most of the time time testing scale models works. Applying to dynamic analysis sometimes has short comings.

 

[piercas]

Dimensional analysis is a powerful tool in fluid mechanics because it allows us to simplify complex systems and focus on the most important variables. By grouping variables into dimensionless parameters, we can reduce the number of variables we need to consider and make the problem more manageable. This is especially useful in experimental studies, where it can be difficult to control and measure all the variables involved.

The reason for using dimensionless parameters is because they are independent of the specific units used to measure the variables. This means that the results will be the same regardless of whether we use metric or imperial units, for example. This makes it easier to compare and generalize results from different experiments.

In addition, many physical laws and relationships in fluid mechanics are expressed in terms of dimensionless parameters. For example, the Reynolds number is a dimensionless parameter that represents the ratio of inertial forces to viscous forces in a fluid flow. This parameter is used to predict the transition from laminar to turbulent flow, which is a fundamental concept in fluid mechanics.

Experimenting on randomly selected variables without considering their dimensionless relationships may lead to incorrect or inconsistent results. Dimensionless parameters provide a more systematic and comprehensive approach to understanding fluid mechanics phenomena. They also help us to identify similarities between different systems and make predictions based on those similarities.

In summary, dimensional analysis is a crucial tool in fluid mechanics as it allows us to simplify complex systems and focus on the most important variables. Dimensionless parameters are essential in this process as they provide a consistent and universal way of representing physical relationships and allow for easier comparison and generalization of results.