Why use non-dimensional coefficients?

Question

Aerospace engineering textbooks make frequent use of non-dimensional coefficients like $C_L$, $C_D$ and a whole host of others. Usually, for equations of motion, it is preferred to replace variables with non-dimensional coefficients. However, fundamentally I do not understand why.

I have two questions:

1. Why do we make aircraft parameters non-dimensional ($L$ to $C_L$, $M$ to $C_m$)?

   An often-heard response is you need the coefficients in order to compare different aircraft; $L$ and $D$ are really dependent on aircraft dimensions and flight conditions. However, as far as I know, coefficients $C_L$ and $C_D$ are also dependent on aircraft dimensions and flight conditions! Where is the inherent superiority of one over the other?

2. Does non-dimensionalizing make the coefficient independent of anything (weight, aircraft dimensions, dynamic pressure, Mach number etc.)?

   Some textbooks state that non-dimensional coefficients are independent of weight and aircraft dimensions but do not give an argument for why that should be the case.

Answer 1

There are two major benefits to dimensional analysis (non-dimensional coefficients) according to Frank M. White, Fluid Mechanics, 2nd Ed. My answer is heavily paraphrased from this source.

1. They allow scientists/engineers to reduce the number of experiments required to explore a given phenomenon.
2. They provide scaling laws allowing experiments to be performed on small, cheap scale models which can then be extended to full-size applications.

Reducing the number of experiments

If we wanted to design an experiment to measure the force on a cylinder immersed in a moving fluid, and we knew it depends on the following parameters: $$ F = f(L, V, \rho, \mu) $$ Then if we want to properly identify the phenomenon we would need to explore a rather large parameter space. You would want to be able to fit a curve to the data, which is going to require at least 10 data points for each variable. So we would need to design an experiment with 10 values of $L$, and for each value of $L$ we would also have 10 values of $V$, and for each value of $V$ 10 values of $\rho$... and so on. This would be time consuming and expensive. However, the equation can be reduced using dimensional analysis to get: $$ \frac{F}{\rho V^2 L^2} = g\left( \frac{\rho VL}{\mu}\right)$$ $$ C_F = g\left( Re\right) $$ Where $C_F$ is the dimensionless force coefficient and $Re$ is the Reynolds number. Now we only need to explore around 10 values of $Re$ to start to get a basic characterization of the phenomenon. This is a greatly simplified example (there is more to fluid forces than the Reynolds number) but it serves to illustrate the point.

Furthermore, we can use non-dimensional coefficients to compare different geometries. Consider, for example, the Reynolds number. The Reynolds number can be calculated for many geometries (inside a pipe, on an aircraft wing, around the body of a submarine), different flow conditions (internal vs. external flow) and different scales, yet it can tell us important information about the nature of the flow (laminar vs. turbulent vs. transition).

Scaling laws

Non-dimensional coefficients are also useful because they allow easy comparison between engineering cases at different scales. They allow us to establish a condition of similarity between a model and a full-scale prototype.

Due to this property (independence from scale), non-dimensional coefficients are used to design scale-model tests. If you make a 1:10 model of an airplane and put it in a wind tunnel, the layperson might think you would just scale the design air velocity by the same factor (and they would be wrong). Similarly if you want to design an RC model airplane, you can't just scale down a Boeing 747 and have it fly! Here is a Wikipedia article on the subject.

Using the example function $f$ given above, we can achieve what is called similarity if the Reynolds numbers of a model and a full-size prototype are equal.

If $Re_m = Re_p$ then $C_{Fm} = C_{Fp}$

Using the definition of the force coefficient from before, we get the following scaling law:

$$ \frac{F_p}{F_m} = \frac{\rho_p}{\rho_m}\left(\frac{V_p}{V_m}\right)^2\left( \frac{L_p}{L_m}\right)^2$$

So if you were to make a model at 1:10 scale, $L_p/L_m = 10$. Assuming that the velocity of the fluid $V$ and the density of the fluid $\rho$ are identical for both, then the prototype will see 100 times the force compared to the model when subjected to these conditions.

Answer 2

The qualitative behaviour of a physical system often depends on the relative size of two (or more) different effects. For example, in fluid flow, turbulence tends to develop when the force to accelerate the fluid (i.e. its inertia) is big compared with the viscous forces in the flow. In the opposite situation where the inertia forces are small relative to the viscous forces, the flow tends to remain laminar.

There is a similar situation in heat transfer problems, depending on the amount of heat carried by convection in the fluid (which depends on the fluid physically moving around), compared with the amount moved by conduction (which doesn't depend on the fluid motion).

If we invent a simple but "reasonable" way to measure these different effects, the ratio of the two measures will be a non-dimensional number. For the Reynolds number example, a reasonable way to measure the inertia forces is simply mass$\times$acceleration of an element of the fluid, i.e., $\rho L^3 v/t$ where $\rho$ is the density, $L$ is some "average" dimension of the fluid element, $v$ is the velocity, and $t$ some typical time scale in the flow.

Similarly a reasonable way to measure the viscous forces is $\mu(v/L)L^2$, where $\mu$ is the dynamic viscosity, $v/L$ represents the velocity gradient creating the viscous forces, and $L^2$ is the cross sectional area of the fluid element.

Dividing one measure by the other gives $$\frac{\rho\, (L/t)\, L}{\mu}$$ or (since $L/t$ is measuring some sort of velocity - the mean flow velocity through the pipe, for example) $$\text{Re} = \frac{\rho v L}{\mu}$$ where Re = Reynolds' number.

For a particular situation like flow through a pipe, the transition from laminar to turbulent flow typically happens when Re is about $3000$, if we take $L$ to mean the diameter of the pipe.

This is useful, because the result is true in a wide range of flow situations. it doesn't matter if we are talking about low-pressure gas flowing through an $0.1\,\text{mm}$ diameter capillary tube in a vacuum pumping system, or crude oil being pumped through a $1.2\,\text{m}$ diameter long-distance pipeline - just plug in the relevant material properties, diameter, and flow velocity to find Re, and for the same value of Re the flow pattern will be basically the same.

Answer 3

The reason for using non-dimensional coefficients is to be able to manipulate the different possible cases needed to be studied.

By example, to be able to compare the real life scaled airplane to a prototype scaled model. It is extremely useful since it allows for companies and researchers to be able to simply do all the testing on prototypes and evaluate different possible scenarios that could occur in real life.

By doing that, these companies or researchers reduce the amount of real life scaled airplanes for testing to possibly zero, and go straight to manufacturing (in the ideal case).

As mentioned in this link:

"The importance of experiments in Fluid Mechanics needs no additional emphasis. Experiments are required in design and testing of vehicles such as aeroplanes, ships and automobiles, pumps, turbines, fans and other equipment. We also have experiments which are carried out from the point of view of understanding a flow and fundamental phenomena such as turbulence. Needless to say that the experiments have to be planned and executed methodically.

The question we must ask ourselves is :"What are the parameters involved in the given flow? What is their relative significance?" These are some of the questions facing one that conducts any experiment. At the same time, presentation of results in a concise fashion also requires thought and planning. A thoughtless experiment (or for that matter a modern computation) will generate a large body of numbers. It may be difficult to interpret them. A thoughtful presentation will help to render them in a concise and understandable manner. It is here that Dimensional Analysis helps us."