# Why do we use dimensionless expressions in flight mechanics and aerodynamics?

Why do we use non-dimensional expressions in flight mechanics and aerodynamics? We could as well directly calculate forces and moments; would that not be more relevant to the specific problem?

• Yes but in physics meaning what useful of it , you can use for example use force as dimensional formula but in flight mechanics we use dimensionless formula . – SAMER BABIKIR Feb 5 '17 at 13:25
• I have to disagree; the question is clear. We mention the subject of dimensionless coefficients in countless answers (see here for example), but do not yet have one that explains why this is done so. I vote for reopening the question. – Peter Kämpf Feb 5 '17 at 16:41

Here is a dimensionful equation: $$L = \frac{\rho}{2}\cdot v^2\cdot \frac{2\cdot\pi\cdot b^2}{1+\sqrt{1+\left(\frac{b^2}{2\cdot S}\right)^2}}\cdot\alpha$$ Note that all ingredients are physical, measurable values. Now here is the same thing again, now in dimensionless form: $$c_L = \frac{2\cdot\pi\cdot AR}{1+\sqrt{1+\frac{AR^2}{4}}}\cdot\alpha$$ The dimensionless version is independent of speed $v$ and air density $\rho$ and also needs only the aspect ratio $AR$ and the angle of attack $\alpha$ to give a result, while the dimensionful version needs wingspan $b$, wing area $S$ and angle of attack. However, while the dimensionful version yields an actual force, the dimensionless version yields a coefficient which needs to be multiplied with speed, density and wing area again in order to result in a force.

By working in the dimensionless domain, engineers can work on a generalized form of the problem. This was especially helpful in the time before computers: The complex calculation to arrive at a coefficient was needed only once, and computing the dimensionful value from there was trivial.

Also, when comparing between different airplanes, or between wind tunnel models and the real thing, with dimensionless coefficients the numbers will be very similar, which allows much better to error-check results or to make first assumptions on a new design.

However, there are two drawbacks:

1. You need to know the reference values! It is easy to screw up, e.g. when combining values from the wing and the tail which are referenced to their individual area. While lift can be added right away, the lift coefficients must first be converted to a common reference value.
2. Sometimes thinking in coefficients can be misleading: Just consider induced drag. If written in dimensionless coefficients like this $$c_{Di} = \frac{c_L^2}{\pi\cdot AR}$$ you can be forgiven for thinking that induced drag goes down with increasing aspect ratio. But this is wrong! Now the same again with dimensions: $$D_i = \frac{L^2}{\frac{\rho}{2}\cdot v^2\cdot\pi\cdot b^2}$$ This shows that induced drag at the same speed is inversely related to span loading, the amount of lift created per unit of wing span.
• Some of the common values are also dimensionless by nature--for one obvious example, Reynolds number is computed as a ratio in which all the dimensions cancel out, leaving only a dimensionless number. As such, in quite a few cases, you simply can't avoid working with at least a few things that are dimensionless. – Jerry Coffin Feb 6 '17 at 5:31
• @JerryCoffin: But then the Reynolds number was defined in order to make different flow conditions comparable, just like the force and moment coefficients. The difference is that using viscosity, length and density explicitly would needlessly complicate the calculations, but it can be done. – Peter Kämpf Feb 6 '17 at 6:46
• I suppose you could do calculations using dimensional quantities, but it would do more than just complicate calculations--it would hide the applicability of the results (e.g., by knowing that they operate at close to the same Reynolds numbers, you know that calculations for the airfoil of a relatively slow-moving glider apply about equally to a jet engine's turbine blades). – Jerry Coffin Feb 6 '17 at 7:31
• Many thanks that was very helpful. – SAMER BABIKIR Feb 7 '17 at 12:53

As far as aerodynamics are concerned: with dimensionless coefficients (i.e. for lift and drag), statements independent of profile, body size and dynamic pressure can be made for every airfoil and its properties for different flow directions. That allows comparison of different profiles and airfoils - in flight mechanics or aerodynamics, oftentimes it is more interesting to compare airfoils and determine the best, instead of calculating exact forces).

As Stefan also wrote, in science your goal is to ignore e.g. the dimensions your experiment is build in to be able to compare it with other results (regarding similitude theory, which also uses dimensionless numbers). With the gathered dimensionless data it is also much easier to develop theories independtly from any unit system. Furthermore, the variables of fluid dynamics are often connected with power laws. These are much easier to handle mathematically correct if you already got rid of the units. Lastly, when you apply the theory to a specific engineering problem, you can reintroduce your dimensions etc. to get the result.

I would like to answer this question from a historical perspective.

Dimensionless numbers are used much more heavily in fluid dynamics than in other engineering disciplines. For structural mechanics, we happily throw around parameters like Young's Modulus, density, moments of inertia, and expect people to know what these numbers mean, even though they work out completely differently on different scales. We know approximately how things scale in simple objects, and whether we are calculating a landing gear or a wing, the beam bending problem is approximately the same.

Not so in fluid dynamics. Of course, Navier-Stokes, Bernoulli and friends had long ago figured the governing equations for fluid dynamics, but their applicability is much more limited than say, Hooke's law of elasticity. In structural mechanics, one can assume the wing to look approximately the same before and after bending, but in fluid dynamics the initial conditions and geometry tells us nothing about the final results, and a small perturbation on one side may completely change the results on the other side.

These days, we can use CFD (computational fluid dynamics, "computer simulation") tools to get from a set of initial conditions and (dimensionful) properties to a final result. In "the olden days", wind tunnels were used extensively. Since these things are expensive to build and maintain, and difficult to make on a large scale without computers, they were generally kept quite small. As a consequence, tools for scaling were very important - what do small-scale results tell us about the final design?

AS it turns out, dimensionless numbers are very useful for scaling. Since they do not depend on the unit system used, one can infer that they do not depend on the scale of the problem - the number should be the same whether we use a foot-long model or a meter-long one. Flow will become turbulent if $Re>2300$ (by appromation - you will find different values depending on application), and this tells us that, when we have a small model that we test near a Reynolds number of 1000 and all seems well, we may conclude that this also holds for the final airplane if the Reynolds number is around 1000 for that design.

You will note that dimensionless numbers are less interesting in the computerized age (why jump through all these hoops to get all the units out, if you can just plug it into the computer) - for disciplines where FEM etc. is ubiquitous, you will only see them used as a back-of-the-envelope calculation. In fluid dynamics, I should expect them to stick around a bit longer, until CFD becomes powerful enough to readily calculate large models within very short time spans.