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What is meant by the 'natural' time/length scales which some mathematical equations define?

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    $\begingroup$ Can you give some examples? $\endgroup$
    – rob
    Oct 7 '16 at 3:20
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I'd like to briefly expand on the last sentence in tpg1124's answer:

And in really good situations, proper selection of the length and time scales will allow data from different conditions to collapse on top of each other and allow the creation of a simplified theory based on the non-dimensionalization used.

In even better situations, proper selection of length and time scales allow you to actually learn something about the behavior of the system even before you solve the equations. Following is an example.

Consider a damped mass-spring system following the differential equation $$m\frac{d^2x}{dt^2} +\gamma \frac{dx}{dt} + k x = 0.$$ I count five parameters in this problem: the mass $m$, the decay constant $\gamma$, the spring constant $k$, the initial position $x_0$, and the initial velocity $v_0$. We will attempt to reduce the number of parameters by non-dimensionalizing the the equation. First write $$\frac{d^2x}{dt^2} + \frac{\gamma}{m} \frac{dx}{dt} + \frac{k}{m} x = 0,$$ and notice that $m/\gamma$ must have units of time in order for the units to work out. We will choose this quantity as our time-scale, which means that we create a new, unitless time variable, defined by $$\tau = \frac{t}{m/\gamma}.$$ In addition, let's take the initial position $x_0$ as our length scale, in which case we create a new, unitless positions variable, defined by $$\chi = \frac{x}{x_0}.$$

Under this transformation, the equation becomes $$\frac{d^2\chi}{d\tau^2} + \frac{d\chi}{d\tau} + \frac{k m}{\gamma^2} \chi = 0.$$ Something magical has happened. Where before we had five parameters, now we have two, the initial velocity (in properly scaled units), and the compound quantity $k m/\gamma^2$. The initial position is no longer a parameter, because in these units, the initial position is always 1.

Finally, here's the point of all this. Other than the initial velocity, there is exactly one physical parameter that we can vary---$k m/\gamma^2$---and it has a lot of physics in it. The fact that this simplification occurs tells us something interesting: increasing the spring constant by a factor of 2, say, has the same effect as increasing the mass by a factor of 2 or decreasing the drag constant by a factor of $\sqrt{2}$. These three physical parameters are in some sense dependent in this problem.

This makes some amount of sense in a couple of different ways. For instance, an object with a larger mass (all other things being equal) is less likely to be affected by drag than a smaller one. Hence, increasing mass and decreasing drag are equivalent.

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  • $\begingroup$ Plus, if you can combine things in a new way, college kids centuries later will learn how to use the March Number. $\endgroup$
    – tpg2114
    Oct 7 '16 at 4:22
  • $\begingroup$ @tpg2114. That comment probably warrants a separate answer. :) $\endgroup$
    – march
    Oct 7 '16 at 4:24
  • $\begingroup$ It's hard to find gaps in this table to fill in new numbers! $\endgroup$
    – tpg2114
    Oct 7 '16 at 4:29
  • $\begingroup$ @tpg2114. Wow. Just goes to show you how freaking hard fluid dynamics is. $\endgroup$
    – march
    Oct 7 '16 at 4:35
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This generally arises in the context of non-dimensional numbers or when deciding how big or long to run something. Generally, for almost every problem, there is some defining feature in both length and time that defines the dynamics of the system. This is the length or time scale one would choose, which some one could call a "natural" length scale.

Some quick examples:

  1. Flow over a cylinder will have a length scale based on the diameter and a time scale based on the shedding frequency;
  2. Flow over a wing will have a length scale based on the chord of the wing;
  3. Turbulence has many scales -- the large, or integral, length scale and time scale; the smallest scale where the energy is dissipated, the Kolmogorov scale; an intermediate scale, the Taylor microscale;
  4. Combustion has multiple time scales based on the reaction rates of the mechanism and generally has a length scale based on the flame thickness;
  5. Flow over a cavity will depend on the cavity size.

This is, and can never be, by any means an exhaustive list. But natural length or time scales can be found for many problems and can be used to non-dimensionalize the equations and data. And in really good situations, proper selection of the length and time scales will allow data from different conditions to collapse on top of each other and allow the creation of a simplified theory based on the non-dimensionalization used.

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