Scaling in fluid mechanics and other physical discipline

[hanson]

Hi all!
I always come across "scaling" in various subjects.
Like in Fluid mechanics, I am confused about why there is a need to choose some characteristic scales and non-dimensionalize the differential equation?

I am confused about this...can anyone expain me the rationale behind this 'scalin' process and the benefits one can get from it?

Many thanks!

 

[RainmanAero]

Hi all!
I always come across "scaling" in various subjects.
Like in Fluid mechanics, I am confused about why there is a need to choose some characteristic scales and non-dimensionalize the differential equation?

I am confused about this...can anyone expain me the rationale behind this 'scalin' process and the benefits one can get from it?

Many thanks!

Two reasons (but there are more):

1) By non-dimensionalizing your governing equations you are quantifying physical laws independently of whatever units you will eventually choose to build some device or product. This makes the data "portable" to any units systems, and the truths behind the science independent of that choice.

2) Most importantly think of the term "scaling". By non-dimensionalizing you are making it eminently easier to "scale-up" an effect from a small, investigative model all the way up to a full-size model. This is most useful in aircraft and aerodynamics. By quantifying the aerodynamic characteristics in non-dimensional coefficients and stability derivatives, and with respect to non-dimensional flow parameters like Reynolds Number, the results you discover with a model in a wind tunnel are DIRECTLY applicable to the full-scale version of the aircraft.

All-in-all, the best way to describe the advantage is "portability of the results".

Rainman

 

[Clausius2]

3) Scaling also enables you to neglect effects compared with some others in order to "localize" or "magnify" your equations about certain regions of your geometry. That's the aim of the perturbation theory and asymptotic analysis. Identifying disparities on time or length scales helps you to indentify small parameters of your problem and to do asymptotic expansions using asymptotic sequences based on such parameters. The DE you posted in the DE forum may be a non dimensionalized governing equation with $$\epsilon<<1$$ being a ratio of length scales which is just the perturbation parameter of your problem. In practical situations you will find $$\epsilon$$ to be the Reynolds Number (in viscous flow) or its inverse (in potential or boundary layer flow), the Mach Number (in compressible flow), the Strouhal Number (in unsteady flows), the Froude Number (in gravity waves), the Damköhler Number (in reactive flow), or also the thickness of the airfoil (in thin airfoil theory), the Knudsen Number (in rarified flows), a modified ratio of the adiabatic constant (in the Newton-Busseman approximation for hypersonic flows)...
All of them are non dimensional parameters that serve to us to work asymptotic expressions from slight departures of the $$\epsilon=0$$, which is called the leading order state.

To sum up, working out the "power" of each term of the N-S equations gives to us the needed insight to give preference to the important effects, and also gives to us the chance to see how the flow reacts under an small change of a small parameter (perturbation theory).

Now I remember some words that a math professor told us in our first math class in the university: "One of the things we will learn here will be the Taylor expansions, which by the way will be the most frequent mathematical instrument you will use in your future". I didn't understood that at that time, but now I realize that every time you neglect completely an effect in your governing equations, the solution you obtain is a Leading Order term of another longer solution which has into account small perturbations of the neglected effects (and God knows that those effects can have a CAPITAL importance in the physics even if they are present only in a very small quantity, as always happens in real world).