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What does the Reynolds Number of a flow represent physically?

I am having trouble understanding the meaning and the utility of the Reynolds number for a certain flow, could someone please tell me how this type of dimensionless factor is significant and what it tells us about a problem?

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up vote 6 down vote accepted

From the Wikipedia article for Reynolds number:

In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.

In addition to measuring the ratio of inertial to viscous forces in a flow, the incompressible Navier-Stokes equations can be written in non-dimensional form such that the only parameter is the Reynolds number (ignoring body forces). This is very nice because it is the basis for the validity of wind tunnel testing.

Suppose we would like to measure the aerodynamics of the flow around a Boeing 747. Two (at least) options exist:

  1. Build your very own full size 747, instrument it, and fly it. (extremely expensive)
  2. Build a small scale model of a 747, instrument it, test inside a wind tunnel (much less expensive)

But how do we know that the flow we measure in the wind tunnel is what really happens in flight? We match the Reynolds numbers and the exact same equations model both situations--therefore the aerodynamics must be the same. (Ignoring compressibility effects.)

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And what dimensionless numbers rule compressibility effects if I'd like to take them into account? Mach number? –  firtree May 11 '13 at 7:08
    
@firtree The Mach number shows up in the Energy equation with the $Dp/Dt$ and dissipation terms. For low Mach numbers, these terms are negligible. For high Mach numbers these terms increase the spatial temperature gradients and couple the momentum and energy equations. This coupling makes it hard (perhaps impossible) to match Reynolds number and Mach number for compressible experiments. At this point it is up to the experience of the experimentalists to decide the parameters of the tests so that reality is best approximated for scale models. –  OSE May 12 '13 at 20:39
    
So for high Mach numbers scaled experiments are either impossible or (extremely?) inaccurate? Or I got you wrong? –  firtree May 13 '13 at 5:39
    
@firtree Someone please correct me if I'm wrong: I believe for scaled experiments at high Mach numbers, it is most important to match Mach number so the shock structure is the same. If the Mach number is matched the scaled model's Reynolds number will be less than that of the full size model. To approximate higher Reynolds numbers, trip strips are placed where transition from laminar to turbulent flow is expected. If the transition location is unknown, an array of various trip strip locations could be tested. –  OSE May 13 '13 at 14:34
    
Also incompressible measurements can be extended to compressible flow using the Prandtl-Glauert Transformation. –  OSE May 13 '13 at 14:34
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