How can we physically define reynold number variation in terms of characteristic length?

Question

How can you define it physically that greater the characteristic length in $Re=\frac{\rho v L_c}{\mu}$ greater the Reynolds number. Like on wing surface $Re$ is higher than the horizontal tail surface (MiG 29).

Answer 1

The freestream Reynolds number ($Re_c$) of the wing and hstab would be different due to different chord lengths:

$$Re_c=\frac{\rho_\infty V_\infty c}{\mu}$$

where $\rho_\infty$ is the freestream air density, $V_\infty$ is the freestream airspeed and $c$ is the reference chord (of the wing or hstab). I'll discuss the physical significance of reference chord below.

1. Viscous drag

The physical significance on drag is best understood from the flat plate Blasius solution in laminar flow. The overall friction drag coefficient ($C_f$) decreases as a square root of freestream Reynolds number:

$$C_f=\frac{1.328}{\sqrt{Re_c}}$$

So the longer the chord (i.e. higher Re), the less significant the boundary layer viscous effects are to the overall drag. This is further illustrated in the graph below.

_{Image ref: https://www.sciencedirect.com/topics/engineering/friction-drag-coefficient}

Take note, however, that the dimensionalized (total) friction drag still grows since $C_f$ here is non-dimensionalized with respect to the reference chord, $c$.

2. Turbulence transition

Increasing the chord doesn't increase the momentum thickness in absolute terms. However, the relative thickness with respect to the chord is different for different chord lengths:

$$\theta(s) = \frac{0.664\sqrt{\mu}}{\sqrt{\rho_\infty V_\infty}}s^{1/2}$$

where $s$ is the chordwise position from the leading edge.

Since natural transition is a function of the momentum thickness Reynolds number (Ref. Drela, Flight Vehicle Aerodynamics), which increases with respect to the momentum thickness, transition to turbulence will occur at similar chordwise location, $s$, in absolute terms, but earlier in relative terms ($s/c$) for the longer chord.

This has the effect of increasing the stall angle of attack for the larger chord, manifested in the effective increase in the freestream Reynolds number.

Answer 2

This question is not bad, as Reynolds number is determined by velocity, working fluid, and chord length.

Some designers try to make horizontal stabilizers miniature versions of their wings, resulting in safety issues. Some even made the Hstab higher aspect, such as ones linking two fuselage "booms" together, abandoning the robust wedge shaped tails commonly seen on birds and biplanes of 100 years ago.

The result is a tail that stalls before its wing, not very good if you are diving towards earth.

Changing Reynolds number (of a smaller airfoil) can be counter-acted with change in shape to improve safety. Note both the Virgin Galactic "Knight" and the "Spaceship" now both sport rather Spitfire-like separate boom stabilizers.

As far as quantifying and understanding the effect of chord on airfoil properties at various Reynolds numbers, this would be an improvement in predicting the effects of changing Reynolds numbers (now seen on polar graphs). Sort of like the Reynolds number as the molecule, and Velocity, Kinematic Viscosity, and Chord as its atoms.

Of these "atoms", Kinematic Viscosity is constant, chord and velocity are variable. Effects of chord and Velocity changes are worth study in more detail.