# Reynolds Number and Mach Number Effects   Two of the most well known parameters used in aerodynamics are the Reynolds number and the Mach number. For an airfoil, its size is described by a characteristic length based on the chord, c. When the airfoil is moving through a fluid of viscosity д, density p, and sonic velocity a, and with a speed Voo at some relative orientation to the flow, then the method of dimensional analysis shows that the force on the airfoil, F, can be written in functional form as

The combinations pV^c/p, and Voo la are called the Reynolds number (denoted by Re) and Mach number (denoted by M), respectively. Unfortunately, these parameters have both independent and interdependent influences, which complicates the understanding of the problem of finding their effects on the aerodynamic force.

7.3.1 Reynolds Number

The physical significance of the Reynolds number is that it represents the ratio of the inertial forces to the viscous forces in the fluid. This can be seen by writing the Reynolds number as

pVooC pVoociVooc) pV^c2 Inertial force

Ke = ——– = —————– = —————– = —————— . (/.Z)

M Ц’іУооС) At(Voo/c)c2 Viscous force

On the numerator, pV^c2 has units of force and so it represents an inertial force. The coefficient of viscosity, д, is the shear force per unit area per unit velocity gradient. The denominator, therefore, is a viscous force. For an ideal fluid, the Reynolds number is effectively infinite. However, when viscous forces are dominant, the Reynolds number is small in value. For helicopters the range of Reynolds numbers encountered by the rotor is significant, and Reynolds number effects on its aerodynamics are significant. The aerodynamic characteristics of rotor airfoils must be assessed at their actual opera­tional Reynolds numbers and Mach numbers. Figure 7.2 illustrates the operational Reynolds number and Mach number ranges typical of helicopter rotors, both at full scale and model (sub) scale. The maximum lift coefficient, C/max, can be used as one indicator of the sig­nificance of viscous effects. At the low end of the practical Reynolds number range for rotors (i. e., for chord Reynolds numbers in the range 105 < Re < 106), most airfoils have    (b) Turbulent boundary layer 