### What is reduced Reynolds number?

Ashok Gopalarathnam. First created: 2 March 98

Recuded Reynolds number is Re*sqrt(CL). By using reduced Reynolds number for airfoil performance plots, it is possible to take into consideration that the Reynolds number decreases with increasing values of the lift-coeffecient, owing to decreasing flight-speeds.

We know that the airplane lift coeffecient varies inversely with the square of the air speed:

CL = (W/S)/(0.5*rho*V^2),                                         - eqn. 1

where,
CL is the airplane lift coeffecient, approximately equal to the wing lift coeffecient,
rho is the air density, and
V is the flight speed

We also know that for a given chord, Re varies linearly with flight speed:

Re = rho*V*c/mu,                                                        - eqn 2

where,
Re is the Reynolds number at any section whose chord is c, and
mu is the dynamic viscosity of air

We can see from eqn. 2 that the Reynolds number changes as we increase speed from stall speed to cruise speed. So if we want to examine airfoil data, we have to look at stall characteristics at the Re corresponding to stall speeds, the climb performance at Re for climb and cruise perofrmance for Re corresponding to cruise speeds.

To avoid having to look at curves for different Re for different flight conditions, we can instead use reduced Re. From eqn.1 and eqn. 2, we can see that

Reduced Re = Re*sqrt(CL) = (1/mu)*sqrt(2*(W/S)*rho)*c            - eqn 3

From equation 3, for a given atmospheric condition (say sea-level) , wing loading, and chord, the reduced Re is a constant, even though the actual Re changes with flight speed.

The following figure shows the variation of CL and Re for a W/S of 12.85 psf, a chord of 36.2 inches and a reduced Re of 2 million at sea level. As seen from the figure, even though the actual Re varies  with flight speed from about 1 million to about 6 million, the reduced Re is always 2 million.

Fig. 1 Variation of CL and Re with flight speed.
The numerical values used for this plot are available here.

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