The Reynolds Number

About Rowing and Flying

You may know from hard experience that having to push the rowboat through the water with any worthwhile speed is not all that easy. Since a rowboat displacees its total loaded weight in water, you have to move that water bodily aside to get the boat ahead. Water, at say 62 pounds per cubic foot, has a good bit of density. Thus it is somewhat reluctant to move aside just to let you through. Here we come up against the water's inertia, a kind of laziness.

The Air's Viscosity

Like air, water also has viscosity or stickiness. While you are moving your rowboat through the water, the flow of the water creates skin-friction on the wetted hull-surfaces. Thus while rowing you have to overcome two different kinds of forces. There's the inertia, caused by the water's density, and the friction, caused by its stickiness.

Let's say you are moving the boat through the water at a speed of only one mile per hour. So little water is now moving aside that most of your energy goes to overcoming the skin-friction force of the water.

Then you start rowing at a speed of ten miles per hour, the bow-and stern-waves show that you are spending most of your energy moving the water aside. You now have to overcome the inertia of the water caused by its density rather than overcome the friction caused by the water's stickiness. Thus when going slow, the ratio of friction-work to inertia-work is high. When going fast, the ratio of inertia-work to friction-work is high.

The Air's Density

Now when you go out flying, you want your airplane to glide smoothly through the air. The air should flow over wing- and fuselage surfaces with a minimum of disturbance. While air is not nearly as dense as water, (GIVE FIGURE) it does have a certain density. Thus the air is also somewhat reluctant to move aside so your airplane can get through. Thus with your airplane as in the row boat, you also find you are up against the two important characteristics: density (inertia) and viscosity (stickiness).

The British scientist and Engineer Osborne Reynolds discovered these two main parameters years ago during his research into the flow of liquids in pipes. He found that laminar flow or turbulent flow depends only on the ratio of the inertial forces over the friction forces. This ratio of viscosity over density scientists call the "kinematic viscosity." For air at the standard temperature of 59 degrees Fahrenheit at Sea Level the value for this ratio is

0.0000003737 slugs / foot second
-------------------------------------------------- = 0.000156927
0.0023769 slugs / cubic foot

This standard numerical value for the kinematic viscosity, actually the ratio of viscosity over density, and the resulting value for the kinematic viscosity, forms the first part of the famous Reynolds Number formula.

Reynolds also showed that we must take two other factors into account. The first is the velocity or speed with which the air moves over the surface. For your airplane, this is its flying-speed. The second factor is the length dimension of the surface. For the wing, for example, this is the local chord-length. For other parts, it is just their length dimension.

Our calculation of the ratio gives us a simple four-digit number. We can use it without ever having anything to do with the actual values of either the density or the viscosity. Let's see how we get that number:

  Speed x Length
Reynolds Number = -------------------------------------------------
  kinematic viscosity

This is the same as multiplying (Speed x Length) by the product of

u / (1.0 / 0.000156927),

which gives us the number 6372. The complete formula is

Reynolds Number Re = 6378 times V(fps) times L(ength) in feet. It's that simple!


Density and Viscosity

The density, viscosity, and the kinematic viscosity all have in some way the unit (ft) in them. Therefore, in the formula, speed V and length L are respectively in feet per second and in linear feet.

For airspeed in miles per hour, because one mph = 1.46667 fps, we use the number

6378 x 1.46667 = 9354.

For airspeed in knots the constant is 10767. For the dimension in inches, like for the chord-length of model airplane wings, the number to use with speeds in mph is 9354/12 = 780.

Here's a simple example for a wing with a 10 feet chord at 100 mph

flying speed, at Sea Level and "Standard Day" conditions.

Re = 9346 x 100 x 10 = 9346 x 1000 = 9,346,000.

Per 100 mph of flying speed at sea level, the RN is roughly equal to 1 million per foot of length. Thus we can calculate the Reynolds Number for any chordwise position on the wing or any lengthwise point on the fuselage, and for any specific flying speed and density altitude. In general, for full-scale airplanes the nearest 100,000 or even half million figure will do.

In flight the wing's Reynolds Number is of course continuously growing. It starts at zero at the stagnation point in front of the wing's leading edge. While the air flows aft, the Reynolds Number continuously increases to its maximum value at the trailing edge.

With his famous formula, Osborne Reynolds gave us the master key to practical aerodynamics. His work makes it possible for us to make a direct, practical comparison of the boundary layer flow. On the wings and on other parts of our airplanes. Without it, modern aerodynamics would not have been possible. Our hats are off to him!


The Practical Significance of the Reynolds Number.

The resistance experienced by a wing in flight is a function of the Reynolds Number. Normally, the Reynolds Number is the decisive factor in the air-flow in determining whether the inertial effect or the viscous effect wins. Let's take a look at what the Reynolds Number values roughly tell us about airflow and drag.

If the Reynolds Number is large, the viscosity effect is small. For the for us practical values the inertia or density forces dominate, and the parasite drag increases with the square of the velocity. However, although the viscosity is unimportant, it may still affect the very thin boundary layer, leading to the creation of turbulent flow.

Thus the importance of the Reynolds Number is that it tells us the type of flow we can expect. It tells you whether you can hope for having laminar flow over the wing and other parts of your airplane. A low Reynolds Number gives laminar flow while a high Reynolds Number gives turbulent flow. For both a laminar and a turbulent boundary layer increasing Reynolds Number gives lower skin friction drag. However, because of the higher energy loss in the boundary layer, a turbulent layer always has higher skin friction drag.


The Critical Reynolds Number.

Near the wing's leading edge the Reynolds Number is relatively low.

Especially on a smooth wing surface, the boundary layer flow will be laminar at first. With increasing Reynolds Number, further downstream at some chord-wise location it reaches the local critical Reynolds Number.

This is when and where the boundary layer transitions to turbulent flow. The value at which it does so we therefore call the transition or critical Reynolds Number. This is the most important factor in determining transition to turbulence.

The change from laminar-flow conditions to turbulent-flow conditions at the critical Reynolds Number is not definite. The ranges of the Reynolds Number under which laminar- or turbulent-flow conditions exist depends much on the shape and (mostly) on the surface finish. It also depends on such factors as the initial steadiness of flow, absence of vibration, etc.

On the average (poor) wing surface usually it occurs at Reynolds Number values of at least 100,000, and from there up to 500,000. Under ideal conditions it is possible to maintain laminar-flow conditions for relatively very large values of Reynolds Number. However, such boundary-layer flow conditions are unstable. Keeping them stable is the big trick.

The fastest homebuilt airplanes at present obviously have accomplished this, as shown in their very high speeds on relatively low power.


Frictional Resistance (Profile Drag) and the Reynolds Number

The profile drag of a streamlined body varies with the Reynolds Number.

At higher Reynolds Number values the air flow is turbulent. Larg local fluctuations in the airstream velocity occur close to the surface. The irregularity of the turbulent-flow conditions results in much higher energy loss than if the flow were laminar. This accounts for the increased drag accompanying turbulent flow.

In these turbulent flow conditions, some of the pressure force driving the flow speeds up the eddies. Thus the effect of the viscous (sticky) surface flow on the wing's aerodynamic characteristics is to create a drag force. This drag force will consume a good bit of horsepower.


Low Minimum Drag

Low minimum drag at high Reynolds Numbers is the most important aerodynamic characteristic of an airplane. With laminar-flow conditions retained up to higher Reynolds Numbers, the drag will be very much lower.

The state-of-the-art GlassAir, Lancair, and Questar fiberglass homebuilt airplanes are good examples of this. It requires extreme degrees of accuracy of surface finish and laminar

flow conditions. Difficult to achieve, even more difficult to keep. Even slight disturbances like small particles of dust or insects cause the flow to become turbulent. Up goes the drag.

Although skin friction is not the only kind of resistance, it forms a large portion of the total drag. Extreme care in manufacturing a very smooth surface made it possible to keep laminar flow at very high Reynolds Numbers.

The aim is to find some method by which we could avoid turbulent flow altogether. Replacing high-drag turbulent-flow by low-drag laminar-flow at high Reynolds Number reduces the horsepower and fuel requirements a good deal. The improvements in both speed and economy of power are very worthwhile.


Some Reynolds Number Figures.

The Reynolds Number for full-scale flight varies from about 2,000,000 for small slow-speed airplanes to 20,000,000 for large high-speed airplanes. Here are some figures for a couple of light airplanes, at 75% cruise speed. We use the formula Re = 9324 x V(mph) x Av.chd.(ft).

VP-2 9346 x 87 x 4.815 = 3 904 248
BD-5 9346 x 200 x 2.235 = 4 167 828
Cessna 150 9346 x 108 x 4.800 = 4 833 562
Cherokee Cruiser 9346 x 124 x 4.857 = 5 615 547
T-18 9346 x 173 x 4.130 = 6 661 905
Questair Venture 9346 x 280 x 2.62 = 6 856 226
Questair Venture 9346 x 345 x 2.62 = 8 447 850
RV-4 9346 x 175 x 4.780 = 7 799 526
Bonanza V35B 9346 x 181 x 5.400 = 9 113 278

As you go up in altitude, the density of the air decreases. Because of the lower temperatures, the viscosity increases. All this only slightly changes the Reynlds Number.

For any altitude and temperature, under standard atmospheric conditions the air's density and viscosity are constants. Thus when calculating the Reynolds Number for flight at altitude, we must use the proper figures.

Copyright Harmen Koffeman 1992