As an object moves through the atmosphere, the gas molecules
of the atmosphere near the
object are disturbed and move around the object.
Aerodynamic
forces are generated between the gas and the object. The
magnitude of these forces depend on the shape of the object, the
speed of the object,
the mass
of the gas going by the object and on two other important properties
of the gas; the viscosity, or stickiness, of the gas and the
compressibility, or springiness, of the gas. To properly model
these effects, aerodynamicists use
similarity parameters
which are
ratios
of these effects to other forces present in the
problem. If two experiments have the same values for the similarity
parameters, then the relative importance of the forces are being
correctly modeled. Representative values for the properties of
air
are given on another page,
but the actual value of the parameter depends
on the
state of the gas
and on the
altitude.

Aerodynamic forces depend in a complex way on the
viscosity
of the gas.
As an object moves through a gas, the gas molecules stick to
the surface. This creates a layer of air near the surface, called a
boundary layer,
which, in effect, changes the shape of the
object. The flow of gas reacts to the edge of the boundary layer as if
it was the physical surface of the object. To make things more
confusing, the boundary layer may
separate
from the
body and create an effective shape much different from the physical
shape. And to make it even more confusing, the flow conditions in and
near the boundary layer are often unsteady (changing in time).
The boundary layer is very important in determining the
drag
of an object. To determine and predict these conditions,
aerodynamicists rely on
wind tunnel
testing and very sophisticated computer analysis.

The important similarity parameter for viscosity is the
Reynolds number. The Reynolds number expresses the ratio of
inertial (resistant to change or motion) forces to viscous
(heavy and gluey) forces. From a detailed
analysis of the
momentum conservation equation,
the inertial forces are characterized by the product of the
density r times the velocity V times the
gradient of the velocity dV/dx. The viscous forces
are characterized by the dynamic viscosity coefficient mu times
the second gradient of the velocity d^2V/dx^2. The
Reynolds number Re then becomes:

Re = (r * V * dV/dx) / (mu * d^2V/dx^2)

The gradient of the velocity is proportional to the velocity
divided by a length scale L. Similarly, the second derivative
of the velocity is proportional to the velocity divided by the square
of the length scale. Then:

Re = (r * V * V/L) / (mu * V / L^2)

Re = (r * V * L) / mu

The Reynolds number is a dimensionless number. High values of the parameter
(on the order of 10 million) indicate that viscous forces are small and the
flow is essentially inviscid. The
Euler equations
can then be used to model the flow. Low values of the parameter
(on the order of 1 hundred) indicate that viscous forces must be considered.

The Reynolds number can be further simplified if we use the
kinematic viscositynu that is euqal to
the dynamic viscosity divided by the density:

nu = mu / r

Re = V * L / nu

Here's a JavaScript program to calculate the coefficient of viscosity
and the Reynolds number
for different altitude, length, and speed.

Due to IT
security concerns, many users are currently experiencing problems running NASA Glenn
educational applets. The applets are slowly being updated, but it is a lengthy process.
If you are familiar with Java Runtime Environments (JRE), you may want to try downloading
the applet and running it on an Integrated Development Environment (IDE) such as Netbeans or Eclipse.
The following are tutorials for running Java applets on either IDE:
Netbeans Eclipse

Similarity Parameter Calculator

Please Input Altitude, Speed, and Length Scale

Input

feet

mph

feet

Output

Speed/Mach Number

Speed

Speed of Sound

Dynamic Press

Mach #

Compressibility

P static

P total

T static

T total

Viscosity

Density

Dynamic Coef.

Kinematic Coef.

Reynold's #

To change input values, click on the input box (black on white),
backspace over the input value, type in your new value, and
hit the Enter key on the keyboard (this sends your new value to the program).
You will see the output boxes (yellow on black)
change value. You can use either Imperial or Metric units and you can input either the Mach number
or the speed by using the menu buttons. Just click on the menu button and click
on your
selection.
The non-dimensional Mach number and Reynolds number are displayed in
white on blue boxes.
If you are an experienced user of this calculator, you can use a
sleek version
of the program which loads faster on your computer and does not include these instructions.
You can also download your own copy of the program to run off-line by clicking on this button:

For some problems we can divide the Reynolds by the length scale to obtain the
Reynolds number per foot Ref. This is given by:

Ref = V / nu

The Reynolds number per foot (or per meter) is obviously
not a non-dimensional number like the Reynolds number.
You can determine the Reynolds number per foot using the calculator
by specifycing the length scale to be 1 foot.