The aerodynamic drag on an object depends on several
factors, including the
and flow conditions.
All of these factors are related to the value of the drag through the
D = Cd * .5 * rho * V^2 * A
Where D is equal to the drag, rho is the air
density, V is the
velocity , A is a
reference area, and Cd is the
The drag coefficient is a
dimensionless number that characterizes all of the complex factors that affect drag.
The drag coefficient is usually determined
experimentally using a model in a
wind tunnel. In the tunnel, the velocity, density, and size
of the model are known. Measuring the drag then determines the value of the drag
coefficient as given by the above equation. The drag coefficient and the drag equation
can then be used to determine the drag on a similar shaped object at different
flow conditions as long as several flow
similarity parameters are matched. In particular,
Mach number similarity insures that the
compressibility effects are correctly modeled,
Reynolds number similarity insures that the
viscous effects are correctly modeled.
The Reynolds number is the ratio of the
inertia forces to the viscous forces and is given by:
Re = V * rho * l / mu
where l is a reference length, and mu is the
For most aerodynamic
drag coefficient has a nearly constant value across a large range of Reynolds numbers.
But for a simple sphere, the value of the drag coefficient varies widely with
Reynolds number as shown on the figure at the top of this page.
To understand these variations, we are going
to look in some detail at the flow past a cylinder. The two dimensional flow past
a cylinder is very similar to the three dimensional flow past a sphere, but is a little
easier to compute and understand because of the reduced dimensionality.
Flow past a cylinder and a sphere goes through a
number of transitions with velocity as shown in this figure:
In all of the cases presented on this figure,
the density, viscosity, and diameter of the ball are the same. The flow velocity
is gradually increased from the left to increase the Reynolds number. The drag for
each case number is noted on the figure at the top of the page near the drag curve.
Case 1 shows very slow flow in which we have neglected viscosity entirely.
We have an
ideal flow with no
boundary layer along the surface,
completely attached flow (no separation) and
no viscous wake downstream of the cylinder.
Because the flow is symmetric from
upstream to downstream, there is no drag on the cylinder! This strange result is called
d'Alembert's paradox after the early mathematician who studied the problem.
Neglecting viscosity simplifies the analysis,
but this type of flow does not occur in nature where there is always some
small amount of viscosity present in any fluid.
Case 2 illustrates what actually occurs for low velocity. A stable
pair of vortices are formed on the downwind side of the cylinder.
The flow is separated but steady and
the vortices generate a high drag on the cylinder or sphere.
Case 3 shows the flow as velocity is increased.
The downstream vortices become unstable, separate from the body
and are alternately shed downstream.
The wake is very wide and generates a large amount of drag.
The alternate shedding is called the Karman vortex street.
Here is a computer animation of the shedding, with the flow going from right to left.
This type of flow is periodic, it is unsteady but repeats itself at some
time interval. The pressure variation associated with the velocity changes produces a
sound called an aeolian tone. This is the sound
you hear when the wind blows over high-power wires or past tree limbs in
the fall or winter. It is a low frequency (low pitch), haunting tone.
Case 4 shows the flow as the velocity is increased even more.
The periodic flow breaks down into a chaotic wake. The flow in the boundary layer on
the windward side of the cylinder is
laminar and orderly and the chaotic wake is initiated
as the flow turns onto the leeward side of the cylinder. The wake is not quite as wide
as for Case 3, so the drag is slightly less.
Case 5 shows the flow at even higher velocity. The boundary layer
transitions to chaotic turbulent flow with vortices of many different
scales being shed in a turbulent wake from the body. The separation point
is initially slightly downstream from the laminar separation point, so the wake is
initially slightly smaller and the drag is less than the corresponding
laminar drag (Case 4). Increasing velocity eventually brings the turbulent drag
up to and even higher than the laminar drag value, but there is a range of Reynolds
numbers, during transition from laminar to full turbulent,
for which the turbulent drag is less than the laminar drag.
All of the calculations shown for the cylinder have employed a smooth surface. But since
drag depends on the flow in the boundary layer, we can expect some changes with
surface roughness. It is observed
that a roughened cylinder or ball will transition to turbulent flow at a lower
Reynolds number than a smooth cylinder or ball. Drag for a rough surface ball
is shown on the figure at the top of the page by the dashed
line. This effect produces a rather interesting result. There is a small range
of Reynolds numbers for which the drag of a roughened ball is less than the drag
of a smooth ball for the same diameter, velocity and flow conditions. The size and speed
of a golf ball falls within this Reynolds number range. That is why a golf ball has dimples;
the roughened surface causes transition to turbulence that would not occur yet on a smooth
golf ball. The lower drag on the dimpled golf ball allows the ball to fly farther than
a smooth ball of the same speed, diameter and weight.
You can further investigate the effect of airfoil shape and the other
factors affecting drag by using the
FoilSim III Java Applet.
You can also
your own copy of FoilSim to play with
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