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the Overview.

1. Introduction to Drag Forces

Aerodynamic drag forces depend on the size and shape of the object, the
density and viscosity of the fluid medium, and the speed of the fluid flow
around the object. The standard approach [1]
is to make explicit some of the dependences on these variables and hide the rest
in an empirical drag coefficient. (This study will completely ignore the Magnus
Effect, the cause of sideways forces associated with rotation of the moving body
which are essential for a proper treatment of the flight of a baseball.)

The drag force is always in the direction opposite to the velocity (that is,
) and has a magnitude given by

(1)

Here is the density of the medium, L is a ``typical dimension'' of
the object being considered, and CD is the drag coefficient of
that object as a function of the Reynolds number, Re. One should also
note that the velocity v is defined relative to the rest frame of the
medium, which is the frame we use for all of the analyses in this article. More
care is needed if there are windage effects to be considered. Two standard
graphs of measured values for CD are included in Section 1.A.

Jump to
Section 1.A:
Tables and graphs of standard data

What value of L you use is defined by the convention adopted for the
measured values of CD. For example, the area
L2 might be for a sphere or disk. Operationally, one measures
Fdrag for a given set of parameters and deduces the value of
CD by using this formula with a specific definition of
L. You must know what definition was used when CD was
determined in order to use some graphed or tabulated data in the formula.

The Reynolds number is a dimensionless parameter defined by

where is the ``kinematic viscosity'' and = is the viscosity of the fluid medium. Some typical values for the
viscosity are tabulated in
Section 1.A. The point of using the Reynolds number in Eq. (1) is
that the drag coefficient for a particular shape is the same for different
fluids as long as this ratio remains constant. This scaling allows you to relate
data for experiments done with different fluids and objects of different size to
one another.

Our parameterization

For our purposes it will be convenient to isolate the dependence of the drag
force on one dynamical parameter, the velocity v. For a particular fluid
medium (assumed to have a density and viscosity that does not depend on
position, a condition violated if the problem extends over a large range of
altitudes in the atmosphere) and an object of fixed size and shape, we will
define

so that the drag force becomes

(2)

The functional dependence of K(v) can be obtained from the graphs
or the approximate formula provided in Section 1.A, fit to specific data for the
problem of interest, or simply approximated by some expression that allows one
to solve the dynamical equations analytically.

My emphasis here will be on analytic solutions for the free-fall problem for
forces with simple forms for K(v).

In one dimension (with ``down'' positive), the free fall of a body is due the
the net force given by

We discuss the solution of this problem for certain special cases in the
next section. Here we note that the speed when Fnet vanishes
is the terminal velocity, found by solving

for v = Vmax. (This is usually a non-linear
equation, so the secant method or a graphical method would need to be used to
find Vmax.) Knowledge of Vmax can be helpful
when deciding if drag forces will be significant for the speeds typical in some
situations.

Special cases that can be solved analytically - and which also make good
approximations in certain real situations - are K(v) =
K1/v and K(v) = K2. The first
produces a drag force linear in v, typical for very low speeds or high
viscosity (i.e. small Re), while the second produces a quadratic
dependence on the speed as is commonly encountered for larger Re. Both of
these situations can be seen in the CD graphs in Section 1.A and
are explained in Section 1.B.

Jump to
Section 1.B:
Physical rationale for linear and quadratic drag
regimes

Real objects

An object like a baseball, although essentially spherical, has surface
defects (stitching) that make it different from the perfectly smooth sphere used
for the measurements that produce the CD graph in Section 1.A.
Similarly, since we are interested in its behavior at quite large Reynolds
numbers, the special case of the Stokes
formula would not apply even if the ball were smooth. For most real objects,
one has no choice but to use empirically determined parametrizations of the drag
force or of K(v) or do a very sophisticated model calculation of the flow
around the object. An example of data for drag forces on a baseball is given
below.

Figure 1.3:

Graph, adapted from Ref. [2], of
Fdrag versus v
for a baseball (red curve) along
with a linear
approximation (light blue curve) to those data.
The green
circle shows the terminal velocity.

Note that Fdrag is quadratic for low speeds and then goes
through a transition region to a different quadratic dependence at high speeds.
The region of most interest for the game of baseball happens to be where the
forces are particularly complicated, and this is no accident -- it is due to the
design of a ball that can be used to throw curves and other pitches between 70
and 100 mph. These data also tell us that numerical solutions are the only
feasible approach to the analysis of baseball. If I did have to make analytic
estimates, I would use the linear drag approximation that gives the correct
terminal velocity, the light blue line shown in the figure.

Next, the
General Solution for Simple Drag Parametrizations.

Jim Carr
2000-02-17