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the Introduction.

1.B - Physical rationale for linear and quadratic drag regimes

There are simple explanations for these two regions.

Linear:

At low speed (more precisely, at low Reynolds number) the drag force is
essentially linear in v because the viscous forces, which arise as fluid
molecules try to slide past one another in shear flow (i.e. flow parallel to the
surface of the object), are linear in the flow velocity. Drag linear in v
is what we include in physics problems when a damper (viscous shock absorber) is
added to a system, like the classic driven damped harmonic oscillator. It
applies whenever speed is less important than viscosity as measured by
Re.

The theory for this low-Re regime was developed by Stokes, who
obtained

From this we see that Stokes predicts CD =
24/Re, where Re = vD/ and L2 = with D = 2R the diameter of the sphere. This result
translates into our notation, where we want the velocity dependence explicit, if
we define

with the linear drag constant KStokes being given by 24
/D for a sphere of diameter D traveling at very low
speeds. [This constant has to be multiplied by /2 and L2 to obtain the K1 that we
use in our analysis of linear
drag.] The graph in Section 1.A shows that this is only valid for extremely
low speeds or high viscosities where Re is less than 1.

Quadratic:

At higher Reynolds number the pressure forces normal (perpendicular) to the
surface of the object dominate. These are inertial forces that, since pressure
arises from the average KE and KE is quadratic in the velocity, have a
v2 dependence.

Drag crisis:

If you look at the CD graphs for a smooth sphere and
cylinder, you will notice a sharp drop in the drag coefficient that ends the
region with drag that goes like v2. This "drag crisis" is a
result of the onset of turbulent flow. The effects of turbulence for non-smooth
objects is what makes it necessary to use empirical drag forces (or very
sophisticated models) for most interesting real-world problems. For example, a
rough object like a baseball (see graph in the last part of Section 1) will
produce turbulence at lower speeds so the force varies with speed in a
complicated fashion.

Supersonic:

There is a third source of drag, energy that goes into the formation of a
shock wave when the flow goes supersonic. This will not concern us here.

Jim Carr
2000-02-17