this range the particles are larger and denser, and the fall velocity is faster. The physical reason for this

change in the behavior of *C*D is that inertial drag forces have become predominant over the viscous forces,

and the wake behind the particle has become turbulent. At about *Re = 200,000 *the drag coefficient decreases

abruptly. This is the region of very large particles at high fall velocities. Here, not only is the wake turbulent,

but the flow in the boundary layer around the particle is turbulent as well.

(4) In the first region, Stokes found the analytical solution for *C*D as:

24

24 ν

'

'

C*D*

(III-1-8)

(5) This line is labeled "Stokes" in Figure III-1-5. Substituting Equation 1-8 into Equation 1-7 gives the

fall velocity in this region:

ρs

(III-1-9)

&1

18 ν

ρ

(6) Note that in this region the velocity increases as the square of the grain diameter, and is dependent

upon the kinematic viscosity.

(7) For the region of *400 < Re < 200,000*, the approximation *C*D~0.5 is used in Equation 1-7 to obtain:

1

ρs

2

(III-1-10)

&1

Wf ' 1.6 *g D*

ρ

(8) Here it is seen that the fall velocity varies as the square root of the grain diameter and is independent

of the kinematic viscosity.

(9) Similarly, in the region *Re > 200,000*, the approximation *C*D~0.2 is used in Equation 1-7 to obtain:

1

ρs

2

(III-1-11)

&1

' 2.6 *g D*

W*f*

ρ

(10) There is a large transition region between the first two regimes (between *0.5 < Re < 400*). For

quartz spheres falling in water, these Reynolds numbers correspond to grain sizes between about 0.08 mm

and 1.9 mm. Unfortunately, this closely corresponds to all sand particles, as seen in Table III-1-2. Thus, for

very small particles (silts and clays), the fall velocity is proportional to *D*2 and can be calculated from

Equation 1-9. For gravel size particles the fall velocity is proportional to *D* and can be calculated from

Equation 1-10. However, for sand, the size of most interest to coastal engineers, no simple formula is

available. The fall velocity is in a transition region between a *D*2 dependence and a *D* dependence. In this

size range, it is easiest to obtain a fall velocity value from plots such as Figure III-1-6, which show the fall

velocity as a function of grain diameter and water temperature for quartz spheres falling in both water and

air. The vertical and horizontal axes are grain diameter and fall velocity, in millimeters and centimeters per

second, respectively. The short straight lines crossing the curves obliquely are various values of *Re*.

Coastal Sediment Properties

III-1-23

Integrated Publishing, Inc. |