58
2 Water at Rest and in Motion
direction. This may cause the cylinder to oscillate from one side to the other,
particularly when the frequency of these oscillations is very close to the cylinder's natural frequency of oscillation. Such oscillations are not observed for a
sphere.
When motion is very slow, for example, when a small spherical particle is
falling in calm water, the inertial effects are negligible, and the Navier-Stokes
equations (see Appendix C.3.2) can be solved analytically. The following result
was obtained by Stokes in the nineteenth century (see Eq. (C.56):
(2.72)
where Uo is the free stream velocity and v is the coefficient of kinematic viscosity.
By comparing Eq. (2.72) with the general drag form for a sphere (2.71) we
obtain:
(2.73)
Thus:
( D)-l 24
Cd = 24 u:
Re'
(2.74)
in which Re = uoD Iv is the Reynolds number based on the sphere diameter.
This relationship may be used with negligible error up to Re = 0.2. We will
return to this relationship in Sect. 2.6.5 where a determination of terminal
velocity is discussed.
In Fig. 2.25, some data on drag coefficients for various man-made shapes
and marine organisms for Reynolds numbers between 10 4 and 10 6 are shown.
This data is compiled from various sources (Denny, 1988; Vogel, 1994; White,
1994), and therefore they may not be exactly comparable due to variations
in the experimental conditions, and due to use of different definitions of the
surface S. For artificial objects in Fig. 2.25, the surface, S, is the 'frontal' or
projected area of an object - its maximum projection onto the plane normal
to the direction of flow. In the case of marine organisms, it is more relevant to
use another definition of area S, such as 'wetted area', or 'plan form area'. We
will discuss these definitions in Sect. 11.3.
From Fig. 2.25 it is clear that sharp edges always cause flow separation and
high drag. Rounded bodies, such as the ellipsoid, have a drag coefficient which
depends upon the point of separation; both the Reynolds number and the character of the boundary layer are important. A good example of this dependence
are the drag coefficients of automobiles. The famous Model T Ford had a Cd
of the order of 0.8-0.9, while modern cars have an average drag coefficient of
about 0.30, with a constant trend to decrease (White, 1994).
2 Water at Rest and in Motion
direction. This may cause the cylinder to oscillate from one side to the other,
particularly when the frequency of these oscillations is very close to the cylinder's natural frequency of oscillation. Such oscillations are not observed for a
sphere.
When motion is very slow, for example, when a small spherical particle is
falling in calm water, the inertial effects are negligible, and the Navier-Stokes
equations (see Appendix C.3.2) can be solved analytically. The following result
was obtained by Stokes in the nineteenth century (see Eq. (C.56):
(2.72)
where Uo is the free stream velocity and v is the coefficient of kinematic viscosity.
By comparing Eq. (2.72) with the general drag form for a sphere (2.71) we
obtain:
(2.73)
Thus:
( D)-l 24
Cd = 24 u:
Re'
(2.74)
in which Re = uoD Iv is the Reynolds number based on the sphere diameter.
This relationship may be used with negligible error up to Re = 0.2. We will
return to this relationship in Sect. 2.6.5 where a determination of terminal
velocity is discussed.
In Fig. 2.25, some data on drag coefficients for various man-made shapes
and marine organisms for Reynolds numbers between 10 4 and 10 6 are shown.
This data is compiled from various sources (Denny, 1988; Vogel, 1994; White,
1994), and therefore they may not be exactly comparable due to variations
in the experimental conditions, and due to use of different definitions of the
surface S. For artificial objects in Fig. 2.25, the surface, S, is the 'frontal' or
projected area of an object - its maximum projection onto the plane normal
to the direction of flow. In the case of marine organisms, it is more relevant to
use another definition of area S, such as 'wetted area', or 'plan form area'. We
will discuss these definitions in Sect. 11.3.
From Fig. 2.25 it is clear that sharp edges always cause flow separation and
high drag. Rounded bodies, such as the ellipsoid, have a drag coefficient which
depends upon the point of separation; both the Reynolds number and the character of the boundary layer are important. A good example of this dependence
are the drag coefficients of automobiles. The famous Model T Ford had a Cd
of the order of 0.8-0.9, while modern cars have an average drag coefficient of
about 0.30, with a constant trend to decrease (White, 1994).