Page 63 - Mechanical Engineers' Handbook (Volume 4)

52 Fluid Mechanics

The U.S. standard atmosphere is used in measuring altitudes with altimeters (pressure
gauges) and, because the altimeters themselves do not account for variations in the air tem-
perature beneath an aircraft, they read too high in cold weather and too low in warm weather.

3.4 Static Stability

For the atmosphere at rest, if an air mass moves very slowly vertically and remains there,
the atmosphere is neutral. If vertical motion continues, it is unstable; if the air mass moves
to return to its initial position, it is stable. It can be shown that atmospheric stability may
be defined in terms of the polytropic exponent. If n k, the atmosphere is stable (see Table
2); if n k, it is neutral (adiabatic); and if n k, it is unstable.
The stability of a body submerged in a fluid at rest depends on its response to forces
which tend to tip it. If it returns to its original position, it is stable; if it continues to tip, it
is unstable; and if it remains at rest in its tipped position, it is neutral. In Fig. 9 G is the
center of gravity and B is the center of buoyancy. If the body in (a) is tipped to the position
in (b), a couple Wd restores the body toward position (a) and thus the body is stable. If B
were below G and the body displaced, it would move until B becomes above G. Thus stability
requires that G is below B.
Floating bodies may be stable even though the center of buoyancy B is below the center
of gravity G. The center of buoyancy generally changes position when a floating body tips
because of the changing shape of the displaced liquid. The floating body is in equilibrium
in Fig. 10a. In Fig. 10b the center of buoyancy is at B , and the restoring couple rotates the
1
body toward its initial position in Fig. 10a. The intersection of BG is extended and a vertical
line through B is at M, the metacenter, and GM is the metacentric height. The body is stable
1
if M is above G. Thus, the position of B relative to G determines stability of a submerged
body, and the position of M relative to G determines the stability of floating bodies.

4 FLUID KINEMATICS
Fluid flows are classified in many ways. Flow is steady if conditions at a point do not vary
with time, or for turbulent flow, if mean flow parameters do not vary with time. Otherwise
the flow is unsteady. Flow is considered one dimensional if flow parameters are considered
constant throughout a cross section, and variations occur only in the flow direction. Two-
dimensional flow is the same in parallel planes and is not one dimensional. In three-
dimensional flow gradients of flow parameters exist in three mutually perpendicular
directions (x, y, and z). Flow may be rotational or irrotational, depending on whether the

Figure 9 Stability of a submerged body. Figure 10 Floating body.