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058:0160 Chapter 2
Jianming Yang Fall 2012 16
8 Buoyancy and Stability
8.1 Archimedes Principle
= fluid weight above 2ABC fluid weight above 1ADC
= weight of fluid equivalent to body volume
In general,
( = displaced fluid volume).
The line of action is through the centroid of the displaced
volume, which is called the center of buoyancy.
Example: Oscillating floating block
Weight of the block where is displaced water volume by the block and is the specific weight of the liquid, waterline area .
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Instantaneous displaced water volume:
Solution for this homogeneous linear 2nd-order ODE:
Use initial condition ( ) to determine and :
Where the angular frequency
period
Spar Buoy
We can increase period by increasing block mass and/or decreasing waterline area .
http://upload.wikimedia.org/wikipedia/com
mons/0/03/Lateral_view_of_spar-buoy.png
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8.2 Stability: Immersed Bodies
Stable Neutral Unstable
Condition for static equilibrium: (1) Fv=0 and (2) M=0
Condition (2) is met only when C and G coincide, otherwise we can have either a righting
moment (stable) or a heeling moment (unstable) when the body is heeled.
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8.3 Stability: Floating Bodies
For a floating body the situation is slightly more complicated since the center of
buoyancy will generally shift when the body is rotated, depending upon the shape of the
body and the position in which it is floating.
The center of buoyancy (centroid of the displaced volume) shifts laterally to the right for
the case shown because part of the original buoyant volume aOc is transferred to a new
buoyant volume bOd.
The point of intersection of the lines of action of the buoyant force before and after heel
is called the metacenter M and the distance GM is called the metacentric height.
If GM is positive, that is, if M is above G, then the ship is stable;
however, if GM is negative, then the ship is unstable.
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Consider a ship which has taken a small angle of heel
1. evaluate the lateral displacement
of the center of buoyancy,
2. then from trigonometry, we can solve for GM and evaluate the
stability of the ship
Recall that the center of buoyancy is
at the centroid of the displaced
volume of fluid (moment of volume
about y-axis ship centerplane)
This can be evaluated conveniently as follows:
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: moment of before heel (goes to zero due to symmetry of original buoyant
volume about centerplane)
: area moment of inertia of ship waterline about its tilt axis
This equation is used to determine the
stability of floating bodies:
If GM is positive, the body is stable
If GM is negative, the body is unstable
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8.4 Roll
The rotation of a ship about the longitudinal
axis through the center of gravity.
Consider symmetrical ship heeled to a very
small angle . Solve for the subsequent motion due only to hydrostatic and
gravitational forces.
Note: recall that | | , where is the perpendicular distance from to the line of action of :
Angular momentum:
= mass moment of inertia about long axis through
= angular acceleration
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For small :
Definition of radius of gyration:
The solution to equation
is,
where = the initial heel angle, for no initial velocity, the natural frequency
Simple (undamped) harmonic oscillation with period of the motion:
Note that large GM decreases the period of roll, which would make for an uncomfortable
boat ride (high frequency oscillation).
Earlier we found that GM should be positive if a ship is to have transverse stability and,
generally speaking, the stability is increased for larger positive GM. However, the
present example shows that one encounters a design tradeoff since large GM decreases the period of roll, which makes for an uncomfortable ride.
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9 Case (2): Rigid Body Translation or Rotation
In rigid body motion, all particles are in combined translation and/or rotation and
there is no relative motion between particles; consequently, there are no strains or strain
rates and the viscous term drops out of the N-S equation.
from which we see that acts in the direction of , and lines of constant pressure must be perpendicular to this direction (by definition, is perpendicular to const.).
For the general case of rigid body translation/rotation of fluid shown in the figure, if the
center of rotation is at where , the velocity of any arbitrary point is:
where = the angular velocity vector, and the acceleration is:
First term = acceleration of
Second term = centripetal acceleration of relative to
Third term = linear acceleration of due to
Usually, all these terms are not present. In fact, fluids can rarely move in rigid body
motion unless restrained by confining walls for a long time.
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9.1 Uniform Linear Acceleration
[ ]
1. , increase in
2. , decrease in
1. , decrease in
2. and | | , decrease in
3. and | | , increase in
Unit vector in the direction of :
| |
[ ]
Lines of constant pressure are perpendicular to .
Angle between the surface of constant pressure and the axes:
.
In general the rate of increase of pressure in the direction is given by:
[
]
gage pressure
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9.2 Rigid Body Rotation
Consider rotation of the fluid about the axis without any translation.
and
The constant is determined by specifying the pressure at one point; say,
at
(Note: Pressure is linear in and parabolic in )
Curves of constant pressure are given by:
which are paraboloids of revolution, concave upward, with their minimum points on the
axis of rotation.
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The position of the free surface is found, as it is for linear acceleration, by conserving the
volume of fluid.
Unit vector in the direction of :
| |
[ ]
Slope of :
.
( is the angle between the surface of constant pressure and the axis)
i.e.,
(
)
is the equation of surfaces.
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10 Case (3): Pressure Distribution in Irrotational Flow
Potential flow solutions also solutions of NS under such conditions:
1. If viscous effects are neglected, Navier-Stokes equation becomes Euler equation:
(
)
(
)
Vector calculus identity: (
)
2. If ,
(
) ( )
3. Assume a steady flow:
(
)
Consider: perpendicular to , also perpendicular to and .
Stream lines : ; vortex lines :
Therefore,
contains streamlines and vortex lines:
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1. Assuming irrotational flow:
(everywhere same constant)
2. Unsteady irrotational flow
(
)
is a time-dependent constant.
Alternate derivation using streamline coordinates:
[
] [
]
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Time increment:
Space increment:
[
] [
]
: local in the direction of flow
: local normal to the direction of flow
: convective due to convergence/divergence of streamlines
: normal due to streamline curvature
Euler Equation:
Steady flow -direction equation:
(
) , i.e., B=const. along streamline
Steady flow -direction equation:
across streamline
8 Buoyancy and Stability8.1 Archimedes Principle8.2 Stability: Immersed Bodies8.3 Stability: Floating Bodies8.4 Roll
9 Case (2): Rigid Body Translation or Rotation9.1 Uniform Linear Acceleration9.2 Rigid Body Rotation
10 Case (3): Pressure Distribution in Irrotational Flow