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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