8 Buoyancy and Stability

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1 Jianming Yang Fall Buoyancy and Stability 8.1 Archimedes Principle = fluid weight above 2 ABC fluid weight above 1 ADC = 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.

2 058:0160 Chapter 2 Jianming Yang Fall Instantaneous displaced water volume: Solution for this homogeneous linear 2 nd -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. mons/0/03/lateral_view_of_spar-buoy.png

3 Jianming Yang Fall Stability: Immersed Bodies Stable Neutral Unstable Condition for static equilibrium: (1) F v =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.

4 Jianming Yang Fall 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.

5 Jianming Yang Fall 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:

6 058:0160 Chapter 2 Jianming Yang Fall : 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

7 058:0160 Chapter 2 Jianming Yang Fall Roll The rotation of a ship about the longitudinal axis through the center of gravity. Consider symmetrical ship heeled to a very small angle θ. 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

8 058:0160 Chapter 2 Jianming Yang Fall 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.

9 Jianming Yang Fall 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.

10 Jianming Yang Fall 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

11 Jianming Yang Fall 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.

12 Jianming Yang Fall 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.

13 Jianming Yang Fall 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: 2. If, 3. Assume a steady flow: ( ) ( ) Vector calculus identity: ( ) ( ) ( ) ( ) Consider: perpendicular to, also perpendicular to and. Stream lines : ; vortex lines : Therefore, contains streamlines and vortex lines:

14 Jianming Yang Fall Assuming irrotational flow: (everywhere same constant) 2. Unsteady irrotational flow ( ) is a time-dependent constant. Alternate derivation using streamline coordinates: [ ] [ ]

15 Jianming Yang Fall 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

Fig. 9: an immersed body in a fluid, experiences a force equal to the weight of the fluid it displaces.

Fig. 9: an immersed body in a fluid, experiences a force equal to the weight of the fluid it displaces. Buoyancy Archimedes s 1 st laws of buoyancy: A body immersed in a fluid experiences a vertical buoyant force equal to the weight of the fluid it displaces, see Fig. 9 and 10. Fig. 9: an immersed body in

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