How is Buoyancy Calculated on a Submerged Object?
Buoyancy is the upward force on an object that counteracts gravity and other downward forces present. A submerged object in some fluid medium will experience a force due to gravity, a force due to buoyancy, forces due to wave energy, and other forces - the net force will determine whether the object floats or sinks and its rotation.
The design of a product that will be operated underwater, or rely on the fluid medium it is operating in, will incorporate an analysis of the center of gravity and the center of buoyancy in order to control rotation and momentum on a materials design level. Controlling forces using the static design of the product will reduce reliance on the control system upon deployment.
The origin of the coordinate system of underwater physics is located on the surface of the fluid with the z-axis pointed downward. This implies that the force due to buoyancy is directed in the negative z-direction and the force due to gravity is directed in the positive z-direction. An object with a volume, V, submerged in a fluid medium is subjected to a variable force normal to every contoured area changing primarily from water pressure, but other factors can affect the forces felt.
The pressure, P, within a fluid medium at equilibrium due to gravitational forces is defined as , where ρf is the mass density of the fluid, g is the gravitational acceleration, and z is directed in the downward positive direction. Because the formula assumes pressure calculated within a singular fluid medium, at the surface, or where z=0, pressure is equal to zero. As an object increases in depth, the pressure it is subjected to increases. Due to the difference in depth between each surface point on a submersed object, there will be a different magnitude of pressure acting on the top and bottom of the object, causing an upward force.
The Divergence Theorem
The vector field around the surface boundary of an object submerged in a fluid medium (having divergence associated with an outward facing normal vector) can be used to relate the surface flux of the forces acting on the object with the forces acting within the volume of the object. This relationship is defined using the divergence theorem (or Gauss’s theorem):
where the triple integral of the divergence of the vector field over a region S is equal to the flux integral of the vector field over the surface of the region S.
Therefore, the buoyancy force exerted on a submerged object can be represented using the volume of the object in contact with the fluid medium rather than the surface area. Although this will provide a fairly accurate estimate of the buoyancy force in most cases, occasionally the buoyancy force must be calculated for more complicated objects with hard boundaries (such as a largely contoured object or a disk shaped object) rather than closed surfaces. Using the divergence theorem, the buoyancy force is calculated as:
where V is the volume of the submerged part of the object only (hence the phrase “in contact with the fluid medium”). The net force is then equal to the difference between the force due to gravity and the buoyancy force.
If an observed object is not apparently symmetrical about a central plane, it is likely other observed opposing horizontal and vertical forces will cause the object to become unstable rotationally. The center of buoyancy (center of displaced volume of the fluid medium), the center of gravity, and the center of rotation can play a critical role in the stability management of a submerged object. In order to increase the factor of safety for rotational stability, the center of buoyancy must be held above the center of gravity, creating a high center of rotation.
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