CE 3305 Engineering FLUID MECHANICS Lecture 4: Buoyancy and Stability

Outline Forces on curved surfaces Buoyancy Stability

Forces on Submerged Surfaces Recall from last time: Magnitude: Line of Action:

EXAMPLE Consider the following problem:

EXAMPLE State problem, include sketch

EXAMPLE List Known Properties:

EXAMPLE List Unknown Properties:

EXAMPLE Identify Relevant Governing Principles and Equations:

EXAMPLE Solve for the Unknown(s)

EXAMPLE Solve for the Unknown(s)

EXAMPLE Validate/Discuss Results Problem asks for force/length; then asks for the actual forces on the ties. Use static force balance, and divide resultant by spacing to get force/unit width. Use static moment balance to find required forces at top and bottom – this information used to specify bolt strength

FORCES ON SUBMERGED OBJECTS Recall from last time that the force and line of action are determined by (1) integrating the pressure distribution across the area, and by (2) integrating the moment distribution about the free surface. Integral methods will always work, but there is an easier way for reasonably regular geometries.

Alternate method A practical alternate is based on displacement volumes to find components of the resultant force

Alternate method A practical alternate is based on displacement volumes to find components of the resultant force

Alternate method A practical alternate is based on displacement volumes to find components of the resultant force

Alternate method A practical alternate is based on displacement volumes to find components of the resultant force

Alternate method A practical alternate is based on displacement volumes to find components of the resultant force

EXAMPLE Find resultant force on curved surface shown where width is 1 meter into the diagram

Example State problem; include sketch

example List known values

example List unknowns

example Identify Relevant Governing Principles and Equations:

Example Solve for the Unknown(s)

Example Solve for the Unknown(s)

Example Solve for the Unknown(s)

Example Validate/Discuss

buoyancy Objects submerged in a liquid have an upward force applied by the liquid – called buoyant force. Imagine the block shown, 1 cubic meter block, 1 meter deep in a liquid

buoyancy Analyze the forces on the block 1 cubic meter block, 1 meter deep in a liquid Pressure forces (top, sides, and bottom) Weight of the block itself

buoyancy Analyze the forces on the block 1 cubic meter block, 1 meter deep in a liquid Pressure forces (top, sides, and bottom) Weight of the block itself

buoyancy Learn that for any object, the buoyant force will equal the weight of the liquid displaced! Handy because we only need to understand volumes to compute buoyant force.

buoyancy A body floats if the buoyant force equals the weight of the object. Consider the block shown

buoyancy A body floats if the buoyant force equals the weight of the object. Consider the block shown

buoyancy A body floats if the buoyant force equals the weight of the object. Consider the block shown

buoyancy A body floats if the buoyant force equals the weight of the object. Consider the block shown

buoyancy A body floats if the buoyant force equals the weight of the object. Consider the block shown

buoyancy A body floats if the buoyant force equals the weight of the object. Consider the block shown

buoyancy The depth d is called the draft of the vessel. It (d) is important in shallow water operation of vessels and offshore floating platforms.

Stability Determination of whether an object will remained oriented as placed or not is stability analysis Important for cruise ships, ferries, submarines, and platforms. Important for ducks; if a duck is unstable and turns upside down, then it will quack up! The overturning or righting moment depends on the relative position of the weight and buoyant force lines of action

Stability analysis Weight acts through the center of gravity – CG does not change location relative to the object Buoyant force acts through the centroid of the submerged section. The forces create a moment couple that establishes stability

Stability analysis Intersection of a bisector running up through the vessel in the desired condition (a) through the CG and a line of action through the buoyant force vector is called the vessel metacenter. If the metacenter is above the CG the body is stable The righting moment is

Stability analysis The metacentric height GM is obtained from the moment of inertia of the wedge that lifts and sinks

Stability analysis example Problem Statement

Stability analysis example Sketch

Stability analysis example Known 3m X 4m X 2m Draft = 1.2 m Sea water; SG=1.03 Angle = 8 degrees of arc

Stability analysis example Unknown MC height Righting moment

Stability analysis example Equations

Stability analysis example Solution

Stability analysis example Discussion Use tables for finding geometric properties

summary Chapters 1-3 Pressure Hydrostatic equilibrium Hydrostatic equation Pressure distributions and forces Concepts of pressure on curves surfaces Buoyant force and stability

Next Time Quiz #1,Manometry, Euler’s equation