1.2 Fluid statics

Fluid statics explores how fluids behave when they're at rest. It covers pressure, fluid properties, and the forces that act on submerged surfaces. These concepts form the groundwork for aerodynamics and show up directly in aircraft design, from fuel tanks and hydraulic systems to the pitot-static instruments that provide critical flight data.

Pressure in fluids

Pressure describes the force per unit area that a fluid exerts on a surface. It's one of the most fundamental quantities in fluid mechanics, and you'll see it everywhere in aerodynamics.

Hydrostatic pressure

Hydrostatic pressure is the pressure a fluid at rest exerts due to the weight of fluid above a given point. It increases linearly with depth according to:

$p = \rho g h$

where $p$ is pressure, $\rho$ is fluid density, $g$ is gravitational acceleration, and $h$ is the depth below the free surface.

A key property: hydrostatic pressure at a given depth acts equally in all directions. This is Pascal's law, and it's the reason hydraulic systems can transmit force so efficiently. A small force applied over a small area creates a pressure that acts uniformly throughout the fluid, producing a large force on a larger area elsewhere in the system.

Atmospheric pressure

Atmospheric pressure is the pressure exerted by the weight of the air column above a surface. It decreases with altitude because there's less air above you as you go higher.

• Standard sea-level value: 101,325 Pa (1 atm, or 14.7 psi)
• This standard value serves as a reference point for many engineering calculations

In aerodynamics, the variation of atmospheric pressure with altitude directly affects aircraft performance, cabin pressurization requirements, and instrument readings.

Gauge vs. absolute pressure

These two terms describe different reference points for measuring pressure:

• Absolute pressure is measured relative to a perfect vacuum (zero pressure)
• Gauge pressure is measured relative to the local atmospheric pressure

The relationship is straightforward:

$p_{\text{gauge}} = p_{\text{absolute}} - p_{\text{atm}}$

Most pressure gauges read gauge pressure, which works fine for many applications. But aircraft altimeters need absolute pressure measurements, since they determine altitude based on the actual atmospheric pressure at the aircraft's location.

Fluid properties

The physical properties of a fluid govern how it behaves and interacts with surfaces. You need to know these to analyze any fluid system.

Density

Density ($\rho$) is mass per unit volume. For liquids (incompressible fluids), density stays essentially constant. For gases (compressible fluids), density changes with pressure and temperature.

Density drives many fluid behaviors. Buoyancy, for instance, depends entirely on the density difference between an object and the surrounding fluid. The large density difference between air (~1.225 kg/m³ at sea level) and water (~1000 kg/m³) is why aircraft floats and boat hulls work the way they do.

Specific weight

Specific weight ($\gamma$) is the weight per unit volume of a substance. It relates to density by:

$\gamma = \rho g$

You'll use specific weight frequently when calculating hydrostatic pressures and forces on submerged surfaces. For reference, the specific weight of water is approximately 9.81 kN/m³.

Specific gravity

Specific gravity (SG) is the ratio of a substance's density to the density of a reference substance (water for liquids, air for gases). It's dimensionless, which makes it a convenient way to compare densities quickly.

For example, aviation fuel has a specific gravity of about 0.81, meaning it's 81% as dense as water. This matters for fuel system design because the fuel will float on water and behaves differently under pressure than a denser liquid would.

Viscosity

Viscosity measures a fluid's resistance to flow and shear deformation. There are two forms:

• Dynamic (absolute) viscosity ($\mu$): relates shear stress to the velocity gradient in a flowing fluid
• Kinematic viscosity ($\nu$): the ratio of dynamic viscosity to density, $\nu = \mu / \rho$

Higher viscosity means the fluid resists deformation more strongly. In aircraft applications, viscosity influences fuel flow through lines, lubrication system performance, and aerodynamic drag on surfaces.

Forces on submerged surfaces

Any surface submerged in a fluid experiences hydrostatic forces from the surrounding pressure. Calculating these forces correctly is essential for designing structures like tanks, dams, gates, and pressurized aircraft components.

Hydrostatic force on flat surfaces

The total hydrostatic force on a flat submerged surface equals the pressure at the surface's centroid multiplied by the surface area, and it acts perpendicular to the surface:

$F = \rho g h_c A$

where $h_c$ is the depth of the centroid below the free surface and $A$ is the surface area.

The force depends on three things: fluid density, depth of the centroid, and the size of the surface. Notice that the shape of the surface doesn't affect the total force magnitude directly; what matters is where the centroid sits and how large the area is.

Center of pressure

The center of pressure is the single point where the resultant hydrostatic force effectively acts on a submerged surface. For a vertical flat surface, it always sits below the centroid because pressure increases with depth, so the lower portion of the surface carries more load.

The distance from the surface to the center of pressure is:

$y_p = y_c + \frac{I_c}{y_c A}$

where $y_c$ is the distance from the surface to the centroid (measured along the surface), and $I_c$ is the second moment of area about the centroidal axis.

Getting the center of pressure right matters for moment calculations. In aircraft fuel tanks, for example, knowing where the resultant force acts determines where to place baffles and structural supports.

Hydrostatic force on curved surfaces

For curved surfaces, you can't simply multiply pressure at the centroid by area because the pressure direction changes across the surface. Instead, you resolve the resultant force into horizontal and vertical components:

• The horizontal component equals the hydrostatic force on the vertical projection of the curved surface
• The vertical component equals the weight of the fluid directly above (or below) the curved surface, up to the free surface

This decomposition approach is used in designing pipelines, pressure vessels, and aircraft fuselages. A pressurized fuselage, for instance, must withstand the pressure differential between the cabin interior and the lower-pressure atmosphere outside at cruise altitude.

Buoyancy

Buoyancy is the upward force a fluid exerts on any object immersed in it. It's what determines whether something floats, sinks, or hovers neutrally in a fluid.

Archimedes' principle

Archimedes' principle states that the buoyant force on an object equals the weight of the fluid displaced by that object. The buoyant force acts vertically upward through the center of buoyancy, which is the centroid of the displaced fluid volume.

An object's fate in a fluid depends on the comparison between its weight and the buoyant force:

• If the object is denser than the fluid, it sinks
• If it's less dense, it floats (displacing only enough fluid to balance its weight)
• If densities match, it's neutrally buoyant

Buoyant force

The buoyant force is calculated as:

$F_b = \rho_f g V_d$

where $\rho_f$ is the fluid density, $g$ is gravitational acceleration, and $V_d$ is the volume of fluid displaced by the object.

Two things control the buoyant force: the density of the surrounding fluid and how much fluid volume the object displaces. In aircraft design, this calculation is used when sizing floats or pontoons for seaplanes.

Stability of floating bodies

A floating body's stability depends on the relative positions of three points:

• Center of gravity (CG): where the object's weight acts
• Center of buoyancy (CB): where the buoyant force acts (centroid of displaced volume)
• Metacenter (M): the point about which the body tends to rotate when tilted slightly

The metacentric height (GM) is the distance from the CG to the metacenter:

• Positive GM → stable equilibrium (the body rights itself after being tilted)
• Negative GM → unstable (the body will capsize)

For a stable floating body, when it tilts, the CB shifts to create a restoring moment that pushes it back upright. This analysis is critical for designing ships, offshore platforms, and seaplanes.

Fluid statics applications

Manometers

Manometers measure pressure differences using columns of liquid in a U-shaped tube. One side connects to the pressure source, and the other is typically open to the atmosphere.

The pressure difference is found by measuring the height difference between the two liquid columns and applying the hydrostatic equation. Mercury is a common manometer fluid because its high density (SG ≈ 13.6) keeps the column heights manageable. Manometers are used to measure pressures in wind tunnels and to calibrate other pressure sensors.

Hydraulic systems

Hydraulic systems transmit force and control motion using pressurized fluid. A typical system includes a fluid reservoir, pumps, valves, actuators, and connecting lines.

The operating principle comes straight from Pascal's law: pressure applied anywhere in a confined fluid transmits equally throughout. This allows a small input force to generate a much larger output force by using actuators with different cross-sectional areas. Aircraft flight control systems, landing gear, and brakes all rely on hydraulics.

Hydrostatic bearings

Hydrostatic bearings support a load on a thin film of pressurized fluid, supplied through small orifices or porous materials. The fluid film eliminates metal-to-metal contact, providing very low friction and high stiffness.

These bearings appear in high-precision machine tools and telescopes. In aircraft, they're used in fuel pumps and hydraulic actuators to reduce wear and improve efficiency.

Fluid statics in aerodynamics

Importance in aircraft design

Aircraft components experience significant hydrostatic loads. Fuel tanks must handle the weight of fuel at various fill levels and orientations. Hydraulic lines and actuators operate under high internal pressures. The fuselage acts as a pressure vessel, maintaining cabin pressure at altitude.

Cabin pressure is typically regulated to an equivalent altitude of about 6,000 to 8,000 feet, even when the aircraft cruises above 30,000 feet. The resulting pressure differential across the fuselage skin creates substantial structural loads that must be accounted for in design.

Fuel tank pressurization

Aircraft fuel tanks are pressurized to prevent fuel vaporization at high altitudes (where ambient pressure is low) and to ensure consistent fuel delivery to the engines.

Pressurization is typically achieved using bleed air from the engines or the environmental control system. Fluid statics principles determine the required pressure levels and inform the structural design of the tank walls. If pressurization fails, fuel can vaporize and starve the engines, making this a safety-critical system.

Pitot-static systems

Pitot-static systems provide three essential flight measurements: airspeed, altitude, and vertical speed. The system has two pressure inputs:

• A pitot tube facing into the airflow, which captures total pressure (static + dynamic)
• Static ports on the fuselage, which measure static pressure alone

From these two measurements:

1. Airspeed is derived from the difference between total and static pressure (this difference is the dynamic pressure, which relates to velocity)
2. Altitude is determined from the static pressure reading, since atmospheric pressure decreases predictably with altitude
3. Vertical speed is calculated from the rate of change of static pressure

Accurate pressure measurement is critical here. Blocked pitot tubes or static ports have caused serious accidents, which is why understanding the fluid statics behind these instruments matters for both design and maintenance.