1.1 Fluid properties

Density and specific gravity

Density describes how much mass is packed into a given volume, defined as $\rho = \frac{m}{V}$. For air at sea level and standard conditions, density is approximately $1.225 \, kg/m^3$. This value drops significantly with altitude, which directly affects lift and engine performance.

Specific gravity (also called relative density) compares a fluid's density to a reference fluid. For liquids, the reference is water at $4°C$ ($\rho = 1000 \, kg/m^3$); for gases, the reference is typically air at standard conditions. A specific gravity greater than 1 means the substance is denser than the reference fluid.

In aerodynamics, density matters because forces like lift and drag scale directly with it. The dynamic pressure term $\frac{1}{2}\rho V^2$ appears throughout aerodynamic analysis, so getting density right is non-negotiable.

Pressure in static fluids

Absolute vs gauge pressure

• Absolute pressure is the total pressure at a point, measured relative to a perfect vacuum. It's always positive.
• Gauge pressure is measured relative to the local atmospheric pressure: $p_{gauge} = p_{absolute} - p_{atm}$. Gauge pressure can be negative (indicating suction or partial vacuum).
• Standard atmospheric pressure at sea level is approximately $101,325 \, Pa$ ($1 \, atm$).

In aerodynamics, instruments like pitot tubes often measure gauge pressure, but thermodynamic calculations (such as those involving the ideal gas law) require absolute pressure. Mixing these up is a common source of errors.

Hydrostatic pressure variation

Hydrostatic pressure is the pressure a fluid at rest exerts due to its own weight. In a static fluid, pressure increases linearly with depth:

$p = \rho g h$

where $\rho$ is fluid density, $g$ is gravitational acceleration ($9.81 \, m/s^2$), and $h$ is the depth below the free surface.

This relationship assumes constant density, which works well for liquids but only holds for thin layers of gas. For the atmosphere, density changes with altitude, so you need the more general form $dp = -\rho g \, dz$ integrated over height. That's how the International Standard Atmosphere model is built.

Viscosity of fluids

Dynamic vs kinematic viscosity

Viscosity quantifies a fluid's internal resistance to flow. Think of it as internal friction between fluid layers sliding past each other.

• Dynamic (absolute) viscosity $\mu$ measures the shear stress needed to maintain a velocity gradient in the fluid. Units: $Pa \cdot s$. For air at sea level and $15°C$, $\mu \approx 1.789 \times 10^{-5} \, Pa \cdot s$.
• Kinematic viscosity $\nu$ factors out density: $\nu = \frac{\mu}{\rho}$. Units: $m^2/s$. This form appears naturally in the Reynolds number and other dimensionless parameters.

Both increase with temperature for gases (molecules move faster, collide more) but decrease with temperature for liquids. Since aerodynamics deals primarily with air, expect viscosity to rise as temperature increases.

Newtonian vs non-Newtonian fluids

A Newtonian fluid has a linear relationship between shear stress $\tau$ and strain rate $\frac{du}{dy}$:

$\tau = \mu \frac{du}{dy}$

The viscosity $\mu$ stays constant regardless of how fast you shear the fluid. Air and water are both Newtonian under normal conditions.

Non-Newtonian fluids (blood, paint, polymer solutions) have viscosities that change with the applied shear rate. You won't encounter these much in standard aerodynamics, but they matter in specialized applications like de-icing fluids on aircraft surfaces.

Compressibility of fluids

Bulk modulus of elasticity

The bulk modulus $K$ measures how much a fluid resists being compressed:

$K = -V \frac{dp}{dV}$

A large $K$ means the fluid is hard to compress. Water has $K \approx 2.2 \times 10^9 \, Pa$, making it nearly incompressible for most purposes. Air, by contrast, is highly compressible, which becomes critical at higher speeds.

The inverse of the bulk modulus is compressibility: $\beta = \frac{1}{K}$.

Speed of sound

The speed of sound $c$ is how fast small pressure disturbances travel through a fluid:

$c = \sqrt{\frac{K}{\rho}}$

For an ideal gas, this simplifies to:

$c = \sqrt{\gamma R T}$

where $\gamma$ is the ratio of specific heats ($1.4$ for air), $R$ is the specific gas constant ($287 \, J/(kg \cdot K)$ for air), and $T$ is absolute temperature in Kelvin. At sea level and $15°C$, $c \approx 340 \, m/s$.

The Mach number $M = \frac{V}{c}$ tells you whether compressibility effects matter. Below about $M = 0.3$, air behaves as essentially incompressible. Above that, you need to account for density changes in the flow.

Surface tension and capillary effects

Surface tension arises from cohesive forces between molecules at a fluid's surface. Molecules at the surface have no neighbors above them, so the net intermolecular force pulls them inward, creating a "skin" effect.

• Surface tension causes droplet formation, bubble shapes, and the curved meniscus you see in a graduated cylinder.
• Capillary action occurs when adhesive forces between a fluid and a solid surface compete with cohesive forces within the fluid. Water climbs up a narrow glass tube because adhesion to glass exceeds water's internal cohesion.

For large-scale aerodynamic flows, surface tension is negligible. It becomes relevant in micro-scale systems, fuel atomization in engines, and the behavior of liquid films on aircraft surfaces (such as rain or de-icing fluid).

Fluid statics

Buoyancy and Archimedes' principle

When an object is submerged in a fluid, pressure acts on all its surfaces. Since pressure increases with depth, the upward force on the bottom of the object exceeds the downward force on the top. The net result is an upward buoyant force.

Archimedes' principle states that this buoyant force equals the weight of the fluid displaced:

$F_b = \rho_{fluid} \, g \, V_{displaced}$

If the buoyant force exceeds the object's weight, it floats. If not, it sinks. This principle applies to lighter-than-air aircraft (blimps, weather balloons) and also governs fuel behavior in aircraft tanks during maneuvers.

Hydrostatic force on submerged surfaces

A submerged flat surface experiences a distributed pressure load that increases with depth. The total hydrostatic force on a flat surface is:

$F = \rho g \bar{h} A$

where $\bar{h}$ is the depth of the surface's centroid and $A$ is the surface area.

The force doesn't act at the centroid, though. It acts at the center of pressure, which is always deeper than the centroid because pressure increases with depth. Calculating the center of pressure matters for designing fuel tank walls, dam-like structures, and any aircraft component that holds or contacts a static fluid.

Ideal fluid concept

Inviscid flow assumption

An inviscid fluid is a theoretical fluid with zero viscosity. No real fluid is truly inviscid, but this assumption removes the viscous stress terms from the governing equations, making them far easier to solve.

The assumption works well when:

• The Reynolds number is high (inertial forces dominate viscous forces)
• You're analyzing flow away from solid boundaries

Near surfaces, viscous effects create a boundary layer where the inviscid assumption breaks down. That's why aerodynamicists often solve the inviscid outer flow and the viscous boundary layer separately, then match them together.

Potential flow theory and thin airfoil theory both rely on the inviscid assumption and produce surprisingly accurate results for lift prediction, even though they can't predict viscous drag.

Irrotational flow condition

A flow is irrotational if fluid elements translate without spinning about their own axes. Mathematically:

$\nabla \times \vec{V} = 0$

where $\nabla \times \vec{V}$ is the curl of the velocity field, also called vorticity. Zero vorticity means irrotational flow.

When flow is both inviscid and irrotational, you can define a velocity potential $\phi$ such that $\vec{V} = \nabla \phi$. This reduces the vector velocity field to a single scalar function, which is a huge simplification. Combined with incompressibility, this gives Laplace's equation $\nabla^2 \phi = 0$, the foundation of potential flow theory.

Irrotational flow is a good model for the freestream and outer flow regions, but inside boundary layers and wakes, vorticity is generated and the irrotational assumption fails.

Fluid kinematics

Streamlines and pathlines

• Streamlines are curves tangent to the local velocity vector at a single instant. They give you a snapshot of the flow direction everywhere at one moment in time.
• Pathlines trace the actual trajectory of a single fluid particle over time.
• Streaklines (worth knowing) connect all particles that have passed through a particular point. Dye injected into a flow traces a streakline.

In steady flow, all three are identical. In unsteady flow, they diverge, and you need to be careful about which one you're looking at. Wind tunnel smoke visualization typically shows streaklines, which coincide with streamlines only if the flow is steady.

Steady vs unsteady flow

• Steady flow: Flow properties (velocity, pressure, density) at any fixed point don't change with time. Mathematically, $\frac{\partial}{\partial t} = 0$ for all flow variables at every point.
• Unsteady flow: Flow properties at fixed points vary with time.

Many aerodynamic analyses assume steady flow because it simplifies the math considerably. But real phenomena like turbulence, vortex shedding from bluff bodies, flutter, and gust responses are inherently unsteady and require time-dependent analysis.

Laminar vs turbulent flow

Laminar flow consists of smooth, orderly layers sliding past each other with no cross-stream mixing. Turbulent flow is chaotic, with rapid velocity fluctuations and strong mixing between layers.

The Reynolds number determines which regime dominates:

$Re = \frac{\rho V L}{\mu} = \frac{V L}{\nu}$

where $V$ is flow velocity, $L$ is a characteristic length, and $\nu$ is kinematic viscosity. Higher $Re$ favors turbulence.

For flow over a flat plate, transition from laminar to turbulent typically occurs around $Re \approx 5 \times 10^5$. Why does this matter? Turbulent boundary layers produce more skin friction drag but are more resistant to flow separation. Laminar layers have less friction but separate more easily. This tradeoff is central to wing and airfoil design.

Fluid dynamics

Conservation of mass

Mass can't be created or destroyed. For a fluid, this gives the continuity equation. In differential form:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$

For steady, incompressible flow, this simplifies dramatically to:

$\nabla \cdot \vec{V} = 0$

In a one-dimensional duct or streamtube, continuity becomes $\rho_1 A_1 V_1 = \rho_2 A_2 V_2$. For incompressible flow (constant $\rho$), that's just $A_1 V_1 = A_2 V_2$: if the area decreases, velocity increases. This is the principle behind a converging nozzle and the venturi effect.

Conservation of momentum

Newton's second law applied to a fluid gives the Navier-Stokes equations. These relate the rate of change of momentum to the forces acting on a fluid element: pressure forces, viscous forces, and body forces (like gravity).

For an incompressible, Newtonian fluid:

$\rho \frac{D\vec{V}}{Dt} = -\nabla p + \mu \nabla^2 \vec{V} + \rho \vec{g}$

where $\frac{D}{Dt}$ is the material derivative (accounts for both local and convective acceleration).

If you drop the viscous term (inviscid assumption), you get Euler's equations. If you further integrate along a streamline for steady, incompressible, inviscid flow, you arrive at Bernoulli's equation:

$p + \frac{1}{2}\rho V^2 + \rho g z = \text{constant}$

Bernoulli's equation is one of the most used tools in aerodynamics, but remember its assumptions: steady, incompressible, inviscid flow along a streamline.

Conservation of energy

Energy is conserved: it transforms between forms but is never created or destroyed. For fluid flow, the energy equation accounts for kinetic energy, internal (thermal) energy, pressure work, viscous dissipation, and heat transfer.

For steady, adiabatic (no heat transfer), inviscid flow of a compressible gas, the energy equation gives:

$h + \frac{V^2}{2} = h_0 = \text{constant}$

where $h$ is specific enthalpy and $h_0$ is the total (stagnation) enthalpy. This tells you that as a gas accelerates, its temperature drops (kinetic energy increases at the expense of enthalpy), and vice versa.

This relationship is essential for analyzing compressible flows, engine inlets, nozzles, and any situation where significant speed changes cause temperature variations.