1.3 Conservation laws

Conservation laws form the backbone of aerodynamics, governing how fluids behave in motion. Three principles drive everything here: conservation of mass, momentum, and energy. Together, they let you predict fluid flow in scenarios ranging from aircraft design to atmospheric phenomena.

These laws apply whether you're dealing with steady or unsteady flows, compressible or incompressible fluids. They also provide the foundation for computational fluid dynamics (CFD) and for deriving key results like Bernoulli's equation and shock relations.

Conservation of mass

Mass cannot be created or destroyed in a closed system. In fluid dynamics, this means the total mass entering a region must equal the total mass leaving it (plus any accumulation inside). This seemingly simple idea constrains every flow problem you'll encounter.

Continuity equation

The continuity equation is the mathematical statement of mass conservation. In its most general differential form:

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

This relates the local rate of change of density to the divergence of the mass flux. For incompressible flow (constant density), the equation simplifies to:

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

This tells you the velocity field is divergence-free: fluid isn't piling up or depleting anywhere.

Steady vs unsteady flow

• Steady flow: fluid properties at any given point don't change with time. Mathematically, all time derivatives vanish ($\frac{\partial}{\partial t} = 0$), which simplifies the continuity and momentum equations considerably.
• Unsteady flow: fluid properties at a point vary with time. You must retain the time-derivative terms, making the equations harder to solve.

Most introductory aerodynamics problems assume steady flow to keep things tractable.

Compressible vs incompressible flow

• Incompressible flow assumes constant fluid density. This is a valid approximation for low-speed flows where the Mach number is below about 0.3.
• Compressible flow accounts for density changes and becomes necessary at higher Mach numbers (above ~0.3), especially in transonic and supersonic regimes.

The distinction matters because compressibility introduces additional coupling between the energy equation and the other conservation laws. At low speeds you can often solve mass and momentum independently of energy; at high speeds you cannot.

Conservation of momentum

Momentum conservation is Newton's second law applied to a fluid. The net force on a fluid element equals the rate of change of its momentum. This principle connects the forces acting on a fluid (pressure, viscous stress, gravity) to the resulting motion.

Newton's second law

For a point mass: $\vec{F} = m\vec{a}$. In fluid dynamics, you replace mass with density and express acceleration through the material derivative of velocity, giving a force-per-unit-volume formulation.

Momentum equation

The Cauchy momentum equation in differential form is:

$\rho \frac{D\vec{V}}{Dt} = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \vec{g}$

Each term on the right-hand side represents a different category of force. The left-hand side is the material (or substantial) derivative of momentum per unit volume, capturing both local and convective acceleration.

Pressure gradient forces

The term $-\nabla p$ represents forces due to spatial differences in pressure. Fluid accelerates from high-pressure regions toward low-pressure regions. On an airfoil, for example, the pressure distribution over the surface is what generates lift and contributes to drag.

Viscous forces

The term $\nabla \cdot \overline{\overline{\tau}}$ accounts for internal friction between fluid layers moving at different velocities. Here $\overline{\overline{\tau}}$ is the viscous stress tensor. These forces are responsible for skin-friction drag and play a central role in boundary layer behavior.

Body forces

The term $\rho \vec{g}$ represents external forces acting throughout the fluid volume, most commonly gravity. In many aerodynamics problems at typical flight scales, gravitational body forces are small compared to pressure and viscous forces and can be neglected. They become significant in large-scale geophysical flows (ocean currents, atmospheric circulation).

Conservation of energy

The first law of thermodynamics applied to a fluid: energy is neither created nor destroyed, only converted between forms or transferred as heat and work. This equation becomes essential whenever temperature changes, compressibility, or heat transfer matter.

First law of thermodynamics

For a system: $\Delta E = Q - W$, where $\Delta E$ is the change in total energy, $Q$ is heat added to the system, and $W$ is work done by the system. In fluid dynamics, you apply this to a fluid element or control volume, tracking how internal energy, kinetic energy, and potential energy change.

Energy equation

The differential form of the energy equation is:

$\rho \frac{D}{Dt}\left(e + \frac{V^2}{2}\right) = -\nabla \cdot \vec{q} + \nabla \cdot (k\nabla T) + \Phi$

This balances the rate of change of total energy (internal + kinetic) against heat flux, thermal conduction, and viscous dissipation. For incompressible, low-speed flows without significant heating, the energy equation often decouples from mass and momentum. For compressible flows, all three equations are tightly coupled.

Internal energy

Internal energy ($e$) comes from the random thermal motion of molecules. It depends on temperature and is related to the fluid's specific heat properties. Changes in internal energy are significant in compressible flows and any problem involving heat transfer.

Kinetic energy

Kinetic energy ($\frac{1}{2}\rho V^2$ per unit volume) is associated with the bulk motion of the fluid. It becomes a dominant term in high-speed and turbulent flows.

Potential energy

Gravitational potential energy depends on the fluid's elevation in a gravity field. It's important in flows with significant height differences (hydraulic systems, dam spillways) but is often negligible in external aerodynamics.

Heat transfer

Energy transfer driven by temperature differences takes three forms:

• Conduction: energy transfer through molecular interactions, represented by $\nabla \cdot (k\nabla T)$
• Convection: energy carried by bulk fluid motion
• Radiation: electromagnetic energy transfer, relevant at very high temperatures (re-entry vehicles, combustion)

Heat transfer is critical in thermal boundary layers, heat exchangers, and propulsion systems.

Work done by pressure

As fluid expands or compresses under pressure forces, work is done. This is represented by $-\nabla \cdot (p\vec{V})$ in the conservative form of the energy equation. Pressure work is significant in compressible flows and anywhere large pressure variations exist (nozzles, diffusers).

Viscous dissipation

The viscous dissipation function $\Phi$ represents the irreversible conversion of kinetic energy into heat through fluid friction. It's always positive (energy is always lost to heat, never the reverse). Viscous dissipation matters most in flows with large velocity gradients, such as boundary layers at high speeds.

Integral form of conservation laws

Instead of tracking an infinitesimal fluid element, you can apply conservation laws to a finite control volume: a fixed region in space. The integral form relates changes in mass, momentum, and energy inside the control volume to fluxes crossing its boundaries.

This approach is especially practical for engineering analysis of flow through ducts, nozzles, and around bodies.

Control volume analysis

You define a fixed region (the control volume) and its boundary (the control surface). Then you account for:

1. The rate of change of a quantity (mass, momentum, or energy) stored inside the volume
2. The net flux of that quantity across the control surface
3. Any sources or forces acting within the volume

This framework is the workhorse for analyzing turbomachinery, pipe systems, and jet engines.

Reynolds transport theorem

The Reynolds transport theorem bridges the gap between the system (Lagrangian) and control volume (Eulerian) perspectives. For any extensive property $\phi$ per unit mass:

$\frac{D}{Dt} \int_{CV} \rho \phi \, dV = \int_{CV} \frac{\partial (\rho \phi)}{\partial t} \, dV + \int_{CS} \rho \phi (\vec{V} \cdot \vec{n}) \, dA$

The left side is the total rate of change following the fluid. The first integral on the right is the local rate of change inside the control volume. The second integral is the net flux through the control surface. This theorem lets you convert any conservation law from differential to integral form.

Mass flow rate

The mass flow rate through a control surface is:

$\dot{m} = \int_{CS} \rho (\vec{V} \cdot \vec{n}) \, dA$

For uniform flow across a cross-section of area $A$, this simplifies to $\dot{m} = \rho V A$. Mass flow rate is a key design parameter for pipes, valves, turbines, and engines.

Momentum flux

The rate at which momentum crosses a control surface:

$\vec{M} = \int_{CS} \rho \vec{V} (\vec{V} \cdot \vec{n}) \, dA$

Momentum flux is what you use to calculate forces on structures in a flow. For example, the thrust of a jet engine can be found by evaluating the momentum flux difference between the inlet and exhaust.

Energy flux

The rate at which energy crosses a control surface:

$\dot{E} = \int_{CS} \rho \left(e + \frac{V^2}{2} + \frac{p}{\rho}\right) (\vec{V} \cdot \vec{n}) \, dA$

The term $\frac{p}{\rho}$ is the specific flow work (also called pressure work or flow energy). Together, $e + \frac{p}{\rho}$ equals the specific enthalpy $h$, so the integrand is often written as $h + \frac{V^2}{2}$, which is the specific total enthalpy. Energy flux analysis is central to the design of heat exchangers, power plants, and propulsion systems.

Differential form of conservation laws

The differential form applies conservation laws to an infinitesimal fluid element, yielding partial differential equations (PDEs) that describe the detailed, point-by-point behavior of the flow. These PDEs are the foundation of CFD.

Continuity equation in differential form

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

This is the conservative form. It states that any decrease in density at a point must be balanced by a net outflow of mass from that point, and vice versa.

Momentum equations in differential form

In conservative form:

$\frac{\partial (\rho \vec{V})}{\partial t} + \nabla \cdot (\rho \vec{V} \vec{V}) = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \vec{g}$

The left side captures the time rate of change of momentum and the convective transport of momentum. The right side sums the pressure, viscous, and body forces.

Navier-Stokes equations

The Navier-Stokes equations combine the continuity and momentum equations for a viscous, Newtonian fluid. For an incompressible Newtonian fluid, the momentum equation becomes:

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

where $\mu$ is the dynamic viscosity. These equations are notoriously difficult to solve analytically except in simple geometries. Numerical solutions (CFD) are used for most practical problems. The question of whether smooth solutions always exist for the 3D Navier-Stokes equations remains one of the unsolved Millennium Prize problems in mathematics.

Energy equation in differential form

In conservative form:

$\frac{\partial (\rho E)}{\partial t} + \nabla \cdot (\rho \vec{V} E) = -\nabla \cdot (p\vec{V}) - \nabla \cdot \vec{q} + \nabla \cdot (\overline{\overline{\tau}} \cdot \vec{V}) + \rho \vec{g} \cdot \vec{V}$

Here $E = e + \frac{V^2}{2}$ is the total specific energy. Each term on the right accounts for pressure work, heat flux, viscous work, and gravitational work, respectively.

Boundary conditions

To solve the governing PDEs, you need conditions specified at the boundaries of the fluid domain. Without proper boundary conditions, the solution is not unique. The choice of boundary conditions reflects the physics of the problem.

No-slip condition

At a solid surface, the fluid velocity matches the surface velocity. For a stationary wall, this means $\vec{V} = 0$ at the surface. This condition arises from the adhesion of viscous fluid to solid surfaces and is directly responsible for the formation of boundary layers, the thin regions near surfaces where velocity changes rapidly from zero (at the wall) to the freestream value.

Inlet vs outlet conditions

• Inlet conditions specify known quantities at the entrance of the domain: typically velocity, temperature, and sometimes turbulence quantities.
• Outlet conditions specify constraints at the exit, often a fixed pressure or a zero-gradient condition.

Getting these right is critical. Poorly specified inlet or outlet conditions are a common source of error in CFD simulations.

Adiabatic vs isothermal walls

These are thermal boundary conditions at solid surfaces:

• Adiabatic wall: no heat transfer between the fluid and the surface. Mathematically, $\frac{\partial T}{\partial n} = 0$ (zero temperature gradient normal to the wall). This is a good approximation for well-insulated surfaces.
• Isothermal wall: the surface is held at a constant temperature $T = T_{\text{wall}}$. This applies when the wall material has high thermal conductivity or is actively cooled/heated.

The choice depends on the thermal characteristics of both the flow and the solid surface.

Applications of conservation laws

The conservation equations aren't just theoretical constructs. They're the starting point for a range of practical results used daily in aerodynamic design and analysis.

Bernoulli's equation

A simplified form of the momentum equation valid for steady, inviscid, incompressible flow along a streamline:

$\frac{V^2}{2} + \frac{p}{\rho} + gz = \text{constant}$

This relates velocity, pressure, and elevation. It explains why air speeds up over the top of a wing (lower pressure) and slows beneath it (higher pressure). Keep in mind its assumptions: it breaks down when viscosity, compressibility, or unsteadiness are significant.

Isentropic flow

Isentropic means constant entropy. A flow is isentropic when there's no heat transfer, no viscous dissipation, and no shock waves. Under these conditions, the conservation laws yield a set of algebraic relations connecting pressure, density, temperature, and Mach number. These isentropic flow relations are used extensively to analyze compressible flows in nozzles, diffusers, and wind tunnels.

Quasi-one-dimensional flow

This is a simplified model where flow properties vary only along the flow direction, even though the duct cross-section may change. It's valid when area changes are gradual. By applying conservation of mass, momentum, and energy to a variable-area duct, you get the quasi-1D flow equations. These are the basis for analyzing converging-diverging nozzles and predicting where a flow will choke (reach Mach 1).

Shock waves

Shock waves are extremely thin regions (on the order of a few mean free paths) where pressure, density, temperature, and velocity change abruptly. They occur when a supersonic flow is forced to decelerate to subsonic speeds. Across a shock, entropy increases (the process is irreversible), so isentropic relations don't apply.

The Rankine-Hugoniot jump conditions, derived directly from the integral conservation laws applied across the shock, relate the upstream and downstream flow properties.

Expansion waves

Expansion waves are the opposite of shocks: they occur when a supersonic flow accelerates and turns around a convex corner. Unlike shocks, expansion waves are continuous and isentropic. They're analyzed using the method of characteristics, which traces the propagation of small disturbances through the flow using the conservation laws and isentropic relations. A common example is the Prandtl-Meyer expansion fan.