Conservation laws form the backbone of aerodynamics, governing how fluids behave in motion. These principles—mass, momentum, and energy conservation—help us understand and predict fluid flow in various scenarios, from aircraft design to natural phenomena.

By applying these laws, we can analyze complex aerodynamic problems and develop solutions. Whether dealing with steady or unsteady flows, compressible or incompressible fluids, conservation laws provide the foundation for understanding and manipulating fluid behavior in aerospace applications.

Conservation of mass

• Fundamental principle stating that mass cannot be created or destroyed in a closed system
• Applies to both fluid dynamics and aerodynamics, ensuring that the mass of a system remains constant
• Forms the basis for analyzing fluid flow and predicting flow behavior in various scenarios

Continuity equation

• Mathematical expression of conservation of mass in fluid dynamics
• Relates the change in fluid density to the velocity field
• For incompressible flow, simplifies to $\nabla \cdot \vec{V} = 0$, indicating that the velocity field is divergence-free

Steady vs unsteady flow

• Steady flow occurs when fluid properties at a given point do not change with time
• Unsteady flow occurs when fluid properties vary with time
• In steady flow, $\frac{\partial}{\partial t} = 0$, simplifying the continuity and momentum equations

Compressible vs incompressible flow

• Incompressible flow assumes constant fluid density, valid for low-speed flows (Mach number < 0.3)
• Compressible flow considers changes in fluid density, important for high-speed flows (Mach number > 0.3)
• Compressibility effects are crucial in aerodynamics, particularly in transonic and supersonic flight regimes

Conservation of momentum

• Based on Newton's second law, which states that the net force on an object equals its mass times acceleration
• In fluid dynamics, the momentum conservation principle relates forces acting on a fluid to its resulting motion
• Fundamental in understanding the behavior of fluids under the influence of various forces

Newton's second law

• Mathematically expressed as $\vec{F} = m\vec{a}$, where $\vec{F}$ is the net force, $m$ is the mass, and $\vec{a}$ is the acceleration
• In fluid dynamics, the mass is replaced by the fluid density, and acceleration is related to the velocity field

Momentum equation

• Derived from Newton's second law, considering the forces acting on a fluid element
• Relates the fluid velocity, pressure, and density to the forces acting on the fluid
• In differential form, known as the Cauchy momentum equation: $\rho \frac{D\vec{V}}{Dt} = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \vec{g}$

Pressure gradient forces

• Forces arising from pressure differences in the fluid
• Represented by the term $-\nabla p$ in the momentum equation
• Pressure gradient forces drive fluid motion from high-pressure regions to low-pressure regions

Viscous forces

• Forces resulting from fluid viscosity and the relative motion of fluid layers
• Represented by the term $\nabla \cdot \overline{\overline{\tau}}$ in the momentum equation, where $\overline{\overline{\tau}}$ is the viscous stress tensor
• Viscous forces cause fluid friction and contribute to flow resistance

Body forces

• External forces acting on the fluid, such as gravity or electromagnetic forces
• Represented by the term $\rho \vec{g}$ in the momentum equation, where $\vec{g}$ is the acceleration due to the body force
• Body forces can significantly influence fluid motion, especially in large-scale flows (geophysical flows)

Conservation of energy

• Based on the first law of thermodynamics, which states that energy cannot be created or destroyed, only converted from one form to another
• In fluid dynamics, the conservation of energy principle relates the changes in energy within a fluid to the work done and heat transferred
• Essential for analyzing thermal effects, compressibility, and energy transfer in fluid systems

First law of thermodynamics

• Expressed as $\Delta E = Q - W$, where $\Delta E$ is the change in total energy, $Q$ is the heat added to the system, and $W$ is the work done by the system
• In fluid dynamics, the first law is applied to a fluid element or control volume

Energy equation

• Derived from the first law of thermodynamics, accounting for the various forms of energy in a fluid
• Relates the changes in energy to the work done by pressure, viscous dissipation, and heat transfer
• In differential form: $\rho \frac{D}{Dt}(e + \frac{V^2}{2}) = -\nabla \cdot \vec{q} + \nabla \cdot (k\nabla T) + \Phi$

Internal energy

• Energy associated with the random motion of molecules in a fluid
• Depends on the fluid temperature and is related to the fluid's thermal properties
• Changes in internal energy are important in compressible flows and heat transfer problems

Kinetic energy

• Energy associated with the bulk motion of the fluid
• Depends on the fluid velocity and density
• Kinetic energy changes are significant in high-speed flows and turbulent flows

Potential energy

• Energy associated with the fluid's position in a gravitational field
• Depends on the fluid density and the gravitational acceleration
• Potential energy changes are important in flows with significant elevation differences (hydraulic systems)

Heat transfer

• Energy transfer due to temperature differences
• Includes conduction ($\nabla \cdot (k\nabla T)$), convection, and radiation
• Heat transfer is crucial in problems involving thermal boundary layers, heat exchangers, and combustion

Work done by pressure

• Work done by the fluid as it expands or contracts due to pressure forces
• Represented by the term $-\nabla \cdot (p\vec{V})$ in the energy equation
• Pressure work is significant in compressible flows and flows with large pressure variations

Viscous dissipation

• Irreversible conversion of mechanical energy into heat due to fluid friction
• Represented by the term $\Phi$ in the energy equation, which is the viscous dissipation function
• Viscous dissipation is important in high-velocity flows and flows with large velocity gradients

Integral form of conservation laws

• Conservation laws applied to a fixed control volume, rather than an infinitesimal fluid element
• Integral form is useful for analyzing flow through a specific region, such as a pipe or an airfoil
• Relates the changes in mass, momentum, and energy within the control volume to the fluxes across its boundaries

Control volume analysis

• Approach where the conservation laws are applied to a fixed region in space (control volume)
• Fluxes of mass, momentum, and energy across the control volume boundaries are considered
• Useful for analyzing flow through ducts, nozzles, and turbomachinery

Reynolds transport theorem

• Mathematical theorem that relates the change of an extensive property (mass, momentum, energy) within a control volume to the fluxes across its boundaries
• Allows the conversion of conservation laws from the differential form to the integral form
• Expressed as $\frac{D}{Dt} \int_{CV} \rho \phi dV = \int_{CV} \frac{\partial (\rho \phi)}{\partial t} dV + \int_{CS} \rho \phi (\vec{V} \cdot \vec{n}) dA$

Mass flow rate

• The rate at which mass flows through a control surface
• Defined as $\dot{m} = \int_{CS} \rho (\vec{V} \cdot \vec{n}) dA$, where $\rho$ is the fluid density, $\vec{V}$ is the velocity vector, and $\vec{n}$ is the unit normal vector to the control surface
• Mass flow rate is a key parameter in the design of fluid systems (pipes, valves, turbines)

Momentum flux

• The rate at which momentum flows through a control surface
• Defined as $\vec{M} = \int_{CS} \rho \vec{V} (\vec{V} \cdot \vec{n}) dA$, where $\rho$ is the fluid density, $\vec{V}$ is the velocity vector, and $\vec{n}$ is the unit normal vector to the control surface
• Momentum flux is important in the analysis of forces acting on structures immersed in a fluid (buildings, bridges, aircraft)

Energy flux

• The rate at which energy flows through a control surface
• Defined as $\dot{E} = \int_{CS} \rho (e + \frac{V^2}{2} + \frac{p}{\rho}) (\vec{V} \cdot \vec{n}) dA$, where $e$ is the specific internal energy, $\frac{V^2}{2}$ is the specific kinetic energy, and $\frac{p}{\rho}$ is the specific flow work
• Energy flux is crucial in the design of heat exchangers, power plants, and propulsion systems

Differential form of conservation laws

• Conservation laws applied to an infinitesimal fluid element
• Differential form is useful for analyzing the detailed behavior of fluid flow at a specific point
• Provides the basis for computational fluid dynamics (CFD) and the numerical solution of flow problems

Continuity equation in differential form

• Expresses the conservation of mass for an infinitesimal fluid element
• In conservative form: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$, where $\rho$ is the fluid density and $\vec{V}$ is the velocity vector
• Continuity equation is the starting point for deriving other conservation equations

Momentum equations in differential form

• Express the conservation of momentum for an infinitesimal fluid element
• Derived from Newton's second law, considering the forces acting on the fluid element
• In conservative form: $\frac{\partial (\rho \vec{V})}{\partial t} + \nabla \cdot (\rho \vec{V} \vec{V}) = -\nabla p + \nabla \cdot \overline{\overline{\tau}} + \rho \vec{g}$

Navier-Stokes equations

• Set of partial differential equations that describe the motion of viscous fluids
• Obtained by combining the continuity equation and the momentum equations
• Navier-Stokes equations are the foundation of computational fluid dynamics and are used to model a wide range of fluid flow problems

Energy equation in differential form

• Expresses the conservation of energy for an infinitesimal fluid element
• Derived from the first law of thermodynamics, considering the various forms of energy and energy transfer mechanisms
• In conservative form: $\frac{\partial (\rho E)}{\partial t} + \nabla \cdot (\rho \vec{V} E) = -\nabla \cdot (p\vec{V}) - \nabla \cdot \vec{q} + \nabla \cdot (\overline{\overline{\tau}} \cdot \vec{V}) + \rho \vec{g} \cdot \vec{V}$

Boundary conditions

• Conditions specified at the boundaries of a fluid domain to ensure a unique solution to the governing equations
• Boundary conditions are essential for solving fluid flow problems and are based on physical constraints and flow characteristics
• Types of boundary conditions include velocity, pressure, temperature, and stress conditions

No-slip condition

• Velocity boundary condition stating that the fluid velocity at a solid surface is equal to the velocity of the surface
• Applies to viscous flows and is a consequence of fluid adhesion to solid surfaces
• No-slip condition is crucial in the formation of boundary layers and the development of flow separation

Inlet vs outlet conditions

• Inlet conditions specify the fluid properties (velocity, pressure, temperature) at the entrance of a fluid domain
• Outlet conditions specify the fluid properties at the exit of a fluid domain
• Proper specification of inlet and outlet conditions is essential for obtaining accurate solutions to fluid flow problems

Adiabatic vs isothermal walls

• Adiabatic wall condition assumes no heat transfer between the fluid and the solid surface ($\frac{\partial T}{\partial n} = 0$)
• Isothermal wall condition assumes a constant temperature at the solid surface ($T = T_{\text{wall}}$)
• Choice between adiabatic and isothermal wall conditions depends on the thermal characteristics of the flow and the solid surface

Applications of conservation laws

• Conservation laws form the basis for analyzing and solving a wide range of fluid flow problems in aerodynamics and engineering
• Applications include the design of aircraft wings, propulsion systems, wind turbines, and heat exchangers
• Conservation laws are also used to study natural phenomena such as atmospheric flows, ocean currents, and astrophysical flows

Bernoulli's equation

• Simplified form of the momentum equation for steady, inviscid, and incompressible flow along a streamline
• Relates the fluid velocity, pressure, and elevation: $\frac{V^2}{2} + \frac{p}{\rho} + gz = \text{constant}$
• Bernoulli's equation is used to analyze flow through pipes, ducts, and around airfoils

Isentropic flow

• Flow in which the entropy of the fluid remains constant
• Occurs in the absence of heat transfer, viscous dissipation, and shock waves
• Isentropic flow relations are used to analyze compressible flows in nozzles, diffusers, and wind tunnels

Quasi-one-dimensional flow

• Simplified flow model assuming that the flow properties vary only in the flow direction
• Applicable to flows in ducts, nozzles, and channels with gradual area changes
• Quasi-one-dimensional flow equations are derived from the conservation laws and are used to design and analyze fluid systems

Shock waves

• Thin regions of rapid change in fluid properties, such as pressure, density, and velocity
• Occur when a flow transitions from supersonic to subsonic speeds
• Shock waves are governed by the Rankine-Hugoniot jump conditions, which are derived from the conservation laws

Expansion waves

• Regions of gradual change in fluid properties, such as pressure, density, and velocity
• Occur when a flow transitions from subsonic to supersonic speeds
• Expansion waves are described by the method of characteristics, which is based on the conservation laws and the isentropic flow relations