Fundamentals of Aerodynamics

1. Aerodynamics: A Science Driven by Practical Needs and Intellectual Beauty

Fundamentals of Aerodynamics is intended to portray and convey this beauty.

Historical Context. Aerodynamics emerged from practical needs, such as improving ship hull design and enabling manned flight. The study of fluid dynamics has been shaped by historical events, from naval battles to the Wright brothers' wind tunnel experiments, and the space race.

Engineering Applications. Aerodynamics is not just an abstract science; it's an applied discipline with objectives like predicting forces on moving bodies and optimizing internal flows in ducts and engines. The field is driven by practical applications, such as designing efficient aircraft, understanding weather patterns, and improving the performance of engines.

Intellectual Beauty. Beyond its practical applications, aerodynamics is a subject of intellectual beauty, shaped by the contributions of many great minds over centuries. The study of aerodynamics is a journey through the history of science and engineering, revealing the elegance and complexity of fluid motion.

2. Fundamental Aerodynamic Variables: Pressure, Density, Temperature, and Flow Velocity

Pressure is the normal force per unit area exerted on a surface due to the time rate of change of momentum of the gas molecules impacting on (or crossing) that surface.

Defining Aerodynamic Properties. Understanding aerodynamics requires a grasp of key variables: pressure (force per unit area), density (mass per unit volume), temperature (related to molecular kinetic energy), and flow velocity (speed and direction of fluid elements). These properties are defined at a point and can vary throughout the flow field.

Shear Stress. Friction plays a role in a flow, where adjacent streamlines rub against each other, exerting a tangential force. The local shear stress is proportional to the spatial rate of change of velocity normal to the streamline.

Units of Measurement. Aerodynamic calculations rely on consistent units, either the International System of Units (SI) or the English engineering system. Consistent units are essential for accurate calculations and avoiding conversion errors.

3. Aerodynamic Forces and Moments: Integrating Pressure and Shear Stress

The only mechanisms nature has for communicating a force to a body moving through a fluid are pressure and shear stress distributions on the body surface.

Pressure and Shear Stress. Aerodynamic forces and moments on a body are due to the pressure distribution (normal force) and shear stress distribution (tangential force) integrated over the body's surface. These are the only ways nature can exert force on an object moving through a fluid.

Lift and Drag. The resultant aerodynamic force can be resolved into lift (perpendicular to the freestream velocity) and drag (parallel to the freestream velocity). These forces can also be resolved into normal and axial forces relative to the chord line.

Mathematical Integration. The total normal and axial forces, as well as the moment about the leading edge, can be calculated by integrating the pressure and shear stress distributions over the body surface. These integrals demonstrate that aerodynamic forces and moments are a result of these distributions.

4. Dimensionless Coefficients: A Universal Language for Aerodynamics

These are dimensionless force and moment coefficients.

Dimensionless Parameters. Aerodynamicists use dimensionless force and moment coefficients (CL, CD, CN, CA, CM) to generalize aerodynamic forces and moments. These coefficients are defined using freestream dynamic pressure, a reference area, and a reference length.

Reference Values. The reference area and length are chosen to pertain to the given geometric body shape. The particular choice of reference area and length is not critical; however, when using force and moment coefficient data, you must always know what reference quantities the particular data are based upon.

Pressure and Skin Friction Coefficients. Two additional dimensionless quantities of immediate use are the pressure coefficient (Cp) and the skin friction coefficient (cf), which relate local pressure and shear stress to freestream dynamic pressure. These coefficients are useful for integrating over the body to obtain aerodynamic forces and moments.

5. Flow Similarity: Scaling Aerodynamic Phenomena

Two flows will be dynamically similar if: 1. The bodies and any other solid boundaries are geometrically similar for both flows. 2. The similarity parameters are the same for both flows.

Dynamic Similarity. Different flows are dynamically similar if their streamline patterns are geometrically similar, the distributions of nondimensional flow variables are the same, and the force coefficients are the same. Dynamic similarity is the foundation for wind tunnel testing.

Reynolds and Mach Numbers. Two flows are dynamically similar if the bodies are geometrically similar and the Reynolds and Mach numbers are the same for both flows. These are the dominant similarity parameters for many aerodynamic applications.

Wind Tunnel Testing. Wind tunnel testing relies on flow similarity. If a scale model is tested in a wind tunnel, the measured coefficients will be the same as for free flight if the Mach and Reynolds numbers are matched.

6. Fluid Statics: Understanding Buoyancy

Buoyancy force on body = weight of fluid displaced by body

Hydrostatic Equation. In a stagnant fluid, pressure increases with depth according to the hydrostatic equation: dp = -g𝜌dy. This equation governs the variation of atmospheric properties with altitude.

Archimedes' Principle. A body immersed in a fluid experiences a buoyancy force equal to the weight of the fluid displaced by the body. This principle applies to both liquids and gases.

Applications of Buoyancy. Buoyancy is crucial for naval vehicles and lighter-than-air craft. While often negligible in aerodynamics, it's a fundamental force in fluid mechanics.

7. Classifying Fluid Flows: From Continuum to Hypersonic

The term “aerodynamics” is generally used for problems arising from flight and other topics involving the flow of air.

Fluid Dynamics. Fluid dynamics encompasses the study of liquids and gases, with aerodynamics specifically focusing on the flow of air. Fluid dynamics is subdivided into three areas: hydrodynamics (flow of liquids), gas dynamics (flow of gases), and aerodynamics (flow of air).

Flow Types. Aerodynamic flows are classified based on various criteria, including:

• Continuum vs. Free Molecule Flow: Whether the fluid can be treated as a continuous medium.
• Inviscid vs. Viscous Flow: Whether friction is neglected or considered.
• Incompressible vs. Compressible Flow: Whether the density is constant or variable.
• Mach Number Regimes: Subsonic, transonic, supersonic, and hypersonic flows.

Hypersonic Flow. Hypersonic flow, generally defined as any flow above a Mach number of 5, is characterized by thin shock layers, viscous interactions, and chemical reactions.

8. The Boundary Layer: Where Viscosity Dominates

In contrast, a flow that is assumed to involve no friction, thermal conduction, or diffusion is called an inviscid flow.

Viscous vs. Inviscid Flow. Viscous flows account for friction, thermal conduction, and diffusion, while inviscid flows neglect these effects. Although inviscid flows don't exist in nature, they can approximate many practical aerodynamic flows where transport phenomena are small.

Boundary Layer. In high Reynolds number flows, viscous effects are confined to a thin region near the body surface called the boundary layer. The flow outside this layer can be treated as inviscid.

Shear Stress and Velocity Gradients. Friction generates shear stress, which is proportional to the velocity gradient normal to the streamlines. Large velocity gradients within the boundary layer make friction dominant in this region.

9. The Buckingham Pi Theorem: Simplifying Complex Relationships

Then, the physical relation Equation (1.24) may be reexpressed as a relation of (N−K) dimensionless products (called Π products).

Dimensional Analysis. Dimensional analysis is a method to define dimensionless parameters that govern aerodynamic forces and moments. It reduces the number of independent variables in a physical relationship.

Buckingham Pi Theorem. The Buckingham Pi Theorem states that a physical relation involving N physical variables and K fundamental dimensions can be reexpressed as a relation of (N-K) dimensionless products. This theorem simplifies complex problems by reducing the number of independent variables.

Similarity Parameters. Dimensional analysis defines similarity parameters like the Reynolds number (Re) and Mach number (M∞), which govern the flow. These parameters are crucial for ensuring dynamic similarity between different flows.

10. The Center of Pressure: Locating Aerodynamic Forces

It is the location where the resultant of a distributed load effectively acts on the body.

Definition. The center of pressure (xcp) is the point on a body where the resultant aerodynamic force effectively acts. It's the location where the moment due to the distributed load is equivalent to the moment produced by the resultant force.

Calculation. The center of pressure is calculated by equating the moment about the leading edge to the moment produced by the normal force acting at xcp. The center of pressure is not always a convenient concept in aerodynamics.

Alternative Representations. The force-and-moment system on a body can be specified by placing the resultant force at any point, as long as the moment about that point is also given. This allows for flexibility in analyzing aerodynamic forces and moments.

Last updated: April 3, 2025