Fundamental Equations

Some elementary yet important equations are listed herein. Readers must be able to derive them and appreciate the physics of each term for intelligent application

See Symbols and Abbreviations, this volume, pp. xix-xxvii.

to aircraft design. The equations are not derived herein – readers may refer to any introductory aerodynamic textbook for their derivation.

In a flowing fluid, an identifiable physical boundary defined as control volume (CV) (see Figure 10.11) can be chosen to describe mathematically the flow charac­teristics. A CV can be of any shape but the suitable CVs confine several streamlines like well-arranged “spaghetti in a box” in which the ends continue along the stream­line, crossing both cover ends but not the four sides. The conservative laws within the CV for steady flow (independent of time, t) that are valid for both inviscid incom­pressible and compressible flow are provided herein. These can be equated between two stations (e. g., Stations 1 and 2) of a streamline. Inviscid (i. e., ideal) flow under­going a process without any heat transfer is called the isentropic process. During the conceptual study phase, all external flow processes related to aircraft aerodynamics are considered isentropic, making the mathematics simpler. (Combustion in engines is an internal process.)

Newton’s law: applied force, F = mass x acceleration = rate of change of momentum

From kinetics, force = pressure x area

and work = force x distance

Therefore, energy (i. e., rate of work) for the unit mass flow rate m is as follows:

energy = force x (distance/time) = pressure x area x velocity = pAV

mass conservation: mass flow rate m = pAV = constant (3.3)

Momentum conservation: dp = – pVdV (known as Euler’s equation) (3.4)

With viscous terms, it becomes the Navier-Stokes equation. However, friction forces offered by the aircraft body can be accounted for in the inviscid-flow equation as a separate term:

1 2

energy conservation: Cp T + 2 V = constant

When velocity is stagnated to zero (e. g., in the hole of a Pitot tube), then the follow­ing equations can be derived for the isentropic process. The subscript t represents the stagnation property, which is also known as the “total” condition. The equations represent point properties – that is, valid at any point of a streamline (y stands for the ratio of specific heats and M for the Mach number):

The conservation equations yield many other significant equations. In any stream­line of a flow process, the conservation laws exchange pressure energy with the
kinetic energy. In other words, if the velocity at a point is increased, then the pres­sure at that point falls and vice versa (i. e., Bernoulli’s and Euler’s equations). Fol­lowing are a few more important equations. At stagnation, the total pressure, pt, is given.

Bernoulli’s equation: For incompressible isentropic flow,

р/р + V2 /2 = constant = pt (3.10)

Clearly, at any point, if the velocity is increased, then the pressure will fall to maintain conservation. This is the crux of lift generation: The upper surface has lower pressure than the lower surface.

Euler’s equation: For compressible isentropic flow,

There are other important relations using thermodynamic properties, as follows.

From the gas laws (combining Charles’s law and Boyle’s law), the equation for the state of gas for the unit mass is pv = RT, where for air:

R = 287 J/kgK	(3.12)
Cp – Cv = Randy = Cp/Cv	(3.13)
From the energy equation, total temperature:
T T, T(Y – 1)V2 T| V2 t 2 RTy 2CP	(3.14)
Mach number = V/a, where a = speed of sound and
a = Y RT = (dp/dp)isentropic	(3.15)