# Aircraft Drag Formulation

A theoretical overview of drag is provided in this section to show that aircraft geom­etry is not amenable to the equation for an explicit solution. Even so, CFD is yet to achieve an acceptable result for the full aircraft.

Recall the expression in Equation 9.2 for the total aircraft drag, CD, as:

where CDparasite — CDfriction + CDpressure — CDpmin + ACDp

At LRC, when Cdw & 0, the total aircraft drag coefficient is given by:

Cd — CDpmin + ACdp + CDi (9.3)

The general theory of drag on a 2D body (Figure 9.3a) provides the closed – form Equation 9.4. A 2D body has infinite span. In the diagram, airflow is along the x direction and wake depth is shown in the y direction. The wake is formed due to viscous effects immediately behind the body, where integration occurs. The sub­script to denotes the free-stream condition. Consider an arbitrary CV large enough

in the y direction where static pressure is equal to free-stream static pressure (i. e., P = pX). Wake behind a body is due to the viscous effect in which there is a loss of velocity (i. e., momentum) and pressure shown in the figure. Measurement and computation across the wake are performed close to the body; otherwise, the down­stream viscous effect dissipates the wake profile. Integration over the y direction on both sides up to the free-stream value gives:

X (X)

D = Dpress + Dskin = f (Px – p)dy + j pu(Ux – u)dy

— X —X

X

= J [(PxD — p) + pu(Ux — u)] dy (9.4)

— X

An aircraft is a 3D object (Figure 9.3b) with the additional effect of a finite wing span that produces induced drag. In that case, the previous equation can be written as:

x b/2

D = Dskin + Dpress + Di = ff [(Px — p) + pu(Ux — u)]dxdy (9.5)

— x —b/2

where b is the span of the wing in the x direction (i. e., the axis system has changed).

The finite-wing effects on the pressure and velocity distributions result in induced drag Di embedded in the expression on the right-hand side of Equation 9.5. Because the aircraft cruise condition (i. e., LRC) is chosen to operate below Mcrit, the wave drag, Dw, is absent; otherwise, it must be added to the expression. Therefore, Equation 9.5 can be equated with the aircraft drag expression as given in Equa­tion 9.3. Finally, Equation 9.5 can be expressed in non

Unfortunately, the complex 3D geometry of an entire aircraft in Equation

9.3 is not amenable to easy integration. CFD has discretized the flow field into small domains that are numerically integrated, resulting in some errors. Mathemati­cians have successfully managed the error level with sophisticated algorithms (see Chapter 14 for a discussion of CFD). The proven industrial-standard, semi-empirical methods are currently the prevailing practice and are backed up by theories and val­idated by flight tests. CFD assists in the search for improved aerodynamics.