Small-Disturbance Compressible Flows

The full potential equation is very general, but it requires grid-based CFD solution methods which offer little insight into compressible flow behavior. For this reason we will now consider the more restricted class of Small-Disturbance Flows, which in many circumstances can be treated by superposition-based solution methods.

8.5.1 Perturbation velocities

The perturbation velocity is defined in the usual way, as the difference between the local velocity V and the freestream velocity V,. To minimize equation complexity, we will from now on assume that the freestream is along the x axis, so that the angles of attack or sideslip are in the geometry definition. Furthermore, u, v, w will here denote the perturbation velocity components, which are also assumed to be normalized by the freestream speed V,. The local total velocity V and its magnitude V are then expressed as follows.

V, = V, X

V = V, [(1 + u)X + vy + wz] V2 = V ■ V = V,2 [ 1 + 2u + u2+v2+w2 ]

– ?r(V2-V*) = cd{l – (7-l)M2[u + i(u2+t;W)]} (8.34)

The local adiabatic speed of sound, the local Mach number, and the isentropic density and pressure expres­sions can also be expressed in terms of the perturbation velocities as follows.

8.5.2 Small-disturbance approximation

The above restatement of the various flow quantities in terms of perturbation velocities has so far been exact, with no new approximations introduced. We now consider Small-Disturbance Flows, where the condition

u, v,w ^ 1 (8.38)

is assumed to hold. This is generally valid if

• The geometry is slender: t/c ^ 1 for an airfoil, or d/£ ^ 1 for a fuselage.

• The aerodynamic angles are small: a ^ 1 and в ^ 1

Under normal circumstances it is tempting to drop all higher powers of the perturbation velocities like u2,uv, etc. and retain only the linear terms to greatly simplify the flow equations. However, this will be seen to be premature for transonic flows, where some of the nonlinear terms always remain crucial. Hence we will perform the simplification in three steps:

1. First only the cubic and higher terms will be dropped.

2. Next, all the quadratic terms be dropped except the ones which remain indispensable.

3. Next, all the quadratic terms will be dropped, finally giving a linear problem.