Potential Equations

A further simplification to the governing equations of the flow can be achieved by assuming that the flow is irrotational, that is, V x V = 0. In this case the velocity field can be expressed as a gradient of a potential function ф so that

V = V0. (14.32)

Care must be taken with this approach, however, especially given the essentially rotational nature of the flow in the wake. The potential equations can be expressed in many ways, but one form, used commonly to model compressible flows, is

where V — I V|. This form of the potential equation is usually referred to as the full-potential equation because no terms have been neglected in its derivation from the Euler equations – see Sankar & Prichard (1985), Strawn (1986), and Strawn & Caradonna (1987). For tran­sonic flow problems, all the terms are important if the characteristic nonlinear variation of the flow properties with Mach number is to be represented properly. If small disturbances can be assumed then some of the terms are of higher order and can be neglected, however. The transonic small disturbance (TSD) equations are a subset of the full potential equations and can be written in the form

Афц+Вфхі = Ех+фгі+СфуУ+ cross terms and low-order derivatives. (14.34)

The nonlinearity of the problem is contained in the Fx term, which involves фх and ф2. Historically this was one of the first flow equations ever to be tackled by numerical means, launching the field known today as CFD – see Caradonna & Isom (1972) and Caradonna

(1990) .

If all the higher-order terms are neglected, the TSD equation reduces to a wave equation of the form

4-0« = V2*, (14.35)

aX>

where Яоо is the free-stream speed of sound. A final simplification is to assume incom­pressible flow (a0о —» oo), in which case the velocity potential is governed by Laplace’s equation, that is,

V20 = 0. (14.36)

A feature of this equation is that it is linear and so it allows complex flows to be synthesized by combining more elementary flows. This equation underlies the vortex methods de­scribed in Section 14.4 as well as the surface singularity methods described in the following section.