TWO-DIMENSIONAL POINT SINGULARITY ELEMENTS

These elements are probably the simplest and easiest to use and also the most efficient in terms of computational effort. Consequently, even when higher – order elements are used, if the point of interest is considered to be far from the element, then point elements can be used to describe the “far-field” effect. The three point elements that will be discussed are source, doublet, and vortex, and their formulation is given in the following sections.

10.1.1 Two-Dimensional Point Source

Consider a point source singularity at (x0, z0), with a strength a, as shown in Fig. 10.2. The increment to the velocity potential at a point P (following Section 3.7) is then

ф(*> z) = TIln V(* ~ *<>)* + (z – z0)2 (10.1)

and after differentiation of the potential, the velocity component increments are

ЭФ a x — x0

Эх 2л (x – x0)2 + (z – z0)2 ЭФ a z — Zq

dz 2л (x – x0)2 + (z – Zq)2

10.1.2 Two-Dimensional Point Doublet

Consider a doublet that is oriented in the z direction [ц = (0, /і)] as in Section 3.7. If the doublet is located at the point (x0, z0), then its incremental influence on the velocity potential at point P (Fig. 10.2) is

FIGURE 10.2 FIGURE 10.3

The influence of a point singularity element Transformation from panel to global coord-

at point P. inate system.

and the velocity component increments are

__дф _jl (x ~ Xp)(z – Zq)

Эх n[(x-x0)2 + (z-Zo)2]2 _ Эф (x ~ Xpf – (z – Zo)2

dz 2я [(x – x0)2 + (z – z0)2]2

In the case when the basic singularity element is given in a system (x*, z*), which is rotated by the argle /3 relative to the (x, z) system, as shown in Fig. 10.3, then the velocity components can be found by the transformation