Aerodynamic Modeling – Vorticity and Source Lumping

The general velocity field representation (2.3) via the source and vorticity fields is primarily conceptual, since the volume integrals for the Vr and V components are impractical to evaluate in numerical appli­cations. However, the representation (2.3) can be approximated and greatly simplified by the process of lumping, which is based on the approximation that the kernel function is constant along some small interval of one of the s£n coordinates. This allows the source and vorticity volume distributions to be in effect con­centrated into surfaces or sheets, and then possibly further concentrated into lines (or filaments), and then possibly even points. This process, illustrated in Figure 2.3, is the basis of aerodynamic modeling.

Note that at each lumping stage the singularity geometry becomes simplified, but the resulting velocity field becomes more singular and less realistic near the sheet, filament, or point (which is the origin of the name singularity). However, sufficiently far away the actual and approximated velocity fields become the same.

The lumping operation uses the curvilinear s£n coordinates, defined such that s£ lie on the surface of the sheet, filaments, or points, and n is normal to this surface. For simplicity, the curvatures of the s£n coordinate

lines are assumed to be sufficiently small so that they form a local effectively Cartesian system. A volume element is then simply ds dl dn, and a surface element is ds dl, so that the Jacobian factors which should appear in these elements are assumed to be unity and hence omitted for simplicity. This approximation does not adversely affect the effectiveness of the lumping concept for most aerodynamic applications.

Figure 2.3: Lumping of source and vorticity volume distributions into sheets and then lines. Source lines can be further lumped into source points. The evaluation of the velocity at any field point r then becomes progressively simplified. Lumping is the basis of aerodynamic modeling.