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Sheets
In the first simplification stage we neglect the kernel function’s n dependence by assuming a representative integration point r'(s, i,n) — r'(s, e,no) at some fixed no location, indicated in Figure 2.3. The simplified kernel function can then be removed from the n integral, allowing the a or ш distribution to be integrated or lumped in n across the layer thickness from щ to? г2, thus defining the sheet strengths Л(s, e) and The
volume integrals in the velocity superposition (2.3) then become the simpler surface integrals over the sheet coordinates s and l, with r/(s, i) now denoting the integration points on the sheet.
|
Vj (r) — VA(r) = |
si |
A(s, i) = |
П2 a(s, i,n) dn ni |
(2.15) |
|
|
Vw(r) — Vy (r) = |
si |
b*,;~rfdsd< : |
Y(s, i) = |
П2 ш(s, l,n) dn ni |
(2.16) |
The resulting velocities Va and VY are now discontinuous across the sheets, but this does not cause any problems in practice. Note also that outside the original source or vorticity volume, Va is very nearly the same as the actual V<t, and VY is very nearly the same as the actual Vu, as Figure 2.3 suggests. Sheets are extensively used in aerodynamic modeling and computation, and will be discussed in more detail later.
