Neumann Boundary Condition

In this case it is required that ЭФ*/дп will be specified on the solid boundary SB, e. g.:

У(Ф + Ф„) • n = 0 (9.4)

where Ф is the perturbation potential consisting of the two integral terms in Eq. (9.2a). From this point and on, for convenience, the velocity potential will be split such that Ф„ is the free-stream velocity potential (Eq. 9.3) relative to the origin of the coordinates attached to the surface SB. The second boundary condition (at the distant, outer boundaries of the flow) requires that the flow disturbance, due to the body’s motion through the fluid, should diminish far from the body,

lim УФ = 0 (9.5)

where r = (x, y,z). This condition is automatically met by all the singular solutions considered here. To satisfy the boundary condition in Eq. (9.4)

directly, we use the velocity field due to the singularity distribution of Eq.

УФ*(х, у, z) = – f 4л Jb

dS + V*!)» (9.6)

(9.2)

:

If the singularity distribution strengths о and p are known, then Eq. (9.6) describes the velocity field everywhere (of course, special treatment is needed when the velocity is evaluated on the surface SB). Substitution of Eq. (9.6) into the boundary condition in Eq. (9.4) results in:

fr~f ‘lVfT'(")ldS-r" f oV(“)d5+V$4’n=0 (9-7>

14я Jbody+wake 1дпгП 4Л )ъойу W J

This equation is the basis for many numerical solutions and should hold for every point on the surface SB. For example, a certain number of points (called collocation points) can be selected on the surface SB. The boundary condition of Eq. (9.7) is then specified at each of these points in terms of the unknown singularities at all the collocation points. This approach reduces the integral equation (Eq. (9.7)) to a set of algebraic equations. As noted in Chapter 3, the solution at this point is not unique, and the combination of sources and doublets must be specified.

Note that if for an enclosed boundary (e. g., SB) ЭФ*/Эп =0, as required by the boundary condition in Eq. (9.4), then the potential inside the body (without internal singularities) will not change (Lamb,9 1 p. 41):

Ф* = const.

which constant could be selected also as zero. This observation is important since it allows us to specify the boundary condition (Eq. (9.4)) in terms of the potential inside SB, which is the Dirichlet problem (or Dirichlet boundary condition).