W.W.Shultz

University of Michigan, USA

1. Higher-order panels are known to be more susceptible to singularities. Have you noticed such problems at the corners? What constraints do you put on the patches?

2. The lines midway between cylinders look like appropriate locations to apply periodic boundary conditions if the wavenumber in the in-line cylinder direction is properly chosen. Then, couldn't waves come in many directions?

1. We do not place any constraints on the solution at the boundaries between contiguous patches. We find that the potential is practically continuous at these boundaries, even in cases where there is an external corner flow. Figure 12, reproduced from [16], illustrates this in the case of the streaming flow past a cube (without a free surface). The equipotential lines in this figure appear to be smooth and continuous within graphical accuracy, except for a small discontinuity which is evident near the corner where the three edges meet.

2. For a long array, the solution is nearly periodic along the array, with a constant phase shift between adjacent cylinders. Away from the resonant peaks, this phase shift is governed by the longitudinal component of the incident-wave wavenumber and by the cylinder spacing, as assumed by Linton & Evans [23]. However, at the resonant peaks, where the “sloshing modes” are dominant, the phase shift is either zero (Dirchlet modes), or 180° (Neumann modes), in accordance with the requirement that the sloshing modes be continuous between adjacent cylinders. In the latter case, the direction of the incident waves is irrelevant, as suggested by Professor Schultz.

[23] Linton, C.M. & Evans, D.V., “The interaction of waves with a row of circular cylinders,” J. Fluid Mech., 251, 1993, pp. 687–708.

FIGURE 12: Equipotential contours on an octant of a translating cube. Each side of the octant is a patch and discretized by 8×8 panels/patch. The lower figure is a close-up of the corner showing continuity across patches.