Time-Marching Free-Vortex Wakes

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approximation to the rotor wake problem, and also with the fewest restrictions in application. In a time-marching algorithm the time evolution of the position vectors of each wake Lagrangian marker can be expressed by

As shown previously in Section 10.7.3 a second-order, five-point central difference ap­proximation can be used to find the spatial (J/W) derivative, and the right-hand side of Eq. 10.118 can be represented discretely as F (r(jfb), ^). Time-marching algorithms can also take advantage of the numerical improvements produced by a predictor-corrector se­quence. Although any initial wake geometry can be specified from which to march the solution, the initial or starting induced velocity field is usually a relaxation solution. This helps minimize nonphysical transients (and numerical costs) associated with defining the initial condition. Unfortunately time-marching methods have proven susceptible to insta-

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uniuvo ivouiviii^ uv/ui uxv iintiuwxvxi vi uuiuvuvui iiuvx vuu uviujl vj |^ow jLiiiugrvui cv juviouuiuu (2000b, 2001a)], thereby reducing the confidence that a physically realistic wake solution has been obtained. Properly distinguishing between the known physical instabilities of rotor wakes and those that are numerical in origin have proven to be a major hindrance in the development of reliable and robust time-accurate vortex wake models for helicopter rotor applications.

Bhagwat & Leishman (2001a) have used a five-point central difference scheme for both the spatial derivative and the temporal derivative in Eq. 10.54 during the predictor step, and a five-point central difference scheme for the spatial derivative and a three-point second-order backward difference scheme for the temporal derivative in the corrector step. This stencil is then swept over the computational domain – see Fig. 10.46. A Taylor’s series expansion, around the cell evaluation point shows that leading terms in the expansions of the difference approximations are 0(А^|) and O(AV^), so the time-marching approximation is second – order accurate in both space f/b and time frw.