Acceleration Methods

Various ideas have been used to decrease these computational times and make the FVM more attractive for various helicopter applications. Often, methods can be used to differentiate between vortex elements in the “near field” and “far field,” the latter which have a smaller influence and can be either excluded from the calculation altogether or can be included by lumping together the induced effects of several “far” elements. Miller & Bliss (1993) have employed an analytical and numerical matching technique, in which a near field solution for the vortex near the core is matched to a far field solution. Substantial reductions in computational cost are possible. An extension of this approach is discussed by Miller (1993). Although successful in decreasing execution times, the resulting higher-order velocity field errors associated with the approximations may still undermine the accuracy of the wake predictions. Linear interpolation of the positions of the wake markers, such as discussed by Brooks et al. (1996), simply to effect a larger number of points does not improve the net accuracy of the wake solution. Sarpkaya (1989) discusses other methods that can potentially be used to reduce the computational time of vortex methods.

Bagai & Leishman (1998) have approached the problem through adaptive refinement of the finite-difference grid used to solve the governing equations of the vortex wake, with inter­polation of known information onto intermediate points in the wake. Reductions in compu­tational effort of over an order of magnitude are possible. The underlying goal is to perform fewer explicit induced velocity calculations using the Biot-Savart law, while retaining the

accuracy and fidelity of the wake solutions. For equal discretization (Aif/b = Ax(/W) with Щь = 2n/Aif/b azimuthal steps per rotor revolution, the number of vortex segments in each wake turn is — N^b = N. Therefore, the number of Biot-Savart calculations

required is Nh N^h = NbN2. (Without periodicity assumptions this becomes N%N3.)

Velocity field interpolation reduces the effective number of Biot-Savart calculations by using both “free” and “pseudo-free” Lagrangian markers. For the case where Д Jsw > Afrb, the induced velocities are calculated explicitly only at the free Lagrangian markers spaced at Д ijsw, and by linear interpolation of the velocity field at the pseudo-free markers. Therefore, the number of Biot-Savart evaluations is decreased by a factor corresponding to the ratio of the discretization Af/W/Афь. For example, with Atyw = 2 Aj/b the number of Biot-Savart calculations is reduced by a factor of 4 to (Nb TV3)/4. In a time-marching approach this in­terpolation scheme is applied after each time-step, whereas using the relaxation algorithm the interpolation is applied over the entire domain with a wake iteration.

In addition to the overall lower cost of a relaxation algorithm, another advantage is that azimuthal interpolation can be used where Afrb > A(rw. The basic approach is similar to that described for vortex filament interpolation, the difference being that the interpolation is performed at a constant vortex age along the blade azimuth. This results in a computa­tional saving of a factor of Af/b/Afrw in the number of Biot-Savart velocity evaluation points. However, the total number of wake markers remains unchanged. For example, with Афь = 2 Alfw times the number of Biot-Savart calculations is reduced by half to (Nb N*)/2. The blade attachment boundary condition (i. e., the first Lagrangian marker) needs special treatment. Azimuthal interpolation, however, is a redundant acceleration technique for the time-marching approach (see Section 10.8.4) because it is always necessary to use small time (azimuthal) steps to preserve accuracy in time.