Cost of the Free-Wake Solution

To understand the potential numerical cost of a FVM it is possible to estimate the number of Biot-Savart-like velocity evaluations required for a typical wake solution. Let Nfo = 360°/ДчД be the number of discrete azimuthal grid points. Likewise, for a vortex filament that is n rotor revolutions old, the number of free collocation points (= number of free vortex elements on that filament) is N^w = n 360° / Afw. Therefore, a total of Biot-Savart evaluations must be performed for each free vortex filament at each of N^b locations to account for the total self-induced velocities at each collocation point from every other vortex element. If the wake is modeled using only a single free vortex filament from each blade tip, this results in N фь velocity field evaluations to define the rotor wake at all azimuth angles. For a rotor with Nb blades, a further Nb (Nb — 1) N^b evaluations must be performed to account for mutually induced effects. Therefore, the total number of Biot-Savart evaluations required per free-vortex wake computation, Ne, is given by the equation

МЕ = (1 + Мь(Мь-1т^М^. (10.95)

For equal step sizes, =>■ N^w — nN^b, and so the total number of evaluations

becomes

Afe = (1 + Nb(Nb – 1 ))n2Nlt. (10.96)

For a typical four-bladed helicopter rotor modeled with three revolutions of free-tip vortices and equal discretization step sizes of 10 degrees, Eq. 10.96 suggests that the Biot-Savart integral must be evaluated over 1.8 x 106 times to cover the entire computational domain just one time. Doubling the resolution (i. e., using step sizes half the original size, such that Дijfb = 5°) requires eight times that number, or over 14.5 x 10° velocity evaluations. For additional free vortex filaments, the mutual interactions among all the additional free vortices must also be computed leading to a substantial computational effort in a FVM for routine use.