1Global coordinate system 2-------------------------------------------------------------------------------- 3 4The spatial coordinates are bound to the aircraft: 5x : frontal 6y : vertical 7z : spanwise left (horizontal) 8 9 ^ ____ 10 y |/ / 11 | / 12 /| / __ 13 _____/ | /_____/ _/ x 14 / | /__/ -------------> 15 \_____ _______/ 16 \ \ 17 \ \ \ 18 \ \ \ 19 \_\_\ 20 \ z 21 \| 22 23x-y is the wing symmetry plane, while the wing lies along z axis. The origin is 24the center of the wing. 25 26Maths & Physics 27-------------------------------------------------------------------------------- 28The method is based on approximation the flowfield around the wing by the 29potential flow induced by an ensemble of horseshoe vortices, with bound 30segments aligned along the wing's centerline, and the free segments shedding 31in the streamwise direction. Their strengths, i.e. circulations, are unknown 32and need to be determined using the flow equations. With known strengths, 33the flowfield velocity can be calculated at an arbitrary point using 34Biot-Savart law. 35 36The control (unknown) spanwise quantity on i-th vortex is 37 +---------------------+ 38 | g_i = gamma_i/c_i | (1) 39 +---------------------+ 40where gamma_i is the circulation and ch_i is local chord length (this gives 41approximately local cl). 42 43The flow equation to be satisfied locally follows from expressing 44the lift created on a particular section in two ways: 45from the Kutta-Joukowski law (the left-hand side) and 46using the 2D section data (the right-hand side) 47 +-------------------------------------------------------+ 48 | rho * v * gamma = 1/2 * rho * v^2 * cl * c | (2) 49 +-------------------------------------------------------+ 50 51giving 52 +----------------------------+ 53 | g = v * cl / 2 | (3) 54 +----------------------------+ 55 56where 57 +--------------------------+ 58 | cl = cl(alfa) | 59 | alfa = atan(vy/vx) | (4) 60 | v = sqrt(vx^2 + vy^2) | 61 +--------------------------+ 62 63Numerics 64-------------------------------------------------------------------------------- 65Using Biot-Savart law, it is possible to express the induced velocity at each 66collocation point caused by unit circulation on each vortex, thus obtain the 67"influence" tensor. Thus, from g_i, vx_i and vy_i are obtained by applying 68appropriate influence matrices. atan(vy_i/vx_i) then gives alfa_i - see (4) 69(at this point, nonlinearity is introduced). cl_i is then obtained from alfa_i 70by interpolating the provided 2D section data (a combination of spanwise and 71angle-wise interpolation is used). Finally, (3) closes the cycle, arriving 72at g_i again. 73In this fashion, for any global angle of attack we obtain a system of nonlinear 74equations. This parametric system is solved by starting at a low angle of attack 75and tracking the nonlinear solution to higher angles while possible, using a 76predictor-corrector strategy. 77The 2D section lifts are interpolated in two dimensions: by spanwise coordinate 78(combining different datasets) and by angle of attack. Instead of a naive 79interpolation, a "feature" interpolation is done: at first, each lifting line is 80analyzed for a zero-lift and maximum-lift angle. The lifting line is then scaled 81to 0,1, and these three features (zero-lift angle, max-lift angle, and scaled 82lifting line) are interpolated spanwise by linear interpolation. 83 84