## File: airfoil.dem

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gnuplot 3.5beta6.340-5
 `123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122123124125126127128129130131132133134135136137138139140141142143144145146147148149150151152153154155156157158159160161162163164165166167168169170` ``````# # \$Id: airfoil.dem,v 1.3 1993/10/07 16:53:28 alex Exp \$ # # This demo shows how to use bezier splines to define NACA four # series airfoils and complex variables to define Joukowski # Airfoils. It will be expanded after overplotting in implemented # to plot Coefficient of Pressure as well. # Alex Woo, Dec. 1992 # # The definitions below follows: "Bezier presentation of airfoils", # by Wolfgang Boehm, Computer Aided Geometric Design 4 (1987) pp 17-22. # # Gershon Elber, Nov. 1992 # # m = percent camber # p = percent chord with maximum camber pause 0 "NACA four series airfoils by bezier splines" pause 0 "Will add pressure distribution later with Overplotting" mm = 0.6 # NACA6xxx thick = 0.09 # nine percent NACAxx09 pp = 0.4 # NACAx4xx # Combined this implies NACA6409 airfoil # # Airfoil thickness function. # set xlabel "NACA6409 -- 9% thick, 40% max camber, 6% camber" x0 = 0.0 y0 = 0.0 x1 = 0.0 y1 = 0.18556 x2 = 0.03571 y2 = 0.34863 x3 = 0.10714 y3 = 0.48919 x4 = 0.21429 y4 = 0.58214 x5 = 0.35714 y5 = 0.55724 x6 = 0.53571 y6 = 0.44992 x7 = 0.75000 y7 = 0.30281 x8 = 1.00000 y8 = 0.01050 # # Directly defining the order 8 Bezier basis function for a faster evaluation. # bez_d4_i0(x) = (1 - x)**4 bez_d4_i1(x) = 4 * (1 - x)**3 * x bez_d4_i2(x) = 6 * (1 - x)**2 * x**2 bez_d4_i3(x) = 4 * (1 - x)**1 * x**3 bez_d4_i4(x) = x**4 bez_d8_i0(x) = (1 - x)**8 bez_d8_i1(x) = 8 * (1 - x)**7 * x bez_d8_i2(x) = 28 * (1 - x)**6 * x**2 bez_d8_i3(x) = 56 * (1 - x)**5 * x**3 bez_d8_i4(x) = 70 * (1 - x)**4 * x**4 bez_d8_i5(x) = 56 * (1 - x)**3 * x**5 bez_d8_i6(x) = 28 * (1 - x)**2 * x**6 bez_d8_i7(x) = 8 * (1 - x) * x**7 bez_d8_i8(x) = x**8 m0 = 0.0 m1 = 0.1 m2 = 0.1 m3 = 0.1 m4 = 0.0 mean_y(t) = m0 * mm * bez_d4_i0(t) + \ m1 * mm * bez_d4_i1(t) + \ m2 * mm * bez_d4_i2(t) + \ m3 * mm * bez_d4_i3(t) + \ m4 * mm * bez_d4_i4(t) p0 = 0.0 p1 = pp / 2 p2 = pp p3 = (pp + 1) / 2 p4 = 1.0 mean_x(t) = p0 * bez_d4_i0(t) + \ p1 * bez_d4_i1(t) + \ p2 * bez_d4_i2(t) + \ p3 * bez_d4_i3(t) + \ p4 * bez_d4_i4(t) z_x(x) = x0 * bez_d8_i0(x) + x1 * bez_d8_i1(x) + x2 * bez_d8_i2(x) + \ x3 * bez_d8_i3(x) + x4 * bez_d8_i4(x) + x5 * bez_d8_i5(x) + \ x6 * bez_d8_i6(x) + x7 * bez_d8_i7(x) + x8 * bez_d8_i8(x) z_y(x, tk) = \ y0 * tk * bez_d8_i0(x) + y1 * tk * bez_d8_i1(x) + y2 * tk * bez_d8_i2(x) + \ y3 * tk * bez_d8_i3(x) + y4 * tk * bez_d8_i4(x) + y5 * tk * bez_d8_i5(x) + \ y6 * tk * bez_d8_i6(x) + y7 * tk * bez_d8_i7(x) + y8 * tk * bez_d8_i8(x) # # Given t value between zero and one, the airfoild curve is defined as # # c(t) = mean(t1(t)) +/- z(t2(t)) n(t1(t)), # # where n is the unit normal to the mean line. See the above paper for more. # # Unfortunately, the parametrization of c(t) is not the same for mean(t1) # and z(t2). The mean line (and its normal) can assume linear function t1 = t, # -1 # but the thickness z_y is, in fact, a function of z_x (t). Since it is # not obvious how to compute this inverse function analytically, we instead # replace t in c(t) equation above by z_x(t) to get: # # c(z_x(t)) = mean(z_x(t)) +/- z(t) n(z_x(t)), # # and compute and display this instead. Note we also ignore n(t) and assumes # n(t) is constant in the y direction, # airfoil_y1(t, thick) = mean_y(z_x(t)) + z_y(t, thick) airfoil_y2(t, thick) = mean_y(z_x(t)) - z_y(t, thick) airfoil_y(t) = mean_y(z_x(t)) airfoil_x(t) = mean_x(z_x(t)) set nogrid set nozero set parametric set xrange [-0.1:1.1] set yrange [-0.1:.7] set trange [ 0.0:1.0] set title "NACA6409 Airfoil" plot airfoil_x(t), airfoil_y(t) title "mean line" w l 2, \ airfoil_x(t), airfoil_y1(t, thick) title "upper surface" w l 1, \ airfoil_x(t), airfoil_y2(t, thick) title "lower surface" w l 1 pause -1 "Press Return" mm = 0.0 pp = .5 thick = .12 set title "NACA0012 Airfoil" set xlabel "12% thick, no camber -- classical test case" plot airfoil_x(t), airfoil_y(t) title "mean line" w l 2, \ airfoil_x(t), airfoil_y1(t, thick) title "upper surface" w l 1, \ airfoil_x(t), airfoil_y2(t, thick) title "lower surface" w l 1 pause -1 "Press Return" set title "" set xlab "" set key set parametric set samples 100 set isosamples 10 set data style lines set function style lines pause 0 "Joukowski Airfoil using Complex Variables" set title "Joukowski Airfoil using Complex Variables" 0,0 set time set yrange [-.2 : 1.8] set trange [0: 2*pi] set xrange [-.6:.6] zeta(t) = -eps + (a+eps)*exp(t*{0,1}) eta(t) = zeta(t) + a*a/zeta(t) eps = 0.06 a =.250 set xlabel "eps = 0.06 real" plot real(eta(t)),imag(eta(t)) pause -1 "Press Return" eps = 0.06*{1,-1} set xlabel "eps = 0.06 + i0.06" plot real(eta(t)),imag(eta(t)) pause -1 "Press Return" set title "" set xlabel "" set notime ``````