Aircraft Drag Build-Up and Trim Drag: Method

Aircraft Drag Build-Up and Trim Drag

What the tool computes

The zero-lift drag coefficient of a subsonic airplane by the component build-up of the USAF Stability and Control DATCOM (1978), the induced drag and the drag polar, the trim drag of the horizontal tail at the entered centre of gravity, and the transonic drag rise. Every component row gives a flat-plate skin-friction coefficient at its own Reynolds number, a form factor for the pressure drag of its shape, an interference factor and its wetted area; the row products are summed over the wing area, the miscellaneous items and an excrescence allowance are added, and the page prints the table, the equivalent flat-plate area and the share of each item. The library behind the page is checked against the four DATCOM sample problems, the NASA CR-151971 example and an independent numpy / scipy implementation (test-scripts/test_drag_build_up.php, 867 checks).

Skin friction

The Reynolds number of a component is Re = ρVl/μ on its reference length, the mean aerodynamic chord of an exposed surface or the length of a body, with the density and the Sutherland viscosity of the 1976 Standard Atmosphere at the entered altitude. The turbulent flat-plate mean coefficient is the Prandtl–Schlichting law C_F = 0.455/(log₁₀Re)^2.58, which is the M = 0 line of DATCOM figure 4.1.5.1-26 (the chart readings printed in the DATCOM sample problems are reproduced within 1 %), the Kármán–Schoenherr law 0.242/√C_F = log₁₀(Re C_F), which NASA CR-151971 uses and which lies 1–2 % lower, or the power law 0.044 Re^−1/6 that NASA RP-1060 gives for Reynolds numbers of 10⁶ to 10⁸. NACA TN 2305 notes that Schlichting's older 0.472 form sits about 4 % above the Kármán–Schoenherr line; the tool finds 5–6 % between 10⁶ and 3·10⁷. A laminar run to the transition point x_tr (a fraction of the reference length) is credited by the momentum balance C_F = C_Ft(Re) − x_tr[C_Ft(x_trRe) − C_Fl(x_trRe)] with the Blasius laminar coefficient 1.328/√Re; DATCOM assumes a fully turbulent layer on aircraft and so does the default.

Compressibility is applied as a factor on the incompressible coefficient of an insulated plate. The default is the interpolation formula of NACA TN 2305 eq. (10a), C_F = 0.472/[(log₁₀Re)^2.58(1 + (γ−1)M²/2)^0.467], whose Mach factor (1 + 0.2M²)^−0.467 represents the extended Frankl–Voishel analysis within 2 % up to M = 4; the alternative is the T′ reference-temperature method of Sommer and Short (NACA TN 3391 eq. 2): T′/T₁ = 1 + 0.035M₁² + 0.45(T_w/T₁ − 1) with the adiabatic wall temperature at the recovery factor 0.89, Re′ = Re/[(T′/T₁)(μ′/μ₁)], C_F = C_F′(Re′)·T₁/T′. The two agree within 0.01 at M 0.8 and both give the 0.966 at M 0.6 that NASA CR-151971 reads from its chart.

Roughness: DATCOM figure 4.1.5.1-27 gives a cutoff Reynolds number as a function of l/k, the reference length over the equivalent sand-roughness height; when the flight Reynolds number exceeds it, the friction coefficient is read at the cutoff instead (a rough plate's friction stops falling with Re). The tool uses the power law Re_cut = 45.0 (l/k)^1.053 fitted to the four readings of that chart printed in the DATCOM sample problems (within 6 % of each); Raymer's published fit of the same chart, 38.21 (l/k)^1.053, has the same exponent and a constant 15 % lower. The roughness heights are DATCOM table 4.1.5.1-A (smooth 0; polished metal or wood 0.02–0.08 mil; natural sheet metal 0.16; smooth matte paint 0.25; camouflage paint 0.40, mass-sprayed 1.20; dip-galvanized 6; cast iron 10 mil).

Form factors and interference

Surfaces: DATCOM equation 4.1.5.1-a, C_D0 = C_F[1 + L(t/c) + 100(t/c)⁴] R_LS S_wet/S, with L = 1.2 when the maximum thickness lies at or behind 0.30 c and 2.0 ahead of it, and the lifting-surface correlation factor R_LS of figure 4.1.5.1-28b, a function of the Mach number and the cosine of the sweep of the maximum-thickness line (0.81 at cos Λ = 0.5 to 1.08 at an unswept wing for M ≤ 0.25, rising to 1.09–1.35 at M 0.9). The chart was read into a table at every 0.05 of cos Λ for its four Mach curves; the eight R_LS values printed in the DATCOM sample problems are reproduced within 1 %. Below cos Λ = 0.5 (sweeps above 60°) the table is clamped and the row is flagged; DATCOM's dashed curves for the inboard panels of cranked wings are not used.

Bodies: DATCOM equation 4.2.3.1-a, C_D0 = C_F[1 + 60/f³ + 0.0025 f] S_S/S with the fineness ratio f = l/d (an equivalent diameter √(4A/π) for other cross sections), the last term being the pressure drag; a blunt base adds 0.029 (d_b/d)³/√C_Df,b on the frontal area (equation 4.2.3.1-b, C_Df,b the body drag exclusive of the base on that area). The 60/f³ term is the pressure drag of a body closed at both ends and overstates a short flow-through nacelle (f ≈ 2 gives 8.5), so a nacelle row takes 1 + 0.35/f, the form Raymer quotes for nacelles, and its wetted area is the open cylinder πdl.

Interference: each row carries an entered factor Q (1.0 none; the default's 1.05 on the tails and 1.3 on the wing-mounted nacelles are illustrative). DATCOM Section 4.3.3.1 instead multiplies the wing and body by a correlation factor R_WB read from figure 4.3.3.1-37 against the fuselage Reynolds number and the Mach number (1.07 to 0.87, the curves meeting at 1.0 near Re 7·10⁷); the checkbox applies it, the table was read from the chart and reproduces the four values printed in the DATCOM sample problems within 0.3 %. DATCOM warns that the correlation was built from ogive-cylinder bodies of high fineness ratio and has no data below M 0.25. Section 4.5.3.1 adds the tail panels as exposed components with no further factor, which is what the tool does.

Wetted areas

A blank wetted area is filled from the geometry: a surface by 2 S_exp(1 + 0.25 t/c), which is NASA RP-1060's 2 × planform (its eq. 3.25) with the usual thickness term; a fuselage by Loftin's πd²(l/d − 1), the area of a cylinder closed by a hemisphere and a cone of height 2d (RP-1060 eq. 3.25), or by Torenbeek's πdl(1 − 2/f)^2/3(1 + 1/f²); a nacelle by πdl. The page marks computed areas with an asterisk and prints the equivalent skin-friction coefficient C_D0S/S_wet, Loftin's skin-friction parameter, as the overall measure of cleanness.

Miscellaneous drag and excrescences

Items are entered as increments on the wing area or as drag areas D/q; the default rows (cooling air 0.0015, antennas and gaps 0.0010, retracted gear 0) are illustrative allowances for a light twin. For the orders of magnitude, NACA WR L-489 measured item by item on 1940 service airplanes cooling-duct and leakage increments of 0.001–0.003 and fixed-gear increments of 0.002–0.015 on the wing area, and NACA Report 518 the drag of nonretractable gears. The excrescence allowance is a percentage of the summed component drag; NASA CR-151971 correlates the flight-test minimum drag of fighters at 28 % above the computed friction drag (its figure 9 slope 1.284), the upper end for an airplane with gun ports, blisters and antennas.

Induced drag and the polar

K = 1/(π e AR) with the entered Oswald factor, or 1/(π e_LL AR) + K_v with the span efficiency e_LL = 1/(1 + δ) of the entered straight-tapered planform from the Prandtl lifting line (the Airfoil and Wing Lift tool's routine, 40 Multhopp stations, untwisted, a₀ = 2π) and an entered viscous drag-due-to-lift factor K_v. The lifting-line value (0.97–0.99 for ordinary tapers) is inviscid and ignores the fuselage, so an airplane Oswald factor of 0.7–0.85 is the usual choice; the page prints both. The polar optima are the closed forms: C_L = √(C_D0/K) and (L/D)_max = 1/(2√(C_D0K)) where the parasite and induced drag are equal, C_L = √(3C_D0/K) at minimum power, and the speeds √(2W/(ρSC_L)) at the entered weight and altitude, which the Aircraft Performance tool takes further.

Trim drag

The horizontal tail must balance the pitching moment about the centre of gravity. In the Longitudinal Static Stability tool's convention (positions h as fractions of the mean aerodynamic chord from its leading edge, the tail arm l_t from the wing aerodynamic centre to the tail aerodynamic centre, the tail at the dynamic-pressure ratio η_t) the balance is C_mac + C_m0,f + (h − h_ac,wb) C_Lw − η_t V_H C_Lt = 0, where the fuselage moment slope is carried as a forward shift of the aerodynamic centre, h_ac,wb = h_ac − C_mα,f/a_w, and V_H = (S_t/S)(l_t − (h − h_ac)c̄)/c̄ is the tail volume about the c.g.; the lift is C_L = C_Lw + η_t(S_t/S) C_Lt. The two linear equations give the wing and tail lift coefficients directly; the stability tool's full trim, which also solves the angle of attack and elevator angle with the downwash, gives the same loads (the test checks it to 10⁻⁹), because the loads are fixed by equilibrium and the downwash only sets the elevator angle needed. The trim drag is the induced drag of the trimmed airplane less that of the untrimmed one at the same lift: C_D,trim = K(C_Lw² − C_L²) + K_t η_t(S_t/S) C_Lt² with K_t = 1/(π e_t AR_t); a tail download makes the wing carry more than the weight, which is the first term, and the tail's own induced drag is the second. The tail's profile drag is in C_D0 already; its lift-dependent part is neglected. The page sweeps the trim drag against C_L for several c.g. positions and against the c.g. at the flight C_L, and prints the stick-fixed neutral point h_np = h_ac + [η_t V_H,ac a_t(1 − dε/dα) − C_mα,f]/a, the stability tool's exact form (the fuselage term divides by the airplane slope a, not by a_w: an earlier draft of this tool had it wrong and the cross-check caught it), with the tail slope from NACA Report 688 when none is entered.

Transonic drag rise

Korn's equation M_dd = κ_A/cos Λ − (t/c)/cos²Λ − C_L/(10 cos³Λ), with the technology factor κ_A entered (0.87 for conventional NACA 6-series sections and 0.95 for supercritical sections are the values commonly quoted with it, as in Mason's notes), gives the drag-divergence Mach number from the wing row's thickness ratio and sweep and the flight C_L; Lock's fourth-power law C_Dw = 20(M − M_cr)⁴ draws the rise above a critical Mach number M_cr = M_dd − (0.1/80)^1/3 = M_dd − 0.1077, so that the slope dC_D/dM is 0.1 at M_dd, the usual definition of drag divergence. Both are design approximations with named constants, used as mathematics; DATCOM's own transonic method (an initial rise of 0.002 at M 0.7 faired to the supersonic value) is cited, not implemented.

Acceptance cases

DATCOM 4.1.5.1 sample problem (cranked wing at M 0.13): the printed chart readings entered reproduce C_D0 = 0.00834 (test value 0.0086); the tool's formulas give 0.00843. DATCOM 4.2.3.1 (elliptic body at M 0.8): 0.0875 and base drag 0.0122 on the frontal area. DATCOM 4.3.3.1 (ogee wing-body at M 0.7): 0.0152 with R_WB 0.955; the scan's OCR reads the base-area ratio as 0.02176 where the page prints 0.0276 (= π0.875²/4 ÷ 21.75). DATCOM 4.5.3.1 (wing-body-tail at M 0.4, 0.6, 0.7): 0.0130, 0.0129, 0.0131 from the printed readings; its printed S_B = 1.083 in² is inconsistent with πd²/4 = 1.981 in² for its d = 1.588 in, so the formula route is compared without the base term (within 1.5 %). NASA CR-151971 A-4F and RA-5C (M 0.6): the Kármán–Schoenherr readings at eight Reynolds numbers within 1.2 % (one, the RA-5C fuselage at Re 9.5·10⁷, is 5 % above the law), the component products and the sums 0.01328 and 0.01281; the A-4F fuselage term printed as 0.00585 is 0.00546 from its own printed factors (a misprint carried into the printed total).

Units and conventions

Lengths, areas, forces and the airspeed go through the unit dictionary (knots and m/s for the speed); the roughness height is a plain number in mils (0.001 in = 25.4 μm); coefficients, degrees and percentages are plain; the equivalent flat-plate area is printed in the display area unit. Every coefficient is referred to the wing area except the base drag, which the DATCOM equation defines on the frontal area and the tool converts. No factor of safety or load level enters: the tool produces aerodynamic coefficients, not strength checks.

Limits

Subsonic only (the DATCOM transonic and supersonic methods are not implemented; the drag rise is an estimate); straight-tapered planforms for the lifting line and R_LS; no tail-body, nacelle-wing or slipstream interference beyond the entered Q factors and the wing-body correlation; no lift-dependent profile drag; no flaps, gear or ground effect except as entered increments. The default airplane is illustrative and not a published design.