Zero-Lift Drag Coefficient — theory

Overview

The drag coefficient has a major effect on rocket performance, and the effect is largest for the smallest rockets because of square–cube scaling. Physically there are four sources of zero-lift drag:

1. Skin friction — viscous shear in the boundary layer, laminar or turbulent;
2. Wave drag — an unavoidable consequence of supersonic flow;
3. Base drag — low pressure behind a blunt base of the body or a fin;
4. Parasite drag — all the small things: joints, lugs, roughness.

Induced drag is not addressed, because sounding rockets fly at very nearly zero lift. All coefficients are referenced to the body cross-section S_{ref}=\pi d^2/4, with

Skin-friction drag

Reynolds number

For a station x on the body and flight Mach number M,

with p_\infty in lb/ft² and T in °R — an atmospheric property, so only pressure, temperature and Mach number are needed from a trajectory or flight data.

Laminar and turbulent friction

Whether the surface is flat or cylindrical hardly matters. The Blasius laminar friction coefficient is

and for incompressible turbulent boundary layers the memo uses the simpler, almost equally accurate alternative to Schoenherr’s law,

Transition

Natural transition is assumed to occur instantaneously at a transition Reynolds number. For rocket bodies of revolution below hypersonic speed the memo recommends Re_T=400{,}000; for fins Re_T=25{,}000, because leading edges that are not smoothly rounded to a near-hemicylindrical contour trip early. (The body is stabilised by its growing radius, which thins the boundary layer.) A boundary-layer trip can also be specified; transition is taken at the trip or the natural point, whichever is closer to the tip. The maximum laminar run is

Wetted area and the body friction coefficient

Along the body dS_{wet}/dx=2\pi R(x). Only a few points of the wetted-area curve are needed if they are accurate; between them a straight line is adequate (memo Fig. 1, reproduced in the calculator). The laminar wetted area lamS_{wet} is read from the curve at l_T; if l_T\ge L the whole body is laminar. The area-averaged friction coefficient is

where the turbulent term uses the Reynolds number based on the overall length. For a tangent ogive nose the wetted area has the closed form of the Nose Cone calculator.

Fins

Fins are treated the same way, remembering that both sides are wetted. Each strip is assumed to have the same span-wise width and the average chord \bar c=(c_R+c_T)/2. With a laminar run x_T=\mu Re_{T}/\rho U shorter than \bar c,

where the last term removes the double-counting of turbulent friction over the laminar part; otherwise the fin is fully laminar, C_f=1.328/\sqrt{Re_{\bar c}}. The fin friction drag is

Compressibility

Following Hoerner (pp. 15-9 and 17-4) for modest heating and an adiabatic wall, the laminar friction is left unchanged and the turbulent friction is reduced by

Each friction drag is also multiplied by an empirical factor (1-0.09M^2).

Wave drag

Wave drag cannot be estimated with simple classical methods until the flow is well-developed supersonic. For the body of revolution the memo uses the second-order shock-expansion method (Syvertson & Dennis, NACA Report 1328); very blunt parts — hemispherical caps, rounded fin edges — use modified Newtonian theory. In this calculator the nose and fin wave drag are inputs, to be taken from those tools; the memo’s examples show an ogive wave drag falling from about 0.68 at Mach 1.5 to 0.15 at Mach 4, and fin leading-edge wave drag of 0.07–0.09.

Base drag

Hoerner (p. 3-19) gives the incompressible base-pressure coefficient in terms of the fore-body friction drag:

For well-developed supersonic flow the base drag is about 65 % of vacuum:

In transonic flow the base drag is independent of Mach number, equal to the incompressible value plus an increment,

extended until it meets the quasi-vacuum curve at

The memo refers the increment to below the fore-body critical Mach number (about 0.8); the calculator switches at M=1.

Power-on

Power-on base drag is complicated and depends on nozzle exit pressure. Hoerner (p. 20-16) suggests that, based on the unpowered base area, it could be 1.5 times the power-off value for well-developed nozzle flow, because the jet entrains air from the base region. The memo recommends applying this to the annulus outside the nozzle:

For a 4-inch nozzle on a 6-inch body, 1.5(1-4^2/6^2)=0.833: power-on base drag is still 83 % of the power-off value.

Fin airfoil pressure and base drag

Flat-plate fin base drag at subsonic speeds (Hoerner p. 3-21) is

with C_f the friction coefficient of one side of one panel. A single-wedge airfoil adds base area at its trailing edge. The crude pressure-drag model treats leading and trailing edges as either blunt or sharp:

The blunt-leading-edge, sharp-trailing-edge case has no subsonic pressure drag (d’Alembert’s paradox). The Newtonian drag of a hemicylindrical leading edge of thickness t, swept at \Lambda, is evaluated with the normal Mach number M\cos\Lambda:

Slender-airfoil wave drag is neglected here. The sweep is taken as \tan^{-1}\big((c_R-c_T)/b\big), i.e. an unswept trailing edge.

Riding lugs

Two riding lugs hold the rocket on its rail. Hoerner gives an incompressible drag coefficient of 0.76–0.8 on the projected frontal area, so C_DS\approx0.8(2Rh). A compressible estimate uses the Newtonian stagnation-pressure coefficient,

the second being the Rayleigh pitot formula. The cylindrical Newtonian model gives C_DS=\tfrac43RhC_{PT}; to match Hoerner’s data the calculator uses

The calculator evaluates this stagnation-pressure coefficient in cpt().

Flat-faced nose

Flat faces are common, especially at high angle of attack. The streamline that reaches the stagnation point passes through a normal shock and is then compressed isentropically, so the face drag coefficient is the stagnation-pressure coefficient above (within 4 % of wind-tunnel data). Subsonically the memo uses

which meets the supersonic value at M=1 (1.275 against C_{PT}(1)=1.276). The face area is \pi d_f^2/4.

Parasite drag

Parasite drag covers sources too small to justify individual analysis: skin joints, roughness, antennas. Many could be analysed with the methods above, but that is usually a poor use of effort, so it is normal to add about 5 % to the itemised total.

Totals

Power-on, the body base term is replaced by its power-on value. The calculator also reports the totals with the parasite allowance. With the memo equations power-on drag can exceed power-off for a small nozzle, because of the factor 1.5.

Assumptions and limits

• Single stage, single fin set, cylindrical body without a boat-tail (base area equals reference area).
• Zero angle of attack; no induced drag.
• Instantaneous transition at a fixed Reynolds number or a trip.
• Supersonic body and fin wave drag are inputs from other tools.
• The (1-0.09M^2) friction factor and the quasi-vacuum base law become increasingly uncertain above about Mach 2.5; the calculator limits Mach to 3.2.

Worked example

Body 6.14 in diameter, 148 in long with a 36 in tangent ogive and a 1.5 in flat face; four fins (11.8/4.5 in chords, 5.84 in span, 1/8 in thick, blunt edges); 0.4 in nozzle; two lugs of 0.36 in². Mach 0.9, 499.17 °R, 2124 lb/ft²; body Re_T=200{,}000, fin Re_T=25{,}000.

1. \rho a/\mu=7.459\times10^6 /ft; the laminar run is only 0.030 ft, so 4.6 in² of the 2625 in² body is laminar.
2. Body C_f=0.00195 → C_{D_f}=0.160; fins C_f=0.00315 → 0.038.
3. Body base 0.0697 power-off; fin base 0.091; fin leading edges 0.028; lugs 0.024.
4. Flat face 0.069.
5. Totals: 0.479 power-off, 0.514 power-on.

Equations ↔ code

Verification: tests/drag.test.js reproduces the reference example to 10⁻¹², including the stagnation-pressure values. It also checks each method note listed below.

Notes on the method

• Flat face. The subsonic face drag uses memo eq. (22), 0.85(1+\tfrac14(M^2+M^4)). It is continuous with the supersonic value at Mach 1 (1.275 against 1.276). The face term is counted once in the totals.
• Fin laminar friction. The laminar term is weighted by the laminar fraction x_T/\bar c, as in the fin equation above.
• Power-on base drag. The factor 1.5 of memo eq. (14) is applied to the annulus outside the nozzle.
• Laminar wetted area. The laminar area is interpolated on the full wetted-area curve, so transition may lie anywhere on the body, not only on the nose.
• Cone nose. The cone wetted area \pi R(R+\sqrt{R^2+L^2}) includes the base disc.
• Single fin set. A second fin set is not modelled; the wedge base area uses the fin count of the one set.

References

1. C. P. Hoult, “Drag Coefficient (rev. 3.4),” RS&T memo, 8 April 2016. View in library
2. S. F. Hoerner, “Aerodynamic Drag,” published by the author, 1965 (cited by the memo; widely known as Fluid-Dynamic Drag).
3. Schoenherr, “Resistance of Plates,” S.N.A.M.E., 1932.
4. C. A. Syvertson and D. H. Dennis, “A Second-Order Shock-Expansion Method Applicable to Bodies of Revolution Near Zero Lift,” NACA Report 1328, 1957.
5. C. P. Hoult, “Modified Newtonian Aerodynamics (rev. 4),” RS&T memo, 6 April 2016. View in library
6. Ames Research Staff, “Equations, Tables and Charts for Compressible Flow,” NACA Report 1135, 1953.
7. Source file: DRAG_COEFFICIENT3.4.xls (rsandt.com).