Subsonic Centre of Pressure — theory

Overview

Static pitch stability depends on where the aerodynamic normal force acts — the centre of pressure (CP) — relative to the centre of gravity. The memo is blunt about folklore alternatives such as balancing a cardboard cut-out or whirling the rocket on a string. For years the amateur community has treated the Barrowman equations — slender-body loading for the body plus Diederich’s approximate lifting-surface theory for the fins — as the standard for subsonic CP estimation.

The method keeps Barrowman’s body and fin models and improves three things:

• a tangent hemispherical nose cap with its own normal-force model when the profile does not start at zero radius;
• full Prandtl–Glauert compressibility in Diederich’s fin formula;
• the full slender-body wing–body and body–wing interference of NACA Report 1307.

Everything is linear in angle of attack: each part contributes a normal-force coefficient slope C_{N\alpha} (per radian, on S_{ref}=\pi d^2/4) acting at its own station.

Body of revolution — slender-body theory

Munk’s slender-body theory gives the normal force per unit length as proportional to the rate of change of cross-section area, dC_N/dx=(2/S_{ref})\,dA/dx\cdot\alpha. For a segment between profile points i-1 and i:

Its centre of pressure follows from the moment of dA/dx; with the frustum volume V_i,

Cylinders (A_i=A_{i-1}) carry no normal force; a boat-tail (decreasing area) carries negative normal force. The body totals are

For a pointed nose this sums to the classic slender-body result C_{N\alpha}=2A_{base}/S_{ref}=2.

Nose shapes

Instead of typing profile points, choose a nose shape. Its radius r(x) at distance x aft of the tip is generated from the nose length L_n and the body radius R (40 points, closer together near the tip), followed by the cylinder and an optional conical boat-tail. With t=x/L_n:

In slender-body theory every pointed nose has C_{N\alpha}=2 (on the base area); the shape only moves where that force acts — a cone at 2L_n/3 from the tip, a tangent ogive at about 0.466\,L_n. Shape definitions follow G. A. Crowell, The Descriptive Geometry of Nose Cones (1996).

Hemispherical nose cap

If the first profile radius r_0 is not zero, a hemisphere is fitted tangent to the first segment. Its tangency angle and radius are

The memo starts from the incompressible pressure distribution on a sphere, C_p=1-\tfrac94\sin^2\theta, measures \theta from the displaced stagnation point (\theta'=\theta-\alpha\cos\varphi), keeps first-order terms in \alpha and integrates over the cap up to \theta_T:

A full hemisphere on a body of the same radius gives C_{N\alpha}=9/8, well below the slender-body value of 2. The cap CP behind its own tip is

and, remarkably, the same result follows from modified Newtonian flow — the cap CP is independent of Mach number. In body stations (station 0 at the tangency point) it lies at R_h(\cos\theta_T-1)+x_{cp}. A sphere’s critical Mach number is about 0.41, so below that a hemispherical tip has no wave-drag penalty.

If the tip is a flat face instead, the cap’s normal force is zero and a small flat-face pitching-moment term is added (see Notes on the method).

Isolated fins — Diederich with compressibility

Opposite panels form a wing. For one panel with root chord c_R, tip chord c_T, exposed semispan b, root and tip leading-edge stations x_R, x_T (and a boat-tail half-angle \beta under the root, usually zero):

With the Prandtl–Glauert factor \beta_\Lambda=\sqrt{1-M^2\cos^2\Lambda_c}, Diederich’s compressible plan-form parameter and lift slope (on the area of the panel pair) are

For N panels on the body reference area,

The fin CP is a quarter of the mean aerodynamic chord behind its leading edge:

Fin–body interference (NACA Report 1307)

The body increases the lift on the fins, and the fins induce lift on the body. With r the body radius at the quarter root chord, s=r+b the fin tip radius and \tau=r/s, the slender-body factors are

which satisfy K_{W(B)}+K_{B(W)}=(1+\tau)^2, tending to 1 and 0 with no body and to 2 and 2 for vanishing fins. The tail-assembly normal force and moment are

where the body carry-over lift acts at (NACA 1307 eq. D6)

For short fins (s<2r) the square root and the inverse hyperbolic cosine are both imaginary but their products are real: \sqrt{s(s-2r)}\cosh^{-1}y\to-\sqrt{s(2r-s)}\cos^{-1}y and \cosh^{-1}y/\sqrt{s(s-2r)}\to\cos^{-1}y/\sqrt{s(2r-s)}. The calculator uses this analytic continuation, which is continuous at s=2r, where the real form is undefined.

The calculator also reports the single-panel normal force per radian of fin deflection, C_{N\delta}=(k_{W(B)}+k_{B(W)})\,C_{N\alpha,W}\,S/S_{ref}, with k_{W(B)} from NACA 1307 eq. (19) and k_{B(W)}=K_{W(B)}-k_{W(B)}, eq. (33).

Vehicle CP and static margin

measured aft of the nose tip (station 0). With the CG at X_{CG}, the static margin in calibres is

Positive means the CP is behind the CG and the rocket is statically stable. A margin of 1–2 calibres is common amateur practice.

Worst-case margin and the verdict

The margin changes during the flight, so the calculator does not judge a single number. It evaluates every combination of

• Mach 0 (leaving the rail) and your highest flight Mach number — compressibility moves the fin CP;
• the CG at launch and, if you give it, the CG at burnout — the CG usually moves forward as propellant burns, but not always;
• with and without the body carry-over lift of NACA 1307. Barrowman’s own method uses the smaller factor 1 + r/(r + s) instead, so a rocket that only passes with it is relying on lift most tools do not count.

The lowest of these is the worst-case static margin, and the verdict uses it:

These limits are common amateur practice, not a published standard: the 1-calibre floor and the 1–2 calibre band are widely used rules of thumb that this site has not traced to a primary source. Above about 4 calibres the rocket is very stable but weathercocks strongly into the wind; the calculator notes that but does not mark it unsafe.

When the worst case misses the target, the calculator gives two fixes: the forward CG shift that reaches the target (\Delta X_{CG}=(SM_{target}-SM)\,d), and the exposed semispan that reaches it with the other fin dimensions held, found by bisection over the full worst-case search.

Assumptions and limits

• Subsonic flow below the critical Mach number; small angles of attack; linear aerodynamics.
• One fin set. The two-stage fin–fin vortex-interference model of the 1307 Supplement memo is not included.
• Slender-body theory for the body: no viscous cross-flow lift, which grows at larger angles of attack.
• Fins are flat, trapezoidal and planar, mounted on the body radius at their quarter root chord.

Worked example (Gold Rush III, ignition)

Mach 0.05; 4.12 in diameter, 93.16 in long, blunt flat tip of 0.05 in radius; six fins with c_R=10, c_T=3, b=4 in, root LE at 83.16 in and tip LE at 90.16 in. The 17.5 in profile point is left out, so 15 → 20 in is a single frustum:

1. Body: C_{N\alpha,body}=1.9994, moment 18.883 in.
2. Fins: A\!R=1.2308, \Lambda_c=52.7°, F=2.0299, C_{N\alpha,W}=1.5946, C_{N\alpha,F}=9.3295; x_F=87.814 in.
3. Interference with \tau=2.06/6.06: K_{W(B)}=1.2907, K_{B(W)}=0.5047; carry-over CP 87.134 in.
4. Tail: C_{N\alpha,tail}=16.750 at 87.623 in.
5. Vehicle: C_{N\alpha}=18.750 /rad, X_{CP}=79.286 in from the tip.

Equations ↔ code

Verification: tests/subsonic-cp.test.js reproduces the reference example (every body segment, every fin and interference quantity and the vehicle totals) to 10⁻¹², checks the interference factors against the limit and the identity K_{W(B)}+K_{B(W)}=(1+\tau)^2, and checks that the carry-over CP is continuous through s=2r.

Notes on the method

• One fin set. Only a single fin set is modelled. Fin–fin vortex interference for two-stage rockets is not included.
• Short fins. For s<2r the carry-over CP uses its analytic continuation (see Interference).
• Body radius at the fins and boat-tail angle. The body radius at the quarter root chord is interpolated from the profile. The boat-tail angle under the fins is an input.
• Station-0 area. The cross-section at station 0 is taken as half the tip disc, A_0=\pi r_0^2/2, so the first segment’s normal force is 2(A_1-A_0/2)/S_{ref}. Neither memo explains this choice.
• Flat-face moment term. For a flat tip, C_{m\alpha,flat}=-0.0054\cdot0.85\left(1+(M^2+M^4)/4\right)(2r_0/d) is added to the moment sum (in inches). It is dimensionally inconsistent but negligible (−1×10⁻⁴ in for Gold Rush III).

References

1. C. P. Hoult, “Barrowman Equations,” RS&T memo, 21 March 2010. View in library
2. J. S. Barrowman, “The Practical Calculation of the Aerodynamic Characteristics of Slender Finned Vehicles,” M.S. thesis, Catholic University of America, 1967. View in library
3. C. P. Hoult, “Normal Force Acting on a Hemispherical Nose Tip,” RS&T memo, 19 March 2010. View in library
4. F. W. Diederich, “A Plan-Form Parameter for Correlating Certain Aerodynamic Characteristics of Swept Wings,” NACA TN-2335, 1951.
5. W. C. Pitts, J. N. Nielsen and G. E. Kaattari, “Lift and Center of Pressure of Wing-Body-Tail Combinations at Subsonic, Transonic, and Supersonic Speeds,” NACA Report 1307, 1953.
6. M. M. Munk, “The Aerodynamic Forces on Airship Hulls,” NACA Report 184, 1924.
7. C. P. Hoult, “N.A.C.A. Report 1307 Supplement for Downwash-Dependent Fin-Fin Interference,” RS&T memo, 2014. View in library
8. Source file: SUBSONIC BARROWMAN EQUATIONS2.4.xls (rsandt.com).