Nose Cone Design — Theory & documentation

Overview

The nose is drawn from its shape family, length L and base radius R. At the flight condition (Mach number M, angle of attack \alpha, altitude) the calculator works out, along the nose and in total:

the surface pressure: subsonic by Hoult’s line-source model, supersonic by cone flow;
the drag coefficient: pressure plus turbulent skin friction, on the base area A=\pi R^2;
the tip shock waves and the Mach angle;
the lift (normal force), its centre of pressure, and the shear and bending it builds up towards the shoulder;
the stagnation temperature and pressure at the tip.

The dynamic pressure is q=\tfrac12\rho V^2=\tfrac12\gamma pM^2, with the air at the chosen altitude from Hoult’s atmosphere (standard sea-level temperature, latitude 45°). Lengths are in inches internally; any unit can be displayed.

The flight is split into three ranges. Below the critical Mach number M_{cr} the subsonic model applies. From Mach 1.1 the supersonic model applies. In between, the transonic range, no simple method is reliable, and the calculator shows “—”.

Nose shapes

The profile r(x) runs from the tip (x=0) to the shoulder (x=L). The definitions are those collected by Crowell:

Conical: r=Rx/L.
Bi-conic: a cone of length L_1 to radius R_1, then a frustum to R at L.
Tangent ogive: a circular arc of radius \rho_t=(R^2+L^2)/2R that meets the body without a kink: r=\sqrt{\rho_t^2-(L-x)^2}+R-\rho_t.
Secant ogive: an arc of a chosen radius \rho with L/2<\rho; it does not meet the body tangentially. Set \alpha=\tan^{-1}(R/L)-\cos^{-1}\!\big(\sqrt{L^2+R^2}/2\rho\big); then r=\sqrt{\rho^2-(x-\rho\cos\alpha)^2}+\rho\sin\alpha. For \rho>\rho_t it is a tangent ogive with its base cut off; for \rho<\rho_t it bulges (the Honest John nose).
Elliptical: r=R\sqrt{1-(1-x/L)^2}, a half prolate spheroid.
Power series: r=R(x/L)^n, 0<n\le1. Here n=1 is a cone and n=\tfrac12 the blunt “parabolic” model-rocket nose.
Parabolic series: r=R\,\dfrac{2(x/L)-K'(x/L)^2}{2-K'}. Here K'=0 is a cone and K'=1 the full parabola, tangent to the body.
Haack series: r=\dfrac{R}{\sqrt\pi}\sqrt{\theta-\tfrac12\sin2\theta+C\sin^3\theta}, with \theta=\cos^{-1}(1-2x/L). C=0 is the Von Kármán (least drag for a given length and diameter); C=\tfrac13 is the LV-Haack (least drag for a given length and volume).
Superellipse: (r/R)^n+(1-x/L)^n=1, the family of the subsonic optimum (below).

Rounded tip. A sphere of radius r_n centred on the axis is fitted tangent to the profile where r\sqrt{1+r'^2}=r_n. The profile ahead of that point is replaced by the spherical cap, so the nose gets slightly shorter. Crowell reports that bluffness 2r_n/2R up to about 0.2 costs almost no drag.

Measures. Volume, wetted area and their running totals come from frustum sums along a 240-station profile (denser near the tip). The shell mass is m\approx S\,t\,\rho_m. The “Profile coordinates” plot exports the radius at every chosen step for templates and CAD.

Tangent ogive in closed form: Geometry and the defining circle

Place the origin at the centre of the defining circle, with x along the rocket axis measured forward from the base plane and y perpendicular. The circle of radius a is

x^2+y^2=a^2 .

The centre lies in the base plane, a distance a-R below the axis, so the arc is horizontal (tangent to the tube) at x=0. The tip lies on the axis, y=a-R, at x=L. Substituting gives the defining radius:

L^2+(a-R)^2=a^2\quad\Longrightarrow\quad a=\frac{L^2+R^2}{2R}.

Two further quantities follow directly. The fineness ratio is L/2R. The surface makes an angle with the axis at the tip of

\theta_{tip}=\sin^{-1}\!\frac{L}{a},

since the tangent at the tip is perpendicular to the radius drawn to it.

Symbol	Meaning
R	Base radius
L	Nose length, tip to base
a	Defining-circle radius
\Delta x	Profile increment
x	Axial coordinate forward of the base
r(x)	Local body radius
S	Wetted (lateral) area
M	First moment of wetted area about the base
\bar x	Area centroid, body station aft of the tip

Subsonic pressure (Hoult’s line-source model)

Why the nose pressure matters

A rolling rocket sees its body-fixed misalignments as torques at the roll frequency. When the roll rate nears the pitch frequency, the angle of attack grows. Above a modest angle of attack the boundary layer separates and sheds a pair of vortices along the body; when they reach the fins they induce a roll torque that can overpower the fin cant and lock the roll rate in. A smooth, continuously falling pressure along the body helps keep the boundary layer attached; an adverse gradient — pressure rising aft — encourages separation.

The circumferential pressure from cross flow, C_p=\sin^2\alpha(1-4\sin^2\theta), does not depend on the body shape. The longitudinal pressure does, and the nose designer controls it.

Munk airship theory

The flow round a body of revolution at zero angle of attack is represented by sources and sinks along its axis. A source of strength \sigma\,d\zeta at station \zeta induces at the point (z,r) the velocities

q_r=\frac{1}{4\pi}\int\frac{r\,\sigma\,d\zeta}{[(z-\zeta)^2+r^2]^{3/2}},\qquad q_z=\frac{1}{4\pi}\int\frac{(z-\zeta)\,\sigma\,d\zeta}{[(z-\zeta)^2+r^2]^{3/2}}

For a slender body the flow from nearby sources is nearly radial, so flow tangency at the surface r_B(z) gives \sigma=2\pi U_\infty r_B\,dr_B/dz. Linearising Bernoulli’s equation, C_p\simeq-2q_z/U_\infty:

C_{p0}(z)=-\int\frac{(z-\zeta)\,r_B\,\frac{dr_B}{d\zeta}}{\left[(z-\zeta)^2+r_B^2\right]^{3/2}}\,d\zeta

Numerically the body radius is sampled at equal stations \Delta z apart and r\,dr/dz=r_2(r_3-r_1)/(z_3-z_1) from a parabola through three neighbouring stations. The sum runs over every station of body and wake. The method is poor near a blunt tip and close to a blunt base; elsewhere it gives a good picture of the pressure.

Closing the body: the wake

A source distribution must close, or the body would grow without end. Behind the base the flow forms a wake that narrows to a cylinder. Its diameter follows from two relations: the drag impulse equals the momentum lost by the air entering the cylindrical wake, and Bernoulli’s equation holds from the base (at the base pressure) to the wake, where the pressure has recovered. Solving them,

D_w=D\sqrt{\frac{C_D}{2\left(1-\sqrt{C_{Db}}\right)}}

where C_D is the forebody drag coefficient including the base and C_{Db} the base drag coefficient, both on the body cross-section. The wake converges as a cone of half angle \varepsilon; lacking better data, \varepsilon\approx0.1 rad.

Compressibility and the critical Mach number

The Kármán–Tsien rule scales the incompressible pressure to a subcritical Mach number:

C_p=\frac{C_{p0}}{\beta+\frac{C_{p0}}{2}(1-\beta)},\qquad\beta=\sqrt{1-M_\infty^2}

The critical Mach number is the lowest free-stream Mach number at which the surface flow somewhere reaches sonic speed. There the corrected minimum pressure equals the sonic pressure coefficient

C_p^*=\frac{2}{\gamma M^2}\left[\left(\frac{2+(\gamma-1)M^2}{\gamma+1}\right)^{\frac{\gamma}{\gamma-1}}-1\right]

and the calculator solves for that Mach number. Above it a supersonic pocket forms and the correction no longer applies. For the incompressible minimum of the memo’s example, C_{p0}=-0.0566, the critical Mach number is about 0.92.

Adverse pressure gradient

For most nose shapes the pressure coefficient bottoms out near −0.06 just behind the shoulder, where the air speeds up to round the corner, and then rises along the cylinder. The calculator reports the steepest rise per body diameter behind the minimum. Reshaping the nose — a gentler shoulder, a longer nose — cannot remove the dip, but the memo shows it can roughly halve the adverse gradient. Use the custom profile to explore faired shapes, and compare the standard shapes in the plot.

Axial force on the nose

The pressure on a slice of nose surface gives the axial force dF=2\pi r_Bp\,dr_B. With p=p_\infty+qC_p the hydrostatic part integrates to p_\infty times the base area (it cancels over a whole body but matters for internal loads) and the hydrodynamic part is

F=2\pi q\int_0^{L_n}C_p\,r_B\frac{dr_B}{dz}\,dz,\qquad C_A=\frac{F}{q\pi R^2}

The first element at a pointed tip contributes \pi qC_{p1}r_1^2. In subsonic flow the nose axial force can be negative — the suction on the shoulder pulls the nose forward.

The optimum subsonic nose

What makes a nose “optimum”

Air must speed up to get round a nose, so the pressure behind the shoulder dips below ambient and then rises again along the cylinder. That rise is an adverse pressure gradient: at angle of attack it helps the boundary layer separate into a vortex pair, and the vortices can induce roll torques on the fins and lock the roll rate in. The deeper the dip, the steeper the rise — and the lower the critical Mach number, above which a shock forms behind the shoulder and makes separation worse.

The nose shape does not change the cross-flow pressure, but it does set the longitudinal pressure. Following Felkel, the single figure of merit is the minimum incompressible pressure coefficient behind the shoulder: the closer it is to zero, the better. The calculator places each nose on a long cylinder and computes its zero-angle-of-attack pressure by Munk’s line-source method, exactly as the Nose Pressure Distribution calculator does.

The superellipse family

Felkel found his optimum noses by cut and try. A simpler family matches them: the superellipse nose, an ellipse with its exponent 2 replaced by n:

\left(\frac{r}{R}\right)^n+\left(1-\frac{z}{L_n}\right)^n=1

n=1 is a cone and n=2 an ellipse. Small exponents give a sharp shoulder, which speeds the flow up there; large ones a blunt tip region, which speeds it up near the tip. In between, the two suction peaks are equal, and that exponent is the optimum. Because the minimum pressure jumps from one peak to the other there, the curve of C_{p0,min} against n has a kink at the optimum; the calculator scans 1<n\le4 and refines the kink by golden-section search.

The optimum exponent depends on the fineness ratio: about 1.75 at L_n/D=3, 1.85 at L_n/D=6, rising slowly for longer noses. The value 1.6 suggested in the original notes is a little blunt-shouldered for this method.

The calculator reports the optimum exponent for your length and diameter. The button “Use the subsonic optimum shape” switches the nose to it.

Supersonic tip: Mach angle and shock waves

Above Mach 1 a pressure disturbance cannot travel upstream. The weakest disturbance, a sound wave, lies on the Mach cone \mu=\sin^{-1}(1/M). A pointed tip turns the flow, which takes a shock wave. Just at the tip the flow is the same as for the initially tangent cone (NACA 1135). So the calculator solves the flow past a cone with the tip’s half-angle.

Behind a conical shock of angle \beta the oblique-shock relations of NACA 1135 give the flow deflection and state. Between the shock and the cone the flow is conical, a function of the polar angle only, and obeys the Taylor–Maccoll equation:

\tfrac{\gamma-1}{2}\left(1-V_r^2-V_\theta^2\right)\left(2V_r+V_\theta\cot\theta+V_r''\right)-V_\theta\left(V_rV_\theta+V_\theta V_r''\right)=0,\qquad V_\theta=V_r'

Velocities are scaled by the limiting speed. The equation is integrated inward from the shock (RK4) until V_\theta=0, which is the cone surface. A bisection on \beta then finds the shock for a given cone. The largest cone angle reached on the weak branch is the attachment limit: blunter tips carry a detached bow shock, drawn only as a sketch. The equation is the standard form, derived from the Euler equations for conical flow; it was not read in a primary source here. The solution is checked against NACA 1135:

Figure 4, the attachment limit: 9.7° at Mach 1.05, 19.5° at Mach 1.2 and 30.5° at Mach 1.5 (chart: about 10°, 20° and 30.5°); 57.3° as M\to\infty (report: 57.5°, γ = 1.405).
Charts 5 and 6, shock angle and surface pressure: at a 32° cone, C_p = 0.72 at Mach 2 and 0.64 at Mach 3 (chart: 0.73 and 0.65).

At an angle of attack the windward and leeward tip shocks are estimated from cones of half-angle \theta_{tip}\pm\alpha, measured from the oncoming flow. This is the tangent-cone idea applied at incidence: an estimate. When \theta_{tip}-\alpha\le0 the leeward side sees an expansion. The tip half-angle is taken as the cone through the tip and the profile at 2 % of the length, which is what a real, slightly rounded tip presents. A rounded tip always carries a detached bow shock.

Prandtl–Meyer expansion along the nose (shock–expansion)

Behind the tip shock the surface of a curved nose turns away from the flow, so the flow expands and speeds up. The calculator follows it as in shock–expansion theory. The flow starts with the cone solution at the tip: surface Mach number M_c and pressure p_c. At each station of local slope \theta it then takes

\nu(M)=\nu(M_c)+(\theta_{tip}-\theta),\qquad \frac{p}{p_c}=\left(\frac{1+\frac{\gamma-1}{2}M_c^2}{1+\frac{\gamma-1}{2}M^2}\right)^{\frac{\gamma}{\gamma-1}}

Here \nu is the Prandtl–Meyer angle (NACA 1135 eqs. 171–173). Where the nose meets the body at an angle (cone, bi-conic, secant ogive), the flow turns through a corner expansion fan of that angle. The drawing shows these waves at zero incidence: Mach waves leaving the curving surface and the fan at the shoulder. The plot compares the resulting pressure with the tangent-cone estimate.

This is the planar (two-dimensional) relation applied along a body of revolution. It ignores the extra relief of the axisymmetric flow, so it tends to over-expand near the shoulder of an ogive. For a cone the two methods agree exactly. Treat it as a second estimate, not a better one. The generalized shock–expansion method for bodies was not read here.

Drag

Skin friction. The turbulent flat-plate law of Hoult’s drag memo, with its compressibility factor, is applied at the Reynolds number on the nose length:

C_f=\frac{0.427\,(1+0.15M^2)^{-0.58}}{(\log_{10}Re-0.407)^{2.64}},\qquad C_{D,f}=C_f\frac{S}{A}

Subsonic pressure drag is the axial force of Hoult’s model (above). It is nearly zero and often slightly negative, because the shoulder suction pulls the nose forward. Crowell’s summary agrees: below Mach 0.8 the nose pressure drag is essentially zero and friction dominates.

Supersonic pressure (wave) drag uses the tangent-cone estimate. Each surface element takes the pressure of a cone with its local slope, from the same cone-flow solution. Where the slope is too steep for an attached shock, Lees’ modified Newtonian pressure is used instead, C_p=C_{p,max}\sin^2\theta, with C_{p,max} the stagnation (pitot) value of Hoult’s drag memo. Then

C_{D,p}=\frac{1}{A}\sum C_{p,i}\,\pi\left(r_{i+1}^2-r_i^2\right)

This is exact for a cone. For curved noses it is an engineering estimate that was not checked against a source here, so treat it as about ±20 %. As one cross-check, it gives 0.047 for a 5:1 Von Kármán nose at Mach 2, against about 0.04 from slender-body theory. The transonic range is left blank.

Comparing shapes. The drag plot puts the common shapes, at your length and diameter, side by side. It does not reproduce every point of Crowell’s qualitative ranking: near Mach 1–1.5, where the estimates are weakest, the inviscid tangent-cone estimate favours slender cones. Crowell’s ranking from wind-tunnel data for Mach 0.8–2 runs as follows: Von Kármán and the ½-power series are best through the transonic range; blunt shapes are fine subsonic; ogives and cones are the weakest near Mach 1; cones improve at high Mach.

Lift, centre of pressure and bending

By slender-body theory (Munk; Hoult’s “Barrowman Equations”) a station of the nose carries a normal force per unit length proportional to the growth of its cross-section:

\frac{dN}{dx}=2q\alpha\frac{dA}{dx}\quad\Rightarrow\quad C_{N\alpha}=2,\qquad x_{cp}=L-\frac{V}{A}

The slope is per radian, on the base area, and the same for every pointed nose (Crowell; Barrowman). NACA 1135 chart 8 shows cones up to about 15° within roughly 10 % of 2 at supersonic speed. Integrating towards the shoulder gives the shear V(x)=2q\alpha A(x) and the bending moment M_b(x)=2q\alpha\,V_{ol}(x). Here V_{ol}(x) is the nose volume ahead of station x. These are aerodynamic loads only, before the nose’s inertia is taken off. They are linear in \alpha, so above about 10° the non-linear cross-flow lift is missing and the loads are underestimated.

Crowell’s centre-of-pressure fractions follow from V/A, measured from the base: cone L/3, tangent ogive about 0.534L, Von Kármán 0.500L. The tests reproduce them. Two printed values need correcting. Crowell’s LV-Haack “0.437L” is measured from the tip; from the base it is 0.563L. His elliptical “3L/2” should read 2L/3 from the base.

Stagnation temperature and pressure

Where the air is brought to rest at the tip its temperature rises to the stagnation value (NACA 1135, energy equation):

T_0=T\left(1+\tfrac{\gamma-1}{2}M^2\right)

This is the upper limit of the skin temperature. The boundary layer recovers a little less, and a short boost heats the skin far less because the wall needs time to warm up. The calculator does not model heating over time. So the verdict only flags as marginal a stagnation temperature above your material’s limit (DECISIONS D27). It also gives the Mach number at which T_0 reaches the limit. The stagnation pressure is p_0=p+C_{p,max}\,q, using the isentropic value below Mach 1 and the pitot (normal-shock) value above.

Verdict. The verdict is marginal when any of these holds:

the stagnation temperature is above your material limit;
the windward tip shock is detached (higher drag and heating at the tip);
the flight is in the transonic range, where shock waves form on the nose.

Otherwise it is safe. The fixes are solved:

the Mach number for your material;
the shortest nose of this shape whose windward tip shock stays attached (bisection);
the subsonic optimum superellipse with its critical Mach number.

Verification

Shapes: every profile meets r(0)=0, r(L)=R. Volume, wetted area and centroid reproduce Hoult’s ogive memo example (R=3 in, L=30 in: S = 379.25 in²). The centre-of-pressure fractions above match Crowell.
Cone flow: the NACA 1135 checks above.
Subsonic: Hoult’s field tool and Felkel’s Table 1, unchanged from the former calculators.

Notes on the method

Axial force. The field tool’s drag-area column omits the factor 2\pi on the body elements and sums only the first 15 in of the nose; the calculator integrates the memo’s formula over the whole nose.
Field point. As in the field tool, the distance from each source uses the body radius at the source station; this is the usual slender-body simplification.
Plume. The wake model is for a coasting (non-thrusting) rocket; a motor plume changes the flow behind the base.

References

G. A. Crowell Sr., “The Descriptive Geometry of Nose Cones,” 1996.
Ames Research Staff, “Equations, Tables, and Charts for Compressible Flow,” NACA Report 1135, 1953.
C. P. Hoult, “Ogive Nose,” RS&T memo, 6 May 2011. View in library
C. P. Hoult, “Nose Pressure Distribution (rev. 4),” RS&T, 2012. View in library
R. J. Felkel, “Effect of Different Nose Profiles on Subsonic Pressure Coefficients,” AIAA student paper, 2011. View in library
C. P. Hoult, “Compressibility Effects on a Minimum Pressure Gradient Nose,” RS&T, 2010. View in library
C. P. Hoult, “Barrowman Equations,” RS&T, 2010. View in library
C. P. Hoult, “Drag Coefficient (rev. 3.4),” RS&T, 2016. View in library
C. P. Hoult, “Modified Newtonian Aerodynamics,” RS&T. View in library
M. M. Munk, NACA TN-192, 1924 (through Hoult).

Nose Cone Design — theory