Modified Newtonian Drag — theory

Supersonic pressure drag of blunt elements. Based on C. P. Hoult, Modified Newtonian Aerodynamics (rev. 4), RS&T memo, 06 April 2016.

Open the calculatorOriginal memo (PDF)

Overview

At supersonic speed the blunt parts of a rocket — a rounded nose tip, a flat face, the rounded leading edges of the fins — are locally hypersonic. The usual test is the hypersonic similarity parameter H=\sqrt{M_\infty^2-1}\,/(L/D)\ge 1, which a blunt element with small fineness L/D passes at modest Mach numbers. The drag of such elements is estimated with Newtonian impact theory, using the stagnation pressure coefficient proposed by Lees.

The calculator evaluates four elements and adds their drag areas:

  • a hemispherical nose cap (all or part of a hemisphere);
  • a conical segment (a pointed cone or a frustum, e.g. behind the cap of a sphere-cone nose);
  • a flat face;
  • hemicylindrical (rounded) fin leading edges.

Every coefficient is referred to the body cross-section S_{ref}=\pi R_{ref}^2, so the nose and fin results can be entered directly as the wave-drag inputs of the drag build-up.

Stagnation pressure coefficient

The streamline that reaches the stagnation point crosses a normal shock and is then compressed isentropically to rest. The Rayleigh pitot formula (NACA Report 1135, eq. 100) gives the stagnation pressure:

\frac{p_T}{p_\infty}=\left[\tfrac12(\gamma+1)M_\infty^2\right]^{\frac{\gamma}{\gamma-1}}\left[\frac{\gamma+1}{2\gamma M_\infty^2-(\gamma-1)}\right]^{\frac{1}{\gamma-1}}\qquad(M_\infty>1)

and, by definition of the pressure coefficient,

C_{PT}=\frac{p_T-p_\infty}{\tfrac12\gamma p_\infty M_\infty^2}=\frac{2}{\gamma M_\infty^2}\left(\frac{p_T}{p_\infty}-1\right)

With \gamma=1.4, C_{PT} rises from 1.276 at Mach 1 towards 1.839 at very high Mach number. Below Mach 1 there is no shock, and the calculator shows the isentropic value p_T/p_\infty=(1+\tfrac{\gamma-1}{2}M_\infty^2)^{\gamma/(\gamma-1)}, which joins the pitot value at Mach 1.

For a swept leading edge only the Mach number normal to the edge, M_N=M_\infty\cos\Lambda, matters. The cross-flow C_{PT} uses the same formula with M_N in place of M_\infty.

Modified Newtonian theory. Impact theory assumes that air striking the surface loses its normal momentum, which gives C_P=2\cos^2\theta, where \theta is the angle between the surface normal and the free stream. Lees’ modification replaces the 2 by C_{PT}: C_P=C_{PT}\cos^2\theta.

Hemispherical nose cap

Take a cap of radius R that covers the sphere from the stagnation point to the angle \theta_M (90° for a full hemisphere). Integrating the axial component of the pressure over the cap gives the drag area:

(C_DS)_{cap}=\pi R^2\,\frac{C_{PT}}{2}\,\left(1-\cos^4\theta_M\right)

For a full hemisphere this is C_D=C_{PT}/2 on the cap’s frontal area, about 0.92 at high Mach number, in line with measured sphere drag.

At a small angle of attack \alpha the local angle becomes \theta-\alpha\cos\varphi. Integrating round the circumference and then over \theta gives the normal-force slope:

C_{N\alpha}S_{ref}=\pi R^2\,\frac{C_{PT}}{2}\,\sin^4\theta_M

The pitching moment about the tip, divided by the normal force, gives the centre of pressure behind the tip. The memo’s eq. (17) subtracts terms that nearly cancel for a small cap. Writing it in u=1-\cos\theta_M turns the bracket into \tfrac23u^3-\tfrac34u^4+\tfrac15u^5, and

\frac{X_{CP,cap}}{R}=\frac{4u\left(\frac23-\frac34u+\frac15u^2\right)}{(2-u)^2}

This gives 7R/15 for a full hemisphere and tends to R\theta_M^2/3 for a shallow cap.

A cap tangent to a cone of half-angle \delta has \theta_M=90^\circ-\delta and base radius R\sin\theta_M.

Conical segment

On a cone of half-angle \delta the surface normal makes the angle \theta=90^\circ-\delta with the flow, so

C_{PC}=C_{PT}\sin^2\delta,\qquad (C_DS)_{cone}=C_{PC}\,\pi\left(r_2^2-r_1^2\right)

where r_1 and r_2 are the forward and aft radii of the segment and \tan\delta=(r_2-r_1)/L_C. At a small angle of attack the pressure is C_{PT}(\sin^2\delta+2\alpha\cos\delta\sin\delta\cos\varphi). Integrating its normal component gives dN=2\pi q r\alpha C_{PT}\cos\delta\sin\delta\,dx. Over the segment this becomes:

C_{N\alpha}S_{ref}=\pi C_{PT}\sin\delta\cos\delta\,L_C\,(r_1+r_2),\qquad X_{CP,cone}=\frac{L_C(r_1+2r_2)}{3(r_1+r_2)}

For a pointed cone (r_1=0) these reduce to the memo’s C_{N\alpha}S_{ref}=\pi L_C^2C_{PT}\sin^2\delta and X_{CP}=2L_C/3. Results for a frustum add up exactly with those for the cone tip it replaces.

Flat face

Supersonic wind-tunnel data for flat faces fall within 4 % of the stagnation pressure coefficient, C_{D,F}=C_{PT}, on the face area. Below Mach 1 the subsonic model from Hoerner, ch. 15, is used:

C_{D,F}=0.85\left[1+\tfrac14\left(M_\infty^2+M_\infty^4\right)\right]\qquad(M_\infty<1)

The two meet at Mach 1 to within 0.05 %. A flat face has no normal-force slope.

Hemicylindrical fin leading edge

Take a rounded leading edge of radius R_{LE} and length l on a flat-plate fin. With Newtonian pressure and the cross-flow dynamic pressure, its drag area normal to the edge is \tfrac83R_{LE}l. Referring it to the free stream adds a factor \cos^2\Lambda. Taking the stream-wise component adds another \cos\Lambda. Replacing the Newtonian 2 by the cross-flow C_{PT}(M_N) then gives

(C_DS)_{LE}=\tfrac43\,R_{LE}\,l\,C_{PT}(M_N)\cos^3\Lambda\qquad\text{per fin}

The memo’s eq. (22) carries an extra factor \pi. The derivation in its eqs. (18)–(21) does not produce one, and the memo’s own example plot agrees with the form above.

Below Mach 1

Newtonian theory does not apply to subsonic flow. The memo suggests fairing the curves to zero at the critical Mach number: about 0.6 for a sphere and 0.47 for a circular cylinder in cross flow. The calculator fairs the cap and cone values linearly from zero at M_\infty=0.6 to their Mach 1 values. It fairs the leading-edge value from zero at M_N=0.47 to its value at M_N=1. The faired range is an engineering estimate and is flagged on the page.

Totals

C_{D,nose}=\frac{(C_DS)_{cap}+(C_DS)_{cone}+(C_DS)_F}{S_{ref}},\qquad C_{D,LE}=\frac{N\,(C_DS)_{LE}}{S_{ref}},\qquad C_D=C_{D,nose}+C_{D,LE}

Enter C_{D,nose} and C_{D,LE} as the nose and fin wave-drag inputs of the drag build-up. Its own leading-edge term covers the blunt-edge case in a simpler form, so use one or the other, not both.

Verification

tests/newtonian.test.js checks the following:

  • the stagnation pressure coefficient and the leading-edge drag area against the memo’s tabulated example, from Mach 1 to 4;
  • the cap drag area, normal-force slope and centre of pressure against direct numerical integration of the Newtonian pressure, for cap angles from 1° to 90°;
  • the pointed-cone forms, and that frustum results add up to the whole cone;
  • continuity at Mach 1.

Notes on the method

  • Cap centre of pressure. Evaluated in the cancellation-free form above, which is algebraically identical to the memo’s eq. (17).
  • Leading-edge drag. Uses \tfrac43R_{LE}lC_{PT}\cos^3\Lambda, without the stray \pi of eq. (22).
  • Conical frustum. The memo’s pointed-cone normal-force result is extended to a frustum by integrating the same dN from r_1 to r_2.
  • Subsonic fairing. The linear fairing to the critical Mach number follows the memo’s suggestion to “fair by eye”.

References

  1. C. P. Hoult, “Modified Newtonian Aerodynamics (rev. 4),” RS&T memo, 06 April 2016. View in library
  2. Ames Research Staff, “Equations, Tables and Charts for Compressible Flow,” NACA Report 1135, 1953.
  3. S. F. Hoerner, Fluid-Dynamic Drag, self-published, 1965.
  4. J. F. Campbell and D. T. Howell, “Supersonic Aerodynamics of Large-Angle Cones,” NASA TN D-4719, 1968.