Barrowman Method of Calculating Normal Force

     [Equation 1]

where:
Nc = normal force, lbf or Newtons. Subscript c refers to the particular component of interest (such as nosecone or fin).
q = dynamic pressure = ½ ρ V², lbf/in² or N/m²
A = reference area = ¼ π D² where D = nosecone base diameter, in. or m.
V = vehicle velocity, ft/sec or m/s
(C_{N α})= slope of the normal force coefficient at α = 0 (∂C_N/∂α | α=0) , per radian
α = effective angle of attack, radians

As is seen by the last two terms of Equation 1, angle of attack is multiplied by the slope of the normal force coefficient. Slope is defined as the rise over run of a straight line tangent to the curve, at α=0, of the normal force coefficient versus angle of attack. This concept is illustrated in Figure 1. The slope of the straight line is change (Δ), or rise, in force coefficient over the corresponding change, or run, in attack angle. The curve of the function is seen to be nonlinear, with the slope gradually increasing as alpha increases. This demonstrates why this method of determining the normal force is valid for "small" angles of attack.

Figure 1: Concept of slope of force coefficient

The derivation of slope of the normal force coefficient at α = 0 ( C_{N α}) for the various rocket components is given in Resource 7. The derivations are worth reviewing to gain insight into the method and its assumptions.

Key assumptions applicable to the Barrowman method:

1. Flow over rocket is smooth and frictionless.
   In reality, there will be some friction, as air has viscosity. Flow will not be perfectly smooth, as all EX rockets have some protruberances such as fasteners, joints, etc. The net affect can be considered to be minor with respect to accuracy of the method.

2. Flow over rocket is subsonic.
   As indicated in Resource 9, the Barrowman method can be used for velocities somewhat greater than mach one. However, the results generally become unconservative. To compensate, larger safety margins should be used when applying the results to supersonic rocket design. Resource 9 provides methods to deal with higher mach numbers.

3. Angle of attack is small.
   As shown in Figure 1, the relationship between force coefficient and angle of attack is nonlinear. The Barrowman method utilizes slope at zero degree angle of attack. As angle of attack increases, the slope changes at an increasing rate. The degree to which the slope changes will be different for each rocket component and is not feasible to quantify. Barrowman suggests 10 degrees as a maximum angle of attack for the method to remain accurate. Resource 9 discusses how to deal with larger angles of attack.

4. Nosecone tip is sharp
   If a nosecone has a rounded, rather than sharp tip, base the calculations on the nosecone geometry assuming the rounded tip is not present.

5. Rocket finset consists of either 3 or 4 fins
   Originally the method was derived for rockets with 3 or 4 fins. However, Barrowman expanded the method to include 6 fin rockets.

6. Fins are thin flat plates
   Aerofoiled fins possess a centre of pressure that is at a location different than that of flat plates. This should be borne in mind when assessing a rocket with aerofoiled fins.

The expressions for normal force coefficient slope at α = 0 for various rocket components are summarized below. Units are per radian.

Nosecone    (C_{N α})_N = 2

Body    (C_{N α})_B = 0

Conical Shoulder   

Conical Transition or Boattail   

S₁, S₂ = cross-sectional area, as indicated (in² or mm²).
d = reference length = diameter at base of nosecone (in. or mm)
The expressions for conical shoulder and conical boattail are identical. It is important to note that, for a boattail, S₂ is less than S₁. As such, the value of (C_{N α})_B is negative, meaning the normal force acts in an opposite sense to a conical shoulder.
The expression for conical shoulder/boattail can be simplified by incorporating the expressions for areas S₁ = ¼ π d₁² and S₂ = ¼ π d₂², giving:

Finset   

where N = number of fins in finset (3, 4 or 6). Theta (θ) and l are the mid-chord sweep angle and mid-chord length, respectively. Interference coefficient f=1 for 3 or 4 fins, f=0.5 for 6 fins

The normal force resultant, as calculated above, acts at the centre of pressure (C.P.) of each of the components. Resource 7 provides the derivations. The expressions for the C.P. location of various rocket components are summarized below. Units of inches or mm.

Nosecone    

Note that if the nosecone tip is radiused, rather than sharp, the reference line from which the nosecone length and C.P. location is obtained by extrapolating the profile of the nosecone curve as shown.

Conical Transition 

where d₁ is the forward diameter of the transition, and d₂ is the aft diameter of the transition.

Finset   

Example

Click for larger image.

Use units of Newtons (N.) and millimetres (mm)

Angle of attack, α = 10 degrees = 0.1745 radians
Reference diameter, d = 75 mm
Reference area, A = ¼ π 75² = 4418 mm²
Nosecone profile is tangent ogive
V = 300 metres/second
Air density = 1.10 kg/cu.metre
The dynamic pressure is therefore:

Use units of Newtons (N.) and millimetres (mm), therefore q = 0.0495 N/mm²

For any nosecone, (C_{N α})_N = 2 (per radian) and for any rocket body (C_{N α})_B = 0

For the conical shoulder

For the finset

For the boattail

The normal force acting at each component is now calculated

At the nosecone C.P

At the conical shoulder C.P

At the boattail C.P

At the finset C.P

The centre of pressure locations are next calculated

For the nosecone

For the conical shoulder

For the boattail

For the finset

Results summary:

Important note: These are instantaeous forces that are reacted solely by mass inertia of the rocket acting under lateral and rotational accelerations.

Resources:
Res.7 The Theoretical Prediction of the Center of Pressure by James S. Barrowman and Judith A. Barrowman, presented as a Research and Development Project at NARAM-8
Res.8 BARROWMAN.XLS by R.Nakka, Excel spreadsheet for calculating rocket centre of pressure using Barrowman method.
Res.9 Rocketry Aerodynamics by Rick Newlands, Aspire Space